vibrations Lab report

FREE UNDAMPED AND DAMPED VIBRATIONS Lab Report. ABSTRACT A mechanical system is said to be vibrating when its component part are undergoing periodic oscillations about a central statical equilibrium position. Any system can be caused to vibrate by externally applying forces due to its inherent mass and elasticity.

OBJECTIVES

  

To investigate the response and behaviour of a pendulum system undergoing free vibrations with and without viscous damping. To determine values of damping coefficient ‘C’ and damping ratio for a set of damper setting. To verify the suitability of the mathematical method used in determining the damping values.

DESCRIPTION OF EQUIPMENTS

1. The oscilloscope: is basically a graph-displaying device - it draws a graph of an electrical signal. It is used in observing constantly varying signal voltages, usually as a two-dimensional graph of one or more electrical potential differences using the vertical 'Y' axis, plotted as a function of time.

Fig 1: An Oscilloscope.

2. Potentiometer: is a three-terminal resistor with a sliding contact that forms an adjustable voltage divider. It is a simple electro-mechanical transducer. It converts rotary or linear motion from the operator into a change of resistance. It has terminals which can be connected to a signal amplifying or display unit as the case may be (sound.westhost.com).

Fig 2: The Potentiometer.

3. Variable damping unit: this is used to set varying damping values been used for the experiment.

Fig 3: Damper setting unit.

4. Digital weighing scale: a measurement device used to measure the weight or mass of an object or substances. Most digital scales make their measurements based on an internal strain gauge, a thin foil piece that conducts electricity and is sensitive to deformation is attached with some adhesive to a flexible surface. When weight is applied to the digital scale, various mechanisms within the digital scale ensure the weight is evenly distributed on the strain gauge. The weight

bends the flexible surface, deforming the foil piece, which alters the flow of electrical current (www.wisegeek.com).

Fig 4: The weighing scale.

5. Helical Spring: an elastic body, whose function is to distort when loaded and to recover its original shape when the load is removed. It is made up of a wire coiled in the form of a helix and are primarily intended for compressive or tensile loads (engg-learning.blogspot.com).

Fig 5: Attached helical spring.

6. Signal Amplifier: is an electronic device that increases the power of a signal. It does this by taking energy from a power supply and controlling the output to match the input signal shape but with a larger amplitude. In this sense, an amplifier modulates the output of the power supply (en.wikipedia.org).

Fig 6: The signal amplifier.

The mass is mounted at one end of the arm of the horizontally mounted pendulum with the other end connected by a helical spring to a fixed support. A rotary potentiometer is mounted on the hinged end of the pendulum. The potentiometer receives the displacement signal as the pendulum swings and this signal is transmitted through a signal amplifier connected to the oscilloscope to be displayed.

Fig 7: Set up of the equipment for the laboratory exercise.

THEORETICAL ANALYSIS If a mechanical system is displaced from its equilibrium position and then released, the restoring force will bring about return towards the equilibrium position. This is referred to as “Free Vibration”. This type of vibration arises from an initial impact energy that is continually changing from potential to kinetic form. In a free vibration, the system is said to vibrate at its natural frequency. However, due to various causes there will be some dissipation of mechanical energy during each cycle of vibration and this effect is called “Damping.” (Ryder and Bennett, 1990). Theoretically, an un-damped free vibration system continues vibrating once it is started. This experiment examines the effect of damping and the level of damping on the behaviour of a pendulum. Vibration can be classified in several ways, the important ones includes: 

Free and Forced Vibration: in this type of vibration, no external force acts on the system as the system is left to vibrate on its own after an initial disturbance. E.g.: the oscillation of a

simple pendulum. Similarly, when the system is subjected to an external force (often, a repeating type of force. E.g.: the oscillation of machines like diesel engines. 

Undamped and Damped Vibration: if no energy is lost or dissipated in friction or resistance during oscillation, the vibration is known as undamped vibration. Should any energy be lost in its way, it is called damped vibration. (S.S Rao. 2011).

Other ways of classifying vibration are; linear and non-linear vibration, deterministic and random vibration.

The simplest possible vibratory system consist of a mass attached by means of a spring to an immovable support as shown below. The mass is constrained to translational motion in the direction of the axis so that its change of position from an initial reference is described fully by the value of a single quantity . This is called a ‘single degree of freedom’ (R.E Blake, 2002).

Fig 8: Undamped single degree of freedom system. The differential equation of motion of mass, m for the undamped system is:

The angular natural frequency is given by: Here:

k – spring stiffness

√

̈

rad/sec

m – mass.

In order to simplify the mathematics involved, the damping is modelled as a viscous damping depending on the magnitude of damping. A damped system can be under-damped, critically damped or over-damped.

Fig 9: Damped system model.

For a damped system, the corresponding equation of motion of mass is given by: ̈

̇

Under-damped System: this occurs when the damping of the system is less than critical, ζ 1. The result is an exponential decay with no oscillations but it will take longer to reach the rest position than with critical damping. An over-damped doorcloser will take longer to close than a critically damped door would.

Fig 12: Typical response to an Over-damped system (D.V Hutton 1981).

Logarithmic Decrement: the displacement of an underdamped system is a sinusoidal oscillation with decaying amplitude. A quite useful property of an underdamped system can be obtained by comparing the amplitudes of any two successive cycles of the displacement (D.V. Hutton, 1981).

( Here:

)

is the period of the motion

Substitution of T and

in the equation above:

√

The term

√

is called the logarithmic decrement of the response.

Mathematical model To determine the values of spring stiffness, K and pendulum arm mass, Ma using the following equations:

√

√

Where: Large Mass, ML

Small Mass, MS

FL – Natural frequency with ML attached to arm FS – Natural frequency with MS attached to arm K – Spring stiffness

Ma – Pendulum arm mass.

(√

) --------------- (1)

(√

) --------------- (2)

From equation (1) Square both side (

)

(

) (

(

)

)

(

)

( )

From equation (2) Square both sides (

)

(

) (

)

(

)

(

)

( )

( ) (

)

(

)

(

)

(

)

(

(

)

(

)

)

Therefore;

Part B: To find the value of damper coefficient . (

)

But

√

(

)

√

√ (

)√

(

) (

)

(

)

(

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(

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(

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( (

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) (

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(

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√ √(

)

EXPERIMENTAL METHOD AND CALCULATIONS Part A. i. ii. iii. iv. v.

At commencing the experiment, the value (weight) of the small mass and large masses were ascertained. One of the available masses was attached to the end of the tube which forms the pendulum arm. The oscilloscope is set to ‘single arm mode.’ After a suitable voltage sensitivity setting was achieved, the pendulum is made to swing freely against the spring. The vibration was noted and the frequency reading was recorded from the slope. The procedure was repeated twice and the average value for the frequency was rightly noted. Similarly, the values of the frequency was determined for using the second mass and the also when attached with no mass.

Part B i. ii.

With initial apparatus set up in ‘part A’ previously, the large mass was attached and also the damper unit to pendulum arm. The damper unit was set to 1 on the damper scale and the scope was adjusted to read zero voltage by moving the vertical cursor and aligning the horizontal cursor on the input signal trace.

iii.

The pendulum was made to swing freely against the spring with the resulting trace on the scope fixed using the store facility. The voltage reading which represented the peak value of the waveforms were noted. This step is repeated for damper scale setting of 3 and 5. The damper was set to a higher scale and the corresponding system response was observed.

iv. v.

Experimental Values

1

2

3

Average

Natural Frequency, fL with mL attached to arm (Hz)

3.73

3.68

3.73

3.71

Natural Frequency, fS with mS attached to arm (Hz)

4.31

4.39

4.39

4.36

Natural Frequency, f0 without mass

5.68

5.68

5.68

5.68

Calculations

(

) (

Sub for (

( (

)

) )

= 0.0825 in equation (4) )

(

)

K = 39.47 × 4.362 × (0.0825 + 0.0647) K = 39.47 × 19.01 × 0.1472 K = 110.45 Kg/m2

F0 – natural frequency without mass = 5.68Hz To check for the value of the natural frequency F0 of the arm:

(√

)

(√

)

F0 = 0.159 × 36.59 F0 = 5.82Hz

Successive amplitude ratio for damper setting 1

Average value of amplitude ratio for setting 1

Successive amplitude ratio for damper setting

Average value of amplitude ratio for setting 3

Successive amplitude ratio for damper setting 5

Average value of amplitude ratio for setting 5

To determine the values of the damping ratio, ε for each damper setting. √

(

) (

)

Damping setting = 1.

√

(

) (

)

√

Damper setting = 3.

√

(

) (

)

√

Damper setting = 5.

√

(

) (

)

√

To determine the values of the damping coefficient, c for each damper setting √(

)

M = MA + ML M = 0.0825 + 0.1208 = 0.2033Kg

Damper setting = 1. √

√ Damper setting = 3.

√

√ Damper setting = 5.

√

√

RESULTS AND ANALYSIS.

Damper Setting

1

3

5

Experimental

X1

X2

X3

20.00

16.0

15.3

13.3

37.3

28.0

26.0

24.0

26.0

19.3

18.6

17.3

36.0

24.0

22.0

19.3

23.3

16.6

15.3

14.0

24.0

16.6

16.0

14.6

22.0

13.3

12.6

11.3

20.6

14.0

12.6

11.3

25.3

18.6

15.3

13.3

Calculated Values

Values F0 = 5.68

X4

F0 = 5.82 MA = 0.0825Kg/m

X1

X2

X3

X4

93.3

63.3

59.9

54.6

83.3

57.2

53.3

47.9

67.9

45.9

40.5

35.9

Damper Setting

Damping coefficient, c

Damping ratio,

1

0.218

0.023

3

0.294

0.031

5

0.332

0.035

2

K = 110.45Kg/m

Analysis:

A vibratory system is a dynamic one which for which the variables such as the

excitations (input) and responses (output) are time dependent. The response of a vibrating system generally depends on the initial conditions as well as any form of external excitations (S.S Rao, 2011). The vibrations which occur in a mechanical equipment most often results from forces which arise from the functional operation of the equipment (D.V. Hutton, 1981). Therefore, analysing a vibrating system will involve setting up a mathematical model, deriving and solving equations pertaining to the model, interpreting the results and assumptions and reanalyse or redesign if need be.

DISCUSSION. The experimental and calculated results does not differ much as there is a marginal error of about 2.4% (that is:(

)

). This might be as a result of some system imbalance

and/or hysteresis. Similarly, the discrepancy might be a result of some error in calculations. As the experiment was been carried out with different damper settings, the frequency of oscillation of the pendulum reduces as the damper setting increases. When the damper was set to a high setting, the pendulum simply remained stationary because the there was no room for any form of displacement of the pendulum except there is additional force exerted to overcome the damping force in place. The mathematical model used for the system is a valid one for determining the damping values as it was not too complex. Starting with an initial elementary model of differential equation of motion and then developed gradually and refined to accommodate the other input components and details to closely calculate and observe the system behaviour. Advantages of damping Dampers dissipate energy within a system by converting it to heat. If designed properly, damping forces can be completely out of phase with structural stress. Thus, the right damper can reduce stress and deflection simultaneously. Without damping, there would be no suspension in cars. With no suspensions, it would be twice as dangerous to drive in a car as it is today. The handling of the wheels would be extremely difficult, and breaking will be much uneasy. Damping is employed across different areas aside the automotive industry which includes but not limited to Aerospace and Defense, Heavy Industry steel Mills, Aluminium Mills, Shipbuilders Offshore Oil Drilling, Civil Engineering buildings, Bridges and Stadiums Towers. Similarly, Damping can cause additional friction losses and heat build-up in certain machineries which a bit of disadvantage.

CONCLUSION Set objectives for this laboratory exercise was achieved with a good knowledge about the subject matter of pendulum behaviour and response when subjected to free oscillatory motions. The theory of free vibrational motion with and without viscous damping was studied and appropriate mathematical model was used to calculate the value of the spring stiffness, K the natural frequency, when the large mass was attached, also the value of the natural frequency, with no mass attached. The damping coefficient was also calculated for different damping setting. Damping is very useful and it should be incorporated in the design of systems or mechanism subjected to vibrations and shock as it helps to minimize fatigue and failure. The right damper will reduce stress and deflection.

REFERENCES:    



David H. Hutton, Applied Mechanical Vibrations (1981), McGraw-Hill Series. London. G.H. Ryder and M.D. Bennett, Mechanics of Machines (2nd Ed), Macmillan: Hong-Kong. Singiresu S. Rao, Mechanical Vibrations (5th Ed.) 2011, Pearson: Singapore. Douglas P. Taylor, The Application of Energy Dissipating Damping devices to an Engineered Structure or Mechanism [Online].http://www.shockandvibration.comn Accessed: 20th March 2014. JDJ: Vibration Types [Online] http://www.mcasco.com/Answers/qa_vtype.htm accessed: 20th March 2014.