2485:
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1034:
890:. A mass suspended from a spring, for example, might, if pulled and released, bounce up and down. On each bounce, the system tends to return to its equilibrium position, but overshoots it. Sometimes losses (e.g. frictional) damp the system and can cause the oscillations to gradually decay in amplitude towards zero or
1138:
faster than it is being supplied. A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sine wave. A given sinusoidal waveform may be of intermediate phase, having both sine and cosine components. The
3634:
When an object is falling through the air, the only force opposing its freefall is air resistance. An object falling through water or oil would slow down at a greater rate, until eventually reaching a steady-state velocity as the drag force comes into equilibrium with the force from gravity. This is
1142:
The most common form of damping, which is usually assumed, is the form found in linear systems. This form is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve. That is, when you connect the maximum point of each successive curve, the result
3619:
2725:. Situations where overdamping is practical tend to have tragic outcomes if overshooting occurs, usually electrical rather than mechanical. For example, landing a plane in autopilot: if the system overshoots and releases landing gear too late, the outcome would be a disaster.
3268:
1780:. In general, systems with higher damping ratios (one or greater) will demonstrate more of a damping effect. Underdamped systems have a value of less than one. Critically damped systems have a damping ratio of exactly 1, or at least very close to it.
1855:
3497:
2315:
3816:
lean and steer perturbations die away in a seemingly damped fashion. However, the system has no true damping and conserves energy. The energy in the lean and steer oscillations is transferred to the forward speed rather than being
3043:
2472:
1006:
Commonly, the mass tends to overshoot its starting position, and then return, overshooting again. With each overshoot, some energy in the system is dissipated, and the oscillations die towards zero. This case is called
1684:
3685:, which changes viscosity when subjected to a magnetic field. In this case, Magnetorheological damping may be considered an interdisciplinary form of damping with both viscous and magnetic damping mechanisms.
2846:
2644:
3512:
1956:
2079:
1216:
87:
3671:
directly opposing the oscillating movement, creating a resistive force. In other words, the resistance caused by magnetic forces slows a system down. An example of this concept being applied is the
111:
4122:
Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver; Dugard, Luc (2010-01-01), Savaresi, Sergio M.; Poussot-Vassal, Charles; Spelta, Cristiano; Sename, Oliver (eds.),
1013:
Between the overdamped and underdamped cases, there exists a certain level of damping at which the system will just fail to overshoot and will not make a single oscillation. This case is called
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2127:
3124:
2195:
1620:
1101:
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2006:
1409:
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1535:
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988:
Where the spring–mass system is completely lossless, the mass would oscillate indefinitely, with each bounce of equal height to the last. This hypothetical case is called
2875:
2719:
1804:
3780:
J. P. Meijaard; J. M. Papadopoulos; A. Ruina & A. L. Schwab (2007). "Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review".
1783:
The damping ratio provides a mathematical means of expressing the level of damping in a system relative to critical damping. For a damped harmonic oscillator with mass
2906:
2755:
2520:
1322:
1296:
3422:
2972:
2926:
1351:
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1439:
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886:
measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of
1248:
2320:
This equation is more general than just the mass–spring system, and also applies to electrical circuits and to other domains. It can be solved with the approach
2567:, as expected. This case is extremely rare in the natural world with the closest examples being cases where friction was purposefully reduced to minimal values.
1274:
777:
2488:
Phase portrait of damped oscillator, with increasing damping strength. It starts at undamped, proceeds to underdamped, then critically damped, then overdamped.
2932:
of the oscillations. A lower damping ratio implies a lower decay rate, and so very underdamped systems oscillate for long times. For example, a high quality
3651:(AC) use resistors to damp the electric current, since they are periodic. Dimmer switches or volume knobs are examples of damping in an electrical system.
1017:. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.
366:
4194:
2980:
828:. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include
485:
2761:. This turns out to be a desirable outcome in many cases where engineering design of a damped oscillator is required (e.g., a door closing mechanism).
2484:
4204:
3954:
458:
2399:
1146:
732:
4143:
4106:
3938:
3873:
1625:
848:
2577:
is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like
2481:
satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes:
3614:{\displaystyle \zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}}
2790:
972:. The physical quantity that is oscillating varies greatly, and could be the swaying of a tall building in the wind, or the speed of an
440:
2580:
1798:, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:
3782:
2936:, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer.
1866:
1769:
791:
4045:
3993:
3852:
770:
2695:
is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for
2017:
1027:
3663:
which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component of
1122:
case of damped second-order systems, or underdamped second-order differential equations. Damped sine waves are commonly seen in
864:
106:
43:
4209:
4174:
4061:
101:
361:
1143:
resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as:
356:
4214:
3048:
2090:
763:
750:
511:
434:
199:
4199:
430:
231:
3263:{\displaystyle \delta =\ln {\frac {x_{1}}{x_{3}}}=\ln {\frac {x_{2}}{x_{4}}}=\ln {\frac {x_{1}-x_{2}}{x_{3}-x_{4}}}}
650:
539:
465:
325:
258:
3678:
3664:
1487:
404:
976:, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.
3830:
3682:
2151:
680:
524:
20:
1591:
1040:
859:. Damping not based on energy loss can be important in other oscillating systems such as those that occur in
3405:(PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the
894:. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next.
670:
630:
394:
795:
984:
Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds.
969:
965:
961:
868:
852:
675:
492:
4123:
2649:
2525:
1968:
1497:
is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to
952:
The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include
875:, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.
2974:. The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks:
2952:
999:
fluid, the mass could slowly return to its rest position without ever overshooting. This case is called
685:
660:
346:
164:
1371:
1454:
3791:
1500:
957:
705:
700:
665:
573:
569:
561:
551:
341:
334:
90:
1540:
3648:
2326:
2146:
1850:{\displaystyle \zeta ={\frac {c}{c_{c}}}={\frac {\text{actual damping}}{\text{critical damping}}},}
1777:
1695:
1131:
995:
If the system contained high losses, for example if the spring–mass experiment were conducted in a
953:
480:
421:
399:
144:
139:
134:
34:
3807:
3672:
3394:
2854:
2698:
2522:
corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like
1765:
1588:
is a non-dimensional characterization of the decay rate relative to the frequency, approximately
887:
856:
821:
610:
351:
226:
194:
154:
3492:{\displaystyle \mathrm {PO} =100\exp \left({-{\frac {\zeta \pi }{\sqrt {1-\zeta ^{2}}}}}\right)}
2310:{\displaystyle {\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{n}{\frac {dx}{dt}}+\omega _{n}^{2}x=0.}
4139:
4102:
4087:"Quasi-Steady Herschel-Bulkley Analysis of Electro- and Magneto-Rheological Flow Mode Dampers"
4041:
3989:
3934:
3869:
3848:
2929:
2888:
2878:
2734:
2499:
2130:
1354:
1307:
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577:
534:
529:
470:
246:
236:
129:
3928:
2957:
2911:
1336:
4131:
4094:
3799:
3762:
3714:
2137:
The damping ratio is dimensionless, being the ratio of two coefficients of identical units.
1571:
1414:
715:
695:
640:
635:
581:
556:
411:
269:
214:
189:
3718:
3357:
3330:
3303:
3276:
1139:
term "damped sine wave" describes all such damped waveforms, whatever their initial phase.
3706:
1792:
1748:
1224:
829:
809:
710:
655:
605:
600:
519:
3930:
Process control engineering: a textbook for chemical, mechanical and electrical engineers
3795:
4135:
3636:
3390:
2386:
1773:
1259:
973:
836:
737:
645:
546:
263:
209:
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4188:
3890:
3668:
3398:
1448:
883:
625:
452:
16:
Influence on an oscillating physical system which reduces or prevents its oscillation
3811:
3741:, which is the term used in the study of vibration to denote a dissipation of energy
3038:{\displaystyle \zeta ={\frac {\delta }{\sqrt {\delta ^{2}+\left(2\pi \right)^{2}}}}}
3660:
1298:
is the decay rate, in the reciprocal of the time units of the independent variable
690:
615:
304:
184:
2757:
is the border between the overdamped and underdamped cases, and is referred to as
4086:
2933:
1127:
891:
825:
4098:
2944:
475:
149:
4163:
4012:
2467:{\displaystyle s=-\omega _{n}\left(\zeta \pm i{\sqrt {1-\zeta ^{2}}}\right).}
3968:
2882:
1494:
1411:, the number of cycles per time unit. It is expressed in inverse time units
1365:
1115:
844:
833:
497:
3803:
3753:
Crandall, S. H. (January 1970). "The role of damping in vibration theory".
1735:
is another non-dimensional characterization of the amount of damping; high
790:
3659:
Kinetic energy that causes oscillations is dissipated as heat by electric
1033:
19:
This article is about damping in oscillatory systems. For other uses, see
2774:
1689:
1118:
whose amplitude approaches zero as time increases. It corresponds to the
872:
840:
416:
299:
274:
2197:
and the definition of the damping ratio above, we can rewrite this as:
1679:{\displaystyle \zeta =\lambda /{\sqrt {\lambda ^{2}+\omega ^{2}}}<1}
1123:
996:
860:
242:
159:
3983:
3639:, which for example is applied in automatic doors or anti-slam doors.
4038:
Principles of
Electrical, Electronics and Instrumentation Engineering
2951:
For underdamped vibrations, the damping ratio is also related to the
1135:
817:
448:
294:
204:
3397:
refers to an output exceeding its final, steady-state value. For a
2841:{\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{n}}.}
2943:
2639:{\textstyle \exp \left(i\omega _{n}{\sqrt {1-\zeta ^{2}}}t\right)}
2483:
1442:
789:
284:
279:
221:
3889:
Rahman, J.; Mushtaq, M.; Ali, A.; Anjam, Y.N; Nazir, S. (2014).
1951:{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+c{\frac {dx}{dt}}+kx=0}
902:
289:
252:
4091:
Electro-Rheological Fluids and
Magneto-Rheological Suspensions
1752:
The effect of varying damping ratio on a second-order system.
1037:
Plot of a damped sinusoidal wave represented as the function
4124:"CHAPTER 2 - Semi-Active Suspension Technologies and Models"
2074:{\displaystyle c_{c}=2m{\sqrt {\frac {k}{m}}}=2m\omega _{n}}
3412:
The percentage overshoot (PO) is related to damping ratio (
1211:{\displaystyle y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )}
3409:
is just the maximum value of the step response minus one.
1485:, the time for the amplitude to decrease by the factor of
3891:"Modelling damped mass spring system in MATHLAB Simulink"
82:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
3955:"Dynamics and Vibrations: Notes: Free Damped Vibrations"
3506:) that yields a given percentage overshoot is given by:
3988:(Eighth ed.). NY: Wiley. p. §7.3 p. 236–237.
2877:(that is, when the system is underdamped), it has two
2583:
2154:
1962:
and the corresponding critical damping coefficient is
3837:. OpenStax – via University of Central Florida.
3515:
3425:
3360:
3333:
3306:
3279:
3127:
3051:
2983:
2960:
2914:
2891:
2857:
2793:
2784:, and exponential decay rate α are related such that
2737:
2701:
2652:
2528:
2502:
2402:
2329:
2206:
2093:
2020:
1971:
1869:
1807:
1698:
1628:
1594:
1574:
1543:
1503:
1457:
1417:
1374:
1339:
1310:
1284:
1262:
1227:
1149:
1043:
46:
3847:
Douglas C. Giancoli (2000). . Prentice Hall. p. 387
3327:
are amplitudes of two successive positive peaks and
897:
The damping ratio is a system parameter, denoted by
1739:
indicates slow damping relative to the oscillation.
4128:Semi-Active Suspension Control Design for Vehicles
3895:Journal of Faculty of Engineering & Technology
3613:
3491:
3373:
3346:
3319:
3292:
3262:
3091:{\displaystyle \delta =\ln {\frac {x_{0}}{x_{1}}}}
3090:
3037:
2966:
2920:
2900:
2869:
2840:
2749:
2713:
2673:
2638:
2559:
2514:
2466:
2366:
2309:
2189:
2122:{\displaystyle \omega _{n}={\sqrt {\frac {k}{m}}}}
2121:
2073:
2000:
1950:
1849:
1727:
1678:
1614:
1580:
1557:
1529:
1477:
1433:
1403:
1345:
1316:
1290:
1268:
1242:
1210:
1095:
81:
4130:, Boston: Butterworth-Heinemann, pp. 15–39,
3381:are amplitudes of two successive negative peaks.
4085:LEE, DUG-YOUNG; WERELEY, NORMAN M. (June 2000).
3655:Magnetic damping and Magnetorheological damping
1772:. It is particularly important in the study of
4062:"Eddy Currents and Magnetic Damping | Physics"
4013:"damping | Definition, Types, & Examples"
3982:Kuo, Benjamin C & Golnaraghi M F (2003).
771:
8:
3866:Introduction to Mechatronics and Measurement
3115:are amplitudes of any two successive peaks.
1764:(Greek letter zeta), that characterizes the
855:, and absorption and scattering of light in
3829:Urone, Paul Peter; Hinrichs, Roger (2016).
1770:second-order ordinary differential equation
2477:Two such solutions, for the two values of
2190:{\textstyle \omega _{n}={\sqrt {{k}/{m}}}}
778:
764:
25:
3591:
3589:
3576:
3563:
3540:
3538:
3522:
3514:
3475:
3454:
3450:
3426:
3424:
3365:
3359:
3338:
3332:
3311:
3305:
3284:
3278:
3251:
3238:
3226:
3213:
3206:
3189:
3179:
3173:
3156:
3146:
3140:
3126:
3080:
3070:
3064:
3050:
3026:
3000:
2990:
2982:
2959:
2913:
2890:
2856:
2827:
2818:
2800:
2792:
2736:
2700:
2651:
2620:
2608:
2602:
2582:
2545:
2527:
2501:
2448:
2436:
2416:
2401:
2352:
2328:
2292:
2287:
2260:
2254:
2232:
2214:
2207:
2205:
2180:
2175:
2170:
2168:
2159:
2153:
2107:
2098:
2092:
2065:
2040:
2025:
2019:
1988:
1976:
1970:
1913:
1898:
1880:
1873:
1868:
1860:where the system's equation of motion is
1834:
1823:
1814:
1806:
1708:
1697:
1662:
1649:
1643:
1638:
1627:
1604:
1593:
1573:
1547:
1542:
1519:
1502:
1467:
1456:
1422:
1416:
1384:
1373:
1338:
1309:
1283:
1276:is the initial amplitude of the envelope;
1261:
1226:
1172:
1148:
1063:
1042:
63:
57:
48:
47:
45:
3734:An Introduction to Mechanical Vibrations
3707:"A Dictionary of Mechanical Engineering"
1747:
1032:
3705:Escudier, Marcel; Atkins, Tony (2019).
3694:
1615:{\displaystyle \zeta =\lambda /\omega }
1250:is the instantaneous amplitude at time
1096:{\displaystyle y(t)=e^{-t}\cos(2\pi t)}
97:
33:
871:). Damping is not to be confused with
4007:
4005:
3736:. John Wiley & Sons. p. 37.
3647:Electrical systems that operate with
460:Newton's law of universal gravitation
7:
3719:10.1093/acref/9780198832102.001.0001
3700:
3698:
2908:; that is, the decay rate parameter
1361:Other important parameters include:
2674:{\displaystyle \ 0\leq \zeta <1}
2560:{\displaystyle \exp(i\omega _{n}t)}
2001:{\displaystyle c_{c}=2{\sqrt {km}}}
1760:is a parameter, usually denoted by
441:Mechanics of planar particle motion
49:
4195:Dimensionless numbers of mechanics
4136:10.1016/b978-0-08-096678-6.00002-x
3783:Proceedings of the Royal Society A
3595:
3592:
3544:
3541:
3430:
3427:
14:
4040:. S. chand Limited. p. 338.
3927:Ming Rao and Haiming Qiu (1993).
2145:Using the natural frequency of a
1404:{\displaystyle f=\omega /(2\pi )}
1478:{\displaystyle \tau =1/\lambda }
1028:Damped wave (radio transmission)
745:
744:
731:
64:
4205:Ordinary differential equations
3502:Conversely, the damping ratio (
2851:When a second-order system has
1530:{\displaystyle \ln(2)/\lambda }
3914:Circuits, Signals, and Systems
3755:Journal of Sound and Vibration
3118:As shown in the right figure:
2554:
2535:
2339:
2333:
1776:. It is also important in the
1722:
1713:
1558:{\displaystyle 0.693/\lambda }
1516:
1510:
1398:
1389:
1237:
1231:
1205:
1190:
1159:
1153:
1090:
1078:
1053:
1047:
1:
4093:. WORLD SCIENTIFIC: 579–586.
3868:(3rd ed.). McGraw Hill.
3831:"16.7 Damped Harmonic Motion"
3767:10.1016/s0022-460x(70)80105-5
3643:Damping in electrical systems
2367:{\displaystyle x(t)=Ce^{st},}
1728:{\displaystyle Q=1/(2\zeta )}
367:Koopman–von Neumann mechanics
3864:Alciatore, David G. (2007).
435:Non-inertial reference frame
2870:{\displaystyle \zeta <1}
2714:{\displaystyle \zeta >1}
362:Appell's equation of motion
232:Inertial frame of reference
4231:
4099:10.1142/9789812793607_0066
3679:Magnetorheological Dampers
1025:
18:
4066:courses.lumenlearning.com
3985:Automatic control systems
3933:. CRC Press. p. 96.
3665:electromagnetic induction
3625:Examples and applications
3683:Magnetorheological fluid
2901:{\displaystyle -\alpha }
2750:{\displaystyle \zeta =1}
2721:, and is referred to as
2681:, and is referred to as
2646:. This case occurs for
2515:{\displaystyle \zeta =0}
1744:Damping ratio definition
1317:{\displaystyle \varphi }
1291:{\displaystyle \lambda }
1026:Not to be confused with
525:Rotating reference frame
357:Hamilton–Jacobi equation
21:Damping (disambiguation)
4168:Encyclopædia Britannica
4017:Encyclopedia Britannica
2967:{\displaystyle \delta }
2928:represents the rate of
2921:{\displaystyle \alpha }
2881:poles that each have a
1537:which is approximately
1346:{\displaystyle \omega }
905:"), that can vary from
466:Newton's laws of motion
326:Newton's laws of motion
3912:William McC. Siebert.
3804:10.1098/rspa.2007.1857
3615:
3493:
3375:
3348:
3321:
3294:
3264:
3092:
3039:
2968:
2948:
2922:
2902:
2871:
2842:
2751:
2715:
2675:
2640:
2561:
2516:
2489:
2468:
2368:
2311:
2191:
2123:
2075:
2002:
1952:
1851:
1787:, damping coefficient
1753:
1729:
1680:
1616:
1582:
1581:{\displaystyle \zeta }
1559:
1531:
1479:
1435:
1434:{\displaystyle t^{-1}}
1405:
1347:
1324:is the phase angle at
1318:
1292:
1270:
1244:
1212:
1103:
1097:
970:electrical engineering
966:structural engineering
962:mechanical engineering
869:Suspension (mechanics)
853:electronic oscillators
805:
493:Simple harmonic motion
406:Euler's laws of motion
200:D'Alembert's principle
83:
4210:Mathematical analysis
4036:Gupta, B. R. (2001).
3675:on roller coasters.
3616:
3494:
3376:
3374:{\displaystyle x_{4}}
3349:
3347:{\displaystyle x_{2}}
3322:
3320:{\displaystyle x_{3}}
3295:
3293:{\displaystyle x_{1}}
3265:
3093:
3040:
2969:
2953:logarithmic decrement
2947:
2940:Logarithmic decrement
2923:
2903:
2872:
2843:
2769:factor and decay rate
2752:
2716:
2685:(e.g., bungee cable).
2676:
2641:
2562:
2517:
2487:
2469:
2369:
2312:
2192:
2124:
2076:
2003:
1953:
1852:
1751:
1730:
1681:
1617:
1583:
1560:
1532:
1480:
1436:
1406:
1348:
1319:
1293:
1271:
1245:
1213:
1098:
1036:
793:
347:Hamiltonian mechanics
165:Statistical mechanics
84:
4173:OpenStax, College. "
3969:"Damping Evaluation"
3667:where they set up a
3513:
3423:
3403:percentage overshoot
3385:Percentage overshoot
3358:
3331:
3304:
3277:
3125:
3049:
2981:
2958:
2912:
2889:
2855:
2791:
2735:
2699:
2650:
2581:
2526:
2500:
2400:
2327:
2204:
2152:
2091:
2018:
1969:
1867:
1805:
1696:
1626:
1592:
1572:
1541:
1501:
1455:
1415:
1372:
1337:
1308:
1282:
1260:
1243:{\displaystyle y(t)}
1225:
1147:
1041:
958:chemical engineering
570:Angular acceleration
562:Rotational frequency
342:Lagrangian mechanics
335:Analytical mechanics
91:Second law of motion
44:
4215:Classical mechanics
3796:2007RSPSA.463.1955M
3790:(2084): 1955–1982.
3649:alternating current
2297:
2147:harmonic oscillator
1778:harmonic oscillator
1132:harmonic oscillator
1116:sinusoidal function
954:control engineering
857:optical oscillators
422:Harmonic oscillator
400:Equations of motion
35:Classical mechanics
29:Part of a series on
4200:Engineering ratios
3971:. 19 October 2015.
3611:
3489:
3371:
3344:
3317:
3290:
3260:
3088:
3035:
2964:
2949:
2918:
2898:
2867:
2838:
2747:
2711:
2671:
2636:
2557:
2512:
2496:Is the case where
2490:
2464:
2364:
2307:
2283:
2187:
2119:
2071:
1998:
1948:
1847:
1766:frequency response
1754:
1725:
1676:
1612:
1578:
1555:
1527:
1475:
1431:
1401:
1343:
1314:
1288:
1266:
1240:
1208:
1104:
1093:
888:static equilibrium
861:biological systems
822:oscillating system
806:
796:spring–mass system
738:Physics portal
352:Routhian mechanics
227:Frame of reference
79:
4145:978-0-08-096678-6
4108:978-981-02-4258-9
3940:978-2-88124-628-9
3875:978-0-07-296305-2
3681:(MR Dampers) use
3609:
3608:
3602:
3551:
3482:
3481:
3258:
3195:
3162:
3086:
3033:
3032:
2930:exponential decay
2879:complex conjugate
2833:
2813:
2759:critically damped
2728:Critically damped
2655:
2626:
2454:
2278:
2239:
2185:
2131:natural frequency
2117:
2116:
2050:
2049:
1996:
1931:
1905:
1842:
1841:
1838:
1829:
1668:
1355:angular frequency
1269:{\displaystyle A}
980:Oscillation cases
929:critically damped
788:
787:
535:Centrifugal force
530:Centripetal force
486:Euler's equations
471:Relative velocity
247:Moment of inertia
77:
51:
4222:
4155:
4154:
4153:
4152:
4119:
4113:
4112:
4082:
4076:
4075:
4073:
4072:
4058:
4052:
4051:
4033:
4027:
4026:
4024:
4023:
4009:
4000:
3999:
3979:
3973:
3972:
3965:
3959:
3958:
3951:
3945:
3944:
3924:
3918:
3917:
3909:
3903:
3902:
3886:
3880:
3879:
3861:
3855:
3845:
3839:
3838:
3826:
3820:
3819:
3777:
3771:
3770:
3761:(1): 3–18, IN1.
3750:
3744:
3743:
3732:Steidel (1971).
3729:
3723:
3722:
3711:Oxford Reference
3702:
3620:
3618:
3617:
3612:
3610:
3607:
3603:
3598:
3590:
3581:
3580:
3568:
3567:
3558:
3557:
3556:
3552:
3547:
3539:
3523:
3498:
3496:
3495:
3490:
3488:
3484:
3483:
3480:
3479:
3464:
3463:
3455:
3433:
3380:
3378:
3377:
3372:
3370:
3369:
3353:
3351:
3350:
3345:
3343:
3342:
3326:
3324:
3323:
3318:
3316:
3315:
3299:
3297:
3296:
3291:
3289:
3288:
3269:
3267:
3266:
3261:
3259:
3257:
3256:
3255:
3243:
3242:
3232:
3231:
3230:
3218:
3217:
3207:
3196:
3194:
3193:
3184:
3183:
3174:
3163:
3161:
3160:
3151:
3150:
3141:
3097:
3095:
3094:
3089:
3087:
3085:
3084:
3075:
3074:
3065:
3044:
3042:
3041:
3036:
3034:
3031:
3030:
3025:
3021:
3005:
3004:
2995:
2991:
2973:
2971:
2970:
2965:
2927:
2925:
2924:
2919:
2907:
2905:
2904:
2899:
2876:
2874:
2873:
2868:
2847:
2845:
2844:
2839:
2834:
2832:
2831:
2819:
2814:
2812:
2801:
2780:, damping ratio
2756:
2754:
2753:
2748:
2720:
2718:
2717:
2712:
2680:
2678:
2677:
2672:
2653:
2645:
2643:
2642:
2637:
2635:
2631:
2627:
2625:
2624:
2609:
2607:
2606:
2566:
2564:
2563:
2558:
2550:
2549:
2521:
2519:
2518:
2513:
2473:
2471:
2470:
2465:
2460:
2456:
2455:
2453:
2452:
2437:
2421:
2420:
2389:constants, with
2373:
2371:
2370:
2365:
2360:
2359:
2316:
2314:
2313:
2308:
2296:
2291:
2279:
2277:
2269:
2261:
2259:
2258:
2240:
2238:
2237:
2236:
2223:
2219:
2218:
2208:
2196:
2194:
2193:
2188:
2186:
2184:
2179:
2174:
2169:
2164:
2163:
2128:
2126:
2125:
2120:
2118:
2109:
2108:
2103:
2102:
2080:
2078:
2077:
2072:
2070:
2069:
2051:
2042:
2041:
2030:
2029:
2007:
2005:
2004:
1999:
1997:
1989:
1981:
1980:
1957:
1955:
1954:
1949:
1932:
1930:
1922:
1914:
1906:
1904:
1903:
1902:
1889:
1885:
1884:
1874:
1856:
1854:
1853:
1848:
1843:
1840:critical damping
1839:
1836:
1835:
1830:
1828:
1827:
1815:
1793:spring constant
1734:
1732:
1731:
1726:
1712:
1685:
1683:
1682:
1677:
1669:
1667:
1666:
1654:
1653:
1644:
1642:
1621:
1619:
1618:
1613:
1608:
1587:
1585:
1584:
1579:
1564:
1562:
1561:
1556:
1551:
1536:
1534:
1533:
1528:
1523:
1484:
1482:
1481:
1476:
1471:
1440:
1438:
1437:
1432:
1430:
1429:
1410:
1408:
1407:
1402:
1388:
1352:
1350:
1349:
1344:
1330:
1323:
1321:
1320:
1315:
1301:
1297:
1295:
1294:
1289:
1275:
1273:
1272:
1267:
1253:
1249:
1247:
1246:
1241:
1217:
1215:
1214:
1209:
1183:
1182:
1108:damped sine wave
1102:
1100:
1099:
1094:
1071:
1070:
1022:Damped sine wave
1015:critical damping
948:
937:
926:
915:
900:
841:surface friction
832:in a fluid (see
810:physical systems
804:
780:
773:
766:
753:
748:
747:
740:
736:
735:
641:Johann Bernoulli
636:Daniel Bernoulli
557:Tangential speed
461:
437:
412:Fictitious force
407:
259:Mechanical power
249:
190:Angular momentum
88:
86:
85:
80:
78:
76:
68:
67:
58:
53:
52:
26:
4230:
4229:
4225:
4224:
4223:
4221:
4220:
4219:
4185:
4184:
4159:
4158:
4150:
4148:
4146:
4121:
4120:
4116:
4109:
4084:
4083:
4079:
4070:
4068:
4060:
4059:
4055:
4048:
4035:
4034:
4030:
4021:
4019:
4011:
4010:
4003:
3996:
3981:
3980:
3976:
3967:
3966:
3962:
3953:
3952:
3948:
3941:
3926:
3925:
3921:
3911:
3910:
3906:
3888:
3887:
3883:
3876:
3863:
3862:
3858:
3846:
3842:
3835:College Physics
3828:
3827:
3823:
3779:
3778:
3774:
3752:
3751:
3747:
3731:
3730:
3726:
3704:
3703:
3696:
3691:
3657:
3645:
3635:the concept of
3632:
3627:
3585:
3572:
3559:
3534:
3524:
3511:
3510:
3471:
3456:
3446:
3421:
3420:
3387:
3361:
3356:
3355:
3334:
3329:
3328:
3307:
3302:
3301:
3280:
3275:
3274:
3247:
3234:
3233:
3222:
3209:
3208:
3185:
3175:
3152:
3142:
3123:
3122:
3114:
3107:
3076:
3066:
3047:
3046:
3014:
3010:
3009:
2996:
2979:
2978:
2956:
2955:
2942:
2910:
2909:
2887:
2886:
2853:
2852:
2823:
2805:
2789:
2788:
2771:
2733:
2732:
2731:The case where
2697:
2696:
2648:
2647:
2616:
2598:
2594:
2590:
2579:
2578:
2541:
2524:
2523:
2498:
2497:
2444:
2426:
2422:
2412:
2398:
2397:
2348:
2325:
2324:
2270:
2262:
2250:
2228:
2224:
2210:
2209:
2202:
2201:
2155:
2150:
2149:
2143:
2094:
2089:
2088:
2061:
2021:
2016:
2015:
1972:
1967:
1966:
1923:
1915:
1894:
1890:
1876:
1875:
1865:
1864:
1819:
1803:
1802:
1746:
1694:
1693:
1658:
1645:
1624:
1623:
1590:
1589:
1570:
1569:
1568:Damping ratio:
1539:
1538:
1499:
1498:
1453:
1452:
1418:
1413:
1412:
1370:
1369:
1335:
1334:
1325:
1306:
1305:
1299:
1280:
1279:
1258:
1257:
1251:
1223:
1222:
1168:
1145:
1144:
1112:damped sinusoid
1059:
1039:
1038:
1031:
1024:
982:
943:
932:
921:
910:
898:
830:viscous damping
816:is the loss of
799:
784:
743:
730:
729:
722:
721:
720:
595:
587:
586:
566:
520:Circular motion
514:
504:
503:
502:
459:
429:
426:
405:
384:
376:
375:
372:
371:
329:
319:
311:
310:
309:
268:
264:Mechanical work
257:
241:
179:
171:
170:
169:
124:
116:
93:
69:
59:
42:
41:
24:
17:
12:
11:
5:
4228:
4226:
4218:
4217:
4212:
4207:
4202:
4197:
4187:
4186:
4183:
4182:
4171:
4157:
4156:
4144:
4114:
4107:
4077:
4053:
4046:
4028:
4001:
3994:
3974:
3960:
3946:
3939:
3919:
3904:
3881:
3874:
3856:
3840:
3821:
3772:
3745:
3724:
3693:
3692:
3690:
3687:
3656:
3653:
3644:
3641:
3631:
3628:
3626:
3623:
3622:
3621:
3606:
3601:
3597:
3594:
3588:
3584:
3579:
3575:
3571:
3566:
3562:
3555:
3550:
3546:
3543:
3537:
3533:
3530:
3527:
3521:
3518:
3500:
3499:
3487:
3478:
3474:
3470:
3467:
3462:
3459:
3453:
3449:
3445:
3442:
3439:
3436:
3432:
3429:
3391:control theory
3386:
3383:
3368:
3364:
3341:
3337:
3314:
3310:
3287:
3283:
3271:
3270:
3254:
3250:
3246:
3241:
3237:
3229:
3225:
3221:
3216:
3212:
3205:
3202:
3199:
3192:
3188:
3182:
3178:
3172:
3169:
3166:
3159:
3155:
3149:
3145:
3139:
3136:
3133:
3130:
3112:
3105:
3099:
3098:
3083:
3079:
3073:
3069:
3063:
3060:
3057:
3054:
3029:
3024:
3020:
3017:
3013:
3008:
3003:
2999:
2994:
2989:
2986:
2963:
2941:
2938:
2917:
2897:
2894:
2866:
2863:
2860:
2849:
2848:
2837:
2830:
2826:
2822:
2817:
2811:
2808:
2804:
2799:
2796:
2770:
2764:
2763:
2762:
2746:
2743:
2740:
2729:
2726:
2710:
2707:
2704:
2689:
2686:
2670:
2667:
2664:
2661:
2658:
2634:
2630:
2623:
2619:
2615:
2612:
2605:
2601:
2597:
2593:
2589:
2586:
2571:
2568:
2556:
2553:
2548:
2544:
2540:
2537:
2534:
2531:
2511:
2508:
2505:
2494:
2475:
2474:
2463:
2459:
2451:
2447:
2443:
2440:
2435:
2432:
2429:
2425:
2419:
2415:
2411:
2408:
2405:
2375:
2374:
2363:
2358:
2355:
2351:
2347:
2344:
2341:
2338:
2335:
2332:
2318:
2317:
2306:
2303:
2300:
2295:
2290:
2286:
2282:
2276:
2273:
2268:
2265:
2257:
2253:
2249:
2246:
2243:
2235:
2231:
2227:
2222:
2217:
2213:
2183:
2178:
2173:
2167:
2162:
2158:
2142:
2139:
2135:
2134:
2133:of the system.
2115:
2112:
2106:
2101:
2097:
2082:
2081:
2068:
2064:
2060:
2057:
2054:
2048:
2045:
2039:
2036:
2033:
2028:
2024:
2009:
2008:
1995:
1992:
1987:
1984:
1979:
1975:
1960:
1959:
1947:
1944:
1941:
1938:
1935:
1929:
1926:
1921:
1918:
1912:
1909:
1901:
1897:
1893:
1888:
1883:
1879:
1872:
1858:
1857:
1846:
1837:actual damping
1833:
1826:
1822:
1818:
1813:
1810:
1774:control theory
1745:
1742:
1741:
1740:
1724:
1721:
1718:
1715:
1711:
1707:
1704:
1701:
1687:
1675:
1672:
1665:
1661:
1657:
1652:
1648:
1641:
1637:
1634:
1631:
1611:
1607:
1603:
1600:
1597:
1577:
1566:
1554:
1550:
1546:
1526:
1522:
1518:
1515:
1512:
1509:
1506:
1492:
1474:
1470:
1466:
1463:
1460:
1446:
1428:
1425:
1421:
1400:
1397:
1394:
1391:
1387:
1383:
1380:
1377:
1359:
1358:
1342:
1332:
1313:
1303:
1287:
1277:
1265:
1255:
1239:
1236:
1233:
1230:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1181:
1178:
1175:
1171:
1167:
1164:
1161:
1158:
1155:
1152:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1069:
1066:
1062:
1058:
1055:
1052:
1049:
1046:
1023:
1020:
1019:
1018:
1011:
1004:
993:
981:
978:
974:electric motor
786:
785:
783:
782:
775:
768:
760:
757:
756:
755:
754:
741:
724:
723:
719:
718:
713:
708:
703:
698:
693:
688:
683:
678:
673:
668:
663:
658:
653:
648:
643:
638:
633:
628:
623:
618:
613:
608:
603:
597:
596:
593:
592:
589:
588:
585:
584:
565:
564:
559:
554:
549:
547:Coriolis force
544:
543:
542:
532:
527:
522:
516:
515:
510:
509:
506:
505:
501:
500:
495:
490:
489:
488:
483:
473:
468:
463:
456:
445:
444:
443:
438:
425:
424:
419:
414:
409:
402:
397:
392:
386:
385:
382:
381:
378:
377:
374:
373:
370:
369:
364:
359:
354:
349:
344:
338:
332:
330:
323:
320:
317:
316:
313:
312:
308:
307:
302:
297:
292:
287:
282:
277:
272:
266:
261:
255:
250:
239:
234:
229:
224:
219:
218:
217:
212:
202:
197:
192:
187:
181:
180:
177:
176:
173:
172:
168:
167:
162:
157:
152:
147:
142:
137:
132:
126:
125:
122:
121:
118:
117:
115:
114:
109:
104:
98:
95:
94:
89:
75:
72:
66:
62:
56:
38:
37:
31:
30:
15:
13:
10:
9:
6:
4:
3:
2:
4227:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
4192:
4190:
4180:
4176:
4172:
4169:
4165:
4161:
4160:
4147:
4141:
4137:
4133:
4129:
4125:
4118:
4115:
4110:
4104:
4100:
4096:
4092:
4088:
4081:
4078:
4067:
4063:
4057:
4054:
4049:
4047:9788121901031
4043:
4039:
4032:
4029:
4018:
4014:
4008:
4006:
4002:
3997:
3995:0-471-13476-7
3991:
3987:
3986:
3978:
3975:
3970:
3964:
3961:
3956:
3950:
3947:
3942:
3936:
3932:
3931:
3923:
3920:
3915:
3908:
3905:
3900:
3896:
3892:
3885:
3882:
3877:
3871:
3867:
3860:
3857:
3854:
3853:0-13-021517-1
3850:
3844:
3841:
3836:
3832:
3825:
3822:
3818:
3813:
3809:
3805:
3801:
3797:
3793:
3789:
3785:
3784:
3776:
3773:
3768:
3764:
3760:
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3746:
3742:
3740:
3735:
3728:
3725:
3720:
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3712:
3708:
3701:
3699:
3695:
3688:
3686:
3684:
3680:
3676:
3674:
3670:
3669:magnetic flux
3666:
3662:
3661:eddy currents
3654:
3652:
3650:
3642:
3640:
3638:
3629:
3624:
3604:
3599:
3586:
3582:
3577:
3573:
3569:
3564:
3560:
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3509:
3508:
3507:
3505:
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3418:
3417:
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3408:
3404:
3400:
3396:
3392:
3384:
3382:
3366:
3362:
3339:
3335:
3312:
3308:
3285:
3281:
3252:
3248:
3244:
3239:
3235:
3227:
3223:
3219:
3214:
3210:
3203:
3200:
3197:
3190:
3186:
3180:
3176:
3170:
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3164:
3157:
3153:
3147:
3143:
3137:
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3131:
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3119:
3116:
3111:
3104:
3081:
3077:
3071:
3067:
3061:
3058:
3055:
3052:
3027:
3022:
3018:
3015:
3011:
3006:
3001:
2997:
2992:
2987:
2984:
2977:
2976:
2975:
2961:
2954:
2946:
2939:
2937:
2935:
2931:
2915:
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2892:
2884:
2880:
2864:
2861:
2858:
2835:
2828:
2824:
2820:
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2809:
2806:
2802:
2797:
2794:
2787:
2786:
2785:
2783:
2779:
2777:
2768:
2765:
2760:
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2738:
2730:
2727:
2724:
2708:
2705:
2702:
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2690:
2687:
2684:
2668:
2665:
2662:
2659:
2656:
2632:
2628:
2621:
2617:
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2595:
2591:
2587:
2584:
2576:
2572:
2569:
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2538:
2532:
2529:
2509:
2506:
2503:
2495:
2492:
2491:
2486:
2482:
2480:
2461:
2457:
2449:
2445:
2441:
2438:
2433:
2430:
2427:
2423:
2417:
2413:
2409:
2406:
2403:
2396:
2395:
2394:
2392:
2388:
2384:
2380:
2361:
2356:
2353:
2349:
2345:
2342:
2336:
2330:
2323:
2322:
2321:
2304:
2301:
2298:
2293:
2288:
2284:
2280:
2274:
2271:
2266:
2263:
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2251:
2247:
2244:
2241:
2233:
2229:
2225:
2220:
2215:
2211:
2200:
2199:
2198:
2181:
2176:
2171:
2165:
2160:
2156:
2148:
2140:
2138:
2132:
2113:
2110:
2104:
2099:
2095:
2087:
2086:
2085:
2066:
2062:
2058:
2055:
2052:
2046:
2043:
2037:
2034:
2031:
2026:
2022:
2014:
2013:
2012:
1993:
1990:
1985:
1982:
1977:
1973:
1965:
1964:
1963:
1945:
1942:
1939:
1936:
1933:
1927:
1924:
1919:
1916:
1910:
1907:
1899:
1895:
1891:
1886:
1881:
1877:
1870:
1863:
1862:
1861:
1844:
1831:
1824:
1820:
1816:
1811:
1808:
1801:
1800:
1799:
1797:
1796:
1790:
1786:
1781:
1779:
1775:
1771:
1767:
1763:
1759:
1758:damping ratio
1750:
1743:
1738:
1719:
1716:
1709:
1705:
1702:
1699:
1691:
1688:
1673:
1670:
1663:
1659:
1655:
1650:
1646:
1639:
1635:
1632:
1629:
1622:, or exactly
1609:
1605:
1601:
1598:
1595:
1575:
1567:
1552:
1548:
1544:
1524:
1520:
1513:
1507:
1504:
1496:
1493:
1490:
1489:
1472:
1468:
1464:
1461:
1458:
1450:
1449:Time constant
1447:
1444:
1426:
1423:
1419:
1395:
1392:
1385:
1381:
1378:
1375:
1367:
1364:
1363:
1362:
1356:
1340:
1333:
1328:
1311:
1304:
1285:
1278:
1263:
1256:
1234:
1228:
1221:
1220:
1219:
1202:
1199:
1196:
1193:
1187:
1184:
1179:
1176:
1173:
1169:
1165:
1162:
1156:
1150:
1140:
1137:
1133:
1130:, wherever a
1129:
1125:
1121:
1117:
1113:
1109:
1087:
1084:
1081:
1075:
1072:
1067:
1064:
1060:
1056:
1050:
1044:
1035:
1029:
1021:
1016:
1012:
1010:
1005:
1002:
998:
994:
991:
987:
986:
985:
979:
977:
975:
971:
967:
963:
959:
955:
950:
946:
941:
935:
930:
924:
919:
913:
908:
904:
895:
893:
889:
885:
884:dimensionless
881:
880:damping ratio
876:
874:
870:
866:
862:
858:
854:
850:
846:
842:
838:
835:
831:
827:
823:
819:
815:
811:
802:
797:
792:
781:
776:
774:
769:
767:
762:
761:
759:
758:
752:
742:
739:
734:
728:
727:
726:
725:
717:
714:
712:
709:
707:
704:
702:
699:
697:
694:
692:
689:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
652:
649:
647:
644:
642:
639:
637:
634:
632:
629:
627:
624:
622:
619:
617:
614:
612:
609:
607:
604:
602:
599:
598:
591:
590:
583:
579:
575:
571:
568:
567:
563:
560:
558:
555:
553:
550:
548:
545:
541:
538:
537:
536:
533:
531:
528:
526:
523:
521:
518:
517:
513:
508:
507:
499:
496:
494:
491:
487:
484:
482:
479:
478:
477:
474:
472:
469:
467:
464:
462:
457:
454:
450:
447:
446:
442:
439:
436:
432:
428:
427:
423:
420:
418:
415:
413:
410:
408:
403:
401:
398:
396:
393:
391:
388:
387:
380:
379:
368:
365:
363:
360:
358:
355:
353:
350:
348:
345:
343:
340:
339:
337:
336:
331:
328:
327:
322:
321:
315:
314:
306:
303:
301:
298:
296:
293:
291:
288:
286:
283:
281:
278:
276:
273:
271:
267:
265:
262:
260:
256:
254:
251:
248:
244:
240:
238:
235:
233:
230:
228:
225:
223:
220:
216:
213:
211:
208:
207:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
182:
175:
174:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
127:
120:
119:
113:
110:
108:
105:
103:
100:
99:
96:
92:
73:
70:
60:
54:
40:
39:
36:
32:
28:
27:
22:
4178:
4167:
4149:, retrieved
4127:
4117:
4090:
4080:
4069:. Retrieved
4065:
4056:
4037:
4031:
4020:. Retrieved
4016:
3984:
3977:
3963:
3949:
3929:
3922:
3916:. MIT Press.
3913:
3907:
3898:
3894:
3884:
3865:
3859:
3843:
3834:
3824:
3815:
3787:
3781:
3775:
3758:
3754:
3748:
3738:
3737:
3733:
3727:
3710:
3677:
3658:
3646:
3637:viscous drag
3633:
3630:Viscous drag
3503:
3501:
3413:
3411:
3406:
3402:
3388:
3272:
3117:
3109:
3102:
3100:
2950:
2850:
2781:
2775:
2772:
2766:
2758:
2722:
2692:
2682:
2574:
2478:
2476:
2390:
2382:
2378:
2376:
2319:
2144:
2136:
2083:
2010:
1961:
1859:
1794:
1788:
1784:
1782:
1761:
1757:
1755:
1736:
1486:
1360:
1326:
1141:
1119:
1111:
1107:
1105:
1014:
1009:underdamped.
1008:
1000:
989:
983:
951:
944:
939:
933:
928:
922:
917:
911:
906:
896:
879:
877:
813:
807:
800:
794:Underdamped
580: /
576: /
574:displacement
572: /
433: /
395:Displacement
389:
333:
324:
318:Formulations
305:Virtual work
245: /
185:Acceleration
178:Fundamentals
3817:dissipated.
2934:tuning fork
2683:underdamped
2570:Underdamped
2393:satisfying
1128:engineering
1120:underdamped
918:underdamped
826:dissipation
716:von Neumann
383:Core topics
4189:Categories
4151:2023-07-15
4071:2021-06-09
4022:2021-06-09
3689:References
3399:step input
2723:overdamped
2688:Overdamped
2141:Derivation
1134:is losing
1001:overdamped
940:overdamped
927:) through
849:resistance
651:d'Alembert
631:Maupertuis
594:Scientists
476:Rigid body
150:Kinematics
3583:
3561:π
3532:
3526:−
3517:ζ
3473:ζ
3469:−
3461:π
3458:ζ
3452:−
3444:
3407:overshoot
3395:overshoot
3245:−
3220:−
3204:
3171:
3138:
3129:δ
3062:
3053:δ
3019:π
2998:δ
2993:δ
2985:ζ
2962:δ
2916:α
2896:α
2893:−
2883:real part
2859:ζ
2825:ω
2821:α
2795:ζ
2739:ζ
2703:ζ
2663:ζ
2660:≤
2618:ζ
2614:−
2600:ω
2588:
2543:ω
2533:
2504:ζ
2446:ζ
2442:−
2431:±
2428:ζ
2414:ω
2410:−
2385:are both
2285:ω
2252:ω
2248:ζ
2157:ω
2096:ω
2063:ω
1809:ζ
1720:ζ
1660:ω
1647:λ
1636:λ
1630:ζ
1610:ω
1602:λ
1596:ζ
1576:ζ
1553:λ
1525:λ
1508:
1495:Half-life
1473:λ
1459:τ
1424:−
1396:π
1382:ω
1366:Frequency
1341:ω
1312:φ
1286:λ
1203:φ
1200:−
1194:ω
1188:
1177:λ
1174:−
1085:π
1076:
1065:−
892:attenuate
845:radiation
696:Liouville
578:frequency
498:Vibration
215:potential
140:Continuum
135:Celestial
112:Textbooks
3812:18309860
2493:Undamped
1690:Q factor
990:undamped
907:undamped
873:friction
751:Category
676:Hamilton
661:Lagrange
656:Clairaut
621:Horrocks
582:velocity
552:Pendulum
540:reactive
512:Rotation
481:dynamics
431:Inertial
417:Friction
300:Velocity
275:Momentum
155:Kinetics
145:Dynamics
123:Branches
107:Timeline
4175:Physics
4164:Damping
3792:Bibcode
2387:complex
2129:is the
1353:is the
1218:where:
1124:science
997:viscous
834:viscous
814:damping
711:Koopman
671:Poisson
666:Laplace
611:Huygens
606:Galileo
451: (
390:Damping
243:Inertia
237:Impulse
210:kinetic
160:Statics
130:Applied
102:History
4142:
4105:
4044:
3992:
3937:
3872:
3851:
3810:
3739:damped
3673:brakes
3416:) by:
3401:, the
3273:where
3101:where
3045:where
2778:factor
2654:
2377:where
2084:where
1791:, and
1136:energy
968:, and
947:> 1
925:< 1
820:of an
818:energy
803:< 1
749:
701:Appell
686:Cauchy
681:Jacobi
626:Halley
616:Newton
601:Kepler
453:linear
449:Motion
295:Torque
270:Moment
205:Energy
195:Couple
4179:Lumen
3808:S2CID
1768:of a
1545:0.693
1443:hertz
1441:, or
1114:is a
938:) to
882:is a
867:(ex.
865:bikes
798:with
706:Gibbs
691:Routh
646:Euler
285:Speed
280:Space
222:Force
4140:ISBN
4103:ISBN
4042:ISBN
3990:ISBN
3935:ISBN
3870:ISBN
3849:ISBN
3108:and
2862:<
2773:The
2706:>
2666:<
2381:and
1756:The
1671:<
1126:and
903:zeta
878:The
863:and
837:drag
290:Time
253:Mass
4177:".
4166:".
4132:doi
4095:doi
3800:doi
3788:463
3763:doi
3715:doi
3600:100
3549:100
3441:exp
3438:100
3389:In
2885:of
2691:If
2585:exp
2573:If
2530:exp
2011:or
1329:= 0
1185:cos
1110:or
1073:cos
949:).
936:= 1
916:),
914:= 0
851:in
839:),
824:by
808:In
4191::
4138:,
4126:,
4101:.
4089:.
4064:.
4015:.
4004:^
3897:.
3893:.
3833:.
3814:.
3806:.
3798:.
3786:.
3759:11
3757:.
3713:.
3709:.
3697:^
3574:ln
3529:ln
3393:,
3354:,
3300:,
3201:ln
3168:ln
3135:ln
3059:ln
2305:0.
1958:.
1692::
1505:ln
1451::
1368::
1106:A
964:,
960:,
956:,
901:("
847:,
843:,
812:,
4181:.
4170:.
4162:"
4134::
4111:.
4097::
4074:.
4050:.
4025:.
3998:.
3957:.
3943:.
3901:.
3899:2
3878:.
3802::
3794::
3769:.
3765::
3721:.
3717::
3605:)
3596:O
3593:P
3587:(
3578:2
3570:+
3565:2
3554:)
3545:O
3542:P
3536:(
3520:=
3504:ζ
3486:)
3477:2
3466:1
3448:(
3435:=
3431:O
3428:P
3414:ζ
3367:4
3363:x
3340:2
3336:x
3313:3
3309:x
3286:1
3282:x
3253:4
3249:x
3240:3
3236:x
3228:2
3224:x
3215:1
3211:x
3198:=
3191:4
3187:x
3181:2
3177:x
3165:=
3158:3
3154:x
3148:1
3144:x
3132:=
3113:1
3110:x
3106:0
3103:x
3082:1
3078:x
3072:0
3068:x
3056:=
3028:2
3023:)
3016:2
3012:(
3007:+
3002:2
2988:=
2865:1
2836:.
2829:n
2816:=
2810:Q
2807:2
2803:1
2798:=
2782:ζ
2776:Q
2767:Q
2745:1
2742:=
2709:1
2693:s
2669:1
2657:0
2633:)
2629:t
2622:2
2611:1
2604:n
2596:i
2592:(
2575:s
2555:)
2552:t
2547:n
2539:i
2536:(
2510:0
2507:=
2479:s
2462:.
2458:)
2450:2
2439:1
2434:i
2424:(
2418:n
2407:=
2404:s
2391:s
2383:s
2379:C
2362:,
2357:t
2354:s
2350:e
2346:C
2343:=
2340:)
2337:t
2334:(
2331:x
2302:=
2299:x
2294:2
2289:n
2281:+
2275:t
2272:d
2267:x
2264:d
2256:n
2245:2
2242:+
2234:2
2230:t
2226:d
2221:x
2216:2
2212:d
2182:m
2177:/
2172:k
2166:=
2161:n
2114:m
2111:k
2105:=
2100:n
2067:n
2059:m
2056:2
2053:=
2047:m
2044:k
2038:m
2035:2
2032:=
2027:c
2023:c
1994:m
1991:k
1986:2
1983:=
1978:c
1974:c
1946:0
1943:=
1940:x
1937:k
1934:+
1928:t
1925:d
1920:x
1917:d
1911:c
1908:+
1900:2
1896:t
1892:d
1887:x
1882:2
1878:d
1871:m
1845:,
1832:=
1825:c
1821:c
1817:c
1812:=
1795:k
1789:c
1785:m
1762:ζ
1737:Q
1723:)
1717:2
1714:(
1710:/
1706:1
1703:=
1700:Q
1686:.
1674:1
1664:2
1656:+
1651:2
1640:/
1633:=
1606:/
1599:=
1565:.
1549:/
1521:/
1517:)
1514:2
1511:(
1491:.
1488:e
1469:/
1465:1
1462:=
1445:.
1427:1
1420:t
1399:)
1393:2
1390:(
1386:/
1379:=
1376:f
1357:.
1331:;
1327:t
1302:;
1300:t
1264:A
1254:;
1252:t
1238:)
1235:t
1232:(
1229:y
1206:)
1197:t
1191:(
1180:t
1170:e
1166:A
1163:=
1160:)
1157:t
1154:(
1151:y
1091:)
1088:t
1082:2
1079:(
1068:t
1061:e
1057:=
1054:)
1051:t
1048:(
1045:y
1030:.
1003:.
992:.
945:ζ
942:(
934:ζ
931:(
923:ζ
920:(
912:ζ
909:(
899:ζ
801:ζ
779:e
772:t
765:v
455:)
74:t
71:d
65:p
61:d
55:=
50:F
23:.
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