Mechanical free damped oscillations

THE AIM is to determine the logarithmic decrement and the damping factor of mechanical oscillations.

INSTRUMENTATION AND APPLIANCES: spring pendulum, stopwatch, set of loads.

Short theory

Oscillatory motion of mechanical system, as a rule, occurs in the presence of friction, resulting in the transformation of mechanical energy of oscillations into heat. Ideal oscillations will continue for ever without change in amplitude. Friction, however, produces damped oscillations. In this case it is possible to write the displacement equations in the form

x = Acoswt, (18.1)

but A is understood to decrease with time.

Figure 18.1

To determine in what a way A(t) depends on time, the frictional force must be known for every instant of time during which oscillations occur. A simplifying assumption is that the frictional force is proportional to the velocity of motion

f_fr= - r v (18.2)

where the coefficient r is known as the resistance constant.

In the presence of friction the oscillating body is under the action of two forces:

restoring force

F = - kx = - m w₀²x (18.3)

and the force of friction

f_fr= - r v (18.4)

Applying Newton's second law we obtain:

ma = - kx - r v (18.5)

By substitution, it is not difficult to show that this equation is satisfied by the equation

x = A₀e- (r/2m) tcoswt . (18.6)

Here, A₀ is the amplitude at the instant of time t = 0.

It should be noted that the ratio of two successive amplitudes is a constant. Thus, the expressions for the amplitude after (n – 1) and n periods, respectively, are

A_n-1= A₀e- [r/2m](n-1) T (18.7)

A_n = A₀e- [r/2m] n T

Let as divide the former relation by the latter. The ratio

(18.8)

does not depend on n. The rate of damping is sometimes expressed by the logarithmic decrement λ

(18.9)

Calculate the velocity and acceleration of motion expressed by the formula

(18.10)

(18.11)

Then we obtain

( -mw² +mb² + k - rb ) sinw t + ( 2mbw - rw ) cosw t = 0

This equation holds good at any instant (at any combinations of sinw t and cosw t), which is possible if the coefficients before sine and cosine equal zero. Then

2mb =r ; m(w² + b² ) = k = mw₀²;

(18.12)

Here ω is the frequency of damped oscillations.

Experimental part

The damping factor is

(18.13)

1. Make the spring pendulum in oscillation motion.

2. Measure the amplitudes of first - A₁ and of fifths - A₅ oscillations.

3. Calculate the period of vibration by the formula

T = t/n , (18.14)

where t is the time of "n" oscillations ; "n" is the number of oscillations.

4. Calculate the logarithmic decrement and the damping factor in accordance with the relations

(18.15)

(18.16)

5. Calculate Dl/l , Db/b , Dl and Db for five measurements. Calculate the half-width of the confidence interval for A₁ ,A₅ and t using the main errors of measurements.

6. Put down the date of measurements and the results of calculation in the table 1.

7. Repeat the measurements for another mass.

8. Put down the date of measurements in table 1.

9. Make analyses of the experimental results.

Table 18.1

Control questions

1. What oscillations is damped?

2. Write down differential equation of damped oscillations and specify physical value.

3. What law does displacement of damped oscillations change on from time? Write equation of damped oscillations and draw its graph.

4. How does amplitude of damped oscillations depend from time?

5. Give determination and write down expression for logarithmic decrement.

6. What is named times of relaxation?

Translator: S.P. Lushchin, the reader, candidate of physical and mathematical sciences.

Reviewer: S.V. Loskutov, professor, doctor of physical and mathematical sciences.

Approved by the chair of physics. Protocol № 6 from 30.03.2009 .

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