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

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 = A cos w t, (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) tcosw t. (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.