14168:
13376:
14163:{\displaystyle {\begin{aligned}\mathbf {v} _{P}&=^{-1}\mathbf {P} (t)\\&={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}={\begin{bmatrix}{\dot {A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}A&\mathbf {d} \\0&1\end{bmatrix}}^{-1}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\&={\begin{bmatrix}{\dot {A}}&{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}A^{-1}{\begin{bmatrix}1&-\mathbf {d} \\0&A\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\&={\begin{bmatrix}{\dot {A}}A^{-1}&-{\dot {A}}A^{-1}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\&={\begin{bmatrix}{\dot {A}}A^{\text{T}}&-{\dot {A}}A^{\text{T}}\mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}\\\mathbf {v} _{P}&=\mathbf {P} .\end{aligned}}}
744:
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9162:
1045:
9853:
1029:
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757:
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8599:
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1873:
10811:
15794:, or slider, requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body, and a plane parallel to this line in the moving body maintain contact with a similar parallel plane in the fixed body. This imposes five constraints on the relative movement of the links, which therefore has one degree of freedom. This degree of freedom is the distance of the slide along the line.
984:
5255:
8842:
8363:
9152:
2871:{\displaystyle \mathbf {\bar {a}} ={\frac {\Delta \mathbf {\bar {v}} }{\Delta t}}={\frac {\Delta {\bar {v}}_{x}}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta {\bar {v}}_{y}}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta {\bar {v}}_{z}}{\Delta t}}{\hat {\mathbf {z} }}={\bar {a}}_{x}{\hat {\mathbf {x} }}+{\bar {a}}_{y}{\hat {\mathbf {y} }}+{\bar {a}}_{z}{\hat {\mathbf {z} }}\,}
3287:
12929:
14802:
7655:
8132:
15216:
13335:{\displaystyle \mathbf {v} _{P}=\mathbf {p} ={\begin{bmatrix}\mathbf {v} _{P}\\0\end{bmatrix}}=\left({\frac {d}{dt}}{\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}\right){\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}{\dot {A}}(t)&{\dot {\mathbf {d} }}(t)\\0&0\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.}
3066:
2392:
7885:
10159:
5033:
12680:
8624:
10413:
10764:
2137:{\displaystyle \mathbf {\bar {v}} ={\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {\Delta x}{\Delta t}}{\hat {\mathbf {x} }}+{\frac {\Delta y}{\Delta t}}{\hat {\mathbf {y} }}+{\frac {\Delta z}{\Delta t}}{\hat {\mathbf {z} }}={\bar {v}}_{x}{\hat {\mathbf {x} }}+{\bar {v}}_{y}{\hat {\mathbf {y} }}+{\bar {v}}_{z}{\hat {\mathbf {z} }}\,}
9407:
8961:
3073:
12708:
5613:
7441:
14795:
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2218:
15739:
This is the case where bodies are connected by an idealized cord that remains in tension and cannot change length. The constraint is that the sum of lengths of all segments of the cord is the total length, and accordingly the time derivative of this sum is zero. A dynamic problem of this type is the
2540:
The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to
9812:
is the study of the properties of figures that remain the same while the space is transformed in various ways—more technically, it is the study of invariants under a set of transformations. These transformations can cause the displacement of the triangle in the plane, while leaving the vertex angle
8594:{\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.}
9577:
7734:
15797:
A cylindrical joint requires that a line, or axis, in the moving body remain co-linear with a line in the fixed body. It is a combination of a revolute joint and a sliding joint. This joint has two degrees of freedom. The position of the moving body is defined by both the rotation about and slide
9959:
1066:
Particle kinematics is the study of the trajectory of particles. The position of a particle is defined as the coordinate vector from the origin of a coordinate frame to the particle. For example, consider a tower 50 m south from your home, where the coordinate frame is centered at your home,
965:
Kinematic and cinématique are related to the French word cinéma, but neither are directly derived from it. However, they do share a root word in common, as cinéma came from the shortened form of cinématographe, "motion picture projector and camera", once again from the Greek word for movement and
1090:
In the most general case, a three-dimensional coordinate system is used to define the position of a particle. However, if the particle is constrained to move within a plane, a two-dimensional coordinate system is sufficient. All observations in physics are incomplete without being described with
15786:
A revolute pair, or hinged joint, requires a line, or axis, in the moving body to remain co-linear with a line in the fixed body, and a plane perpendicular to this line in the moving body maintain contact with a similar perpendicular plane in the fixed body. This imposes five constraints on the
9805:
to each part and determining how the various reference frames move relative to each other. If the structural stiffness of the parts are sufficient, then their deformation can be neglected and rigid transformations can be used to define this relative movement. This reduces the description of the
15027:{\displaystyle ={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega {\dot {\mathbf {d} }}+{\ddot {\mathbf {d} }}\\0&0\end{bmatrix}}={\begin{bmatrix}{\dot {\Omega }}&-{\dot {\Omega }}\mathbf {d} -\Omega \mathbf {v} _{O}+\mathbf {A} _{O}\\0&0\end{bmatrix}}}
12507:
3291:
Thus, acceleration is the first derivative of the velocity vector and the second derivative of the position vector of that particle. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
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4317:
5897:
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10587:
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836:
of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within
15583:
Kinematic constraints are constraints on the movement of components of a mechanical system. Kinematic constraints can be considered to have two basic forms, (i) constraints that arise from hinges, sliders and cam joints that define the construction of the system, called
6058:
12403:
12305:
11124:
5026:
15773:. He distinguished between higher pairs which were said to have line contact between the two links and lower pairs that have area contact between the links. J. Phillips shows that there are many ways to construct pairs that do not fit this simple classification.
5755:
1861:
as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle
5250:{\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\int _{0}^{t}\mathbf {v} (\tau )\,{\text{d}}\tau =\mathbf {r} _{0}+\int _{0}^{t}\left(\mathbf {v} _{0}+\mathbf {a} \tau \right){\text{d}}\tau =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}.}
3815:
8837:{\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\theta ){\hat {\mathbf {r} }}+(a+v\omega ){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.}
12211:
5492:
5336:
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14258:
7192:
1748:
7964:
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A lower pair is an ideal joint, or holonomic constraint, that maintains contact between a point, line or plane in a moving solid (three-dimensional) body to a corresponding point line or plane in the fixed solid body. There are the following cases:
7730:
5428:
1233:
15813:
Generally speaking, a higher pair is a constraint that requires a curve or surface in the moving body to maintain contact with a curve or surface in the fixed body. For example, the contact between a cam and its follower is a higher pair called a
9460:
9147:{\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}.}
15698:
3282:{\displaystyle \mathbf {a} =\lim _{(\Delta t)^{2}\to 0}{\frac {\Delta \mathbf {r} }{(\Delta t)^{2}}}={\frac {{\text{d}}^{2}\mathbf {r} }{{\text{d}}t^{2}}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.}
12924:{\displaystyle \mathbf {p} =^{-1}\mathbf {P} (t)={\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}={\begin{bmatrix}A(t)^{\text{T}}&-A(t)^{\text{T}}\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {P} (t)\\1\end{bmatrix}}.}
6783:
6162:
3390:
8329:
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11624:
7650:{\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.}
5770:
8225:
7267:
12452:
and acceleration of points in a moving body as they trace trajectories in three-dimensional space. This is particularly important for the center of mass of a body, which is used to derive equations of motion using either
9245:
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15232:
14327:
8127:{\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\theta {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.}
1590:
of the position vector provide a quantitative measure of direction. In general, an object's position vector will depend on the frame of reference; different frames will lead to different values for the position vector.
15211:{\displaystyle ^{2}={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +\mathbf {v} _{O}\\0&0\end{bmatrix}}^{2}={\begin{bmatrix}\Omega ^{2}&-\Omega ^{2}\mathbf {d} +\Omega \mathbf {v} _{O}\\0&0\end{bmatrix}}.}
4618:
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4112:
4009:
10576:
3061:{\displaystyle \mathbf {a} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {{\text{d}}\mathbf {v} }{{\text{d}}t}}=a_{x}{\hat {\mathbf {x} }}+a_{y}{\hat {\mathbf {y} }}+a_{z}{\hat {\mathbf {z} }}.}
2387:{\displaystyle \mathbf {v} =\lim _{\Delta t\to 0}{\frac {\Delta \mathbf {r} }{\Delta t}}={\frac {{\text{d}}\mathbf {r} }{{\text{d}}t}}=v_{x}{\hat {\mathbf {x} }}+v_{y}{\hat {\mathbf {y} }}+v_{z}{\hat {\mathbf {z} }}.}
15588:, and (ii) constraints imposed on the velocity of the system such as the knife-edge constraint of ice-skates on a flat plane, or rolling without slipping of a disc or sphere in contact with a plane, which are called
15345:
11860:
9716:
12439:
is the constant angular acceleration. Although position in space and velocity in space are both true vectors (in terms of their properties under rotation), as is angular velocity, angle itself is not a true vector.
12122:
2398:
to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.
5486:
7880:{\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega {\hat {\mathbf {r} }}.}
5940:
12309:
10154:{\displaystyle ={\begin{bmatrix}A(\phi )&\mathbf {d} \\\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}.}
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and the distances between vertices unchanged. Kinematics is often described as applied geometry, where the movement of a mechanical system is described using the rigid transformations of
Euclidean geometry.
9788:
5642:
3650:
3566:
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14506:
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Figure 2: Velocity and acceleration for nonuniform circular motion: the velocity vector is tangential to the orbit, but the acceleration vector is not radially inward because of its tangential component
11708:
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1584:
12675:{\displaystyle \mathbf {P} (t)=\mathbf {p} ={\begin{bmatrix}\mathbf {P} \\1\end{bmatrix}}={\begin{bmatrix}A(t)&\mathbf {d} (t)\\0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {p} \\1\end{bmatrix}}.}
98:
12126:
15569:
14567:
10408:{\displaystyle \mathbf {P} =\mathbf {r} ={\begin{bmatrix}\cos \phi &-\sin \phi &d_{x}\\\sin \phi &\cos \phi &d_{y}\\0&0&1\end{bmatrix}}{\begin{bmatrix}x\\y\\1\end{bmatrix}}.}
5262:
14173:
7076:
1632:
9641:
8925:
7891:
10759:{\displaystyle \mathbf {v} _{P}={\dot {\mathbf {r} }}(t)={\dot {\mathbf {d} }}(t)=\mathbf {v} _{O},\quad \mathbf {a} _{P}={\ddot {\mathbf {r} }}(t)={\ddot {\mathbf {d} }}(t)=\mathbf {a} _{O},}
7660:
5341:
1129:
6939:
8884:
10958:
9402:{\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=r\omega {\hat {\mathbf {\theta } }}=v{\hat {\mathbf {\theta } }},}
3665:
6248:
16136:
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1484:
14312:
10852:
Rotational or angular kinematics is the description of the rotation of an object. In what follows, attention is restricted to simple rotation about an axis of fixed orientation. The
6252:
This reduces the parametric equations of motion of the particle to a
Cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that
2530:
15631:
5432:
A relationship between velocity, position and acceleration without explicit time dependence can be had by solving the average acceleration for time and substituting and simplifying
9806:
motion of the various parts of a complicated mechanical system to a problem of describing the geometry of each part and geometric association of each part relative to other parts.
1390:
1359:
1328:
832:
since it can be studied without considering the mass of a body or the forces acting upon it. A kinematics problem begins by describing the geometry of the system and declaring the
2167:
3306:
12999:
6852:
4893:
4864:
122:
1627:
15926:. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the
6576:
1506:
8242:
5608:{\displaystyle \left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =\left(\mathbf {v} -\mathbf {v} _{0}\right)\cdot {\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\ ,}
15729:
11521:
1054:, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.
828:
that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of both applied and pure
1124:
14790:{\displaystyle \mathbf {A} _{P}={\frac {d}{dt}}\mathbf {v} _{P}={\frac {d}{dt}}\left(\mathbf {P} \right)=\mathbf {P} +{\dot {\mathbf {P} }}=\mathbf {P} +\mathbf {P} .}
8147:
6875:
6374:
6348:
6314:
6185:
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2213:
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6543:
5633:
15801:
A spherical joint, or ball joint, requires that a point in the moving body maintain contact with a point in the fixed body. This joint has three degrees of freedom.
9176:
12070:
The equations of translational kinematics can easily be extended to planar rotational kinematics for constant angular acceleration with simple variable exchanges:
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1835:
1806:
1777:
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6440:
4919:
10479:
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6063:
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In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors (
2492:
1454:
1434:
1414:
1293:
1273:
1253:
11801:
9661:
6676:
12073:
5437:
9572:{\displaystyle \mathbf {a} _{P}={\frac {{\text{d}}(v{\hat {\mathbf {\theta } }})}{{\text{d}}t}}=a{\hat {\mathbf {\theta } }}-v\theta {\hat {\mathbf {r} }}.}
11157:
4347:
3841:
2409:
1102:
to the particle. It expresses both the distance of the point from the origin and its direction from the origin. In three dimensions, the position vector
11452:
12023:
11946:
3402:
is a vector that defines the position of one point relative to another. It is the difference in position of the two points. The position of one point
9721:
15460:
14455:
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1511:
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15856:
2889:
The acceleration of the particle is the limit of the average acceleration as the time interval approaches zero, which is the time derivative,
15804:
A planar joint requires that a plane in the moving body maintain contact with a plane in fixed body. This joint has three degrees of freedom.
15528:
14523:
9927:
can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2
4818:{\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}=\left(a_{C_{x}}-a_{B_{x}},a_{C_{y}}-a_{B_{y}},a_{C_{z}}-a_{B_{z}}\right)}
4312:{\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}=\left(v_{A_{x}}-v_{B_{x}},v_{A_{y}}-v_{B_{y}},v_{A_{z}}-v_{B_{z}}\right)}
788:
9856:
The movement of each of the components of the
Boulton & Watt Steam Engine (1784) is modeled by a continuous set of rigid displacements.
5892:{\displaystyle 2\left|\mathbf {r} -\mathbf {r} _{0}\right|\left|\mathbf {a} \right|\cos \alpha =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.}
9584:
3416:
377:
7408:{\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},}
496:
15335:{\displaystyle \mathbf {A} _{P}={\dot {\Omega }}(\mathbf {P} -\mathbf {d} )+\mathbf {A} _{O}+\Omega ^{2}(\mathbf {P} -\mathbf {d} ),}
14441:{\displaystyle \mathbf {v} _{P}=(\mathbf {P} -\mathbf {d} )+{\dot {\mathbf {d} }}=\omega \times \mathbf {R} _{P/O}+\mathbf {v} _{O},}
3577:
3493:
2494:
is the arc-length measured along the trajectory of the particle. This arc-length must always increase as the particle moves. Hence,
2215:
approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector,
10903:
1060:
Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2-d space, but a plane in any higher dimension.
469:
15446:{\displaystyle \mathbf {A} _{P}=\alpha \times \mathbf {R} _{P/O}+\omega \times \omega \times \mathbf {R} _{P/O}+\mathbf {A} _{O},}
15744:. Another example is a drum turned by the pull of gravity upon a falling weight attached to the rim by the inextensible cord. An
6190:
10464:, the motion is called pure translation. In this case, the trajectory of every point in the body is an offset of the trajectory
8343:
might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector
5259:
Additional relations between displacement, velocity, acceleration, and time can be derived. Since the acceleration is constant,
905:
used to describe motion. In engineering, for instance, kinematic analysis may be used to find the range of movement for a given
6053:{\displaystyle |\mathbf {v} |^{2}=|\mathbf {v} _{0}|^{2}+2\left|\mathbf {a} \right|\left|\mathbf {r} -\mathbf {r} _{0}\right|.}
54:
15787:
relative movement of the links, which therefore has one degree of freedom, which is pure rotation about the axis of the hinge.
14263:
12398:{\displaystyle \omega _{\mathrm {f} }^{2}=\omega _{\mathrm {i} }^{2}+2\alpha (\theta _{\mathrm {f} }-\theta _{\mathrm {i} }).}
16855:
16775:
16748:
16553:
16276:
16247:
16218:
12300:{\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }={\tfrac {1}{2}}(\omega _{\mathrm {f} }+\omega _{\mathrm {i} })t}
11119:{\displaystyle ={\begin{bmatrix}\cos(\theta (t))&-\sin(\theta (t))\\\sin(\theta (t))&\cos(\theta (t))\end{bmatrix}},}
16868:
16645:
5021:{\displaystyle \mathbf {v} (t)=\mathbf {v} _{0}+\int _{0}^{t}\mathbf {a} \,{\text{d}}\tau =\mathbf {v} _{0}+\mathbf {a} t.}
9860:
The position of one component of a mechanical system relative to another is defined by introducing a reference frame, say
4827:
Alternatively, this same result could be obtained by computing the second time derivative of the relative position vector
9903:
A smooth curve from one position to another in this configuration space is a continuous set of displacements, called the
8893:
451:
5750:{\displaystyle 2\left(\mathbf {r} -\mathbf {r} _{0}\right)\cdot \mathbf {a} =|\mathbf {v} |^{2}-|\mathbf {v} _{0}|^{2}.}
12934:
16931:
16906:
16848:
The
Machines of Leonardo Da Vinci and Franz Reuleaux, Kinematics of Machines from the Renaissance to the 20th Century
16834:
16721:
16688:
16657:
16611:
16580:
16387:
16172:
16145:
6880:
3810:{\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}=\left(x_{A}-x_{B},y_{A}-y_{B},z_{A}-z_{B}\right)}
781:
8849:
4321:
Alternatively, this same result could be obtained by computing the time derivative of the relative position vector
17:
743:
16470:
12206:{\displaystyle \theta _{\mathrm {f} }-\theta _{\mathrm {i} }=\omega _{\mathrm {i} }t+{\tfrac {1}{2}}\alpha t^{2}}
117:
868:, kinematics is used to describe the motion of systems composed of joined parts (multi-link systems) such as an
16886:
16304:
15852:
15831:
6350:
by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the
5331:{\displaystyle \mathbf {a} ={\frac {\Delta \mathbf {v} }{\Delta t}}={\frac {\mathbf {v} -\mathbf {v} _{0}}{t}}}
1459:
372:
112:
15818:. Similarly, the contact between the involute curves that form the meshing teeth of two gears are cam joints.
14253:{\displaystyle ={\begin{bmatrix}\Omega &-\Omega \mathbf {d} +{\dot {\mathbf {d} }}\\0&0\end{bmatrix}}}
7187:{\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},}
6255:
1743:{\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},}
9896:
2497:
15931:
16926:
16545:
Robot manipulators: mathematics, programming, and control : the computer control of robot manipulators
16526:
Geometry: the study of properties of given elements that remain invariant under specified transformations.
10805:
7959:{\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\theta {\hat {\mathbf {r} }}.}
1364:
1333:
1302:
774:
761:
522:
445:
367:
210:
11871:: the oriented distance from a selected origin on the rotational axis to a point of an object is a vector
9169:
A special case of a particle trajectory on a circular cylinder occurs when there is no movement along the
7725:{\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.}
5423:{\displaystyle \mathbf {r} (t)=\mathbf {r} _{0}+\left({\frac {\mathbf {v} +\mathbf {v} _{0}}{2}}\right)t.}
1228:{\displaystyle \mathbf {r} =(x,y,z)=x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }},}
1079:= (0 m, −50 m, 0 m). If the tower is 50 m high, and this height is measured along the
16409:
12494:
10450:
9802:
7223:
2147:
441:
242:
6187:
can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so
220:
661:
550:
476:
336:
269:
16803:
as the variable of integration, although that can be confused with
Lagrange's notation for derivatives
15927:
15855:
of a kinematic chain is computed from the number of links and the number and type of joints using the
4869:
4840:
2394:
Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is
16483:
13366:
9876:
defines the relative position of the two components. A displacement consists of the combination of a
6818:
1601:
415:
12931:
This expression uses the fact that the transpose of a rotation matrix is also its inverse, that is:
9718:
so the radial and tangential acceleration components for circular trajectories are also written as
9165:
Each particle on the wheel travels in a planar circular trajectory (Kinematics of
Machinery, 1876).
8887:
691:
535:
16680:
8886:
acts toward the center of curvature of the path at that point on the path, is commonly called the
6548:
1489:
16603:
16439:
16335:
15848:
6997:
6993:
1863:
934:
906:
641:
405:
16362:
15705:
15457:
is the angular acceleration vector obtained from the derivative of the angular velocity matrix;
3838:
The velocity of one point relative to another is simply the difference between their velocities
681:
15693:{\displaystyle {\boldsymbol {v}}_{G}(t)={\boldsymbol {\Omega }}\times {\boldsymbol {r}}_{G/O}.}
12454:
10422:
9881:
7005:
7001:
1105:
942:
857:
686:
503:
16765:
16713:
16628:
16320:
16264:
16206:
10864:
This allows the description of a rotation as the angular position of a planar reference frame
8239:
varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes:
6857:
6353:
6330:
6296:
6167:
5908:
5764:
2195:
2172:
16818:
16235:
10818:
points up for counterclockwise rotation and down for clockwise rotation, as specified by the
10426:
10418:
9412:
8928:
6445:
3399:
1296:
696:
671:
357:
175:
15996:
6788:
6518:
5618:
4921:
are known, the first integration yields the velocity of the particle as a function of time.
16001:
15876:
15589:
12458:
9655:
2546:
978:
918:
892:
884:
716:
676:
584:
580:
572:
562:
352:
345:
101:
6778:{\textstyle A={\frac {1}{2}}BH={\frac {1}{2}}att={\frac {1}{2}}at^{2}={\frac {at^{2}}{2}}}
6647:
6157:{\displaystyle |\mathbf {a} |=a,|\mathbf {v} |=v,|\mathbf {r} -\mathbf {r} _{0}|=\Delta r}
3385:{\displaystyle |\mathbf {a} |=|{\dot {\mathbf {v} }}|={\frac {{\text{d}}v}{{\text{d}}t}}.}
1811:
1782:
1753:
711:
8:
16921:
16511:
16356:
16106:
16076:
16021:
16011:
9161:
6621:
6419:
4898:
1044:
842:
813:
491:
432:
410:
155:
150:
145:
45:
16674:
16597:
15972:, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm",
16896:
16081:
16066:
16051:
16041:
16031:
15628:
of its angular velocity with a vector from the point of contact to the center of mass:
15613:
15601:
14452:
is the angular velocity vector obtained from the components of the matrix ; the vector
8939:
If the trajectory of the particle is constrained to lie on a cylinder, then the radius
6601:
6581:
6498:
6478:
6399:
6379:
2477:
1439:
1419:
1399:
1278:
1258:
1238:
914:
838:
833:
621:
362:
237:
205:
165:
15826:
10161:
These homogeneous transforms perform rigid transformations on the points in the plane
2545:
of a particle is the vector defined by the rate of change of the velocity vector. The
913:
to design a mechanism for a desired range of motion. In addition, kinematics applies
16895:, featuring movies and photos of hundreds of working models of mechanical systems at
16851:
16830:
16771:
16744:
16717:
16706:
16684:
16653:
16607:
16576:
16549:
16528:
16383:
16300:
16272:
16243:
16214:
16168:
16141:
16091:
16071:
16016:
16006:
15934:, the two ternary links do not have a common joint and are connected by binary links.
9798:
3825:
1028:
1012:
922:
902:
888:
631:
588:
545:
540:
481:
257:
247:
140:
8324:{\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.}
16086:
16061:
16046:
16036:
15909:
15585:
11619:{\displaystyle \mathbf {A} _{P}={\ddot {P}}(t)=\mathbf {P} +{\dot {\mathbf {P} }},}
10784:
are the velocity and acceleration, respectively, of the origin of the moving frame
10580:
Thus, for bodies in pure translation, the velocity and acceleration of every point
9904:
9821:
9430:
1867:
1587:
1095:
910:
898:
873:
853:
726:
706:
651:
646:
592:
567:
422:
280:
225:
200:
31:
16339:
9852:
891:, simplifying the derivation of the equations of motion. They are also central to
16843:
16823:
Companion
Encyclopedia of the History and Philosophy of the Mathematical Sciences
16738:
16570:
16543:
16377:
16294:
16189:
16162:
16131:
15923:
15843:
15791:
11999:
11922:
10819:
9928:
9915:
The motion of a body consists of a continuous set of rotations and translations.
9829:
8220:{\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.}
1099:
821:
721:
666:
616:
611:
530:
16900:
8943:
is constant and the velocity and acceleration vectors simplify. The velocity of
16626:
16595:
16507:
16056:
15838:
15770:
15766:
15761:
15621:
877:
748:
656:
557:
274:
16892:
16881:
16471:
https://www.mathsisfun.com/algebra/trig-area-triangle-without-right-angle.html
16458:
16433:
9240:{\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }},}
6319:
16915:
15969:
15625:
11783:{\displaystyle ={\begin{bmatrix}0&-\alpha \\\alpha &0\end{bmatrix}},}
11426:{\displaystyle ={\begin{bmatrix}0&-\omega \\\omega &0\end{bmatrix}},}
6948:
824:(objects), and systems of bodies (groups of objects) without considering the
636:
463:
27:
Branch of physics describing the motion of objects without considering forces
12473:
of a mechanical system is defined by the set of rotations and translations
6060:
This can be simplified using the notation for the magnitudes of the vectors
2541:
define velocity, can be applied to the velocity to define acceleration. The
16864:
15991:
15863:
of kinematic chains that have a given degree of freedom, which is known as
9820:(two dimensional space). Rigid transformations are those that preserve the
7438:) can be simplified by introducing the radial and tangential unit vectors,
7207:
7050:
plane. In this case, its velocity and acceleration take a convenient form.
4613:{\displaystyle \mathbf {a} _{B}=\left(a_{B_{x}},a_{B_{y}},a_{B_{z}}\right)}
4510:{\displaystyle \mathbf {a} _{C}=\left(a_{C_{x}},a_{C_{y}},a_{C_{z}}\right)}
4107:{\displaystyle \mathbf {v} _{B}=\left(v_{B_{x}},v_{B_{y}},v_{B_{z}}\right)}
4004:{\displaystyle \mathbf {v} _{A}=\left(v_{A_{x}},v_{A_{y}},v_{A_{z}}\right)}
3296:
2542:
1844:
The distance travelled is always greater than or equal to the displacement.
1837:
describe each coordinate of the particle's position as a function of time.
1393:
865:
849:
701:
626:
315:
195:
16767:
Mechanism design:enumeration of kinematic structures according to function
16404:
10571:{\displaystyle \mathbf {r} (t)=\mathbf {p} =\mathbf {d} (t)+\mathbf {p} .}
3483:
which is the difference between the components of their position vectors.
11855:{\displaystyle \alpha ={\frac {{\text{d}}^{2}\theta }{{\text{d}}t^{2}}}.}
9711:{\displaystyle \omega ={\dot {\theta }},\quad \alpha ={\ddot {\theta }},}
6987:
5636:
829:
809:
12443:
12117:{\displaystyle \omega _{\mathrm {f} }=\omega _{\mathrm {i} }+\alpha t\!}
5639:, which is appropriate as the products are scalars rather than vectors.
2406:
of an object is the magnitude of its velocity. It is a scalar quantity:
979:
Kinematics of a particle trajectory in a non-rotating frame of reference
5481:{\displaystyle t={\frac {\mathbf {v} -\mathbf {v} _{0}}{\mathbf {a} }}}
4411:
which is the difference between the components of their accelerations.
1840:
486:
15968:
For larger chains and their linkage topologies, see R. P. Sunkari and
6545:. Now let's find the top area (a triangle). The area of a triangle is
5767:
for more details) and the vectors by their magnitudes, in which case:
16826:
15702:
For the case of an object that does not tip or turn, this reduces to
508:
15769:
called the ideal connections between components that form a machine
11231:{\displaystyle \mathbf {v} _{P}={\dot {\mathbf {P} }}=\mathbf {p} .}
9816:
The coordinates of points in a plane are two-dimensional vectors in
4404:{\displaystyle \mathbf {a} _{C/B}=\mathbf {a} _{C}-\mathbf {a} _{B}}
3905:
which is the difference between the components of their velocities.
3898:{\displaystyle \mathbf {v} _{A/B}=\mathbf {v} _{A}-\mathbf {v} _{B}}
3830:
3473:{\displaystyle \mathbf {r} _{A/B}=\mathbf {r} _{A}-\mathbf {r} _{B}}
2467:{\displaystyle v=|\mathbf {v} |={\frac {{\text{d}}s}{{\text{d}}t}},}
1456:
coordinate axes, respectively. The magnitude of the position vector
16137:
A Treatise on the
Analytical Dynamics of Particles and Rigid Bodies
16101:
16026:
15986:
15860:
15749:
15741:
15617:
12449:
9877:
9809:
9792:
1854:
861:
427:
310:
285:
11998:
and sense determined by the direction of rotation as given by the
11864:
The description of rotation then involves these three quantities:
8235:) is not constrained to lie on a circular cylinder, so the radius
1629:, which defines the curve traced by the moving particle, given by
16096:
12435:
are, respectively, the initial and final angular velocities, and
11492:{\displaystyle \omega ={\frac {{\text{d}}\theta }{{\text{d}}t}}.}
7010:
It is often convenient to formulate the trajectory of a particle
3834:
Relative velocities between two particles in classical mechanics.
2395:
805:
400:
253:
170:
16191:
Elements of
Mechanics Including Kinematics, Kinetics and Statics
12060:{\displaystyle \alpha ={\frac {{\text{d}}\omega }{{\text{d}}t}}}
11983:{\displaystyle \omega ={\frac {{\text{d}}\theta }{{\text{d}}t}}}
10810:
2532:
is non-negative, which implies that speed is also non-negative.
12469:
In order to define these formulas, the movement of a component
9824:
between any two points. The set of rigid transformations in an
9783:{\displaystyle a_{r}=-r\omega ^{2},\quad a_{\theta }=r\alpha .}
3645:{\displaystyle \mathbf {r} _{B}=\left(x_{B},y_{B},z_{B}\right)}
3561:{\displaystyle \mathbf {r} _{A}=\left(x_{A},y_{A},z_{A}\right)}
869:
817:
459:
305:
215:
13020:) is obtained as the time derivative of this position vector,
10766:
where the dot denotes the derivative with respect to time and
7073:) traces its trajectory, which is a curve in space, given by:
1075:-axis, then the coordinate vector to the base of the tower is
16799:
is used as the variable of integration, some authors may use
16513:
The
Kinematics of Machinery: Outlines of a Theory of Machines
15506:{\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,}
14501:{\displaystyle \mathbf {R} _{P/O}=\mathbf {P} -\mathbf {d} ,}
13342:
The dot denotes the derivative with respect to time; because
11898:) on a plane perpendicular to the axis of rotation. Then the
9837:
2403:
1083:-axis, then the coordinate vector to the top of the tower is
983:
967:
954:
945:
825:
295:
290:
232:
12421:
are, respectively, the initial and final angular positions,
11990:
The angular velocity is represented in Figure 1 by a vector
11126:
is the rotation matrix that defines the angular position of
9868:
on the other. The rigid transformation, or displacement, of
2549:
of a particle over a time interval is defined as the ratio.
15944:= 10 : eight-bar linkage with 16 different topologies;
14589:
is obtained as the time derivative of its velocity vector:
14317:
Multiplying by the operator , the formula for the velocity
11703:{\displaystyle \mathbf {A} _{P}=\mathbf {P} +\mathbf {P} ,}
11355:{\displaystyle \mathbf {v} _{P}=\mathbf {P} =\mathbf {P} ,}
9923:
The combination of a rotation and translation in the plane
5759:
The dot product can be replaced by the cosine of the angle
1579:{\displaystyle |\mathbf {r} |={\sqrt {x^{2}+y^{2}+z^{2}}}.}
300:
263:
15954:= 13 : ten-bar linkage with 230 different topologies;
7415:
where the constant distance from the center is denoted as
11887:) has some projection (or, equivalently, some component)
10799:
93:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
30:"Kinematic" redirects here. For the Australian band, see
16627:
William Thomson Kelvin & Peter Guthrie Tait (1894).
16596:
William Thomson Kelvin & Peter Guthrie Tait (1894).
16375:
16269:
Mechanical Systems, Classical Models: Particle Mechanics
16358:
A History of European Thought in the Nineteenth Century
16130:
15564:{\displaystyle \mathbf {A} _{O}={\ddot {\mathbf {d} }}}
14562:{\displaystyle \mathbf {v} _{O}={\dot {\mathbf {d} }},}
13349:
This formula can be modified to obtain the velocity of
3303:| of its acceleration vector. It is a scalar quantity:
887:, are used to describe the movement of components in a
16903:
of classic texts on mechanical design and engineering.
15571:
is the acceleration of the origin of the moving frame
15134:
15066:
14934:
14832:
14194:
14084:
13985:
13938:
13833:
13786:
13745:
13676:
13629:
13582:
13525:
13487:
13306:
13232:
13201:
13140:
13082:
12886:
12801:
12770:
12646:
12590:
12559:
12481:) assembled into the homogeneous transformation =. If
12250:
12179:
11743:
11386:
10995:
10374:
10270:
10184:
define the coordinates of points in a reference frame
10049:
9997:
7236:
that moves only on the surface of a circular cylinder
6988:
Particle trajectories in cylindrical-polar coordinates
6941:. This equation is applicable when the final velocity
6883:
6821:
6679:
6551:
6258:
5218:
4837:
Assuming that the initial conditions of the position,
4344:
is simply the difference between their accelerations.
1857:
of a particle is a vector quantity that describes the
16652:. Research & Education Association. p. 613.
16187:
15964:= 16 : twelve-bar linkage with 6,856 topologies.
15708:
15634:
15531:
15463:
15348:
15235:
15040:
14805:
14595:
14526:
14458:
14330:
14266:
14176:
13379:
13026:
12937:
12711:
12510:
12444:
Point trajectories in body moving in three dimensions
12312:
12218:
12129:
12076:
12026:
11949:
11804:
11716:
11632:
11524:
11455:
11368:
11256:
11160:
10968:
10906:
10590:
10482:
10441:, translations become affine linear transformations.
10222:
10204:
and rotated by the angle φ relative to the x-axis of
9962:
9724:
9664:
9587:
9463:
9415:
9273:
9179:
8964:
8896:
8852:
8627:
8366:
8245:
8150:
7971:
7894:
7737:
7663:
7657:
and their time derivatives from elementary calculus:
7444:
7270:
7079:
6860:
6791:
6650:
6624:
6604:
6584:
6521:
6501:
6481:
6448:
6422:
6402:
6382:
6356:
6333:
6299:
6193:
6170:
6066:
5943:
5911:
5773:
5645:
5621:
5495:
5440:
5344:
5265:
5036:
4927:
4901:
4872:
4843:
4634:
4527:
4424:
4350:
4128:
4021:
3918:
3844:
3668:
3580:
3496:
3419:
3309:
3076:
2895:
2555:
2500:
2480:
2412:
2221:
2198:
2175:
2150:
1876:
1814:
1785:
1756:
1635:
1604:
1514:
1492:
1462:
1442:
1422:
1402:
1367:
1336:
1305:
1281:
1261:
1241:
1132:
1108:
57:
16893:
Kinematic Models for Design Digital Library (KMODDL)
16887:
Physclips: Mechanics with animations and video clips
16672:
15930:, the two ternary links have a common joint; in the
11921:) in a known rotation sense (typically given by the
11518:
is obtained as the time derivative of the velocity,
9636:{\displaystyle a_{r}=-v\theta ,\quad a_{\theta }=a,}
5338:
can be substituted into the above equation to give:
2169:
is the displacement vector during the time interval
16204:
16160:
14799:This equation can be expanded firstly by computing
11994:pointing along the axis of rotation with magnitude
8920:{\displaystyle v\omega {\hat {\mathbf {\theta } }}}
5030:A second integration yields its path (trajectory),
987:Kinematic quantities of a classical particle: mass
16705:
16568:
16531:. Merriam-Webster on-line dictionary. 31 May 2023.
15723:
15692:
15563:
15505:
15445:
15334:
15210:
15026:
14789:
14561:
14500:
14440:
14306:
14252:
14162:
13334:
12993:
12923:
12690:= (X, Y, Z), which is hopefully clear in context.
12674:
12397:
12299:
12205:
12116:
12059:
11982:
11854:
11782:
11702:
11618:
11491:
11425:
11354:
11230:
11118:
10952:
10758:
10570:
10407:
10153:
9782:
9710:
9635:
9571:
9421:
9401:
9239:
9146:
8919:
8878:
8836:
8593:
8323:
8219:
8126:
7958:
7879:
7724:
7649:
7407:
7186:
6933:
6869:
6846:
6807:
6777:
6665:
6636:
6610:
6590:
6570:
6537:
6507:
6487:
6467:
6434:
6408:
6388:
6368:
6342:
6308:
6285:
6242:
6179:
6156:
6052:
5929:
5891:
5749:
5627:
5607:
5480:
5422:
5330:
5249:
5020:
4913:
4887:
4858:
4817:
4612:
4509:
4403:
4311:
4106:
4003:
3897:
3809:
3644:
3560:
3472:
3384:
3281:
3060:
2870:
2524:
2486:
2466:
2386:
2207:
2184:
2161:
2136:
1829:
1800:
1771:
1742:
1621:
1578:
1500:
1478:
1448:
1428:
1408:
1384:
1353:
1322:
1287:
1267:
1247:
1227:
1118:
92:
16708:Mathematical Thought from Ancient to Modern Times
15859:. This formula can also be used to enumerate the
15851:and robots are examples of kinematic chains. The
15748:problem (i.e. not kinematic) of this type is the
15513:is the relative position vector (the position of
12990:
12697:can be inverted to compute the coordinate vector
12113:
11242:and write this as an operation on the trajectory
3410:is simply the difference between their positions
2583:
2562:
1883:
1866:) by the time interval. This ratio is called the
16913:
16907:Micro-Inch Positioning with Kinematic Components
16869:"Foundations and goals of analytical kinematics"
16770:(illustrated ed.). CRC Press. p. 121.
16743:(reprint ed.). Cambridge University Press.
16262:
16233:
9793:Point trajectories in a body moving in the plane
6934:{\textstyle \Delta r=v_{0}t+{\frac {at^{2}}{2}}}
3086:
2905:
2231:
16506:
15898:= 1 : a two-bar linkage that is the lever;
9864:, on one that moves relative to a fixed frame,
9156:
8612:, which is the time derivative of the velocity
7264:plane can be used to define the trajectory as,
7252:with the axis of the cylinder. Then, the angle
7065:. As the particle moves, its coordinate vector
15616:without slipping obeys the condition that the
12500:, then the trajectory of this point traced in
11906:from a reference axis (typically the positive
11238:It is convenient to eliminate the coordinates
10433:. However, using projective geometry, so that
8879:{\displaystyle -v\theta {\hat {\mathbf {r} }}}
16817:Koetsier, Teun (1994), "§8.3 Kinematics", in
16213:(12th ed.). Prentice Hall. p. 298.
15604:exactly constrains all 6 degrees of freedom.
10953:{\displaystyle \mathbf {P} (t)=\mathbf {p} ,}
933:The term kinematic is the English version of
782:
16703:
16575:. Cambridge, England: Cambridge University.
16242:(2nd ed.). Cambridge University Press.
12448:Important formulas in kinematics define the
12008:: the magnitude of the angular acceleration
4628:is the difference between their components:
4122:is the difference between their components:
3662:is the difference between their components:
1598:of a particle is a vector function of time,
16459:https://www.youtube.com/watch?v=jLJLXka2wEM
16313:
16292:
15607:
12682:This notation does not distinguish between
11433:is known as the angular velocity matrix of
9847:
6243:{\displaystyle v^{2}=v_{0}^{2}+2a\Delta r.}
5765:Geometric interpretation of the dot product
3394:
1071:-axis and north is in the direction of the
841:, not kinematics. For further details, see
15592:. The following are some common examples.
12012:is the rate at which the angular velocity
11935:is the rate at which the angular position
10421:. This formulation is necessary because a
9258:are constants. In this case, the velocity
7244:) = constant, it is possible to align the
1067:such that east is in the direction of the
953:("movement, motion"), itself derived from
901:is the process of measuring the kinematic
789:
775:
16643:
16620:
16140:. Cambridge University Press. Chapter 1.
10165:= 1, that is, on points with coordinates
9828:-dimensional space is called the special
8950:is the time derivative of the trajectory
8352:is the time derivative of the trajectory
7053:Recall that the trajectory of a particle
6316:is the area under a velocity–time graph.
6286:{\textstyle \Delta r=\int v\,{\text{d}}t}
6274:
5098:
4980:
2867:
2133:
1870:over that time interval and is defined as
1479:{\displaystyle \left|\mathbf {r} \right|}
1022:, always points radially from the origin.
16816:
16736:
16124:
15830:Illustration of a four-bar linkage from
15825:
15578:
14307:{\displaystyle ={\dot {A}}A^{\text{T}},}
10809:
10796:is constant, so its derivative is zero.
9918:
9851:
9160:
6947:
6495:here is different from the acceleration
6318:
4331:
3829:
1839:
982:
16889:from the University of New South Wales.
16562:
16369:
16286:
16207:"Kinematics and kinetics of a particle"
15669:
15637:
10856:-axis has been chosen for convenience.
10196:is displaced by the translation vector
2525:{\displaystyle {\text{d}}s/{\text{d}}t}
1094:The position vector of a particle is a
1038:, always tangent to the path of motion.
883:Geometric transformations, also called
14:
16914:
16666:
16637:
16602:. Cambridge University Press. p.
16589:
16473:Area of Triangles Without Right Angles
16334:
11790:is the angular acceleration matrix of
10814:Figure 1: The angular velocity vector
10800:Rotation of a body around a fixed axis
9654:The notation for angular velocity and
6962:that increases the rate of rotation: d
6515:). This means that the bottom area is
2192:. In the limit that the time interval
16697:
16382:. Dover Publications. preface, p. 5.
16341:Essai sur la Philosophie des Sciences
16181:
15595:
13346:is constant, its derivative is zero.
1848:
1486:gives the distance between the point
1385:{\displaystyle {\hat {\mathbf {z} }}}
1354:{\displaystyle {\hat {\mathbf {y} }}}
1323:{\displaystyle {\hat {\mathbf {x} }}}
1087:= (0 m, −50 m, 50 m).
471:Newton's law of universal gravitation
16842:
16763:
16541:
16354:
16319:J. M. McCarthy and G. S. Soh, 2010,
16154:
16112:Chebychev–Grübler–Kutzbach criterion
15837:Rigid bodies ("links") connected by
15734:
12693:This equation for the trajectory of
11445:is the time derivative of the angle
7061:measured in a fixed reference frame
7057:is defined by its coordinate vector
3819:
16867:(1913) D.H. Delphenich translator,
16712:. Oxford University Press. p.
16167:. Taylor & Francis. p. 1.
15821:
13373:into the velocity equation yields:
10444:
2162:{\displaystyle \Delta \mathbf {r} }
856:and collections of such bodies. In
452:Mechanics of planar particle motion
60:
24:
16810:
16402:
15755:
15301:
15254:
15170:
15153:
15138:
15077:
15069:
14974:
14957:
14940:
14872:
14855:
14838:
14349:
14270:
14205:
14197:
12383:
12368:
12339:
12319:
12285:
12270:
12240:
12225:
12166:
12151:
12136:
12098:
12083:
11723:
11686:
11677:
11654:
11593:
11570:
11372:
11338:
10449:If a rigid body moves so that its
8934:
7042:)) using polar coordinates in the
6884:
6861:
6334:
6300:
6259:
6231:
6171:
6148:
5287:
5277:
3133:
3120:
3093:
2936:
2926:
2909:
2736:
2712:
2683:
2659:
2630:
2606:
2591:
2574:
2262:
2252:
2235:
2199:
2176:
2151:
2002:
1994:
1965:
1957:
1928:
1920:
1905:
1895:
25:
16943:
16875:
16740:Freedom in Machinery, Volumes 1–2
16376:O. Bottema & B. Roth (1990).
15997:Affine geometry § Kinematics
15875:The planar one degree-of-freedom
15220:The formula for the acceleration
14576:
11879:) locating the point. The vector
11501:
10460:= 0) relative to the fixed frame
10417:Homogeneous transforms represent
9938:) and the 2×1 translation vector
15976:#41, pp. 1021–1030 (2006).
15660:
15551:
15534:
15496:
15488:
15466:
15430:
15407:
15372:
15351:
15322:
15314:
15287:
15275:
15267:
15238:
15175:
15163:
15090:
15081:
14994:
14979:
14967:
14896:
14879:
14865:
14780:
14754:
14722:
14702:
14671:
14628:
14598:
14546:
14529:
14491:
14483:
14461:
14425:
14402:
14381:
14367:
14359:
14333:
14314:is the angular velocity matrix.
14220:
14209:
14149:
14122:
14088:
14049:
14038:
13942:
13903:
13892:
13790:
13757:
13697:
13633:
13591:
13546:
13492:
13459:
13386:
13310:
13262:
13205:
13158:
13087:
13070:
13029:
12994:{\displaystyle ^{\text{T}}=I.\!}
12890:
12848:
12774:
12749:
12713:
12650:
12608:
12563:
12547:
12512:
11693:
11667:
11635:
11603:
11583:
11527:
11345:
11328:
11259:
11221:
11180:
11163:
10943:
10908:
10743:
10719:
10693:
10676:
10660:
10636:
10610:
10593:
10561:
10544:
10536:
10516:
10484:
10258:
10247:
10224:
10024:
10015:
9979:
9887:The set of all displacements of
9797:The movement of components of a
9556:
9466:
9338:
9318:
9276:
9224:
9204:
9181:
9131:
9084:
9029:
9009:
8967:
8866:
8821:
8759:
8719:
8672:
8630:
8578:
8531:
8508:
8456:
8431:
8411:
8369:
8308:
8279:
8247:
8204:
8175:
8152:
8084:
8045:
7943:
7864:
7757:
7676:
7634:
7593:
7531:
7490:
7449:
7430:The cylindrical coordinates for
7392:
7363:
7319:
7272:
7171:
7142:
7113:
7081:
6847:{\textstyle {\frac {at^{2}}{2}}}
6130:
6121:
6097:
6073:
6032:
6023:
6009:
5976:
5950:
5864:
5838:
5812:
5792:
5783:
5722:
5696:
5683:
5664:
5655:
5583:
5574:
5552:
5543:
5530:
5511:
5502:
5472:
5460:
5451:
5394:
5385:
5364:
5346:
5312:
5303:
5281:
5267:
5230:
5201:
5186:
5161:
5147:
5112:
5085:
5056:
5038:
5008:
4994:
4976:
4947:
4929:
4888:{\displaystyle \mathbf {v} _{0}}
4875:
4859:{\displaystyle \mathbf {r} _{0}}
4846:
4675:
4660:
4637:
4530:
4427:
4391:
4376:
4353:
4169:
4154:
4131:
4024:
3921:
3885:
3870:
3847:
3709:
3694:
3671:
3583:
3499:
3460:
3445:
3422:
3337:
3316:
3266:
3239:
3212:
3171:
3124:
3078:
3045:
3018:
2991:
2957:
2930:
2897:
2857:
2821:
2785:
2749:
2696:
2643:
2580:
2559:
2425:
2371:
2344:
2317:
2283:
2256:
2223:
2155:
2123:
2087:
2051:
2015:
1978:
1941:
1899:
1880:
1727:
1698:
1669:
1637:
1606:
1521:
1494:
1468:
1372:
1341:
1310:
1212:
1192:
1172:
1134:
1111:
1043:
1027:
1011:
941:, which he constructed from the
756:
755:
742:
75:
16789:
16757:
16730:
16535:
16520:
16500:
16476:
16464:
16452:
16442:from the original on 2021-11-13
16426:
16417:
16396:
16348:
16211:Engineering Mechanics: Dynamics
15808:
15612:An object that rolls against a
13353:by operating on its trajectory
10788:. Recall the coordinate vector
10673:
9757:
9686:
9613:
7544:
4620:then the acceleration of point
3299:of an object is the magnitude |
2535:
1622:{\displaystyle \mathbf {r} (t)}
909:and, working in reverse, using
16630:Elements of Natural Philosophy
16599:Elements of Natural Philosophy
16510:; Kennedy, Alex B. W. (1876),
16461:Crash course physics integrals
16361:. Blackwood, London. pp.
16328:
16256:
16227:
16198:
16188:Thomas Wallace Wright (1896).
15887:hinges or sliding joints are:
15653:
15647:
15326:
15310:
15279:
15263:
15048:
15041:
14821:
14806:
14776:
14770:
14767:
14761:
14750:
14735:
14715:
14709:
14698:
14683:
14667:
14661:
14569:is the velocity of the origin
14371:
14355:
14352:
14346:
14273:
14267:
14183:
14177:
14145:
14139:
14098:
14092:
13952:
13946:
13800:
13794:
13643:
13637:
13469:
13463:
13446:
13442:
13436:
13430:
13427:
13424:
13418:
13403:
13361:) measured in the fixed frame
13278:
13272:
13253:
13247:
13168:
13162:
13152:
13146:
13066:
13063:
13057:
13042:
12978:
12975:
12969:
12963:
12954:
12950:
12944:
12938:
12900:
12894:
12858:
12852:
12838:
12831:
12814:
12807:
12759:
12753:
12736:
12732:
12726:
12720:
12618:
12612:
12602:
12596:
12543:
12540:
12534:
12528:
12522:
12516:
12485:is the coordinates of a point
12389:
12359:
12291:
12261:
11732:
11717:
11689:
11683:
11680:
11674:
11663:
11648:
11596:
11590:
11579:
11564:
11558:
11552:
11375:
11369:
11341:
11335:
11324:
11312:
11305:
11299:
11296:
11293:
11287:
11272:
11217:
11214:
11208:
11193:
11102:
11099:
11093:
11087:
11076:
11073:
11067:
11061:
11048:
11045:
11039:
11033:
11019:
11016:
11010:
11004:
10984:
10981:
10975:
10969:
10939:
10936:
10930:
10924:
10918:
10912:
10830:) changes with time at a rate
10735:
10729:
10709:
10703:
10652:
10646:
10626:
10620:
10554:
10548:
10532:
10529:
10526:
10520:
10506:
10500:
10494:
10488:
10254:
10251:
10237:
10231:
10188:coincident with a fixed frame
10009:
10003:
9986:
9983:
9969:
9963:
9643:are called, respectively, the
9560:
9537:
9507:
9501:
9487:
9390:
9370:
9342:
9322:
9228:
9208:
9191:
9185:
9135:
9108:
9088:
9061:
9033:
9013:
8911:
8870:
8825:
8798:
8787:
8772:
8763:
8752:
8737:
8723:
8696:
8676:
8582:
8558:
8552:
8535:
8524:
8512:
8485:
8460:
8435:
8415:
8312:
8301:
8295:
8283:
8272:
8266:
8257:
8251:
8208:
8197:
8191:
8179:
8162:
8156:
8115:
8088:
8055:
8049:
8032:
7995:
7947:
7911:
7868:
7848:
7818:
7812:
7798:
7761:
7713:
7680:
7638:
7627:
7624:
7618:
7612:
7597:
7586:
7583:
7577:
7571:
7553:
7535:
7524:
7521:
7515:
7509:
7494:
7483:
7480:
7474:
7468:
7453:
7396:
7385:
7379:
7367:
7356:
7353:
7347:
7341:
7323:
7312:
7309:
7303:
7297:
7282:
7276:
7175:
7164:
7158:
7146:
7135:
7129:
7117:
7106:
7100:
7091:
7085:
6141:
6116:
6102:
6092:
6078:
6068:
5988:
5970:
5956:
5945:
5876:
5858:
5844:
5833:
5734:
5716:
5702:
5691:
5356:
5350:
5095:
5089:
5048:
5042:
4939:
4933:
4336:The acceleration of one point
3348:
3329:
3321:
3311:
3270:
3243:
3216:
3140:
3130:
3109:
3100:
3090:
3049:
3022:
2995:
2915:
2861:
2841:
2825:
2805:
2789:
2769:
2753:
2722:
2700:
2669:
2647:
2616:
2430:
2420:
2375:
2348:
2321:
2241:
2127:
2107:
2091:
2071:
2055:
2035:
2019:
1982:
1945:
1824:
1818:
1795:
1789:
1766:
1760:
1731:
1720:
1714:
1702:
1691:
1685:
1673:
1662:
1656:
1647:
1641:
1616:
1610:
1526:
1516:
1376:
1345:
1314:
1216:
1196:
1176:
1159:
1141:
1091:respect to a reference frame.
13:
1:
16673:Irving Porter Church (1908).
16322:Geometric Design of Linkages,
16240:Dynamics of Multibody Systems
16118:
15832:Kinematics of Machinery, 1876
15776:
13003:
12464:
12016:changes with respect to time
11939:changes with respect to time
6618:is the height. In this case,
6571:{\textstyle {\frac {1}{2}}BH}
2882:is the average velocity and Δ
1098:drawn from the origin of the
378:Koopman–von Neumann mechanics
16882:Java applet of 1D kinematics
16650:The Mechanics Problem Solver
16548:. MIT Press, Cambridge, MA.
16205:Russell C. Hibbeler (2009).
16161:Joseph Stiles Beggs (1983).
15974:Mechanism and Machine Theory
14581:The acceleration of a point
10806:Rotation around a fixed axis
9801:are analyzed by attaching a
9157:Planar circular trajectories
8603:Similarly, the acceleration
6416:is the height. In this case
4521:has acceleration components
4418:has acceleration components
1501:{\displaystyle \mathbf {r} }
928:
446:Non-inertial reference frame
7:
16569:R. Douglas Gregory (2006).
16410:Online Etymology Dictionary
16299:. Oxford University Press.
16194:. E and FN Spon. Chapter 1.
15979:
15870:
11902:of that point is the angle
11137:
10892:are related to coordinates
10859:
10192:. Then, when the origin of
8227:In general, the trajectory
6323:Velocity Time physics graph
4114:then the velocity of point
3654:then the position of point
373:Appell's equation of motion
243:Inertial frame of reference
10:
16948:
15759:
15724:{\displaystyle v=r\omega }
13008:The velocity of the point
10803:
10584:in the body are given by:
10437:is considered a subset of
10200:relative to the origin of
6991:
4340:relative to another point
3823:
3406:relative to another point
3400:A relative position vector
968:
955:
946:
852:to describe the motion of
29:
16271:. Springer. p. 287.
16263:P. P. Teodorescu (2007).
16234:Ahmed A. Shabana (2003).
15590:non-holonomic constraints
14170:The matrix is given by:
10208:, the new coordinates in
7427:) is a function of time.
5763:between the vectors (see
1119:{\displaystyle {\bf {r}}}
16932:Mechanisms (engineering)
16676:Mechanics of Engineering
16529:"Definition of geometry"
16485:kinematics.gif (508×368)
15841:("joints") are known as
15608:Rolling without slipping
15229:can now be obtained as:
10900:by the matrix equation:
9848:Displacements and motion
8888:centripetal acceleration
7256:around this axis in the
7248:axis of the fixed frame
6877:results in the equation
6870:{\displaystyle \Delta r}
6854:results in the equation
6369:{\displaystyle A\cdot B}
6343:{\displaystyle \Delta r}
6309:{\displaystyle \Delta r}
6180:{\displaystyle \Delta r}
5930:{\displaystyle \cos 0=1}
4015:has velocity components
3912:has velocity components
3574:has position components
3490:has position components
3395:Relative position vector
2208:{\displaystyle \Delta t}
2185:{\displaystyle \Delta t}
536:Rotating reference frame
368:Hamilton–Jacobi equation
16764:Tsai, Lung-Wen (2001).
16737:Phillips, Jack (2007).
16132:Edmund Taylor Whittaker
15517:relative to the origin
14512:relative to the origin
12493:measured in the moving
11931:: the angular velocity
11134:as a function of time.
9422:{\displaystyle \omega }
6998:Curvilinear coordinates
6994:Generalized coordinates
6468:{\displaystyle B=v_{0}}
477:Newton's laws of motion
337:Newton's laws of motion
16819:Grattan-Guinness, Ivor
16542:Paul, Richard (1981).
16379:Theoretical Kinematics
16236:"Reference kinematics"
15834:
15725:
15694:
15565:
15507:
15447:
15336:
15212:
15028:
14791:
14563:
14502:
14442:
14308:
14254:
14164:
13336:
12995:
12925:
12676:
12399:
12301:
12207:
12118:
12061:
11984:
11856:
11784:
11704:
11620:
11493:
11427:
11356:
11232:
11120:
10954:
10849:
10760:
10572:
10419:affine transformations
10409:
10155:
9857:
9784:
9712:
9637:
9573:
9441:axis of the cylinder.
9423:
9403:
9241:
9166:
9148:
8921:
8880:
8838:
8595:
8325:
8221:
8128:
7960:
7881:
7726:
7651:
7409:
7188:
7006:Frenet-Serret formulas
7002:Orthogonal coordinates
6984:
6935:
6871:
6848:
6809:
6808:{\displaystyle v_{0}t}
6779:
6667:
6638:
6612:
6592:
6572:
6539:
6538:{\displaystyle tv_{0}}
6509:
6489:
6469:
6436:
6410:
6390:
6370:
6344:
6324:
6310:
6287:
6244:
6181:
6158:
6054:
5931:
5893:
5751:
5629:
5628:{\displaystyle \cdot }
5609:
5482:
5424:
5332:
5251:
5022:
4915:
4889:
4860:
4819:
4614:
4511:
4405:
4313:
4108:
4005:
3899:
3835:
3811:
3646:
3562:
3474:
3386:
3283:
3062:
2886:is the time interval.
2872:
2526:
2488:
2468:
2388:
2209:
2186:
2163:
2138:
1845:
1831:
1802:
1773:
1744:
1623:
1580:
1502:
1480:
1450:
1430:
1410:
1386:
1355:
1324:
1289:
1269:
1249:
1229:
1120:
1004:
858:mechanical engineering
848:Kinematics is used in
504:Simple harmonic motion
417:Euler's laws of motion
211:D'Alembert's principle
94:
16829:, pp. 994–1001,
16704:Morris Kline (1990).
15829:
15726:
15695:
15586:holonomic constraints
15579:Kinematic constraints
15566:
15508:
15448:
15337:
15213:
15029:
14792:
14564:
14503:
14443:
14309:
14255:
14165:
13337:
13012:along its trajectory
12996:
12926:
12677:
12400:
12302:
12208:
12119:
12062:
11985:
11910:-axis) to the vector
11857:
11785:
11705:
11621:
11494:
11428:
11357:
11233:
11121:
10955:
10813:
10761:
10573:
10427:linear transformation
10410:
10156:
9919:Matrix representation
9855:
9785:
9713:
9649:tangential components
9638:
9574:
9424:
9404:
9242:
9164:
9149:
8929:Coriolis acceleration
8922:
8881:
8839:
8596:
8326:
8222:
8136:Using this notation,
8129:
7961:
7882:
7727:
7652:
7410:
7189:
6951:
6936:
6872:
6849:
6810:
6780:
6668:
6639:
6613:
6593:
6573:
6540:
6510:
6490:
6470:
6437:
6411:
6391:
6371:
6345:
6322:
6311:
6288:
6245:
6182:
6159:
6055:
5932:
5894:
5752:
5630:
5610:
5483:
5425:
5333:
5252:
5023:
4916:
4890:
4861:
4820:
4615:
4512:
4406:
4332:Relative acceleration
4314:
4109:
4006:
3900:
3833:
3812:
3647:
3563:
3475:
3387:
3295:The magnitude of the
3284:
3063:
2873:
2527:
2489:
2469:
2389:
2210:
2187:
2164:
2139:
1843:
1832:
1803:
1774:
1745:
1624:
1581:
1503:
1481:
1451:
1431:
1411:
1387:
1356:
1325:
1297:Cartesian coordinates
1290:
1270:
1250:
1230:
1121:
986:
885:rigid transformations
816:, that describes the
358:Hamiltonian mechanics
176:Statistical mechanics
95:
16423:Crash course physics
16293:A. Biewener (2003).
16002:Analytical mechanics
15706:
15632:
15529:
15521:of the moving frame
15461:
15346:
15233:
15038:
14803:
14593:
14524:
14516:of the moving frame
14456:
14328:
14264:
14174:
13377:
13024:
12935:
12709:
12686:= (X, Y, Z, 1), and
12508:
12459:Lagrange's equations
12310:
12216:
12127:
12074:
12024:
12006:Angular acceleration
11947:
11802:
11714:
11630:
11522:
11506:The acceleration of
11453:
11366:
11254:
11158:
10966:
10904:
10868:relative to a fixed
10588:
10480:
10220:
9960:
9722:
9662:
9658:is often defined as
9656:angular acceleration
9585:
9461:
9413:
9271:
9177:
8962:
8894:
8850:
8625:
8364:
8243:
8148:
7969:
7892:
7735:
7661:
7442:
7268:
7232:Consider a particle
7077:
6881:
6858:
6819:
6789:
6677:
6666:{\displaystyle H=at}
6648:
6622:
6602:
6582:
6549:
6519:
6499:
6479:
6446:
6420:
6400:
6380:
6354:
6331:
6297:
6256:
6191:
6168:
6064:
5941:
5909:
5771:
5643:
5619:
5493:
5438:
5342:
5263:
5034:
4925:
4899:
4870:
4841:
4632:
4525:
4422:
4348:
4126:
4019:
3916:
3842:
3666:
3578:
3494:
3417:
3307:
3074:
2893:
2553:
2547:average acceleration
2498:
2478:
2410:
2219:
2196:
2173:
2148:
1874:
1830:{\displaystyle z(t)}
1812:
1801:{\displaystyle y(t)}
1783:
1772:{\displaystyle x(t)}
1754:
1633:
1602:
1512:
1490:
1460:
1440:
1420:
1400:
1365:
1334:
1303:
1279:
1259:
1239:
1130:
1126:can be expressed as
1106:
1050:Acceleration vector
919:mechanical advantage
917:to the study of the
581:Angular acceleration
573:Rotational frequency
353:Lagrangian mechanics
346:Analytical mechanics
102:Second law of motion
55:
16927:Classical mechanics
16516:, London: Macmillan
16355:Merz, John (1903).
16336:Ampère, André-Marie
16325:Springer, New York.
16107:Integral kinematics
16077:Kinematic synthesis
16022:Classical mechanics
16012:Celestial mechanics
15932:Stephenson topology
15867:in machine design.
14508:is the position of
13365:. Substituting the
12455:Newton's second law
12349:
12329:
10876:-axis. Coordinates
10822:. Angular position
10472:) of the origin of
10180:In particular, let
9897:configuration space
9433:of the unit vector
6637:{\displaystyle B=t}
6435:{\displaystyle A=t}
6221:
5139:
5083:
4974:
4914:{\displaystyle t=0}
911:kinematic synthesis
843:analytical dynamics
814:classical mechanics
433:Harmonic oscillator
411:Equations of motion
46:Classical mechanics
40:Part of a series on
16897:Cornell University
16644:M. Fogiel (1980).
16082:Kinetics (physics)
16067:Kinematic coupling
16052:Inverse kinematics
16042:Forward kinematics
16032:Dynamics (physics)
15835:
15721:
15690:
15602:kinematic coupling
15596:Kinematic coupling
15561:
15503:
15443:
15332:
15208:
15199:
15114:
15024:
15018:
14920:
14787:
14559:
14498:
14438:
14304:
14250:
14244:
14160:
14158:
14110:
14073:
13964:
13927:
13812:
13775:
13721:
13655:
13609:
13570:
13511:
13332:
13323:
13295:
13218:
13185:
13106:
12991:
12921:
12912:
12875:
12787:
12672:
12663:
12635:
12576:
12395:
12333:
12313:
12297:
12259:
12203:
12188:
12114:
12057:
11980:
11852:
11780:
11771:
11700:
11616:
11489:
11423:
11414:
11352:
11228:
11150:, its velocity in
11116:
11107:
10950:
10872:about this shared
10850:
10756:
10568:
10405:
10396:
10363:
10151:
10142:
10035:
9858:
9780:
9708:
9633:
9569:
9419:
9399:
9237:
9167:
9144:
8917:
8876:
8834:
8591:
8321:
8217:
8144:) takes the form,
8124:
7956:
7877:
7722:
7647:
7405:
7184:
6985:
6931:
6867:
6844:
6805:
6775:
6663:
6634:
6608:
6588:
6568:
6535:
6505:
6485:
6465:
6432:
6406:
6386:
6366:
6340:
6325:
6306:
6283:
6240:
6207:
6177:
6154:
6050:
5927:
5889:
5747:
5625:
5605:
5478:
5420:
5328:
5247:
5227:
5125:
5069:
5018:
4960:
4911:
4885:
4856:
4815:
4624:relative to point
4610:
4507:
4401:
4309:
4118:relative to point
4104:
4001:
3895:
3836:
3807:
3658:relative to point
3642:
3558:
3470:
3382:
3279:
3116:
3058:
2922:
2868:
2522:
2484:
2464:
2384:
2248:
2205:
2182:
2159:
2134:
1849:Velocity and speed
1846:
1827:
1798:
1769:
1740:
1619:
1576:
1498:
1476:
1446:
1426:
1406:
1382:
1351:
1320:
1285:
1265:
1245:
1225:
1116:
1005:
915:algebraic geometry
899:Kinematic analysis
834:initial conditions
749:Physics portal
363:Routhian mechanics
238:Frame of reference
90:
16857:978-1-4020-5598-0
16777:978-0-8493-0901-4
16750:978-0-521-67331-0
16679:. Wiley. p.
16555:978-0-262-16082-7
16403:Harper, Douglas.
16344:. Chez Bachelier.
16296:Animal Locomotion
16278:978-1-4020-5441-9
16249:978-0-521-54411-5
16220:978-0-13-607791-6
16092:Orbital mechanics
16072:Kinematic diagram
16017:Centripetal force
16007:Applied mechanics
15853:degree of freedom
15735:Inextensible cord
15558:
15260:
14963:
14946:
14903:
14886:
14861:
14844:
14818:
14747:
14729:
14695:
14654:
14624:
14585:in a moving body
14553:
14448:where the vector
14388:
14298:
14288:
14227:
14056:
14034:
14024:
14007:
13997:
13910:
13875:
13845:
13704:
13688:
13553:
13537:
13415:
13367:inverse transform
13269:
13244:
13133:
13054:
12960:
12844:
12820:
12258:
12187:
12055:
12049:
12039:
11978:
11972:
11962:
11847:
11834:
11818:
11729:
11660:
11610:
11576:
11549:
11484:
11478:
11468:
11362:where the matrix
11284:
11205:
11187:
11146:does not move in
10726:
10700:
10643:
10617:
10456:does not rotate (
9799:mechanical system
9702:
9680:
9651:of acceleration.
9563:
9540:
9521:
9515:
9504:
9485:
9457:is now given by:
9444:The acceleration
9393:
9373:
9345:
9325:
9304:
9298:
9293:
9231:
9211:
9138:
9111:
9091:
9064:
9036:
9016:
8995:
8989:
8984:
8914:
8873:
8828:
8801:
8766:
8726:
8699:
8679:
8658:
8652:
8647:
8585:
8555:
8538:
8515:
8488:
8463:
8438:
8418:
8397:
8391:
8386:
8360:), which yields:
8315:
8286:
8211:
8182:
8118:
8091:
8069:
8063:
8052:
8030:
8019:
8006:
7998:
7979:
7950:
7928:
7922:
7914:
7901:
7871:
7851:
7832:
7826:
7815:
7796:
7785:
7772:
7764:
7745:
7716:
7697:
7691:
7683:
7670:
7641:
7600:
7556:
7538:
7497:
7456:
7399:
7370:
7326:
7178:
7149:
7120:
6929:
6842:
6773:
6735:
6713:
6694:
6611:{\displaystyle H}
6591:{\displaystyle B}
6560:
6508:{\displaystyle a}
6488:{\displaystyle A}
6409:{\displaystyle B}
6396:is the width and
6389:{\displaystyle A}
6278:
5601:
5597:
5476:
5408:
5326:
5294:
5226:
5176:
5102:
4984:
3826:Relative velocity
3820:Relative velocity
3377:
3371:
3361:
3344:
3273:
3246:
3219:
3193:
3180:
3162:
3150:
3085:
3052:
3025:
2998:
2972:
2966:
2954:
2943:
2904:
2864:
2844:
2828:
2808:
2792:
2772:
2756:
2743:
2725:
2703:
2690:
2672:
2650:
2637:
2619:
2598:
2586:
2565:
2517:
2504:
2487:{\displaystyle s}
2459:
2453:
2443:
2378:
2351:
2324:
2298:
2292:
2280:
2269:
2230:
2130:
2110:
2094:
2074:
2058:
2038:
2022:
2009:
1985:
1972:
1948:
1935:
1912:
1886:
1734:
1705:
1676:
1588:direction cosines
1571:
1449:{\displaystyle z}
1429:{\displaystyle y}
1409:{\displaystyle x}
1379:
1348:
1317:
1288:{\displaystyle z}
1268:{\displaystyle y}
1248:{\displaystyle x}
1219:
1199:
1179:
923:mechanical system
889:mechanical system
804:is a subfield of
799:
798:
546:Centrifugal force
541:Centripetal force
497:Euler's equations
482:Relative velocity
258:Moment of inertia
88:
62:
16:(Redirected from
16939:
16861:
16844:Moon, Francis C.
16839:
16804:
16793:
16782:
16781:
16761:
16755:
16754:
16734:
16728:
16727:
16711:
16701:
16695:
16694:
16670:
16664:
16663:
16641:
16635:
16634:
16624:
16618:
16617:
16593:
16587:
16586:
16566:
16560:
16559:
16539:
16533:
16532:
16524:
16518:
16517:
16504:
16498:
16497:
16495:
16493:
16480:
16474:
16468:
16462:
16456:
16450:
16449:
16448:
16447:
16430:
16424:
16421:
16415:
16414:
16400:
16394:
16393:
16373:
16367:
16366:
16352:
16346:
16345:
16332:
16326:
16317:
16311:
16310:
16290:
16284:
16282:
16260:
16254:
16253:
16231:
16225:
16224:
16202:
16196:
16195:
16185:
16179:
16178:
16158:
16152:
16151:
16128:
16087:Motion (physics)
16047:Four-bar linkage
16037:Fictitious force
15910:four-bar linkage
15857:mobility formula
15844:kinematic chains
15822:Kinematic chains
15730:
15728:
15727:
15722:
15699:
15697:
15696:
15691:
15686:
15685:
15681:
15672:
15663:
15646:
15645:
15640:
15624:is equal to the
15570:
15568:
15567:
15562:
15560:
15559:
15554:
15549:
15543:
15542:
15537:
15512:
15510:
15509:
15504:
15499:
15491:
15483:
15482:
15478:
15469:
15452:
15450:
15449:
15444:
15439:
15438:
15433:
15424:
15423:
15419:
15410:
15389:
15388:
15384:
15375:
15360:
15359:
15354:
15341:
15339:
15338:
15333:
15325:
15317:
15309:
15308:
15296:
15295:
15290:
15278:
15270:
15262:
15261:
15253:
15247:
15246:
15241:
15217:
15215:
15214:
15209:
15204:
15203:
15184:
15183:
15178:
15166:
15161:
15160:
15146:
15145:
15125:
15124:
15119:
15118:
15099:
15098:
15093:
15084:
15056:
15055:
15033:
15031:
15030:
15025:
15023:
15022:
15003:
15002:
14997:
14988:
14987:
14982:
14970:
14965:
14964:
14956:
14948:
14947:
14939:
14925:
14924:
14905:
14904:
14899:
14894:
14888:
14887:
14882:
14877:
14868:
14863:
14862:
14854:
14846:
14845:
14837:
14820:
14819:
14811:
14796:
14794:
14793:
14788:
14783:
14757:
14749:
14748:
14740:
14731:
14730:
14725:
14720:
14705:
14697:
14696:
14688:
14679:
14675:
14674:
14655:
14653:
14642:
14637:
14636:
14631:
14625:
14623:
14612:
14607:
14606:
14601:
14568:
14566:
14565:
14560:
14555:
14554:
14549:
14544:
14538:
14537:
14532:
14507:
14505:
14504:
14499:
14494:
14486:
14478:
14477:
14473:
14464:
14447:
14445:
14444:
14439:
14434:
14433:
14428:
14419:
14418:
14414:
14405:
14390:
14389:
14384:
14379:
14370:
14362:
14342:
14341:
14336:
14324:takes the form:
14313:
14311:
14310:
14305:
14300:
14299:
14296:
14290:
14289:
14281:
14259:
14257:
14256:
14251:
14249:
14248:
14229:
14228:
14223:
14218:
14212:
14169:
14167:
14166:
14161:
14159:
14152:
14131:
14130:
14125:
14115:
14114:
14091:
14078:
14077:
14058:
14057:
14052:
14047:
14041:
14036:
14035:
14032:
14026:
14025:
14017:
14009:
14008:
14005:
13999:
13998:
13990:
13973:
13969:
13968:
13945:
13932:
13931:
13912:
13911:
13906:
13901:
13895:
13890:
13889:
13877:
13876:
13868:
13860:
13859:
13847:
13846:
13838:
13821:
13817:
13816:
13793:
13780:
13779:
13760:
13739:
13738:
13726:
13725:
13706:
13705:
13700:
13695:
13690:
13689:
13681:
13664:
13660:
13659:
13636:
13623:
13622:
13614:
13613:
13594:
13575:
13574:
13555:
13554:
13549:
13544:
13539:
13538:
13530:
13516:
13515:
13501:
13500:
13495:
13475:
13462:
13457:
13456:
13417:
13416:
13408:
13395:
13394:
13389:
13341:
13339:
13338:
13333:
13328:
13327:
13313:
13300:
13299:
13271:
13270:
13265:
13260:
13246:
13245:
13237:
13223:
13222:
13208:
13195:
13191:
13190:
13189:
13161:
13134:
13132:
13121:
13111:
13110:
13096:
13095:
13090:
13073:
13056:
13055:
13047:
13038:
13037:
13032:
13000:
12998:
12997:
12992:
12962:
12961:
12958:
12930:
12928:
12927:
12922:
12917:
12916:
12893:
12880:
12879:
12851:
12846:
12845:
12842:
12822:
12821:
12818:
12792:
12791:
12777:
12752:
12747:
12746:
12716:
12681:
12679:
12678:
12673:
12668:
12667:
12653:
12640:
12639:
12611:
12581:
12580:
12566:
12550:
12515:
12404:
12402:
12401:
12396:
12388:
12387:
12386:
12373:
12372:
12371:
12348:
12343:
12342:
12328:
12323:
12322:
12306:
12304:
12303:
12298:
12290:
12289:
12288:
12275:
12274:
12273:
12260:
12251:
12245:
12244:
12243:
12230:
12229:
12228:
12212:
12210:
12209:
12204:
12202:
12201:
12189:
12180:
12171:
12170:
12169:
12156:
12155:
12154:
12141:
12140:
12139:
12123:
12121:
12120:
12115:
12103:
12102:
12101:
12088:
12087:
12086:
12066:
12064:
12063:
12058:
12056:
12054:
12050:
12047:
12044:
12040:
12037:
12034:
11989:
11987:
11986:
11981:
11979:
11977:
11973:
11970:
11967:
11963:
11960:
11957:
11929:Angular velocity
11900:angular position
11869:Angular position
11861:
11859:
11858:
11853:
11848:
11846:
11845:
11844:
11835:
11832:
11829:
11825:
11824:
11819:
11816:
11812:
11789:
11787:
11786:
11781:
11776:
11775:
11731:
11730:
11722:
11709:
11707:
11706:
11701:
11696:
11670:
11662:
11661:
11653:
11644:
11643:
11638:
11625:
11623:
11622:
11617:
11612:
11611:
11606:
11601:
11586:
11578:
11577:
11569:
11551:
11550:
11542:
11536:
11535:
11530:
11498:
11496:
11495:
11490:
11485:
11483:
11479:
11476:
11473:
11469:
11466:
11463:
11441:. The parameter
11432:
11430:
11429:
11424:
11419:
11418:
11361:
11359:
11358:
11353:
11348:
11331:
11323:
11322:
11286:
11285:
11277:
11268:
11267:
11262:
11237:
11235:
11234:
11229:
11224:
11207:
11206:
11198:
11189:
11188:
11183:
11178:
11172:
11171:
11166:
11125:
11123:
11122:
11117:
11112:
11111:
10959:
10957:
10956:
10951:
10946:
10911:
10847:
10765:
10763:
10762:
10757:
10752:
10751:
10746:
10728:
10727:
10722:
10717:
10702:
10701:
10696:
10691:
10685:
10684:
10679:
10669:
10668:
10663:
10645:
10644:
10639:
10634:
10619:
10618:
10613:
10608:
10602:
10601:
10596:
10577:
10575:
10574:
10569:
10564:
10547:
10539:
10519:
10487:
10445:Pure translation
10414:
10412:
10411:
10406:
10401:
10400:
10368:
10367:
10343:
10342:
10307:
10306:
10261:
10250:
10227:
10160:
10158:
10157:
10152:
10147:
10146:
10122:
10121:
10086:
10085:
10040:
10039:
10027:
10018:
9982:
9789:
9787:
9786:
9781:
9767:
9766:
9753:
9752:
9734:
9733:
9717:
9715:
9714:
9709:
9704:
9703:
9695:
9682:
9681:
9673:
9642:
9640:
9639:
9634:
9623:
9622:
9597:
9596:
9578:
9576:
9575:
9570:
9565:
9564:
9559:
9554:
9542:
9541:
9536:
9531:
9522:
9520:
9516:
9513:
9510:
9506:
9505:
9500:
9495:
9486:
9483:
9480:
9475:
9474:
9469:
9453:of the particle
9436:
9431:angular velocity
9428:
9426:
9425:
9420:
9408:
9406:
9405:
9400:
9395:
9394:
9389:
9384:
9375:
9374:
9369:
9364:
9352:
9348:
9347:
9346:
9341:
9336:
9327:
9326:
9321:
9316:
9305:
9303:
9299:
9296:
9291:
9290:
9285:
9284:
9279:
9246:
9244:
9243:
9238:
9233:
9232:
9227:
9222:
9213:
9212:
9207:
9202:
9184:
9153:
9151:
9150:
9145:
9140:
9139:
9134:
9129:
9126:
9125:
9113:
9112:
9107:
9102:
9093:
9092:
9087:
9082:
9079:
9078:
9066:
9065:
9060:
9055:
9043:
9039:
9038:
9037:
9032:
9027:
9018:
9017:
9012:
9007:
8996:
8994:
8990:
8987:
8982:
8981:
8976:
8975:
8970:
8926:
8924:
8923:
8918:
8916:
8915:
8910:
8905:
8885:
8883:
8882:
8877:
8875:
8874:
8869:
8864:
8843:
8841:
8840:
8835:
8830:
8829:
8824:
8819:
8816:
8815:
8803:
8802:
8797:
8792:
8768:
8767:
8762:
8757:
8733:
8729:
8728:
8727:
8722:
8717:
8714:
8713:
8701:
8700:
8695:
8690:
8681:
8680:
8675:
8670:
8659:
8657:
8653:
8650:
8645:
8644:
8639:
8638:
8633:
8600:
8598:
8597:
8592:
8587:
8586:
8581:
8576:
8573:
8572:
8557:
8556:
8551:
8546:
8540:
8539:
8534:
8529:
8517:
8516:
8511:
8506:
8503:
8502:
8490:
8489:
8484:
8479:
8476:
8465:
8464:
8459:
8454:
8445:
8441:
8440:
8439:
8434:
8429:
8420:
8419:
8414:
8409:
8398:
8396:
8392:
8389:
8384:
8383:
8378:
8377:
8372:
8330:
8328:
8327:
8322:
8317:
8316:
8311:
8306:
8288:
8287:
8282:
8277:
8250:
8226:
8224:
8223:
8218:
8213:
8212:
8207:
8202:
8184:
8183:
8178:
8173:
8155:
8133:
8131:
8130:
8125:
8120:
8119:
8114:
8109:
8106:
8105:
8093:
8092:
8087:
8082:
8070:
8068:
8064:
8061:
8058:
8054:
8053:
8048:
8043:
8031:
8028:
8025:
8020:
8018:
8017:
8016:
8007:
8004:
8001:
8000:
7999:
7994:
7989:
7986:
7985:
7980:
7977:
7973:
7965:
7963:
7962:
7957:
7952:
7951:
7946:
7941:
7929:
7927:
7923:
7920:
7917:
7916:
7915:
7910:
7905:
7902:
7899:
7896:
7886:
7884:
7883:
7878:
7873:
7872:
7867:
7862:
7853:
7852:
7847:
7842:
7833:
7831:
7827:
7824:
7821:
7817:
7816:
7811:
7806:
7797:
7794:
7791:
7786:
7784:
7783:
7782:
7773:
7770:
7767:
7766:
7765:
7760:
7755:
7752:
7751:
7746:
7743:
7739:
7731:
7729:
7728:
7723:
7718:
7717:
7712:
7707:
7698:
7696:
7692:
7689:
7686:
7685:
7684:
7679:
7674:
7671:
7668:
7665:
7656:
7654:
7653:
7648:
7643:
7642:
7637:
7632:
7602:
7601:
7596:
7591:
7558:
7557:
7552:
7547:
7540:
7539:
7534:
7529:
7499:
7498:
7493:
7488:
7458:
7457:
7452:
7447:
7414:
7412:
7411:
7406:
7401:
7400:
7395:
7390:
7372:
7371:
7366:
7361:
7328:
7327:
7322:
7317:
7275:
7229:, respectively.
7193:
7191:
7190:
7185:
7180:
7179:
7174:
7169:
7151:
7150:
7145:
7140:
7122:
7121:
7116:
7111:
7084:
6944:
6940:
6938:
6937:
6932:
6930:
6925:
6924:
6923:
6910:
6902:
6901:
6876:
6874:
6873:
6868:
6853:
6851:
6850:
6845:
6843:
6838:
6837:
6836:
6823:
6814:
6812:
6811:
6806:
6801:
6800:
6784:
6782:
6781:
6776:
6774:
6769:
6768:
6767:
6754:
6749:
6748:
6736:
6728:
6714:
6706:
6695:
6687:
6672:
6670:
6669:
6664:
6643:
6641:
6640:
6635:
6617:
6615:
6614:
6609:
6598:is the base and
6597:
6595:
6594:
6589:
6577:
6575:
6574:
6569:
6561:
6553:
6544:
6542:
6541:
6536:
6534:
6533:
6514:
6512:
6511:
6506:
6494:
6492:
6491:
6486:
6474:
6472:
6471:
6466:
6464:
6463:
6441:
6439:
6438:
6433:
6415:
6413:
6412:
6407:
6395:
6393:
6392:
6387:
6375:
6373:
6372:
6367:
6349:
6347:
6346:
6341:
6315:
6313:
6312:
6307:
6292:
6290:
6289:
6284:
6279:
6276:
6249:
6247:
6246:
6241:
6220:
6215:
6203:
6202:
6186:
6184:
6183:
6178:
6163:
6161:
6160:
6155:
6144:
6139:
6138:
6133:
6124:
6119:
6105:
6100:
6095:
6081:
6076:
6071:
6059:
6057:
6056:
6051:
6046:
6042:
6041:
6040:
6035:
6026:
6016:
6012:
5997:
5996:
5991:
5985:
5984:
5979:
5973:
5965:
5964:
5959:
5953:
5948:
5936:
5934:
5933:
5928:
5904:
5898:
5896:
5895:
5890:
5885:
5884:
5879:
5873:
5872:
5867:
5861:
5853:
5852:
5847:
5841:
5836:
5819:
5815:
5806:
5802:
5801:
5800:
5795:
5786:
5762:
5756:
5754:
5753:
5748:
5743:
5742:
5737:
5731:
5730:
5725:
5719:
5711:
5710:
5705:
5699:
5694:
5686:
5678:
5674:
5673:
5672:
5667:
5658:
5634:
5632:
5631:
5626:
5614:
5612:
5611:
5606:
5599:
5598:
5593:
5592:
5591:
5586:
5577:
5571:
5566:
5562:
5561:
5560:
5555:
5546:
5533:
5525:
5521:
5520:
5519:
5514:
5505:
5487:
5485:
5484:
5479:
5477:
5475:
5470:
5469:
5468:
5463:
5454:
5448:
5429:
5427:
5426:
5421:
5413:
5409:
5404:
5403:
5402:
5397:
5388:
5382:
5373:
5372:
5367:
5349:
5337:
5335:
5334:
5329:
5327:
5322:
5321:
5320:
5315:
5306:
5300:
5295:
5293:
5285:
5284:
5275:
5270:
5256:
5254:
5253:
5248:
5243:
5242:
5233:
5228:
5219:
5210:
5209:
5204:
5195:
5194:
5189:
5177:
5174:
5172:
5168:
5164:
5156:
5155:
5150:
5138:
5133:
5121:
5120:
5115:
5103:
5100:
5088:
5082:
5077:
5065:
5064:
5059:
5041:
5027:
5025:
5024:
5019:
5011:
5003:
5002:
4997:
4985:
4982:
4979:
4973:
4968:
4956:
4955:
4950:
4932:
4920:
4918:
4917:
4912:
4894:
4892:
4891:
4886:
4884:
4883:
4878:
4865:
4863:
4862:
4857:
4855:
4854:
4849:
4824:
4822:
4821:
4816:
4814:
4810:
4809:
4808:
4807:
4806:
4789:
4788:
4787:
4786:
4769:
4768:
4767:
4766:
4749:
4748:
4747:
4746:
4729:
4728:
4727:
4726:
4709:
4708:
4707:
4706:
4684:
4683:
4678:
4669:
4668:
4663:
4654:
4653:
4649:
4640:
4619:
4617:
4616:
4611:
4609:
4605:
4604:
4603:
4602:
4601:
4584:
4583:
4582:
4581:
4564:
4563:
4562:
4561:
4539:
4538:
4533:
4516:
4514:
4513:
4508:
4506:
4502:
4501:
4500:
4499:
4498:
4481:
4480:
4479:
4478:
4461:
4460:
4459:
4458:
4436:
4435:
4430:
4410:
4408:
4407:
4402:
4400:
4399:
4394:
4385:
4384:
4379:
4370:
4369:
4365:
4356:
4318:
4316:
4315:
4310:
4308:
4304:
4303:
4302:
4301:
4300:
4283:
4282:
4281:
4280:
4263:
4262:
4261:
4260:
4243:
4242:
4241:
4240:
4223:
4222:
4221:
4220:
4203:
4202:
4201:
4200:
4178:
4177:
4172:
4163:
4162:
4157:
4148:
4147:
4143:
4134:
4113:
4111:
4110:
4105:
4103:
4099:
4098:
4097:
4096:
4095:
4078:
4077:
4076:
4075:
4058:
4057:
4056:
4055:
4033:
4032:
4027:
4010:
4008:
4007:
4002:
4000:
3996:
3995:
3994:
3993:
3992:
3975:
3974:
3973:
3972:
3955:
3954:
3953:
3952:
3930:
3929:
3924:
3904:
3902:
3901:
3896:
3894:
3893:
3888:
3879:
3878:
3873:
3864:
3863:
3859:
3850:
3816:
3814:
3813:
3808:
3806:
3802:
3801:
3800:
3788:
3787:
3775:
3774:
3762:
3761:
3749:
3748:
3736:
3735:
3718:
3717:
3712:
3703:
3702:
3697:
3688:
3687:
3683:
3674:
3651:
3649:
3648:
3643:
3641:
3637:
3636:
3635:
3623:
3622:
3610:
3609:
3592:
3591:
3586:
3567:
3565:
3564:
3559:
3557:
3553:
3552:
3551:
3539:
3538:
3526:
3525:
3508:
3507:
3502:
3479:
3477:
3476:
3471:
3469:
3468:
3463:
3454:
3453:
3448:
3439:
3438:
3434:
3425:
3391:
3389:
3388:
3383:
3378:
3376:
3372:
3369:
3366:
3362:
3359:
3356:
3351:
3346:
3345:
3340:
3335:
3332:
3324:
3319:
3314:
3288:
3286:
3285:
3280:
3275:
3274:
3269:
3264:
3261:
3260:
3248:
3247:
3242:
3237:
3234:
3233:
3221:
3220:
3215:
3210:
3207:
3206:
3194:
3192:
3191:
3190:
3181:
3178:
3175:
3174:
3169:
3168:
3163:
3160:
3156:
3151:
3149:
3148:
3147:
3128:
3127:
3118:
3115:
3108:
3107:
3081:
3067:
3065:
3064:
3059:
3054:
3053:
3048:
3043:
3040:
3039:
3027:
3026:
3021:
3016:
3013:
3012:
3000:
2999:
2994:
2989:
2986:
2985:
2973:
2971:
2967:
2964:
2961:
2960:
2955:
2952:
2949:
2944:
2942:
2934:
2933:
2924:
2921:
2900:
2877:
2875:
2874:
2869:
2866:
2865:
2860:
2855:
2852:
2851:
2846:
2845:
2837:
2830:
2829:
2824:
2819:
2816:
2815:
2810:
2809:
2801:
2794:
2793:
2788:
2783:
2780:
2779:
2774:
2773:
2765:
2758:
2757:
2752:
2747:
2744:
2742:
2734:
2733:
2732:
2727:
2726:
2718:
2710:
2705:
2704:
2699:
2694:
2691:
2689:
2681:
2680:
2679:
2674:
2673:
2665:
2657:
2652:
2651:
2646:
2641:
2638:
2636:
2628:
2627:
2626:
2621:
2620:
2612:
2604:
2599:
2597:
2589:
2588:
2587:
2579:
2572:
2567:
2566:
2558:
2531:
2529:
2528:
2523:
2518:
2515:
2513:
2505:
2502:
2493:
2491:
2490:
2485:
2473:
2471:
2470:
2465:
2460:
2458:
2454:
2451:
2448:
2444:
2441:
2438:
2433:
2428:
2423:
2393:
2391:
2390:
2385:
2380:
2379:
2374:
2369:
2366:
2365:
2353:
2352:
2347:
2342:
2339:
2338:
2326:
2325:
2320:
2315:
2312:
2311:
2299:
2297:
2293:
2290:
2287:
2286:
2281:
2278:
2275:
2270:
2268:
2260:
2259:
2250:
2247:
2226:
2214:
2212:
2211:
2206:
2191:
2189:
2188:
2183:
2168:
2166:
2165:
2160:
2158:
2143:
2141:
2140:
2135:
2132:
2131:
2126:
2121:
2118:
2117:
2112:
2111:
2103:
2096:
2095:
2090:
2085:
2082:
2081:
2076:
2075:
2067:
2060:
2059:
2054:
2049:
2046:
2045:
2040:
2039:
2031:
2024:
2023:
2018:
2013:
2010:
2008:
2000:
1992:
1987:
1986:
1981:
1976:
1973:
1971:
1963:
1955:
1950:
1949:
1944:
1939:
1936:
1934:
1926:
1918:
1913:
1911:
1903:
1902:
1893:
1888:
1887:
1879:
1868:average velocity
1836:
1834:
1833:
1828:
1807:
1805:
1804:
1799:
1778:
1776:
1775:
1770:
1749:
1747:
1746:
1741:
1736:
1735:
1730:
1725:
1707:
1706:
1701:
1696:
1678:
1677:
1672:
1667:
1640:
1628:
1626:
1625:
1620:
1609:
1585:
1583:
1582:
1577:
1572:
1570:
1569:
1557:
1556:
1544:
1543:
1534:
1529:
1524:
1519:
1508:and the origin.
1507:
1505:
1504:
1499:
1497:
1485:
1483:
1482:
1477:
1475:
1471:
1455:
1453:
1452:
1447:
1435:
1433:
1432:
1427:
1415:
1413:
1412:
1407:
1391:
1389:
1388:
1383:
1381:
1380:
1375:
1370:
1360:
1358:
1357:
1352:
1350:
1349:
1344:
1339:
1329:
1327:
1326:
1321:
1319:
1318:
1313:
1308:
1294:
1292:
1291:
1286:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1221:
1220:
1215:
1210:
1201:
1200:
1195:
1190:
1181:
1180:
1175:
1170:
1137:
1125:
1123:
1122:
1117:
1115:
1114:
1047:
1034:Velocity vector
1031:
1018:Position vector
1015:
971:
970:
958:
957:
949:
948:
893:dynamic analysis
854:celestial bodies
791:
784:
777:
764:
759:
758:
751:
747:
746:
652:Johann Bernoulli
647:Daniel Bernoulli
568:Tangential speed
472:
448:
423:Fictitious force
418:
270:Mechanical power
260:
201:Angular momentum
99:
97:
96:
91:
89:
87:
79:
78:
69:
64:
63:
37:
36:
32:Kinematic (band)
21:
18:Exact constraint
16947:
16946:
16942:
16941:
16940:
16938:
16937:
16936:
16912:
16911:
16878:
16858:
16837:
16825:, vol. 2,
16813:
16811:Further reading
16808:
16807:
16794:
16790:
16785:
16778:
16762:
16758:
16751:
16735:
16731:
16724:
16702:
16698:
16691:
16671:
16667:
16660:
16646:"Problem 17-11"
16642:
16638:
16625:
16621:
16614:
16594:
16590:
16583:
16567:
16563:
16556:
16540:
16536:
16527:
16525:
16521:
16505:
16501:
16491:
16489:
16482:
16481:
16477:
16469:
16465:
16457:
16453:
16445:
16443:
16435:2.4 Integration
16432:
16431:
16427:
16422:
16418:
16401:
16397:
16390:
16374:
16370:
16353:
16349:
16333:
16329:
16318:
16314:
16307:
16291:
16287:
16279:
16261:
16257:
16250:
16232:
16228:
16221:
16203:
16199:
16186:
16182:
16175:
16159:
16155:
16148:
16129:
16125:
16121:
16116:
15982:
15924:six-bar linkage
15908:= 4 : the
15879:assembled from
15873:
15839:kinematic pairs
15824:
15811:
15798:along the axis.
15792:prismatic joint
15779:
15771:kinematic pairs
15764:
15758:
15756:Kinematic pairs
15737:
15707:
15704:
15703:
15677:
15673:
15668:
15667:
15659:
15641:
15636:
15635:
15633:
15630:
15629:
15610:
15598:
15581:
15550:
15548:
15547:
15538:
15533:
15532:
15530:
15527:
15526:
15495:
15487:
15474:
15470:
15465:
15464:
15462:
15459:
15458:
15434:
15429:
15428:
15415:
15411:
15406:
15405:
15380:
15376:
15371:
15370:
15355:
15350:
15349:
15347:
15344:
15343:
15321:
15313:
15304:
15300:
15291:
15286:
15285:
15274:
15266:
15252:
15251:
15242:
15237:
15236:
15234:
15231:
15230:
15228:
15198:
15197:
15192:
15186:
15185:
15179:
15174:
15173:
15162:
15156:
15152:
15147:
15141:
15137:
15130:
15129:
15120:
15113:
15112:
15107:
15101:
15100:
15094:
15089:
15088:
15080:
15072:
15062:
15061:
15060:
15051:
15047:
15039:
15036:
15035:
15017:
15016:
15011:
15005:
15004:
14998:
14993:
14992:
14983:
14978:
14977:
14966:
14955:
14954:
14949:
14938:
14937:
14930:
14929:
14919:
14918:
14913:
14907:
14906:
14895:
14893:
14892:
14878:
14876:
14875:
14864:
14853:
14852:
14847:
14836:
14835:
14828:
14827:
14810:
14809:
14804:
14801:
14800:
14779:
14753:
14739:
14738:
14721:
14719:
14718:
14701:
14687:
14686:
14670:
14660:
14656:
14646:
14641:
14632:
14627:
14626:
14616:
14611:
14602:
14597:
14596:
14594:
14591:
14590:
14579:
14545:
14543:
14542:
14533:
14528:
14527:
14525:
14522:
14521:
14490:
14482:
14469:
14465:
14460:
14459:
14457:
14454:
14453:
14429:
14424:
14423:
14410:
14406:
14401:
14400:
14380:
14378:
14377:
14366:
14358:
14337:
14332:
14331:
14329:
14326:
14325:
14323:
14295:
14291:
14280:
14279:
14265:
14262:
14261:
14243:
14242:
14237:
14231:
14230:
14219:
14217:
14216:
14208:
14200:
14190:
14189:
14175:
14172:
14171:
14157:
14156:
14148:
14132:
14126:
14121:
14120:
14117:
14116:
14109:
14108:
14102:
14101:
14087:
14080:
14079:
14072:
14071:
14066:
14060:
14059:
14048:
14046:
14045:
14037:
14031:
14027:
14016:
14015:
14010:
14004:
14000:
13989:
13988:
13981:
13980:
13971:
13970:
13963:
13962:
13956:
13955:
13941:
13934:
13933:
13926:
13925:
13920:
13914:
13913:
13902:
13900:
13899:
13891:
13882:
13878:
13867:
13866:
13861:
13852:
13848:
13837:
13836:
13829:
13828:
13819:
13818:
13811:
13810:
13804:
13803:
13789:
13782:
13781:
13774:
13773:
13768:
13762:
13761:
13756:
13751:
13741:
13740:
13731:
13727:
13720:
13719:
13714:
13708:
13707:
13696:
13694:
13693:
13691:
13680:
13679:
13672:
13671:
13662:
13661:
13654:
13653:
13647:
13646:
13632:
13625:
13624:
13615:
13608:
13607:
13602:
13596:
13595:
13590:
13588:
13578:
13577:
13576:
13569:
13568:
13563:
13557:
13556:
13545:
13543:
13542:
13540:
13529:
13528:
13521:
13520:
13510:
13509:
13503:
13502:
13496:
13491:
13490:
13483:
13482:
13473:
13472:
13458:
13449:
13445:
13407:
13406:
13396:
13390:
13385:
13384:
13380:
13378:
13375:
13374:
13322:
13321:
13315:
13314:
13309:
13302:
13301:
13294:
13293:
13288:
13282:
13281:
13261:
13259:
13258:
13256:
13236:
13235:
13228:
13227:
13217:
13216:
13210:
13209:
13204:
13197:
13196:
13184:
13183:
13178:
13172:
13171:
13157:
13155:
13136:
13135:
13125:
13120:
13119:
13115:
13105:
13104:
13098:
13097:
13091:
13086:
13085:
13078:
13077:
13069:
13046:
13045:
13033:
13028:
13027:
13025:
13022:
13021:
13006:
12957:
12953:
12936:
12933:
12932:
12911:
12910:
12904:
12903:
12889:
12882:
12881:
12874:
12873:
12868:
12862:
12861:
12847:
12841:
12837:
12823:
12817:
12813:
12797:
12796:
12786:
12785:
12779:
12778:
12773:
12766:
12765:
12748:
12739:
12735:
12712:
12710:
12707:
12706:
12662:
12661:
12655:
12654:
12649:
12642:
12641:
12634:
12633:
12628:
12622:
12621:
12607:
12605:
12586:
12585:
12575:
12574:
12568:
12567:
12562:
12555:
12554:
12546:
12511:
12509:
12506:
12505:
12495:reference frame
12467:
12446:
12434:
12427:
12420:
12413:
12382:
12381:
12377:
12367:
12366:
12362:
12344:
12338:
12337:
12324:
12318:
12317:
12311:
12308:
12307:
12284:
12283:
12279:
12269:
12268:
12264:
12249:
12239:
12238:
12234:
12224:
12223:
12219:
12217:
12214:
12213:
12197:
12193:
12178:
12165:
12164:
12160:
12150:
12149:
12145:
12135:
12134:
12130:
12128:
12125:
12124:
12097:
12096:
12092:
12082:
12081:
12077:
12075:
12072:
12071:
12046:
12045:
12036:
12035:
12033:
12025:
12022:
12021:
12000:right-hand rule
11969:
11968:
11959:
11958:
11956:
11948:
11945:
11944:
11923:right-hand rule
11916:
11893:
11840:
11836:
11831:
11830:
11820:
11815:
11814:
11813:
11811:
11803:
11800:
11799:
11770:
11769:
11764:
11758:
11757:
11749:
11739:
11738:
11721:
11720:
11715:
11712:
11711:
11692:
11666:
11652:
11651:
11639:
11634:
11633:
11631:
11628:
11627:
11602:
11600:
11599:
11582:
11568:
11567:
11541:
11540:
11531:
11526:
11525:
11523:
11520:
11519:
11504:
11475:
11474:
11465:
11464:
11462:
11454:
11451:
11450:
11413:
11412:
11407:
11401:
11400:
11392:
11382:
11381:
11367:
11364:
11363:
11344:
11327:
11315:
11311:
11276:
11275:
11263:
11258:
11257:
11255:
11252:
11251:
11220:
11197:
11196:
11179:
11177:
11176:
11167:
11162:
11161:
11159:
11156:
11155:
11140:
11106:
11105:
11079:
11052:
11051:
11022:
10991:
10990:
10967:
10964:
10963:
10942:
10907:
10905:
10902:
10901:
10862:
10831:
10820:right-hand rule
10808:
10802:
10783:
10774:
10747:
10742:
10741:
10718:
10716:
10715:
10692:
10690:
10689:
10680:
10675:
10674:
10664:
10659:
10658:
10635:
10633:
10632:
10609:
10607:
10606:
10597:
10592:
10591:
10589:
10586:
10585:
10560:
10543:
10535:
10515:
10483:
10481:
10478:
10477:
10451:reference frame
10447:
10395:
10394:
10388:
10387:
10381:
10380:
10370:
10369:
10362:
10361:
10356:
10351:
10345:
10344:
10338:
10334:
10332:
10321:
10309:
10308:
10302:
10298:
10296:
10282:
10266:
10265:
10257:
10246:
10223:
10221:
10218:
10217:
10141:
10140:
10135:
10130:
10124:
10123:
10117:
10113:
10111:
10100:
10088:
10087:
10081:
10077:
10075:
10061:
10045:
10044:
10034:
10033:
10028:
10023:
10020:
10019:
10014:
10012:
9993:
9992:
9978:
9961:
9958:
9957:
9954:
9947:
9929:rotation matrix
9921:
9850:
9830:Euclidean group
9803:reference frame
9795:
9762:
9758:
9748:
9744:
9729:
9725:
9723:
9720:
9719:
9694:
9693:
9672:
9671:
9663:
9660:
9659:
9618:
9614:
9592:
9588:
9586:
9583:
9582:
9581:The components
9555:
9553:
9552:
9532:
9530:
9529:
9512:
9511:
9496:
9494:
9493:
9482:
9481:
9479:
9470:
9465:
9464:
9462:
9459:
9458:
9452:
9434:
9414:
9411:
9410:
9385:
9383:
9382:
9365:
9363:
9362:
9337:
9335:
9334:
9317:
9315:
9314:
9310:
9306:
9295:
9294:
9289:
9280:
9275:
9274:
9272:
9269:
9268:
9266:
9257:
9223:
9221:
9220:
9203:
9201:
9200:
9180:
9178:
9175:
9174:
9159:
9130:
9128:
9127:
9121:
9117:
9103:
9101:
9100:
9083:
9081:
9080:
9074:
9070:
9056:
9054:
9053:
9028:
9026:
9025:
9008:
9006:
9005:
9001:
8997:
8986:
8985:
8980:
8971:
8966:
8965:
8963:
8960:
8959:
8949:
8937:
8935:Constant radius
8906:
8904:
8903:
8895:
8892:
8891:
8865:
8863:
8862:
8851:
8848:
8847:
8820:
8818:
8817:
8811:
8807:
8793:
8791:
8790:
8758:
8756:
8755:
8718:
8716:
8715:
8709:
8705:
8691:
8689:
8688:
8671:
8669:
8668:
8664:
8660:
8649:
8648:
8643:
8634:
8629:
8628:
8626:
8623:
8622:
8621:, is given by:
8620:
8611:
8577:
8575:
8574:
8568:
8564:
8547:
8545:
8544:
8530:
8528:
8527:
8507:
8505:
8504:
8498:
8494:
8480:
8478:
8477:
8472:
8455:
8453:
8452:
8430:
8428:
8427:
8410:
8408:
8407:
8403:
8399:
8388:
8387:
8382:
8373:
8368:
8367:
8365:
8362:
8361:
8351:
8307:
8305:
8304:
8278:
8276:
8275:
8246:
8244:
8241:
8240:
8203:
8201:
8200:
8174:
8172:
8171:
8151:
8149:
8146:
8145:
8110:
8108:
8107:
8101:
8097:
8083:
8081:
8080:
8060:
8059:
8044:
8042:
8041:
8027:
8026:
8024:
8012:
8008:
8003:
8002:
7990:
7988:
7987:
7981:
7976:
7975:
7974:
7972:
7970:
7967:
7966:
7942:
7940:
7939:
7919:
7918:
7906:
7904:
7903:
7898:
7897:
7895:
7893:
7890:
7889:
7863:
7861:
7860:
7843:
7841:
7840:
7823:
7822:
7807:
7805:
7804:
7793:
7792:
7790:
7778:
7774:
7769:
7768:
7756:
7754:
7753:
7747:
7742:
7741:
7740:
7738:
7736:
7733:
7732:
7708:
7706:
7705:
7688:
7687:
7675:
7673:
7672:
7667:
7666:
7664:
7662:
7659:
7658:
7633:
7631:
7630:
7592:
7590:
7589:
7548:
7546:
7545:
7530:
7528:
7527:
7489:
7487:
7486:
7448:
7446:
7445:
7443:
7440:
7439:
7391:
7389:
7388:
7362:
7360:
7359:
7318:
7316:
7315:
7271:
7269:
7266:
7265:
7224:reference frame
7170:
7168:
7167:
7141:
7139:
7138:
7112:
7110:
7109:
7080:
7078:
7075:
7074:
7008:
6990:
6978:
6961:
6942:
6919:
6915:
6911:
6909:
6897:
6893:
6882:
6879:
6878:
6859:
6856:
6855:
6832:
6828:
6824:
6822:
6820:
6817:
6816:
6796:
6792:
6790:
6787:
6786:
6763:
6759:
6755:
6753:
6744:
6740:
6727:
6705:
6686:
6678:
6675:
6674:
6649:
6646:
6645:
6623:
6620:
6619:
6603:
6600:
6599:
6583:
6580:
6579:
6552:
6550:
6547:
6546:
6529:
6525:
6520:
6517:
6516:
6500:
6497:
6496:
6480:
6477:
6476:
6459:
6455:
6447:
6444:
6443:
6421:
6418:
6417:
6401:
6398:
6397:
6381:
6378:
6377:
6355:
6352:
6351:
6332:
6329:
6328:
6298:
6295:
6294:
6275:
6257:
6254:
6253:
6216:
6211:
6198:
6194:
6192:
6189:
6188:
6169:
6166:
6165:
6140:
6134:
6129:
6128:
6120:
6115:
6101:
6096:
6091:
6077:
6072:
6067:
6065:
6062:
6061:
6036:
6031:
6030:
6022:
6021:
6017:
6008:
6004:
5992:
5987:
5986:
5980:
5975:
5974:
5969:
5960:
5955:
5954:
5949:
5944:
5942:
5939:
5938:
5910:
5907:
5906:
5902:
5880:
5875:
5874:
5868:
5863:
5862:
5857:
5848:
5843:
5842:
5837:
5832:
5811:
5807:
5796:
5791:
5790:
5782:
5781:
5777:
5772:
5769:
5768:
5760:
5738:
5733:
5732:
5726:
5721:
5720:
5715:
5706:
5701:
5700:
5695:
5690:
5682:
5668:
5663:
5662:
5654:
5653:
5649:
5644:
5641:
5640:
5620:
5617:
5616:
5587:
5582:
5581:
5573:
5572:
5570:
5556:
5551:
5550:
5542:
5541:
5537:
5529:
5515:
5510:
5509:
5501:
5500:
5496:
5494:
5491:
5490:
5471:
5464:
5459:
5458:
5450:
5449:
5447:
5439:
5436:
5435:
5398:
5393:
5392:
5384:
5383:
5381:
5377:
5368:
5363:
5362:
5345:
5343:
5340:
5339:
5316:
5311:
5310:
5302:
5301:
5299:
5286:
5280:
5276:
5274:
5266:
5264:
5261:
5260:
5238:
5234:
5229:
5217:
5205:
5200:
5199:
5190:
5185:
5184:
5173:
5160:
5151:
5146:
5145:
5144:
5140:
5134:
5129:
5116:
5111:
5110:
5099:
5084:
5078:
5073:
5060:
5055:
5054:
5037:
5035:
5032:
5031:
5007:
4998:
4993:
4992:
4981:
4975:
4969:
4964:
4951:
4946:
4945:
4928:
4926:
4923:
4922:
4900:
4897:
4896:
4879:
4874:
4873:
4871:
4868:
4867:
4866:, and velocity
4850:
4845:
4844:
4842:
4839:
4838:
4833:
4802:
4798:
4797:
4793:
4782:
4778:
4777:
4773:
4762:
4758:
4757:
4753:
4742:
4738:
4737:
4733:
4722:
4718:
4717:
4713:
4702:
4698:
4697:
4693:
4692:
4688:
4679:
4674:
4673:
4664:
4659:
4658:
4645:
4641:
4636:
4635:
4633:
4630:
4629:
4597:
4593:
4592:
4588:
4577:
4573:
4572:
4568:
4557:
4553:
4552:
4548:
4547:
4543:
4534:
4529:
4528:
4526:
4523:
4522:
4494:
4490:
4489:
4485:
4474:
4470:
4469:
4465:
4454:
4450:
4449:
4445:
4444:
4440:
4431:
4426:
4425:
4423:
4420:
4419:
4395:
4390:
4389:
4380:
4375:
4374:
4361:
4357:
4352:
4351:
4349:
4346:
4345:
4334:
4327:
4296:
4292:
4291:
4287:
4276:
4272:
4271:
4267:
4256:
4252:
4251:
4247:
4236:
4232:
4231:
4227:
4216:
4212:
4211:
4207:
4196:
4192:
4191:
4187:
4186:
4182:
4173:
4168:
4167:
4158:
4153:
4152:
4139:
4135:
4130:
4129:
4127:
4124:
4123:
4091:
4087:
4086:
4082:
4071:
4067:
4066:
4062:
4051:
4047:
4046:
4042:
4041:
4037:
4028:
4023:
4022:
4020:
4017:
4016:
3988:
3984:
3983:
3979:
3968:
3964:
3963:
3959:
3948:
3944:
3943:
3939:
3938:
3934:
3925:
3920:
3919:
3917:
3914:
3913:
3889:
3884:
3883:
3874:
3869:
3868:
3855:
3851:
3846:
3845:
3843:
3840:
3839:
3828:
3822:
3796:
3792:
3783:
3779:
3770:
3766:
3757:
3753:
3744:
3740:
3731:
3727:
3726:
3722:
3713:
3708:
3707:
3698:
3693:
3692:
3679:
3675:
3670:
3669:
3667:
3664:
3663:
3631:
3627:
3618:
3614:
3605:
3601:
3600:
3596:
3587:
3582:
3581:
3579:
3576:
3575:
3547:
3543:
3534:
3530:
3521:
3517:
3516:
3512:
3503:
3498:
3497:
3495:
3492:
3491:
3464:
3459:
3458:
3449:
3444:
3443:
3430:
3426:
3421:
3420:
3418:
3415:
3414:
3397:
3368:
3367:
3358:
3357:
3355:
3347:
3336:
3334:
3333:
3328:
3320:
3315:
3310:
3308:
3305:
3304:
3265:
3263:
3262:
3256:
3252:
3238:
3236:
3235:
3229:
3225:
3211:
3209:
3208:
3202:
3198:
3186:
3182:
3177:
3176:
3170:
3164:
3159:
3158:
3157:
3155:
3143:
3139:
3129:
3123:
3119:
3117:
3103:
3099:
3089:
3077:
3075:
3072:
3071:
3070:Alternatively,
3044:
3042:
3041:
3035:
3031:
3017:
3015:
3014:
3008:
3004:
2990:
2988:
2987:
2981:
2977:
2963:
2962:
2956:
2951:
2950:
2948:
2935:
2929:
2925:
2923:
2908:
2896:
2894:
2891:
2890:
2856:
2854:
2853:
2847:
2836:
2835:
2834:
2820:
2818:
2817:
2811:
2800:
2799:
2798:
2784:
2782:
2781:
2775:
2764:
2763:
2762:
2748:
2746:
2745:
2735:
2728:
2717:
2716:
2715:
2711:
2709:
2695:
2693:
2692:
2682:
2675:
2664:
2663:
2662:
2658:
2656:
2642:
2640:
2639:
2629:
2622:
2611:
2610:
2609:
2605:
2603:
2590:
2578:
2577:
2573:
2571:
2557:
2556:
2554:
2551:
2550:
2538:
2514:
2509:
2501:
2499:
2496:
2495:
2479:
2476:
2475:
2450:
2449:
2440:
2439:
2437:
2429:
2424:
2419:
2411:
2408:
2407:
2370:
2368:
2367:
2361:
2357:
2343:
2341:
2340:
2334:
2330:
2316:
2314:
2313:
2307:
2303:
2289:
2288:
2282:
2277:
2276:
2274:
2261:
2255:
2251:
2249:
2234:
2222:
2220:
2217:
2216:
2197:
2194:
2193:
2174:
2171:
2170:
2154:
2149:
2146:
2145:
2122:
2120:
2119:
2113:
2102:
2101:
2100:
2086:
2084:
2083:
2077:
2066:
2065:
2064:
2050:
2048:
2047:
2041:
2030:
2029:
2028:
2014:
2012:
2011:
2001:
1993:
1991:
1977:
1975:
1974:
1964:
1956:
1954:
1940:
1938:
1937:
1927:
1919:
1917:
1904:
1898:
1894:
1892:
1878:
1877:
1875:
1872:
1871:
1851:
1813:
1810:
1809:
1784:
1781:
1780:
1755:
1752:
1751:
1726:
1724:
1723:
1697:
1695:
1694:
1668:
1666:
1665:
1636:
1634:
1631:
1630:
1605:
1603:
1600:
1599:
1565:
1561:
1552:
1548:
1539:
1535:
1533:
1525:
1520:
1515:
1513:
1510:
1509:
1493:
1491:
1488:
1487:
1467:
1463:
1461:
1458:
1457:
1441:
1438:
1437:
1421:
1418:
1417:
1401:
1398:
1397:
1371:
1369:
1368:
1366:
1363:
1362:
1340:
1338:
1337:
1335:
1332:
1331:
1309:
1307:
1306:
1304:
1301:
1300:
1280:
1277:
1276:
1260:
1257:
1256:
1240:
1237:
1236:
1211:
1209:
1208:
1191:
1189:
1188:
1171:
1169:
1168:
1133:
1131:
1128:
1127:
1110:
1109:
1107:
1104:
1103:
1100:reference frame
1064:
1063:
1062:
1061:
1057:
1056:
1055:
1048:
1040:
1039:
1032:
1024:
1023:
1016:
999:, acceleration
981:
966:from the Greek
931:
812:, developed in
795:
754:
741:
740:
733:
732:
731:
606:
598:
597:
577:
531:Circular motion
525:
515:
514:
513:
470:
440:
437:
416:
395:
387:
386:
383:
382:
340:
330:
322:
321:
320:
279:
275:Mechanical work
268:
252:
190:
182:
181:
180:
135:
127:
104:
80:
74:
70:
68:
59:
58:
56:
53:
52:
35:
28:
23:
22:
15:
12:
11:
5:
16945:
16935:
16934:
16929:
16924:
16910:
16909:
16904:
16901:e-book library
16890:
16884:
16877:
16876:External links
16874:
16873:
16872:
16862:
16856:
16840:
16835:
16812:
16809:
16806:
16805:
16787:
16786:
16784:
16783:
16776:
16756:
16749:
16729:
16722:
16696:
16689:
16665:
16658:
16636:
16633:. p. 296.
16619:
16612:
16588:
16581:
16561:
16554:
16534:
16519:
16499:
16475:
16463:
16451:
16425:
16416:
16395:
16388:
16368:
16347:
16327:
16312:
16305:
16285:
16277:
16255:
16248:
16226:
16219:
16197:
16180:
16173:
16153:
16146:
16122:
16120:
16117:
16115:
16114:
16109:
16104:
16099:
16094:
16089:
16084:
16079:
16074:
16069:
16064:
16059:
16057:Jerk (physics)
16054:
16049:
16044:
16039:
16034:
16029:
16024:
16019:
16014:
16009:
16004:
15999:
15994:
15989:
15983:
15981:
15978:
15966:
15965:
15955:
15945:
15935:
15913:
15899:
15872:
15869:
15865:type synthesis
15823:
15820:
15810:
15807:
15806:
15805:
15802:
15799:
15795:
15788:
15778:
15775:
15762:Kinematic pair
15760:Main article:
15757:
15754:
15736:
15733:
15720:
15717:
15714:
15711:
15689:
15684:
15680:
15676:
15671:
15666:
15662:
15658:
15655:
15652:
15649:
15644:
15639:
15622:center of mass
15609:
15606:
15597:
15594:
15580:
15577:
15557:
15553:
15546:
15541:
15536:
15502:
15498:
15494:
15490:
15486:
15481:
15477:
15473:
15468:
15442:
15437:
15432:
15427:
15422:
15418:
15414:
15409:
15404:
15401:
15398:
15395:
15392:
15387:
15383:
15379:
15374:
15369:
15366:
15363:
15358:
15353:
15331:
15328:
15324:
15320:
15316:
15312:
15307:
15303:
15299:
15294:
15289:
15284:
15281:
15277:
15273:
15269:
15265:
15259:
15256:
15250:
15245:
15240:
15224:
15207:
15202:
15196:
15193:
15191:
15188:
15187:
15182:
15177:
15172:
15169:
15165:
15159:
15155:
15151:
15148:
15144:
15140:
15136:
15135:
15133:
15128:
15123:
15117:
15111:
15108:
15106:
15103:
15102:
15097:
15092:
15087:
15083:
15079:
15076:
15073:
15071:
15068:
15067:
15065:
15059:
15054:
15050:
15046:
15043:
15021:
15015:
15012:
15010:
15007:
15006:
15001:
14996:
14991:
14986:
14981:
14976:
14973:
14969:
14962:
14959:
14953:
14950:
14945:
14942:
14936:
14935:
14933:
14928:
14923:
14917:
14914:
14912:
14909:
14908:
14902:
14898:
14891:
14885:
14881:
14874:
14871:
14867:
14860:
14857:
14851:
14848:
14843:
14840:
14834:
14833:
14831:
14826:
14823:
14817:
14814:
14808:
14786:
14782:
14778:
14775:
14772:
14769:
14766:
14763:
14760:
14756:
14752:
14746:
14743:
14737:
14734:
14728:
14724:
14717:
14714:
14711:
14708:
14704:
14700:
14694:
14691:
14685:
14682:
14678:
14673:
14669:
14666:
14663:
14659:
14652:
14649:
14645:
14640:
14635:
14630:
14622:
14619:
14615:
14610:
14605:
14600:
14578:
14575:
14558:
14552:
14548:
14541:
14536:
14531:
14497:
14493:
14489:
14485:
14481:
14476:
14472:
14468:
14463:
14437:
14432:
14427:
14422:
14417:
14413:
14409:
14404:
14399:
14396:
14393:
14387:
14383:
14376:
14373:
14369:
14365:
14361:
14357:
14354:
14351:
14348:
14345:
14340:
14335:
14321:
14303:
14294:
14287:
14284:
14278:
14275:
14272:
14269:
14247:
14241:
14238:
14236:
14233:
14232:
14226:
14222:
14215:
14211:
14207:
14204:
14201:
14199:
14196:
14195:
14193:
14188:
14185:
14182:
14179:
14155:
14151:
14147:
14144:
14141:
14138:
14135:
14133:
14129:
14124:
14119:
14118:
14113:
14107:
14104:
14103:
14100:
14097:
14094:
14090:
14086:
14085:
14083:
14076:
14070:
14067:
14065:
14062:
14061:
14055:
14051:
14044:
14040:
14030:
14023:
14020:
14014:
14011:
14003:
13996:
13993:
13987:
13986:
13984:
13979:
13976:
13974:
13972:
13967:
13961:
13958:
13957:
13954:
13951:
13948:
13944:
13940:
13939:
13937:
13930:
13924:
13921:
13919:
13916:
13915:
13909:
13905:
13898:
13894:
13888:
13885:
13881:
13874:
13871:
13865:
13862:
13858:
13855:
13851:
13844:
13841:
13835:
13834:
13832:
13827:
13824:
13822:
13820:
13815:
13809:
13806:
13805:
13802:
13799:
13796:
13792:
13788:
13787:
13785:
13778:
13772:
13769:
13767:
13764:
13763:
13759:
13755:
13752:
13750:
13747:
13746:
13744:
13737:
13734:
13730:
13724:
13718:
13715:
13713:
13710:
13709:
13703:
13699:
13692:
13687:
13684:
13678:
13677:
13675:
13670:
13667:
13665:
13663:
13658:
13652:
13649:
13648:
13645:
13642:
13639:
13635:
13631:
13630:
13628:
13621:
13618:
13612:
13606:
13603:
13601:
13598:
13597:
13593:
13589:
13587:
13584:
13583:
13581:
13573:
13567:
13564:
13562:
13559:
13558:
13552:
13548:
13541:
13536:
13533:
13527:
13526:
13524:
13519:
13514:
13508:
13505:
13504:
13499:
13494:
13489:
13488:
13486:
13481:
13478:
13476:
13474:
13471:
13468:
13465:
13461:
13455:
13452:
13448:
13444:
13441:
13438:
13435:
13432:
13429:
13426:
13423:
13420:
13414:
13411:
13405:
13402:
13399:
13397:
13393:
13388:
13383:
13382:
13331:
13326:
13320:
13317:
13316:
13312:
13308:
13307:
13305:
13298:
13292:
13289:
13287:
13284:
13283:
13280:
13277:
13274:
13268:
13264:
13257:
13255:
13252:
13249:
13243:
13240:
13234:
13233:
13231:
13226:
13221:
13215:
13212:
13211:
13207:
13203:
13202:
13200:
13194:
13188:
13182:
13179:
13177:
13174:
13173:
13170:
13167:
13164:
13160:
13156:
13154:
13151:
13148:
13145:
13142:
13141:
13139:
13131:
13128:
13124:
13118:
13114:
13109:
13103:
13100:
13099:
13094:
13089:
13084:
13083:
13081:
13076:
13072:
13068:
13065:
13062:
13059:
13053:
13050:
13044:
13041:
13036:
13031:
13005:
13002:
12989:
12986:
12983:
12980:
12977:
12974:
12971:
12968:
12965:
12956:
12952:
12949:
12946:
12943:
12940:
12920:
12915:
12909:
12906:
12905:
12902:
12899:
12896:
12892:
12888:
12887:
12885:
12878:
12872:
12869:
12867:
12864:
12863:
12860:
12857:
12854:
12850:
12840:
12836:
12833:
12830:
12827:
12824:
12816:
12812:
12809:
12806:
12803:
12802:
12800:
12795:
12790:
12784:
12781:
12780:
12776:
12772:
12771:
12769:
12764:
12761:
12758:
12755:
12751:
12745:
12742:
12738:
12734:
12731:
12728:
12725:
12722:
12719:
12715:
12671:
12666:
12660:
12657:
12656:
12652:
12648:
12647:
12645:
12638:
12632:
12629:
12627:
12624:
12623:
12620:
12617:
12614:
12610:
12606:
12604:
12601:
12598:
12595:
12592:
12591:
12589:
12584:
12579:
12573:
12570:
12569:
12565:
12561:
12560:
12558:
12553:
12549:
12545:
12542:
12539:
12536:
12533:
12530:
12527:
12524:
12521:
12518:
12514:
12466:
12463:
12445:
12442:
12432:
12425:
12418:
12411:
12394:
12391:
12385:
12380:
12376:
12370:
12365:
12361:
12358:
12355:
12352:
12347:
12341:
12336:
12332:
12327:
12321:
12316:
12296:
12293:
12287:
12282:
12278:
12272:
12267:
12263:
12257:
12254:
12248:
12242:
12237:
12233:
12227:
12222:
12200:
12196:
12192:
12186:
12183:
12177:
12174:
12168:
12163:
12159:
12153:
12148:
12144:
12138:
12133:
12112:
12109:
12106:
12100:
12095:
12091:
12085:
12080:
12068:
12067:
12053:
12043:
12032:
12029:
12003:
11976:
11966:
11955:
11952:
11926:
11914:
11891:
11851:
11843:
11839:
11828:
11823:
11810:
11807:
11779:
11774:
11768:
11765:
11763:
11760:
11759:
11756:
11753:
11750:
11748:
11745:
11744:
11742:
11737:
11734:
11728:
11725:
11719:
11699:
11695:
11691:
11688:
11685:
11682:
11679:
11676:
11673:
11669:
11665:
11659:
11656:
11650:
11647:
11642:
11637:
11626:which becomes
11615:
11609:
11605:
11598:
11595:
11592:
11589:
11585:
11581:
11575:
11572:
11566:
11563:
11560:
11557:
11554:
11548:
11545:
11539:
11534:
11529:
11503:
11500:
11488:
11482:
11472:
11461:
11458:
11422:
11417:
11411:
11408:
11406:
11403:
11402:
11399:
11396:
11393:
11391:
11388:
11387:
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10804:Main article:
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3824:Main article:
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1058:
1049:
1042:
1041:
1033:
1026:
1025:
1017:
1010:
1009:
1008:
1007:
1006:
980:
977:
975:("to write").
930:
927:
925:or mechanism.
878:human skeleton
797:
796:
794:
793:
786:
779:
771:
768:
767:
766:
765:
752:
735:
734:
730:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
607:
604:
603:
600:
599:
596:
595:
576:
575:
570:
565:
560:
558:Coriolis force
555:
554:
553:
543:
538:
533:
527:
526:
521:
520:
517:
516:
512:
511:
506:
501:
500:
499:
494:
484:
479:
474:
467:
456:
455:
454:
449:
436:
435:
430:
425:
420:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
381:
380:
375:
370:
365:
360:
355:
349:
343:
341:
334:
331:
328:
327:
324:
323:
319:
318:
313:
308:
303:
298:
293:
288:
283:
277:
272:
266:
261:
250:
245:
240:
235:
230:
229:
228:
223:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
168:
163:
158:
153:
148:
143:
137:
136:
133:
132:
129:
128:
126:
125:
120:
115:
109:
106:
105:
100:
86:
83:
77:
73:
67:
49:
48:
42:
41:
26:
9:
6:
4:
3:
2:
16944:
16933:
16930:
16928:
16925:
16923:
16920:
16919:
16917:
16908:
16905:
16902:
16898:
16894:
16891:
16888:
16885:
16883:
16880:
16879:
16870:
16866:
16863:
16859:
16853:
16849:
16845:
16841:
16838:
16836:0-415-09239-6
16832:
16828:
16824:
16820:
16815:
16814:
16802:
16798:
16792:
16788:
16779:
16773:
16769:
16768:
16760:
16752:
16746:
16742:
16741:
16733:
16725:
16723:0-19-506136-5
16719:
16715:
16710:
16709:
16700:
16692:
16690:1-110-36527-6
16686:
16682:
16678:
16677:
16669:
16661:
16659:0-87891-519-2
16655:
16651:
16647:
16640:
16632:
16631:
16623:
16615:
16613:1-57392-984-0
16609:
16605:
16601:
16600:
16592:
16584:
16582:0-521-82678-0
16578:
16574:
16573:
16565:
16557:
16551:
16547:
16546:
16538:
16530:
16523:
16515:
16514:
16509:
16503:
16487:
16486:
16479:
16472:
16467:
16460:
16455:
16441:
16437:
16436:
16429:
16420:
16412:
16411:
16406:
16399:
16391:
16389:0-486-66346-9
16385:
16381:
16380:
16372:
16364:
16360:
16359:
16351:
16343:
16342:
16337:
16331:
16324:
16323:
16316:
16308:
16302:
16298:
16297:
16289:
16280:
16274:
16270:
16266:
16259:
16251:
16245:
16241:
16237:
16230:
16222:
16216:
16212:
16208:
16201:
16193:
16192:
16184:
16176:
16174:0-89116-355-7
16170:
16166:
16165:
16157:
16149:
16147:0-521-35883-3
16143:
16139:
16138:
16133:
16127:
16123:
16113:
16110:
16108:
16105:
16103:
16100:
16098:
16095:
16093:
16090:
16088:
16085:
16083:
16080:
16078:
16075:
16073:
16070:
16068:
16065:
16063:
16062:Kepler's laws
16060:
16058:
16055:
16053:
16050:
16048:
16045:
16043:
16040:
16038:
16035:
16033:
16030:
16028:
16025:
16023:
16020:
16018:
16015:
16013:
16010:
16008:
16005:
16003:
16000:
15998:
15995:
15993:
15990:
15988:
15985:
15984:
15977:
15975:
15971:
15970:L. C. Schmidt
15963:
15959:
15956:
15953:
15949:
15946:
15943:
15939:
15936:
15933:
15929:
15928:Watt topology
15925:
15922:= 7 : a
15921:
15917:
15914:
15911:
15907:
15903:
15900:
15897:
15893:
15890:
15889:
15888:
15886:
15882:
15878:
15868:
15866:
15862:
15858:
15854:
15850:
15846:
15845:
15840:
15833:
15828:
15819:
15817:
15803:
15800:
15796:
15793:
15789:
15785:
15784:
15783:
15774:
15772:
15768:
15763:
15753:
15751:
15747:
15743:
15732:
15718:
15715:
15712:
15709:
15700:
15687:
15682:
15678:
15674:
15664:
15656:
15650:
15642:
15627:
15626:cross product
15623:
15619:
15615:
15605:
15603:
15593:
15591:
15587:
15576:
15574:
15555:
15544:
15539:
15524:
15520:
15516:
15500:
15492:
15484:
15479:
15475:
15471:
15456:
15440:
15435:
15425:
15420:
15416:
15412:
15402:
15399:
15396:
15393:
15390:
15385:
15381:
15377:
15367:
15364:
15361:
15356:
15329:
15318:
15305:
15297:
15292:
15282:
15271:
15257:
15248:
15243:
15227:
15223:
15218:
15205:
15200:
15194:
15189:
15180:
15167:
15157:
15149:
15142:
15131:
15126:
15121:
15115:
15109:
15104:
15095:
15085:
15074:
15063:
15057:
15052:
15044:
15019:
15013:
15008:
14999:
14989:
14984:
14971:
14960:
14951:
14943:
14931:
14926:
14921:
14915:
14910:
14900:
14889:
14883:
14869:
14858:
14849:
14841:
14829:
14824:
14815:
14812:
14797:
14784:
14773:
14764:
14758:
14744:
14741:
14732:
14726:
14712:
14706:
14692:
14689:
14680:
14676:
14664:
14657:
14650:
14647:
14643:
14638:
14633:
14620:
14617:
14613:
14608:
14603:
14588:
14584:
14574:
14572:
14556:
14550:
14539:
14534:
14519:
14515:
14511:
14495:
14487:
14479:
14474:
14470:
14466:
14451:
14435:
14430:
14420:
14415:
14411:
14407:
14397:
14394:
14391:
14385:
14374:
14363:
14343:
14338:
14320:
14315:
14301:
14292:
14285:
14282:
14276:
14245:
14239:
14234:
14224:
14213:
14202:
14191:
14186:
14180:
14153:
14142:
14136:
14134:
14127:
14111:
14105:
14095:
14081:
14074:
14068:
14063:
14053:
14042:
14028:
14021:
14018:
14012:
14001:
13994:
13991:
13982:
13977:
13975:
13965:
13959:
13949:
13935:
13928:
13922:
13917:
13907:
13896:
13886:
13883:
13879:
13872:
13869:
13863:
13856:
13853:
13849:
13842:
13839:
13830:
13825:
13823:
13813:
13807:
13797:
13783:
13776:
13770:
13765:
13753:
13748:
13742:
13735:
13732:
13728:
13722:
13716:
13711:
13701:
13685:
13682:
13673:
13668:
13666:
13656:
13650:
13640:
13626:
13619:
13616:
13610:
13604:
13599:
13585:
13579:
13571:
13565:
13560:
13550:
13534:
13531:
13522:
13517:
13512:
13506:
13497:
13484:
13479:
13477:
13466:
13453:
13450:
13439:
13433:
13421:
13412:
13409:
13400:
13398:
13391:
13372:
13368:
13364:
13360:
13356:
13352:
13347:
13345:
13329:
13324:
13318:
13303:
13296:
13290:
13285:
13275:
13266:
13250:
13241:
13238:
13229:
13224:
13219:
13213:
13198:
13192:
13186:
13180:
13175:
13165:
13149:
13143:
13137:
13129:
13126:
13122:
13116:
13112:
13107:
13101:
13092:
13079:
13074:
13060:
13051:
13048:
13039:
13034:
13019:
13015:
13011:
13001:
12987:
12984:
12981:
12972:
12966:
12947:
12941:
12918:
12913:
12907:
12897:
12883:
12876:
12870:
12865:
12855:
12834:
12828:
12825:
12810:
12804:
12798:
12793:
12788:
12782:
12767:
12762:
12756:
12743:
12740:
12729:
12723:
12717:
12704:
12700:
12696:
12691:
12689:
12685:
12669:
12664:
12658:
12643:
12636:
12630:
12625:
12615:
12599:
12593:
12587:
12582:
12577:
12571:
12556:
12551:
12537:
12531:
12525:
12519:
12504:is given by:
12503:
12499:
12496:
12492:
12488:
12484:
12480:
12476:
12472:
12462:
12460:
12456:
12451:
12441:
12438:
12431:
12424:
12417:
12410:
12405:
12392:
12378:
12374:
12363:
12356:
12353:
12350:
12345:
12334:
12330:
12325:
12314:
12294:
12280:
12276:
12265:
12255:
12252:
12246:
12235:
12231:
12220:
12198:
12194:
12190:
12184:
12181:
12175:
12172:
12161:
12157:
12146:
12142:
12131:
12110:
12107:
12104:
12093:
12089:
12078:
12051:
12041:
12030:
12027:
12019:
12015:
12011:
12007:
12004:
12001:
11997:
11993:
11974:
11964:
11953:
11950:
11942:
11938:
11934:
11930:
11927:
11924:
11920:
11913:
11909:
11905:
11901:
11897:
11890:
11886:
11882:
11878:
11874:
11870:
11867:
11866:
11865:
11862:
11849:
11841:
11837:
11826:
11821:
11808:
11805:
11797:
11793:
11777:
11772:
11766:
11761:
11754:
11751:
11746:
11740:
11735:
11726:
11697:
11671:
11657:
11645:
11640:
11613:
11607:
11587:
11573:
11561:
11555:
11546:
11543:
11537:
11532:
11517:
11513:
11509:
11499:
11486:
11480:
11470:
11459:
11456:
11448:
11444:
11440:
11436:
11420:
11415:
11409:
11404:
11397:
11394:
11389:
11383:
11378:
11349:
11332:
11319:
11316:
11308:
11302:
11290:
11281:
11278:
11269:
11264:
11249:
11245:
11241:
11225:
11211:
11202:
11199:
11190:
11184:
11173:
11168:
11153:
11149:
11145:
11142:If the point
11135:
11133:
11129:
11113:
11108:
11096:
11090:
11084:
11081:
11070:
11064:
11058:
11055:
11042:
11036:
11030:
11027:
11024:
11013:
11007:
11001:
10998:
10992:
10987:
10978:
10972:
10960:
10947:
10933:
10927:
10921:
10915:
10899:
10895:
10891:
10887:
10883:
10879:
10875:
10871:
10867:
10857:
10855:
10846:
10842:
10838:
10834:
10829:
10825:
10821:
10817:
10812:
10807:
10797:
10795:
10791:
10787:
10782:
10778:
10773:
10769:
10753:
10748:
10738:
10732:
10723:
10712:
10706:
10697:
10686:
10681:
10670:
10665:
10655:
10649:
10640:
10629:
10623:
10614:
10603:
10598:
10583:
10578:
10565:
10557:
10551:
10540:
10523:
10512:
10509:
10503:
10497:
10491:
10475:
10471:
10467:
10463:
10459:
10455:
10452:
10442:
10440:
10436:
10432:
10428:
10424:
10420:
10415:
10402:
10397:
10391:
10384:
10377:
10371:
10364:
10358:
10353:
10348:
10339:
10335:
10329:
10326:
10323:
10318:
10315:
10312:
10303:
10299:
10293:
10290:
10287:
10284:
10279:
10276:
10273:
10267:
10262:
10243:
10240:
10234:
10228:
10215:
10212:of points in
10211:
10207:
10203:
10199:
10195:
10191:
10187:
10183:
10178:
10176:
10172:
10168:
10164:
10148:
10143:
10137:
10132:
10127:
10118:
10114:
10108:
10105:
10102:
10097:
10094:
10091:
10082:
10078:
10072:
10069:
10066:
10063:
10058:
10055:
10052:
10046:
10041:
10036:
10030:
10006:
10000:
9994:
9989:
9975:
9972:
9966:
9955:
9948:
9941:
9937:
9933:
9930:
9926:
9916:
9914:
9910:
9906:
9902:
9898:
9894:
9890:
9885:
9883:
9879:
9875:
9871:
9867:
9863:
9854:
9845:
9843:
9841:
9835:
9831:
9827:
9823:
9819:
9814:
9811:
9807:
9804:
9800:
9790:
9777:
9774:
9771:
9768:
9763:
9759:
9754:
9749:
9745:
9741:
9738:
9735:
9730:
9726:
9705:
9699:
9696:
9690:
9687:
9683:
9677:
9674:
9668:
9665:
9657:
9652:
9650:
9646:
9630:
9627:
9624:
9619:
9615:
9610:
9607:
9604:
9601:
9598:
9593:
9589:
9579:
9566:
9549:
9546:
9543:
9533:
9526:
9523:
9517:
9497:
9490:
9476:
9471:
9456:
9451:
9447:
9442:
9440:
9432:
9416:
9396:
9386:
9379:
9376:
9366:
9359:
9356:
9353:
9349:
9331:
9328:
9311:
9307:
9300:
9286:
9281:
9267:is given by:
9265:
9261:
9254:
9250:
9234:
9217:
9214:
9197:
9194:
9188:
9172:
9163:
9154:
9141:
9122:
9118:
9114:
9104:
9097:
9094:
9075:
9071:
9067:
9057:
9050:
9047:
9044:
9040:
9022:
9019:
9002:
8998:
8991:
8977:
8972:
8957:
8953:
8946:
8942:
8932:
8930:
8907:
8900:
8897:
8889:
8859:
8856:
8853:
8844:
8831:
8812:
8808:
8804:
8794:
8784:
8781:
8778:
8775:
8769:
8749:
8746:
8743:
8740:
8734:
8730:
8710:
8706:
8702:
8692:
8685:
8682:
8665:
8661:
8654:
8640:
8635:
8619:
8615:
8610:
8606:
8601:
8588:
8569:
8565:
8561:
8548:
8541:
8521:
8518:
8499:
8495:
8491:
8481:
8473:
8469:
8466:
8449:
8446:
8442:
8424:
8421:
8404:
8400:
8393:
8379:
8374:
8359:
8355:
8350:
8346:
8342:
8338:
8334:
8318:
8298:
8292:
8289:
8269:
8263:
8260:
8254:
8238:
8234:
8230:
8214:
8194:
8188:
8185:
8168:
8165:
8159:
8143:
8139:
8134:
8121:
8111:
8102:
8098:
8094:
8077:
8074:
8071:
8065:
8038:
8035:
8021:
8013:
8009:
7991:
7982:
7953:
7936:
7933:
7930:
7924:
7907:
7887:
7874:
7857:
7854:
7844:
7837:
7834:
7828:
7808:
7801:
7787:
7779:
7775:
7748:
7719:
7709:
7702:
7699:
7693:
7644:
7621:
7615:
7609:
7606:
7603:
7580:
7574:
7568:
7565:
7562:
7559:
7549:
7541:
7518:
7512:
7506:
7503:
7500:
7477:
7471:
7465:
7462:
7459:
7437:
7433:
7428:
7426:
7422:
7418:
7402:
7382:
7376:
7373:
7350:
7344:
7338:
7335:
7332:
7329:
7306:
7300:
7294:
7291:
7288:
7285:
7279:
7263:
7259:
7255:
7251:
7247:
7243:
7239:
7235:
7230:
7228:
7225:
7221:
7217:
7213:
7209:
7205:
7201:
7197:
7181:
7161:
7155:
7152:
7132:
7126:
7123:
7103:
7097:
7094:
7088:
7072:
7068:
7064:
7060:
7056:
7051:
7049:
7045:
7041:
7037:
7033:
7029:
7025:
7021:
7017:
7013:
7007:
7003:
6999:
6995:
6982:
6977:
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2005:
1997:
1988:
1968:
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1070:
1053:
1046:
1037:
1030:
1021:
1014:
1002:
998:
994:
990:
985:
976:
974:
963:
962:("to move").
961:
952:
944:
940:
936:
926:
924:
920:
916:
912:
908:
904:
900:
896:
894:
890:
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823:
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815:
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792:
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778:
773:
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769:
763:
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739:
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723:
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718:
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688:
685:
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658:
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645:
643:
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638:
635:
633:
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625:
623:
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618:
615:
613:
610:
609:
602:
601:
594:
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586:
582:
579:
578:
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569:
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559:
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552:
549:
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379:
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366:
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356:
354:
351:
350:
348:
347:
342:
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333:
332:
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317:
314:
312:
309:
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276:
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227:
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222:
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186:
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177:
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119:
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111:
110:
108:
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103:
84:
81:
71:
65:
51:
50:
47:
44:
43:
39:
38:
33:
19:
16865:Eduard Study
16850:. Springer.
16847:
16822:
16800:
16796:
16791:
16766:
16759:
16739:
16732:
16707:
16699:
16675:
16668:
16649:
16639:
16629:
16622:
16598:
16591:
16571:
16564:
16544:
16537:
16522:
16512:
16508:Reuleaux, F.
16502:
16490:. Retrieved
16484:
16478:
16466:
16454:
16444:, retrieved
16434:
16428:
16419:
16408:
16398:
16378:
16371:
16357:
16350:
16340:
16330:
16321:
16315:
16295:
16288:
16268:
16265:"Kinematics"
16258:
16239:
16229:
16210:
16200:
16190:
16183:
16163:
16156:
16135:
16126:
15992:Acceleration
15973:
15967:
15961:
15957:
15951:
15947:
15941:
15937:
15919:
15915:
15905:
15901:
15895:
15891:
15884:
15880:
15874:
15864:
15842:
15836:
15815:
15812:
15809:Higher pairs
15780:
15765:
15745:
15738:
15701:
15611:
15599:
15582:
15572:
15522:
15518:
15514:
15454:
15225:
15221:
15219:
14798:
14586:
14582:
14580:
14577:Acceleration
14570:
14517:
14513:
14509:
14449:
14318:
14316:
13370:
13362:
13358:
13354:
13350:
13348:
13343:
13017:
13013:
13009:
13007:
12702:
12698:
12694:
12692:
12687:
12683:
12501:
12497:
12490:
12486:
12482:
12478:
12474:
12470:
12468:
12447:
12436:
12429:
12422:
12415:
12408:
12406:
12069:
12017:
12013:
12009:
12005:
11995:
11991:
11940:
11936:
11932:
11928:
11918:
11911:
11907:
11903:
11899:
11895:
11888:
11884:
11880:
11876:
11872:
11868:
11863:
11795:
11791:
11515:
11511:
11507:
11505:
11502:Acceleration
11446:
11442:
11438:
11437:relative to
11434:
11247:
11243:
11239:
11154:is given by
11151:
11147:
11143:
11141:
11131:
11130:relative to
11127:
10961:
10897:
10896:= (X, Y) in
10893:
10889:
10885:
10881:
10877:
10873:
10869:
10865:
10863:
10853:
10851:
10844:
10840:
10836:
10832:
10827:
10823:
10815:
10793:
10789:
10785:
10780:
10776:
10771:
10767:
10581:
10579:
10473:
10469:
10465:
10461:
10457:
10453:
10448:
10438:
10434:
10430:
10416:
10213:
10209:
10205:
10201:
10197:
10193:
10189:
10185:
10181:
10179:
10174:
10170:
10166:
10162:
9950:
9943:
9939:
9935:
9931:
9924:
9922:
9912:
9911:relative to
9908:
9900:
9892:
9891:relative to
9888:
9886:
9873:
9872:relative to
9869:
9865:
9861:
9859:
9839:
9833:
9825:
9817:
9815:
9808:
9796:
9653:
9648:
9644:
9580:
9454:
9449:
9445:
9443:
9438:
9263:
9259:
9252:
9248:
9170:
9168:
8955:
8951:
8944:
8940:
8938:
8845:
8617:
8613:
8608:
8604:
8602:
8357:
8353:
8348:
8344:
8340:
8336:
8332:
8236:
8232:
8228:
8141:
8137:
8135:
7888:
7435:
7431:
7429:
7424:
7420:
7416:
7261:
7257:
7253:
7249:
7245:
7241:
7237:
7233:
7231:
7226:
7222:axes of the
7219:
7215:
7211:
7208:unit vectors
7203:
7199:
7195:
7070:
7066:
7062:
7058:
7054:
7052:
7047:
7043:
7039:
7035:
7031:
7027:
7023:
7019:
7015:
7011:
7009:
6980:
6975:
6971:
6967:
6963:
6958:
6954:
6945:is unknown.
6327:We can take
6326:
6251:
5900:
5758:
5635:denotes the
5489:
5434:
5431:
5258:
5029:
4836:
4828:
4826:
4625:
4621:
4518:
4415:
4413:
4341:
4337:
4335:
4322:
4320:
4119:
4115:
4012:
3909:
3907:
3837:
3659:
3655:
3653:
3571:
3569:
3487:
3485:
3482:
3407:
3403:
3398:
3300:
3297:acceleration
3294:
3290:
3069:
2888:
2883:
2879:
2543:acceleration
2539:
2536:Acceleration
2401:
1864:displacement
1858:
1852:
1595:
1593:
1394:unit vectors
1093:
1089:
1084:
1080:
1076:
1072:
1068:
1065:
1051:
1035:
1019:
1000:
996:
992:
988:
972:
964:
959:
950:
938:
932:
897:
882:
866:biomechanics
850:astrophysics
847:
801:
800:
591: /
587: /
585:displacement
583: /
444: /
406:Displacement
344:
335:
329:Formulations
316:Virtual work
256: /
196:Acceleration
189:Fundamentals
160:
15746:equilibrium
11449:, that is:
10423:translation
9882:translation
9437:around the
8890:. The term
5905:) is 0, so
5637:dot product
995:, velocity
991:, position
939:cinématique
935:A.M. Ampère
874:robotic arm
830:mathematics
820:of points,
810:mathematics
727:von Neumann
394:Core topics
16922:Kinematics
16916:Categories
16572:Chapter 16
16492:3 November
16446:2021-07-04
16306:019850022X
16164:Kinematics
16119:References
15883:links and
15861:topologies
15849:Mechanisms
15777:Lower pair
7210:along the
6992:See also:
4517:and point
4011:and point
3570:and point
1596:trajectory
1396:along the
903:quantities
802:Kinematics
662:d'Alembert
642:Maupertuis
605:Scientists
487:Rigid body
161:Kinematics
16827:Routledge
15816:cam joint
15719:ω
15665:×
15661:Ω
15556:¨
15493:−
15403:×
15400:ω
15397:×
15394:ω
15368:×
15365:α
15319:−
15302:Ω
15272:−
15258:˙
15255:Ω
15171:Ω
15154:Ω
15150:−
15139:Ω
15078:Ω
15075:−
15070:Ω
14975:Ω
14972:−
14961:˙
14958:Ω
14952:−
14944:˙
14941:Ω
14901:¨
14884:˙
14873:Ω
14870:−
14859:˙
14856:Ω
14850:−
14842:˙
14839:Ω
14816:˙
14745:˙
14727:˙
14693:˙
14551:˙
14488:−
14398:×
14395:ω
14386:˙
14364:−
14350:Ω
14286:˙
14271:Ω
14225:˙
14206:Ω
14203:−
14198:Ω
14054:˙
14022:˙
14013:−
13995:˙
13908:˙
13884:−
13873:˙
13864:−
13854:−
13843:˙
13754:−
13733:−
13702:˙
13686:˙
13617:−
13551:˙
13535:˙
13451:−
13413:˙
13267:˙
13242:˙
13052:˙
12826:−
12741:−
12379:θ
12375:−
12364:θ
12357:α
12335:ω
12315:ω
12281:ω
12266:ω
12236:θ
12232:−
12221:θ
12191:α
12162:ω
12147:θ
12143:−
12132:θ
12108:α
12094:ω
12079:ω
12042:ω
12028:α
11965:θ
11951:ω
11827:θ
11806:α
11762:α
11755:α
11752:−
11727:˙
11724:Ω
11687:Ω
11678:Ω
11658:˙
11655:Ω
11608:˙
11594:Ω
11574:˙
11571:Ω
11547:¨
11471:θ
11457:ω
11405:ω
11398:ω
11395:−
11373:Ω
11339:Ω
11317:−
11282:˙
11203:˙
11185:˙
11091:θ
11085:
11065:θ
11059:
11037:θ
11031:
11025:−
11008:θ
11002:
10724:¨
10698:¨
10641:˙
10615:˙
10476:that is:
10425:is not a
10330:ϕ
10327:
10319:ϕ
10316:
10294:ϕ
10291:
10285:−
10280:ϕ
10277:
10241:ϕ
10109:ϕ
10106:
10098:ϕ
10095:
10073:ϕ
10070:
10064:−
10059:ϕ
10056:
10007:ϕ
9973:ϕ
9775:α
9764:θ
9746:ω
9739:−
9700:¨
9697:θ
9688:α
9678:˙
9675:θ
9666:ω
9620:θ
9608:θ
9602:−
9561:^
9550:θ
9544:−
9538:^
9534:θ
9502:^
9498:θ
9417:ω
9391:^
9387:θ
9371:^
9367:θ
9360:ω
9343:^
9323:^
9229:^
9209:^
9136:^
9109:^
9105:θ
9089:^
9062:^
9058:θ
9051:ω
9034:^
9014:^
8912:^
8908:θ
8901:ω
8871:^
8860:θ
8854:−
8846:The term
8826:^
8799:^
8795:θ
8785:ω
8764:^
8750:θ
8744:−
8724:^
8697:^
8693:θ
8677:^
8583:^
8553:^
8549:θ
8536:^
8513:^
8486:^
8482:θ
8474:ω
8461:^
8436:^
8416:^
8313:^
8284:^
8209:^
8180:^
8116:^
8112:θ
8099:ω
8095:−
8089:^
8078:α
8075:−
8050:^
8039:θ
8036:−
7996:^
7992:θ
7948:^
7937:θ
7934:−
7912:^
7908:θ
7869:^
7858:ω
7855:−
7849:^
7845:θ
7838:α
7813:^
7809:θ
7802:ω
7762:^
7714:^
7710:θ
7703:ω
7681:^
7639:^
7616:θ
7610:
7598:^
7575:θ
7569:
7563:−
7554:^
7550:θ
7536:^
7513:θ
7507:
7495:^
7472:θ
7466:
7454:^
7397:^
7368:^
7345:θ
7339:
7324:^
7301:θ
7295:
7176:^
7147:^
7118:^
6885:Δ
6862:Δ
6785:. Adding
6361:⋅
6335:Δ
6301:Δ
6269:∫
6260:Δ
6232:Δ
6172:Δ
6149:Δ
6126:−
6028:−
5916:
5855:−
5827:α
5824:
5788:−
5713:−
5680:⋅
5660:−
5623:⋅
5568:⋅
5548:−
5527:⋅
5507:−
5456:−
5308:−
5288:Δ
5278:Δ
5179:τ
5166:τ
5127:∫
5105:τ
5093:τ
5071:∫
4987:τ
4962:∫
4791:−
4751:−
4711:−
4671:−
4414:If point
4387:−
4285:−
4245:−
4205:−
4165:−
3908:If point
3881:−
3790:−
3764:−
3738:−
3705:−
3486:If point
3456:−
3342:˙
3271:^
3244:^
3217:^
3134:Δ
3121:Δ
3110:→
3094:Δ
3050:^
3023:^
2996:^
2937:Δ
2927:Δ
2916:→
2910:Δ
2862:^
2842:¯
2826:^
2806:¯
2790:^
2770:¯
2754:^
2737:Δ
2723:¯
2713:Δ
2701:^
2684:Δ
2670:¯
2660:Δ
2648:^
2631:Δ
2617:¯
2607:Δ
2592:Δ
2584:¯
2575:Δ
2563:¯
2376:^
2349:^
2322:^
2263:Δ
2253:Δ
2242:→
2236:Δ
2200:Δ
2177:Δ
2152:Δ
2128:^
2108:¯
2092:^
2072:¯
2056:^
2036:¯
2020:^
2003:Δ
1995:Δ
1983:^
1966:Δ
1958:Δ
1946:^
1929:Δ
1921:Δ
1906:Δ
1896:Δ
1884:¯
1859:direction
1732:^
1703:^
1674:^
1377:^
1346:^
1315:^
1217:^
1197:^
1177:^
929:Etymology
907:mechanism
707:Liouville
589:frequency
509:Vibration
226:potential
151:Continuum
146:Celestial
123:Textbooks
16846:(2007).
16440:archived
16405:"cinema"
16338:(1834).
16134:(1904).
16102:Velocity
16027:Distance
15987:Absement
15980:See also
15877:linkages
15871:Examples
15767:Reuleaux
15750:catenary
15742:pendulum
15618:velocity
13004:Velocity
12465:Position
12450:velocity
11138:Velocity
10860:Position
9878:rotation
9822:distance
9810:Geometry
7206:are the
4895:at time
1855:velocity
1392:are the
1295:are the
862:robotics
839:kinetics
762:Category
687:Hamilton
672:Lagrange
667:Clairaut
632:Horrocks
593:velocity
563:Pendulum
551:reactive
523:Rotation
492:dynamics
442:Inertial
428:Friction
311:Velocity
286:Momentum
166:Kinetics
156:Dynamics
134:Branches
118:Timeline
16899:and an
16821:(ed.),
16488:(Image)
16438:, MIT,
16097:Statics
15620:of its
15614:surface
15525:); and
9956:), as:
9429:is the
2878:where Δ
2396:tangent
876:or the
806:physics
722:Koopman
682:Poisson
677:Laplace
622:Huygens
617:Galileo
462: (
401:Damping
254:Inertia
248:Impulse
221:kinetic
171:Statics
141:Applied
113:History
16854:
16833:
16795:While
16774:
16747:
16720:
16687:
16656:
16610:
16579:
16552:
16386:
16303:
16275:
16246:
16217:
16171:
16144:
15960:= 12,
15950:= 10,
15453:where
14520:; and
14260:where
11798:, and
11710:where
10962:where
10177:, 1).
9905:motion
9880:and a
9645:radial
9409:where
9247:where
9173:axis:
8339:, and
8331:Where
7419:, and
7202:, and
7194:where
7004:, and
6578:where
6376:where
6164:where
5937:, and
5615:where
5600:
2474:where
2144:where
1808:, and
1750:where
1436:, and
1275:, and
1235:where
1096:vector
973:grapho
969:γρᾰ́φω
960:kinein
956:κινεῖν
951:kinema
947:κίνημα
870:engine
864:, and
826:forces
822:bodies
818:motion
760:
712:Appell
697:Cauchy
692:Jacobi
637:Halley
627:Newton
612:Kepler
464:linear
460:Motion
306:Torque
281:Moment
216:Energy
206:Couple
15940:= 8,
15918:= 6,
15904:= 4,
15894:= 2,
12407:Here
11514:) in
10888:) in
10839:) = d
7018:) = (
6475:(the
2404:speed
943:Greek
921:of a
717:Gibbs
702:Routh
657:Euler
296:Speed
291:Space
233:Force
16852:ISBN
16831:ISBN
16772:ISBN
16745:ISBN
16718:ISBN
16685:ISBN
16654:ISBN
16608:ISBN
16577:ISBN
16550:ISBN
16494:2023
16384:ISBN
16301:ISBN
16273:ISBN
16244:ISBN
16215:ISBN
16169:ISBN
16142:ISBN
15034:and
13369:for
12705:as:
12428:and
12414:and
10775:and
9647:and
9251:and
7218:and
6815:and
6644:and
6442:and
2402:The
1853:The
1594:The
1586:The
1361:and
1299:and
872:, a
808:and
301:Time
264:Mass
16714:472
16681:111
15342:or
12701:in
12489:in
12457:or
11794:on
11250:),
11082:cos
11056:sin
11028:sin
10999:cos
10880:= (
10792:in
10429:of
10324:cos
10313:sin
10288:sin
10274:cos
10169:= (
10103:cos
10092:sin
10067:sin
10053:cos
9942:= (
9907:of
9899:of
9838:SE(
9832:on
8958:),
7607:cos
7566:sin
7504:sin
7463:cos
7336:sin
7292:cos
7034:),
7026:),
6970:= |
6673:or
6293:or
5913:cos
5821:cos
4832:B/A
4326:B/A
3087:lim
2906:lim
2232:lim
937:'s
16918::
16801:t′
16716:.
16683:.
16648:.
16606:.
16407:.
16267:.
16238:.
16209:.
15847:.
15790:A
15752:.
15731:.
15600:A
15575:.
14573:.
12461:.
12020::
11943::
11925:).
10884:,
10843:/d
10474:M,
10173:,
9949:,
9913:F.
9901:M.
9884:.
9866:F,
9844:.
8931:.
8335:,
7214:,
7198:,
7196:x̂
7000:,
6996:,
6979:|/
6966:/d
4834:.
4328:.
1779:,
1416:,
1330:,
1255:,
895:.
880:.
860:,
845:.
16871:.
16860:.
16797:τ
16780:.
16753:.
16726:.
16693:.
16662:.
16616:.
16604:4
16585:.
16558:.
16496:.
16413:.
16392:.
16365:.
16363:5
16309:.
16283:.
16281:.
16252:.
16223:.
16177:.
16150:.
15962:j
15958:N
15952:j
15948:N
15942:j
15938:N
15920:j
15916:N
15912:;
15906:j
15902:N
15896:j
15892:N
15885:j
15881:N
15716:r
15713:=
15710:v
15688:.
15683:O
15679:/
15675:G
15670:r
15657:=
15654:)
15651:t
15648:(
15643:G
15638:v
15573:M
15552:d
15545:=
15540:O
15535:A
15523:M
15519:O
15515:P
15501:,
15497:d
15489:P
15485:=
15480:O
15476:/
15472:P
15467:R
15455:α
15441:,
15436:O
15431:A
15426:+
15421:O
15417:/
15413:P
15408:R
15391:+
15386:O
15382:/
15378:P
15373:R
15362:=
15357:P
15352:A
15330:,
15327:)
15323:d
15315:P
15311:(
15306:2
15298:+
15293:O
15288:A
15283:+
15280:)
15276:d
15268:P
15264:(
15249:=
15244:P
15239:A
15226:P
15222:A
15206:.
15201:]
15195:0
15190:0
15181:O
15176:v
15168:+
15164:d
15158:2
15143:2
15132:[
15127:=
15122:2
15116:]
15110:0
15105:0
15096:O
15091:v
15086:+
15082:d
15064:[
15058:=
15053:2
15049:]
15045:S
15042:[
15020:]
15014:0
15009:0
15000:O
14995:A
14990:+
14985:O
14980:v
14968:d
14932:[
14927:=
14922:]
14916:0
14911:0
14897:d
14890:+
14880:d
14866:d
14830:[
14825:=
14822:]
14813:S
14807:[
14785:.
14781:P
14777:]
14774:S
14771:[
14768:]
14765:S
14762:[
14759:+
14755:P
14751:]
14742:S
14736:[
14733:=
14723:P
14716:]
14713:S
14710:[
14707:+
14703:P
14699:]
14690:S
14684:[
14681:=
14677:)
14672:P
14668:]
14665:S
14662:[
14658:(
14651:t
14648:d
14644:d
14639:=
14634:P
14629:v
14621:t
14618:d
14614:d
14609:=
14604:P
14599:A
14587:B
14583:P
14571:O
14557:,
14547:d
14540:=
14535:O
14530:v
14518:M
14514:O
14510:P
14496:,
14492:d
14484:P
14480:=
14475:O
14471:/
14467:P
14462:R
14450:ω
14436:,
14431:O
14426:v
14421:+
14416:O
14412:/
14408:P
14403:R
14392:=
14382:d
14375:+
14372:)
14368:d
14360:P
14356:(
14353:]
14347:[
14344:=
14339:P
14334:v
14322:P
14319:v
14302:,
14297:T
14293:A
14283:A
14277:=
14274:]
14268:[
14246:]
14240:0
14235:0
14221:d
14214:+
14210:d
14192:[
14187:=
14184:]
14181:S
14178:[
14154:.
14150:P
14146:]
14143:S
14140:[
14137:=
14128:P
14123:v
14112:]
14106:1
14099:)
14096:t
14093:(
14089:P
14082:[
14075:]
14069:0
14064:0
14050:d
14043:+
14039:d
14033:T
14029:A
14019:A
14006:T
14002:A
13992:A
13983:[
13978:=
13966:]
13960:1
13953:)
13950:t
13947:(
13943:P
13936:[
13929:]
13923:0
13918:0
13904:d
13897:+
13893:d
13887:1
13880:A
13870:A
13857:1
13850:A
13840:A
13831:[
13826:=
13814:]
13808:1
13801:)
13798:t
13795:(
13791:P
13784:[
13777:]
13771:A
13766:0
13758:d
13749:1
13743:[
13736:1
13729:A
13723:]
13717:0
13712:0
13698:d
13683:A
13674:[
13669:=
13657:]
13651:1
13644:)
13641:t
13638:(
13634:P
13627:[
13620:1
13611:]
13605:1
13600:0
13592:d
13586:A
13580:[
13572:]
13566:0
13561:0
13547:d
13532:A
13523:[
13518:=
13513:]
13507:0
13498:P
13493:v
13485:[
13480:=
13470:)
13467:t
13464:(
13460:P
13454:1
13447:]
13443:)
13440:t
13437:(
13434:T
13431:[
13428:]
13425:)
13422:t
13419:(
13410:T
13404:[
13401:=
13392:P
13387:v
13371:p
13363:F
13359:t
13357:(
13355:P
13351:P
13344:p
13330:.
13325:]
13319:1
13311:p
13304:[
13297:]
13291:0
13286:0
13279:)
13276:t
13273:(
13263:d
13254:)
13251:t
13248:(
13239:A
13230:[
13225:=
13220:]
13214:1
13206:p
13199:[
13193:)
13187:]
13181:1
13176:0
13169:)
13166:t
13163:(
13159:d
13153:)
13150:t
13147:(
13144:A
13138:[
13130:t
13127:d
13123:d
13117:(
13113:=
13108:]
13102:0
13093:P
13088:v
13080:[
13075:=
13071:p
13067:]
13064:)
13061:t
13058:(
13049:T
13043:[
13040:=
13035:P
13030:v
13018:t
13016:(
13014:P
13010:P
12988:.
12985:I
12982:=
12979:]
12976:)
12973:t
12970:(
12967:A
12964:[
12959:T
12955:]
12951:)
12948:t
12945:(
12942:A
12939:[
12919:.
12914:]
12908:1
12901:)
12898:t
12895:(
12891:P
12884:[
12877:]
12871:1
12866:0
12859:)
12856:t
12853:(
12849:d
12843:T
12839:)
12835:t
12832:(
12829:A
12819:T
12815:)
12811:t
12808:(
12805:A
12799:[
12794:=
12789:]
12783:1
12775:p
12768:[
12763:=
12760:)
12757:t
12754:(
12750:P
12744:1
12737:]
12733:)
12730:t
12727:(
12724:T
12721:[
12718:=
12714:p
12703:M
12699:p
12695:P
12688:P
12684:P
12670:.
12665:]
12659:1
12651:p
12644:[
12637:]
12631:1
12626:0
12619:)
12616:t
12613:(
12609:d
12603:)
12600:t
12597:(
12594:A
12588:[
12583:=
12578:]
12572:1
12564:P
12557:[
12552:=
12548:p
12544:]
12541:)
12538:t
12535:(
12532:T
12529:[
12526:=
12523:)
12520:t
12517:(
12513:P
12502:F
12498:M
12491:B
12487:P
12483:p
12479:t
12477:(
12475:d
12471:B
12437:α
12433:f
12430:ω
12426:i
12423:ω
12419:f
12416:θ
12412:i
12409:θ
12393:.
12390:)
12384:i
12369:f
12360:(
12354:2
12351:+
12346:2
12340:i
12331:=
12326:2
12320:f
12295:t
12292:)
12286:i
12277:+
12271:f
12262:(
12256:2
12253:1
12247:=
12241:i
12226:f
12199:2
12195:t
12185:2
12182:1
12176:+
12173:t
12167:i
12158:=
12152:i
12137:f
12111:t
12105:+
12099:i
12090:=
12084:f
12052:t
12048:d
12038:d
12031:=
12018:t
12014:ω
12010:α
12002:.
11996:ω
11992:Ω
11975:t
11971:d
11961:d
11954:=
11941:t
11937:θ
11933:ω
11919:t
11917:(
11915:⊥
11912:r
11908:x
11904:θ
11896:t
11894:(
11892:⊥
11889:r
11885:t
11883:(
11881:r
11877:t
11875:(
11873:r
11850:.
11842:2
11838:t
11833:d
11822:2
11817:d
11809:=
11796:F
11792:M
11778:,
11773:]
11767:0
11747:0
11741:[
11736:=
11733:]
11718:[
11698:,
11694:P
11690:]
11684:[
11681:]
11675:[
11672:+
11668:P
11664:]
11649:[
11646:=
11641:P
11636:A
11614:,
11604:P
11597:]
11591:[
11588:+
11584:P
11580:]
11565:[
11562:=
11559:)
11556:t
11553:(
11544:P
11538:=
11533:P
11528:A
11516:F
11512:t
11510:(
11508:P
11487:.
11481:t
11477:d
11467:d
11460:=
11447:θ
11443:ω
11439:F
11435:M
11421:,
11416:]
11410:0
11390:0
11384:[
11379:=
11376:]
11370:[
11350:,
11346:P
11342:]
11336:[
11333:=
11329:P
11325:]
11320:1
11313:)
11309:t
11306:(
11303:A
11300:[
11297:]
11294:)
11291:t
11288:(
11279:A
11273:[
11270:=
11265:P
11260:v
11248:t
11246:(
11244:P
11240:p
11226:.
11222:p
11218:]
11215:)
11212:t
11209:(
11200:A
11194:[
11191:=
11181:P
11174:=
11169:P
11164:v
11152:F
11148:M
11144:p
11132:F
11128:M
11114:,
11109:]
11103:)
11100:)
11097:t
11094:(
11088:(
11077:)
11074:)
11071:t
11068:(
11062:(
11049:)
11046:)
11043:t
11040:(
11034:(
11020:)
11017:)
11014:t
11011:(
11005:(
10993:[
10988:=
10985:]
10982:)
10979:t
10976:(
10973:A
10970:[
10948:,
10944:p
10940:]
10937:)
10934:t
10931:(
10928:A
10925:[
10922:=
10919:)
10916:t
10913:(
10909:P
10898:F
10894:P
10890:M
10886:y
10882:x
10878:p
10874:z
10870:F
10866:M
10854:z
10848:.
10845:t
10841:θ
10837:t
10835:(
10833:ω
10828:t
10826:(
10824:θ
10816:Ω
10794:M
10790:p
10786:M
10781:O
10777:a
10772:O
10768:v
10754:,
10749:O
10744:a
10739:=
10736:)
10733:t
10730:(
10720:d
10713:=
10710:)
10707:t
10704:(
10694:r
10687:=
10682:P
10677:a
10671:,
10666:O
10661:v
10656:=
10653:)
10650:t
10647:(
10637:d
10630:=
10627:)
10624:t
10621:(
10611:r
10604:=
10599:P
10594:v
10582:P
10566:.
10562:p
10558:+
10555:)
10552:t
10549:(
10545:d
10541:=
10537:p
10533:]
10530:)
10527:)
10524:t
10521:(
10517:d
10513:,
10510:0
10507:(
10504:T
10501:[
10498:=
10495:)
10492:t
10489:(
10485:r
10470:t
10468:(
10466:d
10462:F
10458:θ
10454:M
10439:R
10435:R
10431:R
10403:.
10398:]
10392:1
10385:y
10378:x
10372:[
10365:]
10359:1
10354:0
10349:0
10340:y
10336:d
10304:x
10300:d
10268:[
10263:=
10259:r
10255:]
10252:)
10248:d
10244:,
10238:(
10235:T
10232:[
10229:=
10225:P
10214:M
10210:F
10206:F
10202:F
10198:d
10194:M
10190:F
10186:M
10182:r
10175:y
10171:x
10167:r
10163:z
10149:.
10144:]
10138:1
10133:0
10128:0
10119:y
10115:d
10083:x
10079:d
10047:[
10042:=
10037:]
10031:1
10025:0
10016:d
10010:)
10004:(
10001:A
9995:[
9990:=
9987:]
9984:)
9980:d
9976:,
9970:(
9967:T
9964:[
9953:y
9951:d
9946:x
9944:d
9940:d
9936:φ
9934:(
9932:A
9925:R
9909:M
9893:F
9889:M
9874:F
9870:M
9862:M
9842:)
9840:n
9834:R
9826:n
9818:R
9778:.
9772:r
9769:=
9760:a
9755:,
9750:2
9742:r
9736:=
9731:r
9727:a
9706:,
9691:=
9684:,
9669:=
9631:,
9628:a
9625:=
9616:a
9611:,
9605:v
9599:=
9594:r
9590:a
9567:.
9557:r
9547:v
9527:a
9524:=
9518:t
9514:d
9508:)
9491:v
9488:(
9484:d
9477:=
9472:P
9467:a
9455:P
9450:P
9446:a
9439:z
9435:θ
9397:,
9380:v
9377:=
9357:r
9354:=
9350:)
9339:z
9332:z
9329:+
9319:r
9312:r
9308:(
9301:t
9297:d
9292:d
9287:=
9282:P
9277:v
9264:P
9260:v
9256:0
9253:z
9249:r
9235:,
9225:z
9218:z
9215:+
9205:r
9198:r
9195:=
9192:)
9189:t
9186:(
9182:r
9171:z
9142:.
9132:z
9123:z
9119:v
9115:+
9098:v
9095:=
9085:z
9076:z
9072:v
9068:+
9048:r
9045:=
9041:)
9030:z
9023:z
9020:+
9010:r
9003:r
8999:(
8992:t
8988:d
8983:d
8978:=
8973:P
8968:v
8956:t
8954:(
8952:r
8948:P
8945:v
8941:r
8898:v
8867:r
8857:v
8832:.
8822:z
8813:z
8809:a
8805:+
8788:)
8782:v
8779:+
8776:a
8773:(
8770:+
8760:r
8753:)
8747:v
8741:a
8738:(
8735:=
8731:)
8720:z
8711:z
8707:v
8703:+
8686:v
8683:+
8673:r
8666:v
8662:(
8655:t
8651:d
8646:d
8641:=
8636:P
8631:a
8618:P
8614:v
8609:P
8605:a
8589:.
8579:z
8570:z
8566:v
8562:+
8559:)
8542:+
8532:r
8525:(
8522:v
8519:=
8509:z
8500:z
8496:v
8492:+
8470:r
8467:+
8457:r
8450:v
8447:=
8443:)
8432:z
8425:z
8422:+
8412:r
8405:r
8401:(
8394:t
8390:d
8385:d
8380:=
8375:P
8370:v
8358:t
8356:(
8354:r
8349:P
8345:v
8341:z
8337:θ
8333:r
8319:.
8309:z
8302:)
8299:t
8296:(
8293:z
8290:+
8280:r
8273:)
8270:t
8267:(
8264:r
8261:=
8258:)
8255:t
8252:(
8248:r
8237:R
8233:t
8231:(
8229:r
8215:.
8205:z
8198:)
8195:t
8192:(
8189:z
8186:+
8176:r
8169:r
8166:=
8163:)
8160:t
8157:(
8153:r
8142:t
8140:(
8138:r
8122:.
8103:2
8085:r
8072:=
8066:t
8062:d
8056:)
8046:r
8033:(
8029:d
8022:=
8014:2
8010:t
8005:d
7983:2
7978:d
7954:.
7944:r
7931:=
7925:t
7921:d
7900:d
7875:.
7865:r
7835:=
7829:t
7825:d
7819:)
7799:(
7795:d
7788:=
7780:2
7776:t
7771:d
7758:r
7749:2
7744:d
7720:.
7700:=
7694:t
7690:d
7677:r
7669:d
7645:.
7635:y
7628:)
7625:)
7622:t
7619:(
7613:(
7604:+
7594:x
7587:)
7584:)
7581:t
7578:(
7572:(
7560:=
7542:,
7532:y
7525:)
7522:)
7519:t
7516:(
7510:(
7501:+
7491:x
7484:)
7481:)
7478:t
7475:(
7469:(
7460:=
7450:r
7436:t
7434:(
7432:r
7425:t
7423:(
7421:θ
7417:r
7403:,
7393:z
7386:)
7383:t
7380:(
7377:z
7374:+
7364:y
7357:)
7354:)
7351:t
7348:(
7342:(
7333:r
7330:+
7320:x
7313:)
7310:)
7307:t
7304:(
7298:(
7289:r
7286:=
7283:)
7280:t
7277:(
7273:r
7262:y
7260:–
7258:x
7254:θ
7250:F
7246:z
7242:t
7240:(
7238:r
7234:P
7227:F
7220:z
7216:y
7212:x
7204:ẑ
7200:ŷ
7182:,
7172:z
7165:)
7162:t
7159:(
7156:z
7153:+
7143:y
7136:)
7133:t
7130:(
7127:y
7124:+
7114:x
7107:)
7104:t
7101:(
7098:x
7095:=
7092:)
7089:t
7086:(
7082:r
7071:t
7069:(
7067:r
7063:F
7059:r
7055:P
7048:Y
7046:–
7044:X
7040:t
7038:(
7036:z
7032:t
7030:(
7028:y
7024:t
7022:(
7020:x
7016:t
7014:(
7012:r
6983:.
6981:R
6976:θ
6972:a
6968:t
6964:ω
6959:θ
6955:a
6943:v
6927:2
6921:2
6917:t
6913:a
6907:+
6904:t
6899:0
6895:v
6891:=
6888:r
6865:r
6840:2
6834:2
6830:t
6826:a
6803:t
6798:0
6794:v
6771:2
6765:2
6761:t
6757:a
6751:=
6746:2
6742:t
6738:a
6733:2
6730:1
6725:=
6722:t
6719:t
6716:a
6711:2
6708:1
6703:=
6700:H
6697:B
6692:2
6689:1
6684:=
6681:A
6661:t
6658:a
6655:=
6652:H
6632:t
6629:=
6626:B
6606:H
6586:B
6566:H
6563:B
6558:2
6555:1
6531:0
6527:v
6523:t
6503:a
6483:A
6461:0
6457:v
6453:=
6450:B
6430:t
6427:=
6424:A
6404:B
6384:A
6364:B
6358:A
6338:r
6304:r
6281:t
6277:d
6272:v
6266:=
6263:r
6238:.
6235:r
6229:a
6226:2
6223:+
6218:2
6213:0
6209:v
6205:=
6200:2
6196:v
6175:r
6152:r
6146:=
6142:|
6136:0
6131:r
6122:r
6117:|
6113:,
6110:v
6107:=
6103:|
6098:v
6093:|
6089:,
6086:a
6083:=
6079:|
6074:a
6069:|
6048:.
6044:|
6038:0
6033:r
6024:r
6019:|
6014:|
6010:a
6006:|
6002:2
5999:+
5994:2
5989:|
5982:0
5977:v
5971:|
5967:=
5962:2
5957:|
5951:v
5946:|
5925:1
5922:=
5919:0
5903:α
5887:.
5882:2
5877:|
5870:0
5865:v
5859:|
5850:2
5845:|
5839:v
5834:|
5830:=
5817:|
5813:a
5809:|
5804:|
5798:0
5793:r
5784:r
5779:|
5775:2
5761:α
5745:.
5740:2
5735:|
5728:0
5723:v
5717:|
5708:2
5703:|
5697:v
5692:|
5688:=
5684:a
5676:)
5670:0
5665:r
5656:r
5651:(
5647:2
5603:,
5595:2
5589:0
5584:v
5579:+
5575:v
5564:)
5558:0
5553:v
5544:v
5539:(
5535:=
5531:a
5523:)
5517:0
5512:r
5503:r
5498:(
5473:a
5466:0
5461:v
5452:v
5445:=
5442:t
5418:.
5415:t
5411:)
5406:2
5400:0
5395:v
5390:+
5386:v
5379:(
5375:+
5370:0
5365:r
5360:=
5357:)
5354:t
5351:(
5347:r
5324:t
5318:0
5313:v
5304:v
5297:=
5291:t
5282:v
5272:=
5268:a
5245:.
5240:2
5236:t
5231:a
5224:2
5221:1
5215:+
5212:t
5207:0
5202:v
5197:+
5192:0
5187:r
5182:=
5175:d
5170:)
5162:a
5158:+
5153:0
5148:v
5142:(
5136:t
5131:0
5123:+
5118:0
5113:r
5108:=
5101:d
5096:)
5090:(
5086:v
5080:t
5075:0
5067:+
5062:0
5057:r
5052:=
5049:)
5046:t
5043:(
5039:r
5016:.
5013:t
5009:a
5005:+
5000:0
4995:v
4990:=
4983:d
4977:a
4971:t
4966:0
4958:+
4953:0
4948:v
4943:=
4940:)
4937:t
4934:(
4930:v
4909:0
4906:=
4903:t
4881:0
4876:v
4852:0
4847:r
4829:r
4812:)
4804:z
4800:B
4795:a
4784:z
4780:C
4775:a
4771:,
4764:y
4760:B
4755:a
4744:y
4740:C
4735:a
4731:,
4724:x
4720:B
4715:a
4704:x
4700:C
4695:a
4690:(
4686:=
4681:B
4676:a
4666:C
4661:a
4656:=
4651:B
4647:/
4643:C
4638:a
4626:B
4622:C
4607:)
4599:z
4595:B
4590:a
4586:,
4579:y
4575:B
4570:a
4566:,
4559:x
4555:B
4550:a
4545:(
4541:=
4536:B
4531:a
4519:B
4504:)
4496:z
4492:C
4487:a
4483:,
4476:y
4472:C
4467:a
4463:,
4456:x
4452:C
4447:a
4442:(
4438:=
4433:C
4428:a
4416:C
4397:B
4392:a
4382:C
4377:a
4372:=
4367:B
4363:/
4359:C
4354:a
4342:B
4338:C
4323:r
4306:)
4298:z
4294:B
4289:v
4278:z
4274:A
4269:v
4265:,
4258:y
4254:B
4249:v
4238:y
4234:A
4229:v
4225:,
4218:x
4214:B
4209:v
4198:x
4194:A
4189:v
4184:(
4180:=
4175:B
4170:v
4160:A
4155:v
4150:=
4145:B
4141:/
4137:A
4132:v
4120:B
4116:A
4101:)
4093:z
4089:B
4084:v
4080:,
4073:y
4069:B
4064:v
4060:,
4053:x
4049:B
4044:v
4039:(
4035:=
4030:B
4025:v
4013:B
3998:)
3990:z
3986:A
3981:v
3977:,
3970:y
3966:A
3961:v
3957:,
3950:x
3946:A
3941:v
3936:(
3932:=
3927:A
3922:v
3910:A
3891:B
3886:v
3876:A
3871:v
3866:=
3861:B
3857:/
3853:A
3848:v
3804:)
3798:B
3794:z
3785:A
3781:z
3777:,
3772:B
3768:y
3759:A
3755:y
3751:,
3746:B
3742:x
3733:A
3729:x
3724:(
3720:=
3715:B
3710:r
3700:A
3695:r
3690:=
3685:B
3681:/
3677:A
3672:r
3660:B
3656:A
3639:)
3633:B
3629:z
3625:,
3620:B
3616:y
3612:,
3607:B
3603:x
3598:(
3594:=
3589:B
3584:r
3572:B
3555:)
3549:A
3545:z
3541:,
3536:A
3532:y
3528:,
3523:A
3519:x
3514:(
3510:=
3505:A
3500:r
3488:A
3466:B
3461:r
3451:A
3446:r
3441:=
3436:B
3432:/
3428:A
3423:r
3408:B
3404:A
3380:.
3374:t
3370:d
3364:v
3360:d
3353:=
3349:|
3338:v
3330:|
3326:=
3322:|
3317:a
3312:|
3301:a
3277:.
3267:z
3258:z
3254:a
3250:+
3240:y
3231:y
3227:a
3223:+
3213:x
3204:x
3200:a
3196:=
3188:2
3184:t
3179:d
3172:r
3166:2
3161:d
3153:=
3145:2
3141:)
3137:t
3131:(
3125:r
3113:0
3105:2
3101:)
3097:t
3091:(
3083:=
3079:a
3056:.
3046:z
3037:z
3033:a
3029:+
3019:y
3010:y
3006:a
3002:+
2992:x
2983:x
2979:a
2975:=
2969:t
2965:d
2958:v
2953:d
2946:=
2940:t
2931:v
2919:0
2913:t
2902:=
2898:a
2884:t
2880:v
2858:z
2849:z
2839:a
2832:+
2822:y
2813:y
2803:a
2796:+
2786:x
2777:x
2767:a
2760:=
2750:z
2740:t
2730:z
2720:v
2707:+
2697:y
2687:t
2677:y
2667:v
2654:+
2644:x
2634:t
2624:x
2614:v
2601:=
2595:t
2581:v
2569:=
2560:a
2520:t
2516:d
2511:/
2507:s
2503:d
2482:s
2462:,
2456:t
2452:d
2446:s
2442:d
2435:=
2431:|
2426:v
2421:|
2417:=
2414:v
2382:.
2372:z
2363:z
2359:v
2355:+
2345:y
2336:y
2332:v
2328:+
2318:x
2309:x
2305:v
2301:=
2295:t
2291:d
2284:r
2279:d
2272:=
2266:t
2257:r
2245:0
2239:t
2228:=
2224:v
2203:t
2180:t
2156:r
2124:z
2115:z
2105:v
2098:+
2088:y
2079:y
2069:v
2062:+
2052:x
2043:x
2033:v
2026:=
2016:z
2006:t
1998:z
1989:+
1979:y
1969:t
1961:y
1952:+
1942:x
1932:t
1924:x
1915:=
1909:t
1900:r
1890:=
1881:v
1862:(
1825:)
1822:t
1819:(
1816:z
1796:)
1793:t
1790:(
1787:y
1767:)
1764:t
1761:(
1758:x
1738:,
1728:z
1721:)
1718:t
1715:(
1712:z
1709:+
1699:y
1692:)
1689:t
1686:(
1683:y
1680:+
1670:x
1663:)
1660:t
1657:(
1654:x
1651:=
1648:)
1645:t
1642:(
1638:r
1617:)
1614:t
1611:(
1607:r
1574:.
1567:2
1563:z
1559:+
1554:2
1550:y
1546:+
1541:2
1537:x
1531:=
1527:|
1522:r
1517:|
1495:r
1473:|
1469:r
1465:|
1444:z
1424:y
1404:x
1373:z
1342:y
1311:x
1283:z
1263:y
1243:x
1223:,
1213:z
1206:z
1203:+
1193:y
1186:y
1183:+
1173:x
1166:x
1163:=
1160:)
1157:z
1154:,
1151:y
1148:,
1145:x
1142:(
1139:=
1135:r
1112:r
1085:r
1081:z
1077:r
1073:y
1069:x
1052:a
1036:v
1020:r
1003:.
1001:a
997:v
993:r
989:m
790:e
783:t
776:v
466:)
85:t
82:d
76:p
72:d
66:=
61:F
34:.
20:)
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