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The system is much more complex than a simple pendulum, as the properties of the spring add an extra dimension of freedom to the system. For example, when the spring compresses, the shorter radius causes the spring to move faster due to the conservation of
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177:. It is also possible that the spring has a range that is overtaken by the motion of the pendulum, making it practically neutral to the motion of the pendulum.
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2267:
Holovatsky V., Holovatskyi I., Holovatska Ya., Struk Ya. Oscillations of the resonant elastic pendulum. Physics and
Educational Technology, 2023, 1, 10–17,
661:
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1291:{\displaystyle {\partial L \over \partial \theta }-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {\theta }}}=0}
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2111:
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1985:
2771:
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2501:
3006:
2180:
1197:{\displaystyle {\partial L \over \partial x}-{\operatorname {d} \over \operatorname {d} t}{\partial L \over \partial {\dot {x}}}=0}
2688:
1045:{\displaystyle L={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})-{\frac {1}{2}}kx^{2}+gm(l_{0}+x)\cos \theta }
1926:
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456:, on the other hand, is determined by the height of the mass. For a given angle and displacement, the potential energy is:
3115:
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1933:. Furthermore, one can use analytical methods to study the intriguing phenomenon of order-chaos-order in this system.
655:
to the other variables, the velocity is written as a combination of a movement along and perpendicular to the spring:
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553:
1915:{\displaystyle {\ddot {\theta }}=-{\frac {g}{l_{0}+x}}\sin \theta -{\frac {2{\dot {x}}}{l_{0}+x}}{\dot {\theta }}}
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256:
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1758:{\displaystyle -gm(l_{0}+x)\sin \theta -m(l_{0}+x)^{2}{\ddot {\theta }}-2m(l_{0}+x){\dot {x}}{\dot {\theta }}=0}
376:
2585:
2530:
2058:
2225:"Continuation of Periodic Solutions of Dissipative and Conservative Systems: Application to Elastic Pendulum"
148:, there also appears to be regular motion. The motion of an elastic pendulum is governed by a set of coupled
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Anurag, Anurag; Basudeb, Mondal; Bhattacharjee, Jayanta Kumar; Chakraborty, Sagar (2020).
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Please expand the article to include this information. Further details may exist on the
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1567:{\displaystyle {\ddot {x}}=(l_{0}+x){\dot {\theta }}^{2}-{\frac {k}{m}}x+g\cos \theta }
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27:"Spring pendulum" redirects here. Not to be confused with the one-dimensional vertical
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2264:(interactive animation), Wolfram Demonstrations Project, published February 19, 2019.
2181:"Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum"
2160:
2009:"Stability Condition for Vertical Oscillation of 3-dim Heavy Spring Elastic Pendulum"
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763:{\displaystyle T={\frac {1}{2}}m({\dot {x}}^{2}+(l_{0}+x)^{2}{\dot {\theta }}^{2})}
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2129:"Understanding the order-chaos-order transition in the planar elastic pendulum"
1429:{\displaystyle m(l_{0}+x){\dot {\theta }}^{2}-kx+gm\cos \theta -m{\ddot {x}}=0}
17:
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132:. For specific energy values, the system demonstrates all the hallmarks of
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90:
152:.This behavior suggests a complex interplay between energy states and
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37:
2272:
http://journals.vnu.volyn.ua/index.php/physics/article/view/1093
2280:
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62:
about the characteristics of chaotic motion in the system, cf.
2104:
Computer Aided
Graphing and Simulation Tools for AutoCAD Users
44:
121:
so that the resulting motion contains elements of both a
1925:
The elastic pendulum is now described with two coupled
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2 DOF elastic pendulum with polar coordinate plots.
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1104:, the equations of motion can be found using two
2106:(1st ed.). Boca Raton, Florida: CRC Press.
2089:The Swinging Spring: Regular and Chaotic Motion
2002:
2000:
1998:
522:{\displaystyle V_{g}=-gm(l_{0}+x)\cos \theta }
370:is the potential energy of the spring itself:
232:. The angle of oscillation of the pendulum is
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419:{\displaystyle V_{k}={\frac {1}{2}}kx^{2}}
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117:where a piece of mass is connected to a
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33:Simple harmonic motion#Mass on a spring
601:{\displaystyle T={\frac {1}{2}}mv^{2}}
2269:https://doi.org/10.32782/pet-2023-1-2
2262:"Oscillations of an elastic pendulum"
7:
2260:Holovatsky V., Holovatska Y. (2019)
2232:Mathematical Problems in Engineering
2439:Measure-preserving dynamical system
1986:"Dynamics of the Elastic Pendulum"
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25:
3007:Oleksandr Mykolayovych Sharkovsky
1789:{\displaystyle {\ddot {\theta }}}
212:and can be stretched by a length
2537:
2529:
2072:Hill, Christian (19 July 2017).
559:The kinetic energy is given by:
49:
1927:ordinary differential equations
821:{\displaystyle L=T-V_{k}-V_{g}}
185:The spring has the rest length
150:ordinary differential equations
2772:Rabinovich–Fabrikant equations
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64:Double pendulum#Chaotic motion
1:
2188:Regular and Chaotic Dynamics
2016:Regular and Chaotic Dynamics
2507:Poincaré recurrence theorem
2153:10.1016/j.physd.2019.132256
1460:{\displaystyle {\ddot {x}}}
773:So the Lagrangian becomes:
160:Analysis and interpretation
144:.At very low and very high
3137:
2502:Poincaré–Bendixson theorem
1984:Xiao, Qisong; et al.
554:gravitational acceleration
452:The potential energy from
26:
2854:Swinging Atwood's machine
2527:
2497:Krylov–Bogolyubov theorem
2374:
2208:10.1134/S1560354708030027
2102:Simionescu, P.A. (2014).
2036:10.1134/S1560354708030027
2762:Lotka–Volterra equations
2586:Synchronization of chaos
2389:axiom A dynamical system
2057:Sivasrinivas, Kolukula.
1106:Euler-Lagrange equations
449:is the spring constant.
2747:Double scroll attractor
2512:Stable manifold theorem
2419:False nearest neighbors
2223:Pokorny, Pavel (2009).
2179:Pokorny, Pavel (2008).
2007:Pokorny, Pavel (2008).
1590:{\displaystyle \theta }
1097:{\displaystyle \theta }
635:of the mass. To relate
245:{\displaystyle \theta }
31:with gravity, cf. also
2787:Van der Pol oscillator
2767:Mackey–Glass equations
2399:Box-counting dimension
1953:Pendulum (mathematics)
1929:. These can be solved
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60:is missing information
42:
2937:Svetlana Jitomirskaya
2844:Multiscroll attractor
2689:Interval exchange map
2642:Dyadic transformation
2627:Complex quadratic map
2469:Topological conjugacy
2404:Correlation dimension
2379:Anosov diffeomorphism
2074:"The spring pendulum"
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306:{\displaystyle L=T-V}
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205:{\displaystyle l_{0}}
167:
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3116:Mathematical physics
2947:Edward Norton Lorenz
1803:
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2907:Mitchell Feigenbaum
2849:Population dynamics
2834:Hénon–Heiles system
2694:Irrational rotation
2647:Dynamical billiards
2632:Coupled map lattice
2492:Liouville's theorem
2424:Hausdorff dimension
2409:Conservative system
2394:Bifurcation diagram
2245:10.1155/2009/104547
2200:2008RCD....13..155P
2145:2020PhyD..40232256A
2028:2008RCD....13..155P
1056:Equations of motion
3085:Santa Fe Institute
2952:Aleksandr Lyapunov
2782:Three-body problem
2669:Gingerbreadman map
2556:Bifurcation theory
2434:Lyapunov stability
1958:Spring-mass system
1948:Duffing oscillator
1912:
1786:
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1062:degrees of freedom
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142:initial conditions
130:spring-mass system
43:
29:spring-mass system
3111:Dynamical systems
3093:
3092:
2957:Benoît Mandelbrot
2922:Martin Gutzwiller
2912:Peter Grassberger
2795:
2794:
2777:Rössler attractor
2525:
2524:
2429:Invariant measure
2351:Lyapunov exponent
2113:978-1-4822-5290-3
2059:"Spring Pendulum"
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1314:{\displaystyle x}
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648:{\displaystyle v}
624:{\displaystyle v}
583:
545:{\displaystyle g}
442:{\displaystyle k}
401:
353:{\displaystyle V}
329:{\displaystyle T}
271:{\displaystyle L}
225:{\displaystyle x}
99:dynamical systems
97:, in the area of
87:
86:
16:(Redirected from
3128:
3065:Butterfly effect
2977:Itamar Procaccia
2927:Brosl Hasslacher
2824:Elastic pendulum
2752:Duffing equation
2699:Kaplan–Yorke map
2617:Arnold's cat map
2605:
2581:Stability theory
2566:Dynamical system
2561:Control of chaos
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2449:Poincaré section
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2997:Caroline Series
2892:Mary Cartwright
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2819:Double pendulum
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2659:Exponential map
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2487:Ergodic theorem
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2464:Stable manifold
2454:Recurrence plot
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2172:Further reading
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154:system dynamics
127:one-dimensional
123:simple pendulum
115:physical system
111:swinging spring
107:spring pendulum
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18:Spring pendulum
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3080:Predictability
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105:(also called
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58:This article
56:
52:
47:
46:
40:
34:
30:
19:
3106:Chaotic maps
3027:Mary Tsingou
2992:David Ruelle
2987:Otto Rössler
2932:Michel HĂ©non
2902:Leon O. Chua
2859:Tilt-A-Whirl
2829:FPUT problem
2823:
2714:Standard map
2709:Logistic map
2534:
2308:Chaos theory
2235:
2231:
2191:
2187:
2136:
2132:
2122:
2103:
2097:
2088:
2082:
2067:
2052:
2019:
2015:
1924:
1767:
1576:
1438:
1300:
1059:
772:
610:
558:
531:
451:
428:
366:
315:
254:
184:
171:
110:
106:
102:
88:
78:October 2019
75:
59:
3012:Nina Snaith
3002:Yakov Sinai
2887:Rufus Bowen
2637:Duffing map
2622:Baker's map
2547:Theoretical
2459:SRB measure
2366:Phase space
2336:Bifurcation
1931:numerically
368:Hooke's law
95:mathematics
3100:Categories
3070:Complexity
2967:Edward Ott
2814:Convection
2739:Continuous
2414:Ergodicity
2139:: 132256.
1964:References
1796:isolated:
1467:isolated:
257:Lagrangian
181:Lagrangian
3121:Pendulums
2982:Mary Rees
2942:Bryna Kra
2875:theorists
2684:Ikeda map
2674:HĂ©non map
2664:Gauss map
2346:Limit set
2331:Attractor
2161:209905775
2133:Physica D
1907:˙
1904:θ
1874:˙
1859:−
1856:θ
1853:
1822:−
1813:¨
1810:θ
1781:¨
1778:θ
1744:˙
1741:θ
1732:˙
1695:−
1689:¨
1686:θ
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1645:θ
1642:
1608:−
1585:θ
1562:θ
1559:
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1484:¨
1452:¨
1415:¨
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1400:θ
1397:
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1364:˙
1361:θ
1274:˙
1271:θ
1265:∂
1257:∂
1245:
1233:−
1227:θ
1224:∂
1216:∂
1180:˙
1171:∂
1163:∂
1151:
1139:−
1130:∂
1122:∂
1092:θ
1060:With two
1040:θ
1037:
977:−
962:˙
959:θ
911:˙
876:˙
873:θ
864:θ
855:˙
806:−
793:−
746:˙
743:θ
695:˙
517:θ
514:
480:−
298:−
240:θ
138:sensitive
70:talk page
3058:articles
2800:Physical
2719:Tent map
2609:Discrete
2549:branches
2479:Theorems
2315:Concepts
2238:: 1–15.
2216:56090968
2044:56090968
1937:See also
1577:And for
633:velocity
3056:Related
2864:Weather
2802:systems
2595:Chaotic
2341:Fractal
2196:Bibcode
2141:Bibcode
2024:Bibcode
631:is the
552:is the
454:gravity
360:is the
336:is the
136:and is
113:) is a
91:physics
2962:Hee Oh
2597:maps (
2444:Mixing
2214:
2159:
2110:
2042:
1064:, for
611:where
532:where
429:where
316:where
146:energy
125:and a
119:spring
2873:Chaos
2652:outer
2356:Orbit
2228:(PDF)
2212:S2CID
2184:(PDF)
2157:S2CID
2040:S2CID
2012:(PDF)
1989:(PDF)
101:, an
2599:list
2323:Core
2236:2009
2108:ISBN
1301:For
1084:and
340:and
278:is:
255:The
93:and
2240:doi
2204:doi
2149:doi
2137:402
2032:doi
1850:sin
1639:sin
1556:cos
1394:cos
1034:cos
511:cos
364:.
252:.
140:to
109:or
89:In
3102::
2234:.
2230:.
2210:.
2202:.
2192:13
2190:.
2186:.
2155:.
2147:.
2135:.
2131:.
2038:.
2030:.
2020:13
2018:.
2014:.
1997:^
1972:^
1597::
1321::
1108::
556:.
156:.
2601:)
2300:e
2293:t
2286:v
2248:.
2242::
2218:.
2206::
2198::
2163:.
2151::
2143::
2116:.
2091:.
2076:.
2061:.
2046:.
2034::
2026::
1991:.
1895:x
1892:+
1887:0
1883:l
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1865:2
1844:x
1841:+
1836:0
1832:l
1827:g
1819:=
1753:0
1750:=
1729:x
1723:)
1720:x
1717:+
1712:0
1708:l
1704:(
1701:m
1698:2
1678:2
1674:)
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1658:l
1654:(
1651:m
1636:)
1633:x
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1621:l
1617:(
1614:m
1611:g
1553:g
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1542:m
1539:k
1529:2
1512:)
1509:x
1506:+
1501:0
1497:l
1493:(
1490:=
1481:x
1449:x
1424:0
1421:=
1412:x
1406:m
1391:m
1388:g
1385:+
1382:x
1379:k
1371:2
1354:)
1351:x
1348:+
1343:0
1339:l
1335:(
1332:m
1309:x
1286:0
1283:=
1260:L
1248:t
1242:d
1238:d
1219:L
1192:0
1189:=
1177:x
1166:L
1154:t
1148:d
1144:d
1133:x
1125:L
1072:x
1031:)
1028:x
1025:+
1020:0
1016:l
1012:(
1009:m
1006:g
1003:+
998:2
994:x
990:k
985:2
982:1
974:)
969:2
950:2
946:)
942:x
939:+
934:0
930:l
926:(
923:+
918:2
908:x
901:(
898:m
893:2
890:1
885:=
882:]
867:,
861:,
852:x
846:,
843:x
840:[
837:L
814:g
810:V
801:k
797:V
790:T
787:=
784:L
758:)
753:2
734:2
730:)
726:x
723:+
718:0
714:l
710:(
707:+
702:2
692:x
685:(
682:m
677:2
674:1
669:=
666:T
643:v
619:v
594:2
590:v
586:m
581:2
578:1
573:=
570:T
540:g
508:)
505:x
502:+
497:0
493:l
489:(
486:m
483:g
477:=
472:g
468:V
437:k
412:2
408:x
404:k
399:2
396:1
391:=
386:k
382:V
348:V
324:T
301:V
295:T
292:=
289:L
266:L
220:x
198:0
194:l
80:)
76:(
72:.
66:.
35:.
20:)
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