solving a 1 DoF MSS-Spring-Damper when the spring stiffness is a function of displacement

Hello guys,

I'm trying to solve a 1 DoF SMD system whith an external force (as a function of time) and a step spring stiffness as a function of the displacement.

Actually, after reaching a specific displacement, there would be a secondary spring (in parallel) so the total stiffness is afunction of displacement.

Any help would be appreciated.

[Sam Chak]

@reza ahmadian,

Can you show the mathematical equation for the 1-DoF SMD system that you are trying to solve?

How is the secondary spring suddenly brought into the 1-DoF SMD system without an external force?

[reza ahmadian]

Dear Sam,

Thank for replying,

The equation will be something like :

F(t)=m*x(2)+b*x(1)+k1*x+k2*f1(x)

where f1(x)= x-x1 if x>x1

=0 if x<=x1

x1,x,b,k1 and k2 are constant parameters

There is an external force. the fact is if that force exceed a specific value, the system will reach the engaging point over time, where the secondary spring will get involved.

Answer 1

[Sam Chak]

Hi @reza ahmadian

If the system is stiff, then you can try using the ode15s() solver.

In the following example, the external force f(t) is assumed a unit step function.

% Parameters
m      = 1;
b      = 2;
k1     = 1;
k2     = 1;
xd     = 0.5;

% Settings
% x(1) is the position of the spring
% x(2) is the time derivative of x(1)
odefun = @(t, x) [x(2); 
                 (heaviside(t) - b*x(2) - k1*x(1) - k2*((x(1) - xd).*heaviside(x(1) - xd)))/m];
tspan  = [0 15];
x0     = [0 0];

% Solving and plotting
[t, x] = ode15s(odefun, tspan, x0);
plot(t, x), grid on
xlabel('t'), ylabel('\bf{x}')
legend('x_1', 'x_2', 'location', 'east')

[reza ahmadian]

Thanks Sam,

That helped me alot, although I managed to model the problem by Simulink, the answers looks ok now