Oscillations and Waves > Forced and damped oscillations > Problem 3.5.18

Statement

If at time a damped oscillator is in its equilibrium position and its velocity is , then its coordinate at time is given by the formula

where , , and , and are the stiffness, mass, and damping coefficient of the oscillator, respectively. Show that the properties of the oscillator described in problems 3.5.12 and 3.5.15 do not contradict this statement.

Solution

Connection to problem 3.5.12:

Problem 3.5.12 considers two oscillators with identical ratios and .

From the formula for the coordinate , it is evident that it depends only on the initial velocity , the natural frequency , and the damping coefficient .

Since the parameters and of the two oscillators coincide, under identical initial conditions (), the equation will yield identically equal dependencies . There is no contradiction.

Connection to problem 3.5.15:

Let's find the velocity of the oscillator by taking the time derivative of the coordinate :

The oscillator passes through the equilibrium position when , which corresponds to the condition . At these moments, .

Substituting these values into the velocity expression, we obtain the magnitude of the velocity at the moments of passing through equilibrium:

According to the condition of problem 3.5.15, after a time , the velocity decreases by a factor of . We can write this as:

From this, we obtain the decay factor over one period :

Let's determine the velocity after a time :

Similarly, after a time :

The obtained analytical expressions strictly show that the velocity at the equilibrium positions decreases in a geometric progression. There is no contradiction with the conclusions of problem 3.5.15.

Answer

See the solution.