E5_ME3112-1_Vibration-Measurement_Report.pdf

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E5 –

ME3112-1Vibration Measurement

(E1-02-03)

10-Oct-2011

Name: ChenHaojin

Matric Number: U096128E

Group: 3C1



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Objective:

1. To familiarize with the techniques in measuring dynamic quantities as well as using the

related equipments.

2.

To determine the resonance frequencies and the corresponding mode-shapes of avibrating beam with several different techniques.

Results:

Table 1

ModeNodes Position (m) Experiment

Error %

CROFrequency

(Hz)

StroboscopeFrequency

(Hz)

TheoreticalFrequency

(Hz)Theoretical Experimental

1 --- --- --- --- --- 4.53

2 0.394 0.372 5.50% 26.11 25.18 28.41

3 0.238 | 0.428 0.237 | 0.413 0.42% | 3.50% 74.77 76.87 79.55

4 0.166 | 0.308 | 0.442 0.171 | 0.304 | 0.43 -3.0% | 1.3% | 2.7% 148.10 148.00 155.89

5 --- --- --- --- --- 257.66



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-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

X/L

A m p l i t u d e

Graph 1: Mode Shape

Table 2

X (m) X/Ly

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

0.000 0.0 0.0000 0.0000 0.0000 0.0000 0.0000

0.048 0.1 0.0335 0.1859 0.4560 0.7701 1.0745

0.095 0.2 0.1277 0.6072 1.2080 1.5077 1.3193

0.143 0.3 0.2729 1.0695 1.5090 0.8680 -0.4226

0.190 0.4 0.4597 1.4083 1.0428 -0.6300 -1.3933

0.238 0.5 0.6790 1.5098 0.0190 -1.4101 0.0007

0.285 0.6 0.9222 1.3269 -0.9916 -0.6404 1.3970

0.333 0.7 1.1817 0.8833 -1.4101 0.8320 0.4372

0.380 0.8 1.4509 0.2652 -0.9973 1.3969 -1.2600

0.428 0.9 1.7248 -0.4001 0.0021 0.4366 -0.8314

0.475 1.0 2.0001 -0.9723 1.0014 -1.0004 0.9998



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Sample Calculation:

Given:

E = Young’s Modulus = 220 GPa

I = Area Moment of Inertia of Beam about Neutral Axis = 12

3bt

L = Span of beam = 0.475 m

m = Mass per unit length = bt

= Density = 7903 kg/m3

t = Thickness = 0.0012 m

b = Breadth = 0.03 m

1. For Mode 2 (i=2), Theoretical Natural Frequency:

21

2

2

22

2

m

EI

L f

21

0012.003.07903

)0012.003.012

1(10220

475.02

694.439

2

2

= 28.406979 Hz

≈ 28.41 Hz

2. For Mode 2 (i=2) and L

x= 0.1, Amplitude of Vibration:

L

x L xa

L x

L x y ii

iii

sinsinhcoscosh

00.1ia For i > 1

)1.0694.4sin()1.0694.4sinh(00.1)1.0694.4cos()1.0694.4cosh( y

≈ 0.1859



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Discussion:

1. Resonance frequency a. What is resonance

Resonance is the tendency of a system to oscillate at greater amplitude at some frequenciesthan at others. These are known as the system’s resonant frequencies.

At these frequencies, even small periodic driving forces can produce large amplitude

oscillations, because the system stores vibration energy.

Resonance occurs when a system is able to store and easily transfer energy between two or

more different storage modes.

(From Wikipedia)

b. Under what condition does resonance happen

Each object tends to have a number of natural frequencies like 1st natural frequency, 2nd

natural frequency and so on.

These natural frequencies are largely dependent on the shape and material property of the

object and degrees of freedom in the object.

Basically it is related to the supports to the object as well as medium in which the object is in

(like air, water).

Resonance happens when a driving force’s frequency on an object is equivalent to the natural

frequency of the object.

c. What is the significance of resonance frequency in the design of building or structure

When resonance occurs, the building and/or structure (like bridges) will vibrate in a large

scale which may cause damage, even destruction.

Therefore, when doing the design of building and/or structure, designer should consider

about the resonance effect and determine the materials and shapes of the building so that the

natural frequency is different from oscillate frequency of winds and earthquakes etc.

By doing so, the building and/or structure can avoid the resonance damaging.



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2. Mode shapes

a. How the mode shapes are characterized by its number of nodes and node position

According to the fomular of Mode shape:

)L

sinL

sinh(aL

cosL

cosh ii

i

i x x x x y i

The mode shape is like a sinusoidal curve, the nodes happen when the amplitude is zero, and

number of nodes can dertermine how many wave heats along the lenght.

The position of nodes can dertermine where the mode does not vibrate.

They are illutrated in the figure below:

Acturally the mode shape is determined by number of nodes and node position.

b. The definition of node and anti-node

A node is a point along a standing wave where the wave has minimal amplitude.

The opposite of a node is an anti-node, a point where the amplitude of the standing wave is a

maximum. These occur midway between the nodes.

(From Wikipedia)

http://en.wikipedia.org/wiki/Standing_wave

http://en.wikipedia.org/wiki/Amplitude

http://en.wikipedia.org/wiki/Amplitude

http://en.wikipedia.org/wiki/Standing_wave



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c. Classical spring-mass damper system

To reduce the amplitude in anti-node (maximum amplitude), a proper damper should beinstalled to both absorb resonant frequencies and thus dissipate the absorbed energy.

An ideal mass – spring – damper system with mass m (in kilograms), spring

constant k (in newtons per meter) and viscous damper of damping coefficient c (in newton-

seconds per meter or kilograms per second) is subject to an oscillatory force

(From Wikipedia)

The damping types is dertemined by damping ratio:

A critical damping: damping ratio = 1A over-damping: damping ratio > 1

Under damping: damping ratio < 1

The layout of a mass-spring-damper is below:

http://en.wikipedia.org/wiki/Hooke%27s_law

http://en.wikipedia.org/wiki/Kilograms

http://en.wikipedia.org/wiki/Spring_constant


http://en.wikipedia.org/wiki/Newton_(unit)

http://en.wikipedia.org/wiki/Viscous

http://en.wikipedia.org/wiki/Newton-second



http://en.wikipedia.org/wiki/Second

http://en.wikipedia.org/wiki/Second




http://en.wikipedia.org/wiki/Viscous

http://en.wikipedia.org/wiki/Newton_(unit)




http://en.wikipedia.org/wiki/Hooke%27s_law



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3. Simple comparison of the 5 modes

a. Pitch of sound

The higher the mode (mode 5 is highest in this case), the higher/sharper the pitch of sound.

Reason:

Higher the mode have higher frequency, thus have higher energy so have higher pitch of

sound.

b. Number of nodes

Mode 1: 0 node (theoretical, but we didn’t done this, reason is given in next question) Mode 2: 1 node

Mode 3: 2 node

Mode 4: 3 node

Mode 5: 4 node (theoretical, but we didn’t done this, reason is given in next question)

c. Amplitude of vibration

Mode 1 (maximum) > Mode 2 > Mode 3 > Mode 4 > Mode 5 (minimum)

4. Why mode 1 and mode 5 are not done in the experiment

Reason:

Mode 1 has no anti-node and node; it has least energy but maximum vibration

Mode 5 should have 4 nodes, but the maximum amplitude should be very difficult to observe

(since it was already very hard to observe for mode 4).



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5. Why the CRO fr equency is always lower than the resonance natur al f requency

Reason:

The loaded mass (transducer) will also decrease the frequency of the vibration system, but it

is neglected for the CRO frequency.

6. Exper imental errors

i. The loaded mass decrease the nature frequency of the whole system. That is also one

of the reasons why the experimental frequencies are a bit smaller than the theoretical

ones.ii. Experimental error while reading the node position. Because the beam was vibrating

and we can only decide the position roughly through our judgment. And the more the

nodes, the more difficult to do the reading.

iii. Experimental error while using stroboscope method to tell the beam vibrating at its

resonant frequency. That is depended on our own judgment since the illuminated line

could not stop moving completely.

iv. The beam was bent before the experiment which leads to inaccuracies in the results

we get.

Conclusion:

1. Through this experiment, we have a better understanding about resonance.

2. Learned how to use an accelerometer and stroboscope to measure the nature frequency of

the ruler at different modes.

3. Being familiarized with the techniques in measuring dynamic quantities.

Documents

E5_ME3112-1_Vibration-Measurement_Report.pdf