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Experiment Title : Free Vibration of CantileverCourse : UEME1143 DynamicsProgram : BIName of Student : TAN WEI TIENStudent ID No : 1700723Year and Trimester : Y2SDate of Experiment : 22 January 2020Name of Lecturer : Dr. Ooi Jong BoonIntroductionA cantilever beam system is a system where one end of ...

Description

Experiment Title

:

Free Vibration of Cantilever

Course

:

UEME1143 Dynamics

Program

:

BI

Name of Student

:

TAN WEI TIEN

Student ID No

:

1700723

Year and Trimester :

Y2S3

Date of Experiment :

22 January 2020

Name of Lecturer :

Dr. Ooi Jong Boon

Introduction A cantilever beam system is a system where one end of the system is rigidly fixed to a support and the other end is free to move. When given an excitation and left to vibrate on its own, the frequency at which a cantilever beam will oscillate is its natural frequency. This is known as free vibration. The system of a cantilever beam subjected to free vibration is considered as continuous system. The mass of the beam is assumed to be distributed along with the stiffness of the shaft. (Free Vibration of a Cantilever Beam, n.d.) In a viscously damped vibration system, the mechanical energy is dissipated by frictional forces present within the mechanical system. It is common and is formed in many engineering systems such as shock absorbers. The viscous damping force is proportional to the first power of the velocity across the damper, and it always opposes the motion, so that the damping force is a linear continuous function of the velocity. (Viscous Damping, n.d.) The value of natural frequency depends only on system parameters of mass and stiffness. When a real system is approximated to a simple cantilever beam, some assumptions are made for modelling and analysis (for undamped system): i. ii. iii.

The mass of the whole system is considered to be lumped at the free end of the beam No energy consuming element (damping) is present in the system i.e. undamped vibration The complex cross section and type of material of the real system has been simplified to equate to a cantilever beam

Apparatus and Materials 1. 2. 3. 4. 5. 6. 7. 8.

Cantilever beam apparatus Computer DC-7104 Controller software Strain gauge Strain recorder DC-104R Compact flash memory card Viscous damper Ruler (1 meter)

Methodology

1. The computer and the strain recorder are switched on. 2. The strain recorder application software is started. 3. The viscous damper is removed from the beam. 4. The beam is displaced and held at ymax by -20 mm, -15 mm, -10 mm, -5 mm, 0 mm, 5 mm, 10 mm, 15 mm and 20 mm and the strain recorder reading for each displacement value is recorded manually from the “Numerical Monitor” screen of the application software. 5. The graph of the relationship between the displacement (of the free end of the beam) and the strain recorder reading is plotted using a spreadsheet. 6. The beam is displaced by 30 mm and left to vibrate on its own. 7. The strain recorder reading is recorded by clicking on the “Play” and “Stop” button. 8. The recorded file is retrieved by clicking on the “Read USB” button. 9. The graph of the beam displacement versus the time, it is plotted. 10. The experiment is repeated using beam displacement of 50 mm. 11. The viscous damper is connected and steps 6 to 10 are repeated using beam displacement of 30 mm and 50 mm, respectively.

Results Free vibration Displacement (mm) -20 -15 -10 -5 0 5 10 15 20

Strain 1

Strain 2

Strain 3

Average strain

-262 -170 -142 -68 -2 80 170 248 324

-264 -168 -140 -66 2 78 176 246 330

-270 -164 -136 -70 0 74 178 248 332

-265 -167 -139 -68 0 77 174 247 329

Displacement (mm) vs Average Strain Displacement

25 20 15 10 5 0 -300

-200

-100

-5 -10 -15 -20 -25

Viscously Damped Vibration

0

100

200

300

Average Strain

400

=============

Calculations Mass of cantilever beam, mbeam = 298g Mass of damper, mdamper = 113.77g Length of beam, L = 925mm Width of beam, b = 19.09mm Height of beam, h = 6.35mm Modulus of Elasticity of Aluminum, E = 70GPa Free vibration   Gradient of the graph, m =   



()

m = () m = 0.0889 Since the equation of a straight line is y = mx Therefore, the equation of the linear graph is y=0.0889x

Viscously Damped Vibration (Sample calculation) i. Displacement of 30mm without damper Moment of inertia of the beam, I

=

 

=

(.)(.)





= 4.073 x 10 -10 m4 Equivalent mass of cantilever beam, meq



=  mbeam 

=  (0.298kg) m eq Stiffness of cantilever beam, k

= 0.0702kg

=



=

()(. )



(.)

= 108.071 N/m

Experimental value of natural frequency,

ωn =

 



ω n = .. ω n = 39.270 rad/s

Theoretical value of natural frequency, ωn

=

= ωn Percentage error

=

|..|

= . = 0.09%

x 100

$. .

= 39.236 rad/s

|&'()'*+,-'.('+)/,-| '.('+)/,-

!

 "#

x 100

ii. Displacement of 50mm without damper Experimental value of natural frequency, ωn Theoretical value of natural frequency, ωn Percentage error = 3.32%

= 40.537 rad/s = 39.236 rad/s

iii. Displacement of 30mm with damper Natural frequency of damped vibration (30mm), Natural frequency of free vibration (30mm),

ωd = 21.299 rad/s ωn = 39.270 rad/s

From the equation, = 01 − ζ ωn ωd 45  ) 46

ζ

= 1−(

ζ

= 1 − ( .)

ζ

= 0.840

.

iv. Displacement of 50mm with damper Natural frequency of damped vibration (50mm), Natural frequency of free vibration (50mm), ζ

= 0.856

ωd = 20.944 rad/s ωn = 40.537 rad/s

Discussion

Comparison of experimental and theoretical values of the natural frequency of the beam in free vibration Natural frequency, ωn (rad/s)

40.8

- : Experimental - : Theoretical

40.537

40.6 40.4 40.2 40 39.8 39.6

39.27

39.4

39.236

39.2

39.236

39 0

10

20

30

40

50

60

Initial displacement (mm)

From the comparison above, the natural frequency of the beam has increased from 39.270 rad/s to 40.537 rad/s. The theoretical value of the natural frequency is 39.236 rad/s for both initial displacement of 30mm and 50mm. The theoretical value remains the same for both 30mm and 50mm because the stiffness and mass of the beam is constant throughout the experiment. Initial displacement does not affect the natural frequency of the beam. This is because natural frequency is only affected by mass of the beam, mbeam and stiffness of the beam, k. The lower the stiffness, the lower the natural frequency. As for mass, a heavier beam will decrease its natural frequency. Adding weight to a spring system will also lower the natural frequency. The natural frequency is also affected by moving the location of the weight, if there is any attached. (Practical Solutions to Machinery and Maintenance Vibration Problems, n.d.) However, the initial displacement will affect the amplitude at which the beam vibrates. For example, the amplitude of the beam is greater for an initial displacement of 50mm than that of 30mm. Theoretically, the amplitude of the displacement of the beam should be the same for vibration without damper. From the experimental graph, the amplitude decreases slightly. This is due to air resistance. The friction between the beam and the air causes some of the mechanical energy to be dissipated, and thus gradually decreasing the amplitude of displacement from one oscillation to the next. The mechanical energy dissipated in a viscously damped vibration system is higher due to the presence of a viscous damper, which is water in this experiment.

Underdamped Critically damped Overdamped

There are three types of damping in a mechanical vibration system, namely underdamping, overdamping and critical damping. The damping ratio of an underdamped system is always less than 1. Applications of an underdamped system include the string of a guitar. If not, the sound produced will be dull. A diving board is also underdamped. Otherwise, the diver might as well just dive from the edge of the pool. In an overdamping system, the system returns to equilibrium position very slowly and without oscillating. It has a damping ratio greater than 1. For example, the actuator of a door swings shut slowly without swinging back and forth. The toilet flush handle is also one of the applications of an overdamping system. Lastly, critical damping is when the system returns to equilibrium position in the shortest time possible, without any oscillation. The damping ratio is equal to 1. Instruments such as speedometer of a car is critically damped so that when the car accelerates, the meter will change quickly. It does not oscillate and therefore the driver will not get confused. Another example is a gun, which returns to its neutral position in the shortest amount of time between firing.

Conclusion In conclusion, the equation of the linear graph of Strain against Displacement is y=0.0889x. The natural frequency of the cantilever beam is 39.270 rad/s with a percentage error of 0.09% when the initial displacement is 30mm without damper. When the initial displacement is increased to 50mm, the natural frequency is 40.537 rad/s with 3.32% error. The natural frequency of the beam in viscously damped vibration when displaced by 30mm is 21.299 rad/s with damping ratio of 0.840. The natural frequency is 20.944 rad/s with a damping ratio of 0.856 when the beam is displaced by 50mm in a viscously damped vibration. It is an underdamped vibration system where the damping ratio is always less than 1.

References 1. Free Vibration of a Cantilever Beam. (n.d.). Retrieved from AMRITA: http://vlab.amrita.edu/?sub=62&brch=175&sim=1080&cnt=1 2. Practical Solutions to Machinery and Maintenance Vibration Problems. (n.d.). Retrieved from Update International: http://updateinternational.com/Book/VibrationBook2e.htm 3. Viscous Damping. (n.d.). Retrieved from Control Theory Pro: http://wikis.controltheorypro.com/Viscous_Damping...