vibrations lab 1.pdf

8/2/2019 Vibrations Lab 1.pdf

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Purpose

The purpose of this experiment was to familiarize us with some basic instruments used in vibrations. The

main focus in this lab was to learn to calibrate an accelerometer, to get familiarize with the dynamic

signal analyzer, and to make some fundamental measurements.

Introduction

In this experiment, three different lengths of both steel and aluminum were attached to a clamp on one

end. On the other end, the dynamic frequency analyzer was attached and it was connected to the

computer. At the end where the frequency analyzer was attached, a shake was given and the damping

frequency graph was observed on the computer.

The measurements taken off the computer were written down in tables B and C and the calculations

were performed.

Firstly, the actual frequency and the actual time period of all the six configurations were written down

separately. Then to find the theoretical frequency, the following formula was used:

( ) (Eq. 1)Where, E is the elastic modulus

I is the mass moment of inertia

m is the mass density

L is the length of the beam

Elastic Modulus was found from the table A. Mass moment of inertia (I) was found from the following

formula where base (b) and height (h) were found from the table A:

(Eq. 2)Mass density (m) was found from the following formula where mb is the mass of the beam and L is the

length of the beam:

(Eq. 3)And finally, the value 3.56 in (Eq. 1) was found from the derivation of the mass effective (meff.) that was

incorporated in formula.


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In the second part of the lab, we were asked to find the damping coefficient:

(Eq. 4)

where is the damping ratio and is the critical damping coefficient.Now to find the damping ratio, the following formula must be used:

(Eq. 5)Where is the logarithmic decrement that can be found by Also to find the critical damping the following formula must be used:

(Eq. 6)Where m is the mass effective and is the frequency.


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Table A

Beam Type STEEL ALUMINUMProperties

Width [in] 1 1

Height [in] 0.25 0.25

Elastic Modulus [lb/in2] 3.0E+07 1.0E+07

Density [lb/in3] 0.284 0.098

STEEL

Table B

Beam type Steel

Length (in) 12 24 36

Frequency (Hz) 48 12 5.6

Amplitude 1 (g) 0.2594 0.7478 1.092

Amplitude 2 (g) 0.2450 0.6888 1.073

For Length 12 in:

Actual Values: fn = 48 Hz

T = 1/fn = 0.02083 s

Theoretical Values:

( )

( )

[ ]

( )


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For Length 24 in:

Actual values: fn = 12 Hz

T = 1/fn = 0.04167 s

Theoretical Values:

( ) For length 36 in:

Actual values: fn = 5.6 Hz

T = 1/fn = 0.1786 s

Theoretical values:

(

) There was an average error of about 14% when compared with the values of Tables A and B with the

theoretical calculated values. One of the reasons for this discrepancy would be the standing waves.

Since the steel beam was of only one length from which the three lengths were measured and clamped

at those lengths, there was the remaining length that went behind the clamp. When a pulse was given

to the one end of the steel beam, the wave may have passed through the clamp to the other side of the

beam and may have created the standing waves. Another reason for this discrepancy could be that the

clamp would not have been rigidly attached to the table. The loose attachment of the clamp to the table

may have let the entire table vibrate when the pulse was given to one end of the beam.


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ALUMINUM

Table C

Beam type Aluminum

Length (in) 12 24 36

Frequency (Hz) 56 12.4 6Amplitude 1 (g) 0.1513 1.024 1.277

Amplitude 2 (g) 0.1196 0.9726 1.245

For Length 12 in:

Actual Values: fn = 56 Hz

T = 1/fn = 0.01786 s

Theoretical Values:

( )

( )

[ ]

( )


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For Length 24 in:

Actual values: fn = 12.4 Hz

T = 1/fn = 0.08064 s

Theoretical Values:

( ) For length 36 in:

Actual values: fn = 6 Hz

T = 1/fn = 0.1667 s

Theoretical values:

( )

As for the Aluminum, there was an average error of about 8%. This discrepancy may have caused due to

the same reasons as for the steel beam.


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As clearly seen in the graph, the relationship for the frequency graph is non-linear and the time period

graph is (almost) linear. These graphs are non-linear because of the equations of the line that they

follow. Let us first observe the frequency graph first. The equation of this graph has a final value with

unit 1/s. This shows that the graph of this equation would be a hyperbola and this can be seen clearly in

the frequency vs length graph. As for time period vs length graph, the equation for time period is the

inverse of the frequency equation which makes the graph linear. This can be clearly seen in the actual

values. The theoretical value graph of time period is non-linear because of the discrepancies discussed in

the previous discussion of actual and theoretical frequencies.

0

10

20

30

40

50

60

70

0 10 20 30 40

Frequency(Hz)

Length (in)

Frequency (Hz) vs. Length (in) graph for Steel

Actual

Theoretical

Linear (Actual)

Linear (Theoretical)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 10 20 30 40

TimeP

eriod(s)

Length (in)

Time Period (s) vs. Length (in) graph for Steel

Actual

Theoretical

Linear (Actual)



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As discussed for frequency and time period graphs of steel, the same reasons are valid for the both

graphs of aluminum. The only difference is its time period versus length graph. In this time period graph

the actual graph line and the theoretical graph line is almost on top of each other. This simply implies

that the experiment performed for aluminum achieved maximum accuracy.

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0

10

20

30

40

50

60

0 10 20 30 40

Frequency(Hz)

Length (in)

Frequency (Hz) vs. Length (in) graph for

Aluminum

Actual

Theoretical

Linear (Actual)


0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 10 20 30 40

TimePeriod(s)

Length (in)

Time Period (s) vs. Length (in) graph for

Aluminum

Actual

Theoretical

Linear (Actual)



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For 12 in steel:

Logarithmic decrement Damping ratio

Mass effective Critical Damping Coefficient Damping Coefficient For 24 in steel:

()

For 36 in Steel: ()


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For 12 in Aluminum:

()

For 24 in Aluminum:

()

For 36 in Aluminum:

()

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