8.2. Differential Equation for Own (Free) Harmonic Vibrations

O wn (free) vibrations take place after excitation of vibrations without external influence. Let us find the law of motion for body with mass m (see model system in Fig. 8.2).

Figure 8.2

We consider the case when the friction does not occur. For an ideal spring the extension is directly proportional to the applied force. The magnitude of the deformation is proportional to the force producing it. In accordance with Newton’s second law:

Hence the differential equation of free harmonic vibrations is given by:

As k>0, m>0, it is possible to introduce ₀² = k/m. The solution of the differential equation

is given by:

where ₀ is the cyclic frequency of free harmonic vibrations, A is the amplitude, ₀ is the initial phase.

The phase speed:

8.3. Energy of Harmonic Vibrations

The kinetic energy of the material point performing a harmonic vibration :

Note, that the kinetic energy periodically oscillates between 0 and kA²/2 with the cyclic frequency 2₀ and the amplitude kA²/4 above the average meaning kA²/4.

The potential energy of the material point performing the harmonic vibration is:

The potential energy oscillates too between 0 and kA²/2 with the cyclic frequency 2₀ and the amplitude kA²/4 above the average meaning kA²/4.

Values of W_k and W_p periodically oscillate with the shift of phases , but at any moment of the time their sum stays constant (Fig.8.3):

It’s easy to check up this statement:

Figure 8.3

In the process of harmonic vibrations the potential energy transforms into the kinetic energy and vice versa. In the return point and when passing the states of equilibrium an energy of one type equals to zero while an energy of other type is maximal.

8.4. Damped Vibrations

The vibration damping is a process of reducing the amplitude with the time caused by energy losses of the vibration system.

All vibrations in nature are really the damped vibrations and, as WA², when W0, then A0 too.

Reasons of the vibration damping:

1. A friction.

2. Internal damping, viscous damping.

3. Generation of sound waves in media.

The law of the vibration damping depends on the mechanical properties of the vibration system.

Only the linear systems will be considered further, namely the systems with parameters not alternating with time.

Let us consider the vibration system shown in Fig. 8.4. In this case it’s necessary to take into account two forces: the elastic force F = - kx and the force of the viscous friction F_fr = - bv, where b is the coefficient of a friction.

F igure 8.4

Newton’s second law for this case can be written as:

and can be transformed into the differential equation of damped vibrations:

where 2 = b/m, ₀² =k/m,  is called the coefficient of damping.

The solution of this equation is (Fig.8.5)

where _demp=2_demp=2/T_demp = (₀² - ²)^1/2. It is possible to use the term occurrence to describe the damped vibrations only nominally, because the damped vibrations are not strictly periodic. It is explained by the fact that the amplitude meaning at the certain moment of the time does not repeat at other moment of the time due to the factor exp(-t) (Fig. 8.5). The amplitude of damped vibrations: A = A₀e^-t.

Characteristics of the vibration damping:

1 ) The time of relaxation =1/ characterizes an alteration of the amplitude with the time. Let t=, then A=A₀e^-t=A₀/e. Hence, time of the relaxation is the time during which the amplitude is decreased in e-times. Figure 8.5

2) The decrement of damping  is a value that demonstrates how much the amplitude of the vibration decreases during one period of time (Fig. 8.5)

Since then

3) The logarithmic decrement of damping: