Solving Problems in Dynamics and Vibrations Using MATLAB ...

38
6. A Half Cylinder rolling on a Horizontal Plane
Example
The general governing differential equation of motion of the above problem is
2
2
3mr
..
.
( − 2mer
sin β ) β − mer cos β β = wecos
β . (1)
2
4r
Solve the above differential equation for m = 5 Kg; e = m; r = 0.1 m.
3π
Solution
To solve the differential equation, convert the equation into two first order differential equation.
With
β
.
β
=
=
y(1);
y(2);
Equation (1) reduces to the following equations
.
y(1)
= y(2);
.
[ wecos(
y(1))
+ mer cos( y(1))(
y(2)^2)]
y(2)
=
[1.5mr^2
− 2mer
sin( y(1))]
MATLAB Code
The function file ‘tofro.m’ contains the systems derivatives in the form of a vector.
function yp = tofro(t,y)
m=5;
r=0.1;
e=(4*r)/(3*pi);
g=9.81;
yp=[y(2);((m*g)*e*cos(y(1))+m*e*r*cos(y(1))*(y(2)^2))/(1.5*m*r*r
-2*m*e*r*sin(y(1)))]
The main code, which is given below calls the function ‘tofro.m’, which contains the system
differential equation and plots the displacement with respect to time. Since there is no damping
present, the plot resembles a sinusoid of constant amplitude.
tspan=[0 4];