Vibration Measurement

Presented byMr.Kiratikorn Kaewpankan553040033-2

Mechanical Engineering KhonKaen University

ObjectiveTo determine the damped natural

frequency and damping ratio of vibration system by testing.

To determine the damped and undamped natural frequency of vibration system by theory.

Compare frequency from testing and theory.

Experimental equipment.

1. Vibration system

2. Vibration measurement

3. Computer and software (Vibview)

How to Experiment1. Turn on vibration measurement and setting.

2. Install the sensor at the end of the beam.

3. Use rubber hammer to tap on the beam and wait until the scope shown result in graph.

4. Record the graph and use Vipview to change the graph to the number that use to plot.

Beam

Free Body Diagram

Theory

Mass moment of inertia

𝐽𝑏=13𝑚𝑏𝑏

2𝐽 s=𝑚𝑠(𝑠¿¿2+h2)¿

∑ 𝑀𝑜= 𝐽 �̈�𝐽 �̈�=−𝑘1 𝑦1𝑙1𝜃−𝑘2 𝑦2 𝑙2𝜃+𝑐1 �̈�+𝑐2 �̈�

𝐹=𝑘 𝑦𝐽 �̈�+𝑘1 𝑙1

2𝜃+𝑘2 𝑙22𝜃=0

ωω

From

Change

We can find

=full volume of beam

=inner volume of beam

− 𝐽 𝜔2+(𝑘¿¿1 𝑙12+𝑘2 𝑙2

2)=0¿

𝜔𝑛=√ (𝑘¿¿1 𝑙12+𝑘2 𝑙2

2)𝐽

¿

(ω

Now we want to find but this vibration system can’t be find pure ∑ 𝑀𝑜=0

𝐹 1𝑙1 cos𝜃+𝐹2 𝑙2𝑐𝑜𝑠𝜃=𝑚𝑤𝑤 cos𝜃

𝐹=𝑘𝑦𝑘1 𝑦1𝑙1𝑐𝑜𝑠 𝜃+𝑘2 𝑦2 𝑙2𝑐𝑜𝑠𝜃=𝑚𝑤𝑤 cos𝜃

𝑘1 𝑦1𝑙1+𝑘2 𝑦2𝑙2=𝑚𝑤𝑤

𝑦 𝑥

𝑙𝑥= 𝑦𝑤

Similar triangles

𝑘1𝑙1𝑦𝑤𝑙1+𝑘2𝑙2

𝑦𝑤𝑙2=𝑚𝑤𝑤

𝑘1𝑙12+𝑘2𝑙2

2=𝑚𝑤𝑤𝑤𝑦

Using = 10 kg = 98.1 N

𝑘1𝑙12+𝑘2𝑙2

2=2275 . 470𝑁 .𝑚

ResultsSection 1. I’m use the average of 5graph to fine natural frequency and damping ratio

When n= cycle of vibration V0 = maximum velocity=

t0, tn=time for vibration

we use for five graph

𝑓 𝑑 𝑒=27 .5229+27 .5229+27 .1003+27 .1257+27 .2537

5

𝑓 𝑑𝑒=27 .3111𝐻𝑧

𝜔𝑑=2𝜋 𝑓 𝑑𝜔𝑑𝑒=2𝜋 (27 .3111𝐻𝑧 )=171 .6007𝑟𝑎𝑑 /𝑠

= 0.1145Logarithmic decrement

=𝛿

√(2𝜋)2+𝛿2Damping Ratio

0 .1145

√(2𝜋)2+0 .11452=0 .0182=

Find Damping Ratio ()

Section 2. From Theory

𝜔𝑛=√ (𝑘¿¿1 𝑙12+𝑘2 𝑙2

2)𝐽

¿

𝑘1𝑙12+𝑘2𝑙2

2=2275.470𝑁 .𝑚

= = = )() = =

We can change Undamped into Damped

𝜔𝑑𝑡=𝜔𝑛√1−2From testing

= 27.9246 Hz

𝐸𝑟𝑟𝑜𝑟=| 𝑓 𝑑𝑡− 𝑓 𝑑

𝑓 𝑑𝑡|×100%¿|27.9246Hz−27.3111Hz27.9246Hz |×100%=2.1969%

Error between and and

Bonus >> No weight. How about frequency ?

𝐽=[ 13 0 .23377614 (0 .598)2]New J ¿0 .027866 𝑘𝑔 .𝑚2

Change Undamped into Damped

𝜔𝑑𝑡=𝜔𝑛√1−2

𝜔𝑑𝑡= (287 .203 rad /s )×√1−0 .01822

𝜔𝑑𝑡=287 .155 rad /s

= 45.702 Hz

SummaryFrom testingDamped natural frequency () = 27.3111 Hz Damping ratio =0.0182

From TheoryUndamped Natural frequency () =27.9292 Hzdamped Natural frequency () =27.9249 Hz

Bonus no weightUndamped Natural frequency () = 45.470 Hzdamped Natural frequency () = 45.702 Hz

Error of Damped natural frequency between testing and theory =2.19 %

Discussion