761:
1808:
4818:
4028:
2590:
2913:
2753:
3491:
3867:
This approximation can be better understood by looking at the potential curve of the system. By thinking of the potential curve as a hill, in which, if one placed a ball anywhere on the curve, the ball would roll down with the slope of the potential curve. This is true due to the relationship between
2917:
Depending on the starting point of the masses, this system has 2 possible frequencies (or a combination of the two). If the masses are started with their displacements in the same direction, the frequency is that of a single mass system, because the middle spring is never extended. If the two masses
735:
oscillators, different directions have different constants of restoring forces. The solution is similar to isotropic oscillators, but there is a different frequency in each direction. Varying the frequencies relative to each other can produce interesting results. For example, if the frequency in one
780:
which continually convert some of the energy stored in the oscillator into heat in the environment. This is called damping. Thus, oscillations tend to decay with time unless there is some net source of energy into the system. The simplest description of this decay process can be illustrated by
722:
2064:
2386:
2760:
3766:
259:
stored in the spring at the extremes of its path. The spring-mass system illustrates some common features of oscillation, namely the existence of an equilibrium and the presence of a restoring force which grows stronger the further the system deviates from equilibrium.
784:
Damped oscillators are created when a resistive force is introduced, which is dependent on the first derivative of the position, or in this case velocity. The differential equation created by Newton's second law adds in this resistive force with an arbitrary constant
2928:
Coupled oscillators are a common description of two related, but different phenomena. One case is where both oscillations affect each other mutually, which usually leads to the occurrence of a single, entrained oscillation state, where both oscillate with a
3241:
1819:. More complicated systems have more degrees of freedom, for example, two masses and three springs (each mass being attached to fixed points and to each other). In such cases, the behavior of each variable influences that of the others. This leads to a
2597:
3248:
1629:
1721:
492:
of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between the positive and negative amplitude forever without friction.
2371:
560:
1100:
3863:
1842:
3662:
1516:
3576:
3669:
2189:
1831:
in 1665. The apparent motions of the compound oscillations typically appears very complicated but a more economic, computationally simpler and conceptually deeper description is given by resolving the motion into
1326:
934:
1162:
3253:
2252:
1785:, that is, when the driving frequency is equal to the natural frequency of the system. When this occurs, the denominator of the amplitude is minimized, which maximizes the amplitude of the oscillations.
373:
3073:
3078:
2765:
2602:
2391:
565:
1395:
2585:{\displaystyle {\begin{aligned}m_{1}=m_{2}=m,\;\;k_{1}=k_{2}=k_{3}=k,\\M={\begin{bmatrix}m&0\\0&m\end{bmatrix}},\;\;k={\begin{bmatrix}2k&-k\\-k&2k\end{bmatrix}}\end{aligned}}}
1839:
The simplest form of coupled oscillators is a 3 spring, 2 mass system, where masses and spring constants are the same. This problem begins with deriving Newton's second law for both masses.
2908:{\displaystyle {\begin{aligned}&\left(3k-m\omega ^{2}\right)\left(k-m\omega ^{2}\right)=0\\&\omega _{1}={\sqrt {\frac {k}{m}}},\;\;\omega _{2}={\sqrt {\frac {3k}{m}}}\end{aligned}}}
4039:
The mathematics of oscillation deals with the quantification of the amount that a sequence or function tends to move between extremes. There are several related notions: oscillation of a
852:
472:
4788:
553:
3921:
970:
2116:
978:
411:
3773:
3066:
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1823:
of the oscillations of the individual degrees of freedom. For example, two pendulum clocks (of identical frequency) mounted on a common wall will tend to synchronise. This
1521:
3583:
1634:
2945:
In physics, a system with a set of conservative forces and an equilibrium point can be approximated as a harmonic oscillator near equilibrium. An example of this is the
1402:
2748:{\displaystyle {\begin{aligned}\left(k-M\omega ^{2}\right)a&=0\\{\begin{bmatrix}2k-m\omega ^{2}&-k\\-k&2k-m\omega ^{2}\end{bmatrix}}&=0\end{aligned}}}
2933:. Another case is where one external oscillation affects an internal oscillation, but is not affected by this. In this case the regions of synchronization, known as
2257:
297:
3925:
By thinking of the potential in this way, one will see that at any local minimum there is a "well" in which the ball would roll back and forth (oscillate) between
3495:
The system will undergo oscillations near the equilibrium point. The force that creates these oscillations is derived from the effective potential constant above:
3486:{\displaystyle {\begin{aligned}\gamma _{\text{eff}}&=\left.{\frac {d^{2}U}{dr^{2}}}\right|_{r=r_{0}}=U_{0}\left\\&={\frac {114U_{0}}{r^{2}}}\end{aligned}}}
4749:
3498:
1242:
1816:
859:
760:
717:{\displaystyle {\begin{aligned}x(t)&=A_{x}\cos(\omega t-\delta _{x}),\\y(t)&=A_{y}\cos(\omega t-\delta _{y}),\\&\;\,\vdots \end{aligned}}}
2059:{\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1}+k_{2})x_{1}+k_{2}x_{2}\\m_{2}{\ddot {x}}_{2}=k_{2}x_{1}-(k_{2}+k_{3})x_{2}\end{cases}}}
505:
oscillator, where the restoring force is proportional to the displacement from equilibrium with the same restorative constant in all directions.
308:
224:
is added to the system, the point of equilibrium is shifted. The time taken for an oscillation to occur is often referred to as the oscillatory
2121:
49:
1794:
1736:
is the transient solution to the differential equation. The transient solution can be found by using the initial conditions of the system.
2925:, where the oscillation alternates between the elongation of a vertical spring and the rotation of an object at the end of that spring.
1739:
Some systems can be excited by energy transfer from the environment. This transfer typically occurs where systems are embedded in some
4831:
4507:
212:
on the mass, tending to bring it back to equilibrium. However, in moving the mass back to the equilibrium position, it has acquired
4772:
4725:
4430:
4032:
1797:
Two pendulums with the same period fixed on a string act as pair of coupled oscillators. The oscillation alternates between the two.
792:
3761:{\displaystyle \omega _{0}={\sqrt {\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}}={\sqrt {\frac {114U_{0}}{r^{2}m_{\text{eff}}}}}}
2194:
1105:
418:
736:
direction is twice that of another, a figure eight pattern is produced. If the ratio of frequencies is irrational, the motion is
4609:
4535:
4445:
4204:
508:
4604:
3871:
1807:
2921:
More special cases are the coupled oscillators where energy alternates between two forms of oscillation. Well-known is the
4522:
4517:
4485:
4371:
2071:
4337:
1331:
501:
In two or three dimensions, harmonic oscillators behave similarly to one dimension. The simplest example of this is an
4494:
4450:
4327:
4104:
769:
38:
31:
2952:
216:
which keeps it moving beyond that position, establishing a new restoring force in the opposite sense. If a constant
4455:
4244:
4468:
4381:
4098:
4022:
3236:{\displaystyle {\begin{aligned}{\frac {dU}{dr}}&=0=U_{0}\left\\\Rightarrow r&\approx r_{0}\end{aligned}}}
4013:
number of normal modes and their oscillations occur in the form of waves that can characteristically propagate.
42:
2946:
755:
4822:
270:
4615:
4578:
4249:
4219:
4052:
4048:
737:
244:
231:
The systems where the restoring force on a body is directly proportional to its displacement, such as the
205:
180:
72:
4690:
4512:
4352:
4279:
4229:
4199:
4194:
4152:
939:
777:
83:. Oscillations can be used in physics to approximate complex interactions, such as those between atoms.
1624:{\displaystyle A={\sqrt {\frac {f_{0}^{2}}{(\omega _{0}^{2}-\omega ^{2})^{2}+4\beta ^{2}\omega ^{2}}}}}
378:
740:. This motion is periodic on each axis, but is not periodic with respect to r, and will never repeat.
4851:
4675:
4635:
4489:
4332:
4164:
4134:
2922:
232:
156:
3955:
3928:
1851:
1716:{\displaystyle \delta =\tan ^{-1}\left({\frac {2\beta \omega }{\omega _{0}^{2}-\omega ^{2}}}\right)}
4440:
4189:
4174:
4159:
4119:
4084:
3998:
1763:
1744:
1228:
749:
236:
80:
53:
4743:
4599:
4366:
4342:
4299:
4262:
4209:
2918:
are started in opposite directions, the second, faster frequency is the frequency of the system.
1828:
1223:
302:
208:
state when the spring is static. If the system is displaced from the equilibrium, there is a net
103:
75:) or between two or more different states. Familiar examples of oscillation include a swinging
4828:
4768:
4731:
4721:
4680:
4670:
4620:
4588:
4224:
4179:
4114:
4079:
2366:{\displaystyle k={\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\-k_{2}&k_{2}+k_{3}\end{bmatrix}}}
1824:
1802:
1175:
is the damping coefficient. There are 3 categories of damped oscillators: under-damped, where
1168:
201:
123:
64:
557:
This produces a similar solution, but now there is a different equation for every direction.
4660:
4655:
4630:
4573:
4553:
4386:
4347:
4322:
4303:
4239:
4184:
4139:
4129:
4094:
256:
240:
160:
131:
1095:{\displaystyle x(t)=e^{-\beta t}\left(C_{1}e^{\omega _{1}t}+C_{2}e^{-\omega _{1}t}\right),}
4835:
4435:
4285:
4090:
4074:
4006:
3858:{\displaystyle \omega _{0}={\sqrt {\left.{\frac {d^{2}U}{dr^{2}}}\right\vert _{r=r_{0}}}}}
3580:
This differential equation can be re-written in the form of a simple harmonic oscillator:
1759:
204:. Such a system may be approximated on an air table or ice surface. The system is in an
193:
148:
1222:
In addition, an oscillating system may be subject to some external force, as when an AC
4563:
4558:
4405:
4400:
4307:
4289:
264:
252:
168:
152:
95:
87:
3245:
The second derivative is then found, and used to be the effective potential constant:
4845:
4568:
4548:
3657:{\displaystyle {\ddot {r}}+{\frac {\gamma _{\text{eff}}}{m_{\text{eff}}}}(r-r_{0})=0}
2934:
1766:, leading to a still greater displacement. At sufficiently large displacements, the
1226:
is connected to an outside power source. In this case the oscillation is said to be
415:
The solution to this differential equation produces a sinusoidal position function:
48:
4473:
4109:
3997:
As the number of degrees of freedom becomes arbitrarily large, a system approaches
3980:
1793:
1748:
1511:{\displaystyle x(t)=A\cos(\omega t-\delta )+A_{tr}\cos(\omega _{1}t-\delta _{tr}),}
1770:
of the wing dominates to provide the restoring force that enables an oscillation.
4650:
4625:
4234:
4214:
4124:
4044:
1833:
489:
91:
147:
Oscillation, especially rapid oscillation, may be an undesirable phenomenon in
4416:
4410:
732:
127:
4735:
4685:
4665:
4640:
4594:
1778:
1767:
1236:
140:
135:
99:
17:
4817:
3571:{\displaystyle F=-\gamma _{\text{eff}}(r-r_{0})=m_{\text{eff}}{\ddot {r}}}
4715:
4583:
4056:
4040:
4010:
2937:, can lead to highly complex phenomena as for instance chaotic dynamics.
1752:
773:
502:
213:
76:
4031:
Oscillation of a sequence (shown in blue) is the difference between the
2184:{\displaystyle M={\begin{bmatrix}m_{1}&0\\0&m_{2}\end{bmatrix}}}
4376:
4169:
248:
221:
115:
107:
764:
Phase portrait of damped oscillator, with increasing damping strength.
4645:
4296:
at frequencies 10 Hz – 10 Hz in the transient regime)
197:
190:
186:
119:
111:
1321:{\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=f(t),}
90:
in virtually every area of science: for example the beating of the
4293:
4275:
4027:
4026:
4002:
1806:
1792:
1740:
759:
217:
164:
47:
929:{\displaystyle {\ddot {x}}+2\beta {\dot {x}}+\omega _{0}^{2}x=0,}
247:. In the spring-mass system, oscillations occur because, at the
3992:
1815:
The harmonic oscillator and the systems it models have a single
1755:
68:
235:
of the spring-mass system, are described mathematically by the
86:
Oscillations occur not only in mechanical systems but also in
2594:
These matrices can now be plugged into the general solution.
1157:{\textstyle \omega _{1}={\sqrt {\beta ^{2}-\omega _{0}^{2}}}}
2757:
The determinant of this matrix yields a quadratic equation.
2247:{\displaystyle x={\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}}
1762:
of the wing on the air flow and a consequential increase in
1235:
The simplest example of this is a spring-mass system with a
163:. In these cases it is called chattering or flapping, as in
3794:
3275:
2052:
1811:
Experimental Setup of
Huygens synchronization of two clocks
368:{\displaystyle {\ddot {x}}=-{\frac {k}{m}}x=-\omega ^{2}x,}
71:, of some measure about a central value (often a point of
27:
Repetitive variation of some measure about a central value
4001:; examples include a string or the surface of a body of
789:. This example assumes a linear dependence on velocity.
144:
is precisely used to describe a mechanical oscillation.
3070:
The equilibrium points of the function are then found:
1167:
The exponential term outside of the parenthesis is the
2656:
2535:
2488:
2272:
2209:
2136:
1108:
942:
772:. This means there are dissipative processes such as
381:
3958:
3931:
3874:
3776:
3672:
3586:
3501:
3251:
3076:
2955:
2763:
2600:
2389:
2260:
2197:
2124:
2074:
2068:
The equations are then generalized into matrix form.
1845:
1758:(from its equilibrium) results in an increase in the
1637:
1524:
1405:
1334:
1245:
981:
862:
795:
563:
511:
421:
311:
273:
3979:. This approximation is also useful for thinking of
1751:
occurs when an arbitrarily small displacement of an
3971:
3944:
3915:
3857:
3760:
3656:
3570:
3485:
3235:
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2747:
2584:
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2183:
2110:
2058:
1715:
1623:
1510:
1389:
1320:
1156:
1094:
964:
928:
846:
716:
547:
466:
405:
367:
291:
4612:for computing periodicity in unevenly spaced data
1390:{\displaystyle f(t)=f_{0}\cos(\omega t+\delta ).}
30:"Oscillator" redirects here. For other uses, see
4051:at a point, and oscillation of a function on an
267:states that the restoring force of a spring is:
185:The simplest mechanical oscillating system is a
37:"Vibrating" redirects here. For other uses, see
4838: – a chapter from an online textbook
4765:Sync: The Emerging Science of Spontaneous Order
4591:for computing periodicity in evenly spaced data
1781:occurs in a damped driven oscillator when ω = ω
3666:Thus, the frequency of small oscillations is:
781:oscillation decay of the harmonic oscillator.
4165:Armstrong (or Tickler or Meissner) oscillator
847:{\displaystyle m{\ddot {x}}+b{\dot {x}}+kx=0}
8:
467:{\displaystyle x(t)=A\cos(\omega t-\delta )}
305:, the differential equation can be derived:
130:in the brain, and the periodic swelling of
4748:: CS1 maint: location missing publisher (
2870:
2869:
2523:
2522:
2427:
2426:
856:This equation can be rewritten as before:
705:
4377:gonadotropin releasing hormone pulsations
3963:
3957:
3936:
3930:
3875:
3873:
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3834:
3821:
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2196:
2167:
2143:
2131:
2123:
2085:
2084:
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2043:
2030:
2017:
2001:
1991:
1978:
1967:
1966:
1959:
1945:
1935:
1922:
1909:
1896:
1877:
1866:
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1858:
1846:
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1700:
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1538:
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1493:
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1333:
1291:
1286:
1268:
1267:
1247:
1246:
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1146:
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1128:
1122:
1113:
1107:
1073:
1065:
1055:
1037:
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1022:
1001:
980:
952:
941:
908:
903:
885:
884:
864:
863:
861:
818:
817:
800:
799:
794:
706:
686:
658:
619:
591:
564:
562:
534:
533:
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510:
420:
393:
388:
380:
353:
330:
313:
312:
310:
272:
4703:
548:{\displaystyle {\vec {F}}=-k{\vec {r}}}
263:In the case of the spring-mass system,
251:equilibrium displacement, the mass has
4741:
3916:{\displaystyle {\frac {dU}{dt}}=-F(r)}
2383:can be substituted into the matrices.
1743:flow. For example, the phenomenon of
768:All real-world oscillator systems are
3987:Continuous systems – waves
480:is the frequency of the oscillation,
7:
4767:. Hyperion Press. pp. 106–109.
4709:
4707:
975:This produces the general solution:
4282:with frequency of order 10 Hz)
2949:, where the potential is given by:
965:{\textstyle 2\beta ={\frac {b}{m}}}
2111:{\displaystyle F=M{\ddot {x}}=kx,}
406:{\textstyle \omega ={\sqrt {k/m}}}
25:
4431:Atlantic multidecadal oscillation
4306:, as well as to a usual model in
4033:limit superior and limit inferior
4816:
159:), where the aim is convergence
4610:Least-squares spectral analysis
4205:Extended interaction oscillator
4047:, oscillation of a real-valued
3061:{\displaystyle U(r)=U_{0}\left}
2941:Small oscillation approximation
1201:; and critically damped, where
4292:(pulsation of output power of
3972:{\displaystyle r_{\text{max}}}
3945:{\displaystyle r_{\text{min}}}
3910:
3904:
3645:
3626:
3540:
3521:
3404:
3398:
3361:
3355:
3206:
2965:
2959:
2036:
2010:
1915:
1889:
1582:
1550:
1502:
1470:
1445:
1430:
1415:
1409:
1381:
1366:
1344:
1338:
1312:
1306:
991:
985:
770:thermodynamically irreversible
692:
670:
644:
638:
625:
603:
577:
571:
539:
518:
461:
446:
431:
425:
1:
4508:Belousov–Zhabotinsky reaction
4536:Cellular Automata oscillator
4486:Neutral particle oscillation
4446:El Niño-Southern Oscillation
4372:Insulin release oscillations
4005:. Such systems have (in the
3868:potential energy and force.
4720:. Mill Valley, California.
4495:Quantum harmonic oscillator
4451:Pacific decadal oscillation
4328:Bacterial Circadian Rhythms
4105:Quantum harmonic oscillator
497:Two-dimensional oscillators
175:Simple harmonic oscillation
39:Vibrations (disambiguation)
32:Oscillator (disambiguation)
4868:
4789:"23.7: Small Oscillations"
4456:Quasi-biennial oscillation
4150:
4020:
3990:
1800:
753:
747:
237:simple harmonic oscillator
178:
118:, vibration of strings in
36:
29:
4763:Strogatz, Steven (2003).
4605:Madden–Julian oscillation
4382:Pilot-induced oscillation
4099:Neutron-star oscillations
4089:Oscillations in the Sun (
4023:Oscillation (mathematics)
1399:This gives the solution:
4714:Taylor, John R. (2005).
4523:Bray–Liebhafsky reaction
4518:Briggs–Rauscher reaction
4302:may refer to an optical
255:which is converted into
67:variation, typically in
56:is an oscillatory system
43:Vibrate (disambiguation)
4338:Lotka–Volterra equation
2947:Lennard-Jones potential
756:Anti-vibration compound
727:Anisotropic oscillators
4616:Oscillator phase noise
4579:Earthquake engineering
4424:Climate and geophysics
4250:Wien bridge oscillator
4220:Phase-shift oscillator
4036:
3973:
3946:
3917:
3859:
3762:
3658:
3572:
3487:
3237:
3062:
2909:
2749:
2586:
2367:
2248:
2185:
2112:
2060:
1827:was first observed by
1812:
1798:
1717:
1625:
1512:
1391:
1322:
1158:
1096:
966:
930:
848:
765:
718:
549:
484:is the amplitude, and
468:
407:
369:
293:
245:simple harmonic motion
181:Simple harmonic motion
57:
4691:Vibrator (mechanical)
4513:Mercury beating heart
4490:neutrino oscillations
4280:electromagnetic field
4230:Relaxation oscillator
4200:Electronic oscillator
4195:Delay-line oscillator
4170:Astable multivibrator
4153:Electronic oscillator
4030:
3974:
3947:
3918:
3860:
3763:
3659:
3573:
3488:
3238:
3063:
2910:
2750:
2587:
2368:
2249:
2186:
2113:
2061:
1810:
1796:
1718:
1626:
1513:
1392:
1323:
1188:; over-damped, where
1159:
1097:
967:
931:
849:
778:electrical resistance
763:
719:
550:
469:
408:
370:
294:
292:{\displaystyle F=-kx}
126:, periodic firing of
106:population cycles in
63:is the repetitive or
51:
4825:at Wikimedia Commons
4676:Structural stability
4636:Reciprocating motion
4333:Circadian oscillator
4135:Wilberforce pendulum
3956:
3929:
3872:
3774:
3770:Or, in general form
3670:
3584:
3499:
3249:
3074:
2953:
2931:compromise frequency
2923:Wilberforce pendulum
2761:
2598:
2387:
2258:
2195:
2122:
2072:
1843:
1789:Coupled oscillations
1635:
1522:
1403:
1332:
1243:
1106:
979:
940:
860:
793:
561:
509:
419:
379:
309:
271:
157:sliding mode control
94:(for circulation),
4717:Classical mechanics
4441:Climate oscillation
4394:Economic and social
4190:Colpitts oscillator
4175:Blocking oscillator
4160:Alternating current
4120:Torsional vibration
4085:Helmholtz resonator
3421:
3378:
3183:
3149:
1725:The second term of
1692:
1567:
1548:
1296:
1218:Driven oscillations
1151:
913:
750:Harmonic oscillator
744:Damped oscillations
303:Newton's second law
243:motion is known as
81:alternating current
4834:2010-12-14 at the
4793:Physics LibreTexts
4600:Hidden oscillation
4480:Quantum mechanical
4367:Neural oscillation
4353:Segmentation clock
4343:Neural oscillation
4300:Quantum oscillator
4263:Crystal oscillator
4257:Electro-mechanical
4210:Hartley oscillator
4115:String instruments
4037:
3969:
3942:
3913:
3855:
3758:
3654:
3568:
3483:
3481:
3407:
3364:
3233:
3231:
3169:
3135:
3058:
2905:
2903:
2745:
2743:
2725:
2582:
2580:
2572:
2513:
2363:
2357:
2244:
2238:
2181:
2175:
2108:
2056:
2051:
1829:Christiaan Huygens
1813:
1799:
1713:
1678:
1621:
1553:
1534:
1508:
1387:
1318:
1282:
1154:
1137:
1092:
962:
926:
899:
844:
766:
714:
712:
545:
464:
403:
365:
289:
124:string instruments
58:
54:spring–mass system
4821:Media related to
4681:Tuned mass damper
4671:Strange attractor
4621:Periodic function
4589:Fourier transform
4360:Human oscillation
4245:Vačkář oscillator
4225:Pierce oscillator
4180:Butler oscillator
4080:Foucault pendulum
3966:
3939:
3893:
3853:
3828:
3756:
3755:
3751:
3710:
3709:
3706:
3696:
3624:
3621:
3611:
3596:
3565:
3553:
3518:
3477:
3309:
3263:
3099:
3041:
3006:
2899:
2898:
2864:
2863:
2093:
1975:
1874:
1817:degree of freedom
1803:Injection locking
1707:
1619:
1618:
1276:
1255:
1152:
960:
893:
872:
826:
808:
542:
521:
401:
338:
321:
16:(Redirected from
4859:
4820:
4804:
4803:
4801:
4800:
4785:
4779:
4778:
4760:
4754:
4753:
4747:
4739:
4711:
4661:Signal generator
4656:Self-oscillation
4631:Quasiperiodicity
4574:Dynamical system
4554:Beat (acoustics)
4387:Voice production
4348:Oscillating gene
4323:Circadian rhythm
4304:local oscillator
4278:(oscillation of
4240:Royer oscillator
4185:Clapp oscillator
4140:Lever escapement
4130:Vibrating string
4110:Playground swing
4095:asteroseismology
4035:of the sequence.
3978:
3976:
3975:
3970:
3968:
3967:
3964:
3951:
3949:
3948:
3943:
3941:
3940:
3937:
3922:
3920:
3919:
3914:
3894:
3892:
3884:
3876:
3864:
3862:
3861:
3856:
3854:
3852:
3851:
3850:
3849:
3833:
3829:
3827:
3826:
3825:
3812:
3808:
3807:
3797:
3791:
3786:
3785:
3767:
3765:
3764:
3759:
3757:
3754:
3753:
3752:
3749:
3743:
3742:
3732:
3731:
3730:
3717:
3716:
3711:
3708:
3707:
3704:
3698:
3697:
3694:
3688:
3687:
3682:
3681:
3663:
3661:
3660:
3655:
3644:
3643:
3625:
3623:
3622:
3619:
3613:
3612:
3609:
3603:
3598:
3597:
3589:
3577:
3575:
3574:
3569:
3567:
3566:
3558:
3555:
3554:
3551:
3539:
3538:
3520:
3519:
3516:
3492:
3490:
3489:
3484:
3482:
3478:
3476:
3475:
3466:
3465:
3464:
3451:
3443:
3439:
3435:
3434:
3433:
3420:
3415:
3391:
3390:
3377:
3372:
3346:
3345:
3333:
3332:
3331:
3330:
3314:
3310:
3308:
3307:
3306:
3293:
3289:
3288:
3278:
3265:
3264:
3261:
3242:
3240:
3239:
3234:
3232:
3228:
3227:
3201:
3197:
3196:
3195:
3182:
3177:
3162:
3161:
3148:
3143:
3123:
3122:
3100:
3098:
3090:
3082:
3067:
3065:
3064:
3059:
3057:
3053:
3052:
3051:
3046:
3042:
3037:
3036:
3027:
3017:
3016:
3011:
3007:
3002:
3001:
2992:
2980:
2979:
2914:
2912:
2911:
2906:
2904:
2900:
2894:
2886:
2885:
2880:
2879:
2865:
2856:
2855:
2850:
2849:
2839:
2829:
2825:
2824:
2823:
2800:
2796:
2795:
2794:
2767:
2754:
2752:
2751:
2746:
2744:
2730:
2729:
2722:
2721:
2680:
2679:
2633:
2629:
2628:
2627:
2591:
2589:
2588:
2583:
2581:
2577:
2576:
2518:
2517:
2463:
2462:
2450:
2449:
2437:
2436:
2416:
2415:
2403:
2402:
2382:
2378:
2372:
2370:
2369:
2364:
2362:
2361:
2354:
2353:
2341:
2340:
2329:
2328:
2312:
2311:
2297:
2296:
2284:
2283:
2253:
2251:
2250:
2245:
2243:
2242:
2235:
2234:
2221:
2220:
2190:
2188:
2187:
2182:
2180:
2179:
2172:
2171:
2148:
2147:
2117:
2115:
2114:
2109:
2095:
2094:
2086:
2065:
2063:
2062:
2057:
2055:
2054:
2048:
2047:
2035:
2034:
2022:
2021:
2006:
2005:
1996:
1995:
1983:
1982:
1977:
1976:
1968:
1964:
1963:
1950:
1949:
1940:
1939:
1927:
1926:
1914:
1913:
1901:
1900:
1882:
1881:
1876:
1875:
1867:
1863:
1862:
1764:lift coefficient
1735:
1722:
1720:
1719:
1714:
1712:
1708:
1706:
1705:
1704:
1691:
1686:
1676:
1665:
1656:
1655:
1630:
1628:
1627:
1622:
1620:
1617:
1616:
1615:
1606:
1605:
1590:
1589:
1580:
1579:
1566:
1561:
1547:
1542:
1533:
1532:
1517:
1515:
1514:
1509:
1501:
1500:
1482:
1481:
1463:
1462:
1396:
1394:
1393:
1388:
1359:
1358:
1327:
1325:
1324:
1319:
1295:
1290:
1278:
1277:
1269:
1257:
1256:
1248:
1213:
1200:
1187:
1174:
1163:
1161:
1160:
1155:
1153:
1150:
1145:
1133:
1132:
1123:
1118:
1117:
1101:
1099:
1098:
1093:
1088:
1084:
1083:
1082:
1078:
1077:
1060:
1059:
1047:
1046:
1042:
1041:
1027:
1026:
1012:
1011:
971:
969:
968:
963:
961:
953:
935:
933:
932:
927:
912:
907:
895:
894:
886:
874:
873:
865:
853:
851:
850:
845:
828:
827:
819:
810:
809:
801:
788:
723:
721:
720:
715:
713:
701:
691:
690:
663:
662:
624:
623:
596:
595:
554:
552:
551:
546:
544:
543:
535:
523:
522:
514:
487:
483:
479:
473:
471:
470:
465:
412:
410:
409:
404:
402:
397:
389:
374:
372:
371:
366:
358:
357:
339:
331:
323:
322:
314:
298:
296:
295:
290:
257:potential energy
239:and the regular
196:subject to only
132:Cepheid variable
21:
4867:
4866:
4862:
4861:
4860:
4858:
4857:
4856:
4842:
4841:
4836:Wayback Machine
4813:
4808:
4807:
4798:
4796:
4787:
4786:
4782:
4775:
4762:
4761:
4757:
4740:
4728:
4713:
4712:
4705:
4700:
4695:
4544:
4532:
4527:
4503:
4482:
4465:
4460:
4436:Chandler wobble
4426:
4421:
4396:
4391:
4362:
4357:
4318:
4313:
4286:Oscillator Toda
4271:
4259:
4254:
4155:
4149:
4144:
4091:helioseismology
4075:Double pendulum
4070:
4065:
4025:
4019:
4007:classical limit
3995:
3989:
3959:
3954:
3953:
3932:
3927:
3926:
3885:
3877:
3870:
3869:
3841:
3817:
3813:
3799:
3798:
3793:
3792:
3777:
3772:
3771:
3744:
3734:
3733:
3722:
3718:
3699:
3689:
3673:
3668:
3667:
3635:
3614:
3604:
3582:
3581:
3546:
3530:
3511:
3497:
3496:
3480:
3479:
3467:
3456:
3452:
3441:
3440:
3422:
3379:
3351:
3347:
3337:
3322:
3298:
3294:
3280:
3279:
3274:
3273:
3266:
3256:
3247:
3246:
3230:
3229:
3219:
3212:
3203:
3202:
3184:
3150:
3128:
3124:
3114:
3101:
3091:
3083:
3072:
3071:
3028:
3022:
3021:
2993:
2987:
2986:
2985:
2981:
2971:
2951:
2950:
2943:
2902:
2901:
2887:
2871:
2841:
2837:
2836:
2815:
2805:
2801:
2786:
2773:
2769:
2759:
2758:
2742:
2741:
2731:
2724:
2723:
2713:
2699:
2690:
2689:
2681:
2671:
2652:
2648:
2647:
2637:
2619:
2609:
2605:
2596:
2595:
2579:
2578:
2571:
2570:
2562:
2553:
2552:
2544:
2531:
2512:
2511:
2506:
2500:
2499:
2494:
2484:
2474:
2473:
2454:
2441:
2428:
2407:
2394:
2385:
2384:
2380:
2376:
2356:
2355:
2345:
2332:
2330:
2320:
2314:
2313:
2303:
2298:
2288:
2275:
2268:
2256:
2255:
2237:
2236:
2226:
2223:
2222:
2212:
2205:
2193:
2192:
2174:
2173:
2163:
2161:
2155:
2154:
2149:
2139:
2132:
2120:
2119:
2070:
2069:
2050:
2049:
2039:
2026:
2013:
1997:
1987:
1965:
1955:
1952:
1951:
1941:
1931:
1918:
1905:
1892:
1864:
1854:
1847:
1841:
1840:
1805:
1791:
1784:
1776:
1760:angle of attack
1726:
1696:
1677:
1666:
1660:
1644:
1633:
1632:
1607:
1597:
1581:
1571:
1549:
1520:
1519:
1489:
1473:
1451:
1401:
1400:
1350:
1330:
1329:
1241:
1240:
1239:driving force.
1220:
1212:
1202:
1199:
1189:
1186:
1176:
1172:
1124:
1109:
1104:
1103:
1069:
1061:
1051:
1033:
1028:
1018:
1017:
1013:
997:
977:
976:
938:
937:
858:
857:
791:
790:
786:
758:
752:
746:
729:
711:
710:
699:
698:
682:
654:
647:
632:
631:
615:
587:
580:
559:
558:
507:
506:
499:
485:
481:
477:
417:
416:
377:
376:
349:
307:
306:
269:
268:
210:restoring force
183:
177:
161:to stable state
149:process control
96:business cycles
88:dynamic systems
46:
35:
28:
23:
22:
15:
12:
11:
5:
4865:
4863:
4855:
4854:
4844:
4843:
4840:
4839:
4826:
4812:
4811:External links
4809:
4806:
4805:
4780:
4773:
4755:
4726:
4702:
4701:
4699:
4696:
4694:
4693:
4688:
4683:
4678:
4673:
4668:
4663:
4658:
4653:
4648:
4643:
4638:
4633:
4628:
4623:
4618:
4613:
4607:
4602:
4597:
4592:
4586:
4581:
4576:
4571:
4566:
4564:Critical speed
4561:
4559:BIBO stability
4556:
4551:
4545:
4543:
4540:
4539:
4538:
4531:
4528:
4526:
4525:
4520:
4515:
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4448:
4443:
4438:
4433:
4427:
4425:
4422:
4420:
4419:
4414:
4408:
4406:Generation gap
4403:
4401:Business cycle
4397:
4395:
4392:
4390:
4389:
4384:
4379:
4374:
4369:
4363:
4361:
4358:
4356:
4355:
4350:
4345:
4340:
4335:
4330:
4325:
4319:
4317:
4314:
4312:
4311:
4308:quantum optics
4297:
4290:self-pulsation
4283:
4272:
4270:
4267:
4266:
4265:
4258:
4255:
4253:
4252:
4247:
4242:
4237:
4232:
4227:
4222:
4217:
4212:
4207:
4202:
4197:
4192:
4187:
4182:
4177:
4172:
4167:
4162:
4156:
4151:Main article:
4148:
4145:
4143:
4142:
4137:
4132:
4127:
4122:
4117:
4112:
4107:
4102:
4087:
4082:
4077:
4071:
4069:
4066:
4064:
4061:
4021:Main article:
4018:
4015:
3991:Main article:
3988:
3985:
3962:
3935:
3912:
3909:
3906:
3903:
3900:
3897:
3891:
3888:
3883:
3880:
3848:
3844:
3840:
3837:
3832:
3824:
3820:
3816:
3811:
3806:
3802:
3795:
3789:
3784:
3780:
3747:
3741:
3737:
3729:
3725:
3721:
3714:
3702:
3692:
3685:
3680:
3676:
3653:
3650:
3647:
3642:
3638:
3634:
3631:
3628:
3617:
3607:
3601:
3595:
3592:
3564:
3561:
3549:
3545:
3542:
3537:
3533:
3529:
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3523:
3514:
3510:
3507:
3504:
3474:
3470:
3463:
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3438:
3432:
3429:
3425:
3419:
3414:
3410:
3406:
3403:
3400:
3397:
3394:
3389:
3386:
3382:
3376:
3371:
3367:
3363:
3360:
3357:
3354:
3350:
3344:
3340:
3336:
3329:
3325:
3321:
3318:
3313:
3305:
3301:
3297:
3292:
3287:
3283:
3276:
3272:
3269:
3267:
3259:
3255:
3254:
3226:
3222:
3218:
3215:
3213:
3211:
3208:
3205:
3204:
3200:
3194:
3191:
3187:
3181:
3176:
3172:
3168:
3165:
3160:
3157:
3153:
3147:
3142:
3138:
3134:
3131:
3127:
3121:
3117:
3113:
3110:
3107:
3104:
3102:
3097:
3094:
3089:
3086:
3080:
3079:
3056:
3050:
3045:
3040:
3035:
3031:
3025:
3020:
3015:
3010:
3005:
3000:
2996:
2990:
2984:
2978:
2974:
2970:
2967:
2964:
2961:
2958:
2942:
2939:
2935:Arnold Tongues
2897:
2893:
2890:
2883:
2878:
2874:
2868:
2862:
2859:
2853:
2848:
2844:
2840:
2838:
2835:
2832:
2828:
2822:
2818:
2814:
2811:
2808:
2804:
2799:
2793:
2789:
2785:
2782:
2779:
2776:
2772:
2768:
2766:
2740:
2737:
2734:
2732:
2728:
2720:
2716:
2712:
2709:
2706:
2703:
2700:
2698:
2695:
2692:
2691:
2688:
2685:
2682:
2678:
2674:
2670:
2667:
2664:
2661:
2658:
2657:
2655:
2650:
2649:
2646:
2643:
2640:
2638:
2636:
2632:
2626:
2622:
2618:
2615:
2612:
2608:
2604:
2603:
2575:
2569:
2566:
2563:
2561:
2558:
2555:
2554:
2551:
2548:
2545:
2543:
2540:
2537:
2536:
2534:
2529:
2526:
2521:
2516:
2510:
2507:
2505:
2502:
2501:
2498:
2495:
2493:
2490:
2489:
2487:
2482:
2479:
2476:
2475:
2472:
2469:
2466:
2461:
2457:
2453:
2448:
2444:
2440:
2435:
2431:
2425:
2422:
2419:
2414:
2410:
2406:
2401:
2397:
2393:
2392:
2375:The values of
2360:
2352:
2348:
2344:
2339:
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2327:
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2319:
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2306:
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2299:
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2287:
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2278:
2274:
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2271:
2266:
2263:
2241:
2233:
2229:
2225:
2224:
2219:
2215:
2211:
2210:
2208:
2203:
2200:
2178:
2170:
2166:
2162:
2160:
2157:
2156:
2153:
2150:
2146:
2142:
2138:
2137:
2135:
2130:
2127:
2107:
2104:
2101:
2098:
2092:
2089:
2083:
2080:
2077:
2053:
2046:
2042:
2038:
2033:
2029:
2025:
2020:
2016:
2012:
2009:
2004:
2000:
1994:
1990:
1986:
1981:
1974:
1971:
1962:
1958:
1954:
1953:
1948:
1944:
1938:
1934:
1930:
1925:
1921:
1917:
1912:
1908:
1904:
1899:
1895:
1891:
1888:
1885:
1880:
1873:
1870:
1861:
1857:
1853:
1852:
1850:
1801:Main article:
1790:
1787:
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1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1357:
1353:
1349:
1346:
1343:
1340:
1337:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1294:
1289:
1285:
1281:
1275:
1272:
1266:
1263:
1260:
1254:
1251:
1219:
1216:
1210:
1197:
1184:
1169:decay function
1149:
1144:
1140:
1136:
1131:
1127:
1121:
1116:
1112:
1091:
1087:
1081:
1076:
1072:
1068:
1064:
1058:
1054:
1050:
1045:
1040:
1036:
1031:
1025:
1021:
1016:
1010:
1007:
1004:
1000:
996:
993:
990:
987:
984:
959:
956:
951:
948:
945:
925:
922:
919:
916:
911:
906:
902:
898:
892:
889:
883:
880:
877:
871:
868:
843:
840:
837:
834:
831:
825:
822:
816:
813:
807:
804:
798:
748:Main article:
745:
742:
728:
725:
709:
704:
702:
700:
697:
694:
689:
685:
681:
678:
675:
672:
669:
666:
661:
657:
653:
650:
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640:
637:
634:
633:
630:
627:
622:
618:
614:
611:
608:
605:
602:
599:
594:
590:
586:
583:
581:
579:
576:
573:
570:
567:
566:
541:
538:
532:
529:
526:
520:
517:
498:
495:
463:
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
427:
424:
400:
396:
392:
387:
384:
364:
361:
356:
352:
348:
345:
342:
337:
334:
329:
326:
320:
317:
288:
285:
282:
279:
276:
253:kinetic energy
189:attached to a
179:Main article:
176:
173:
169:route flapping
153:control theory
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4864:
4853:
4850:
4849:
4847:
4837:
4833:
4830:
4827:
4824:
4819:
4815:
4814:
4810:
4794:
4790:
4784:
4781:
4776:
4774:0-786-86844-9
4770:
4766:
4759:
4756:
4751:
4745:
4737:
4733:
4729:
4727:1-891389-22-X
4723:
4719:
4718:
4710:
4708:
4704:
4697:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4662:
4659:
4657:
4654:
4652:
4649:
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4639:
4637:
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4629:
4627:
4624:
4622:
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4614:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4569:Cycle (music)
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4549:Antiresonance
4547:
4546:
4541:
4537:
4534:
4533:
4529:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4505:
4500:
4496:
4493:
4491:
4487:
4484:
4483:
4479:
4475:
4472:
4470:
4469:Neutron stars
4467:
4466:
4462:
4457:
4454:
4452:
4449:
4447:
4444:
4442:
4439:
4437:
4434:
4432:
4429:
4428:
4423:
4418:
4415:
4412:
4409:
4407:
4404:
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4399:
4398:
4393:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4364:
4359:
4354:
4351:
4349:
4346:
4344:
4341:
4339:
4336:
4334:
4331:
4329:
4326:
4324:
4321:
4320:
4315:
4309:
4305:
4301:
4298:
4295:
4291:
4287:
4284:
4281:
4277:
4274:
4273:
4268:
4264:
4261:
4260:
4256:
4251:
4248:
4246:
4243:
4241:
4238:
4236:
4233:
4231:
4228:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4206:
4203:
4201:
4198:
4196:
4193:
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4183:
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4178:
4176:
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4171:
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4146:
4141:
4138:
4136:
4133:
4131:
4128:
4126:
4123:
4121:
4118:
4116:
4113:
4111:
4108:
4106:
4103:
4100:
4096:
4092:
4088:
4086:
4083:
4081:
4078:
4076:
4073:
4072:
4067:
4062:
4060:
4058:
4054:
4050:
4046:
4042:
4034:
4029:
4024:
4016:
4014:
4012:
4008:
4004:
4000:
3994:
3986:
3984:
3982:
3981:Kepler orbits
3960:
3933:
3923:
3907:
3901:
3898:
3895:
3889:
3886:
3881:
3878:
3865:
3846:
3842:
3838:
3835:
3830:
3822:
3818:
3814:
3809:
3804:
3800:
3787:
3782:
3778:
3768:
3745:
3739:
3735:
3727:
3723:
3719:
3712:
3700:
3690:
3683:
3678:
3674:
3664:
3651:
3648:
3640:
3636:
3632:
3629:
3615:
3605:
3599:
3593:
3590:
3578:
3562:
3559:
3547:
3543:
3535:
3531:
3527:
3524:
3512:
3508:
3505:
3502:
3493:
3472:
3468:
3461:
3457:
3453:
3447:
3445:
3436:
3430:
3427:
3423:
3417:
3412:
3408:
3401:
3395:
3392:
3387:
3384:
3380:
3374:
3369:
3365:
3358:
3352:
3348:
3342:
3338:
3334:
3327:
3323:
3319:
3316:
3311:
3303:
3299:
3295:
3290:
3285:
3281:
3270:
3268:
3257:
3243:
3224:
3220:
3216:
3214:
3209:
3198:
3192:
3189:
3185:
3179:
3174:
3170:
3166:
3163:
3158:
3155:
3151:
3145:
3140:
3136:
3132:
3129:
3125:
3119:
3115:
3111:
3108:
3105:
3103:
3095:
3092:
3087:
3084:
3068:
3054:
3048:
3043:
3038:
3033:
3029:
3023:
3018:
3013:
3008:
3003:
2998:
2994:
2988:
2982:
2976:
2972:
2968:
2962:
2956:
2948:
2940:
2938:
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2932:
2926:
2924:
2919:
2915:
2895:
2891:
2888:
2881:
2876:
2872:
2866:
2860:
2857:
2851:
2846:
2842:
2833:
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2820:
2816:
2812:
2809:
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2802:
2797:
2791:
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2783:
2780:
2777:
2774:
2770:
2755:
2738:
2735:
2733:
2726:
2718:
2714:
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2701:
2696:
2693:
2686:
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2644:
2641:
2639:
2634:
2630:
2624:
2620:
2616:
2613:
2610:
2606:
2592:
2573:
2567:
2564:
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2549:
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2524:
2519:
2514:
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2503:
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2404:
2399:
2395:
2373:
2358:
2350:
2346:
2342:
2337:
2333:
2325:
2321:
2317:
2308:
2304:
2300:
2293:
2289:
2285:
2280:
2276:
2269:
2264:
2261:
2239:
2231:
2227:
2217:
2213:
2206:
2201:
2198:
2176:
2168:
2164:
2158:
2151:
2144:
2140:
2133:
2128:
2125:
2105:
2102:
2099:
2096:
2090:
2087:
2081:
2078:
2075:
2066:
2044:
2040:
2031:
2027:
2023:
2018:
2014:
2007:
2002:
1998:
1992:
1988:
1984:
1979:
1972:
1969:
1960:
1956:
1946:
1942:
1936:
1932:
1928:
1923:
1919:
1910:
1906:
1902:
1897:
1893:
1886:
1883:
1878:
1871:
1868:
1859:
1855:
1848:
1837:
1835:
1830:
1826:
1822:
1818:
1809:
1804:
1795:
1788:
1786:
1780:
1773:
1771:
1769:
1765:
1761:
1757:
1754:
1750:
1746:
1742:
1737:
1733:
1729:
1723:
1709:
1701:
1697:
1693:
1688:
1683:
1679:
1673:
1670:
1667:
1661:
1657:
1652:
1649:
1645:
1641:
1638:
1612:
1608:
1602:
1598:
1594:
1591:
1586:
1576:
1572:
1568:
1563:
1558:
1554:
1544:
1539:
1535:
1528:
1525:
1505:
1497:
1494:
1490:
1486:
1483:
1478:
1474:
1467:
1464:
1459:
1456:
1452:
1448:
1442:
1439:
1436:
1433:
1427:
1424:
1421:
1418:
1412:
1406:
1397:
1384:
1378:
1375:
1372:
1369:
1363:
1360:
1355:
1351:
1347:
1341:
1335:
1315:
1309:
1303:
1300:
1297:
1292:
1287:
1283:
1279:
1273:
1270:
1264:
1261:
1258:
1252:
1249:
1238:
1233:
1231:
1230:
1225:
1217:
1215:
1209:
1205:
1196:
1192:
1183:
1179:
1170:
1165:
1147:
1142:
1138:
1134:
1129:
1125:
1119:
1114:
1110:
1089:
1085:
1079:
1074:
1070:
1066:
1062:
1056:
1052:
1048:
1043:
1038:
1034:
1029:
1023:
1019:
1014:
1008:
1005:
1002:
998:
994:
988:
982:
973:
957:
954:
949:
946:
943:
923:
920:
917:
914:
909:
904:
900:
896:
890:
887:
881:
878:
875:
869:
866:
854:
841:
838:
835:
832:
829:
823:
820:
814:
811:
805:
802:
796:
782:
779:
775:
771:
762:
757:
751:
743:
741:
739:
738:quasiperiodic
734:
726:
724:
707:
703:
695:
687:
683:
679:
676:
673:
667:
664:
659:
655:
651:
649:
641:
635:
628:
620:
616:
612:
609:
606:
600:
597:
592:
588:
584:
582:
574:
568:
555:
536:
530:
527:
524:
515:
504:
496:
494:
491:
474:
458:
455:
452:
449:
443:
440:
437:
434:
428:
422:
413:
398:
394:
390:
385:
382:
362:
359:
354:
350:
346:
343:
340:
335:
332:
327:
324:
318:
315:
304:
299:
286:
283:
280:
277:
274:
266:
261:
258:
254:
250:
246:
242:
238:
234:
229:
227:
223:
219:
215:
211:
207:
203:
199:
195:
192:
188:
182:
174:
172:
170:
167:chatter, and
166:
162:
158:
154:
150:
145:
143:
142:
137:
133:
129:
125:
121:
117:
113:
110:, geothermal
109:
105:
104:predator–prey
101:
97:
93:
89:
84:
82:
78:
74:
70:
66:
62:
55:
50:
44:
40:
33:
19:
4797:. Retrieved
4795:. 2020-07-01
4792:
4783:
4764:
4758:
4716:
4474:Cyclic Model
4463:Astrophysics
4045:real numbers
4038:
3996:
3924:
3866:
3769:
3665:
3579:
3494:
3244:
3069:
2944:
2930:
2927:
2920:
2916:
2756:
2593:
2374:
2067:
1838:
1834:normal modes
1820:
1814:
1777:
1749:aerodynamics
1738:
1731:
1727:
1724:
1398:
1234:
1227:
1221:
1207:
1203:
1194:
1190:
1181:
1177:
1166:
974:
855:
783:
767:
730:
556:
500:
475:
414:
300:
262:
230:
225:
209:
184:
146:
139:
85:
60:
59:
52:An undamped
4852:Oscillation
4823:Oscillation
4651:Seasonality
4626:Phase noise
4235:RLC circuit
4215:Oscillistor
4125:Tuning fork
4017:Mathematics
733:anisotropic
490:phase shift
265:Hooke's law
206:equilibrium
138:. The term
128:nerve cells
92:human heart
73:equilibrium
61:Oscillation
18:Oscillators
4829:Vibrations
4799:2022-04-21
4698:References
4417:News cycle
4411:Malthusian
4316:Biological
4147:Electrical
4093:), stars (
4068:Mechanical
3999:continuity
1825:phenomenon
1237:sinusoidal
754:See also:
122:and other
4744:cite book
4686:Vibration
4666:Squegging
4641:Resonator
4595:Frequency
4530:Computing
4413:economics
3899:−
3779:ω
3691:γ
3675:ω
3633:−
3606:γ
3594:¨
3563:¨
3528:−
3513:γ
3509:−
3428:−
3393:−
3385:−
3258:γ
3217:≈
3207:⇒
3190:−
3156:−
3130:−
3019:−
2873:ω
2843:ω
2817:ω
2810:−
2788:ω
2781:−
2715:ω
2708:−
2694:−
2684:−
2673:ω
2666:−
2621:ω
2614:−
2557:−
2547:−
2318:−
2301:−
2091:¨
2008:−
1973:¨
1887:−
1872:¨
1779:Resonance
1774:Resonance
1768:stiffness
1698:ω
1694:−
1680:ω
1674:ω
1671:β
1658:
1650:−
1639:δ
1609:ω
1599:β
1573:ω
1569:−
1555:ω
1491:δ
1487:−
1475:ω
1468:
1443:δ
1440:−
1434:ω
1428:
1379:δ
1370:ω
1364:
1284:ω
1274:˙
1265:β
1253:¨
1139:ω
1135:−
1126:β
1111:ω
1071:ω
1067:−
1035:ω
1006:β
1003:−
947:β
901:ω
891:˙
882:β
870:¨
824:˙
806:¨
708:⋮
684:δ
680:−
674:ω
668:
617:δ
613:−
607:ω
601:
540:→
528:−
519:→
503:isotropic
459:δ
456:−
450:ω
444:
383:ω
351:ω
347:−
328:−
319:¨
301:By using
281:−
155:(e.g. in
141:vibration
136:astronomy
134:stars in
100:economics
4846:Category
4832:Archived
4736:55729992
4584:Feedback
4542:See also
4501:Chemical
4063:Examples
4057:open set
4053:interval
4049:function
4041:sequence
4011:infinite
1821:coupling
1753:aircraft
774:friction
241:periodic
233:dynamics
220:such as
214:momentum
77:pendulum
65:periodic
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4771:
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45:.
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20:)
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