Question

For the example problem 2.2.4 (page 20, textbook or page 3, ClassNote Wk#3), the system is oscillating without any external excitation. Parameters are given as follows. (Picture Posted)

is a CV 2 Ca CL

initial conditions: $\theta ^{^{\cdot }}$ = 0.5 rad/sec, $\theta$ (0) = 2 degrees

Mass of the bar: 5kg_m

Moment of inertia of the bar about O: J_o= 0.1 kg-m²

Spring constant: k = 150 N/m

Dimentions: a= 0.5 m; b= 1.5 m; c= 0.5 m

Using the MATLAB, plot the angular displacement, angular velocity and angular acceleration of the system.

Note: Use an appropriate time vector to show at least 3 cycles.

Answer 1

Governing Equation:

Given: $J_0=0.1$ kg-m² , $k=150$ N/m , $a=0.5$ m, $b=1.5$ m, $c=0.5$ , $m=5$ kg,

$\dot{\theta }\left ( 0 \right )=0.5$ rad/s and $\theta \left ( 0 \right )=2^0=2 \times \frac{\pi}{180}=0.035$

Converting the second order differential equation to first order for MATLAB calculation, Let

$x=\begin{bmatrix} \theta \\ \dot \theta \end{bmatrix}$

Then,

$\dot x=\begin{bmatrix} 0 & 1\\ -k\left ( a^2+b^2 \right )/J_0 & 0 \end{bmatrix}\begin{bmatrix} \theta \\ \dot \theta \end{bmatrix}$ or $\dot x=\begin{bmatrix} 0 & 1\\ -k\left ( a^2+b^2 \right )/J_0 & 0 \end{bmatrix}x$

Also, $x_0=\begin{bmatrix} 0.035\\ 0.5 \end{bmatrix}$

Time Period for one cycle $T=\frac{1}{f}=2 \pi \sqrt{\frac{J_0}{k\left ( a^2 + b^2\right )}}=2 \times 3.142 \times \sqrt{\frac{0.1}{150\left ( 0.5^2 + 1.5^2\right )}}=0.103$

MATLAB CODE

clear
clc

k=150
a=0.5
b=1.5
J0=1

tspan = 0:0.001:1;
y0 = [0.035, 0.5];
opts = odeset('RelTol',1e-2,'AbsTol',1e-4);
sol = ode45(@(t,y) odefun(t,y,k,a,b,J0), tspan, y0, opts);

t = linspace(0,1,1000);
y = deval(sol,t);
% plot(t,y(1,:)) %Plot angular displacement vs t
% plot(t,y(2,:)) %Plot angular velocity vs t

acc=-k*(a^2+b^2)/J0*y(1,:);
plot(t,acc) % Angular acceleration vs Time

function dydt = odefun(t,y,k,a,b,J0)
dydt = zeros(2,1);
dydt(1) = y(2);
dydt(2) = -k*(a^2+b^2)/J0*y(1);
end