Eastern Mediterranean University Department of Mechanical Engineering Laboratory Handout

COURSE : Dynamics of Machinery MENG 331

Semester: Spring (2012-2013)

Name of Experiment: Damped Free Vibrations

Instructor: Assist. Prof. Dr. Mostafa Ranjbar

Submitted by:

Student No:

Group No:

Date of experiment:

Date of submission:

------------------------------------------------------------------------------------------------------------EVALUATION

Activity During Experiment & Procedure 30 %

Data , Results & Graphs 35 %

Discussion, Conclusion & Answer to Questions 30 %

Neat and tidy report writing 5 %

Overall Mark

Name of evaluator: -------------------------------------

DAMPED FREE VIBRATIONS OBJECTIVE

1. To determine the viscous damping coefficient C as a function of the dashpot position

in a certain dynamic system.

2. To learn how to estimate the damping coefficient C of a dynamic system experimentally.

A) DESCRIPTION AND THEORY: All systems possessing mass and elasticity are capable of executing free vibrations, i.e.,

vibrations that take place with the absence of external excitation. The system shown in

Figure 1 is an example here on. If such an ideal undamped and frictionless system is

given a small displacement, it will continue to oscillate without stopping. Such an ideal

system does not exist in the real life due to the existence of internal friction between the

molecules of the beam’s material, due to the friction between the oscillating beam and

surrounding air and due to the friction at the supports of the beam. One can easily notice

that any system, when given a small displacement, its oscillatory motion will decay until

it completely dies out after a while. The rate of decay can be increased by introducing a

dashpot with a damping constant C.

Consider the dynamic system shown in Figure 1. If the beam is pulled down and released,

the equation of the angular motion becomes

where

ωn : Natural frequency of the system

k : Spring stiffness

IA : Mass moment of inertia about point A

ζ : Damping ratio

The response of the system can be found using the solution to the governing differential

equation, which is given by:

Plotting the response using Matlab, the shape of the oscillatory motion, the system

undergoes, is shown in Figure 2.

A convenient way of measuring the amount of damping present in the dynamic system is

to measure the decay of the oscillations. The larger the damping, the greater will be the

rate of decay, which is expressed as logarithmic decrement. The logarithmic decrement

is defined as the natural logarithm of the ratio of any two successive amplitudes, i.e.

δ = ln(θ1/θ2) , as shown in Fig.2. The logarithmic decrement can be found by any of the

following equations:

Both of these equations can be used to evaluate the damping coefficient of any free

vibrating system with damping. However, the second equation is more accurate, since the

determination of δ is very sensitive to any inaccuracy in the amplitude measurements.

The amount of damping in a dynamic system C is usually assessed by any of the

following equations

where τd is the damped time period. Both methods if used to determine the damping

constant C are supposed to giver similar results.

EXPERIMENTAL PROCEDURES: APPARATUS

A rectangular beam, which supported at one end (A) by a trunion, pivoted in ball

bearings and all located in a fixed housing. The outer end of the beam is supported by a

helical spring of stiffness k. The free vibrations of the system are damped by means of a

dashpot, fixed to the base by sliding bracket. The dashpot consists of a transparent

cylindrical container filled with oil. Inside the container, there are two discs each with

several orifices. The two discs can be rotated relatively to each other to change the

damping characteristics of the dashpot

The amplitude time recording is provided by the chart recorder, which is clamped to the

right hand upright. The unit consists of a slowly rotating drum driven by a synchronous

motor. The motor is operated from the Auxiliary supply on the speed control unit. A roll

of recording paper is fitted adjacent to the drum and is wound a round the drum so that

the paper is driven at constant speed. A felt-tipped pen is fitted to the free end of the

beam and the drum may be adjusted in the horizontal plane so that the pen just touches

the paper. The paper is guided vertically downwards by a small attachable weight. By

switching on the motor, a trace can be obtained showing the oscillations of the end of the

beam.

EXPERIMENTAL PROCEDURES AND REQUIREMENTS:1. Record all dimensions needed for your analysis.

2. Pull the beam downward with the dashpot unattached and release it, while the

drum is rotating. Record the total time required for the motion.

3. Clamp the dashpot at a certain distance h along the beam and pull down on the

free end of the beam and release it. Draw the decaying curve on the recording

drum, while it is rotating.

4. Vary the damping characteristics of the system by moving the dashpot to a new

position and obtain a new record of the decaying curve.

5. Repeat steps 3 & 4 to get the records for 8 different positions of the dashpot.

6. From the resulting plot in step # 2 above, obtain the undamped natural time period

n and based on that calculate the natural (undamped) frequency of the system

ωn=2π/τn.

7. From the resulting plots in steps (3 through 5 above), obtain the damped natural

frequencies ωd = 2π/d of the system in each case (for each h length). The damped

period can be determined by using the decaying curve drawn on the recording

drum. Assuming that the length of n oscillations is measured to be x and the drum

speed v is known in inch/sec, then the damped period can be found as:

8. Determine the damping constant C of the system in each of the cases using both

equations mentioned earlier for calculating the damping constant. (Hint: You

might want to use

to determine the spring stiffness k. Note that the mass of the beam is known or

can be determined.

9. Use the following Matlab code as a guide to compare your experimental results

with the calculated results using the code.

EXPERIMENTAL RESULTS:

Parameter L l m IA

Value

Table 1: Values of apparatus parameters

Un-Damped

Parametersτn ωn m k

Table 2: The undamped parameter (damper is unattached)

Run. No. h τn Δ ζ ωd

Table 3: Obtained and calculated values for listed parameters

RESULTS and DISCUSSION:1. What would be the reason of discrepancy in values of C between using

,

Answer 1:

2. Which one of the equations;

Would you use to get a more accurate account of damping presence in the system?

Answer 2:

3. How would you find the spring stiffness constant k?

Answer 3:

4. what suggestions would you make to improve the accuracy of the experiment?

Answer 4:

.