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901:. 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
1149:
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
3645:
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
1153:
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
3630:
2736:. 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.
3279:
1791:. 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.
1866:
3508:
2326:
3827:
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
3054:
2483:
1017:
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
1695:
3696:, 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.
2857:
2655:
3523:
1967:
2090:
1227:
98:
3682:
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
122:
4133:
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.),
1024:
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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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
2886:
2730:
1815:
3791:
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".
1794:
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
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2531:
1333:
1307:
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measure describing how oscillations in a system decay after a disturbance. Many systems exhibit oscillatory behavior when they are disturbed from their position of
1259:
2331:
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
2578:, 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.
1285:
788:
2499:
Phase portrait of damped oscillator, with increasing damping strength. It starts at undamped, proceeds to underdamped, then critically damped, then overdamped.
2943:
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
3662:(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.
1028:. The key difference between critical damping and overdamping is that, in critical damping, the system returns to equilibrium in the minimum amount of time.
377:
4205:
2991:
839:. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include
496:
2772:. 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).
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469:
2410:
1157:
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is a pair of complex values, then each complex solution term is a decaying exponential combined with an oscillatory portion that looks like
2492:
satisfying the equation, can be combined to make the general real solutions, with oscillatory and decaying properties in several regimes:
3625:{\displaystyle \zeta ={\frac {-\ln \left({\frac {\rm {PO}}{100}}\right)}{\sqrt {\pi ^{2}+\ln ^{2}\left({\frac {\rm {PO}}{100}}\right)}}}}
2801:
983:. 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
451:
2591:
1809:, it can be defined as the ratio of the damping coefficient in the system's differential equation to the critical damping coefficient:
3793:
2947:, which has a very low damping ratio, has an oscillation that lasts a long time, decaying very slowly after being struck by a hammer.
1877:
1780:
802:
4056:
4004:
3863:
781:
2706:
is a pair of real values, then the solution is simply a sum of two decaying exponentials with no oscillation. This case occurs for
2028:
1038:
3674:
which are induced by passing through a magnet's poles, either by a coil or aluminum plate. Eddy currents are a key component of
1133:
case of damped second-order systems, or underdamped second-order differential equations. Damped sine waves are commonly seen in
875:
117:
54:
4220:
4185:
4072:
112:
372:
1154:
resembles an exponential decay function. The general equation for an exponentially damped sinusoid may be represented as:
367:
4225:
3059:
2101:
774:
761:
522:
445:
210:
4210:
441:
242:
3274:{\displaystyle \delta =\ln {\frac {x_{1}}{x_{3}}}=\ln {\frac {x_{2}}{x_{4}}}=\ln {\frac {x_{1}-x_{2}}{x_{3}-x_{4}}}}
661:
550:
476:
336:
269:
3689:
3675:
1498:
415:
987:, but a normalised, or non-dimensionalised approach can be convenient in describing common aspects of behavior.
3841:
3693:
2162:
691:
535:
31:
1602:
1051:
870:. Damping not based on energy loss can be important in other oscillating systems such as those that occur in
3416:(PO) is the maximum value minus the step value divided by the step value. In the case of the unit step, the
905:. The damping ratio is a measure describing how rapidly the oscillations decay from one bounce to the next.
681:
641:
405:
806:
995:
Depending on the amount of damping present, a system exhibits different oscillatory behaviors and speeds.
980:
976:
972:
879:
863:
686:
503:
4134:
2660:
2536:
1979:
1508:
is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to
963:
The behaviour of oscillating systems is often of interest in a diverse range of disciplines that include
886:, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping.
2985:. The damping ratio can be found for any two peaks, even if they are not adjacent. For adjacent peaks:
2963:
1010:
fluid, the mass could slowly return to its rest position without ever overshooting. This case is called
696:
671:
357:
175:
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1511:
968:
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352:
345:
101:
1551:
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2337:
2157:
1861:{\displaystyle \zeta ={\frac {c}{c_{c}}}={\frac {\text{actual damping}}{\text{critical damping}}},}
1788:
1706:
1142:
1006:
If the system contained high losses, for example if the spring–mass experiment were conducted in a
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491:
432:
410:
155:
150:
145:
45:
3818:
3683:
3405:
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2709:
2533:
corresponds to the undamped simple harmonic oscillator, and in that case the solution looks like
1776:
1599:
is a non-dimensional characterization of the decay rate relative to the frequency, approximately
898:
867:
832:
621:
362:
237:
205:
165:
3503:{\displaystyle \mathrm {PO} =100\exp \left({-{\frac {\zeta \pi }{\sqrt {1-\zeta ^{2}}}}}\right)}
2321:{\displaystyle {\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{n}{\frac {dx}{dt}}+\omega _{n}^{2}x=0.}
4150:
4113:
4098:"Quasi-Steady Herschel-Bulkley Analysis of Electro- and Magneto-Rheological Flow Mode Dampers"
4052:
4000:
3945:
3880:
3859:
2940:
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2745:
2510:
2141:
1365:
1318:
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257:
247:
140:
3939:
2968:
2922:
1347:
4142:
4105:
3810:
3773:
3725:
2148:
The damping ratio is dimensionless, being the ratio of two coefficients of identical units.
1582:
1425:
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706:
651:
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592:
567:
422:
280:
225:
200:
3729:
3368:
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3314:
3287:
1150:
term "damped sine wave" describes all such damped waveforms, whatever their initial phase.
3717:
1803:
1759:
1235:
840:
820:
721:
666:
616:
611:
530:
3941:
Process control engineering: a textbook for chemical, mechanical and electrical engineers
3806:
4146:
3647:
3401:
2397:
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1270:
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847:
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1459:
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636:
463:
27:
Influence on an oscillating physical system which reduces or prevents its oscillation
3822:
3752:, which is the term used in the study of vibration to denote a dissipation of energy
3049:{\displaystyle \zeta ={\frac {\delta }{\sqrt {\delta ^{2}+\left(2\pi \right)^{2}}}}}
3671:
1309:
is the decay rate, in the reciprocal of the time units of the independent variable
701:
626:
315:
195:
2768:
is the border between the overdamped and underdamped cases, and is referred to as
4097:
2944:
1138:
902:
836:
4109:
2955:
486:
160:
4174:
4023:
2478:{\displaystyle s=-\omega _{n}\left(\zeta \pm i{\sqrt {1-\zeta ^{2}}}\right).}
3979:
2893:
1505:
1422:, the number of cycles per time unit. It is expressed in inverse time units
1376:
1126:
855:
844:
508:
3814:
3764:
Crandall, S. H. (January 1970). "The role of damping in vibration theory".
1746:
is another non-dimensional characterization of the amount of damping; high
801:
17:
3670:
Kinetic energy that causes oscillations is dissipated as heat by electric
1044:
30:
This article is about damping in oscillatory systems. For other uses, see
2785:
1700:
1129:
whose amplitude approaches zero as time increases. It corresponds to the
883:
851:
427:
310:
285:
2208:
and the definition of the damping ratio above, we can rewrite this as:
1690:{\displaystyle \zeta =\lambda /{\sqrt {\lambda ^{2}+\omega ^{2}}}<1}
1134:
1007:
871:
253:
170:
3994:
3650:, which for example is applied in automatic doors or anti-slam doors.
4049:
Principles of
Electrical, Electronics and Instrumentation Engineering
2962:
For underdamped vibrations, the damping ratio is also related to the
1146:
828:
459:
305:
215:
3408:
refers to an output exceeding its final, steady-state value. For a
2852:{\displaystyle \zeta ={\frac {1}{2Q}}={\alpha \over \omega _{n}}.}
2954:
2650:{\textstyle \exp \left(i\omega _{n}{\sqrt {1-\zeta ^{2}}}t\right)}
2494:
1453:
800:
295:
290:
232:
3900:
Rahman, J.; Mushtaq, M.; Ali, A.; Anjam, Y.N; Nazir, S. (2014).
1962:{\displaystyle m{\frac {d^{2}x}{dt^{2}}}+c{\frac {dx}{dt}}+kx=0}
913:
300:
263:
4102:
Electro-Rheological Fluids and
Magneto-Rheological Suspensions
1763:
The effect of varying damping ratio on a second-order system.
1048:
Plot of a damped sinusoidal wave represented as the function
4135:"CHAPTER 2 - Semi-Active Suspension Technologies and Models"
2085:{\displaystyle c_{c}=2m{\sqrt {\frac {k}{m}}}=2m\omega _{n}}
3423:
The percentage overshoot (PO) is related to damping ratio (
1222:{\displaystyle y(t)=Ae^{-\lambda t}\cos(\omega t-\varphi )}
3420:
is just the maximum value of the step response minus one.
1496:, the time for the amplitude to decrease by the factor of
3902:"Modelling damped mass spring system in MATHLAB Simulink"
93:{\displaystyle {\textbf {F}}={\frac {d\mathbf {p} }{dt}}}
3966:"Dynamics and Vibrations: Notes: Free Damped Vibrations"
3517:) that yields a given percentage overshoot is given by:
3999:(Eighth ed.). NY: Wiley. p. §7.3 p. 236–237.
2888:(that is, when the system is underdamped), it has two
2594:
2165:
1973:
and the corresponding critical damping coefficient is
3848:. OpenStax – via University of Central Florida.
3526:
3436:
3371:
3344:
3317:
3290:
3138:
3062:
2994:
2971:
2925:
2902:
2868:
2804:
2795:, and exponential decay rate α are related such that
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2712:
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2513:
2413:
2340:
2217:
2104:
2031:
1982:
1880:
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1321:
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1273:
1238:
1160:
1054:
57:
3858:
Douglas C. Giancoli (2000). . Prentice Hall. p. 387
3338:
are amplitudes of two successive positive peaks and
908:
The damping ratio is a system parameter, denoted by
1750:
indicates slow damping relative to the oscillation.
4139:Semi-Active Suspension Control Design for Vehicles
3906:Journal of Faculty of Engineering & Technology
3624:
3502:
3384:
3357:
3330:
3303:
3273:
3102:{\displaystyle \delta =\ln {\frac {x_{0}}{x_{1}}}}
3101:
3048:
2977:
2931:
2911:
2880:
2851:
2760:
2724:
2684:
2649:
2570:
2525:
2477:
2377:
2320:
2200:
2133:{\displaystyle \omega _{n}={\sqrt {\frac {k}{m}}}}
2132:
2084:
2011:
1961:
1860:
1738:
1689:
1625:
1591:
1568:
1540:
1488:
1444:
1414:
1356:
1327:
1301:
1279:
1253:
1221:
1106:
92:
4141:, Boston: Butterworth-Heinemann, pp. 15–39,
3392:are amplitudes of two successive negative peaks.
4096:LEE, DUG-YOUNG; WERELEY, NORMAN M. (June 2000).
3666:Magnetic damping and Magnetorheological damping
1783:. It is particularly important in the study of
4073:"Eddy Currents and Magnetic Damping | Physics"
4024:"damping | Definition, Types, & Examples"
3993:Kuo, Benjamin C & Golnaraghi M F (2003).
782:
8:
3877:Introduction to Mechatronics and Measurement
3126:are amplitudes of any two successive peaks.
1775:(Greek letter zeta), that characterizes the
866:, and absorption and scattering of light in
3840:Urone, Paul Peter; Hinrichs, Roger (2016).
1781:second-order ordinary differential equation
2488:Two such solutions, for the two values of
2201:{\textstyle \omega _{n}={\sqrt {{k}/{m}}}}
789:
775:
36:
3602:
3600:
3587:
3574:
3551:
3549:
3533:
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3465:
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3011:
3001:
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2811:
2803:
2747:
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2662:
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2613:
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2447:
2427:
2412:
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2218:
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2118:
2109:
2103:
2076:
2051:
2036:
2030:
1999:
1987:
1981:
1924:
1909:
1891:
1884:
1879:
1871:where the system's equation of motion is
1845:
1834:
1825:
1817:
1719:
1708:
1673:
1660:
1654:
1649:
1638:
1615:
1604:
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1558:
1553:
1530:
1513:
1478:
1467:
1433:
1427:
1395:
1384:
1349:
1320:
1294:
1287:is the initial amplitude of the envelope;
1272:
1237:
1183:
1159:
1074:
1053:
74:
68:
59:
58:
56:
3745:An Introduction to Mechanical Vibrations
3718:"A Dictionary of Mechanical Engineering"
1758:
1043:
3716:Escudier, Marcel; Atkins, Tony (2019).
3705:
1626:{\displaystyle \zeta =\lambda /\omega }
1261:is the instantaneous amplitude at time
1107:{\displaystyle y(t)=e^{-t}\cos(2\pi t)}
108:
44:
882:). Damping is not to be confused with
4018:
4016:
3747:. John Wiley & Sons. p. 37.
3658:Electrical systems that operate with
471:Newton's law of universal gravitation
7:
3730:10.1093/acref/9780198832102.001.0001
3711:
3709:
2919:; that is, the decay rate parameter
1372:Other important parameters include:
2685:{\displaystyle \ 0\leq \zeta <1}
2571:{\displaystyle \exp(i\omega _{n}t)}
2012:{\displaystyle c_{c}=2{\sqrt {km}}}
1771:is a parameter, usually denoted by
452:Mechanics of planar particle motion
60:
4206:Dimensionless numbers of mechanics
4147:10.1016/b978-0-08-096678-6.00002-x
3794:Proceedings of the Royal Society A
3606:
3603:
3555:
3552:
3441:
3438:
25:
4051:. S. chand Limited. p. 338.
3938:Ming Rao and Haiming Qiu (1993).
2156:Using the natural frequency of a
1415:{\displaystyle f=\omega /(2\pi )}
1489:{\displaystyle \tau =1/\lambda }
1039:Damped wave (radio transmission)
756:
755:
742:
75:
4216:Ordinary differential equations
3513:Conversely, the damping ratio (
2862:When a second-order system has
1541:{\displaystyle \ln(2)/\lambda }
3925:Circuits, Signals, and Systems
3766:Journal of Sound and Vibration
3129:As shown in the right figure:
2565:
2546:
2350:
2344:
1787:. It is also important in the
1733:
1724:
1569:{\displaystyle 0.693/\lambda }
1527:
1521:
1409:
1400:
1248:
1242:
1216:
1201:
1170:
1164:
1101:
1089:
1064:
1058:
1:
4104:. WORLD SCIENTIFIC: 579–586.
3879:(3rd ed.). McGraw Hill.
3842:"16.7 Damped Harmonic Motion"
3778:10.1016/s0022-460x(70)80105-5
3654:Damping in electrical systems
2378:{\displaystyle x(t)=Ce^{st},}
1739:{\displaystyle Q=1/(2\zeta )}
378:Koopman–von Neumann mechanics
3875:Alciatore, David G. (2007).
446:Non-inertial reference frame
2881:{\displaystyle \zeta <1}
2725:{\displaystyle \zeta >1}
373:Appell's equation of motion
243:Inertial frame of reference
4242:
4110:10.1142/9789812793607_0066
3690:Magnetorheological Dampers
1036:
29:
4077:courses.lumenlearning.com
3996:Automatic control systems
3944:. CRC Press. p. 96.
3676:electromagnetic induction
3636:Examples and applications
3694:Magnetorheological fluid
2912:{\displaystyle -\alpha }
2761:{\displaystyle \zeta =1}
2732:, and is referred to as
2692:, and is referred to as
2657:. This case occurs for
2526:{\displaystyle \zeta =0}
1755:Damping ratio definition
1328:{\displaystyle \varphi }
1302:{\displaystyle \lambda }
1037:Not to be confused with
536:Rotating reference frame
368:Hamilton–Jacobi equation
32:Damping (disambiguation)
4179:Encyclopædia Britannica
4028:Encyclopedia Britannica
2978:{\displaystyle \delta }
2939:represents the rate of
2932:{\displaystyle \alpha }
2892:poles that each have a
1548:which is approximately
1357:{\displaystyle \omega }
916:"), that can vary from
477:Newton's laws of motion
337:Newton's laws of motion
3923:William McC. Siebert.
3815:10.1098/rspa.2007.1857
3626:
3504:
3386:
3359:
3332:
3305:
3275:
3103:
3050:
2979:
2959:
2933:
2913:
2882:
2853:
2762:
2726:
2686:
2651:
2572:
2527:
2500:
2479:
2379:
2322:
2202:
2134:
2086:
2013:
1963:
1862:
1798:, damping coefficient
1764:
1740:
1691:
1627:
1593:
1592:{\displaystyle \zeta }
1570:
1542:
1490:
1446:
1445:{\displaystyle t^{-1}}
1416:
1358:
1335:is the phase angle at
1329:
1303:
1281:
1255:
1223:
1114:
1108:
981:electrical engineering
977:structural engineering
973:mechanical engineering
880:Suspension (mechanics)
864:electronic oscillators
816:
504:Simple harmonic motion
417:Euler's laws of motion
211:D'Alembert's principle
94:
4221:Mathematical analysis
4047:Gupta, B. R. (2001).
3686:on roller coasters.
3627:
3505:
3387:
3385:{\displaystyle x_{4}}
3360:
3358:{\displaystyle x_{2}}
3333:
3331:{\displaystyle x_{3}}
3306:
3304:{\displaystyle x_{1}}
3276:
3104:
3051:
2980:
2964:logarithmic decrement
2958:
2951:Logarithmic decrement
2934:
2914:
2883:
2854:
2780:factor and decay rate
2763:
2727:
2696:(e.g., bungee cable).
2687:
2652:
2573:
2528:
2498:
2480:
2380:
2323:
2203:
2135:
2087:
2014:
1964:
1863:
1762:
1741:
1692:
1628:
1594:
1571:
1543:
1491:
1447:
1417:
1359:
1330:
1304:
1282:
1256:
1224:
1109:
1047:
804:
358:Hamiltonian mechanics
176:Statistical mechanics
95:
4184:OpenStax, College. "
3980:"Damping Evaluation"
3678:where they set up a
3524:
3434:
3414:percentage overshoot
3396:Percentage overshoot
3369:
3342:
3315:
3288:
3136:
3060:
2992:
2969:
2923:
2900:
2866:
2802:
2746:
2710:
2661:
2592:
2537:
2511:
2411:
2338:
2215:
2163:
2102:
2029:
1980:
1878:
1816:
1707:
1637:
1603:
1583:
1552:
1512:
1466:
1426:
1383:
1348:
1319:
1293:
1271:
1254:{\displaystyle y(t)}
1236:
1158:
1052:
969:chemical engineering
581:Angular acceleration
573:Rotational frequency
353:Lagrangian mechanics
346:Analytical mechanics
102:Second law of motion
55:
4226:Classical mechanics
3807:2007RSPSA.463.1955M
3801:(2084): 1955–1982.
3660:alternating current
2308:
2158:harmonic oscillator
1789:harmonic oscillator
1143:harmonic oscillator
1127:sinusoidal function
965:control engineering
868:optical oscillators
433:Harmonic oscillator
411:Equations of motion
46:Classical mechanics
40:Part of a series on
4211:Engineering ratios
3982:. 19 October 2015.
3622:
3500:
3382:
3355:
3328:
3301:
3271:
3099:
3046:
2975:
2960:
2929:
2909:
2878:
2849:
2758:
2722:
2682:
2647:
2568:
2523:
2507:Is the case where
2501:
2475:
2375:
2318:
2294:
2198:
2130:
2082:
2009:
1959:
1858:
1777:frequency response
1765:
1736:
1687:
1623:
1589:
1566:
1538:
1486:
1442:
1412:
1354:
1325:
1299:
1277:
1251:
1219:
1115:
1104:
899:static equilibrium
872:biological systems
833:oscillating system
817:
807:spring–mass system
749:Physics portal
363:Routhian mechanics
238:Frame of reference
90:
4156:978-0-08-096678-6
4119:978-981-02-4258-9
3951:978-2-88124-628-9
3886:978-0-07-296305-2
3692:(MR Dampers) use
3620:
3619:
3613:
3562:
3493:
3492:
3269:
3206:
3173:
3097:
3044:
3043:
2941:exponential decay
2890:complex conjugate
2844:
2824:
2770:critically damped
2739:Critically damped
2666:
2637:
2465:
2289:
2250:
2196:
2142:natural frequency
2128:
2127:
2061:
2060:
2007:
1942:
1916:
1853:
1852:
1849:
1840:
1679:
1366:angular frequency
1280:{\displaystyle A}
991:Oscillation cases
940:critically damped
799:
798:
546:Centrifugal force
541:Centripetal force
497:Euler's equations
482:Relative velocity
258:Moment of inertia
88:
62:
16:(Redirected from
4233:
4166:
4165:
4164:
4163:
4130:
4124:
4123:
4093:
4087:
4086:
4084:
4083:
4069:
4063:
4062:
4044:
4038:
4037:
4035:
4034:
4020:
4011:
4010:
3990:
3984:
3983:
3976:
3970:
3969:
3962:
3956:
3955:
3935:
3929:
3928:
3920:
3914:
3913:
3897:
3891:
3890:
3872:
3866:
3856:
3850:
3849:
3837:
3831:
3830:
3788:
3782:
3781:
3772:(1): 3–18, IN1.
3761:
3755:
3754:
3743:Steidel (1971).
3740:
3734:
3733:
3722:Oxford Reference
3713:
3631:
3629:
3628:
3623:
3621:
3618:
3614:
3609:
3601:
3592:
3591:
3579:
3578:
3569:
3568:
3567:
3563:
3558:
3550:
3534:
3509:
3507:
3506:
3501:
3499:
3495:
3494:
3491:
3490:
3475:
3474:
3466:
3444:
3391:
3389:
3388:
3383:
3381:
3380:
3364:
3362:
3361:
3356:
3354:
3353:
3337:
3335:
3334:
3329:
3327:
3326:
3310:
3308:
3307:
3302:
3300:
3299:
3280:
3278:
3277:
3272:
3270:
3268:
3267:
3266:
3254:
3253:
3243:
3242:
3241:
3229:
3228:
3218:
3207:
3205:
3204:
3195:
3194:
3185:
3174:
3172:
3171:
3162:
3161:
3152:
3108:
3106:
3105:
3100:
3098:
3096:
3095:
3086:
3085:
3076:
3055:
3053:
3052:
3047:
3045:
3042:
3041:
3036:
3032:
3016:
3015:
3006:
3002:
2984:
2982:
2981:
2976:
2938:
2936:
2935:
2930:
2918:
2916:
2915:
2910:
2887:
2885:
2884:
2879:
2858:
2856:
2855:
2850:
2845:
2843:
2842:
2830:
2825:
2823:
2812:
2791:, damping ratio
2767:
2765:
2764:
2759:
2731:
2729:
2728:
2723:
2691:
2689:
2688:
2683:
2664:
2656:
2654:
2653:
2648:
2646:
2642:
2638:
2636:
2635:
2620:
2618:
2617:
2577:
2575:
2574:
2569:
2561:
2560:
2532:
2530:
2529:
2524:
2484:
2482:
2481:
2476:
2471:
2467:
2466:
2464:
2463:
2448:
2432:
2431:
2400:constants, with
2384:
2382:
2381:
2376:
2371:
2370:
2327:
2325:
2324:
2319:
2307:
2302:
2290:
2288:
2280:
2272:
2270:
2269:
2251:
2249:
2248:
2247:
2234:
2230:
2229:
2219:
2207:
2205:
2204:
2199:
2197:
2195:
2190:
2185:
2180:
2175:
2174:
2139:
2137:
2136:
2131:
2129:
2120:
2119:
2114:
2113:
2091:
2089:
2088:
2083:
2081:
2080:
2062:
2053:
2052:
2041:
2040:
2018:
2016:
2015:
2010:
2008:
2000:
1992:
1991:
1968:
1966:
1965:
1960:
1943:
1941:
1933:
1925:
1917:
1915:
1914:
1913:
1900:
1896:
1895:
1885:
1867:
1865:
1864:
1859:
1854:
1851:critical damping
1850:
1847:
1846:
1841:
1839:
1838:
1826:
1804:spring constant
1745:
1743:
1742:
1737:
1723:
1696:
1694:
1693:
1688:
1680:
1678:
1677:
1665:
1664:
1655:
1653:
1632:
1630:
1629:
1624:
1619:
1598:
1596:
1595:
1590:
1575:
1573:
1572:
1567:
1562:
1547:
1545:
1544:
1539:
1534:
1495:
1493:
1492:
1487:
1482:
1451:
1449:
1448:
1443:
1441:
1440:
1421:
1419:
1418:
1413:
1399:
1363:
1361:
1360:
1355:
1341:
1334:
1332:
1331:
1326:
1312:
1308:
1306:
1305:
1300:
1286:
1284:
1283:
1278:
1264:
1260:
1258:
1257:
1252:
1228:
1226:
1225:
1220:
1194:
1193:
1119:damped sine wave
1113:
1111:
1110:
1105:
1082:
1081:
1033:Damped sine wave
1026:critical damping
959:
948:
937:
926:
911:
852:surface friction
843:in a fluid (see
821:physical systems
815:
791:
784:
777:
764:
759:
758:
751:
747:
746:
652:Johann Bernoulli
647:Daniel Bernoulli
568:Tangential speed
472:
448:
423:Fictitious force
418:
270:Mechanical power
260:
201:Angular momentum
99:
97:
96:
91:
89:
87:
79:
78:
69:
64:
63:
37:
21:
4241:
4240:
4236:
4235:
4234:
4232:
4231:
4230:
4196:
4195:
4170:
4169:
4161:
4159:
4157:
4132:
4131:
4127:
4120:
4095:
4094:
4090:
4081:
4079:
4071:
4070:
4066:
4059:
4046:
4045:
4041:
4032:
4030:
4022:
4021:
4014:
4007:
3992:
3991:
3987:
3978:
3977:
3973:
3964:
3963:
3959:
3952:
3937:
3936:
3932:
3922:
3921:
3917:
3899:
3898:
3894:
3887:
3874:
3873:
3869:
3857:
3853:
3846:College Physics
3839:
3838:
3834:
3790:
3789:
3785:
3763:
3762:
3758:
3742:
3741:
3737:
3715:
3714:
3707:
3702:
3668:
3656:
3646:the concept of
3643:
3638:
3596:
3583:
3570:
3545:
3535:
3522:
3521:
3482:
3467:
3457:
3432:
3431:
3398:
3372:
3367:
3366:
3345:
3340:
3339:
3318:
3313:
3312:
3291:
3286:
3285:
3258:
3245:
3244:
3233:
3220:
3219:
3196:
3186:
3163:
3153:
3134:
3133:
3125:
3118:
3087:
3077:
3058:
3057:
3025:
3021:
3020:
3007:
2990:
2989:
2967:
2966:
2953:
2921:
2920:
2898:
2897:
2864:
2863:
2834:
2816:
2800:
2799:
2782:
2744:
2743:
2742:The case where
2708:
2707:
2659:
2658:
2627:
2609:
2605:
2601:
2590:
2589:
2552:
2535:
2534:
2509:
2508:
2455:
2437:
2433:
2423:
2409:
2408:
2359:
2336:
2335:
2281:
2273:
2261:
2239:
2235:
2221:
2220:
2213:
2212:
2166:
2161:
2160:
2154:
2105:
2100:
2099:
2072:
2032:
2027:
2026:
1983:
1978:
1977:
1934:
1926:
1905:
1901:
1887:
1886:
1876:
1875:
1830:
1814:
1813:
1757:
1705:
1704:
1669:
1656:
1635:
1634:
1601:
1600:
1581:
1580:
1579:Damping ratio:
1550:
1549:
1510:
1509:
1464:
1463:
1429:
1424:
1423:
1381:
1380:
1346:
1345:
1336:
1317:
1316:
1310:
1291:
1290:
1269:
1268:
1262:
1234:
1233:
1179:
1156:
1155:
1123:damped sinusoid
1070:
1050:
1049:
1042:
1035:
993:
954:
943:
932:
921:
909:
841:viscous damping
827:is the loss of
810:
795:
754:
741:
740:
733:
732:
731:
606:
598:
597:
577:
531:Circular motion
525:
515:
514:
513:
470:
440:
437:
416:
395:
387:
386:
383:
382:
340:
330:
322:
321:
320:
279:
275:Mechanical work
268:
252:
190:
182:
181:
180:
135:
127:
104:
80:
70:
53:
52:
35:
28:
23:
22:
15:
12:
11:
5:
4239:
4237:
4229:
4228:
4223:
4218:
4213:
4208:
4198:
4197:
4194:
4193:
4182:
4168:
4167:
4155:
4125:
4118:
4088:
4064:
4057:
4039:
4012:
4005:
3985:
3971:
3957:
3950:
3930:
3915:
3892:
3885:
3867:
3851:
3832:
3783:
3756:
3735:
3704:
3703:
3701:
3698:
3667:
3664:
3655:
3652:
3642:
3639:
3637:
3634:
3633:
3632:
3617:
3612:
3608:
3605:
3599:
3595:
3590:
3586:
3582:
3577:
3573:
3566:
3561:
3557:
3554:
3548:
3544:
3541:
3538:
3532:
3529:
3511:
3510:
3498:
3489:
3485:
3481:
3478:
3473:
3470:
3464:
3460:
3456:
3453:
3450:
3447:
3443:
3440:
3402:control theory
3397:
3394:
3379:
3375:
3352:
3348:
3325:
3321:
3298:
3294:
3282:
3281:
3265:
3261:
3257:
3252:
3248:
3240:
3236:
3232:
3227:
3223:
3216:
3213:
3210:
3203:
3199:
3193:
3189:
3183:
3180:
3177:
3170:
3166:
3160:
3156:
3150:
3147:
3144:
3141:
3123:
3116:
3110:
3109:
3094:
3090:
3084:
3080:
3074:
3071:
3068:
3065:
3040:
3035:
3031:
3028:
3024:
3019:
3014:
3010:
3005:
3000:
2997:
2974:
2952:
2949:
2928:
2908:
2905:
2877:
2874:
2871:
2860:
2859:
2848:
2841:
2837:
2833:
2828:
2822:
2819:
2815:
2810:
2807:
2781:
2775:
2774:
2773:
2757:
2754:
2751:
2740:
2737:
2721:
2718:
2715:
2700:
2697:
2681:
2678:
2675:
2672:
2669:
2645:
2641:
2634:
2630:
2626:
2623:
2616:
2612:
2608:
2604:
2600:
2597:
2582:
2579:
2567:
2564:
2559:
2555:
2551:
2548:
2545:
2542:
2522:
2519:
2516:
2505:
2486:
2485:
2474:
2470:
2462:
2458:
2454:
2451:
2446:
2443:
2440:
2436:
2430:
2426:
2422:
2419:
2416:
2386:
2385:
2374:
2369:
2366:
2362:
2358:
2355:
2352:
2349:
2346:
2343:
2329:
2328:
2317:
2314:
2311:
2306:
2301:
2297:
2293:
2287:
2284:
2279:
2276:
2268:
2264:
2260:
2257:
2254:
2246:
2242:
2238:
2233:
2228:
2224:
2194:
2189:
2184:
2178:
2173:
2169:
2153:
2150:
2146:
2145:
2144:of the system.
2126:
2123:
2117:
2112:
2108:
2093:
2092:
2079:
2075:
2071:
2068:
2065:
2059:
2056:
2050:
2047:
2044:
2039:
2035:
2020:
2019:
2006:
2003:
1998:
1995:
1990:
1986:
1971:
1970:
1958:
1955:
1952:
1949:
1946:
1940:
1937:
1932:
1929:
1923:
1920:
1912:
1908:
1904:
1899:
1894:
1890:
1883:
1869:
1868:
1857:
1848:actual damping
1844:
1837:
1833:
1829:
1824:
1821:
1785:control theory
1756:
1753:
1752:
1751:
1735:
1732:
1729:
1726:
1722:
1718:
1715:
1712:
1698:
1686:
1683:
1676:
1672:
1668:
1663:
1659:
1652:
1648:
1645:
1642:
1622:
1618:
1614:
1611:
1608:
1588:
1577:
1565:
1561:
1557:
1537:
1533:
1529:
1526:
1523:
1520:
1517:
1503:
1485:
1481:
1477:
1474:
1471:
1457:
1439:
1436:
1432:
1411:
1408:
1405:
1402:
1398:
1394:
1391:
1388:
1370:
1369:
1353:
1343:
1324:
1314:
1298:
1288:
1276:
1266:
1250:
1247:
1244:
1241:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1192:
1189:
1186:
1182:
1178:
1175:
1172:
1169:
1166:
1163:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1080:
1077:
1073:
1069:
1066:
1063:
1060:
1057:
1034:
1031:
1030:
1029:
1022:
1015:
1004:
992:
989:
985:electric motor
797:
796:
794:
793:
786:
779:
771:
768:
767:
766:
765:
752:
735:
734:
730:
729:
724:
719:
714:
709:
704:
699:
694:
689:
684:
679:
674:
669:
664:
659:
654:
649:
644:
639:
634:
629:
624:
619:
614:
608:
607:
604:
603:
600:
599:
596:
595:
576:
575:
570:
565:
560:
558:Coriolis force
555:
554:
553:
543:
538:
533:
527:
526:
521:
520:
517:
516:
512:
511:
506:
501:
500:
499:
494:
484:
479:
474:
467:
456:
455:
454:
449:
436:
435:
430:
425:
420:
413:
408:
403:
397:
396:
393:
392:
389:
388:
385:
384:
381:
380:
375:
370:
365:
360:
355:
349:
343:
341:
334:
331:
328:
327:
324:
323:
319:
318:
313:
308:
303:
298:
293:
288:
283:
277:
272:
266:
261:
250:
245:
240:
235:
230:
229:
228:
223:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
168:
163:
158:
153:
148:
143:
137:
136:
133:
132:
129:
128:
126:
125:
120:
115:
109:
106:
105:
100:
86:
83:
77:
73:
67:
49:
48:
42:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4238:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4203:
4201:
4191:
4187:
4183:
4180:
4176:
4172:
4171:
4158:
4152:
4148:
4144:
4140:
4136:
4129:
4126:
4121:
4115:
4111:
4107:
4103:
4099:
4092:
4089:
4078:
4074:
4068:
4065:
4060:
4058:9788121901031
4054:
4050:
4043:
4040:
4029:
4025:
4019:
4017:
4013:
4008:
4006:0-471-13476-7
4002:
3998:
3997:
3989:
3986:
3981:
3975:
3972:
3967:
3961:
3958:
3953:
3947:
3943:
3942:
3934:
3931:
3926:
3919:
3916:
3911:
3907:
3903:
3896:
3893:
3888:
3882:
3878:
3871:
3868:
3865:
3864:0-13-021517-1
3861:
3855:
3852:
3847:
3843:
3836:
3833:
3829:
3824:
3820:
3816:
3812:
3808:
3804:
3800:
3796:
3795:
3787:
3784:
3779:
3775:
3771:
3767:
3760:
3757:
3753:
3751:
3746:
3739:
3736:
3731:
3727:
3723:
3719:
3712:
3710:
3706:
3699:
3697:
3695:
3691:
3687:
3685:
3681:
3680:magnetic flux
3677:
3673:
3672:eddy currents
3665:
3663:
3661:
3653:
3651:
3649:
3640:
3635:
3615:
3610:
3597:
3593:
3588:
3584:
3580:
3575:
3571:
3564:
3559:
3546:
3542:
3539:
3536:
3530:
3527:
3520:
3519:
3518:
3516:
3496:
3487:
3483:
3479:
3476:
3471:
3468:
3462:
3458:
3454:
3451:
3448:
3445:
3430:
3429:
3428:
3426:
3421:
3419:
3415:
3411:
3407:
3403:
3395:
3393:
3377:
3373:
3350:
3346:
3323:
3319:
3296:
3292:
3263:
3259:
3255:
3250:
3246:
3238:
3234:
3230:
3225:
3221:
3214:
3211:
3208:
3201:
3197:
3191:
3187:
3181:
3178:
3175:
3168:
3164:
3158:
3154:
3148:
3145:
3142:
3139:
3132:
3131:
3130:
3127:
3122:
3115:
3092:
3088:
3082:
3078:
3072:
3069:
3066:
3063:
3038:
3033:
3029:
3026:
3022:
3017:
3012:
3008:
3003:
2998:
2995:
2988:
2987:
2986:
2972:
2965:
2957:
2950:
2948:
2946:
2942:
2926:
2906:
2903:
2895:
2891:
2875:
2872:
2869:
2846:
2839:
2835:
2831:
2826:
2820:
2817:
2813:
2808:
2805:
2798:
2797:
2796:
2794:
2790:
2788:
2779:
2776:
2771:
2755:
2752:
2749:
2741:
2738:
2735:
2719:
2716:
2713:
2705:
2701:
2698:
2695:
2679:
2676:
2673:
2670:
2667:
2643:
2639:
2632:
2628:
2624:
2621:
2614:
2610:
2606:
2602:
2598:
2595:
2587:
2583:
2580:
2562:
2557:
2553:
2549:
2543:
2540:
2520:
2517:
2514:
2506:
2503:
2502:
2497:
2493:
2491:
2472:
2468:
2460:
2456:
2452:
2449:
2444:
2441:
2438:
2434:
2428:
2424:
2420:
2417:
2414:
2407:
2406:
2405:
2403:
2399:
2395:
2391:
2372:
2367:
2364:
2360:
2356:
2353:
2347:
2341:
2334:
2333:
2332:
2315:
2312:
2309:
2304:
2299:
2295:
2291:
2285:
2282:
2277:
2274:
2266:
2262:
2258:
2255:
2252:
2244:
2240:
2236:
2231:
2226:
2222:
2211:
2210:
2209:
2192:
2187:
2182:
2176:
2171:
2167:
2159:
2151:
2149:
2143:
2124:
2121:
2115:
2110:
2106:
2098:
2097:
2096:
2077:
2073:
2069:
2066:
2063:
2057:
2054:
2048:
2045:
2042:
2037:
2033:
2025:
2024:
2023:
2004:
2001:
1996:
1993:
1988:
1984:
1976:
1975:
1974:
1956:
1953:
1950:
1947:
1944:
1938:
1935:
1930:
1927:
1921:
1918:
1910:
1906:
1902:
1897:
1892:
1888:
1881:
1874:
1873:
1872:
1855:
1842:
1835:
1831:
1827:
1822:
1819:
1812:
1811:
1810:
1808:
1807:
1801:
1797:
1792:
1790:
1786:
1782:
1778:
1774:
1770:
1769:damping ratio
1761:
1754:
1749:
1730:
1727:
1720:
1716:
1713:
1710:
1702:
1699:
1684:
1681:
1674:
1670:
1666:
1661:
1657:
1650:
1646:
1643:
1640:
1633:, or exactly
1620:
1616:
1612:
1609:
1606:
1586:
1578:
1563:
1559:
1555:
1535:
1531:
1524:
1518:
1515:
1507:
1504:
1501:
1500:
1483:
1479:
1475:
1472:
1469:
1461:
1460:Time constant
1458:
1455:
1437:
1434:
1430:
1406:
1403:
1396:
1392:
1389:
1386:
1378:
1375:
1374:
1373:
1367:
1351:
1344:
1339:
1322:
1315:
1296:
1289:
1274:
1267:
1245:
1239:
1232:
1231:
1230:
1213:
1210:
1207:
1204:
1198:
1195:
1190:
1187:
1184:
1180:
1176:
1173:
1167:
1161:
1151:
1148:
1144:
1141:, wherever a
1140:
1136:
1132:
1128:
1124:
1120:
1098:
1095:
1092:
1086:
1083:
1078:
1075:
1071:
1067:
1061:
1055:
1046:
1040:
1032:
1027:
1023:
1021:
1016:
1013:
1009:
1005:
1002:
998:
997:
996:
990:
988:
986:
982:
978:
974:
970:
966:
961:
957:
952:
946:
941:
935:
930:
924:
919:
915:
906:
904:
900:
896:
895:dimensionless
892:
891:damping ratio
887:
885:
881:
877:
873:
869:
865:
861:
857:
853:
849:
846:
842:
838:
834:
830:
826:
822:
813:
808:
803:
792:
787:
785:
780:
778:
773:
772:
770:
769:
763:
753:
750:
745:
739:
738:
737:
736:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
658:
655:
653:
650:
648:
645:
643:
640:
638:
635:
633:
630:
628:
625:
623:
620:
618:
615:
613:
610:
609:
602:
601:
594:
590:
586:
582:
579:
578:
574:
571:
569:
566:
564:
561:
559:
556:
552:
549:
548:
547:
544:
542:
539:
537:
534:
532:
529:
528:
524:
519:
518:
510:
507:
505:
502:
498:
495:
493:
490:
489:
488:
485:
483:
480:
478:
475:
473:
468:
465:
461:
458:
457:
453:
450:
447:
443:
439:
438:
434:
431:
429:
426:
424:
421:
419:
414:
412:
409:
407:
404:
402:
399:
398:
391:
390:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
350:
348:
347:
342:
339:
338:
333:
332:
326:
325:
317:
314:
312:
309:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
282:
278:
276:
273:
271:
267:
265:
262:
259:
255:
251:
249:
246:
244:
241:
239:
236:
234:
231:
227:
224:
222:
219:
218:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
186:
185:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
138:
131:
130:
124:
121:
119:
116:
114:
111:
110:
107:
103:
84:
81:
71:
65:
51:
50:
47:
43:
39:
38:
33:
19:
4189:
4178:
4160:, retrieved
4138:
4128:
4101:
4091:
4080:. Retrieved
4076:
4067:
4048:
4042:
4031:. Retrieved
4027:
3995:
3988:
3974:
3960:
3940:
3933:
3927:. MIT Press.
3924:
3918:
3909:
3905:
3895:
3876:
3870:
3854:
3845:
3835:
3826:
3798:
3792:
3786:
3769:
3765:
3759:
3749:
3748:
3744:
3738:
3721:
3688:
3669:
3657:
3648:viscous drag
3644:
3641:Viscous drag
3514:
3512:
3424:
3422:
3417:
3413:
3399:
3283:
3128:
3120:
3113:
3111:
2961:
2861:
2792:
2786:
2783:
2777:
2769:
2733:
2703:
2693:
2585:
2489:
2487:
2401:
2393:
2389:
2387:
2330:
2155:
2147:
2094:
2021:
1972:
1870:
1805:
1799:
1795:
1793:
1772:
1768:
1766:
1747:
1497:
1371:
1337:
1152:
1130:
1122:
1118:
1116:
1025:
1020:underdamped.
1019:
1011:
1000:
994:
962:
955:
950:
944:
939:
933:
928:
922:
917:
907:
890:
888:
824:
818:
811:
805:Underdamped
591: /
587: /
585:displacement
583: /
444: /
406:Displacement
400:
344:
335:
329:Formulations
316:Virtual work
256: /
196:Acceleration
189:Fundamentals
3828:dissipated.
2945:tuning fork
2694:underdamped
2581:Underdamped
2404:satisfying
1139:engineering
1131:underdamped
929:underdamped
837:dissipation
727:von Neumann
394:Core topics
18:Damped wave
4200:Categories
4162:2023-07-15
4082:2021-06-09
4033:2021-06-09
3700:References
3410:step input
2734:overdamped
2699:Overdamped
2152:Derivation
1145:is losing
1012:overdamped
951:overdamped
938:) through
860:resistance
662:d'Alembert
642:Maupertuis
605:Scientists
487:Rigid body
161:Kinematics
3594:
3572:π
3543:
3537:−
3528:ζ
3484:ζ
3480:−
3472:π
3469:ζ
3463:−
3455:
3418:overshoot
3406:overshoot
3256:−
3231:−
3215:
3182:
3149:
3140:δ
3073:
3064:δ
3030:π
3009:δ
3004:δ
2996:ζ
2973:δ
2927:α
2907:α
2904:−
2894:real part
2870:ζ
2836:ω
2832:α
2806:ζ
2750:ζ
2714:ζ
2674:ζ
2671:≤
2629:ζ
2625:−
2611:ω
2599:
2554:ω
2544:
2515:ζ
2457:ζ
2453:−
2442:±
2439:ζ
2425:ω
2421:−
2396:are both
2296:ω
2263:ω
2259:ζ
2168:ω
2107:ω
2074:ω
1820:ζ
1731:ζ
1671:ω
1658:λ
1647:λ
1641:ζ
1621:ω
1613:λ
1607:ζ
1587:ζ
1564:λ
1536:λ
1519:
1506:Half-life
1484:λ
1470:τ
1435:−
1407:π
1393:ω
1377:Frequency
1352:ω
1323:φ
1297:λ
1214:φ
1211:−
1205:ω
1199:
1188:λ
1185:−
1096:π
1087:
1076:−
903:attenuate
856:radiation
707:Liouville
589:frequency
509:Vibration
226:potential
151:Continuum
146:Celestial
123:Textbooks
3823:18309860
2504:Undamped
1701:Q factor
1001:undamped
918:undamped
884:friction
762:Category
687:Hamilton
672:Lagrange
667:Clairaut
632:Horrocks
593:velocity
563:Pendulum
551:reactive
523:Rotation
492:dynamics
442:Inertial
428:Friction
311:Velocity
286:Momentum
166:Kinetics
156:Dynamics
134:Branches
118:Timeline
4186:Physics
4175:Damping
3803:Bibcode
2398:complex
2140:is the
1364:is the
1229:where:
1135:science
1008:viscous
845:viscous
825:damping
722:Koopman
682:Poisson
677:Laplace
622:Huygens
617:Galileo
462: (
401:Damping
254:Inertia
248:Impulse
221:kinetic
171:Statics
141:Applied
113:History
4153:
4116:
4055:
4003:
3948:
3883:
3862:
3821:
3750:damped
3684:brakes
3427:) by:
3412:, the
3284:where
3112:where
3056:where
2789:factor
2665:
2388:where
2095:where
1802:, and
1147:energy
979:, and
958:> 1
936:< 1
831:of an
829:energy
814:< 1
760:
712:Appell
697:Cauchy
692:Jacobi
637:Halley
627:Newton
612:Kepler
464:linear
460:Motion
306:Torque
281:Moment
216:Energy
206:Couple
4190:Lumen
3819:S2CID
1779:of a
1556:0.693
1454:hertz
1452:, or
1125:is a
949:) to
893:is a
878:(ex.
876:bikes
809:with
717:Gibbs
702:Routh
657:Euler
296:Speed
291:Space
233:Force
4151:ISBN
4114:ISBN
4053:ISBN
4001:ISBN
3946:ISBN
3881:ISBN
3860:ISBN
3119:and
2873:<
2784:The
2717:>
2677:<
2392:and
1767:The
1682:<
1137:and
914:zeta
889:The
874:and
848:drag
301:Time
264:Mass
4188:".
4177:".
4143:doi
4106:doi
3811:doi
3799:463
3774:doi
3726:doi
3611:100
3560:100
3452:exp
3449:100
3400:In
2896:of
2702:If
2596:exp
2584:If
2541:exp
2022:or
1340:= 0
1196:cos
1121:or
1084:cos
960:).
947:= 1
927:),
925:= 0
862:in
850:),
835:by
819:In
4202::
4149:,
4137:,
4112:.
4100:.
4075:.
4026:.
4015:^
3908:.
3904:.
3844:.
3825:.
3817:.
3809:.
3797:.
3770:11
3768:.
3724:.
3720:.
3708:^
3585:ln
3540:ln
3404:,
3365:,
3311:,
3212:ln
3179:ln
3146:ln
3070:ln
2316:0.
1969:.
1703::
1516:ln
1462::
1379::
1117:A
975:,
971:,
967:,
912:("
858:,
854:,
823:,
4192:.
4181:.
4173:"
4145::
4122:.
4108::
4085:.
4061:.
4036:.
4009:.
3968:.
3954:.
3912:.
3910:2
3889:.
3813::
3805::
3780:.
3776::
3732:.
3728::
3616:)
3607:O
3604:P
3598:(
3589:2
3581:+
3576:2
3565:)
3556:O
3553:P
3547:(
3531:=
3515:ζ
3497:)
3488:2
3477:1
3459:(
3446:=
3442:O
3439:P
3425:ζ
3378:4
3374:x
3351:2
3347:x
3324:3
3320:x
3297:1
3293:x
3264:4
3260:x
3251:3
3247:x
3239:2
3235:x
3226:1
3222:x
3209:=
3202:4
3198:x
3192:2
3188:x
3176:=
3169:3
3165:x
3159:1
3155:x
3143:=
3124:1
3121:x
3117:0
3114:x
3093:1
3089:x
3083:0
3079:x
3067:=
3039:2
3034:)
3027:2
3023:(
3018:+
3013:2
2999:=
2876:1
2847:.
2840:n
2827:=
2821:Q
2818:2
2814:1
2809:=
2793:ζ
2787:Q
2778:Q
2756:1
2753:=
2720:1
2704:s
2680:1
2668:0
2644:)
2640:t
2633:2
2622:1
2615:n
2607:i
2603:(
2586:s
2566:)
2563:t
2558:n
2550:i
2547:(
2521:0
2518:=
2490:s
2473:.
2469:)
2461:2
2450:1
2445:i
2435:(
2429:n
2418:=
2415:s
2402:s
2394:s
2390:C
2373:,
2368:t
2365:s
2361:e
2357:C
2354:=
2351:)
2348:t
2345:(
2342:x
2313:=
2310:x
2305:2
2300:n
2292:+
2286:t
2283:d
2278:x
2275:d
2267:n
2256:2
2253:+
2245:2
2241:t
2237:d
2232:x
2227:2
2223:d
2193:m
2188:/
2183:k
2177:=
2172:n
2125:m
2122:k
2116:=
2111:n
2078:n
2070:m
2067:2
2064:=
2058:m
2055:k
2049:m
2046:2
2043:=
2038:c
2034:c
2005:m
2002:k
1997:2
1994:=
1989:c
1985:c
1957:0
1954:=
1951:x
1948:k
1945:+
1939:t
1936:d
1931:x
1928:d
1922:c
1919:+
1911:2
1907:t
1903:d
1898:x
1893:2
1889:d
1882:m
1856:,
1843:=
1836:c
1832:c
1828:c
1823:=
1806:k
1800:c
1796:m
1773:ζ
1748:Q
1734:)
1728:2
1725:(
1721:/
1717:1
1714:=
1711:Q
1697:.
1685:1
1675:2
1667:+
1662:2
1651:/
1644:=
1617:/
1610:=
1576:.
1560:/
1532:/
1528:)
1525:2
1522:(
1502:.
1499:e
1480:/
1476:1
1473:=
1456:.
1438:1
1431:t
1410:)
1404:2
1401:(
1397:/
1390:=
1387:f
1368:.
1342:;
1338:t
1313:;
1311:t
1275:A
1265:;
1263:t
1249:)
1246:t
1243:(
1240:y
1217:)
1208:t
1202:(
1191:t
1181:e
1177:A
1174:=
1171:)
1168:t
1165:(
1162:y
1102:)
1099:t
1093:2
1090:(
1079:t
1072:e
1068:=
1065:)
1062:t
1059:(
1056:y
1041:.
1014:.
1003:.
956:ζ
953:(
945:ζ
942:(
934:ζ
931:(
923:ζ
920:(
910:ζ
812:ζ
790:e
783:t
776:v
466:)
85:t
82:d
76:p
72:d
66:=
61:F
34:.
20:)
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