i

Project

ESA322/EMM342

Dynamic Vibration Absorber

School: Aerospace Engineering

Bil. Name ID Number Signature

1 Ahmad Shaqeer Bin Mohamed Thaheer 111706

2 Hairuniza Binti Ahmed Kutty 111711

3 Nazreen Shah Bin Nasip 111722

4 Shanul Nellisa Binti Sharulnahar 111725

5 Mohammad Iddin Ikram Bin Mohammad Aminuddin 115901

Marks

Organization 20%

Organization 10

Mechanics (grammars & punctuations) 10

Content 80%

Intro/Theory 10

Design (simplicity, originality, CAD) 15

Analysis (FBD, EOM, Solution) 15

Testing (functionality) 15

Results/Discussions - figures/tables 15

Conclusion 7

References 3

Total 100%

ii

CONTENTS

LIST OF FIGURES .............................................................................................................................. ii

LIST OF TABLES ................................................................................................................................ ii

1 INTRODUCTION ......................................................................................................................... 1

2 THEORY ....................................................................................................................................... 2

3 DESIGN ......................................................................................................................................... 5

4 ANALYSIS .................................................................................................................................... 6

5 RESULTS AND DISCUSSION ................................................................................................. 10

6 CONCLUSION ........................................................................................................................... 12

7 REFERENCES ............................................................................................................................ 12

8 APPENDICES ............................................................................................................................. 13

APPENDIX A C.A.D DESIGN ..................................................................................................... 13

APPENDIX B MEASURING SPRING STIFFNESS................................................................... 13

LIST OF FIGURES

Figure 1 The primary system (a) and reduced or simplified system (b). ............................................... 2

Figure 2 With DVA is being added to the system (a) and the new simplified system (b). .................... 2

Figure 3 Free Body Diagram for the primary mass. .............................................................................. 3

Figure 4 Graph of amplitude against frequency ratio ............................................................................ 4

Figure 5 Dynamic Vibration Absorber set up apparatus ........................................................................ 5

Figure 6 Graph of Force against Extension for spring 1 ........................................................................ 6

Figure 7 Graph of Force against Extension for spring 2 ........................................................................ 7

Figure 8 Set-up for data collection to determine spring constant ........................................................ 10

Figure 9 CAD design of the DVA system (a) and closer view of the configurations of the platform

mass (b) ................................................................................................................................................. 13

Figure 10 Determination of spring stiffness by extending the spring .................................................. 13

LIST OF TABLES

Table 1 Determination of spring stiffness 1 ........................................................................................... 6

Table 2 Determination of spring stiffness 2 ........................................................................................... 7

Table 3 Result analysis between primary system and absorber system ............................................... 10

1

1 INTRODUCTION

Dynamic Vibration Absorber (DVA) is basically a type of passive vibration control

system and it is based on the concept of attaching secondary mass to a primary vibrating system

such that the secondary mass dissipates the energy or taking the energy from the primary

system and thus reduces the amplitude of the primary vibration system. In addition, dynamic

vibration absorber is designed so that the natural frequencies of the resulting system are lower

from the excitation frequency. Theoretically, the absorber mass or secondary mass should be

25% from the primary mass.

However, the classic model of the vertical 2DOF spring-mass-damper system with a

DVA is not widely available to demonstrate this phenomenon, so this project targeting in

creating a system with low-cost configurations. It was decided that an electric motor with an

unbalanced mass is the most effective way to generate a periodic applied force in the system

so that the excitation frequency can be controlled. The biggest challenge in making a physical

model of this 2DOF system is constraining the motion.

The DVA has some advantages compared to other vibration suppression techniques

such as it is an external to the machine structure, thus, no re-installation of equipment is needed.

Other than that, the DVA can have a pre-design and tested even before any structural

modifications done to the main product without knowing its mass-elastic properties of the

product. It can be adjusted in the lab environment with predictable field results, therefore, offers

user an economical vibration reduction solution.

2

2 THEORY

Initially, the vibration system is set in a single degree of freedom system as shown in

Figure 1. Then, this system is excited with an unbalance rotating mass until it reaches resonance

condition. Finally, an absorber is applied into the system to absorb the energy from the primary

system i.e. the primary vibration system is transferring its energy to the secondary mass up

until a point in which the amplitude of the primary system becomes or approximately zero.

Consider a vertical spring-mass system is as shown in Figure 1 and unbalance mass is

applied as a harmonic disturbance to a SDOF system.

(a) (b)

Figure 1 The primary system (a) and reduced or simplified system (b).

Initially, the system is in single degree of freedom but then since it is excited in

resonance, an absorber mass should be added to the system and converting the system into two

degree of freedom system. Basically, a vibration absorber is secondary spring-mass system that

being added to the primary system as shown in Figure 2 and design to absorb the input

disturbance by shifting the motion to the new added mass i.e. absorber mass.

(a) (b)

Figure 2 With DVA is being added to the system (a) and the new simplified system (b).

3

Figure 3 Free Body Diagram for the primary mass.

By applying Newtons 2nd law to both primary and secondary mass, then combine these two

equation will yield,

1 1 1 2 2 1 0

2 2 2 2 2

0 sin( )

0 0

m x k k k x F t

m x k k x

(1)

It is assumed that the trial function or steady state response for two degree of freedom is given

by,

1 1

2 2

sin

sin

x X t

x X t

(2)

By differentiating it into 1st and 2nd order,

1st order: 1 1 cosx X t ; 2 2 cosx X t (3)

2nd order: 21 1 sinx X t ; 2

2 2 sinx X t (4)

Substitute equation (3) into equation (1) will gives out,

21 1 2 2 1 01

22 2 2 22

0 sin( )

0 0

m k k k X F tX

m k k XX

(5)

2 1 01 2 22

22 2

sin( )

0

X F tk k m k

Xk k m

(6)

Therefore, through simplification, the amplitude of the system is given by,

2

2 2

1 2 2 2

1 2 1 2 2 2

k mX

k k m k m k

(7)

4

2 0

2 2 2 2

1 2 1 2 2 2

k FX

k k m k m k

(8)

From equation (7), it can be observed that the amplitude of vibration of the primary system,

1X can be zero if the numerator becomes zero,

22 2 0k m (9)

Or,

22 2

2 n

k

m (10)

Based on figure below, when the excitation frequency is equal to the natural frequency

of the absorber, the amplitude of the main system becomes zero even though it is excite by

harmonic motion force. Therefore, the primary mass will be totally damped and transferred to

the secondary mass. The amplitude of the secondary mass is then calculated. But in reality, a

DVA system can only minimize the oscillations unless it is design perfectly.

Figure 4 Graph of amplitude against frequency ratio

5

3 DESIGN

A physical model of the system was developed (CAD design refer to appendix) as can

be seen in Figure 5. The frame are made of square hollow steel where there is a rail track

attached to it. Here, the platform will slide up and down following the rail track. The platform

are made of solid steel and a case fan is used to act as a rotating motor with a unbalanced mass

attach to it. To combine the platform and the case fan, a 5 mm Perspex plate were used. Since

the platform are assembled in a rail concept, thus to reduce frictions, a bearing is used. 2 springs

were used by hooking it up on top and bottom of the plate and a hook is used to help attach the

springs. The electric drive for the case fan is a DC power supply connected to a voltage

regulator since the case fan only supports 12V of power.

Figure 5 Dynamic Vibration Absorber set up apparatus

6

4 ANALYSIS

There 2 types of spring used in the primary system which the stiffness of the springs

are noted as 1k and 2k respectively. For the absorber system, the stiffness of spring is ak .

For spring 1 stiffness, 1k :

Initial displacement, 0x = 7.6 cm

Mass (g) Force (N) Displacement, x (cm) Extension, x (cm) Extension, x (m)

342 3.36 8.40 0.8 0.008

366 3.59 8.60 1.0 0.010

502 4.92 10.0 2.4 0.024

708 6.95 12.1 4.5 0.045

1000 9.81 15.0 7.4 0.074

Table 1 Determination of spring stiffness 1

Figure 6 Graph of Force against Extension for spring 1

Since two spring 1 are used for the SDOF system, therefore 1 is multiplied by 2 due to the

spring position in parallel. Thus, the stiffness of spring 1 is calculated by,

2 11

2 1

2y y

kx x

1

4.92462 3.590462

0.024 0.01k

1 1.1118k N/m

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 2 4 6 8 10 12

Fo

rce,

F (

N)

Extension, x (m)

Graph of Force against Extension

for Spring 1

7

For spring 2 stiffness, 2:

Force (N) Extension, x (cm) Extension, x (m)

5 8.1 0.081

10 9.8 0.098

15 11.7 0.117

20 13.9 0.139

25 16.2 0.162

30 18.0 0.180

35 19.9 0.199

40 22.2 0.222

Table 2 Determination of spring stiffness 2

Figure 7 Graph of Force against Extension for spring 2

The stiffness of spring 2 is determined by calculating the gradient,

2

25 20

0.162 0.139k

2 217.391k N/m

Therefore, the total stiffness spring of primary system is,

1 2k k k

1.1118 217.391k

218.503k N/m

0

5

10

15

20

25

30

35

40

45

0 0.05 0.1 0.15 0.2 0.25

Fo

rce,

F (

N)

Extension, x (m)

Graph of Force against Extension

for Spring 2

8

For the stiffness of spring in absorber system, :

Initial displacement, 0x = 1.5 cm

Final displacement, fx = 3 cm

Extension, 3 1.5 1.5x cm 0.015 m

Mass, m 100 g 0.1 kg

Gravitational acceleration, g 9.81 ms-2

Force, F mg 0.1 9.81 0.981 N

aF k x

a

Fk

x

0.981

0.015ak

65.4ak N/m

The natural frequency of the primary system and absorber system are calculated as below,

The data taken from the setup of apparatus:

Primary mass, 754 0.745pm g kg

Absorber mass, 101 0.101am g kg

Unbalanced mass, 13 0.013um g kg

For the natural frequency of primary system,

n pp

k

m

218.503

0.754n p

17.0233n p rad/s

For the natural frequency of absorber system,

a

n a

a

k

m

65.4

0.101n a

24.4465n a rad/s

Assume that the speed of fan motor that we take from CPU is 250 RPM and the eccentricity of

the fan motor is 3 cm.

9

1 RPM 0.104719755 rad/s

Therefore, 250 RPM 26.1799 rad/s and eccentricity, 3 0.03e cm m

The force of the motor, 0F , can be calculate:

2

0 pF m e

20 0.754 0.03 (26.1799 )F

0 15.5035F N

The amplitude of primary system, :

2

0

2 2 2

a a

a p a p a

k m FX

k k m k m k

2

2 2 2

65.4 0.101 (17.0233 )

218.5 65.4 0.754 (17.0233 ) 65.4 0.754 17.0233 6

15.5

5.4

035X

0.0392X m

The amplitude of absorber system, :

0

2 2 2

a

a

a a a

k FX

k k m k m k

2 2 265.4 15.5035

218.5 65.4 0.754 17.0233 65.4 0.754 17.0233 65.4aX

0.07095 a

X m

10

5 RESULTS AND DISCUSSION

System Natural frequency (rad/s) Amplitude (m)

Primary 17.0233 -0.03920

Absorber 25.4465 -0.07095

Table 3 Result analysis between primary system and absorber system

Hookes Law, law of elasticity states that any deformation of a spring size or

displacement is directly proportional to the force needed to extend, or compress the spring.

This statement can be expressed mathematically as:

F k x (11)

Where:

F Force applied to the spring

k Spring constant

x Displacement of the spring

To determine the spring constant, some data collection need to be done by conducting a simple

experiment as shown in below. By adding extra mass, the deflection of the spring may be

observed. This data is collected and tabulated in Table 1 and Table 2 in Section 4.

Figure 8 Set-up for data collection to determine spring constant

Figure 6 and Figure 7 shows the graph plotted to determine the spring constant for

spring 1 and spring 2 respectively. As stated above, the relationship of the graph depicts

equation (11), therefore, the spring constant is the slope of the line in the graph plotted.

Initially, the system is a single degree of freedom system, which consists of single

primary mass and rotating unbalance mass, as shown in Figure 1. The presence of unbalance

mass is normally the primary causes of vibrations. This is due to the distribution of the mass

that is uneven, around the axis of rotation that will cause the center of mass to be misaligned

with center of rotation, creating centrifugal force. The unbalance mass effect becomes greater,

as the primary rotating speed increases. The effect of unbalance mass need to be encounter as

it may cause vibration, noise, and may also cause manufacturing defects.

11

The primary natural frequency is the original natural frequency of the system before

attachment of absorber, and absorber natural frequency for the system is measured before it is

attached to primary mass. Table 3 shows the results of the natural frequency and amplitude of

primary system and also absorber. The natural frequency of the absorber is higher compared to

the primary natural absorber. This is because the mass of the absorber is much lower, which

will directly affect the natural frequency. This statement relates to equation (10), which depicts

that the natural frequency is inversely proportional to the mass of the system. In addition, the

single degree of freedom system oscillates with the amplitude of -0.03920 m, and when the

DVA system is applied, the amplitudes of the system changes to -0.07095 m. Adding absorber

increases the resonance frequency, because the system changes from single degree of freedom,

to two degree of freedom system, which will give out two natural frequencies.

12

6 CONCLUSION

Overall, the experimental results were influenced by the unavoidable existence of

friction in the system and heavy platform making it difficult to oscillate. Next, unavailability

of different type of spring stiffness increases the difficulty in finding the perfect combination

for the dynamic vibration absorber. The only parameter that can be changed is the unbalanced

mass but increasing the mass will take up the space in front of the case fan, plus, the fan will

have not enough torque to rotate the fan due to the mass.

Dynamic Vibration Absorber can be utilize to encounter vibration issue. Vibration

neutralizer or dynamic vibration absorber is designed to ensure that the natural frequency of

the system is not the same as the excitation frequency. DVA system actually implemented

another system which consists another spring-mass which will absorb the disturbance by

transferring it into the added mass. Therefore, from a single degree of freedom system

unbalance mass, the system change to two degree of freedom system.

7 REFERENCES

[1] A Hooke's Law Spring. (n.d.). Retrieved from

http://www.4physics.com/phy_demo/HookesLaw/HookesLawLab.html

[2] Dr. Norizham, A. R. (2015). Structural Dynamics ESA 322 & Vibration and Noise

EMM 342. Universiti Sains Malaysia, School of Aerospace Engineering, Pulau

Pinang.

[3] Rao, S. S. (n.d.). Vibration Absorber. In S. S. Rao, Mechanical Vibrations (5th ed.).

[4] Russell, D. A. (n.d.). The Dynamic Vibration Absorber. Retrieved from

http://www.acs.psu.edu/drussell/Demos/absorber/DynamicAbsorber.html

13

8 APPENDICES

APPENDIX A C.A.D DESIGN

(a) (b)

Figure 9 CAD design of the DVA system (a) and closer view of the configurations of the platform mass (b)

APPENDIX B MEASURING SPRING STIFFNESS

Figure 10 Determination of spring stiffness by extending the spring

LIST OF FIGURESLIST OF TABLES1 INTRODUCTION2 THEORY3 DESIGN4 ANALYSIS5 RESULTS AND DISCUSSION6 CONCLUSION7 REFERENCES8 APPENDICESAPPENDIX A C.A.D DESIGNAPPENDIX B MEASURING SPRING STIFFNESS