Project - Position control of mass spring damper system

To obtain the transfer function of a mass-spring-damper system, we can start with the equation of motion for the system:

m * x'' + c * x' + k * x = F

where x is the displacement of the mass, m is the mass, c is the damping coefficient, k is the spring constant, F is the external force applied to the mass, and ' denotes differentiation with respect to time.

Taking the Laplace transform of the above equation and assuming zero initial conditions, we get:

s^2 * M(s) + c * s * X(s) + k * X(s) = F(s)

where M(s), X(s), and F(s) are the Laplace transforms of m, x, and F, respectively. Solving for X(s), we get:

X(s) = F(s) / (s^2 * M(s) + c * s + k)

Substituting the given values of m, c, and k, we get:

X(s) = F(s) / (3.6 * s^2 + 100 * s + 400)

This is the transfer function of the mass-spring-damper system.

Lets us Input the following in a Matlab script to:

clear all;

clc;

% Parameters

m = 3.6; % kg

k = 400; % N/m

c = 100; % Ns/m

zeta = c / (2 * sqrt(k * m));

wn = sqrt(k / m);

Let us provide a some reference gain values (Kp, Kd, Ki):

% Proportional, integral, and derivative gains

Kp = 100;

Ki = 10;

Kd = 50;

Let us use the below simulink model with a Pulse generator as a reference signal:

Let us plot for above gain values we get :

Let us change the gain values to Kp = 1000; Ki = 0; Kd = 100:

Let us change the gain values to Kp = 5000; Ki = 0; Kd = 500:

Let us change the gain values to Kp = 5000; Ki = -10; Kd = 500:

Let us change the gain values to Kp = 7500; Ki = 50; Kd = 550:

Let us change the gain values to Kp = 7500; Ki = 50; Kd = 750:

By further increasing and adjusting the gain values we can make the position plot to follow the reference signal plot.