1505:
750:
1912:
3142:
4967:
1030:
5904:
1042:
9089:
9836:
763:
4527:
4101:. A person on a moving swing can increase the amplitude of the swing's oscillations without any external drive force (pushes) being applied, by changing the moment of inertia of the swing by rocking back and forth ("pumping") or alternately standing and squatting, in rhythm with the swing's oscillations. The varying of the parameters drives the system. Examples of parameters that may be varied are its resonance frequency
1513:
1494:
5454:
6524:
4962:{\displaystyle q_{t}(\tau )={\begin{cases}e^{-\zeta \tau }\left(c_{1}e^{\tau {\sqrt {\zeta ^{2}-1}}}+c_{2}e^{-\tau {\sqrt {\zeta ^{2}-1}}}\right)&\zeta >1{\text{ (overdamping)}}\\e^{-\zeta \tau }(c_{1}+c_{2}\tau )=e^{-\tau }(c_{1}+c_{2}\tau )&\zeta =1{\text{ (critical damping)}}\\e^{-\zeta \tau }\left&\zeta <1{\text{ (underdamping)}}\end{cases}}}
5696:
5224:
5142:
6325:
6137:
6292:
3365:
9912:
is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from
5461:
4159:
Parametric oscillators have been developed as low-noise amplifiers, especially in the radio and microwave frequency range. Thermal noise is minimal, since a reactance (not a resistance) is varied. Another common use is frequency conversion, e.g., conversion from audio to radio frequencies. For
2498:
10222:
When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is
4211:
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing, since the action appears as a time varying modification on a system parameter. This effect is different from
1532:
In real oscillators, friction, or damping, slows the motion of the system. Due to frictional force, the velocity decreases in proportion to the acting frictional force. While in a simple undriven harmonic oscillator the only force acting on the mass is the restoring force, in a damped harmonic
6768:
above). Below is a table showing analogous quantities in four harmonic oscillator systems in mechanics and electronics. If analogous parameters on the same line in the table are given numerically equal values, the behavior of the oscillators – their output waveform, resonant
1518:
1517:
1514:
2636:
1807:
4493:
1519:
3045:
8693:
1006:. The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as
3788:
1520:
4322:
6667:
4986:
5917:
3898:
6144:
3207:
2350:
1679:
1516:
2757:
5449:{\displaystyle q_{s}=Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} q_{s}}{\mathrm {d} \tau }}=i\omega Ae^{i(\omega \tau +\varphi )},\quad {\frac {\mathrm {d} ^{2}q_{s}}{\mathrm {d} \tau ^{2}}}=-\omega ^{2}Ae^{i(\omega \tau +\varphi )}.}
2900:
8918:
2357:
9042:
5804:
2168:
6519:{\displaystyle \tan \varphi =-{\frac {2\zeta \omega }{1-\omega ^{2}}}={\frac {2\zeta \omega }{\omega ^{2}-1}}~~\implies ~~\varphi \equiv \varphi (\zeta ,\omega )=\arctan \left({\frac {2\zeta \omega }{\omega ^{2}-1}}\right)+n\pi .}
4096:
is a driven harmonic oscillator in which the drive energy is provided by varying the parameters of the oscillator, such as the damping or restoring force. A familiar example of parametric oscillation is "pumping" on a playground
1419:
of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.
1189:
5894:
10027:
2513:
1684:
4380:
9609:
9407:
2905:
9222:
8168:
9730:
3659:
1998:= 1): The system returns to steady state as quickly as possible without oscillating (although overshoot can occur if the initial velocity is nonzero). This is often desired for the damping of systems such as doors.
9330:
5691:{\displaystyle -\omega ^{2}Ae^{i(\omega \tau +\varphi )}+2\zeta i\omega Ae^{i(\omega \tau +\varphi )}+Ae^{i(\omega \tau +\varphi )}=(-\omega ^{2}A+2\zeta i\omega A+A)e^{i(\omega \tau +\varphi )}=e^{i\omega \tau }.}
3665:
8310:
3906:
of the oscillation relative to the driving force. The phase value is usually taken to be between −180° and 0 (that is, it represents a phase lag, for both positive and negative values of the arctan argument).
10107:
4227:
7949:
6543:
7063:
6973:
7730:
5217:
3975:
3802:
2073:
8486:
7830:
7780:
104:
7685:
2240:
7018:
6928:
4148:
parametric oscillator oscillates when the diode's capacitance is varied periodically. The circuit that varies the diode's capacitance is called the "pump" or "driver". In microwave electronics,
7904:
1575:
8029:
7989:
10171:. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equation
6749:
5925:
1515:
2643:
4078:) damped harmonic oscillator and represent the systems response to other events that occurred previously. The transient solutions typically die out rapidly enough that they can be ignored.
1852:
1298:
8239:
1904:
1258:
9075:
with a non-vanishing second derivative, one can use the solution to the simple harmonic oscillator to provide an approximate solution for small perturbations around the equilibrium point.
8102:
2805:
9257:
8391:
2116:
11116:
10839:
1477:
885:
4206:
4040:
9489:
9533:
3088:
10561:
10218:
5703:
2214:
10762:
10628:
10427:
10306:
3178:
11369:
11298:
10491:
2128:
8763:
5137:{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau )+i\sin(\omega \tau )=e^{i\omega \tau }.}
128:
1353:
6132:{\displaystyle \left.{\begin{aligned}A^{2}(1-\omega ^{2})^{2}&=\cos ^{2}\varphi \\(2\zeta \omega A)^{2}&=\sin ^{2}\varphi \end{aligned}}\right\}\Rightarrow A^{2}=1.}
11170:
11039:
10985:
10899:
10360:
1088:
9898:
6287:{\displaystyle A=A(\zeta ,\omega )=\operatorname {sgn} \left({\frac {-\sin \varphi }{2\zeta \omega }}\right){\frac {1}{\sqrt {(1-\omega ^{2})^{2}+(2\zeta \omega )^{2}}}}.}
3360:{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {1}{m}}F_{0}\sin(\omega t),}
11331:
9805:
9758:
9663:
9636:
5811:
3525:
8942:
9919:
11228:
10692:
9447:
8731:
7555:
7235:
6881:
3568:
11260:
9825:
9538:
8479:
7423:
6833:
4119:
4076:
3498:
3412:
11196:
9337:
7856:
4139:
3545:
3198:
9778:
8758:
7637:
7485:
4002:
3478:
3392:
1387:
10788:
9073:
8422:
7521:
7309:
10871:
9148:
7373:
7341:
9673:
3573:
11142:
11065:
11011:
10957:
10925:
10721:
10654:
10587:
10517:
10453:
10386:
10332:
10265:
10153:
10129:
9427:
9138:
9118:
8446:
7603:
7579:
7447:
7399:
7285:
7259:
7211:
7187:
7161:
7137:
7113:
7089:
6857:
6811:
4042:, i.e. for significantly underdamped systems. For strongly underdamped systems the value of the amplitude can become quite large near the resonant frequency.
9262:
968:
The boundary solution between an underdamped oscillator and an overdamped oscillator occurs at a particular value of the friction coefficient and is called
9916:
By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
2493:{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x={\frac {F(t)}{m}}.}
1533:
oscillator there is in addition a frictional force which is always in a direction to oppose the motion. In many vibrating systems the frictional force
10043:
8108:
6764:
Harmonic oscillators occurring in a number of areas of engineering are equivalent in the sense that their mathematical models are identical (see
5149:
2008:< 1): The system oscillates (with a slightly different frequency than the undamped case) with the amplitude gradually decreasing to zero. The
794:
10223:
zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.
2631:{\displaystyle {\frac {\mathrm {d} ^{2}z}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} z}{\mathrm {d} t}}+\omega _{0}^{2}z=0,}
1802:{\displaystyle {\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+2\zeta \omega _{0}{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega _{0}^{2}x=0,}
4488:{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=\cos(\omega \tau ).}
11494:
8939:
is arbitrary and thus may be dropped, and a coordinate transformation allows the form of the simple harmonic oscillator to be retrieved:
8245:
6678:
383:
502:
17:
3040:{\displaystyle x(t)=1-e^{-\zeta \omega _{0}t}{\frac {\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right)}{\sin(\varphi )}},}
8322:
The problem of the simple harmonic oscillator occurs frequently in physics, because a mass at equilibrium under the influence of any
1265:
7910:
1201:
11782:
11555:
Roura, P.; Gonzalez, J.A. (2010). "Towards a more realistic description of swing pumping due to the exchange of angular momentum".
7027:
475:
6937:
11396:
8336:
7691:
3917:
2015:
10030:
7786:
7736:
4156:
based parametric oscillators operate in the same fashion. The designer varies a parameter periodically to induce oscillations.
60:
7646:
1431:
837:
3783:{\displaystyle Z_{m}={\sqrt {\left(2\omega _{0}\zeta \right)^{2}+{\frac {1}{\omega ^{2}}}(\omega _{0}^{2}-\omega ^{2})^{2}}}}
6982:
6892:
3125:, i.e. the time necessary to ensure the signal is within a fixed departure from final value, typically within 10%. The term
7865:
3058:
7995:
7955:
987:
10132:
457:
7403:
4317:{\displaystyle {\frac {\mathrm {d} ^{2}q}{\mathrm {d} \tau ^{2}}}+2\zeta {\frac {\mathrm {d} q}{\mathrm {d} \tau }}+q=0}
1504:
1400:, which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass
9847:
gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
6662:{\displaystyle q_{s}(\tau )=A(\zeta ,\omega )\cos(\omega \tau +\varphi (\zeta ,\omega ))=A\cos(\omega \tau +\varphi ).}
11739:
11716:
11697:
11677:
11649:
11615:
787:
8174:
3893:{\displaystyle \varphi =\arctan \left({\frac {2\omega \omega _{0}\zeta }{\omega ^{2}-\omega _{0}^{2}}}\right)+n\pi }
1815:
8688:{\displaystyle V(x)=V(x_{0})+V'(x_{0})\cdot (x-x_{0})+{\tfrac {1}{2}}V''(x_{0})\cdot (x-x_{0})^{2}+O(x-x_{0})^{3}.}
8042:
1864:
9850:
749:
11759:
11431:
9230:
2082:
123:
3200:
of a driven harmonic oscillator. This plot is also called the harmonic oscillator spectrum or motional spectrum.
2345:{\displaystyle F(t)-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}.}
11777:
4506:
378:
118:
11074:
10797:
1674:{\displaystyle F=-kx-c{\frac {\mathrm {d} x}{\mathrm {d} t}}=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}},}
4171:
4161:
4007:
826:
9452:
1524:
Video clip demonstrating a damped harmonic oscillator consisting of a dynamics cart between two springs. An
9494:
4149:
2752:{\displaystyle z(t)=Ae^{-\zeta \omega _{0}t}\sin \left({\sqrt {1-\zeta ^{2}}}\omega _{0}t+\varphi \right),}
780:
767:
528:
451:
373:
216:
10526:
11607:
11426:
10730:
10596:
10395:
10274:
10174:
3148:
2178:
447:
248:
35:
11340:
11269:
10462:
226:
9141:
4497:
The solution to this differential equation contains two parts: the "transient" and the "steady-state".
667:
556:
482:
342:
275:
4328:, since all second-order linear oscillatory systems can be reduced to this form. This is done through
9096:
exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.
2895:{\displaystyle {\frac {F(t)}{m}}={\begin{cases}\omega _{0}^{2}&t\geq 0\\0&t<0\end{cases}}}
1321:
421:
4558:
3133:
refers to the extent the response falls below final value for times following the response maximum.
2838:
11151:
11020:
10966:
10880:
10341:
7289:
3796:
2223:
Driven harmonic oscillators are damped oscillators further affected by an externally applied force
1315:. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its
697:
541:
11309:
9783:
9736:
9641:
9614:
6672:
4153:
3503:
3446:
647:
411:
4983:
method" by solving the auxiliary equation below and then finding the real part of its solution:
1911:
687:
11498:
9671:
7451:
3663:
2234:
1862:
1569:
1199:
1082:
1019:
908:
816:
692:
509:
11213:
10661:
9432:
8700:
7531:
7220:
6866:
3553:
1429:
1263:
835:
11411:
11245:
9810:
9146:
8451:
7427:
7408:
6818:
6304:. This amplitude function is particularly important in the analysis and understanding of the
5799:{\displaystyle -\omega ^{2}A+2\zeta i\omega A+A=e^{-i\varphi }=\cos \varphi -i\sin \varphi .}
4329:
4104:
4093:
4087:
4048:
3483:
3427:
3397:
1194:
702:
677:
363:
181:
11728:
11181:
7841:
4124:
3550:
The steady-state solution is proportional to the driving force with an induced phase change
3530:
3183:
11787:
11669:
11564:
11529:
9843:
When a spring is stretched or compressed by a mass, the spring develops a restoring force.
9763:
8736:
7615:
7456:
3980:
3792:
3456:
3370:
3107:
2163:{\displaystyle Q=2\pi \times {\frac {\text{energy stored}}{\text{energy lost per cycle}}}.}
1358:
1010:
and radio circuits. They are the source of virtually all sinusoidal vibrations and waves.
722:
682:
590:
586:
578:
568:
358:
351:
107:
10773:
9049:
8913:{\displaystyle V(x)=V(x_{0})+{\tfrac {1}{2}}V''(x_{0})\cdot (x-x_{0})^{2}+O(x-x_{0})^{3}.}
8398:
7497:
7294:
6786:
975:
If an external time-dependent force is present, the harmonic oscillator is described as a
717:
8:
10850:
7350:
7345:
7318:
3419:
808:
497:
416:
161:
156:
151:
51:
27:
Physical system that responds to a restoring force inversely proportional to displacement
11568:
11533:
6781:
11792:
11580:
11127:
11050:
10996:
10942:
10910:
10706:
10639:
10572:
10502:
10438:
10371:
10317:
10250:
10138:
10114:
9412:
9123:
9103:
8431:
8323:
7588:
7564:
7432:
7384:
7270:
7244:
7196:
7172:
7146:
7122:
7098:
7074:
6842:
6796:
6759:
6529:
6528:
This phase function is particularly important in the analysis and understanding of the
6305:
1969:
critically determines the behavior of the system. A damped harmonic oscillator can be:
627:
368:
243:
211:
171:
1184:{\displaystyle F=ma=m{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=m{\ddot {x}}=-kx.}
30:
This article is about the harmonic oscillator in classical mechanics. For its uses in
11735:
11712:
11693:
11687:
11673:
11663:
11645:
11621:
11611:
11584:
11576:
9100:
Assuming no damping, the differential equation governing a simple pendulum of length
7093:
3092:
The time an oscillator needs to adapt to changed external conditions is of the order
2076:
2009:
1981:
1855:
1304:
969:
637:
594:
551:
546:
487:
263:
253:
146:
31:
5889:{\displaystyle A(1-\omega ^{2})=\cos \varphi ,\quad 2\zeta \omega A=-\sin \varphi .}
3453:
that is independent of initial conditions and depends only on the driving amplitude
11637:
11572:
11537:
11436:
10164:
9227:
If the maximal displacement of the pendulum is small, we can use the approximation
9037:{\displaystyle V(x)\approx {\tfrac {1}{2}}V''(0)\cdot x^{2}={\tfrac {1}{2}}kx^{2}.}
8330:
7583:
7191:
6977:
6932:
6837:
4980:
1041:
732:
712:
657:
652:
598:
573:
428:
286:
231:
206:
10022:{\displaystyle F(t)=-kx(t)=m{\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}x(t)=ma,}
9449:
are constants that depend on the initial conditions. Using as initial conditions
9093:
7239:
991:
820:
727:
672:
622:
617:
536:
11755:
9604:{\displaystyle \theta (t)=\theta _{0}\cos \left({\sqrt {\frac {g}{l}}}t\right),}
6540:
Combining the amplitude and phase portions results in the steady-state solution
3141:
11797:
11659:
11401:
11391:
10168:
9844:
9402:{\displaystyle \theta (t)=A\cos \left({\sqrt {\frac {g}{l}}}t+\varphi \right),}
7265:
3439:
2502:
This equation can be solved exactly for any driving force, using the solutions
754:
662:
563:
280:
11641:
10040:, and there is no initial velocity, the solution of this equation is given by
9839:
Spring–mass system in equilibrium (A), compressed (B) and stretched (C) states
3438:) and driven spring systems having internal mechanical resistance or external
1029:
11771:
11386:
8923:
8425:
7836:
7379:
3903:
2783:
1915:
1525:
1394:
1055:
961:
954:
946:
930:
642:
469:
9217:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\sin \theta =0.}
8163:{\displaystyle I{\ddot {\theta }}+\Gamma {\dot {\theta }}+\mu \theta =\tau }
1408:, while the amplitude and phase are determined by the starting position and
9725:{\displaystyle \tau =2\pi {\sqrt {\frac {l}{g}}}={\frac {2\pi }{\omega }},}
7215:
6861:
5903:
5808:
Equating the real and imaginary parts results in two independent equations
4098:
3654:{\displaystyle x(t)={\frac {F_{0}}{mZ_{m}\omega }}\sin(\omega t+\varphi ),}
3450:
1984:) to steady state without oscillating. Larger values of the damping ratio
1416:
707:
632:
321:
201:
8326:, in the limit of small motions, behaves as a simple harmonic oscillator.
11520:
Case, W. B. (1996). "The pumping of a swing from the standing position".
11406:
10131:
is the mass on the end of the spring. If the spring itself has mass, its
9088:
7141:
6301:
4213:
3423:
1528:
on top of the cart shows the magnitude and direction of the acceleration.
1054:
A simple harmonic oscillator is an oscillator that is neither driven nor
1003:
915:
893:
1918:
of a damped harmonic oscillator; curves are plotted for three values of
1423:
The potential energy stored in a simple harmonic oscillator at position
9325:{\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\theta =0.}
7117:
912:
492:
166:
9835:
1508:
Phase portrait of damped oscillator, with increasing damping strength.
7610:
7313:
6297:
5907:
4144:
Parametric oscillators are used in many applications. The classical
3911:
3435:
3415:
1316:
1308:
995:
950:
923:
919:
514:
11541:
9669:, the time for one complete oscillation, is given by the expression
1497:
Dependence of the system behavior on the value of the damping ratio
11632:
Hayek, Sabih I. (15 Apr 2003). "Mechanical
Vibration and Damping".
11421:
8305:{\displaystyle C{\ddot {\varphi }}+G{\dot {\varphi }}+\varphi /L=i}
7167:
6887:
5458:
Substituting these quantities into the differential equation gives
4145:
3431:
3418:
for a sinusoidal driving mechanism. This type of system appears in
3129:
refers to the extent the response maximum exceeds final value, and
2122:
1409:
1389:, the number of cycles per unit time. The position at a given time
983:
934:
433:
316:
291:
8481:) to model the behavior of small perturbations from equilibrium.
7559:
7022:
1488:
406:
259:
176:
11625:
10158:
10102:{\displaystyle x(t)=A\cos \left({\sqrt {\frac {k}{m}}}t\right).}
11416:
7526:
6769:
frequency, damping factor, etc. – are the same.
4971:
The transient solution is independent of the forcing function.
2774:
determine the behavior needed to match the initial conditions.
903:
is the only force acting on the system, the system is called a
465:
311:
221:
7944:{\displaystyle {\frac {\Gamma }{2}}{\sqrt {\frac {1}{I\mu }}}}
2640:
and which can be expressed as damped sinusoidal oscillations:
8333:. The potential-energy function of a harmonic oscillator is
7492:
7058:{\displaystyle {\frac {\mathrm {d} \varphi }{\mathrm {d} t}}}
4165:
1007:
301:
296:
238:
6968:{\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} t}}}
6671:
The solution of original universal oscillator equation is a
3145:
Steady-state variation of amplitude with relative frequency
1493:
9666:
7725:{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {\mu }{I}}}}
7069:
5922:
5212:{\displaystyle q_{s}(\tau )=Ae^{i(\omega \tau +\varphi )}.}
4955:
3970:{\textstyle \omega _{r}=\omega _{0}{\sqrt {1-2\zeta ^{2}}}}
2888:
2068:{\textstyle \omega _{1}=\omega _{0}{\sqrt {1-\zeta ^{2}}},}
937:
is also present, the harmonic oscillator is described as a
306:
269:
10163:
In terms of energy, all systems have two types of energy:
7825:{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {1}{LC}}}}
7775:{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {1}{LC}}}}
5910:
of the frequency response of an ideal harmonic oscillator
960:
Decay to the equilibrium position, without oscillations (
941:. Depending on the friction coefficient, the system can:
99:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
9733:
which is a good approximation of the actual period when
9638:
is the largest angle attained by the pendulum (that is,
7680:{\displaystyle {\frac {1}{2\pi }}{\sqrt {\frac {k}{m}}}}
5700:
Dividing by the exponential term on the left results in
9760:
is small. Notice that in this approximation the period
5914:
Squaring both equations and adding them together gives
1197:, we find that the motion is described by the function
10177:
9334:
The general solution to this differential equation is
9007:
8962:
8805:
8580:
8356:
8329:
A conservative force is one that is associated with a
8317:
7013:{\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}}
6923:{\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}}
4045:
The transient solutions are the same as the unforced (
3920:
2181:
2018:
1867:
1818:
1442:
11343:
11312:
11272:
11248:
11216:
11184:
11154:
11130:
11077:
11053:
11023:
10999:
10969:
10945:
10913:
10883:
10853:
10800:
10776:
10733:
10709:
10664:
10642:
10599:
10575:
10529:
10505:
10465:
10441:
10398:
10374:
10344:
10320:
10277:
10253:
10141:
10117:
10046:
9922:
9853:
9813:
9786:
9766:
9739:
9676:
9644:
9617:
9541:
9497:
9455:
9435:
9415:
9340:
9265:
9233:
9151:
9126:
9106:
9052:
8945:
8766:
8739:
8703:
8489:
8454:
8434:
8401:
8339:
8248:
8177:
8111:
8045:
7998:
7958:
7913:
7899:{\displaystyle {\frac {c}{2}}{\sqrt {\frac {1}{km}}}}
7868:
7844:
7789:
7739:
7694:
7649:
7618:
7591:
7567:
7534:
7500:
7459:
7435:
7411:
7387:
7353:
7321:
7297:
7273:
7247:
7223:
7199:
7175:
7149:
7125:
7101:
7077:
7030:
6985:
6940:
6895:
6869:
6845:
6821:
6799:
6681:
6546:
6328:
6147:
5920:
5814:
5706:
5464:
5227:
5152:
4989:
4530:
4383:
4230:
4174:
4127:
4107:
4051:
4010:
4004:) is maximal. This resonance effect only occurs when
3983:
3805:
3668:
3576:
3556:
3533:
3506:
3486:
3459:
3400:
3373:
3210:
3186:
3151:
3061:
2908:
2808:
2646:
2516:
2360:
2243:
2131:
2085:
1687:
1578:
1540:
can be modeled as being proportional to the velocity
1434:
1361:
1355:, the time for a single oscillation or its frequency
1324:
1268:
1204:
1091:
1066:, which pulls the mass in the direction of the point
840:
63:
8024:{\displaystyle {\frac {G}{2}}{\sqrt {\frac {L}{C}}}}
7984:{\displaystyle {\frac {R}{2}}{\sqrt {\frac {C}{L}}}}
9046:Thus, given an arbitrary potential-energy function
6744:{\displaystyle q(\tau )=q_{t}(\tau )+q_{s}(\tau ).}
6675:(sum) of the transient and steady-state solutions:
3449:solution that depends on initial conditions, and a
2079:of the underdamped harmonic oscillator is given by
2012:of the underdamped harmonic oscillator is given by
11727:
11363:
11325:
11292:
11254:
11222:
11190:
11164:
11136:
11110:
11059:
11033:
11005:
10979:
10951:
10919:
10893:
10865:
10833:
10782:
10756:
10715:
10686:
10648:
10622:
10581:
10555:
10511:
10485:
10447:
10421:
10380:
10354:
10326:
10300:
10259:
10212:
10147:
10123:
10101:
10021:
9892:
9819:
9799:
9772:
9752:
9724:
9657:
9630:
9603:
9527:
9483:
9441:
9421:
9401:
9324:
9251:
9216:
9132:
9112:
9067:
9036:
8912:
8752:
8725:
8687:
8473:
8440:
8416:
8385:
8304:
8233:
8162:
8096:
8023:
7983:
7943:
7898:
7850:
7824:
7774:
7724:
7679:
7631:
7597:
7573:
7549:
7515:
7479:
7441:
7417:
7393:
7367:
7335:
7303:
7279:
7253:
7229:
7205:
7181:
7155:
7131:
7107:
7083:
7057:
7012:
6967:
6922:
6875:
6851:
6827:
6805:
6765:
6743:
6661:
6518:
6286:
6131:
5888:
5798:
5690:
5448:
5211:
5136:
4961:
4487:
4316:
4200:
4133:
4113:
4070:
4034:
3996:
3969:
3892:
3782:
3653:
3562:
3539:
3519:
3492:
3472:
3406:
3386:
3359:
3192:
3172:
3082:
3039:
2894:
2751:
2630:
2492:
2344:
2208:
2162:
2110:
2067:
1898:
1846:
1801:
1673:
1471:
1381:
1347:
1292:
1252:
1183:
879:
98:
11769:
8733:is a minimum, the first derivative evaluated at
5221:Its derivatives from zeroth to second order are
4219:
1002:include electrical harmonic oscillators such as
6296:Compare this result with the theory section on
4168:wave into two output waves of lower frequency (
1847:{\textstyle \omega _{0}={\sqrt {\frac {k}{m}}}}
1293:{\displaystyle \omega ={\sqrt {\frac {k}{m}}}.}
11602:Fowles, Grant R.; Cassiday, George L. (1986),
11601:
11456:
10591:Acceleration of gravity at the Earth's surface
8234:{\displaystyle L{\ddot {q}}+R{\dot {q}}+q/C=v}
3910:For a particular driving frequency called the
2218:
1899:{\textstyle \zeta ={\frac {c}{2{\sqrt {mk}}}}}
1253:{\displaystyle x(t)=A\sin(\omega t+\varphi ),}
11554:
10159:Energy variation in the spring–damping system
8395:Given an arbitrary potential-energy function
8097:{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=F}
1482:
1013:
945:Oscillate with a frequency lower than in the
918:about the equilibrium point, with a constant
788:
11709:Physics for Scientists and Engineers: Vol. 1
11686:Serway, Raymond A.; Jewett, John W. (2003).
11685:
8760:must be zero, so the linear term drops out:
9252:{\displaystyle \sin \theta \approx \theta }
8386:{\displaystyle V(x)={\tfrac {1}{2}}kx^{2}.}
3204:In the case of a sinusoidal driving force:
3136:
2111:{\displaystyle \lambda =\omega _{0}\zeta .}
6423:
6419:
4212:regular resonance because it exhibits the
1572:) for damped harmonic oscillators is then
926:(which does not depend on the amplitude).
815:is a system that, when displaced from its
795:
781:
6300:, as well as the "magnitude part" of the
4081:
3117:In electrical engineering, a multiple of
11658:
11468:
11111:{\displaystyle {\mathsf {ML^{2}T^{-2}}}}
10834:{\displaystyle {\mathsf {ML^{2}T^{-2}}}}
9834:
9665:is the amplitude of the pendulum). The
9087:
8034:
7835:
7608:
5902:
4974:
3140:
1910:
1511:
1503:
1492:
1472:{\displaystyle U={\tfrac {1}{2}}kx^{2}.}
880:{\displaystyle {\vec {F}}=-k{\vec {x}},}
11636:. WILEY-VCH Verlag GmbH & Co KGaA.
4201:{\displaystyle \omega _{s},\omega _{i}}
4035:{\displaystyle \zeta <1/{\sqrt {2}}}
3106:. In physics, the adaptation is called
999:
14:
11770:
11706:
11480:
11354:
11351:
11347:
11283:
11280:
11276:
11157:
11101:
11098:
11094:
11088:
11084:
11080:
11026:
10972:
10886:
10824:
10821:
10817:
10811:
10807:
10803:
10747:
10744:
10740:
10736:
10613:
10610:
10606:
10602:
10546:
10543:
10539:
10535:
10532:
10476:
10473:
10469:
10412:
10409:
10405:
10401:
10347:
10291:
10288:
10284:
10280:
10226:
9484:{\displaystyle \theta (0)=\theta _{0}}
5146:Supposing the solution is of the form
2354:It is usually rewritten into the form
2125:of a damped oscillator is defined as
11725:
11631:
11513:
11495:"Two ways of driving a child's swing"
11069:Potential energy stored in oscillator
9830:
9528:{\displaystyle {\dot {\theta }}(0)=0}
6753:
4500:
3083:{\displaystyle \cos \varphi =\zeta .}
2510:) that satisfy the unforced equation
1947:. Time is in units of the decay time
1681:which can be rewritten into the form
477:Newton's law of universal gravitation
11689:Physics for Scientists and Engineers
11548:
11519:
10556:{\displaystyle {\mathsf {MLT^{-2}}}}
10213:{\textstyle U={\frac {1}{2}}kx^{2}.}
2209:{\textstyle Q={\frac {1}{2\zeta }}.}
11486:
11397:Effective mass (spring–mass system)
10757:{\displaystyle {\mathsf {MT^{-2}}}}
10623:{\displaystyle {\mathsf {LT^{-2}}}}
10422:{\displaystyle {\mathsf {MT^{-1}}}}
10301:{\displaystyle {\mathsf {LT^{-2}}}}
9670:
8318:Application to a conservative force
3662:
3445:The general solution is a sum of a
3173:{\displaystyle \omega /\omega _{0}}
2175:is related to the damping ratio by
1861:
1198:
1062:, which experiences a single force
458:Mechanics of planar particle motion
66:
24:
11364:{\displaystyle {\mathsf {T^{-1}}}}
11335:Natural resonant angular frequency
11293:{\displaystyle {\mathsf {T^{-1}}}}
10486:{\displaystyle {\mathsf {T^{-1}}}}
9978:
9966:
9827:represents the angular frequency.
9259:and instead consider the equation
9083:
8130:
7916:
7412:
7045:
7035:
7000:
6990:
6955:
6945:
6910:
6900:
5379:
5356:
5297:
5280:
5051:
5041:
5011:
4995:
4505:The solution based on solving the
4445:
4435:
4405:
4389:
4292:
4282:
4252:
4236:
3282:
3272:
3232:
3216:
2588:
2578:
2538:
2522:
2432:
2422:
2382:
2366:
2322:
2306:
2285:
2275:
1988:return to equilibrium more slowly.
1759:
1749:
1709:
1693:
1651:
1635:
1614:
1604:
1428:
1262:
1131:
1115:
834:
25:
11809:
11749:
9145:
5898:
1073:and depends only on the position
11730:Advanced Engineering Mathematics
11665:Advanced Engineering Mathematics
11492:
10111:Given an ideal massless spring,
9780:is independent of the amplitude
6535:
2795:and a unit step input with
1311:fashion with constant amplitude
1040:
1028:
762:
761:
748:
81:
11783:Ordinary differential equations
11760:The Feynman Lectures on Physics
11711:(4th ed.). W. H. Freeman.
11634:Encyclopedia of Applied Physics
10036:If the initial displacement is
6322:, divide both equations to get
5855:
5350:
5275:
3114:is called the relaxation time.
1965:The value of the damping ratio
1348:{\displaystyle T=2\pi /\omega }
1035:Mass-spring harmonic oscillator
11474:
11462:
11450:
10056:
10050:
10004:
9998:
9953:
9947:
9932:
9926:
9884:
9878:
9863:
9857:
9551:
9545:
9516:
9510:
9465:
9459:
9350:
9344:
9062:
9056:
8987:
8981:
8955:
8949:
8898:
8878:
8863:
8843:
8837:
8824:
8798:
8785:
8776:
8770:
8720:
8707:
8673:
8653:
8638:
8618:
8612:
8599:
8573:
8554:
8548:
8535:
8521:
8508:
8499:
8493:
8411:
8405:
8349:
8343:
7544:
7538:
7510:
7504:
6735:
6729:
6713:
6707:
6691:
6685:
6653:
6638:
6623:
6620:
6608:
6593:
6584:
6572:
6563:
6557:
6451:
6439:
6420:
6269:
6256:
6244:
6224:
6169:
6157:
6120:
6111:
6098:
6086:
6066:
6063:
6050:
6010:
5994:
5958:
5938:
5837:
5818:
5661:
5646:
5635:
5592:
5584:
5569:
5550:
5535:
5504:
5489:
5438:
5423:
5342:
5327:
5267:
5252:
5201:
5186:
5169:
5163:
5109:
5100:
5085:
5076:
4786:
4757:
4738:
4709:
4547:
4541:
4507:ordinary differential equation
4479:
4470:
3769:
3737:
3645:
3630:
3586:
3580:
3394:is the driving amplitude, and
3351:
3342:
3028:
3022:
2918:
2912:
2821:
2815:
2656:
2650:
2478:
2472:
2253:
2247:
1907:is called the "damping ratio".
1244:
1229:
1214:
1208:
868:
847:
13:
1:
11734:(4th ed.). McGraw-Hill.
11595:
11165:{\displaystyle {\mathsf {L}}}
11034:{\displaystyle {\mathsf {T}}}
10980:{\displaystyle {\mathsf {T}}}
10894:{\displaystyle {\mathsf {M}}}
10355:{\displaystyle {\mathsf {L}}}
10336:Peak amplitude of oscillation
10031:Newton's second law of motion
6766:universal oscillator equation
6311:
4326:universal oscillator equation
4220:Universal oscillator equation
4162:Optical parametric oscillator
3977:, the amplitude (for a given
3791:is the absolute value of the
3500:, undamped angular frequency
2777:
1980:> 1): The system returns (
384:Koopman–von Neumann mechanics
11606:(5th ed.), Fort Worth:
9908:is the spring constant, and
9893:{\displaystyle F(t)=-kx(t),}
988:small angles of displacement
982:Mechanical examples include
452:Non-inertial reference frame
7:
11608:Saunders College Publishing
11557:European Journal of Physics
11522:American Journal of Physics
11457:Fowles & Cassiday (1986
11427:Quantum harmonic oscillator
11379:
11326:{\displaystyle \omega _{0}}
10390:Viscous damping coefficient
9800:{\displaystyle \theta _{0}}
9753:{\displaystyle \theta _{0}}
9658:{\displaystyle \theta _{0}}
9631:{\displaystyle \theta _{0}}
9535:, the solution is given by
9078:
4509:is for arbitrary constants
4335:If the forcing function is
3520:{\displaystyle \omega _{0}}
2219:Driven harmonic oscillators
1563:viscous damping coefficient
1077:of the mass and a constant
379:Appell's equation of motion
249:Inertial frame of reference
36:quantum harmonic oscillator
10:
11814:
11668:(3rd ed.), New York:
11577:10.1088/0143-0807/31/5/020
11432:Radial harmonic oscillator
8448:around an energy minimum (
6757:
4085:
2781:
1486:
1483:Damped harmonic oscillator
1017:
1014:Simple harmonic oscillator
905:simple harmonic oscillator
29:
18:Damped harmonic oscillator
11642:10.1002/3527600434.eap231
10242:
10239:
10236:
10233:
9807:. In the above equation,
6775:Translational mechanical
6532:of second-order systems.
6308:of second-order systems.
11443:
11223:{\displaystyle \varphi }
10687:{\displaystyle i^{2}=-1}
9913:flying off to infinity.
9442:{\displaystyle \varphi }
8726:{\displaystyle V(x_{0})}
7550:{\displaystyle \tau (t)}
7230:{\displaystyle \varphi }
6876:{\displaystyle \varphi }
4802: (critical damping)
3914:, or resonant frequency
3797:linear response function
3563:{\displaystyle \varphi }
3527:, and the damping ratio
3137:Sinusoidal driving force
1854:is called the "undamped
1307:, repeating itself in a
1058:. It consists of a mass
819:position, experiences a
542:Rotating reference frame
374:Hamilton–Jacobi equation
11756:The Harmonic Oscillator
11255:{\displaystyle \omega }
10792:Torsion Spring constant
9820:{\displaystyle \omega }
9142:acceleration of gravity
8474:{\displaystyle x=x_{0}}
8035:Differential equation:
7418:{\displaystyle \Gamma }
6828:{\displaystyle \theta }
4377:, the equation becomes
4114:{\displaystyle \omega }
4071:{\displaystyle F_{0}=0}
3493:{\displaystyle \omega }
3407:{\displaystyle \omega }
1568:The balance of forces (
1404:and the force constant
990:), masses connected to
929:If a frictional force (
483:Newton's laws of motion
343:Newton's laws of motion
11365:
11327:
11294:
11256:
11224:
11192:
11191:{\displaystyle \zeta }
11166:
11138:
11112:
11061:
11035:
11007:
10981:
10953:
10921:
10895:
10867:
10835:
10784:
10758:
10717:
10688:
10650:
10624:
10583:
10557:
10513:
10487:
10449:
10423:
10382:
10356:
10328:
10302:
10261:
10214:
10149:
10125:
10103:
10023:
9894:
9840:
9821:
9801:
9774:
9754:
9726:
9659:
9632:
9605:
9529:
9485:
9443:
9423:
9403:
9326:
9253:
9218:
9134:
9114:
9097:
9069:
9038:
8914:
8754:
8727:
8689:
8475:
8442:
8418:
8387:
8306:
8235:
8164:
8098:
8025:
7985:
7945:
7900:
7852:
7851:{\displaystyle \zeta }
7826:
7776:
7726:
7681:
7633:
7599:
7575:
7551:
7517:
7481:
7443:
7419:
7395:
7369:
7337:
7305:
7281:
7255:
7231:
7207:
7183:
7157:
7133:
7109:
7085:
7059:
7014:
6969:
6924:
6877:
6853:
6829:
6807:
6778:Rotational mechanical
6745:
6663:
6520:
6288:
6133:
5911:
5890:
5800:
5692:
5450:
5213:
5138:
4963:
4489:
4318:
4202:
4135:
4134:{\displaystyle \beta }
4115:
4082:Parametric oscillators
4072:
4036:
3998:
3971:
3894:
3784:
3655:
3564:
3541:
3540:{\displaystyle \zeta }
3521:
3494:
3474:
3408:
3388:
3361:
3201:
3194:
3193:{\displaystyle \zeta }
3174:
3084:
3041:
2896:
2753:
2632:
2494:
2346:
2210:
2164:
2112:
2069:
1962:
1900:
1848:
1803:
1675:
1529:
1509:
1501:
1473:
1383:
1349:
1294:
1254:
1185:
1047:Simple harmonic motion
1020:Simple harmonic motion
953:decreasing with time (
933:) proportional to the
909:simple harmonic motion
881:
510:Simple harmonic motion
423:Euler's laws of motion
217:D'Alembert's principle
100:
11778:Mechanical vibrations
11726:Wylie, C. R. (1975).
11707:Tipler, Paul (1998).
11412:Parametric oscillator
11366:
11328:
11295:
11257:
11225:
11193:
11167:
11139:
11113:
11062:
11036:
11008:
10982:
10961:Period of oscillation
10954:
10922:
10896:
10868:
10836:
10785:
10759:
10718:
10689:
10651:
10625:
10584:
10558:
10514:
10488:
10450:
10424:
10383:
10357:
10329:
10303:
10262:
10215:
10150:
10126:
10104:
10024:
9895:
9838:
9822:
9802:
9775:
9773:{\displaystyle \tau }
9755:
9727:
9660:
9633:
9606:
9530:
9486:
9444:
9424:
9404:
9327:
9254:
9219:
9135:
9115:
9091:
9070:
9039:
8915:
8755:
8753:{\displaystyle x_{0}}
8728:
8690:
8476:
8443:
8419:
8388:
8307:
8236:
8165:
8099:
8026:
7986:
7946:
7901:
7853:
7827:
7777:
7727:
7682:
7634:
7632:{\displaystyle f_{n}}
7600:
7576:
7552:
7518:
7482:
7480:{\displaystyle G=1/R}
7444:
7420:
7396:
7370:
7338:
7306:
7282:
7256:
7232:
7208:
7184:
7158:
7134:
7110:
7086:
7060:
7015:
6970:
6925:
6878:
6854:
6830:
6808:
6746:
6664:
6521:
6289:
6134:
5906:
5891:
5801:
5693:
5451:
5214:
5139:
4975:Steady-state solution
4964:
4490:
4330:nondimensionalization
4319:
4203:
4136:
4116:
4094:parametric oscillator
4088:Parametric oscillator
4073:
4037:
3999:
3997:{\displaystyle F_{0}}
3972:
3895:
3785:
3656:
3565:
3542:
3522:
3495:
3475:
3473:{\displaystyle F_{0}}
3409:
3389:
3387:{\displaystyle F_{0}}
3362:
3195:
3175:
3144:
3085:
3042:
2897:
2754:
2633:
2495:
2347:
2211:
2165:
2153:energy lost per cycle
2113:
2070:
1914:
1901:
1849:
1804:
1676:
1523:
1507:
1496:
1474:
1384:
1382:{\displaystyle f=1/T}
1350:
1295:
1255:
1195:differential equation
1186:
1081:. Balance of forces (
882:
364:Hamiltonian mechanics
182:Statistical mechanics
101:
11483:, pp. 369, 389)
11341:
11310:
11270:
11246:
11214:
11182:
11152:
11128:
11075:
11051:
11021:
10997:
10967:
10943:
10911:
10881:
10851:
10798:
10783:{\displaystyle \mu }
10774:
10731:
10707:
10662:
10640:
10597:
10573:
10527:
10503:
10463:
10439:
10396:
10372:
10342:
10318:
10275:
10269:Acceleration of mass
10251:
10175:
10139:
10135:must be included in
10115:
10044:
9920:
9851:
9811:
9784:
9764:
9737:
9674:
9642:
9615:
9539:
9495:
9453:
9433:
9413:
9338:
9263:
9231:
9149:
9124:
9104:
9068:{\displaystyle V(x)}
9050:
8943:
8764:
8737:
8701:
8487:
8452:
8432:
8417:{\displaystyle V(x)}
8399:
8337:
8246:
8175:
8109:
8043:
7996:
7956:
7911:
7866:
7842:
7787:
7737:
7692:
7647:
7616:
7589:
7565:
7532:
7516:{\displaystyle F(t)}
7498:
7457:
7433:
7409:
7385:
7351:
7319:
7304:{\displaystyle \mu }
7295:
7271:
7245:
7221:
7197:
7173:
7147:
7123:
7099:
7075:
7028:
6983:
6938:
6893:
6867:
6843:
6819:
6797:
6787:Parallel RLC circuit
6679:
6544:
6326:
6145:
5918:
5812:
5704:
5462:
5225:
5150:
4987:
4949: (underdamping)
4528:
4381:
4228:
4172:
4125:
4105:
4049:
4008:
3981:
3918:
3803:
3666:
3574:
3554:
3531:
3504:
3484:
3480:, driving frequency
3457:
3398:
3371:
3208:
3184:
3149:
3059:
2906:
2806:
2644:
2514:
2358:
2241:
2179:
2129:
2083:
2016:
1982:exponentially decays
1865:
1816:
1685:
1576:
1432:
1393:also depends on the
1359:
1322:
1266:
1202:
1089:
1085:) for the system is
838:
829:to the displacement
587:Angular acceleration
579:Rotational frequency
359:Lagrangian mechanics
352:Analytical mechanics
108:Second law of motion
61:
11569:2010EJPh...31.1195R
11534:1996AmJPh..64..215C
10866:{\displaystyle m,M}
10227:Definition of terms
7404:Rotational friction
7368:{\displaystyle 1/L}
7346:Magnetic reluctance
7336:{\displaystyle 1/C}
4686: (overdamping)
3873:
3754:
3309:
2855:
2615:
2459:
2235:Newton's second law
1858:of the oscillator",
1786:
1570:Newton's second law
1083:Newton's second law
907:, and it undergoes
813:harmonic oscillator
809:classical mechanics
439:Harmonic oscillator
417:Equations of motion
52:Classical mechanics
46:Part of a series on
11604:Analytic Mechanics
11501:on 9 December 2011
11361:
11323:
11290:
11252:
11220:
11188:
11162:
11134:
11108:
11057:
11031:
11003:
10977:
10949:
10917:
10891:
10863:
10831:
10780:
10754:
10713:
10684:
10646:
10620:
10579:
10553:
10509:
10483:
10445:
10419:
10378:
10352:
10324:
10298:
10257:
10210:
10145:
10121:
10099:
10019:
9890:
9841:
9831:Spring/mass system
9817:
9797:
9770:
9750:
9722:
9655:
9628:
9601:
9525:
9481:
9439:
9419:
9399:
9322:
9249:
9214:
9130:
9110:
9098:
9065:
9034:
9016:
8971:
8910:
8814:
8750:
8723:
8685:
8589:
8471:
8438:
8414:
8383:
8365:
8324:conservative force
8302:
8231:
8160:
8094:
8021:
7981:
7941:
7896:
7848:
7822:
7772:
7722:
7677:
7629:
7611:resonant frequency
7595:
7571:
7547:
7513:
7477:
7439:
7415:
7391:
7365:
7333:
7301:
7277:
7251:
7227:
7203:
7179:
7153:
7129:
7105:
7081:
7055:
7010:
6965:
6920:
6873:
6849:
6825:
6803:
6782:Series RLC circuit
6760:System equivalence
6754:Equivalent systems
6741:
6659:
6530:frequency response
6516:
6306:frequency response
6284:
6129:
6044:
5912:
5886:
5796:
5688:
5446:
5209:
5134:
4959:
4954:
4501:Transient solution
4485:
4314:
4198:
4164:converts an input
4131:
4111:
4068:
4032:
3994:
3967:
3890:
3859:
3780:
3740:
3651:
3560:
3537:
3517:
3490:
3470:
3404:
3384:
3357:
3295:
3202:
3190:
3170:
3080:
3037:
2892:
2887:
2841:
2759:in the case where
2749:
2628:
2601:
2490:
2445:
2342:
2206:
2160:
2108:
2065:
1963:
1896:
1844:
1799:
1772:
1671:
1530:
1510:
1502:
1469:
1451:
1379:
1345:
1290:
1250:
1181:
996:acoustical systems
877:
755:Physics portal
369:Routhian mechanics
244:Frame of reference
96:
11377:
11376:
11264:Angular frequency
11137:{\displaystyle x}
11060:{\displaystyle U}
11006:{\displaystyle t}
10952:{\displaystyle T}
10920:{\displaystyle Q}
10716:{\displaystyle k}
10649:{\displaystyle i}
10582:{\displaystyle g}
10512:{\displaystyle F}
10448:{\displaystyle f}
10381:{\displaystyle c}
10327:{\displaystyle A}
10260:{\displaystyle a}
10192:
10148:{\displaystyle m}
10124:{\displaystyle m}
10086:
10085:
10029:the latter being
9993:
9717:
9699:
9698:
9588:
9587:
9507:
9422:{\displaystyle A}
9380:
9379:
9311:
9298:
9197:
9184:
9133:{\displaystyle g}
9113:{\displaystyle l}
9015:
8970:
8813:
8588:
8441:{\displaystyle x}
8364:
8315:
8314:
8279:
8261:
8208:
8190:
8142:
8124:
8076:
8058:
8019:
8018:
8007:
7979:
7978:
7967:
7939:
7938:
7922:
7894:
7893:
7877:
7820:
7819:
7803:
7770:
7769:
7753:
7720:
7719:
7708:
7675:
7674:
7663:
7598:{\displaystyle i}
7574:{\displaystyle v}
7442:{\displaystyle R}
7394:{\displaystyle c}
7280:{\displaystyle k}
7254:{\displaystyle q}
7206:{\displaystyle L}
7182:{\displaystyle p}
7156:{\displaystyle C}
7132:{\displaystyle L}
7108:{\displaystyle I}
7094:Moment of inertia
7084:{\displaystyle m}
7053:
7008:
6963:
6918:
6852:{\displaystyle q}
6806:{\displaystyle x}
6498:
6429:
6426:
6418:
6415:
6411:
6375:
6279:
6278:
6212:
5394:
5305:
5059:
5026:
4981:complex variables
4950:
4921:
4869:
4803:
4687:
4664:
4618:
4453:
4420:
4300:
4267:
4030:
3965:
3875:
3778:
3735:
3622:
3324:
3290:
3247:
3032:
2988:
2828:
2720:
2596:
2553:
2485:
2440:
2397:
2337:
2293:
2201:
2155:
2154:
2151:
2077:exponential decay
2060:
2010:angular frequency
1992:Critically damped
1894:
1891:
1856:angular frequency
1842:
1841:
1767:
1724:
1666:
1622:
1521:
1450:
1415:The velocity and
1285:
1284:
1163:
1146:
1000:analogous systems
977:driven oscillator
970:critically damped
939:damped oscillator
871:
850:
805:
804:
552:Centrifugal force
547:Centripetal force
503:Euler's equations
488:Relative velocity
264:Moment of inertia
94:
68:
32:quantum mechanics
16:(Redirected from
11805:
11745:
11733:
11722:
11703:
11682:
11655:
11628:
11589:
11588:
11563:(5): 1195–1207.
11552:
11546:
11545:
11517:
11511:
11510:
11508:
11506:
11497:. Archived from
11490:
11484:
11478:
11472:
11466:
11460:
11454:
11437:Elastic pendulum
11370:
11368:
11367:
11362:
11360:
11359:
11358:
11357:
11332:
11330:
11329:
11324:
11322:
11321:
11299:
11297:
11296:
11291:
11289:
11288:
11287:
11286:
11261:
11259:
11258:
11253:
11229:
11227:
11226:
11221:
11197:
11195:
11194:
11189:
11171:
11169:
11168:
11163:
11161:
11160:
11146:Position of mass
11143:
11141:
11140:
11135:
11117:
11115:
11114:
11109:
11107:
11106:
11105:
11104:
11092:
11091:
11066:
11064:
11063:
11058:
11040:
11038:
11037:
11032:
11030:
11029:
11012:
11010:
11009:
11004:
10986:
10984:
10983:
10978:
10976:
10975:
10958:
10956:
10955:
10950:
10926:
10924:
10923:
10918:
10900:
10898:
10897:
10892:
10890:
10889:
10872:
10870:
10869:
10864:
10840:
10838:
10837:
10832:
10830:
10829:
10828:
10827:
10815:
10814:
10789:
10787:
10786:
10781:
10763:
10761:
10760:
10755:
10753:
10752:
10751:
10750:
10722:
10720:
10719:
10714:
10693:
10691:
10690:
10685:
10674:
10673:
10658:Imaginary unit,
10655:
10653:
10652:
10647:
10629:
10627:
10626:
10621:
10619:
10618:
10617:
10616:
10588:
10586:
10585:
10580:
10562:
10560:
10559:
10554:
10552:
10551:
10550:
10549:
10518:
10516:
10515:
10510:
10492:
10490:
10489:
10484:
10482:
10481:
10480:
10479:
10454:
10452:
10451:
10446:
10428:
10426:
10425:
10420:
10418:
10417:
10416:
10415:
10387:
10385:
10384:
10379:
10361:
10359:
10358:
10353:
10351:
10350:
10333:
10331:
10330:
10325:
10307:
10305:
10304:
10299:
10297:
10296:
10295:
10294:
10266:
10264:
10263:
10258:
10231:
10230:
10219:
10217:
10216:
10211:
10206:
10205:
10193:
10185:
10165:potential energy
10154:
10152:
10151:
10146:
10130:
10128:
10127:
10122:
10108:
10106:
10105:
10100:
10095:
10091:
10087:
10078:
10077:
10028:
10026:
10025:
10020:
9994:
9992:
9991:
9990:
9981:
9975:
9974:
9969:
9963:
9899:
9897:
9896:
9891:
9826:
9824:
9823:
9818:
9806:
9804:
9803:
9798:
9796:
9795:
9779:
9777:
9776:
9771:
9759:
9757:
9756:
9751:
9749:
9748:
9732:
9729:
9728:
9723:
9718:
9713:
9705:
9700:
9691:
9690:
9664:
9662:
9661:
9656:
9654:
9653:
9637:
9635:
9634:
9629:
9627:
9626:
9610:
9608:
9607:
9602:
9597:
9593:
9589:
9580:
9579:
9566:
9565:
9534:
9532:
9531:
9526:
9509:
9508:
9500:
9490:
9488:
9487:
9482:
9480:
9479:
9448:
9446:
9445:
9440:
9428:
9426:
9425:
9420:
9408:
9406:
9405:
9400:
9395:
9391:
9381:
9372:
9371:
9331:
9329:
9328:
9323:
9312:
9304:
9299:
9297:
9296:
9295:
9282:
9278:
9277:
9267:
9258:
9256:
9255:
9250:
9224:
9221:
9220:
9215:
9198:
9190:
9185:
9183:
9182:
9181:
9168:
9164:
9163:
9153:
9139:
9137:
9136:
9131:
9119:
9117:
9116:
9111:
9074:
9072:
9071:
9066:
9043:
9041:
9040:
9035:
9030:
9029:
9017:
9008:
9002:
9001:
8980:
8972:
8963:
8938:
8919:
8917:
8916:
8911:
8906:
8905:
8896:
8895:
8871:
8870:
8861:
8860:
8836:
8835:
8823:
8815:
8806:
8797:
8796:
8759:
8757:
8756:
8751:
8749:
8748:
8732:
8730:
8729:
8724:
8719:
8718:
8694:
8692:
8691:
8686:
8681:
8680:
8671:
8670:
8646:
8645:
8636:
8635:
8611:
8610:
8598:
8590:
8581:
8572:
8571:
8547:
8546:
8534:
8520:
8519:
8480:
8478:
8477:
8472:
8470:
8469:
8447:
8445:
8444:
8439:
8426:Taylor expansion
8423:
8421:
8420:
8415:
8392:
8390:
8389:
8384:
8379:
8378:
8366:
8357:
8331:potential energy
8311:
8309:
8308:
8303:
8292:
8281:
8280:
8272:
8263:
8262:
8254:
8240:
8238:
8237:
8232:
8221:
8210:
8209:
8201:
8192:
8191:
8183:
8169:
8167:
8166:
8161:
8144:
8143:
8135:
8126:
8125:
8117:
8103:
8101:
8100:
8095:
8078:
8077:
8069:
8060:
8059:
8051:
8030:
8028:
8027:
8022:
8020:
8011:
8010:
8008:
8000:
7990:
7988:
7987:
7982:
7980:
7971:
7970:
7968:
7960:
7950:
7948:
7947:
7942:
7940:
7937:
7926:
7925:
7923:
7915:
7905:
7903:
7902:
7897:
7895:
7892:
7881:
7880:
7878:
7870:
7857:
7855:
7854:
7849:
7831:
7829:
7828:
7823:
7821:
7818:
7807:
7806:
7804:
7802:
7791:
7781:
7779:
7778:
7773:
7771:
7768:
7757:
7756:
7754:
7752:
7741:
7731:
7729:
7728:
7723:
7721:
7712:
7711:
7709:
7707:
7696:
7686:
7684:
7683:
7678:
7676:
7667:
7666:
7664:
7662:
7651:
7638:
7636:
7635:
7630:
7628:
7627:
7604:
7602:
7601:
7596:
7580:
7578:
7577:
7572:
7556:
7554:
7553:
7548:
7522:
7520:
7519:
7514:
7486:
7484:
7483:
7478:
7473:
7448:
7446:
7445:
7440:
7424:
7422:
7421:
7416:
7400:
7398:
7397:
7392:
7374:
7372:
7371:
7366:
7361:
7342:
7340:
7339:
7334:
7329:
7310:
7308:
7307:
7302:
7290:Torsion constant
7286:
7284:
7283:
7278:
7260:
7258:
7257:
7252:
7236:
7234:
7233:
7228:
7212:
7210:
7209:
7204:
7192:Angular momentum
7188:
7186:
7185:
7180:
7162:
7160:
7159:
7154:
7138:
7136:
7135:
7130:
7114:
7112:
7111:
7106:
7090:
7088:
7087:
7082:
7064:
7062:
7061:
7056:
7054:
7052:
7048:
7042:
7038:
7032:
7019:
7017:
7016:
7011:
7009:
7007:
7003:
6997:
6993:
6987:
6974:
6972:
6971:
6966:
6964:
6962:
6958:
6952:
6948:
6942:
6933:Angular velocity
6929:
6927:
6926:
6921:
6919:
6917:
6913:
6907:
6903:
6897:
6882:
6880:
6879:
6874:
6858:
6856:
6855:
6850:
6834:
6832:
6831:
6826:
6812:
6810:
6809:
6804:
6772:
6771:
6750:
6748:
6747:
6742:
6728:
6727:
6706:
6705:
6668:
6666:
6665:
6660:
6556:
6555:
6525:
6523:
6522:
6517:
6503:
6499:
6497:
6490:
6489:
6479:
6468:
6427:
6424:
6416:
6413:
6412:
6410:
6403:
6402:
6392:
6381:
6376:
6374:
6373:
6372:
6356:
6345:
6321:
6293:
6291:
6290:
6285:
6280:
6277:
6276:
6252:
6251:
6242:
6241:
6223:
6219:
6217:
6213:
6211:
6200:
6186:
6138:
6136:
6135:
6130:
6119:
6118:
6094:
6093:
6084:
6083:
6062:
6061:
6049:
6045:
6035:
6034:
6018:
6017:
5983:
5982:
5966:
5965:
5956:
5955:
5937:
5936:
5895:
5893:
5892:
5887:
5836:
5835:
5805:
5803:
5802:
5797:
5765:
5764:
5719:
5718:
5697:
5695:
5694:
5689:
5684:
5683:
5665:
5664:
5607:
5606:
5588:
5587:
5554:
5553:
5508:
5507:
5477:
5476:
5455:
5453:
5452:
5447:
5442:
5441:
5411:
5410:
5395:
5393:
5392:
5391:
5382:
5376:
5375:
5374:
5365:
5364:
5359:
5352:
5346:
5345:
5306:
5304:
5300:
5294:
5293:
5292:
5283:
5277:
5271:
5270:
5237:
5236:
5218:
5216:
5215:
5210:
5205:
5204:
5162:
5161:
5143:
5141:
5140:
5135:
5130:
5129:
5060:
5058:
5054:
5048:
5044:
5038:
5027:
5025:
5024:
5023:
5014:
5008:
5004:
5003:
4998:
4991:
4968:
4966:
4965:
4960:
4958:
4957:
4951:
4948:
4935:
4931:
4930:
4926:
4922:
4920:
4919:
4904:
4891:
4890:
4878:
4874:
4870:
4868:
4867:
4852:
4839:
4838:
4824:
4823:
4804:
4801:
4782:
4781:
4769:
4768:
4756:
4755:
4734:
4733:
4721:
4720:
4708:
4707:
4688:
4685:
4672:
4668:
4667:
4666:
4665:
4657:
4656:
4647:
4634:
4633:
4621:
4620:
4619:
4611:
4610:
4601:
4591:
4590:
4576:
4575:
4540:
4539:
4494:
4492:
4491:
4486:
4454:
4452:
4448:
4442:
4438:
4432:
4421:
4419:
4418:
4417:
4408:
4402:
4398:
4397:
4392:
4385:
4376:
4361:
4324:is known as the
4323:
4321:
4320:
4315:
4301:
4299:
4295:
4289:
4285:
4279:
4268:
4266:
4265:
4264:
4255:
4249:
4245:
4244:
4239:
4232:
4207:
4205:
4204:
4199:
4197:
4196:
4184:
4183:
4140:
4138:
4137:
4132:
4120:
4118:
4117:
4112:
4077:
4075:
4074:
4069:
4061:
4060:
4041:
4039:
4038:
4033:
4031:
4026:
4024:
4003:
4001:
4000:
3995:
3993:
3992:
3976:
3974:
3973:
3968:
3966:
3964:
3963:
3945:
3943:
3942:
3930:
3929:
3899:
3897:
3896:
3891:
3880:
3876:
3874:
3872:
3867:
3855:
3854:
3844:
3840:
3839:
3823:
3790:
3787:
3786:
3781:
3779:
3777:
3776:
3767:
3766:
3753:
3748:
3736:
3734:
3733:
3721:
3716:
3715:
3710:
3706:
3702:
3701:
3683:
3678:
3677:
3660:
3658:
3657:
3652:
3623:
3621:
3617:
3616:
3603:
3602:
3593:
3569:
3567:
3566:
3561:
3546:
3544:
3543:
3538:
3526:
3524:
3523:
3518:
3516:
3515:
3499:
3497:
3496:
3491:
3479:
3477:
3476:
3471:
3469:
3468:
3413:
3411:
3410:
3405:
3393:
3391:
3390:
3385:
3383:
3382:
3366:
3364:
3363:
3358:
3335:
3334:
3325:
3317:
3308:
3303:
3291:
3289:
3285:
3279:
3275:
3269:
3267:
3266:
3248:
3246:
3245:
3244:
3235:
3229:
3225:
3224:
3219:
3212:
3199:
3197:
3196:
3191:
3179:
3177:
3176:
3171:
3169:
3168:
3159:
3105:
3089:
3087:
3086:
3081:
3046:
3044:
3043:
3038:
3033:
3031:
3014:
3013:
3009:
2999:
2998:
2989:
2987:
2986:
2971:
2957:
2955:
2954:
2950:
2949:
2902:the solution is
2901:
2899:
2898:
2893:
2891:
2890:
2854:
2849:
2829:
2824:
2810:
2801:
2794:
2766:. The amplitude
2765:
2758:
2756:
2755:
2750:
2745:
2741:
2731:
2730:
2721:
2719:
2718:
2703:
2690:
2689:
2685:
2684:
2637:
2635:
2634:
2629:
2614:
2609:
2597:
2595:
2591:
2585:
2581:
2575:
2573:
2572:
2554:
2552:
2551:
2550:
2541:
2535:
2531:
2530:
2525:
2518:
2499:
2497:
2496:
2491:
2486:
2481:
2467:
2458:
2453:
2441:
2439:
2435:
2429:
2425:
2419:
2417:
2416:
2398:
2396:
2395:
2394:
2385:
2379:
2375:
2374:
2369:
2362:
2351:
2349:
2348:
2343:
2338:
2336:
2335:
2334:
2325:
2319:
2315:
2314:
2309:
2302:
2294:
2292:
2288:
2282:
2278:
2272:
2215:
2213:
2212:
2207:
2202:
2200:
2189:
2169:
2167:
2166:
2161:
2156:
2152:
2149:
2148:
2117:
2115:
2114:
2109:
2101:
2100:
2074:
2072:
2071:
2066:
2061:
2059:
2058:
2043:
2041:
2040:
2028:
2027:
1960:
1946:
1945:
1944:
1906:
1903:
1902:
1897:
1895:
1893:
1892:
1884:
1875:
1853:
1851:
1850:
1845:
1843:
1834:
1833:
1828:
1827:
1808:
1806:
1805:
1800:
1785:
1780:
1768:
1766:
1762:
1756:
1752:
1746:
1744:
1743:
1725:
1723:
1722:
1721:
1712:
1706:
1702:
1701:
1696:
1689:
1680:
1678:
1677:
1672:
1667:
1665:
1664:
1663:
1654:
1648:
1644:
1643:
1638:
1631:
1623:
1621:
1617:
1611:
1607:
1601:
1556:
1522:
1479:
1476:
1475:
1470:
1465:
1464:
1452:
1443:
1388:
1386:
1385:
1380:
1375:
1354:
1352:
1351:
1346:
1341:
1300:
1297:
1296:
1291:
1286:
1277:
1276:
1260:
1257:
1256:
1251:
1190:
1188:
1187:
1182:
1165:
1164:
1156:
1147:
1145:
1144:
1143:
1134:
1128:
1124:
1123:
1118:
1111:
1072:
1044:
1032:
887:
884:
883:
878:
873:
872:
864:
852:
851:
843:
797:
790:
783:
770:
765:
764:
757:
753:
752:
658:Johann Bernoulli
653:Daniel Bernoulli
574:Tangential speed
478:
454:
429:Fictitious force
424:
276:Mechanical power
266:
207:Angular momentum
105:
103:
102:
97:
95:
93:
85:
84:
75:
70:
69:
43:
42:
21:
11813:
11812:
11808:
11807:
11806:
11804:
11803:
11802:
11768:
11767:
11765:
11752:
11742:
11719:
11700:
11692:. Brooks/Cole.
11680:
11660:Kreyszig, Erwin
11652:
11618:
11598:
11593:
11592:
11553:
11549:
11542:10.1119/1.18209
11518:
11514:
11504:
11502:
11493:Case, William.
11491:
11487:
11479:
11475:
11467:
11463:
11455:
11451:
11446:
11441:
11382:
11350:
11346:
11345:
11344:
11342:
11339:
11338:
11317:
11313:
11311:
11308:
11307:
11279:
11275:
11274:
11273:
11271:
11268:
11267:
11247:
11244:
11243:
11215:
11212:
11211:
11183:
11180:
11179:
11156:
11155:
11153:
11150:
11149:
11129:
11126:
11125:
11097:
11093:
11087:
11083:
11079:
11078:
11076:
11073:
11072:
11052:
11049:
11048:
11025:
11024:
11022:
11019:
11018:
10998:
10995:
10994:
10971:
10970:
10968:
10965:
10964:
10944:
10941:
10940:
10912:
10909:
10908:
10885:
10884:
10882:
10879:
10878:
10852:
10849:
10848:
10820:
10816:
10810:
10806:
10802:
10801:
10799:
10796:
10795:
10775:
10772:
10771:
10743:
10739:
10735:
10734:
10732:
10729:
10728:
10725:Spring constant
10708:
10705:
10704:
10669:
10665:
10663:
10660:
10659:
10641:
10638:
10637:
10609:
10605:
10601:
10600:
10598:
10595:
10594:
10574:
10571:
10570:
10542:
10538:
10531:
10530:
10528:
10525:
10524:
10504:
10501:
10500:
10472:
10468:
10467:
10466:
10464:
10461:
10460:
10440:
10437:
10436:
10408:
10404:
10400:
10399:
10397:
10394:
10393:
10373:
10370:
10369:
10346:
10345:
10343:
10340:
10339:
10319:
10316:
10315:
10287:
10283:
10279:
10278:
10276:
10273:
10272:
10252:
10249:
10248:
10229:
10201:
10197:
10184:
10176:
10173:
10172:
10161:
10140:
10137:
10136:
10116:
10113:
10112:
10076:
10075:
10071:
10045:
10042:
10041:
9986:
9982:
9977:
9976:
9970:
9965:
9964:
9962:
9921:
9918:
9917:
9852:
9849:
9848:
9833:
9812:
9809:
9808:
9791:
9787:
9785:
9782:
9781:
9765:
9762:
9761:
9744:
9740:
9738:
9735:
9734:
9706:
9704:
9689:
9675:
9672:
9649:
9645:
9643:
9640:
9639:
9622:
9618:
9616:
9613:
9612:
9578:
9577:
9573:
9561:
9557:
9540:
9537:
9536:
9499:
9498:
9496:
9493:
9492:
9475:
9471:
9454:
9451:
9450:
9434:
9431:
9430:
9414:
9411:
9410:
9370:
9369:
9365:
9339:
9336:
9335:
9303:
9291:
9287:
9283:
9273:
9269:
9268:
9266:
9264:
9261:
9260:
9232:
9229:
9228:
9189:
9177:
9173:
9169:
9159:
9155:
9154:
9152:
9150:
9147:
9125:
9122:
9121:
9105:
9102:
9101:
9094:simple pendulum
9086:
9084:Simple pendulum
9081:
9051:
9048:
9047:
9025:
9021:
9006:
8997:
8993:
8973:
8961:
8944:
8941:
8940:
8936:
8926:
8901:
8897:
8891:
8887:
8866:
8862:
8856:
8852:
8831:
8827:
8816:
8804:
8792:
8788:
8765:
8762:
8761:
8744:
8740:
8738:
8735:
8734:
8714:
8710:
8702:
8699:
8698:
8676:
8672:
8666:
8662:
8641:
8637:
8631:
8627:
8606:
8602:
8591:
8579:
8567:
8563:
8542:
8538:
8527:
8515:
8511:
8488:
8485:
8484:
8465:
8461:
8453:
8450:
8449:
8433:
8430:
8429:
8424:, one can do a
8400:
8397:
8396:
8374:
8370:
8355:
8338:
8335:
8334:
8320:
8288:
8271:
8270:
8253:
8252:
8247:
8244:
8243:
8217:
8200:
8199:
8182:
8181:
8176:
8173:
8172:
8134:
8133:
8116:
8115:
8110:
8107:
8106:
8068:
8067:
8050:
8049:
8044:
8041:
8040:
8009:
7999:
7997:
7994:
7993:
7969:
7959:
7957:
7954:
7953:
7930:
7924:
7914:
7912:
7909:
7908:
7885:
7879:
7869:
7867:
7864:
7863:
7843:
7840:
7839:
7811:
7805:
7795:
7790:
7788:
7785:
7784:
7761:
7755:
7745:
7740:
7738:
7735:
7734:
7710:
7700:
7695:
7693:
7690:
7689:
7665:
7655:
7650:
7648:
7645:
7644:
7623:
7619:
7617:
7614:
7613:
7590:
7587:
7586:
7566:
7563:
7562:
7533:
7530:
7529:
7499:
7496:
7495:
7469:
7458:
7455:
7454:
7434:
7431:
7430:
7410:
7407:
7406:
7386:
7383:
7382:
7357:
7352:
7349:
7348:
7325:
7320:
7317:
7316:
7296:
7293:
7292:
7272:
7269:
7268:
7266:Spring constant
7246:
7243:
7242:
7222:
7219:
7218:
7198:
7195:
7194:
7174:
7171:
7170:
7148:
7145:
7144:
7124:
7121:
7120:
7100:
7097:
7096:
7076:
7073:
7072:
7044:
7043:
7034:
7033:
7031:
7029:
7026:
7025:
6999:
6998:
6989:
6988:
6986:
6984:
6981:
6980:
6954:
6953:
6944:
6943:
6941:
6939:
6936:
6935:
6909:
6908:
6899:
6898:
6896:
6894:
6891:
6890:
6868:
6865:
6864:
6844:
6841:
6840:
6820:
6817:
6816:
6798:
6795:
6794:
6762:
6756:
6723:
6719:
6701:
6697:
6680:
6677:
6676:
6551:
6547:
6545:
6542:
6541:
6538:
6485:
6481:
6480:
6469:
6467:
6463:
6398:
6394:
6393:
6382:
6380:
6368:
6364:
6357:
6346:
6344:
6327:
6324:
6323:
6317:
6314:
6272:
6268:
6247:
6243:
6237:
6233:
6218:
6201:
6187:
6185:
6181:
6146:
6143:
6142:
6114:
6110:
6089:
6085:
6079:
6075:
6057:
6053:
6043:
6042:
6030:
6026:
6019:
6013:
6009:
5991:
5990:
5978:
5974:
5967:
5961:
5957:
5951:
5947:
5932:
5928:
5924:
5921:
5919:
5916:
5915:
5901:
5831:
5827:
5813:
5810:
5809:
5754:
5750:
5714:
5710:
5705:
5702:
5701:
5673:
5669:
5642:
5638:
5602:
5598:
5565:
5561:
5531:
5527:
5485:
5481:
5472:
5468:
5463:
5460:
5459:
5419:
5415:
5406:
5402:
5387:
5383:
5378:
5377:
5370:
5366:
5360:
5355:
5354:
5353:
5351:
5323:
5319:
5296:
5295:
5288:
5284:
5279:
5278:
5276:
5248:
5244:
5232:
5228:
5226:
5223:
5222:
5182:
5178:
5157:
5153:
5151:
5148:
5147:
5119:
5115:
5050:
5049:
5040:
5039:
5037:
5019:
5015:
5010:
5009:
4999:
4994:
4993:
4992:
4990:
4988:
4985:
4984:
4977:
4953:
4952:
4947:
4936:
4915:
4911:
4903:
4902:
4898:
4886:
4882:
4863:
4859:
4851:
4850:
4846:
4834:
4830:
4829:
4825:
4813:
4809:
4806:
4805:
4800:
4789:
4777:
4773:
4764:
4760:
4748:
4744:
4729:
4725:
4716:
4712:
4697:
4693:
4690:
4689:
4684:
4673:
4652:
4648:
4646:
4639:
4635:
4629:
4625:
4606:
4602:
4600:
4596:
4592:
4586:
4582:
4581:
4577:
4565:
4561:
4554:
4553:
4535:
4531:
4529:
4526:
4525:
4522:
4515:
4503:
4444:
4443:
4434:
4433:
4431:
4413:
4409:
4404:
4403:
4393:
4388:
4387:
4386:
4384:
4382:
4379:
4378:
4375:
4363:
4353:
4336:
4291:
4290:
4281:
4280:
4278:
4260:
4256:
4251:
4250:
4240:
4235:
4234:
4233:
4231:
4229:
4226:
4225:
4222:
4192:
4188:
4179:
4175:
4173:
4170:
4169:
4126:
4123:
4122:
4106:
4103:
4102:
4090:
4084:
4056:
4052:
4050:
4047:
4046:
4025:
4020:
4009:
4006:
4005:
3988:
3984:
3982:
3979:
3978:
3959:
3955:
3944:
3938:
3934:
3925:
3921:
3919:
3916:
3915:
3868:
3863:
3850:
3846:
3845:
3835:
3831:
3824:
3822:
3818:
3804:
3801:
3800:
3772:
3768:
3762:
3758:
3749:
3744:
3729:
3725:
3720:
3711:
3697:
3693:
3689:
3685:
3684:
3682:
3673:
3669:
3667:
3664:
3612:
3608:
3604:
3598:
3594:
3592:
3575:
3572:
3571:
3555:
3552:
3551:
3532:
3529:
3528:
3511:
3507:
3505:
3502:
3501:
3485:
3482:
3481:
3464:
3460:
3458:
3455:
3454:
3414:is the driving
3399:
3396:
3395:
3378:
3374:
3372:
3369:
3368:
3330:
3326:
3316:
3304:
3299:
3281:
3280:
3271:
3270:
3268:
3262:
3258:
3240:
3236:
3231:
3230:
3220:
3215:
3214:
3213:
3211:
3209:
3206:
3205:
3185:
3182:
3181:
3164:
3160:
3155:
3150:
3147:
3146:
3139:
3103:
3093:
3060:
3057:
3056:
3015:
2994:
2990:
2982:
2978:
2970:
2969:
2965:
2958:
2956:
2945:
2941:
2934:
2930:
2907:
2904:
2903:
2886:
2885:
2874:
2868:
2867:
2856:
2850:
2845:
2834:
2833:
2811:
2809:
2807:
2804:
2803:
2796:
2789:
2786:
2780:
2760:
2726:
2722:
2714:
2710:
2702:
2701:
2697:
2680:
2676:
2669:
2665:
2645:
2642:
2641:
2610:
2605:
2587:
2586:
2577:
2576:
2574:
2568:
2564:
2546:
2542:
2537:
2536:
2526:
2521:
2520:
2519:
2517:
2515:
2512:
2511:
2468:
2466:
2454:
2449:
2431:
2430:
2421:
2420:
2418:
2412:
2408:
2390:
2386:
2381:
2380:
2370:
2365:
2364:
2363:
2361:
2359:
2356:
2355:
2330:
2326:
2321:
2320:
2310:
2305:
2304:
2303:
2301:
2284:
2283:
2274:
2273:
2271:
2242:
2239:
2238:
2237:takes the form
2221:
2193:
2188:
2180:
2177:
2176:
2147:
2130:
2127:
2126:
2096:
2092:
2084:
2081:
2080:
2054:
2050:
2042:
2036:
2032:
2023:
2019:
2017:
2014:
2013:
1958:
1948:
1939:
1937:
1936:
1929:
1919:
1883:
1879:
1874:
1866:
1863:
1832:
1823:
1819:
1817:
1814:
1813:
1781:
1776:
1758:
1757:
1748:
1747:
1745:
1739:
1735:
1717:
1713:
1708:
1707:
1697:
1692:
1691:
1690:
1688:
1686:
1683:
1682:
1659:
1655:
1650:
1649:
1639:
1634:
1633:
1632:
1630:
1613:
1612:
1603:
1602:
1600:
1577:
1574:
1573:
1551:
1545:
1544:of the object:
1539:
1512:
1491:
1485:
1460:
1456:
1441:
1433:
1430:
1371:
1360:
1357:
1356:
1337:
1323:
1320:
1319:
1275:
1267:
1264:
1203:
1200:
1155:
1154:
1139:
1135:
1130:
1129:
1119:
1114:
1113:
1112:
1110:
1090:
1087:
1086:
1067:
1052:
1051:
1050:
1049:
1048:
1045:
1037:
1036:
1033:
1022:
1016:
922:and a constant
863:
862:
842:
841:
839:
836:
821:restoring force
801:
760:
747:
746:
739:
738:
737:
612:
604:
603:
583:
537:Circular motion
531:
521:
520:
519:
476:
446:
443:
422:
401:
393:
392:
389:
388:
346:
336:
328:
327:
326:
285:
281:Mechanical work
274:
258:
196:
188:
187:
186:
141:
133:
110:
86:
80:
76:
74:
65:
64:
62:
59:
58:
39:
28:
23:
22:
15:
12:
11:
5:
11811:
11801:
11800:
11795:
11790:
11785:
11780:
11763:
11762:
11751:
11750:External links
11748:
11747:
11746:
11740:
11723:
11717:
11704:
11698:
11683:
11678:
11656:
11650:
11629:
11616:
11597:
11594:
11591:
11590:
11547:
11528:(3): 215–220.
11512:
11485:
11473:
11469:Kreyszig (1972
11461:
11448:
11447:
11445:
11442:
11440:
11439:
11434:
11429:
11424:
11419:
11414:
11409:
11404:
11402:Fradkin tensor
11399:
11394:
11392:Critical speed
11389:
11383:
11381:
11378:
11375:
11374:
11371:
11356:
11353:
11349:
11336:
11333:
11320:
11316:
11304:
11303:
11300:
11285:
11282:
11278:
11265:
11262:
11251:
11240:
11239:
11236:
11233:
11230:
11219:
11208:
11207:
11204:
11201:
11198:
11187:
11176:
11175:
11172:
11159:
11147:
11144:
11133:
11122:
11121:
11118:
11103:
11100:
11096:
11090:
11086:
11082:
11070:
11067:
11056:
11045:
11044:
11041:
11028:
11016:
11013:
11002:
10991:
10990:
10987:
10974:
10962:
10959:
10948:
10937:
10936:
10933:
10930:
10929:Quality factor
10927:
10916:
10905:
10904:
10901:
10888:
10876:
10873:
10862:
10859:
10856:
10845:
10844:
10841:
10826:
10823:
10819:
10813:
10809:
10805:
10793:
10790:
10779:
10768:
10767:
10764:
10749:
10746:
10742:
10738:
10726:
10723:
10712:
10701:
10700:
10697:
10694:
10683:
10680:
10677:
10672:
10668:
10656:
10645:
10634:
10633:
10630:
10615:
10612:
10608:
10604:
10592:
10589:
10578:
10567:
10566:
10563:
10548:
10545:
10541:
10537:
10534:
10522:
10519:
10508:
10497:
10496:
10493:
10478:
10475:
10471:
10458:
10455:
10444:
10433:
10432:
10429:
10414:
10411:
10407:
10403:
10391:
10388:
10377:
10366:
10365:
10362:
10349:
10337:
10334:
10323:
10312:
10311:
10308:
10293:
10290:
10286:
10282:
10270:
10267:
10256:
10245:
10244:
10241:
10238:
10235:
10228:
10225:
10209:
10204:
10200:
10196:
10191:
10188:
10183:
10180:
10169:kinetic energy
10160:
10157:
10144:
10133:effective mass
10120:
10098:
10094:
10090:
10084:
10081:
10074:
10070:
10067:
10064:
10061:
10058:
10055:
10052:
10049:
10018:
10015:
10012:
10009:
10006:
10003:
10000:
9997:
9989:
9985:
9980:
9973:
9968:
9961:
9958:
9955:
9952:
9949:
9946:
9943:
9940:
9937:
9934:
9931:
9928:
9925:
9904:is the force,
9889:
9886:
9883:
9880:
9877:
9874:
9871:
9868:
9865:
9862:
9859:
9856:
9832:
9829:
9816:
9794:
9790:
9769:
9747:
9743:
9721:
9716:
9712:
9709:
9703:
9697:
9694:
9688:
9685:
9682:
9679:
9652:
9648:
9625:
9621:
9600:
9596:
9592:
9586:
9583:
9576:
9572:
9569:
9564:
9560:
9556:
9553:
9550:
9547:
9544:
9524:
9521:
9518:
9515:
9512:
9506:
9503:
9478:
9474:
9470:
9467:
9464:
9461:
9458:
9438:
9418:
9398:
9394:
9390:
9387:
9384:
9378:
9375:
9368:
9364:
9361:
9358:
9355:
9352:
9349:
9346:
9343:
9321:
9318:
9315:
9310:
9307:
9302:
9294:
9290:
9286:
9281:
9276:
9272:
9248:
9245:
9242:
9239:
9236:
9213:
9210:
9207:
9204:
9201:
9196:
9193:
9188:
9180:
9176:
9172:
9167:
9162:
9158:
9129:
9109:
9085:
9082:
9080:
9077:
9064:
9061:
9058:
9055:
9033:
9028:
9024:
9020:
9014:
9011:
9005:
9000:
8996:
8992:
8989:
8986:
8983:
8979:
8976:
8969:
8966:
8960:
8957:
8954:
8951:
8948:
8934:
8909:
8904:
8900:
8894:
8890:
8886:
8883:
8880:
8877:
8874:
8869:
8865:
8859:
8855:
8851:
8848:
8845:
8842:
8839:
8834:
8830:
8826:
8822:
8819:
8812:
8809:
8803:
8800:
8795:
8791:
8787:
8784:
8781:
8778:
8775:
8772:
8769:
8747:
8743:
8722:
8717:
8713:
8709:
8706:
8684:
8679:
8675:
8669:
8665:
8661:
8658:
8655:
8652:
8649:
8644:
8640:
8634:
8630:
8626:
8623:
8620:
8617:
8614:
8609:
8605:
8601:
8597:
8594:
8587:
8584:
8578:
8575:
8570:
8566:
8562:
8559:
8556:
8553:
8550:
8545:
8541:
8537:
8533:
8530:
8526:
8523:
8518:
8514:
8510:
8507:
8504:
8501:
8498:
8495:
8492:
8468:
8464:
8460:
8457:
8437:
8413:
8410:
8407:
8404:
8382:
8377:
8373:
8369:
8363:
8360:
8354:
8351:
8348:
8345:
8342:
8319:
8316:
8313:
8312:
8301:
8298:
8295:
8291:
8287:
8284:
8278:
8275:
8269:
8266:
8260:
8257:
8251:
8241:
8230:
8227:
8224:
8220:
8216:
8213:
8207:
8204:
8198:
8195:
8189:
8186:
8180:
8170:
8159:
8156:
8153:
8150:
8147:
8141:
8138:
8132:
8129:
8123:
8120:
8114:
8104:
8093:
8090:
8087:
8084:
8081:
8075:
8072:
8066:
8063:
8057:
8054:
8048:
8037:
8036:
8032:
8031:
8017:
8014:
8006:
8003:
7991:
7977:
7974:
7966:
7963:
7951:
7936:
7933:
7929:
7921:
7918:
7906:
7891:
7888:
7884:
7876:
7873:
7860:
7859:
7847:
7833:
7832:
7817:
7814:
7810:
7801:
7798:
7794:
7782:
7767:
7764:
7760:
7751:
7748:
7744:
7732:
7718:
7715:
7706:
7703:
7699:
7687:
7673:
7670:
7661:
7658:
7654:
7641:
7640:
7626:
7622:
7606:
7605:
7594:
7581:
7570:
7557:
7546:
7543:
7540:
7537:
7523:
7512:
7509:
7506:
7503:
7488:
7487:
7476:
7472:
7468:
7465:
7462:
7449:
7438:
7425:
7414:
7401:
7390:
7376:
7375:
7364:
7360:
7356:
7343:
7332:
7328:
7324:
7311:
7300:
7287:
7276:
7262:
7261:
7250:
7237:
7226:
7213:
7202:
7189:
7178:
7164:
7163:
7152:
7139:
7128:
7115:
7104:
7091:
7080:
7066:
7065:
7051:
7047:
7041:
7037:
7020:
7006:
7002:
6996:
6992:
6975:
6961:
6957:
6951:
6947:
6930:
6916:
6912:
6906:
6902:
6884:
6883:
6872:
6859:
6848:
6835:
6824:
6813:
6802:
6790:
6789:
6784:
6779:
6776:
6758:Main article:
6755:
6752:
6740:
6737:
6734:
6731:
6726:
6722:
6718:
6715:
6712:
6709:
6704:
6700:
6696:
6693:
6690:
6687:
6684:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6625:
6622:
6619:
6616:
6613:
6610:
6607:
6604:
6601:
6598:
6595:
6592:
6589:
6586:
6583:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6554:
6550:
6537:
6534:
6515:
6512:
6509:
6506:
6502:
6496:
6493:
6488:
6484:
6478:
6475:
6472:
6466:
6462:
6459:
6456:
6453:
6450:
6447:
6444:
6441:
6438:
6435:
6432:
6422:
6409:
6406:
6401:
6397:
6391:
6388:
6385:
6379:
6371:
6367:
6363:
6360:
6355:
6352:
6349:
6343:
6340:
6337:
6334:
6331:
6313:
6310:
6283:
6275:
6271:
6267:
6264:
6261:
6258:
6255:
6250:
6246:
6240:
6236:
6232:
6229:
6226:
6222:
6216:
6210:
6207:
6204:
6199:
6196:
6193:
6190:
6184:
6180:
6177:
6174:
6171:
6168:
6165:
6162:
6159:
6156:
6153:
6150:
6128:
6125:
6122:
6117:
6113:
6109:
6106:
6103:
6100:
6097:
6092:
6088:
6082:
6078:
6074:
6071:
6068:
6065:
6060:
6056:
6052:
6048:
6041:
6038:
6033:
6029:
6025:
6022:
6020:
6016:
6012:
6008:
6005:
6002:
5999:
5996:
5993:
5992:
5989:
5986:
5981:
5977:
5973:
5970:
5968:
5964:
5960:
5954:
5950:
5946:
5943:
5940:
5935:
5931:
5927:
5926:
5923:
5900:
5899:Amplitude part
5897:
5885:
5882:
5879:
5876:
5873:
5870:
5867:
5864:
5861:
5858:
5854:
5851:
5848:
5845:
5842:
5839:
5834:
5830:
5826:
5823:
5820:
5817:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5763:
5760:
5757:
5753:
5749:
5746:
5743:
5740:
5737:
5734:
5731:
5728:
5725:
5722:
5717:
5713:
5709:
5687:
5682:
5679:
5676:
5672:
5668:
5663:
5660:
5657:
5654:
5651:
5648:
5645:
5641:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5605:
5601:
5597:
5594:
5591:
5586:
5583:
5580:
5577:
5574:
5571:
5568:
5564:
5560:
5557:
5552:
5549:
5546:
5543:
5540:
5537:
5534:
5530:
5526:
5523:
5520:
5517:
5514:
5511:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5484:
5480:
5475:
5471:
5467:
5445:
5440:
5437:
5434:
5431:
5428:
5425:
5422:
5418:
5414:
5409:
5405:
5401:
5398:
5390:
5386:
5381:
5373:
5369:
5363:
5358:
5349:
5344:
5341:
5338:
5335:
5332:
5329:
5326:
5322:
5318:
5315:
5312:
5309:
5303:
5299:
5291:
5287:
5282:
5274:
5269:
5266:
5263:
5260:
5257:
5254:
5251:
5247:
5243:
5240:
5235:
5231:
5208:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5181:
5177:
5174:
5171:
5168:
5165:
5160:
5156:
5133:
5128:
5125:
5122:
5118:
5114:
5111:
5108:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5072:
5069:
5066:
5063:
5057:
5053:
5047:
5043:
5036:
5033:
5030:
5022:
5018:
5013:
5007:
5002:
4997:
4976:
4973:
4956:
4946:
4943:
4940:
4937:
4934:
4929:
4925:
4918:
4914:
4910:
4907:
4901:
4897:
4894:
4889:
4885:
4881:
4877:
4873:
4866:
4862:
4858:
4855:
4849:
4845:
4842:
4837:
4833:
4828:
4822:
4819:
4816:
4812:
4808:
4807:
4799:
4796:
4793:
4790:
4788:
4785:
4780:
4776:
4772:
4767:
4763:
4759:
4754:
4751:
4747:
4743:
4740:
4737:
4732:
4728:
4724:
4719:
4715:
4711:
4706:
4703:
4700:
4696:
4692:
4691:
4683:
4680:
4677:
4674:
4671:
4663:
4660:
4655:
4651:
4645:
4642:
4638:
4632:
4628:
4624:
4617:
4614:
4609:
4605:
4599:
4595:
4589:
4585:
4580:
4574:
4571:
4568:
4564:
4560:
4559:
4557:
4552:
4549:
4546:
4543:
4538:
4534:
4520:
4513:
4502:
4499:
4484:
4481:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4451:
4447:
4441:
4437:
4430:
4427:
4424:
4416:
4412:
4407:
4401:
4396:
4391:
4371:
4351:
4313:
4310:
4307:
4304:
4298:
4294:
4288:
4284:
4277:
4274:
4271:
4263:
4259:
4254:
4248:
4243:
4238:
4224:The equation
4221:
4218:
4195:
4191:
4187:
4182:
4178:
4130:
4110:
4086:Main article:
4083:
4080:
4067:
4064:
4059:
4055:
4029:
4023:
4019:
4016:
4013:
3991:
3987:
3962:
3958:
3954:
3951:
3948:
3941:
3937:
3933:
3928:
3924:
3889:
3886:
3883:
3879:
3871:
3866:
3862:
3858:
3853:
3849:
3843:
3838:
3834:
3830:
3827:
3821:
3817:
3814:
3811:
3808:
3775:
3771:
3765:
3761:
3757:
3752:
3747:
3743:
3739:
3732:
3728:
3724:
3719:
3714:
3709:
3705:
3700:
3696:
3692:
3688:
3681:
3676:
3672:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3620:
3615:
3611:
3607:
3601:
3597:
3591:
3588:
3585:
3582:
3579:
3559:
3536:
3514:
3510:
3489:
3467:
3463:
3440:air resistance
3403:
3381:
3377:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3333:
3329:
3323:
3320:
3315:
3312:
3307:
3302:
3298:
3294:
3288:
3284:
3278:
3274:
3265:
3261:
3257:
3254:
3251:
3243:
3239:
3234:
3228:
3223:
3218:
3189:
3167:
3163:
3158:
3154:
3138:
3135:
3121:is called the
3101:
3079:
3076:
3073:
3070:
3067:
3064:
3036:
3030:
3027:
3024:
3021:
3018:
3012:
3008:
3005:
3002:
2997:
2993:
2985:
2981:
2977:
2974:
2968:
2964:
2961:
2953:
2948:
2944:
2940:
2937:
2933:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2889:
2884:
2881:
2878:
2875:
2873:
2870:
2869:
2866:
2863:
2860:
2857:
2853:
2848:
2844:
2840:
2839:
2837:
2832:
2827:
2823:
2820:
2817:
2814:
2779:
2776:
2748:
2744:
2740:
2737:
2734:
2729:
2725:
2717:
2713:
2709:
2706:
2700:
2696:
2693:
2688:
2683:
2679:
2675:
2672:
2668:
2664:
2661:
2658:
2655:
2652:
2649:
2627:
2624:
2621:
2618:
2613:
2608:
2604:
2600:
2594:
2590:
2584:
2580:
2571:
2567:
2563:
2560:
2557:
2549:
2545:
2540:
2534:
2529:
2524:
2489:
2484:
2480:
2477:
2474:
2471:
2465:
2462:
2457:
2452:
2448:
2444:
2438:
2434:
2428:
2424:
2415:
2411:
2407:
2404:
2401:
2393:
2389:
2384:
2378:
2373:
2368:
2341:
2333:
2329:
2324:
2318:
2313:
2308:
2300:
2297:
2291:
2287:
2281:
2277:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2220:
2217:
2205:
2199:
2196:
2192:
2187:
2184:
2159:
2146:
2143:
2140:
2137:
2134:
2119:
2118:
2107:
2104:
2099:
2095:
2091:
2088:
2064:
2057:
2053:
2049:
2046:
2039:
2035:
2031:
2026:
2022:
1999:
1989:
1956:
1940:1 −
1934:
1927:
1909:
1908:
1890:
1887:
1882:
1878:
1873:
1870:
1859:
1840:
1837:
1831:
1826:
1822:
1798:
1795:
1792:
1789:
1784:
1779:
1775:
1771:
1765:
1761:
1755:
1751:
1742:
1738:
1734:
1731:
1728:
1720:
1716:
1711:
1705:
1700:
1695:
1670:
1662:
1658:
1653:
1647:
1642:
1637:
1629:
1626:
1620:
1616:
1610:
1606:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1561:is called the
1549:
1537:
1487:Main article:
1484:
1481:
1468:
1463:
1459:
1455:
1449:
1446:
1440:
1437:
1378:
1374:
1370:
1367:
1364:
1344:
1340:
1336:
1333:
1330:
1327:
1303:The motion is
1289:
1283:
1280:
1274:
1271:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1180:
1177:
1174:
1171:
1168:
1162:
1159:
1153:
1150:
1142:
1138:
1133:
1127:
1122:
1117:
1109:
1106:
1103:
1100:
1097:
1094:
1046:
1039:
1038:
1034:
1027:
1026:
1025:
1024:
1023:
1018:Main article:
1015:
1012:
966:
965:
958:
892:is a positive
876:
870:
867:
861:
858:
855:
849:
846:
803:
802:
800:
799:
792:
785:
777:
774:
773:
772:
771:
758:
741:
740:
736:
735:
730:
725:
720:
715:
710:
705:
700:
695:
690:
685:
680:
675:
670:
665:
660:
655:
650:
645:
640:
635:
630:
625:
620:
614:
613:
610:
609:
606:
605:
602:
601:
582:
581:
576:
571:
566:
564:Coriolis force
561:
560:
559:
549:
544:
539:
533:
532:
527:
526:
523:
522:
518:
517:
512:
507:
506:
505:
500:
490:
485:
480:
473:
462:
461:
460:
455:
442:
441:
436:
431:
426:
419:
414:
409:
403:
402:
399:
398:
395:
394:
391:
390:
387:
386:
381:
376:
371:
366:
361:
355:
349:
347:
340:
337:
334:
333:
330:
329:
325:
324:
319:
314:
309:
304:
299:
294:
289:
283:
278:
272:
267:
256:
251:
246:
241:
236:
235:
234:
229:
219:
214:
209:
204:
198:
197:
194:
193:
190:
189:
185:
184:
179:
174:
169:
164:
159:
154:
149:
143:
142:
139:
138:
135:
134:
132:
131:
126:
121:
115:
112:
111:
106:
92:
89:
83:
79:
73:
55:
54:
48:
47:
26:
9:
6:
4:
3:
2:
11810:
11799:
11796:
11794:
11791:
11789:
11786:
11784:
11781:
11779:
11776:
11775:
11773:
11766:
11761:
11757:
11754:
11753:
11743:
11741:0-07-072180-7
11737:
11732:
11731:
11724:
11720:
11718:1-57259-492-6
11714:
11710:
11705:
11701:
11699:0-534-40842-7
11695:
11691:
11690:
11684:
11681:
11679:0-471-50728-8
11675:
11671:
11667:
11666:
11661:
11657:
11653:
11651:9783527600434
11647:
11643:
11639:
11635:
11630:
11627:
11623:
11619:
11617:0-03-089725-4
11613:
11609:
11605:
11600:
11599:
11586:
11582:
11578:
11574:
11570:
11566:
11562:
11558:
11551:
11543:
11539:
11535:
11531:
11527:
11523:
11516:
11500:
11496:
11489:
11482:
11477:
11471:, p. 65)
11470:
11465:
11459:, p. 86)
11458:
11453:
11449:
11438:
11435:
11433:
11430:
11428:
11425:
11423:
11420:
11418:
11415:
11413:
11410:
11408:
11405:
11403:
11400:
11398:
11395:
11393:
11390:
11388:
11387:Anharmonicity
11385:
11384:
11372:
11337:
11334:
11318:
11314:
11306:
11305:
11301:
11266:
11263:
11249:
11242:
11241:
11237:
11234:
11231:
11217:
11210:
11209:
11205:
11202:
11200:Damping ratio
11199:
11185:
11178:
11177:
11173:
11148:
11145:
11131:
11124:
11123:
11119:
11071:
11068:
11054:
11047:
11046:
11042:
11017:
11014:
11000:
10993:
10992:
10988:
10963:
10960:
10946:
10939:
10938:
10934:
10931:
10928:
10914:
10907:
10906:
10902:
10877:
10874:
10860:
10857:
10854:
10847:
10846:
10842:
10794:
10791:
10777:
10770:
10769:
10765:
10727:
10724:
10710:
10703:
10702:
10698:
10695:
10681:
10678:
10675:
10670:
10666:
10657:
10643:
10636:
10635:
10631:
10593:
10590:
10576:
10569:
10568:
10564:
10523:
10520:
10506:
10499:
10498:
10494:
10459:
10456:
10442:
10435:
10434:
10430:
10392:
10389:
10375:
10368:
10367:
10363:
10338:
10335:
10321:
10314:
10313:
10309:
10271:
10268:
10254:
10247:
10246:
10232:
10224:
10220:
10207:
10202:
10198:
10194:
10189:
10186:
10181:
10178:
10170:
10166:
10156:
10142:
10134:
10118:
10109:
10096:
10092:
10088:
10082:
10079:
10072:
10068:
10065:
10062:
10059:
10053:
10047:
10039:
10034:
10032:
10016:
10013:
10010:
10007:
10001:
9995:
9987:
9983:
9971:
9959:
9956:
9950:
9944:
9941:
9938:
9935:
9929:
9923:
9914:
9911:
9907:
9903:
9887:
9881:
9875:
9872:
9869:
9866:
9860:
9854:
9846:
9837:
9828:
9814:
9792:
9788:
9767:
9745:
9741:
9731:
9719:
9714:
9710:
9707:
9701:
9695:
9692:
9686:
9683:
9680:
9677:
9668:
9650:
9646:
9623:
9619:
9598:
9594:
9590:
9584:
9581:
9574:
9570:
9567:
9562:
9558:
9554:
9548:
9542:
9522:
9519:
9513:
9504:
9501:
9476:
9472:
9468:
9462:
9456:
9436:
9416:
9396:
9392:
9388:
9385:
9382:
9376:
9373:
9366:
9362:
9359:
9356:
9353:
9347:
9341:
9332:
9319:
9316:
9313:
9308:
9305:
9300:
9292:
9288:
9284:
9279:
9274:
9270:
9246:
9243:
9240:
9237:
9234:
9225:
9223:
9211:
9208:
9205:
9202:
9199:
9194:
9191:
9186:
9178:
9174:
9170:
9165:
9160:
9156:
9143:
9140:is the local
9127:
9107:
9095:
9090:
9076:
9059:
9053:
9044:
9031:
9026:
9022:
9018:
9012:
9009:
9003:
8998:
8994:
8990:
8984:
8977:
8974:
8967:
8964:
8958:
8952:
8946:
8933:
8929:
8925:
8924:constant term
8920:
8907:
8902:
8892:
8888:
8884:
8881:
8875:
8872:
8867:
8857:
8853:
8849:
8846:
8840:
8832:
8828:
8820:
8817:
8810:
8807:
8801:
8793:
8789:
8782:
8779:
8773:
8767:
8745:
8741:
8715:
8711:
8704:
8695:
8682:
8677:
8667:
8663:
8659:
8656:
8650:
8647:
8642:
8632:
8628:
8624:
8621:
8615:
8607:
8603:
8595:
8592:
8585:
8582:
8576:
8568:
8564:
8560:
8557:
8551:
8543:
8539:
8531:
8528:
8524:
8516:
8512:
8505:
8502:
8496:
8490:
8482:
8466:
8462:
8458:
8455:
8435:
8427:
8408:
8402:
8393:
8380:
8375:
8371:
8367:
8361:
8358:
8352:
8346:
8340:
8332:
8327:
8325:
8299:
8296:
8293:
8289:
8285:
8282:
8276:
8273:
8267:
8264:
8258:
8255:
8249:
8242:
8228:
8225:
8222:
8218:
8214:
8211:
8205:
8202:
8196:
8193:
8187:
8184:
8178:
8171:
8157:
8154:
8151:
8148:
8145:
8139:
8136:
8127:
8121:
8118:
8112:
8105:
8091:
8088:
8085:
8082:
8079:
8073:
8070:
8064:
8061:
8055:
8052:
8046:
8039:
8038:
8033:
8015:
8012:
8004:
8001:
7992:
7975:
7972:
7964:
7961:
7952:
7934:
7931:
7927:
7919:
7907:
7889:
7886:
7882:
7874:
7871:
7862:
7861:
7845:
7838:
7837:Damping ratio
7834:
7815:
7812:
7808:
7799:
7796:
7792:
7783:
7765:
7762:
7758:
7749:
7746:
7742:
7733:
7716:
7713:
7704:
7701:
7697:
7688:
7671:
7668:
7659:
7656:
7652:
7643:
7642:
7624:
7620:
7612:
7607:
7592:
7585:
7582:
7568:
7561:
7558:
7541:
7535:
7528:
7524:
7507:
7501:
7494:
7490:
7489:
7474:
7470:
7466:
7463:
7460:
7453:
7450:
7436:
7429:
7426:
7405:
7402:
7388:
7381:
7378:
7377:
7362:
7358:
7354:
7347:
7344:
7330:
7326:
7322:
7315:
7312:
7298:
7291:
7288:
7274:
7267:
7264:
7263:
7248:
7241:
7238:
7224:
7217:
7214:
7200:
7193:
7190:
7176:
7169:
7166:
7165:
7150:
7143:
7140:
7126:
7119:
7116:
7102:
7095:
7092:
7078:
7071:
7068:
7067:
7049:
7039:
7024:
7021:
7004:
6994:
6979:
6976:
6959:
6949:
6934:
6931:
6914:
6904:
6889:
6886:
6885:
6870:
6863:
6860:
6846:
6839:
6836:
6822:
6814:
6800:
6792:
6791:
6788:
6785:
6783:
6780:
6777:
6774:
6773:
6770:
6767:
6761:
6751:
6738:
6732:
6724:
6720:
6716:
6710:
6702:
6698:
6694:
6688:
6682:
6674:
6673:superposition
6669:
6656:
6650:
6647:
6644:
6641:
6635:
6632:
6629:
6626:
6617:
6614:
6611:
6605:
6602:
6599:
6596:
6590:
6587:
6581:
6578:
6575:
6569:
6566:
6560:
6552:
6548:
6536:Full solution
6533:
6531:
6526:
6513:
6510:
6507:
6504:
6500:
6494:
6491:
6486:
6482:
6476:
6473:
6470:
6464:
6460:
6457:
6454:
6448:
6445:
6442:
6436:
6433:
6430:
6407:
6404:
6399:
6395:
6389:
6386:
6383:
6377:
6369:
6365:
6361:
6358:
6353:
6350:
6347:
6341:
6338:
6335:
6332:
6329:
6320:
6316:To solve for
6309:
6307:
6303:
6299:
6294:
6281:
6273:
6265:
6262:
6259:
6253:
6248:
6238:
6234:
6230:
6227:
6220:
6214:
6208:
6205:
6202:
6197:
6194:
6191:
6188:
6182:
6178:
6175:
6172:
6166:
6163:
6160:
6154:
6151:
6148:
6139:
6126:
6123:
6115:
6107:
6104:
6101:
6095:
6090:
6080:
6076:
6072:
6069:
6058:
6054:
6046:
6039:
6036:
6031:
6027:
6023:
6021:
6014:
6006:
6003:
6000:
5997:
5987:
5984:
5979:
5975:
5971:
5969:
5962:
5952:
5948:
5944:
5941:
5933:
5929:
5909:
5905:
5896:
5883:
5880:
5877:
5874:
5871:
5868:
5865:
5862:
5859:
5856:
5852:
5849:
5846:
5843:
5840:
5832:
5828:
5824:
5821:
5815:
5806:
5793:
5790:
5787:
5784:
5781:
5778:
5775:
5772:
5769:
5766:
5761:
5758:
5755:
5751:
5747:
5744:
5741:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5715:
5711:
5707:
5698:
5685:
5680:
5677:
5674:
5670:
5666:
5658:
5655:
5652:
5649:
5643:
5639:
5632:
5629:
5626:
5623:
5620:
5617:
5614:
5611:
5608:
5603:
5599:
5595:
5589:
5581:
5578:
5575:
5572:
5566:
5562:
5558:
5555:
5547:
5544:
5541:
5538:
5532:
5528:
5524:
5521:
5518:
5515:
5512:
5509:
5501:
5498:
5495:
5492:
5486:
5482:
5478:
5473:
5469:
5465:
5456:
5443:
5435:
5432:
5429:
5426:
5420:
5416:
5412:
5407:
5403:
5399:
5396:
5388:
5384:
5371:
5367:
5361:
5347:
5339:
5336:
5333:
5330:
5324:
5320:
5316:
5313:
5310:
5307:
5301:
5289:
5285:
5272:
5264:
5261:
5258:
5255:
5249:
5245:
5241:
5238:
5233:
5229:
5219:
5206:
5198:
5195:
5192:
5189:
5183:
5179:
5175:
5172:
5166:
5158:
5154:
5144:
5131:
5126:
5123:
5120:
5116:
5112:
5106:
5103:
5097:
5094:
5091:
5088:
5082:
5079:
5073:
5070:
5067:
5064:
5061:
5055:
5045:
5034:
5031:
5028:
5020:
5016:
5005:
5000:
4982:
4972:
4969:
4944:
4941:
4938:
4932:
4927:
4923:
4916:
4912:
4908:
4905:
4899:
4895:
4892:
4887:
4883:
4879:
4875:
4871:
4864:
4860:
4856:
4853:
4847:
4843:
4840:
4835:
4831:
4826:
4820:
4817:
4814:
4810:
4797:
4794:
4791:
4783:
4778:
4774:
4770:
4765:
4761:
4752:
4749:
4745:
4741:
4735:
4730:
4726:
4722:
4717:
4713:
4704:
4701:
4698:
4694:
4681:
4678:
4675:
4669:
4661:
4658:
4653:
4649:
4643:
4640:
4636:
4630:
4626:
4622:
4615:
4612:
4607:
4603:
4597:
4593:
4587:
4583:
4578:
4572:
4569:
4566:
4562:
4555:
4550:
4544:
4536:
4532:
4523:
4519:
4512:
4508:
4498:
4495:
4482:
4476:
4473:
4467:
4464:
4461:
4458:
4455:
4449:
4439:
4428:
4425:
4422:
4414:
4410:
4399:
4394:
4374:
4370:
4366:
4359:
4355:
4347:
4343:
4339:
4333:
4331:
4327:
4311:
4308:
4305:
4302:
4296:
4286:
4275:
4272:
4269:
4261:
4257:
4246:
4241:
4217:
4215:
4209:
4193:
4189:
4185:
4180:
4176:
4167:
4163:
4160:example, the
4157:
4155:
4151:
4147:
4142:
4128:
4108:
4100:
4095:
4089:
4079:
4065:
4062:
4057:
4053:
4043:
4027:
4021:
4017:
4014:
4011:
3989:
3985:
3960:
3956:
3952:
3949:
3946:
3939:
3935:
3931:
3926:
3922:
3913:
3908:
3905:
3900:
3887:
3884:
3881:
3877:
3869:
3864:
3860:
3856:
3851:
3847:
3841:
3836:
3832:
3828:
3825:
3819:
3815:
3812:
3809:
3806:
3798:
3794:
3789:
3773:
3763:
3759:
3755:
3750:
3745:
3741:
3730:
3726:
3722:
3717:
3712:
3707:
3703:
3698:
3694:
3690:
3686:
3679:
3674:
3670:
3648:
3642:
3639:
3636:
3633:
3627:
3624:
3618:
3613:
3609:
3605:
3599:
3595:
3589:
3583:
3577:
3557:
3548:
3534:
3512:
3508:
3487:
3465:
3461:
3452:
3448:
3443:
3441:
3437:
3433:
3429:
3425:
3421:
3417:
3401:
3379:
3375:
3354:
3348:
3345:
3339:
3336:
3331:
3327:
3321:
3318:
3313:
3310:
3305:
3300:
3296:
3292:
3286:
3276:
3263:
3259:
3255:
3252:
3249:
3241:
3237:
3226:
3221:
3187:
3165:
3161:
3156:
3152:
3143:
3134:
3132:
3128:
3124:
3123:settling time
3120:
3115:
3113:
3109:
3100:
3096:
3090:
3077:
3074:
3071:
3068:
3065:
3062:
3054:
3052:
3047:
3034:
3025:
3019:
3016:
3010:
3006:
3003:
3000:
2995:
2991:
2983:
2979:
2975:
2972:
2966:
2962:
2959:
2951:
2946:
2942:
2938:
2935:
2931:
2927:
2924:
2921:
2915:
2909:
2882:
2879:
2876:
2871:
2864:
2861:
2858:
2851:
2846:
2842:
2835:
2830:
2825:
2818:
2812:
2799:
2792:
2785:
2784:Step response
2775:
2773:
2769:
2763:
2746:
2742:
2738:
2735:
2732:
2727:
2723:
2715:
2711:
2707:
2704:
2698:
2694:
2691:
2686:
2681:
2677:
2673:
2670:
2666:
2662:
2659:
2653:
2647:
2638:
2625:
2622:
2619:
2616:
2611:
2606:
2602:
2598:
2592:
2582:
2569:
2565:
2561:
2558:
2555:
2547:
2543:
2532:
2527:
2509:
2505:
2500:
2487:
2482:
2475:
2469:
2463:
2460:
2455:
2450:
2446:
2442:
2436:
2426:
2413:
2409:
2405:
2402:
2399:
2391:
2387:
2376:
2371:
2352:
2339:
2331:
2327:
2316:
2311:
2298:
2295:
2289:
2279:
2268:
2265:
2262:
2259:
2256:
2250:
2244:
2236:
2232:
2230:
2226:
2216:
2203:
2197:
2194:
2190:
2185:
2182:
2174:
2170:
2157:
2150:energy stored
2144:
2141:
2138:
2135:
2132:
2124:
2105:
2102:
2097:
2093:
2089:
2086:
2078:
2062:
2055:
2051:
2047:
2044:
2037:
2033:
2029:
2024:
2020:
2011:
2007:
2003:
2000:
1997:
1993:
1990:
1987:
1983:
1979:
1975:
1972:
1971:
1970:
1968:
1955:
1951:
1943:
1933:
1926:
1922:
1917:
1916:Step response
1913:
1905:
1888:
1885:
1880:
1876:
1871:
1868:
1860:
1857:
1838:
1835:
1829:
1824:
1820:
1812:
1811:
1810:
1796:
1793:
1790:
1787:
1782:
1777:
1773:
1769:
1763:
1753:
1740:
1736:
1732:
1729:
1726:
1718:
1714:
1703:
1698:
1668:
1660:
1656:
1645:
1640:
1627:
1624:
1618:
1608:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1571:
1566:
1564:
1560:
1555:
1548:
1543:
1536:
1527:
1526:accelerometer
1506:
1500:
1495:
1490:
1480:
1478:
1466:
1461:
1457:
1453:
1447:
1444:
1438:
1435:
1426:
1421:
1418:
1413:
1411:
1407:
1403:
1399:
1396:
1392:
1376:
1372:
1368:
1365:
1362:
1342:
1338:
1334:
1331:
1328:
1325:
1318:
1314:
1310:
1306:
1301:
1299:
1287:
1281:
1278:
1272:
1269:
1259:
1247:
1241:
1238:
1235:
1232:
1226:
1223:
1220:
1217:
1211:
1205:
1196:
1193:Solving this
1191:
1178:
1175:
1172:
1169:
1166:
1160:
1157:
1151:
1148:
1140:
1136:
1125:
1120:
1107:
1104:
1101:
1098:
1095:
1092:
1084:
1080:
1076:
1070:
1065:
1061:
1057:
1043:
1031:
1021:
1011:
1009:
1005:
1001:
997:
993:
989:
985:
980:
978:
973:
971:
963:
959:
956:
952:
949:case, and an
948:
944:
943:
942:
940:
936:
932:
927:
925:
921:
917:
914:
910:
906:
902:
897:
895:
891:
886:
874:
865:
859:
856:
853:
844:
832:
828:
825:
822:
818:
814:
810:
798:
793:
791:
786:
784:
779:
778:
776:
775:
769:
759:
756:
751:
745:
744:
743:
742:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
704:
701:
699:
696:
694:
691:
689:
686:
684:
681:
679:
676:
674:
671:
669:
666:
664:
661:
659:
656:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
615:
608:
607:
600:
596:
592:
588:
585:
584:
580:
577:
575:
572:
570:
567:
565:
562:
558:
555:
554:
553:
550:
548:
545:
543:
540:
538:
535:
534:
530:
525:
524:
516:
513:
511:
508:
504:
501:
499:
496:
495:
494:
491:
489:
486:
484:
481:
479:
474:
471:
467:
464:
463:
459:
456:
453:
449:
445:
444:
440:
437:
435:
432:
430:
427:
425:
420:
418:
415:
413:
410:
408:
405:
404:
397:
396:
385:
382:
380:
377:
375:
372:
370:
367:
365:
362:
360:
357:
356:
354:
353:
348:
345:
344:
339:
338:
332:
331:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
298:
295:
293:
290:
288:
284:
282:
279:
277:
273:
271:
268:
265:
261:
257:
255:
252:
250:
247:
245:
242:
240:
237:
233:
230:
228:
225:
224:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
199:
192:
191:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
144:
137:
136:
130:
127:
125:
122:
120:
117:
116:
114:
113:
109:
90:
87:
77:
71:
57:
56:
53:
50:
49:
45:
44:
41:
37:
33:
19:
11764:
11729:
11708:
11688:
11664:
11633:
11603:
11560:
11556:
11550:
11525:
11521:
11515:
11503:. Retrieved
11499:the original
11488:
11481:Tipler (1998
11476:
11464:
11452:
10221:
10162:
10110:
10037:
10035:
9915:
9909:
9905:
9901:
9842:
9333:
9226:
9099:
9045:
8931:
8927:
8921:
8696:
8483:
8428:in terms of
8394:
8328:
8321:
7216:Flux linkage
6862:Flux linkage
6763:
6670:
6539:
6527:
6318:
6315:
6295:
6140:
5913:
5807:
5699:
5457:
5220:
5145:
4978:
4970:
4524:
4517:
4510:
4504:
4496:
4372:
4368:
4364:
4357:
4349:
4345:
4341:
4337:
4334:
4325:
4223:
4216:phenomenon.
4210:
4158:
4143:
4121:and damping
4091:
4044:
3909:
3901:
3549:
3451:steady state
3444:
3424:RLC circuits
3203:
3180:and damping
3130:
3126:
3122:
3118:
3116:
3111:
3098:
3094:
3091:
3055:
3050:
3048:
2797:
2790:
2788:In the case
2787:
2771:
2767:
2761:
2639:
2507:
2503:
2501:
2353:
2233:
2228:
2224:
2222:
2172:
2171:
2120:
2005:
2001:
1995:
1991:
1985:
1977:
1973:
1966:
1964:
1953:
1949:
1941:
1931:
1924:
1920:
1567:
1562:
1558:
1553:
1546:
1541:
1534:
1531:
1498:
1424:
1422:
1417:acceleration
1414:
1405:
1401:
1397:
1390:
1312:
1302:
1192:
1078:
1074:
1068:
1063:
1059:
1053:
1004:RLC circuits
981:
976:
974:
967:
964:oscillator).
957:oscillator).
938:
928:
916:oscillations
904:
900:
898:
889:
830:
827:proportional
823:
812:
806:
597: /
593: /
591:displacement
589: /
450: /
438:
412:Displacement
350:
341:
335:Formulations
322:Virtual work
262: /
202:Acceleration
195:Fundamentals
40:
11788:Oscillators
11505:27 November
11407:Normal mode
11232:Phase shift
10521:Drive force
9845:Hooke's law
7452:Conductance
7142:Capacitance
6302:RLC circuit
6141:Therefore,
4979:Apply the "
4214:instability
3049:with phase
2002:Underdamped
955:underdamped
817:equilibrium
733:von Neumann
400:Core topics
11772:Categories
11596:References
10240:Dimensions
10237:Definition
7428:Resistance
7118:Inductance
6312:Phase part
3131:undershoot
3108:relaxation
2782:See also:
2778:Step input
2770:and phase
1974:Overdamped
1309:sinusoidal
962:overdamped
913:sinusoidal
668:d'Alembert
648:Maupertuis
611:Scientists
493:Rigid body
167:Kinematics
11793:Acoustics
11585:122086250
11352:−
11315:ω
11281:−
11250:ω
11218:φ
11186:ζ
11099:−
10822:−
10778:μ
10745:−
10679:−
10611:−
10544:−
10474:−
10457:Frequency
10410:−
10289:−
10243:SI units
10069:
9939:−
9870:−
9815:ω
9789:θ
9768:τ
9742:θ
9715:ω
9711:π
9687:π
9678:τ
9647:θ
9620:θ
9571:
9559:θ
9543:θ
9505:˙
9502:θ
9473:θ
9457:θ
9437:φ
9389:φ
9363:
9342:θ
9314:θ
9280:θ
9247:θ
9244:≈
9241:θ
9238:
9206:θ
9203:
9166:θ
8991:⋅
8959:≈
8885:−
8850:−
8841:⋅
8660:−
8625:−
8616:⋅
8561:−
8552:⋅
8286:φ
8277:˙
8274:φ
8259:¨
8256:φ
8206:˙
8188:¨
8158:τ
8152:θ
8149:μ
8140:˙
8137:θ
8131:Γ
8122:¨
8119:θ
8074:˙
8056:¨
7935:μ
7917:Γ
7846:ζ
7800:π
7750:π
7714:μ
7705:π
7660:π
7609:Undamped
7536:τ
7413:Γ
7314:Elastance
7299:μ
7225:φ
7040:φ
6950:θ
6871:φ
6823:θ
6793:Position
6733:τ
6711:τ
6689:τ
6651:φ
6645:τ
6642:ω
6636:
6618:ω
6612:ζ
6606:φ
6600:τ
6597:ω
6591:
6582:ω
6576:ζ
6561:τ
6511:π
6492:−
6483:ω
6477:ω
6474:ζ
6461:
6449:ω
6443:ζ
6437:φ
6434:≡
6431:φ
6421:⟹
6405:−
6396:ω
6390:ω
6387:ζ
6366:ω
6362:−
6354:ω
6351:ζ
6342:−
6336:φ
6333:
6298:resonance
6266:ω
6263:ζ
6235:ω
6231:−
6209:ω
6206:ζ
6198:φ
6195:
6189:−
6179:
6167:ω
6161:ζ
6108:ω
6105:ζ
6077:ω
6073:−
6051:⇒
6040:φ
6037:
6004:ω
6001:ζ
5988:φ
5985:
5949:ω
5945:−
5908:Bode plot
5881:φ
5878:
5872:−
5863:ω
5860:ζ
5850:φ
5847:
5829:ω
5825:−
5791:φ
5788:
5779:−
5776:φ
5773:
5762:φ
5756:−
5736:ω
5730:ζ
5712:ω
5708:−
5681:τ
5678:ω
5659:φ
5653:τ
5650:ω
5624:ω
5618:ζ
5600:ω
5596:−
5582:φ
5576:τ
5573:ω
5548:φ
5542:τ
5539:ω
5522:ω
5516:ζ
5502:φ
5496:τ
5493:ω
5470:ω
5466:−
5436:φ
5430:τ
5427:ω
5404:ω
5400:−
5385:τ
5340:φ
5334:τ
5331:ω
5314:ω
5302:τ
5265:φ
5259:τ
5256:ω
5199:φ
5193:τ
5190:ω
5167:τ
5127:τ
5124:ω
5107:τ
5104:ω
5098:
5083:τ
5080:ω
5074:
5056:τ
5035:ζ
5017:τ
4939:ζ
4924:τ
4913:ζ
4909:−
4896:
4872:τ
4861:ζ
4857:−
4844:
4821:τ
4818:ζ
4815:−
4792:ζ
4784:τ
4753:τ
4750:−
4736:τ
4705:τ
4702:ζ
4699:−
4676:ζ
4659:−
4650:ζ
4644:τ
4641:−
4613:−
4604:ζ
4598:τ
4573:τ
4570:ζ
4567:−
4545:τ
4477:τ
4474:ω
4468:
4450:τ
4429:ζ
4411:τ
4297:τ
4276:ζ
4258:τ
4190:ω
4177:ω
4150:waveguide
4129:β
4109:ω
4012:ζ
3957:ζ
3950:−
3936:ω
3923:ω
3912:resonance
3888:π
3861:ω
3857:−
3848:ω
3842:ζ
3833:ω
3829:ω
3816:
3807:φ
3793:impedance
3760:ω
3756:−
3742:ω
3727:ω
3704:ζ
3695:ω
3643:φ
3634:ω
3628:
3619:ω
3558:φ
3535:ζ
3509:ω
3488:ω
3447:transient
3436:capacitor
3416:frequency
3402:ω
3346:ω
3340:
3297:ω
3260:ω
3256:ζ
3188:ζ
3162:ω
3153:ω
3127:overshoot
3075:ζ
3069:φ
3066:
3053:given by
3026:φ
3020:
3007:φ
2992:ω
2980:ζ
2976:−
2963:
2943:ω
2939:ζ
2936:−
2928:−
2862:≥
2843:ω
2739:φ
2724:ω
2712:ζ
2708:−
2695:
2678:ω
2674:ζ
2671:−
2603:ω
2566:ω
2562:ζ
2447:ω
2410:ω
2406:ζ
2266:−
2257:−
2198:ζ
2145:×
2142:π
2103:ζ
2094:ω
2087:λ
2052:ζ
2048:−
2034:ω
2021:ω
1869:ζ
1821:ω
1774:ω
1737:ω
1733:ζ
1595:−
1586:−
1343:ω
1335:π
1270:ω
1242:φ
1233:ω
1227:
1170:−
1161:¨
984:pendulums
951:amplitude
924:frequency
920:amplitude
869:→
857:−
848:→
713:Liouville
595:frequency
515:Vibration
232:potential
157:Continuum
152:Celestial
129:Textbooks
11662:(1972),
11626:93085193
11422:Q factor
11380:See also
9120:, where
9079:Examples
8978:″
8821:″
8697:Because
8596:″
8532:′
7168:Momentum
6888:Velocity
4362:, where
4356:) = cos(
4348:) = cos(
4344:) = cos(
4146:varactor
3432:inductor
3428:resistor
3422:-driven
2123:Q factor
1557:, where
1410:velocity
1305:periodic
998:. Other
947:undamped
935:velocity
894:constant
768:Category
693:Hamilton
678:Lagrange
673:Clairaut
638:Horrocks
599:velocity
569:Pendulum
557:reactive
529:Rotation
498:dynamics
448:Inertial
434:Friction
317:Velocity
292:Momentum
172:Kinetics
162:Dynamics
140:Branches
124:Timeline
11565:Bibcode
11530:Bibcode
10843:Nm/rad
7584:Current
7560:Voltage
7380:Damping
7023:Voltage
6978:Current
3902:is the
2800:(0) = 0
1938:√
1489:Damping
992:springs
931:damping
728:Koopman
688:Poisson
683:Laplace
628:Huygens
623:Galileo
468: (
407:Damping
260:Inertia
254:Impulse
227:kinetic
177:Statics
147:Applied
119:History
11738:
11715:
11696:
11676:
11648:
11624:
11614:
11583:
11417:Phasor
11373:rad/s
11302:rad/s
10431:N·s/m
10234:Symbol
9900:where
9667:period
9611:where
9409:where
7527:torque
7525:Drive
7491:Drive
7240:Charge
6838:Charge
6815:Angle
6458:arctan
6428:
6425:
6417:
6414:
3813:arctan
3799:, and
3661:where
3367:where
3110:, and
2793:< 1
1809:where
1317:period
1261:where
1056:damped
1008:clocks
994:, and
986:(with
888:where
766:
718:Appell
703:Cauchy
698:Jacobi
643:Halley
633:Newton
618:Kepler
470:linear
466:Motion
312:Torque
287:Moment
222:Energy
212:Couple
34:, see
11798:Sound
11758:from
11670:Wiley
11581:S2CID
11444:Notes
9144:, is
7493:force
4166:laser
4099:swing
3904:phase
3097:= 1/(
1952:= 1/(
1395:phase
723:Gibbs
708:Routh
663:Euler
302:Speed
297:Space
239:Force
11736:ISBN
11713:ISBN
11694:ISBN
11674:ISBN
11646:ISBN
11622:LCCN
11612:ISBN
11507:2011
11238:rad
11015:Time
10875:Mass
10766:N/m
10632:m/s
10310:m/s
10167:and
9491:and
9429:and
8922:The
7070:Mass
4942:<
4679:>
4516:and
4015:<
2880:<
2121:The
2075:the
811:, a
307:Time
270:Mass
11638:doi
11573:doi
11538:doi
10903:kg
10495:Hz
10066:cos
9568:cos
9360:cos
9235:sin
9200:sin
6633:cos
6588:cos
6330:tan
6192:sin
6176:sgn
6028:sin
5976:cos
5875:sin
5844:cos
5785:sin
5770:cos
5095:sin
5071:cos
4893:sin
4841:cos
4465:cos
4208:).
4154:YAG
3795:or
3625:sin
3337:sin
3063:cos
3017:sin
2960:sin
2764:≤ 1
2692:sin
2231:).
1552:= −
1427:is
1224:sin
1071:= 0
899:If
807:In
11774::
11672:,
11644:.
11620:,
11610:,
11579:.
11571:.
11561:31
11559:.
11536:.
11526:64
11524:.
11206:—
11174:m
11120:J
11043:s
10989:s
10935:—
10699:—
10565:N
10364:m
10155:.
10033:.
9320:0.
9212:0.
9092:A
7858::
7639::
6127:1.
4369:ωt
4367:=
4358:ωτ
4350:ωt
4346:ωt
4332:.
4141:.
4092:A
3570::
3547:.
3442:.
3420:AC
3099:ζω
2802::
1954:ζω
1930:=
1923:=
1565:.
1554:cv
1412:.
979:.
972:.
911::
896:.
833::
11744:.
11721:.
11702:.
11654:.
11640::
11587:.
11575::
11567::
11544:.
11540::
11532::
11509:.
11355:1
11348:T
11319:0
11284:1
11277:T
11235:—
11203:—
11158:L
11132:x
11102:2
11095:T
11089:2
11085:L
11081:M
11055:U
11027:T
11001:t
10973:T
10947:T
10932:—
10915:Q
10887:M
10861:M
10858:,
10855:m
10825:2
10818:T
10812:2
10808:L
10804:M
10748:2
10741:T
10737:M
10711:k
10696:—
10682:1
10676:=
10671:2
10667:i
10644:i
10614:2
10607:T
10603:L
10577:g
10547:2
10540:T
10536:L
10533:M
10507:F
10477:1
10470:T
10443:f
10413:1
10406:T
10402:M
10376:c
10348:L
10322:A
10292:2
10285:T
10281:L
10255:a
10208:.
10203:2
10199:x
10195:k
10190:2
10187:1
10182:=
10179:U
10143:m
10119:m
10097:.
10093:)
10089:t
10083:m
10080:k
10073:(
10063:A
10060:=
10057:)
10054:t
10051:(
10048:x
10038:A
10017:,
10014:a
10011:m
10008:=
10005:)
10002:t
9999:(
9996:x
9988:2
9984:t
9979:d
9972:2
9967:d
9960:m
9957:=
9954:)
9951:t
9948:(
9945:x
9942:k
9936:=
9933:)
9930:t
9927:(
9924:F
9910:x
9906:k
9902:F
9888:,
9885:)
9882:t
9879:(
9876:x
9873:k
9867:=
9864:)
9861:t
9858:(
9855:F
9793:0
9746:0
9720:,
9708:2
9702:=
9696:g
9693:l
9684:2
9681:=
9651:0
9624:0
9599:,
9595:)
9591:t
9585:l
9582:g
9575:(
9563:0
9555:=
9552:)
9549:t
9546:(
9523:0
9520:=
9517:)
9514:0
9511:(
9477:0
9469:=
9466:)
9463:0
9460:(
9417:A
9397:,
9393:)
9386:+
9383:t
9377:l
9374:g
9367:(
9357:A
9354:=
9351:)
9348:t
9345:(
9317:=
9309:l
9306:g
9301:+
9293:2
9289:t
9285:d
9275:2
9271:d
9209:=
9195:l
9192:g
9187:+
9179:2
9175:t
9171:d
9161:2
9157:d
9128:g
9108:l
9063:)
9060:x
9057:(
9054:V
9032:.
9027:2
9023:x
9019:k
9013:2
9010:1
9004:=
8999:2
8995:x
8988:)
8985:0
8982:(
8975:V
8968:2
8965:1
8956:)
8953:x
8950:(
8947:V
8937:)
8935:0
8932:x
8930:(
8928:V
8908:.
8903:3
8899:)
8893:0
8889:x
8882:x
8879:(
8876:O
8873:+
8868:2
8864:)
8858:0
8854:x
8847:x
8844:(
8838:)
8833:0
8829:x
8825:(
8818:V
8811:2
8808:1
8802:+
8799:)
8794:0
8790:x
8786:(
8783:V
8780:=
8777:)
8774:x
8771:(
8768:V
8746:0
8742:x
8721:)
8716:0
8712:x
8708:(
8705:V
8683:.
8678:3
8674:)
8668:0
8664:x
8657:x
8654:(
8651:O
8648:+
8643:2
8639:)
8633:0
8629:x
8622:x
8619:(
8613:)
8608:0
8604:x
8600:(
8593:V
8586:2
8583:1
8577:+
8574:)
8569:0
8565:x
8558:x
8555:(
8549:)
8544:0
8540:x
8536:(
8529:V
8525:+
8522:)
8517:0
8513:x
8509:(
8506:V
8503:=
8500:)
8497:x
8494:(
8491:V
8467:0
8463:x
8459:=
8456:x
8436:x
8412:)
8409:x
8406:(
8403:V
8381:.
8376:2
8372:x
8368:k
8362:2
8359:1
8353:=
8350:)
8347:x
8344:(
8341:V
8300:i
8297:=
8294:L
8290:/
8283:+
8268:G
8265:+
8250:C
8229:v
8226:=
8223:C
8219:/
8215:q
8212:+
8203:q
8197:R
8194:+
8185:q
8179:L
8155:=
8146:+
8128:+
8113:I
8092:F
8089:=
8086:x
8083:k
8080:+
8071:x
8065:c
8062:+
8053:x
8047:m
8016:C
8013:L
8005:2
8002:G
7976:L
7973:C
7965:2
7962:R
7932:I
7928:1
7920:2
7890:m
7887:k
7883:1
7875:2
7872:c
7816:C
7813:L
7809:1
7797:2
7793:1
7766:C
7763:L
7759:1
7747:2
7743:1
7717:I
7702:2
7698:1
7672:m
7669:k
7657:2
7653:1
7625:n
7621:f
7593:i
7569:v
7545:)
7542:t
7539:(
7511:)
7508:t
7505:(
7502:F
7475:R
7471:/
7467:1
7464:=
7461:G
7437:R
7389:c
7363:L
7359:/
7355:1
7331:C
7327:/
7323:1
7275:k
7249:q
7201:L
7177:p
7151:C
7127:L
7103:I
7079:m
7050:t
7046:d
7036:d
7005:t
7001:d
6995:q
6991:d
6960:t
6956:d
6946:d
6915:t
6911:d
6905:x
6901:d
6847:q
6801:x
6739:.
6736:)
6730:(
6725:s
6721:q
6717:+
6714:)
6708:(
6703:t
6699:q
6695:=
6692:)
6686:(
6683:q
6657:.
6654:)
6648:+
6639:(
6630:A
6627:=
6624:)
6621:)
6615:,
6609:(
6603:+
6594:(
6585:)
6579:,
6573:(
6570:A
6567:=
6564:)
6558:(
6553:s
6549:q
6514:.
6508:n
6505:+
6501:)
6495:1
6487:2
6471:2
6465:(
6455:=
6452:)
6446:,
6440:(
6408:1
6400:2
6384:2
6378:=
6370:2
6359:1
6348:2
6339:=
6319:φ
6282:.
6274:2
6270:)
6260:2
6257:(
6254:+
6249:2
6245:)
6239:2
6228:1
6225:(
6221:1
6215:)
6203:2
6183:(
6173:=
6170:)
6164:,
6158:(
6155:A
6152:=
6149:A
6124:=
6121:]
6116:2
6112:)
6102:2
6099:(
6096:+
6091:2
6087:)
6081:2
6070:1
6067:(
6064:[
6059:2
6055:A
6047:}
6032:2
6024:=
6015:2
6011:)
6007:A
5998:2
5995:(
5980:2
5972:=
5963:2
5959:)
5953:2
5942:1
5939:(
5934:2
5930:A
5884:.
5869:=
5866:A
5857:2
5853:,
5841:=
5838:)
5833:2
5822:1
5819:(
5816:A
5794:.
5782:i
5767:=
5759:i
5752:e
5748:=
5745:A
5742:+
5739:A
5733:i
5727:2
5724:+
5721:A
5716:2
5686:.
5675:i
5671:e
5667:=
5662:)
5656:+
5647:(
5644:i
5640:e
5636:)
5633:A
5630:+
5627:A
5621:i
5615:2
5612:+
5609:A
5604:2
5593:(
5590:=
5585:)
5579:+
5570:(
5567:i
5563:e
5559:A
5556:+
5551:)
5545:+
5536:(
5533:i
5529:e
5525:A
5519:i
5513:2
5510:+
5505:)
5499:+
5490:(
5487:i
5483:e
5479:A
5474:2
5444:.
5439:)
5433:+
5424:(
5421:i
5417:e
5413:A
5408:2
5397:=
5389:2
5380:d
5372:s
5368:q
5362:2
5357:d
5348:,
5343:)
5337:+
5328:(
5325:i
5321:e
5317:A
5311:i
5308:=
5298:d
5290:s
5286:q
5281:d
5273:,
5268:)
5262:+
5253:(
5250:i
5246:e
5242:A
5239:=
5234:s
5230:q
5207:.
5202:)
5196:+
5187:(
5184:i
5180:e
5176:A
5173:=
5170:)
5164:(
5159:s
5155:q
5132:.
5121:i
5117:e
5113:=
5110:)
5101:(
5092:i
5089:+
5086:)
5077:(
5068:=
5065:q
5062:+
5052:d
5046:q
5042:d
5032:2
5029:+
5021:2
5012:d
5006:q
5001:2
4996:d
4945:1
4933:]
4928:)
4917:2
4906:1
4900:(
4888:2
4884:c
4880:+
4876:)
4865:2
4854:1
4848:(
4836:1
4832:c
4827:[
4811:e
4798:1
4795:=
4787:)
4779:2
4775:c
4771:+
4766:1
4762:c
4758:(
4746:e
4742:=
4739:)
4731:2
4727:c
4723:+
4718:1
4714:c
4710:(
4695:e
4682:1
4670:)
4662:1
4654:2
4637:e
4631:2
4627:c
4623:+
4616:1
4608:2
4594:e
4588:1
4584:c
4579:(
4563:e
4556:{
4551:=
4548:)
4542:(
4537:t
4533:q
4521:2
4518:c
4514:1
4511:c
4483:.
4480:)
4471:(
4462:=
4459:q
4456:+
4446:d
4440:q
4436:d
4426:2
4423:+
4415:2
4406:d
4400:q
4395:2
4390:d
4373:c
4365:ω
4360:)
4354:τ
4352:c
4342:t
4340:(
4338:f
4312:0
4309:=
4306:q
4303:+
4293:d
4287:q
4283:d
4273:2
4270:+
4262:2
4253:d
4247:q
4242:2
4237:d
4194:i
4186:,
4181:s
4152:/
4066:0
4063:=
4058:0
4054:F
4028:2
4022:/
4018:1
3990:0
3986:F
3961:2
3953:2
3947:1
3940:0
3932:=
3927:r
3885:n
3882:+
3878:)
3870:2
3865:0
3852:2
3837:0
3826:2
3820:(
3810:=
3774:2
3770:)
3764:2
3751:2
3746:0
3738:(
3731:2
3723:1
3718:+
3713:2
3708:)
3699:0
3691:2
3687:(
3680:=
3675:m
3671:Z
3649:,
3646:)
3640:+
3637:t
3631:(
3614:m
3610:Z
3606:m
3600:0
3596:F
3590:=
3587:)
3584:t
3581:(
3578:x
3513:0
3466:0
3462:F
3434:–
3430:–
3426:(
3380:0
3376:F
3355:,
3352:)
3349:t
3343:(
3332:0
3328:F
3322:m
3319:1
3314:=
3311:x
3306:2
3301:0
3293:+
3287:t
3283:d
3277:x
3273:d
3264:0
3253:2
3250:+
3242:2
3238:t
3233:d
3227:x
3222:2
3217:d
3166:0
3157:/
3119:τ
3112:τ
3104:)
3102:0
3095:τ
3078:.
3072:=
3051:φ
3035:,
3029:)
3023:(
3011:)
3004:+
3001:t
2996:0
2984:2
2973:1
2967:(
2952:t
2947:0
2932:e
2925:1
2922:=
2919:)
2916:t
2913:(
2910:x
2883:0
2877:t
2872:0
2865:0
2859:t
2852:2
2847:0
2836:{
2831:=
2826:m
2822:)
2819:t
2816:(
2813:F
2798:x
2791:ζ
2772:φ
2768:A
2762:ζ
2747:,
2743:)
2736:+
2733:t
2728:0
2716:2
2705:1
2699:(
2687:t
2682:0
2667:e
2663:A
2660:=
2657:)
2654:t
2651:(
2648:z
2626:,
2623:0
2620:=
2617:z
2612:2
2607:0
2599:+
2593:t
2589:d
2583:z
2579:d
2570:0
2559:2
2556:+
2548:2
2544:t
2539:d
2533:z
2528:2
2523:d
2508:t
2506:(
2504:z
2488:.
2483:m
2479:)
2476:t
2473:(
2470:F
2464:=
2461:x
2456:2
2451:0
2443:+
2437:t
2433:d
2427:x
2423:d
2414:0
2403:2
2400:+
2392:2
2388:t
2383:d
2377:x
2372:2
2367:d
2340:.
2332:2
2328:t
2323:d
2317:x
2312:2
2307:d
2299:m
2296:=
2290:t
2286:d
2280:x
2276:d
2269:c
2263:x
2260:k
2254:)
2251:t
2248:(
2245:F
2229:t
2227:(
2225:F
2204:.
2195:2
2191:1
2186:=
2183:Q
2173:Q
2158:.
2139:2
2136:=
2133:Q
2106:.
2098:0
2090:=
2063:,
2056:2
2045:1
2038:0
2030:=
2025:1
2006:ζ
2004:(
1996:ζ
1994:(
1986:ζ
1978:ζ
1976:(
1967:ζ
1961:.
1959:)
1957:0
1950:τ
1942:ζ
1935:0
1932:ω
1928:1
1925:ω
1921:μ
1889:k
1886:m
1881:2
1877:c
1872:=
1839:m
1836:k
1830:=
1825:0
1797:,
1794:0
1791:=
1788:x
1783:2
1778:0
1770:+
1764:t
1760:d
1754:x
1750:d
1741:0
1730:2
1727:+
1719:2
1715:t
1710:d
1704:x
1699:2
1694:d
1669:,
1661:2
1657:t
1652:d
1646:x
1641:2
1636:d
1628:m
1625:=
1619:t
1615:d
1609:x
1605:d
1598:c
1592:x
1589:k
1583:=
1580:F
1559:c
1550:f
1547:F
1542:v
1538:f
1535:F
1499:ζ
1467:.
1462:2
1458:x
1454:k
1448:2
1445:1
1439:=
1436:U
1425:x
1406:k
1402:m
1398:φ
1391:t
1377:T
1373:/
1369:1
1366:=
1363:f
1339:/
1332:2
1329:=
1326:T
1313:A
1288:.
1282:m
1279:k
1273:=
1248:,
1245:)
1239:+
1236:t
1230:(
1221:A
1218:=
1215:)
1212:t
1209:(
1206:x
1179:.
1176:x
1173:k
1167:=
1158:x
1152:m
1149:=
1141:2
1137:t
1132:d
1126:x
1121:2
1116:d
1108:m
1105:=
1102:a
1099:m
1096:=
1093:F
1079:k
1075:x
1069:x
1064:F
1060:m
901:F
890:k
875:,
866:x
860:k
854:=
845:F
831:x
824:F
796:e
789:t
782:v
472:)
91:t
88:d
82:p
78:d
72:=
67:F
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.