Question

im having trouble doing this problem with matlab code and we can not use ode45 in our code unfortunitely.

thank you!

T L =undeformed length k =stiffness om The mass m is suspended from an elastic cord with an extensional stiffness k and undeformed length L. If the mass is released from rest at 0 = 60° with the cord unstretched, find the length r of the cord when the position =0 is reached for the first time. The differential equations describing the motion are * = rö' + gcos 0 – (r – L) ä -2re-g sine T Use g = 9.80665 m/s2, k = 40 N/m, L=0.5 m, and m=0.25 kg. Utilize Demo6.m that is made available on Moodle. You are required to solve the problem using the fourth order Runge-Kutta method. Include: a) Program a function for RK4 method (attach whole code into appendix). b) Report the cord length (r) when the angle (theta) is zero. c) Animate the position of the mass against time.

Answer 1

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L = undeformed length K= stiffness Ť = 782+ g coso - Ķ (r-L) - (1) ö= -288 - grino - (2) 9- 9.80665 ms ke 40 NI. L=0.5m m=0.25 kg Initial values. T=0.5m=L o = 60° Y-U-O This problems can be found solved by using MATLAB function ode2 3 or ode45 As it is very difficult to find its analytical solution due to complexity of 2nd order multiple variable problem lale have four veriables to find ro, r, o te all varying with time.

2. first order Above equations can be transformed into Diff equation by i= u = di = 0 = de dizu - (0) Tv - (T) Equation (1) can be written as in die see ru? + g coso - (7-4) Qiji do ö = 40 - do Ji= -200 - g sino live - r using function file and run file solution can be found 1. First save fun Pendulum. m in Matlab 2. save file name run Pendulum. im 3. Run only run Penclud um, in Code

2. first order Above equations can be transformed into Diff equation by i= u = di = 0 = de dizu - (0) Tv - (T) Equation (1) can be written as in die see ru? + g coso - (7-4) Qiji do ö = 40 - do Ji= -200 - g sino live - r using function file and run file solution can be found 1. First save fun Pendulum. m in Matlab 2. save file name run Pendulum. im 3. Run only run Penclud um, in Code

funPendulum.m

function fVal= funPendulum(t,Y)

r=Y(1);

theta=Y(2);

u=Y(3);

v=Y(4);

%constants

K=40;

m=0.25;

g=9.80665;

L=0.5;

% define dy/dt

fVal(1,1)=u;

fVal(2,1)=v;

fVal(3,1)=r*(u^2)+g*cosd(theta)-(K/m)*(r-L);

fVal(4,1)=(-2*u*v-g*sind(theta))/r;

end

runPendulum.m

Y0=[0.5;60;0;0];

tSpan=[0 3.05675];

[time,Ysol]=ode23(@(t,Y) funPendulum(t,Y),tSpan,Y0);

r=Ysol(:,1);

theta=Ysol(:,2);

u=Ysol(:,3);

v=Ysol(:,4);

clf;

plot(time,Ysol);

xlabel('Time(s)');ylabel('r theta u v')

hold on;

plot(3.05675,0.543911,'*b')

[theta,time,r]

e - o to become Zero was found Time needed first for Time = 3.0568 sec. of Gode shows 3 In rundpendulum. m Goloumns. last line timer o 0 0.5 0.0004 O 3.0568 10.5439) So. Ye-o = 0.54 39.m м .