Question

1.2096 Trial and error: by printing out (using any software) the damped free vibration solution under different damping coefficients, please choose the damping coefficient of a spring-mass- damper system with mass of 150 kg and stiffness of 20o0 N/m such that its response will die out after about 2 s, given a zero initial position and an initial velocity of 10 mm/s.

Answer 1

Let the damping coefficient be c.

Then the equation of motion is:

$\bg_white m\ddot{x} + c*\dot{x} + kx = 0$

Applying lapalce transform:

$\bg_white m(s^2*X(s)-sx(0)-\dot{x}(0)) + c*(s*X(s)-x(0)) + k*X(s) = 0$

x(0) = 0

$\bg_white =>ms^2*X(s)+ c*s*X(s) + k*X(s) = m*\dot{x}(0))$

$\bg_white =>X(s) = \frac{m\dot{x}(0)) }{ms^2+ cs + k} = \frac{1.5 }{150s^2+ cs + 2000}$

The matlab code to plot the response for first 3 seconds for various values of damping coefficient is given below:

% Start of the code
close all
t = 0:0.01:3; % Time vector
syms s;
i = 1; % Iteration counter
for c=0:100:1000 %Define various values of damping coefficient
F = 1.5/(150*s^2+c*s+2000); % Laplace fucntion of x
x_il = ilaplace(F); %Inverse laplace of transfer fucntion
x = eval(x_il); % calculate values of x at various time values
plot(t,x); % Plot x as a function of t
hold on
l{i} = strcat('c=',num2str(c),'Ns/m');
i = i+1;
end
legend(l)
xlabel('Time (s)')
ylabel('x(t)')
grid on
% End of code

It can be seen that for a damping coefficient of 800 Ns/m, the response dies out after about 2s.