Exp 3 Force Vibration (Finalized)

LAB DYNAMIC & MACHINES MEC424

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TABLE OF CONTENT

NO CONTENT PAGE

1 ABSTRACT 2

2 INTRODUCTION 3-4

3 THEORY 5-8

4 APPARATUS 9-10

5 PROCEDURE 11

6 RESULT 12-31

7 DISCUSSION AND CONCLUSION 32-33

8 REFERENCES 33

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1.0 ABSTRACT

The experiment is done to determine the natural frequency and resonance of spring in

different damping condition. To run this experiment, we used Control Unit to control the

Universal Vibration System Apparatus. First of all, we adjusted control unit to desired

frequency, 1 Hz to 14 Hz. Then, set the damper to off condition. After that, we set the

unbalance exciter on and set the frequency from 1Hz to 14Hz. And lastly, we recorded the

oscillation produced on drum recorder for each frequency. We collected the data from

different condition of damping which is, no damper, open damper with length 150mm, closed

damper with length 150mm, and lastly, closed damper with length 550mm. The natural

frequency for this experiment is constant which is 8.357 Hz . In order for resonance to

happen, the applied frequency and the natural frequency of the object must be the same. From

this experiment, we found that the resonance will happen if the applied frequency and the

natural frequency of the object is the same. The knowledge of calculating the resonance

frequency and natural frequency is very important in order for us to prevent catastrophic

disaster such as the collapsed of Tacoma narrow bridge in the future.

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2.0INTRODUCTION

Vibration refers to mechanical oscillations about an equilibrium point. The oscillations may be

periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road.

Vibration is occasionally "desirable". For example the motion of a tuning fork, the reed in a woodwind

instrument or harmonica, or the cone of a loudspeaker is desirable vibration, necessary for the correct

functioning of the various devices. More often, vibration is undesirable, wasting energy and creating

unwanted sound – noise. For example, the vibration motions of engines, electric motors, or any

mechanical device in operation are typically unwanted. Such vibrations can be caused by imbalances

in the rotating parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually minimize

unwanted vibrations.

The study of sound and vibration are closely related to sound, or "pressure waves", are

generated by vibrating structures (e.g. vocal cords); these pressure waves can also induce the vibration

of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to

reduce vibration.

They are two types of vibration that is free vibration occur when a mechanical system is set

off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are

pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The

mechanical system will then vibrate at one or more of its "natural frequency" and damp down to zero.

Forced vibration is when an alternating force or motion is applied to a mechanical system.

Examples of this type of vibration include a shaking washing machine due to an imbalance,

transportation vibration (caused by truck engine, springs, road, etc.), or the vibration of a building

during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or

motion applied, with order of magnitude being dependent on the actual mechanical system.

Forced vibration also known as oscillation is vibration that takes place under the excitation of

external forces. The system will vibrate at the excitation frequency when the excitation is oscillatory.

Resonance will occur if the frequency of excitation coincides with one of the natural frequencies of the

system and dangerously large oscillations may result. The failure of major structures such as bridges,

buildings, or airplane wings is an awesome possibility under resonance.

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Free vibration Force vibration

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3.0THEORY

Forced Vibration

Harmonic excitation is often encountered in engineering systems. It is commonly produced by

the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than

periodic or other types of excitation, understanding the behavior of a system undergoing harmonic

excitation is essential in order to comprehend how the system will respond to more general types of

excitation. Harmonic excitation may be in the form of a force or displacement of some point in the

system.[2]

We will first consider a single DOF system with viscous damping, excited by a harmonic force

, as shown in Fig. 7. Its differential equation of motion is found from the free-body

diagram.[2]

(29)

Figure 2: Viscously Damped System with Harmonic Excitation

The solution to this equation consists of two parts, the complementary function, which is the

solution of the homogeneous equation, and the particular integral. The complementary function in

this case, is a damped free vibration. The particular solution to the preceding equation is a steady-state

oscillation of the same frequency w as that of the excitation. We can assume the particular solution to

be of the form:[2]

(30)

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Where X is the amplitude of oscillation and f is the phase of the displacement with respect to the

exciting force. The amplitude and phase in the previous equation are found by substituting Eqn. (30)

into the differential equation (29). Remembering that in harmonic motion the phases of the velocity

and acceleration are ahead of the displacement by 90° and 180°, respectively, the terms of the

differential equation can also be displayed graphically.[2]

Figure 3: Vector Relationship for Forced Vibration with Damping

It is easily seen from this diagram that

(31)

and

(32)

We now express Eqs (31) and (32) in non-dimensional term that enables a concise graphical

presentation of these results. Dividing the numerator and denominator of Eqs. (31) and (32) by k, we

obtain :

(33)

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and

(34)

These equations can be further expressed in terms of the following quantities:

The non-dimensional expressions for the amplitude and phase then become

(35)

and

(36)

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These equations indicate that the non-dimensional amplitude , and the phase f are functions

only of the frequency ratio , and the damping factor z and can be plotted as shown in Fig 9.

Figure 4: Plot of Eqs. (35) and (36)

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4.0 APPARATUS

FIGURE : Universal Vibration System Apparatus (TM 155)

FIGURE: Control Unit (TM 150)

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FIGURE : Universal Vibration System Apparatus (TM 155)

1. Unbalance Exciter

2. Beam

3. Damper

4. Control Unit (TM 150)

5. Mechanical Recorder

6. Spring

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5.0 EXPERIMENTAL PROCEDURES

1) Set up the apparatus and the control unit (TM 150) is switched on.

2) The control unit is adjusted to desired frequency.

3) The length from the damper to the references point is measured.

4) For the first condition, the damper is removed from the beam.

5) The unbalance exciter is switched on and the frequency is set up from 1 Hz until 14Hz with

increment of 1 Hz. When frequency is 8 Hz, the increment will be 0.1Hz until 9Hz.

6) Then, the drum recorder will take an oscillation from the vibration of the stiff beam caused by

the unbalance exciter.

7) The oscillation characteristics is observed.

8) The time taken for 10 oscillations is measured and recorded.

9) The observations is compared with the result on the mechanical recorder.

10) The second condition, the damper is attached to the beam at the length 150mm and the damper

is opened to reduce the damping effect. Then, step 5 to 9 is repeated to get the oscillation data.

11) For the third condition, the damper is attached to the beam at the length 150mm and the

damper is closed. Then, step 5 to 9 is repeated to get the oscillation data.

12) For the fourth condition, the damper is closed but the length is changed to 550mm and step 5

to 9 is repeated.

The data is recorded in the table and the graph is plotted.

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6.0 RESULT

Experiment Damper Lid Damper Distance (mm)

1 No damper No damper

2 Open 150

3 Closed 150

4 Closed 550

Given values:

1. Mass of beam: 1.68kg

2. Mass of imbalance exciter: 0.772kg

3. Length of beam: 0.7m

4. Distance from axis of rotation to spring: 0.65m

5. Distance from axis of rotation to imbalance exciter: 0.35m

6. Spring constant: 3kN/m

7. Damper constant: 5Ns/m (open) or 15Ns/m (closed)

The objective of this experiment is to find the resonance for each damping condition. So, in

order to know whether the frequency of a damping condition is in resonance, we have to use

the equation:

Or frequency ratio, r;

If the above equation is satisfied, resonance occurs.

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Condition(s) Frequency (Hz) Amplitude (mm) Resonance

Frequency (Hz)

No damper

1.0 0.0

9.642

2.0 0.0

3.0 0.0

4.0 0.1

5.0 0.1

6.0 0.2

7.0 0.3

8.0 0.4

8.1 0.5

8.2 0.5

8.3 0.6

8.4 0.7

8.5 0.8

8.6 0.5

8.7 0.9

8.8 1.0

8.9 1.4

9.0 1.4

10.0 0.3

11.0 0.2

12.0 0.1

13.0 0.1

14.0 0.1

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Condition(s) Frequency

(Hz)

Amplitude

(mm)

₰ (zetha) Resonance Frequency

(Hz)

Open damper at

150mm

1.0 0.0

0.04376

8.341

2.0 0.0

3.0 0.0

4.0 0.0

5.0 0.0

6.0 0.0

7.0 0.1

8.0 0.1

8.1 0.1

8.2 0.2

8.3 0.2

8.4 0.2

8.5 0.2

8.6 0.2

8.7 0.2

8.8 0.2

8.9 0.4

9.0 0.4

10.0 0.4

11.0 1.0

12.0 1.0

13.0 1.0

14.0 1.0

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Condition(s) Frequency

(Hz)

Amplitude

(mm)

₰ (zetha) Resonance Frequency

(Hz)

Closed damper

at 150mm

1.0 0.0

0.1313

8.2113

2.0 0.0

3.0 0.0

4.0 0.0

5.0 0.0

6.0 0.0

7.0 0.1

8.0 0.2

8.1 0.2

8.2 0.2

8.3 0.2

8.4 0.3

8.5 0.3

8.6 0.3

8.7 0.3

8.8 0.3

8.9 0.4

9.0 0.4

10.0 0.6

11.0 1.0

12.0 1.1

13.0 1.0

14.0 0.9

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Condition(s) Frequency

(Hz)

Amplitude

(mm)

₰ (zetha) Resonance Frequency

(Hz)

Closed damper

at 550mm

1.0 0.0

0.1323

8.2091

2.0 0.0

3.0 0.0

4.0 0.0

5.0 0.0

6.0 0.0

7.0 0.1

8.0 0.1

8.1 0.1

8.2 0.1

8.3 0.1

8.4 0.1

8.5 0.1

8.6 0.1

8.7 0.1

8.8 0.2

8.9 0.2

9.0 0.2

10.0 0.1

11.0 0.4

12.0 0.2

13.0 0.2

14.0 0.3

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A. No Damped

+ ∑ = : - KlӨ ( ) = Ӫ

Where =

Ӫ + K Ө = 0

Ӫ +

= 0

= √

= √

= √

mg

KlӨ

Ө

750mm

a = 650mm

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B. With Damper (Open) – 150mm

+ ∑ = : - KlӨ - c Ө = Ӫ

Ӫ + c Ө + KlӨ = 0

Ӫ +

Ө +

Ө = 0

= √

Where =

= √

= √

2 ζ =

ζ =

=

a = 650mm

𝑙 = 750mm

W

KlӨ

Ө

CӨ 150mm

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C. With Damper (Closed) – 150mm

+ ∑ = : - KlӨ - c Ө = Ӫ

Ӫ + c Ө + KlӨ = 0

Ӫ +

Ө +

Ө = 0

= √

Where =

= √

= √

2 ζ =

ζ =

=

150mm W

a = 650mm

𝑙 = 750mm

CӨ

KlӨ

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D. With Damper (Closed) – 550mm

+ ∑ = : - KlӨ - c Ө = Ӫ

Ӫ + c Ө + KlӨ = 0

Ӫ +

Ө +

Ө = 0

= √

Where =

= √

= √

2 ζ =

ζ =

=

a = 650mm

𝑙 = 750mm

W

KlӨ

𝑀

CӨ

550mm

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Sample Calculation

A. No Damper

√

=

√

= 8.357 Hz

2. Damping Ratio

ζ =

3. Resonance Frequency

= 8.357 Hz

4. Frequency Ratio

=

= 0.120

5. Imbalance Force

F0 = mass unbalance x a x

= 0.772 x 0.65 x 52.50652

= 1.383 kN

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Frequency

(Hz)

Amplitude

(mm)

(rad/s)

Natural

Frequency

(Rad/s),

Frequency

Ratio

Imbalance

Force (N)

Non-

dimensional

Amplitude

1.0 0.0 6.283

52.5065

0.120

26.3478

0

2.0 0.0 12.566 0.239 0

3.0 0.0 18.850 0.359 0

4.0 0.1 25.133 0.479 0.217

5.0 0.1 31.416 0.598 0.217

6.0 0.2 37.700 0.718 0.434

7.0 0.3 43.982 0.738 0.651

8.0 0.4 50.265 0.957 0.867

8.1 0.5 50.894 0.969 1.084

8.2 0.5 51.522 0.981 1.084

8.3 0.6 52.150 0.993 1.301

8.4 0.7 52.779 1.005 1.518

8.5 0.8 53.407 1.017 1.735

8.6 0.8 54.035 1.029 1.735

8.7 0.9 54.664 1.041 1.952

8.8 1.0 55.292 1.053 2.167

8.9 1.4 55.920 1.065 3.036

9.0 1.4 56.548 1.077 3.036

10.0 0.3 62.832 1.197 0.651

11.0 0.2 69.115 1.316 0.434

12.0 0.1 75.398 1.436 0.217

13.0 0.1 81.681 1.556 0.217

14.0 0.1 87.965 1.675 0.217

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B. Open damper at 150mm

√

=

√

= 8.357 Hz

2. Damping Ratio

ζ =

=

= 0.04376

3. Resonance Frequency

√

= 52.5065√

= 52.4059rad/sec

= 8.341 Hz

4. Frequency Ratio

=

= 0.119891081

5. Imbalance Force

F0 = mass unbalance x a x

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= 0.772 x 0.65 x 52.50652

= 1.383 KN

Frequency

(Hz)

Amplitude

(mm)

(Rad/s)

Resonance

Frequency

(Rad/s),

Frequency

Ratio

Imbalance

Force (N)

Non-

dimensional

Amplitude

1.0 0.0 6.283

52.4059

0.119891081

26.3478

0

2.0 0.0 12.566 0.239782162 0

3.0 0.0 18.850 0.359692325 0

4.0 0.0 25.133 0.479583406 0

5.0 0.0 31.416 0.599474487 0

6.0 0.0 37.700 0.719384649 0

7.0 0.1 43.982 0.839256649 0.217

8.0 0.1 50.265 0.95914773 0.217

8.1 0.1 50.894 0.971150195 0.217

8.2 0.2 51.522 0.983133578 0.434

8.3 0.2 52.150 0.995116962 0.434

8.4 0.2 52.779 1.007119427 0.434

8.5 0.2 53.407 1.019102811 0.434

8.6 0.2 54.035 1.031086194 0.434

8.7 0.2 54.664 1.04308866 0.434

8.8 0.2 55.292 1.055072043 0.434

8.9 0.4 55.920 1.067055427 0.867

9.0 0.4 56.548 1.079038811 0.867

10.0 0.4 62.832 1.198948973 0.867

11.0 1.0 69.115 1.318840054 2.167

12.0 1.0 75.398 1.438731135 2.167

13.0 1.0 81.681 1.558622216 2.167

14.0 1.0 87.965 1.678532379 2.167

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C. Closed damper at 150mm

√

=

√

= 8.357 Hz

2. Damping Ratio

ζ =

=

= 0.1313

3. Resonance Frequency

√

= 52.5065√

= 51.5938 rad/sec

= 8.211 Hz

4. Frequency Ratio

=

= 0.121778198

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5. Imbalance Force

F0 = mass unbalance x a x

= 0.772 x 0.65 x 52.50652

= 1.383 kN

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Frequency

(Hz)

Amplitude

(mm)

(Rad/s)

Natural

Frequency

(Rad/s),

Frequency

Ratio

Imbalance

Force (N)

Non-

dimensional

Amplitude

1.0 0.0 6.283

51.5938

0.121778198

26.3478

0

2.0 0.0 12.566 0.243556396 0

3.0 0.0 18.850 0.365353977 0

4.0 0.0 25.133 0.487132175 0

5.0 0.0 31.416 0.608910373 0

6.0 0.0 37.700 0.730707953 0

7.0 0.1 43.982 0.852466769 0.217

8.0 0.2 50.265 0.974244967 0.434

8.1 0.2 50.894 0.986436355 0.434

8.2 0.2 51.522 0.99860836 0.434

8.3 0.2 52.150 1.010780365 0.434

8.4 0.3 52.779 1.022971752 0.651

8.5 0.3 53.407 1.035143758 0.651

8.6 0.3 54.035 1.047315763 0.651

8.7 0.3 54.664 1.05950715 0.651

8.8 0.3 55.292 1.071679155 0.651

8.9 0.4 55.920 1.08385116 0.867

9.0 0.4 56.548 1.096023166 0.867

10.0 0.6 62.832 1.217820746 1.301

11.0 1.0 69.115 1.339598944 2.167

12.0 1.1 75.398 1.461377142 2.385

13.0 1.0 81.681 1.58315534 2.167

14.0 0.9 87.965 1.704952921 1.952

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D. Closed damper at 550mm

1. Natural Frequency

√

=

√

= 8.357 Hz

2. Damping Ratio

ζ =

=

= 0.1323

3. Resonance Frequency

√

= 52.5065√

= 51.5793 rad/sec

= 8.2091 Hz

4. Frequency Ratio

=

= 0.12181243

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5. Imbalance Force

F0 = mass unbalance x a x

= 0.772 x 0.65 x 52.50652

= 1.383 kN

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Frequency

(Hz)

Amplitude

(mm)

(Rad/s) Natural

Frequency

(Rad/s),

Frequency

Ratio

Imbalance

Force (N)

Non-

dimensional

Amplitude

1.0 0.0 6.283

51.5793

0.12181243

26.3478

0

2.0 0.0 12.566 0.24362487 0

3.0 0.0 18.850 0.36545669 0

4.0 0.0 25.133 0.48726912 0

5.0 0.0 31.416 0.60908155 0

6.0 0.0 37.700 0.73091337 0

7.0 0.1 43.982 0.85270642 0.217

8.0 0.1 50.265 0.97451885 0.217

8.1 0.1 50.894 0.98671366 0.217

8.2 0.1 51.522 0.99888909 0.217

8.3 0.1 52.150 1.01106452 0.217

8.4 0.1 52.779 1.02325933 0.217

8.5 0.1 53.407 1.03543476 0.217

8.6 0.1 54.035 1.04761018 0.217

8.7 0.1 54.664 1.059805 0.217

8.8 0.2 55.292 1.07198043 0.434

8.9 0.2 55.920 1.08415585 0.434

9.0 0.2 56.548 1.09633128 0.434

10.0 0.3 62.832 1.2181631 0.651

11.0 0.4 69.115 1.33997553 0.867

12.0 0.2 75.398 1.46178797 0.434

13.0 0.2 81.681 1.5836004 0.434

14.0 0.3 87.965 1.70543222 0.651

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Graph 1

Graph 2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16

Am

plit

ud

e,a

(mm

)

Frequency, f (Hz)

Frequency Versus Amplitude

No Damper

Open Damper (150mm)

Closed Damper (150mm)

Cloed Damper (550mm)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 2 4 6 8 10 12 14 16

Am

plit

ud

e,a

(mm

)

Frequency, f (Hz)

Amplitude Versus Frequency

No Damper

Open Damper (150mm)

Closed Damper (150mm)

Cloed Damper (550mm)

0

0.5

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2No

n-d

imen

sio

nal

Am

plit

ud

e (x

/(Fo

/k)

Frequency Ratio

Non-dimensional Amplitude Versus Frequency Ratio

No Damper

Open damper at 150mm

Closed damper at 150mm

Closed damper at 550mm

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7.0 DISCUSSION

Based on the set of apparatus that had been used in the experiment, we derive a formula from

the given measurement for no damper and damped. For damped sections, we conduct two types of

experiment which is closed and open damped conditions.. In order to obtain the result, we calculate

the natural frequency, resonance frequency, frequency ratio and non-dimensional amplitude. For

examples, in a state of no damped condition, the value for natural frequency is 8.357 Hz which is same

value with the resonance frequency as the value for damping ratio do not exist. Then from the

resonance frequency and natural frequency, we obtained the frequency ratio for this state. Finally we

calculated the non- dimensional amplitude after we had calculated the imbalance force.

Next, we calculate for the damped conditions. In this state we calculate the natural frequency,

resonance frequency, frequency ratio and non-dimensional amplitude for the three given condition

which is open damped (150mm), closed damped (150mm) and closed damped (550mm). The natural

frequency for the whole given condition is the same which is 8.357 Hz and the reason is because the

distance from the spring to origin of the rod and the length of whole rod to the origin is constant for all

given condition. But the values for resonance frequency are different for closed damped (150mm) and

open damped (150mm) because there are difference in value of damping constant. The damping

constant for open damped (150mm) are 5 N.s/m while for close damped (150mm) are 15 N.s/m. Then

we calculate the frequency ratio and non-dimensional amplitude as the same for no damped condition.

After that, we tabulate all the data into a table which consists of Frequency (Hz), resonance

frequency, frequency ratio, and non-dimensional amplitude. From the obtained results, we plotted

graph amplitude versus frequency and graph of non- dimensional amplitude versus frequency ratio.

Based from the graph of amplitude versus frequency (Graph 1) we can said that the maximum

amplitude only occur at 8-12 Hz while from the graph of non-dimensional amplitude versus frequency

ratio (Graph 2) we can said that the pattern are almost the same as the previous graph. From the

theoretical aspects, our graph are slightly different from the theoretical graph of amplitude versus

frequency and non-dimensional amplitude versus frequency ratio.

The differences that occur maybe due to the systematic and random error. As for random error,

we can said that the elasticity of spring in the machine has decrease from the actual value because it

had been used for many times before. This may affect the oscillation of the rod during the experiment

and this also may result in the variant of data. Random error also occurred when the frequency of the

control unit system become higher. In that condition, the sensitivity of the pencil that is attached to the

graph paper becomes loose and this caused the graph to be inaccurate. However, for the systematic

error, we can say that there are no possibilities of systematic error occurred during this experiment.

There are several precautions that need to be taken in order to improve the accuracy of this

experiment. Firstly, the spring needs to be replaced with a good elasticity of spring. Lastly, the pencil

also must be attached tightly to the holder so that it does not loose from the holder in order for the

graph to be tabulate properly on the graph paper during the high frequency oscillation of the rod.

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CONCLUSION

Based from the experiment, we can say that our experiment is a success although the graph is

slightly different from the theoretical graph. We also manage to achieve the objective of this

experiment which is to determine the resonance of spring in damping condition. In order for the

resonance to happen, the applied frequency and the natural frequency of the object must be the same.

Not just that, we also able to derive the formula that is need to be used in this experiment.

From this experiment also we learned the important of calculating the resonance frequency and

natural frequency so that we can prevent catastrophic disaster such as the collapsed of Tacoma narrow

bridge c in the future.

8.0 REFERENCES

1 - Beer, Johnston, Cornwell, Vector Mechanics for Engineers, Ninth Edition, 2010,Mc Graw Hill

publications.

2 – http ;//www.brown.edu/Departments/Engineering/Courses/En4/Notes/vibrations_forced/vibrations

_forced .html

- http://fsinet.fsid.cvut.cz/en/u2052/node117.html 3

4.Vector dynamics and vibration notes, PM dr.zamri,Faculty Of Mechanical Engineering,Uitm Shah

Alam,2008