Challenge

Project - Position control of mass spring damper system

JJagaveerapandiyan Selvaraj29 Nov 2023

AIM

To Design the transfer function of a spring mass damper system using PID

Description

Spring Mass Damper

Governing Equation

The free-body diagram for this system is shown below. The spring force is proportional to the displacement of the mass, and the viscous damping force is proportional to the velocity of the mass, Both forces oppose the motion of the mass and are, therefore, shown in the negative -direction. Note also that this corresponds to the position of the mass when the spring is unstretched.

`F=ma+cu+kx`

Transfer function

`(x(s))/(F(s)) = 1/(m*s^2 + d*s + k)`

PID

changing equation to PID model

`x=-k/m+1/m(k_p(x_(ref)-x)+k_i(intx_(ref)-x)dt-k_d x`

MATLAB CODE

Main Code

%%%code by Jagaveerapandiyan.S
%%%Design Parameter
clear all;
close all;
clc;
m = 3.6; %kg
k = 400; %N/m 
c = 100; %Ns/m


%% Requirement of requirement responce
wn=sqrt(k/m);
fn=wn/2*pi;
zeta=0.9;
p1=-zeta*wn+wn*sqrt(zeta^2-1);
p2=-zeta*wn-wn*sqrt(zeta^2-1);
P=[p1;p2];
A=[0 1;-k/m -c/m];
B=[0;1/m];
%pole placement Design
K=place(A,B,P);
kp=K(1);
kd=K(2);
ki=750;


err=out.x-out.x_ref % almost need to zero

Simulink

Plot code

%plot
subplot(311)
plot(out.time,out.x,'b')
hold on
plot(out.time,out.x_ref,'r-')
legend('x','x_(referance)')
xlabel('time')
ylabel('x')
subplot(312)
plot(out.time,out.u,'b')
xlabel('time')
ylabel('displacemenr')
subplot(313)
plot(out.time,out.a,'b')
xlabel('time')
ylabel('acceleration')

Plot

ki-0