Determining Suspension System Behaviour

Aim: To determine behaviour of a suspension system

Objective:

To take input from user such as damping coeffiecient, spring stiffness and mass and use that to determine the behaviour of the system, whether it is overdamped, critical damped or underdamped system and plot the response curve.

Theory:

Suspension system has wide application in automobile industries and heavy vibration system. Suspension system damps out the vibration and protects machine against unwanted failure. Suspension system generally includes spring and damper.

According to the application, suspension systems are designed for overdamped, critical damped or underdamped behaviour.

Undamped System:

The system continues to oscillate and never comes to the equilibrium.

Such systems doesn\'t have any practical use.

Underdamped System:

The system oscillates to and fro about the equilibrium position with decreasing amplitude and gradually comes to equilibrium.

Critical Damped System:

The system comes to equilibrium as quickly as possible without oscillating.

Overdamped System:

The system comes to equilibrium without oscillating but with a slower rate depending upon how overdamped the system is.

The mathematical model for the system shown above can be written as,

`(d^2x)/dt + c/m(dx)/dt + k/mx = 0`

where,

c = damping coefficient

k = spring stiffness

m = mass

x = displacement

t = time

Natural frequency of the system is given by,

`omega_n = sqrt(k/m)`

Critical Damping Coefficient is given by,

`C_c = 2momega_n`

Damping Factor,

`zeta = c/C_c`

For Underdamped System,

`zeta<1`

For Critical Damped System,

`zeta=1`

For Overdamed System,

`zeta>1`

For solving the second order differential equation using MATLAB, it is required to first convert into two first order differential equation.

For this considering,

`x = x_1`

`dx_1/dt = x_2`

This simplifies the second order differential equation into two equations as follows:

`dx_1/dt = x_2`

`(dx_2)/dt = - c/mx_2 + k/mx_1`

Now these two equations can be solved using ode45 function of MATLAB.

MATLAB Main Program:

close all
clear all
clc

%Introduction
fprintf(\'This program is designed to determine the behaviour of a Suspension system\')
fprintf(\'\\n\\nInstructions\')
fprintf(\'\\n1)Input field with * mark are neccesary\')
fprintf(\'\\n2)The default value for non-mandatory fields is placed in square brackets []\')
fprintf(\'\\n\\nPlease provide following inputs\') 

% user inputs
c = input(\'\\nEnter Damping Coefficient(N/m/s)* : \');
k = input(\'Enter Spring Stiffness(N/m)* : \');
m = input(\'Enter mass(kg)* : \');
t = input(\'Enter time period(sec.)[5] : \');

% initial Conditions
fprintf(\'\\nProvide initial Conditions\')
d = input(\'\\nDisplacement(m)[0] : \');
v = input(\'Velocity(m/s)[10] : \');
% Default values
if length(d) == 0
    d = 0;
end
if length(v) == 0
    v = 10;
end
% initial Condition array
initial = [d;v];

% gravitational acceleration
g = 9.81;

% time space
if length(t) == 0
    t = 5;
end
time = linspace(0,t,500);

% solving differential equation
[time results] = ode45(@(t,x) ode_func(t, x, c, g, k, m), time, initial);

% Natural frequency
nfreq = (k/m)^0.5;

% critical damping coefficient
Cc = 2*m*nfreq;

% Damping Factor
Damp_fac = c/Cc;

% check system for underdamped, critical damped, overdamped
if Damp_fac>1
    head = \'Overdamped System\';
else if Damp_fac==1
        head = \'Critical Damped System\';
    else if Damp_fac==0
            head = \'Undamped System\';
        else
            head = \'Underdamped System\';
        end
    end
end

% plotting response curve
figure(1)
plot(time, results(:,1),\'linewidth\',2)
hold on
plot(time, results(:,2),\'linewidth\',2)
xlabel(\'time\')
ylabel(\'plot\')
legend(\'Displacement\',\'Velocity\')
title(sprintf(\'System Response - %s\',head))
grid on

ode_func Function:

function [dx_dt] = ode_func(t, x, c, g, k, m)

x1 = x(1);
x2 = x(2);
dx1_dt = x2;
dx2_dt = -(c/m)*x2 - (k/m)*x1;
dx_dt = [dx1_dt; dx2_dt];
end

ode_func converts second order differential equation into two single order differential equation.

Plots:

1) Undamped System

Inputs,

c = 0 N/m/s

k = 100 N/m

m = 25 kg

2) Underdamped System:

Inputs,

c = 50 N/m/s

k = 100 N/m

m = 25 kg

3) Critical Damped System:

Inputs,

c = 100 N/m/s

k = 100 N/m

m = 25 kg

4) Overdamped System:

Inputs,

c = 200 N/m/s

k = 100 N/m

m = 25 kg

Conclusion:

1) Undamped system keeps oscillating and never comes to equilibrium/rest.

2) Underdamped system oscillates to and fro about the equilibrium with decreasing amplitude and gradually comes to rest.

3) Critical damped system comes to rest as quickly as possible without oscillating.

4) Overdamped system comes to rest without oscillating but with a slower rate than critical damped system.