Project - Position control of mass spring damper system

Obtain the transfer function of a mass spring damper system and use it in the model. Add a PID controller to adjust the force on mass so that its position follows a reference signal.

Take m = 3.6; %kg k = 400; %N/m c = 100; %Ns/m

The reference signal is shown in the following figure.

solution:

The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. This model is well-suited for modeling objects with complex material properties such as nonlinearity and viscoelasticity. Packages such as MATLAB may be used to run simulations of such models. As well as engineering simulation, these systems have applications in computer graphics and computer animation.

The spring-mass-damper system consists of a mass with weight (m), a spring with stiffness (k), and a shock absorber with a damping coefficient of (c). This system works based on Newton’s second law, which states that Force is equal to the rate of change of momentum. For a constant mass, force equals mass times acceleration.

Deriving the Mass Spring Damper System based on Newton’s second law F = m*a

where

F = Resultant of all the forces

m = Mass

a = Acceleration

Fig1: Mass Spring Damper System in Rest

k is the spring stiffness

d is the damping coefficient

The above system is in a state of rest. Let's consider a moving Mass Spring Damper System to see forces act during motion. What are the forces acting if the system is moving?

Fig2: Mass Spring Damper System in Motion

Expression for Spring Force and Damping Force

Fs = k*x ----(1)

Fd = d*(dx/dt) ----(2)

Applying F = m*a

ΣF = m*(d^2x/dt^2) (in the positive direction)

− Fs − Fd + F = m*(d^2x/dt^2)

F = m*(d^2x/dt^2) + Fd + Fs ----(3)

Substituting eq(1) and eq(2) in eq(3)

F = m*(d^2x/dt^2) + d*(dx/dt) + k*x

The above equation is the equation of motion (EOM).

Applying Laplace Transform to the above equation

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

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

x(s)/F(s) = 1/(m*s^2 + d*s + k)......transfer fnc

The above equation is the transfer function

Model

The Simulink Model is built to obtain the transfer function of a mass spring damper system. A PID controller is also implemented into the system to adjust the force on the mass so that its position follows a reference signal.

Below are some of the block details used to create the Simulink Model,

• Pulse Generator

(Generate square wave pulses at regular intervals)

The Pulse Generator block generates square wave pulses at regular intervals. The block waveform parameters, Amplitude, Pulse Width, Period, and Phase delay, determine the shape of the output waveform. The following diagram shows how each parameter affects the waveform.

• PID Controller

(Continuous-time or discrete-time PID controller)

This block implements continuous- and discrete-time PID control algorithms and includes advanced features such as anti-windup, external reset, and signal tracking. You can tune the PID gains automatically using the 'Tune...' button (requires Simulink Control Design).

• Transfer Function

(Model linear system by transfer function)

The Transfer Fcn block models a linear system by a transfer function of the Laplace-domain variable s. The block can model single-input single-output (SISO) and single-input multiple-output (SIMO) systems.

A Pulse Generator block is used as a reference Force input signal for the system. A Transfer function block is used for the implementation of the Mass Spring Damper system. The systems-derived equation in the Laplace transform is used in the transfer function. Variables of the same are defined through a Matlab script. The output of the transfer function is the displacement of the system, which is then fed back to the PID Controller as an error signal after subtracting it from the reference signal. Using a Controller the system throws the best tracking response.

Matlab Script

Variables m (mass), k (spring constant), and c (damping constant) used in the transfer function are defined through this Matlab script.

clear
clc
m = 3.6; % Mass [kg] 
k = 400; % Spring constant [N/m] 
c = 100; % Damping constant [Ns/m]

Pulse Generator

The parameters of the Pulse Generator are shown below. It satisfies the given reference signal for the system.

PID Controller

The PID values are selected automatically for the best response through PID Tuner.

Scope Output

The Scope Output is the best tracking response of the system. It is achieved with the help of the PID Controller. Very small Overshoot and Undershoot of the signal can be seen.