# Assignment

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### ENEM14015 Dynamic system control

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Executive Summary
The main objective of this project is to implement the two degree of freedom and construct the three degree of freedom using Matlab Simulink. The transfer function and Laplace modelling equations will be evaluated using two degree of freedom. The simulation of the two degree of freedom gives results to find the magnification factor and transmissibility factor. The First contribution of this project is creating the physical model of motor bike which includes motion stiffness that are used to determine the modelling equations. The simulation of the multiple degree of freedom will be analyzed.

Introduction
The project describes a performance of dynamic System modelling and control of motor bike. The suspension performance and modelling equations will be evaluated in this project. In any variety of automobile system, suspension is most important to maintain the grip on the road. According to newton’s law the two degree of freedom will be analyaed.The development of motorbike model is controlled by some parameters that are stiffness and motion. A motorbike have many advantages like security and comfortability. The variation response of the system will be analyzed and the simulation of multiple degree of freedom will be executed.

Block diagram of Dynamic System

Fig1: Schematic of motorbike
The above diagram displays schematic model of motorbike. This diagram demonstrate the performance of the motor bike. Here the input is a driver, he is controlled by throttle. The two process will be happened in the dynamic system that are actuation and drive torque.

Suspension performance analysis
There are two types of suspension system available.
1) Spring mass    2) Unspring mass

A spring mass is one part of the vehicles. It is used to support the suspension system.
The unspring mass is directly connected to wheel bear, axles and wheel hub. The unwanted and uneven forces on the unspring mass caused by the imperfections of the road.

Suspension system divided into three types.
a) Active Suspension system
b) Semi-active suspension system
c) Passive Suspension system

Active suspension system: The active suspension system is also called as the modern suspension system. In the active suspension system the suspension rate of spring can be varied at any time.

Semi-active suspension system: The semi-active system is also called the adaptive control system. Compared to the active system this system uses less power. If the vehicle running in the normal surface, then the system is to provide the good handling. On the other hand.

Passive Suspension system: The passive suspension system otherwise called as the conventional suspension system.  It is very important and widely used. The passive system is low cost. This is one reason to popular of this system.
Two degree of freedom model
The two degree-of- freedom is used to eliminate the gravitational effects on suspended mass. In degree of freedom two masses are used (m1 and m2). In this case two motion and two stiffness are comes due to when mass is activated.

Fig2: Body diagram of two degree of freedom
The above diagram shows the free body diagram of two degree freedom. Based on this diagram the two degree of freedom expressed as,

The motion equation obtained by adding all the forces of each mass.
Now applying force (F1) to the each mass that is for F1 and F2. Then the equations will be shown in below.

For mass m1:
-------- (1)
For mass m2:
-------------- (2)
Rearrange the expressions then the equations will be,
-----------------------(3)

The above derivations are two degree of freedom modelling equations that are used to draw the diagram of 2DOF. The simulation diagram of the two degree of freedom is executed using these equations.

Simulation of the two-degree-of freedom

The above screenshot shows the Simulink diagram of the two degree of freedom. In this diagram two masses are used. And two integrators will be used to obtain the output. These integrators are connected to scope operator. Two sum operations is used in the DOF system that is used to access the input signal. Then k2 and c2 are connected to the sum operation. Some parameters are used in the two degree of freedom,
Mass1 (m1) = 1/m1
Mass2 (m2) = 1/m2
Using this diagram, the simulation of the two degree of freedom will be executed. Hence applying the values of m1, m2 and k1, k2 and c1, c2. Those values are representing the two degree of freedom waveform.

Simulink output of two degree-of-freedom

The above screenshot displays the output of the two degree of freedom of dynamic system. This result shows two waves that are linearly changed. Compared to single degree of freedom this output have some changes.

Vibration response
Based on the types, Vibration response is divided into two types. That are
Forced vibration
Free vibration

Free vibration means the vibration system started in its initial conditions is known as free vibration. The vibration system with an external force is called as free vibration. When the natural frequency and external frequency are obtained in same time resonance will occur.
The vibration analysis of undamped condition is,

Plot of Magnification Factor

The above screenshot represents the output plot of the transmissibility factor. The transmissibility factor of two degree of freedom will be executed using Matlab code.

Laplace transform of the two degree of freedom
Laplace transform of the equations of motion with c1=0, c2=0, c3=0 and the no damping conditions the equations will be,
------------- (4)

-------------------- (5)
In matrix form it taken to be G11 , G12, G22 and where G11 , G12, G22 are given by,
----------------------- (6)
Transfer Function
The transfer function is ratio of Laplace transform output [X1(s)] to the Laplace transforminput [F1(s)].
G1(s) = X1(s) / F1(s) is given by
---------------------- (7)

Comparison between 1DOF and 2DOF
Difference between single degree of freedom and two degree of freedom is discussed in below.
Compared with the two degree of freedom the Single degree of freedom is easier. Because the single degree of freedom only one mass is used.so the implementation and development of modelling equation process is easy. But in the case of two degree of freedom or multiple degree of freedom many masses are used. The Implementation of the system take more time.
A single degree of freedom mass-spring system have one natural mode of oscillation. But the two degree of freedom has two natural modes.
The output of single degree of freedom is in single wave. is In the case of Two degree of freedom the output is not an exact sine wave and it is not displayed in single wave. It represents in two linear wave.

PART 2

Natural frequency of two degree of freedom
Now consider the mass values m1 = m and m2 = 2m

In matrix form the equation will be,

For non-zero values of A1 and A2 is

So the natural frequency of two degree of freedom is,

It can be written in the matrix form,

Where    m = mass, k = stiffness, x = vector of generate co-ordinate.

Two degree-of-freedom pitch model

The modelling equation of the two DOF for pitch model represented as

Now it can be written in differential equation. Differential equation for pitch θ and bounce z of motorbike is,

Here  is called as coupling co-efficient. If  = 0 in this case no coupling will occurs. Without damping conditions, the equation will be expressed in given below and it gives sinusoidal output. The vertical motion will be represented as,
Z = Z sin ω t
The pitch motion is
ϴ = ϴ sin ω t
Using these analysis, the below parameters are defined

Where,
Kf  = Front ride rate
Kr = Rear ride rate
b= Distance from the front axle to C.G
c= Distance from the rear axle to C.G
Iy = Pitch moment of inertia
k= Radius of gyration
These equations are differentiated twice and substitute these value into equation (2),

Since the terms must be equal to zero. The condition will be shown in below.

The values of this equation are the roots representing the frequency of vibration modes.

The oscillations will be find by using amplitude ratio of equations.

Simulation of three Degree-of-freedom

The above screenshot represents the three degree of freedom simulation diagram. This diagram constructed with three mass subsystem. Each subsystem have three input and two output port. The inputs are force, Cin and Kinand the output ports are X1out and X1dout. Every subsystem have one gain so the total gains are m1, m2, m3. These masses are represents 1/m1, 1/m2, 1/m3.The one step input signal is used in all subsystem. Sum operation will be used in the 2DOF. Motion of the mass is connected to the summation positive port. The scope operator output is used in this model and change that scope operator with its three ports. Each subsystem of X1 out (output) is connected to the scope operator.
The one subsystem of the three degree-of-freedom is given below. This picture represents single degree of freedom.

The subsystem of three degree of freedom

Conclusion
The Method of implementation of multiple degree of freedom has been analyzed by using modelling equations and Matlab Simulink. The suspension performance and two degree-of-freedom unspring mass has been discussed. The Laplace equations and transfer function equations of two degree-of-freedom has been done using the system parameter. The performance of the motor bike has been analyzed and the Transmissibility factor would be plotted using Matlab code.

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