Structural Labs Experiment No. 3

7/31/2019 Structural Labs Experiment No. 3

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NATIONAL UNIVERSITY OF IRELAND GALWAY

4rd Year Civil Engineering

Structures LaboratoryExperiment no. 3 Vibrating Beam

Group 23

Paul Sweeney

Colm Tarmey

Philip Walsh

Fergus Waters

Oisn Callery*

Enda Dunne**Additional Members

16/02/11


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Introduction

The vibration of a structure is the mechanical oscillations or varying displacements that

occur due to the application of dynamic loads. The degree of vibration depends on the mass

along with the elastic and stiffness properties of the material. There are two types of

vibrations, free vibration and forced vibration. Free vibration is when an initial input sets off

vibration in a material. The material is allowed to vibrate freely and will vibrate at one or

more of its natural frequencies until the energy dissipates and the deflections damp down

to zero. Forced vibration is when an alternating force or motion is applied to a material or

structure. The oscillations that occur in a structure may be random or periodic depending

on the type and nature of loading. An example of dynamic loads is wind loading and wave

loading. Wind loading can occur randomly where the peak loads, valley loads and their

sequence is a random process, while wave loading may follow a periodic pattern where the

peak loads are repeated at regular intervals. Other dynamic loads include traffic,

earthquakes and blasts or explosions. Dynamic loads can cause many problems in structures

and structural design. One example is cyclic loading or repeated loading, this can cause

fatigue failure in structures where the periodically repeated stresses induced in the

structure cause failure at stress levels alot lower than the ultimate strength of the material.

Another problem caused by dynamic loading is the phenomenon of resonance and natural

frequency. Resonance is the tendency of a system to oscillate with larger amplitude at some

frequencies than at others, the frequencies at which resonance occurs are known as natural

frequencies. Resonance in a structure can result in large deformations, violent swaying

motions and even catastrophic failure in improperly constructed structures. Therefore in

structural design a great understanding of natural frequency and resonance is required and

structures must be designed to eliminate or if not possible reduce the effect of resonance

by having the natural frequencies of the material not coincide with the frequencies of the

loading. Another important effect in structures subjected to dynamic loading is damping,

damping is the ability of the material to dissipate or respond to the energy induced in the

material by the dynamic load or impact. It involves the reduction of the amplitude of

mechanical oscillations or displacements of the material due to friction and other

resistances. Damping also plays an important role in structural design. It can determine the

suitability of a material for certain loading conditions.


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An example of this would be in extreme loading situation such as an earthquake, for a

structure to withstand an earthquake the material used must be capable of quickly

dissipating the large amounts of energy and reducing the deflections occurring in the

structure.

In the experiment three aspects of beam vibration will be observed including:

(a) Damping(b)Natural Frequency(c) Amplitude and Phase

The remainder of the report will discuss the experimental apparatus, theory and procedures

carried out for the three above parts. The results and data from each experiment will also

be presented in tabular and graphical form, the data collected will also be discussed and

compared to any theoretical values, also any errors or deviations in data will also be

highlighted and examined. Following the results an overall conclusion of the experimental

and report work will be evaluated, the learning outcomes and importance of the experiment

in practical applications will be identified and discussed.

Experimental Apparatus


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The experimental apparatus consists of a simply supported beam and a variable speed

motor which is connected to the beam at mid-span. The motor exerts a time varied or cyclic

force on the beam which causes it to vibrate. The rotational speed of the motor can be

adjusted by a controller and hence the speed or rate of loading and the frequency of thebeams vibration is also adjusted. The motor includes a rotating disk which contains a

circular hole which is positioned on a stationary base that is graduated in 10 degree

increments. As the motor runs and the circular hole rotates, a stroboscope is used to find

the frequency and phase angle of vibration. The stroboscope is adjusted until the circular

hole appears stationary, the phase angle is read from where the hole appears on the

graduated base and the determined from the reading on the digital stroboscope. The

amount of deflection and amplitude of vibration in the beam is measure by a micrometer.

The micrometer and underside of the beam are both connected to a strobe light. The

micrometer is adjusted until it meets the underside of the beam, at the point of contact the

circuit of the strobe light is complete and the strobe light flickers in unison with the

vibration of the beam. At this point the reading on the micrometer is the measurement of

displacement of the beam from its equilibrium position. The beam is also connected to an

oscilloscope, which graphs the motion of vibration or frequency.

(A) Air Damping

The idea of damping is to prevent serious damage or possible collapse of a solid material

due to vibrations. Damping is the gradual dissipation of energy due to vibrations and as a

result reduces the rate of vibration.

In our experiment we applied this knowledge on a vibrating beam, this concept can also be

applied to a standing structure. An impulse force is applied to a beam causing it to oscillate.

The amount of vibration is proportional to the magnitude of the force applied. The damping

effect reduces the amplitude of oscillations or vibrations and eventually brings the beam to

back to its original equilibrium position.

If the damping effect had not been present the beam would continue to vibrate. If the beam

oscillates at the right amplitude it will cause resonance which will could cause the beam to

fail due to violent vibrations throughout the beam. Sufficient damping prevents this

scenario from occurring.


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Experimental Procedure

The level of structural damping is determined by the value of (expressed as a percentage).

The equation for is shown below.

n = the number cycles between two points chosen on the graph.

X0 = the amplitude at the first chosen point.

Xn = the amplitude at the second chosen point.

There are three categories of damping:

Underdamped: < 1, then the amplitude will decrease slowly to zero after theexternal force has been removed.

Critically damped: = 1, the system returns to equilibrium as soon as any vibrationoccurs.

Overdamped: > 1, No vibration will occur in the structure as the damping is verystrong.

Apparatus and Experimental Method:

1) The oscilloscope is connected to the strain gauge on the beam and then adjusted.2) Gauge factor is set on the data acquisition system.3) The data acquisition system RUNS4) An impulse load is applied once to the beam causing a vibration of the beam.5)

Transient deflections for the beam were displayed on the oscilloscope and recordedby a connected computer.

6) The oscilloscope and data acquisition system were removed from the apparatus.7) Using the printout of transient deflections for the system we can calculate .


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Results

Our values Xo and Xn are taken from the transient deflection graph generated. The value for

n is the cycles between these peaks.

n = 17 no. of cycles

Xo = 670 - 555 = 115mm

Xn = 600 555 = 45mm

= 0.00878%

The above table shows the reading and calculation shown damping value for the beam is

0.00878% of critical damping. This suggests that vibrations due to impact loads will dissipate

slowly as was observed in the lab.

0.00878% < 1 therefore it is underdamped and the amplitude will decrease slowly to zero

after the external force has been removed.

Conclusions

The value for obtained was 0.00878 % which is less than one. This indicates that the air

damping in this system causes damping which is less than critical. From observing the

experiment we noticed the beam vibrations gradually decreased towards equilibrium. This

corresponds with graphical results from the oscilloscope as amplitude decreases over the

graph cycles.

Sources of Error for air damping experiment

Measuring off the graph is the main source of error, the quality of the photocopy made it

difficult to judge the measurements. Also if we were to use different values it is very

possible that we generate a different value for .


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Air Damping Oscilloscope Graph

(B) Natural Frequency

The natural frequency is the free oscillation of a material or structure subjected to an

impulse force or load. If a repetitive external load was applied at this frequency the

amplitude of oscillation or vibration would dramatically increase. This behaviour is known as

resonance and has a very important impact in structural design. The natural frequency of a

structure is determined by the stiffness and mass of the material used and all materials

contain more than one natural frequency.

In this experiment the natural frequency of a simply supported beam will be determined by

applying cycle loads on the beam, the frequency of the loading is controlled and can be

adjusted. By adjusting the frequency until resonance is observed the natural frequency of

the beam can be determined.

The results obtained from the experiment will be compared to the theoretical values

calculated from Rayleighs formula and Dunkerleys formula.


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Experimental Procedure

The experimentally calculated natural frequency is compared to the theoretically calculated

results. To calculate the theoretical natural frequency a number of parameters have to be

measured to solve Rayleighs formula and Dunkerleys formula.

Rayleighs Formula:

g = acceleration due to gravity ; yi = deflection ; wi = weight

For the Rayleighs formula the weight of the beam is divided into i components and

dispersed throughout the beam, the ith

deflection corresponding to ith

weight is calculated

and the above formula is solved to find the natural frequency.

Dunkerleys Formula:

E = Youngs Modulus ; I = Moment of Inertia ; L = Beam Length

m1 = Mass of and Damping Assembly ; m2 = Mass of Beam

The above parameters of the beam are measured and substituted into the equation to solve

for angular velocity squared (2). By getting the square root of this and dividing by 2 the

natural frequency is obtained.

Moment of Inertia for a rectangle: Apparatus and Experimental Method

(1)The apparatus was prepared with the variable speed motor on the simply supportedbeam secured and connect to controller.

(2)The length of beam was measured with the depth and thickness measured at threelocations along the beam and the average of each was determined.

(3)The motor was switched on and rotation speed was adjusted until resonance wasobserved, large deflections in the beam occur when the natural frequency of the

beam is reached.


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(4)The motor rotational speed was noted at the point of largest deflections and thenatural frequency of the beam was calculated by dividing the rotational speed (Revs

per minute) by 60 seconds to give the natural frequency of the beam in Hertz.

Results

Measurements

Beam Sizes (mm) Average

b 25.3 25.36 25.28 25.28

d 12.62 12.59 12.59 12.6

L ~ ~ ~ 814

Density of Steel = 7830 Kg/m3

Mass of Motor and Damping Assembly, m1 = 4.55 Kg

Mass of Mass of Beam, m2 = Density x Volume = 0.02528 x 0.0126 x 0.814 x 7830 = 2.03 Kg

Youngs Modulus of Steel, E =200 GPa

Moment of Inertia for a rectangle, I =

= 4.214x10-9

m4

Rayleighs Formula Parameters (obtained from CADS)

W1 = m2g/3 = 6.67N W2 = m2g/3 + m1g = 51.31N W3 = m2g/3 = 6.67N

Y1 = 0.55mm Y2 = 0.79mm Y3 = 0.55mm

Experimentally obtained Natural Frequency fn = 18.8 Hz

Natural Frequency from Dunkerleys Formula = 116.2 rads/sec fn = 18.5 Hz

Natural Frequency from Rayleighs Formula fn = 17.5 Hz

Conclusions

Dunkerleys formula approximation of the natural frequency of the beam is very close to the

experimentally obtained value. There is a difference of 0.3Hz between the two values while

there is larger difference of 1.3Hz between the experimental value and the value gained

from Rayleighs formula. In a large scale structural application these differences would be

considered very small and from a design point of view either of the values calculated would

be more than expectable since such structures are exposed to a wide spectrum of major and

minor loading conditions.


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Sources of Error for Natural Frequency Experiment

Errors in the experiment include possible errors in the measurements of the beams

dimensions. Measurements can easily be misread from the micrometer also only three

measurements were taken for the depth and width to calculate the beams size. If more

measurements were taken a better representation of the beam could have been achieved.

Determining the exact moment of the resonance in the beam can be difficult, the precise

moment of maximum deflections in the beam may not have been accomplished which

would affect the end results. Misuse of the stroboscope and lack of accuracy by the user can

also result in a variation of natural frequencies calculated for the beam.

(C) Amplitude and Phase Angle

Amplitude is the magnitude of oscillation or the measurement of displacement due to

vibration of a structure from its equilibrium position. As mentioned earlier if resonance

occurs within the structure the amplitude of vibration significantly increases. The scale of

amplitude is determined by the energy input into the system and the natural frequency of

the structure. This is another important factor in structural design as large deflects and

excessive swaying may be impractical and may also lead to the collapse of the structure.

The phase angle is the fraction of a wave cycle which has elapsed relative to an arbitrary

point, it is used to describe the motion of the vibration or wave.

In this experiment the amplitude and phase angle are determined at different frequencies

of vibration. The rotational speed of the varied speed motor is adjusted until relatively large

deflections in the beam are observed, the amount of displacement is measured and the

phase angle of the wave is determined for the different vibrations.

Experimental procedure

The experiment is performed on the apparatus that was used in the previous experiments. A

handheld stroboscope is used to calculate the phase angle and the frequency. The damped

lateral vibrations of the beam are examined when the beam reaches resonance and we

observe the following characteristics of the vibrating beam:

frequency phase angle gauge reading


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The phase angle is observed by focusing the stroboscope on the graduated motor disc while

the beam is vibrating. When the rotating hole appears still after the observer has adjusted

the stroboscopes to the suitable frequency, the observer reads off the phase angle position

and the stroboscope records this frequency. These are the values that we require.

Apparatus and Experimental Method

1. Motor controller is connected which allows the frequency of the morter to becontrolled

2. Trigger stobe is connected and the micrometer reads the static bam deflection.3. The motor is switched on and by adjusting the micrometer the connected strobe

flashes when the beam deflection level is reached.

4. Contact is broken by taking away the vernier away.5. Next a handheld stroboscope is is used to determine the phase angle and frequency.6. This method is repeated for eight values of varied frequency.

Our values for the frequency are divided by sixty so that they will be calculated in

Hertz (cycles per second).

Results

Here is a table showing all the calculations from our experiment. These are the values for

which we based our graphs from.

Frequency

Gauge

Reading Amplitude Phase

Frequency

Ratio

0.00 3.34 0.00 0 0

5.28 3.37 0.03 135 0.28

8.64 3.39 0.05 180 0.46

12.53 3.41 0.07 0 0.67

16.75 3.56 0.22 0 1.01

18.90 6.71 3.37 10 1.16

21.80 3.59 0.25 45 1.16

23.37 3.53 0.19 100 1.2425.05 3.53 0.19 90 1.33


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Conclusion

The phase angle increases to 180 degrees and then decreases toward which shows a natural

cycle. The graph of amplitude Vs frequency ratio shows a sudden jump in amplitude as

the frequency of vibration coincides with the natural frequency of the beam. This large

increase in amplitude reassures that the value for natural frequency determined in part B of

the experiment is correct. This knowledge is of knowing the frequency at which resonance

occurs is invaluable in the design of a building, as it can dictate how many storeys high a

building can be or how long a bridge should be in certain regions. For example in regions

where earthquakes occur, the frequency of the earthquake will cause more damage to

buildings at heights that satisfy the natural frequency of the earthquake. A building ten

storeys high may suffer serious damage compared to buildings that are five storeys or more

than twenty storeys high.

0

20

40

60

80

100

120140

160

180

200

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Angle()

Frequency Ratio

Phase angle Vs Frequency Ratio

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Amplitude

Frequency Ratio

Amplitude Vs Frequency Ratio


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Sources of error

The frequency and phase angle of the beam had to be judged by human observation and

judgement which leaves a chance of error occurring.

Documents

Structural Labs Experiment No. 3