Equations of motion for the Pendulum and Augmented-Reality Pendulum sketches

Transcription

1 Equations of motion for the Pendulum and Augmented-Reality Pendulum sketches Ludovico Carbone Issue: Date: February 3, 20 School of Physics and Astronomy University of Birmingham Birmingham, B5 2TT

2 Introduction This document describes the dynamics of a simple pendulum and a double - composite - pendulum systems which are presented in the Pendulum Processing sketch, available at pendulum/. The same set of equations is also used in the sketch Augmented-Reality Pendulum, available at pendulum/. Screenshots of the two sketches are shown in Fig. and Fig. 2. Figure : A screenshot of the Pendulum sketch. Figure 2: A screenshot of the Augmented-Reality Pendulum sketch. page of 5

3 2 Simple Pendulum 2. Constants and definitions A cartoon of the simple pendulum system is shown in Fig. 3-left. The chosen coordinate system is also drawn, with the horizontal axis x directed rightwards and the vertical y axis downwards. The pendulum suspension point has coordinates (x 0, y 0 ), and the suspended mass m has (x, y ). The other physical parameters relevant for the system are: l initial length of pendulum wire at rest (which is defined by the user in the Pendulum sketch) = (x x 0 ) 2 + (y y 0 ) 2 actual length of the suspension wire # while suspended pendulum s mass m residual gas damping coefficient β suspension wire s elastic constants k suspension s wire anelasticity coefficient δk θ is the angle that pendulum # forms with the vertical axis y, for which we can write sin θ = (x x0) and cos θ = (y y0). In the Pendulum sketch the pendulum suspension point can be made to oscillate horizontally with user-defined amplitude A and frequency f. In this case the pendulum s suspension horizontal coordinate x 0 can be written as x 0 (t) = x 0 + A sin(2πft). x! x! y! (x 0,y 0 )! y! (x 0,y 0 )!! "! " (x,y )! (x,y )! m! m!! 2" m 2! (x 2,y 2 )! Figure 3: Cartoon of single pendulum suspension (left) and of the double - composite - pendulum (right). page 2 of 5

4 2.2 Forces 2.2 Forces All forces in this section are written in the form of F = {F x, F y } where F x and F y are the force components along the x and the y axis respectively. The forces acting on the simple pendulum are therefore: Gravity: acts vertically, F g = {0, m g} Elasticity of suspension wire (spring-like force): it acts along the direction of the wire and it can be written as F k = k ( l ) {sin θ, cos θ }. This force can be expressed also in the form F k = k max ( l ) {sin θ, cos θ } such that the elastic force will work only when the wire is stretched ( > l ) and not while it is compressed (this option can be selected by the user in the Augmented Reality Pendulum sketch). Anelasticity of suspension wire (velocity damping-like force): acts along the direction of the wire, F δk = δk d where d = [(x x 0 (t)) (ẋ ẋ 0 ) + (y y 0 ) (ẏ ẏ 0 )] {sin θ, cos θ }. Also this force should take into account of extension/compression of the wire as for the above elastic force Gas damping (velocity damping): assuming small velocity regime i.e. damping linearly proportional to velocity, it can be written as F v = β{v x, v y } = β{ẋ, ẏ }. Note that it is directed in opposite direction with respect to the instantaneous velocity. Diagrams of all the above forces are shown in Fig. 4-left. F k θ m F v2 F k F θ v F θ 2 v m m 2 F k2 F g F k2 F g F g2 θ 2 Figure 4: Single pendulum case: diagram of forces acting on mass m (left). Double pendulum: diagram of forces acting on m (center) and mass m 2 (right). 2.3 Equations of motion Simple pendulum equation of motions can be therefore written as m ẍ = k ( l ) sin θ βẋ δk [(x x 0 (t)) (ẋ ẋ 0 ) + (y y 0 ) (ẏ ẏ 0 )] sin θ () m ÿ = m g k ( l ) cos θ βẏ δk [(x x 0 (t)) (ẋ ẋ 0 ) + (y y 0 ) (ẏ ẏ 0 )] cos θ (2) page 3 of 5

5 3 Double composite pendulum 3. Constants and definitions We consider now the case of a composite pendulum system with a second pendulum #2 hanging from pendulum #, as shown in Fig. 3-right. The relevant physical parameters for pendulum # are the same as above and similarly the ones for pendulum #2 are: pendulum #2 has coordinates (x 2, y 2 ) and it hangs from (x, y ) the length at rest is l 2, the mass m 2, the elastic constants k 2, the anelasticity coefficient δk 2 and the gas damping β 2. d 2 = (x 2 x ) 2 + (y 2 y ) 2 is the actual length of the suspension wire. θ 2 is the angle that pendulum #2 forms with the vertical axis y and, as above, sin θ 2 = (x2 x) d 2 cos θ 2 = (y2 y) d 2 to take into account only the extension part of the force. and 3.2 Forces 3.2. Forces on mass # All forces on mass # described in the previous sections can be expressed as above. The only additional force on mass # comes from the elasticity of pendulum #2 wire which pulls from below. This can be written as F k2 = +k (d 2 l 2 ) {sin θ 2, cos θ 2 } (note the positive sign) and also this force can be expressed as F k2 = +k max (d 2 l 2, 0) {sin θ 2, cos θ 2 } The contribution from anelasticity of wire #2 is neglected for the moment. Diagrams of all the above forces are shown in Fig. 4-centre Forces on mass #2 The forces acting on pendulum #2 follow the usual expressions seen before: Gravity F g2 = m 2 g{0, } Elasticity of wire F k2 = k 2 (d 2 l 2 ) {sin θ 2, cos θ 2 } (or in the form F = k 2 max (d 2 l 2, 0) ) Anelasticity of wire F δk2 = δkd 2 where d 2 = d 2 [(x 2 x ) (ẋ 2 ẋ ) + (y 2 y ) (ẏ 2 ẏ )[ {sin θ 2, cos θ 2 } Residual gas damping F v2 = β{v 2x, v 2y } = β{ẋ 2, ẏ 2 } Diagrams of all the above forces are shown in Fig. 4-right. 3.3 Equations of motion With the above forces, we can finally write the equations of motion for the composite pendulum system as follows: m ẍ = k ( l ) sin θ β ẋ + k 2 (d 2 l 2 ) sin θ 2 δk d ((x x 0 (t)) (ẋ ẋ 0 ) + (y y 0 ) (ẏ ẏ 0 )) sin θ (3) m ÿ = k ( l ) cos θ βẏ + k 2 (d 2 l 2 ) cos θ 2 m g δk d ((x x 0 (t)) (ẋ ẋ 0 ) + (y y 0 ) (ẏ ẏ 0 )) cos θ (4) page 4 of 5

6 m 2 ẍ 2 = k 2 (d 2 l 2 ) sin θ 2 βẋ 2 δk 2 d2 ((x 2 x ) (ẋ 2 ẋ ) + (y 2 y ) (ẏ 2 ẏ )) sin θ (5) m 2 ÿ 2 = k 2 (d 2 l 2 ) cos θ 2 βẏ 2 m 2 g δk 2 d2 ((x 2 x ) (ẋ 2 ẋ ) + (y 2 y ) (ẏ 2 ẏ )) cos θ (6) 4 Leapfrog Method for integration of equations of motion For the solution of the equation of motion the Pendulum and the Augmented-Reality Pendulum sketches make use of the Leapfrog metho, a well know method for the numerical integration of differential equations, which is commonly used to study of the time evolution of physical systems. With this method positions velocities and accelerations are derived at interleaved times (hence leapfrog ), and this proves to provide reliable results as long as the time interval t = (t i+ t i ) between successive samplings i (the leapfrog step) is smaller than any characteristic time in the system. In the Pendulum and Augmented-Reality Pendulum cases this condition can be expressed as t g and t k,2 2π,2 2π m,2 for the different values of k and k 2, m and m 2, and and d 2. In the Pendulum and in the Augmented-Reality Pendulum sketches, the values of velocities are initially set zero (t = 0) and initial values of the coordinates can be arbitrarily chosen. The values of accelerations are derived only at a later stage and therefore can be initially ignored. Then, at any successive times t i, we perform the following iteration:. calculate positions as x (t i+ ) = x (t i ) + ẋ (t i ) (t i+ t i ) (i.e. using the previously calculated value of velocity) 2. calculate accelerations at time t i+ following the equations of motion above, and using the above values of position (x (t i+ )) and velocity (ẋ (t i )) 3. calculate velocities for t i+ as ẋ (t i+ ) = ẍ (t i+ ) + ẋ (t i ) (t i+ t i ) 4. restart from step See for instance integration and references therein. page 5 of 5