Experiment (4)

Simulation of Vibrations Using MATLAB (2) Introduction

In the last experiment, free vibration systems were studied. In this experiment, the forced

vibration of mechanical systems is studied. This could be the model for wide engineering

applications; mostly rotating machines like tires, engines, or any rotor.

Objectives

This experiment aims to:

1- Study the response of systems under harmonic and base excitations.

2- Study the relationship between the natural frequency of the system and the exciter

frequency.

3- Study the resonance behavior of the systems.

Theory

Harmonic Excitation

The model of the single degree of freedom is shown in figure (1) below, where an

external harmonic (periodic) force is applied on the mass.

Figure 1: Single degree of freedom system

Draw the free-body diagram to show that the equation of motion is:

Where

This is a 2nd

order differential equation that can be solved using various techniques.

However, the solution is composed of two parts; homogeneous or called transient and the

particular or called steady-state response. The initial conditions must be applied to the

sum of the two parts of the solution.

Assume that the system is under-damped, and then the final response is:

The general solution is given by:

The response has several cases:

1) Un-damped forced vibrations:

a- Driving frequency is different than and away from natural frequency .

Usually the response in this case is stable with bounded limits. Two distinct

frequencies are embedded in the solution and contributing to the response, see

figure (2).

Figure 2: Case 1-a

b- Driving frequency is close to the natural frequency .

The response in this case is called beating. Two distinct frequencies are

embedded in the solution and contributing to the response, see figure (3).

Figure 3: Case 1-b

c- Driving frequency is equal to the natural frequency .

The response in this case is called resonance. The response in this case is

unstable with unbounded limits. Usually called resonance and it is flutter-like

response in the case of no damping, see figure (4).

Figure 4: Case 1-c 2) Damped forced vibrations: the total response is composed of transient part that die

out with time, and the steady-state part which follows the excitation function and

show up clearly after a few cycles, see figure (5).

Figure 5: Case 2

Usually the steady-state part of the response is more important that transient because it

lasts longer. For that, steady state vibration takes more importance in analysis and design

of machines.

If you take the ratio

, the maximum amplitude of vibration can be rewritten as:

This equation is the normalized version of the original one. It is shown in figure (6):

Figure 6: Normalized amplitude vs. frequency ratio

Many notes can be deduced from this plot, mainly:

1) As the damping ratio is increased, the maximum vibration decreased.

2) The maximum peak vibration occurs a little bit before r=1. (

).

3) A slight shift of the driving frequency can lead to significant reduction in

vibration level.

4) This shows the importance of knowing the natural frequency of the mechanical

system before you design the exciter (e.g. motor) so that you guarantee a driving

speed of motor away from the natural frequency in order to have accepted level of

vibrations (always design away from resonance). This is called modal analysis

where an impulse input is applied to the system and the peaks of the vibration

spectrum are detected in order to deduce the natural frequencies (modes) of the

system.

Figure 7: Phase diagram

Figure (7) above is called the phase diagram. It shows that as the driving frequency

increases, the phase difference between the steady-state response and the driving force

increases from 0 to and is equal to /2 at resonance.

Base Excitation

When a car moves on a rough terrain, it undergoes noticeable vibrations which

sometimes can be severe.

When an airplane moves across air bumps, the wings can be seen clearly vibrating; why?

An operating motor in a factory can cause vibrations that might affect even the furniture

in nearby rooms.

Figure 8: Base excitation examples

All of these are examples of what called base excitation; significant part of vibration

force and vibration displacement is transmitted through the base or support into the

surroundings, see figure (8). This can be modeled as shown in figure (9):

Where; ).

Figure 9: Base excitation model

Because the equation is linear, superposition can be employed. The solution for each part

of the two inputs (the right hand side of the E.O.M.) is similar to what is seen in the

harmonic excitation part.

6)

Where; .

The normalized amplitude (Equations 7 and 8) can be rewritten using trigonometry in a

more compact form and if the frequency ratio

then:

The previous equation is plotted as shown in figure (10).

Figure 10: Normalized amplitude of the displacement vs. frequency ratio

Some notes on the previous plot:

1) The ratio

is the maximum response magnitude to the input displacement

magnitude; i.e. the displacement transmissibility.

2) For

a)

; i.e. the transmitted displacement is amplified.

b) As is increased

is decreased.

3) For

a)

; i.e. the transmitted displacement is attenuated.

b) As is increased

is increased.

Another important quantity that is transmitted from the base to the mass is the force

which is combined of both the spring force and the damper force.

Using the solution x(t) found before, it is possible to derive a new expression of the

transmitted force:

Figure 11: Normalized amplitude of the force vs. frequency ratio

Notes:

1) It is not necessary that the transmitted force is attenuated for .

2) For increased damping, the transmitted force is amplified dramatically for

.

Both displacement transmissibility and force transmissibility are important in design of

vibration absorbers like in car’s suspension system.

Experimental Procedures

Part 1: Harmonic Excitation Simulation

a- Undamped forced vibrations:

1) Open MATLAB software.

2) From New pulldown menu, select Simulink Model. See figure (12).

Figure 12: Open New Simulink Model

3) Open the Simulink Library Browser. See figure (13).

Figure 13: Simulink Library Browser

4) Open the Simscape > Foundation Library > Mechanical > Translational

Elements library as shown in figure (14).

5) Add a Mass, Translational Spring, and Mechanical Translational Reference

blocks into the model window.

6) Add a sensor to measure the speed and position of the mass. Place the Ideal

Translational Motion Sensor block from the Mechanical Sensors library. See

figure (15).

Figure 14: Translational Elements

Figure 15: Mechanical Sensors

7) Add an input force source to the model. Place the Ideal Force Source block

from the Mechanical Sources library. See figure (16).

Figure 16: Mechanical Sources

8) Open the Simulink > Sinks library and copy one Scope block to the model

window. Rename the Scope block to Position. See figure (17).

Figure 17: Simulink sinks library

9) Open the Simulink > Sources library and copy a Sine Wave block to the

model window. See figure (18).

Figure 18: Simulink sources library

10) Open the Simscape > Utilities library, copy a PS-Simulink Converter block,

Simulink-PS Converter block, and Solver Configuration block into the model

window. See figure (19).

Figure 19: Simscape Utilities library

11) Connect the system as shown in figure (20).

Figure 20: Spring-Mass System in Simscape

12) Set the simulation parameters as follows: Force amplitude= 200 N, Mass= 5

kg, Sprig stiffness= 50 N/m.

13) Calculate the natural frequency for this system.

14) Change the force frequency according to table (1) then record the response of

the system.

Table 1: Part 1-a (Undamped case)

Trial # Force frequency ω [rad/s]

1

2 3

3

4

b- Damped forced vibrations:

1) Build the system shown in figure (21).

2) Set the simulation parameters as follows: Force amplitude= 600 N, Mass= 5

kg, Damping coefficient=11.3 N.s/m, Sprig stiffness= 50 N/m.

3) Calculate the natural frequency for this system.

4) Change the force frequency according to table (2) then record the amplitude

and frequency of the displacement.

5) Keep the simulation parameters as in step 2, only change the damping

coefficient to 5 N.s/m.

6) Repeat steps 3 and 4. Record the required data in table (3).

Figure 21: Spring Mass Damper System in Simscape

Table 2: Part 1-b (Damped case, Damping Coefficient= 11.3 N.s/m)

Force frequency

ω [rad/s]

Displacement

amplitude [m] T(s) Displacement

frequency [rad/s]

2.73

Table 3: Part 1-b (Damped case, Damping Coefficient= 5 N.s/m)

Force frequency

ω [rad/s]

Displacement

amplitude [m] T(s) Displacement

frequency [rad/s]

2.73

Part 2: Base Excitation Simulation

1) Open MATLAB software.

2) Open the Simulink > Continuous library and copy a Derivative block to the model

window. See figure (22).

3) Open the Simulink > Signal Routing library and copy a MUX block to the model

window. See figure (23).

4) Build the system shown in figure (24).

Figure 22: Simulink Continuous Library

Figure 23: Simulink Signal Routing Library

Figure 24: Base Excitation (Spring Mass Damper System)

5) Set the simulation parameters as follows:

Mass= 5 kg, Damping coefficient= 2 N.s/m, Sprig stiffness= 50 N/m.

Input displacement amplitude (Y)= 0.1 m.

6) Calculate the damping ratio and the natural frequency, record them in table (4).

7) Change the input frequency according to table (4), then record the required data.

Table 4: Base Excitation data (C= 2 N.s/m)

Input

displacement

frequency

(ωb) [rad/s]

Damping

ratio (ξ)

Natural

frequency

(ωn)

[rad/s]

Output

displacement

frequency

(ω) [rad/s]

Output

displacement

amplitude

(X) [m]

r

Error

%

8) Keep the simulation parameters as in step 5, only change the Damping coefficient

to 18 N.s/m.

9) Repeat steps 6 and 7. Record the required data in table (5).

Table 5: Base Excitation data (C= 18 N.s/m)

Input

displacement

frequency

(ωb) [rad/s]

Damping

ratio (ξ)

Natural

frequency

(ωn)

[rad/s]

Output

displacement

frequency

(ω) [rad/s]

Output

displacement

amplitude

(X) [m]

r

Error

Discussion and Conclusions

According to the harmonic excitation part, answer the questions from 1 to 5:

1) Derive the response of case 1-c (the unstable fluttering case).

2) Obtain the velocity by differentiating the response with respect to time.

3) What is the relation between the velocity phase and driving-force phase?

4) How can this be used to explain the resonance physically?

5) Prove that the maximum peak vibration occurs at .

According to the base excitation part, answer question 6:

6) Show that the transmitted force is in phase with the transmitted displacement.

According to part 1: harmonic excitation simulation, answer question 7:

7) In part a, at which frequency does the resonance occur?

8) In part b, at which frequency does the maximum peak occur?

According to part 2: base excitation simulation, answer the questions from 7 to:

9) How does the amplitude and phase plots change with the change in the parameters

of the system? Justify your answer.