Single and Double Pendulum Models

Transcription

1 Single and Double Pendulum Models Mah 596 Projec Summary Spring 2016 Jarod Har 1 Overview Differen ypes of pendulums are used o model many phenomena in various disciplines. In paricular, single and double pendulums are commonly used in biomechanics engineering o model gai (a person s manner of walking). The differenial equaions ha govern he moion of pendulums can be derived from Newon s laws of physics. This yields a single second order differenial equaion for a single pendulum, and a sysem of wo second order differenial equaions for a double pendulum. Using reducion of order, Euler s mehod for numerical soluions, and Runge-Kua numerical soluion echniques, one can use hese equaions o generae a model for single and double pendulums. The purpose of his projec is o generae a mah model for single and double pendulums. This involves using conceps of physics o derive second order differenial equaions ha describe he moion of pendulums, and simplify hose equaions o sysems of firs order equaions. Once simplified, Euler and Runge-Kua numerical solvers are implemened o describe he approximae moion of he single and double pendulums. 2 Mahemaical and Programming Conen To complee his projec, a background in he following opics is recommended. - Calculus: This projec requires a knowledge of derivaives, ani-derivaives, linear approximaion, and Taylor polynomial approximaions. - Physics: A basic background in composiion and balance of forces is helpful o formulae he differenial equaions ha describe he moion of pendulums. - Differenial equaions: An undersand of firs/second order sysems of ordinary differenial equaions and reducion of order echniques are perinen o his projec. Also a basic undersanding of wha an exac versus numerical soluion is imporan. - Numerical analysis: This projec involves undersanding approximaion of smooh funcion via angen lines and Taylor polynomials, as well as how his can be used o generae approximae soluions o differenial equaions. I is also helpful o undersand how errors in soluions may propagae depending on he lengh of he ime inerval and he size of he ime incremen. - Programming: This projec involves a significan programming aspec. I may involve wriing he code for Runge- Kua mehods of differen orders, as well as undersanding how o display numerical soluions o differenial equaions in a way ha accuraely represens he model. 3 Primary Resources For much of he mahemaical conen lised above, ypical ex books in he perinen area are sufficien. In paricular, mos hings for his projec can be found in jus abou any ordinary differenial equaions exbook, excep possibly for a descripion of Runge-Kua mehods for sysems of equaions. The formulaion of his model is sandard in physics and mah lieraure now. The following are only a few places where hey can be found online. I can be found many oher places.

2 - Gonzáles, G., Louisiana Sae Universiy course Physics 7221 Lecure Noes, - WolframResearch Websie, Formulaion of Double Pendulum Differenial Equaions, scienceworld.wolfram.com/physics/doublependulum.hml 4 Mahemaical Descripion of he Projec We firs discuss he differenial equaions for a single pendulum. Consider a pendulum wih a siff, massless rod wih lengh l ha has a mass m aached o he end. Le θ = θ() be he angle of he rod wih he verical, x = x() be he horizonal displacemen, and y = y() be he verical displacemen a ime as picured below. ` y m x Using rigonomeric ideniies, i follows ha x = lsin(θ) and y = lcos(θ). (1) Noe ha as we have defined he variables here, he posiive y direcion is downwards. This may be a lile counerinuiive, bu we find i convenien for he purpose of his model. The moion of his pendulum is governed by he second order differenial equaion d 2 θ 2 + g sin(θ) = 0, (2) l where g is a he force of graviy. To make a reducion of order, define φ() = dθ, and he second order differenial equaion (2) can be rewrien as he sysem of wo firs order equaion { dθ = φ dφ = g l sin(θ). (3) Wih his definiion φ() = dθ, φ() can be undersood as he angular velociy of he pendulum a ime. Hence forming he iniial value problem relaed o equaion (3) { dθ = φ θ(0) = θ 0 dφ = g l sin(θ) φ(0) = φ 0. (4) corresponds o modeling he pendulum picured above wih iniial angle θ 0 and iniial angular velociy φ 0. Now we consider he formulaion of a double pendulum. Consider a double pendulum wih siff, massless rods of lenghs l 1 and l 2 ha have a masses m 1 and m 2 aached o he ends. Le θ i = θ i (), x i = x i (), and y i = y i () for i = 1,2 be defined as picured below, similar o he way hey were in he single pendulum.

3 1 `1 y 1 m 1 x 1 `2 2 y 2 m 2 x 2 Using rigonomeric relaionships beween hese variables, one can relae hese variables in he following way x 1 = l 1 sin(θ 1 ), y 1 = l 1 cos(θ 1 ), x 2 = x 1 + l 2 sin(θ 2 ), and y 2 = y 1 + cos(θ 2 ). (5) Again noice ha he posiive y 1 and y 2 direcions are oriened downward, like he single pendulum. The moion of his double pendulum is governed by he following pair of second order differenial equaions d Ml 2 θ 1 d 1 + m 2 2 l 2 θ d l 2 θ 1 d 1 cos(θ 2 2 θ 1 ) + l 2 θ ( ) 2 cos(θ 2 θ 1 ) = m 2 l dθ2 2 sin(θ2 θ 1 ) Mgsin(θ 1 ) ( ) 2 = l dθ1 1 sin(θ2 θ 1 ) gsin(θ 2 ), (6) where M = m 1 + m 2 and g again is he force of graviy. To make a reducion of order, define φ 1 () = dθ 1 φ 2 () = dθ 2, which yields he sysem of four firs order equaions dθ 1 = φ 1 dθ 2 = φ 2 Ml 1 dφ 1 + m 2 l 2 dφ 2 l 1 dφ 1 cos(θ 2 θ 1 ) + l 2 dφ 2 cos(θ 2 θ 1 ) = m 2 l 2 φ 2 2 sin(θ 2 θ 1 ) Mgsin(θ 1 ) = l 1 φ 2 1 sin(θ 2 θ 1 ) gsin(θ 2 ) and, (7) Solving for dφ 1 and dφ 2 in he las wo equaions and adding iniial condiions, one can rewrie his sysems and consider he iniial value problem dθ 1 = φ 1 θ 1 (0) = θ 1,0 dθ 2 = φ 2 θ 2 (0) = θ 2,0 dφ 1 = g(m+m 1)sin(θ 1 ) m 2 gsin(θ 1 2θ 2 ) 2sin(θ 1 θ 2 )m 2 (φ 2 1 l 1 cos(θ 1 θ 2 )+φ 2 2 l 2) (8) l 1 (M+m 1 m 2 cos(2θ 1 2θ 2 )) φ 1 (0) = φ 1,0 dφ 2 = 2sin(θ 1 θ 2 )(Mφ 2 1 l 1+gM cos(θ 1 )+φ 2 2 l 2m 2 cos(θ 1 θ 2 )) l 2 (M+m 1 m 2 cos(2θ 1 2θ 2 )) φ 2 (0) = φ 2,0, where θ 1,0 and θ 2,0 denoe he iniial angles of he firs and second rods, and φ 1,0 and φ 2,0 denoe he iniial angular velociies of he firs and second rods. In general, i is no easy (or even possible) o find soluions o equaions (4) and (8) above. Hence we resor o generaing numerical soluions o hese equaions. One can do his in several ways. Maybe he firs wo ha are ypically augh in differenial equaions courses are Euler s mehod and Runge-Kua mehods. Euler s mehod has he benefi ha i is very easy o discuss wih lile background or deph of undersanding of calculus; only an undersanding of linear approximaion. This mehod may work well for he single pendulum, bu i ypically does no

4 perform well when used o model he double pendulum. Hence higher order mehods depending on Taylor polynomial approximaion in place of linear approximaion are beer suied o model he double pendulum. To discuss Euler s mehod consider a firs order iniial value problem given by { dy1 = f 1 (y 1,y 2,) y 1 (0) = y 1,0 dy 2 (9) = f 2 (y 1,y 2,) y 2 (0) = y 2,0 where f 1 and f 2 are smooh funcions. (We will no concern ourselves wih making precise how smooh f 1 and f 2 should be, and simply proceed assuming ha hea are smooh enough for our compuaions o be jusified.) Suppose (ϕ 1 (),ϕ 2 ()) is he (unique) exac soluion o his iniial value problem; ha is, dϕ 1 = f 1 (ϕ 1 (),ϕ 2 (),), dϕ 2 = f 2 (ϕ 1 (),ϕ 2 (),), ϕ 1 (0) = y 1,0, and ϕ 2 (0) = y 2,0. Given a small sep size 0 < h << 1 and using successive angen line approximaion, one can generae a numerical soluion o equaion (9) given by {( k,y 1,k,y 2,k )} k N0, where k = k h for k 0, y 1,k = y 1,k 1 + h f 1 (y 1,k 1,y 2,k 1, k 1 ), and y 2,k = y 2,k 1 + h f 2 (y 1,k 1,y 2,k 1, k 1 ) for k 1. One can apply his mehod o equaion (4) wih y 1 = θ and y 2 = φ o generae numerical soluions o he single pendulum. Some examples are given below. This mehod can also be used o generae a numerical soluion o equaion (8), bu resuls may no be very reliable due o he relaively low accuracy of Euler s mehod. To discuss Runge-Kua s mehod (we will resric ourselves o discussing he fourh order version, which should be sufficien for numerical solving he double pendulum model here), consider a firs order iniial value problem given by dy 1 = f 1 (y 1,y 2,y 3,y 4,) y 1 (0) = y 1,0 dy 2 = f 2 (y 1,y 2,y 3,y 4,) y 2 (0) = y 2,0 dy 3 (10) = f 3 (y 1,y 2,y 3,y 4,) y 3 (0) = y 3,0 dy 4 = f 4 (y 1,y 2,y 3,y 4,) y 4 (0) = y 4,0 Again, we assume ha f 1, f 2, f 3, and f 4 are nice enough funcions so ha our compuaions are jusified, wihou worrying abou echnical deails. One way o adap he fourh order Runge-Kua mehod for he sysem (10) is he following: Define k = h k for k N 0 and y i,k = y i,k 1 + h 6 (A i,k + 2 B i,k + 2 C i,k + D i,k ) for i = 1,2,3,4, where A i,k = f i (y 1,k 1,y 2,k 1,y 3,k 1,y 4,k 1, k 1 ) ( B i,k = f i y 1,k 1 + h 2 A 1,k,y 2,k 1 + h 2 A 2,k,y 3,k 1 + h 2 A 3,k,y 4,k 1 + h 2 A 4,k, k 1 + h ) 2 ( C i,k = f i y 1,k 1 + h 2 B 1,k,y 2,k 1 + h 2 B 2,k,y 3,k 1 + h 2 B 3,k,y 4,k 1 + h 2 B 4,k, k 1 + h ) 2 D i,k = f i (y 1,k 1 + h C 1,k,y 2,k 1 + h C 2,k,y 3,k 1 + h C 3,k,y 4,k 1 + h C 4,k, k ) for k N. Then {( k,y 1,k,y 2,k,y 3,k,y 4,k )} k N0 is a numerical soluion for equaion (10). This numerical soluion can be applied o generae a numerical soluion o equaion (8) using y 1 = θ 1, y 2 = θ 2, y 3 = φ 1, and y 4 = φ 2, and accuracy of he fourh order Runge-Kua solver should be sufficien o generae an accurae model of he double pendulum. Examples of his are provided below.

5 5 Summary of Resuls All resuls below were generaed using eiher Euler s mehod or Runge-Kua mehod of order 4 o find numerical soluions o equaions (4) and (8) wih various iniial condiions. All numerical soluions for he single pendulum (4) are generaed wih Euler s mehod and hose for he double pendulum (8) are generaed wih Runge-Kua order 4 mehods. The following is a numerical soluion o equaion (4) wih g = 9.8, l = 1, θ 0 = 0, and φ 0 = 1. θ φ The following is a numerical soluion o equaion (4) wih g = 9.8, l = 1, θ 0 = π 2, and φ 0 = 0. θ φ The following is a numerical soluion o equaion (8) wih g = 9.8, m 1 = m 2 = l 1 = l 2 = 1, θ 1,0 = π 4, θ 2,0 = π 4, and φ 1,0 = φ 2,0 = 0. θ 1 φ 1 θ 2 φ 2 The following is a numerical soluion o equaion (8) wih wih g = 9.8, m 1 = 1, m 2 = 5, l 1 = 1, l 2 = 1 2, θ 1,0 = θ 2,0 = π 4, φ 1,0 = 1, and φ 2,0 = 4.

6 θ 1 φ 1 θ 2 φ 2 The following is a numerical soluion o equaion (8) wih wih g = 9.8, m 1 = m 2 = 1, l 1 = l 2 = 1, θ 1,0 = π 3, θ 2,0 = 2π 3, and φ 1,0 = φ 2,0 = 0. θ 1 φ 1 θ 2 φ 2 6 Possible Exensions There are a few naural exension for his projec. One is o model pendulums wih more joins. Tha is, one can repea all of he procedures above for a 3, 4, or n pendulum. This would require a more involved formulaion of differenial equaion model; o model an n pendulum, one would require a sysem of n second order differenial equaions or, afer a reducion of order, a sysem of 2n firs order differenial equaions. This would also likely require more accurae numerical solvers. For insance, one can use higher order Runge-Kua solvers or search numerical analysis lieraure for oher high accuracy solvers. Anoher direcion o furher his projec is o work wih invered pendulums. Afer undersanding and modeling sandard pendulums, one can consider he problem of conrolling a pendulum ha is invered. This direcion would involve an inroducion o conrol heory on op of he informaion already conained in his projec, and i also may require more accurae numerical approximaion echniques.

7 7 Noe From he Auhor This is a suden projec from he Mah and Biomedical Research course, augh by he curren auhor Jarod Har, offered a he Universiy of Kansas in he Spring of Some modificaion and addiions were made o he original projec for his summary. The course is suppored by he Iniiaive for Maximizing Suden Developmen (IMSD) hrough an NIH gran NIH-NIGMS 5R25GM The PIs of his IMSD gran are Professors Esela Gavoso (Mahemaics Deparmen) and James Orr (Biology Deparmen). We are happy o share hese projec ideas, and welcome hose who are ineresed o use hem. We d love o hear abou your resuls and exensions relaed o hese projecs, and in some cases, will provide some suppor for he projecs. Please conac Jarod Har a jvhar@ku.edu wih any ypos, errors, quesions, or commens abou his projec summary.