Modelling of Mathematics Function using Matlab & Simulink and Study the Spring Mass Suspension System

Study I - To plot a function yx=(x-1)*(x-3)^2*(x-9)^2 using basic Simulink blocks.

Input :

Here we have to plot the function as given below using basic Matlab Simulink block

y(x) =(x-1)⋅(x-3)2⋅(x-9)2">y(x) =(x−1)⋅(x−3)2⋅(x−9)2y(x) =(x-1)⋅(x-3)2⋅(x-9)2

To make this we have to use various block which is already inbuilt in Matlab Simulink. Various blocks which have been used as input are mentioned below as -

1. Ramp
2. Constant
3. Sum
4. Square
5. Product
6. Display
7. Scope

Description of block :

• Ramp - The Ramp block generates a signal that starts at a specified time and value and changes by a specified rate. The block's Slope, Start time and Initial output parameters determine the characteristics of the output signal.
• Constant - The Constant block generates a real or complex constant value signal.
• Sum - Actually this is the block for addition & subtraction of element but in our case, we used it for subtraction purpose.
• Square - The square block will square the input element.
• Product - This block takes the input element and performs the multiplication of the input provided to it.
• Display - It displays the final result of the model for which the model is been prepared.
• Scope - This block is used to see the result of the model in terms of graph or plot.

With the help of all the above-mentioned blocks, the function is represented in the image below using Matlab Simulink.

Representation :

Fig - Simulink Model of Mathematical Model

Output result using the scope :

Fig - Output nature of a given mathematical model

Here we can see from the graph that as the value of 'x' changes for the time the value of product changes respectively.

Study II - Analyze the system and expalin the spring mass suspension system. Also explain the generation of oscilating velocity source.

1. What is the objective of the simulation? What are the similarities and differences between the three systems understudy?

-> The objective of the simulation is to damp the vibration across the connected velocity source with the use of different suspension system like spring and dampers. There is suspension system arranged in each of the three system between the velocity source which can be any velocity oscillating source and the sprung mass. This is similar model to typical system in car where the sprung mass is dependent on the suspension system and the unsprung mass is attached at the other end of the suspension.

Fig - Suspension System Comparison

Similarities in all the system is the sprung mass and the unsprung mass along with the tire stiffness. The difference is all the three system is in the spring attached between them. In

• In 1st case shock absorber is used to perform the funtion of damping the motion of the force coming onto the sprig through the velocity source.
• In 2nd case the non linear spring is used which exert the non consistent force which mean it will not provide enough force if the displcement goes double in some case for that tranlational damper is attached in combination to give good damping.
• In 3rd case variable spring is attached along with the damper in which some external damping coefficient varriably and get the effect of it on the system.

2.What are your observations from the results?

-> There are basically Goto attached to send signal to & from the block that have specified tag.

The motion sensor is connected in each system as shown below and its output is checked using the scope

Fig - Sensor Sprung Mass SA Fig - Sensor Sprung Mass NL

Fig - Sensor Sprung Mass Var

Reult Observation :

This results shows the position or the displacement of the sprung mass when the system goes under some oscilation. At the start of oscilation the sprung mass position observed to be in between the range of -0.08 to 0 and the respective velocity range is -0.1 to 1. As the time of cycle increased the damping effect also goes better.

In this graph you see the result of frequency of the spring. The variable spring frequency is so high and it seem to be very bad effect on the system. In non linear spring also the deviation of the frequency is high so better is we go with th linear spring of which the force distribution is consistent and damping spring rate is also better.

3.How the oscillating velocity source is created?

->

Oscillatiing velocity is created by giving the sine wave input and pulse generator. an oscillator is an electronic circuit used to generate a continuous output signal. Generally this output signal is in the form of a sinusoid at some predetermined frequency or wavelength set by the resonant components of the circuit. Electrical Waveforms are basically visual representations of the variation of a voltage or current over time. This means that if we plotted these voltage or current variations on a piece of graph paper against a base (x-axis) of time, ( t ) the resulting plot or drawing would represent the shape of a Waveform.

Conclusion -

By solving this Simulink model we can say that how Simulink makes it easier to build a mathematical model in its environment.