Solving Problems in Dynamics and Vibrations Using MATLAB ...

26
3. Coulomb Friction
Example
Solve the following differential equation for θ ≥ 0 .
2
θ ..
.
g
2
(1 + sinθ
) + µ cosθ θ = [cosθ
(1 − µ ) − µ
d
d
l
d
d
µ
d
µ ={0, 0.1,0.2}; l = 0. 25m
. θ (0) = 0; θ(0)
= 0
Solution
.
.
sinθ
]
As explained earlier, to solve the above differential equation by MATLAB, reduce it into two
first order differential equations.
Let θ = y(1)
.
θ = y(2)
The above equation reduces to
.
y (1) = y(2)
.
g
2
y(2)
=
{cos( y(1))[1
− µ
d
] − µ
d
sin( y(1))}
−
l(1
+ µ sin( y(1)))
l(1
+ µ
d
d
1
{ µ
sin( y(1)))
d
cos( y(1))(
y(2)^2)}
MATLAB Code
A M-file must be written to store the derivatives. In this case the file is saved as ‘coulomb.m’.
function yp =coulomb(t,y)
g=9.81;
mu=0;
l=0.25;
k1=1+mu*sin(y(1));
k2=mu*cos(y(1))*(y(2)^2);
k3=cos(y(1))*(1-mu^2);
k4=mu*sin(y(1));
yp=[y(2);((g/l)*((k3-k4)/k1)-(k2/k1))]
Type the following code in a new M-file.