DETERMINATION of DAMPED OSCILLATIONS PARAMETERS

1 Goal of the work:

Studying key parameters and method of description for damped oscillations of mechanical systems.

Main concepts

Real-world oscillatory systems experience different kind of resistances. They loose its energy and with no external energy supply they stop after some finite interval of time. Damped oscillationsare such type of free oscillations, which energy decreases with time.

Mechanical energy of oscillatory system gradually decreases transforming into heat. This process called energy dissipation, and such system – dissipative system.

Besides the restoring (quasi-elastic) force (9) that acts in natural oscillatory systems, in the free linear oscillatory systems a drag force acts:

F_DRAG = – r×υ, (36)

where υ – velocity of pendulum’s moving; r – drag coefficient. “Minus” on the right hand side of (36) means that the drag force always acts in the opposite direction of the velocity.

Thus, for two forces (9) and (36) Newton’s second law for linear damped oscillations will be

ma = —kx—rυ.

In scalars, substituting acceleration of motion a=d²x/dt² and velocity of motion υ = dx/dt and obtain

,

where m – mass of oscillator (a body or a system of oscillating bodies).

Now let’s rearrange this equation and obtain an equation of damped oscillator in differential view:

, (37)

here we introduce designations of variables:

, .

Solution for this differential equation will be the dependence of displacement x from time t, which is termed equations of damped oscillations :

x(t)=A₀×e^–βt×cos(ωt+φ₀₁) or x(t)= A₀×e^–βt×sin(ωt+φ₀₂), (38)

The basic parameters of damping oscillations are:

damping coefficient (damping factor)

(39)

and cyclic frequency of damped oscillations

(40)

here ω₀ – cyclic eigenfrequency (resonant cyclic frequency); A₀e^–βt – exponentially decaying amplitude; A₀ – initial amplitude, determines by energy of the system at instant t = 0.

On the Fig. 2 represented plots of amplitude versus time (dashed line) and displacement versus time (solid line) dependencies.

Damped oscillations are non-periodic: values of oscillating physical quantities (such as displacement, velocity, acceleration) never repeat in damped oscillations process. That is why we can’t use concepts of period and frequency in the way that it been done for periodic (undamped) oscillations.

Conventional period of damped oscillations is such interval of time between to serial states of oscillating system at which oscillating physical quantities vary in the same direction, decreasing or increasing their magnitudes.

Knowing that and ω=2π/T , obtain

. (41)

The conventional period (41) greater than the eigenperiod of T₀ = 2π/ω₀, when no any damping forces are present.

There are different modes of oscillating systems with respect to value of damping coefficient (see Fig.3):

a) β = 0, so r = 0, T=2π/ω₀=T₀ – harmonic oscillations;

b) β<ω₀, ω₀²–β²>0, – almost periodic underdamping mode;

c) β = ω₀, ω₀²–β²=0, T→ ∞ – aperiodic critical mode;

d) β>ω₀, ω₀²–β²<0, T is imaginary – overdamping mode.

Decay decrement is a ratio of two serial amplitudes of the same sign A_t and A_t+T separated in time from each other by the period T:

.

Logarithmic decay decrement is a natural logarithm of this ratio:

. (42)

In practical calculations, for small damping, it is usually suggested that δ = βT₀, where T₀ – period of undamped oscillations of the system.

Denoting a τ – relaxation time as interval of time over which the amplitude of oscillations decreases in e times (e = 2.7183 – is the base of natural logarithm), we can write:

.

Hence, βτ = 1, or

. (43)

Thus, for meaning (physical sense) the damping coefficient β it is inversely proportional time over which amplitude of oscillations decreases in e = 2.7 times.

By the time τ system makes N_e = τ/T oscillations (relaxation number), so, taking into account (42) and (43) we’ll have

. (44)

So, for meaning (physical sense) the logarithmic decay decrement it is inversely proportional to number of oscillations producing by the time over which amplitude of oscillations decreases in e = 2.7 times.

Quality factor of a system is a ratio of coefficients from equation of damped oscillator (37):

.

Taking into account previous equations we obtain

. (45)

Thus, for meaning (physical sense) the quality factor it is proportional to the number of oscillations N_e done by a system by the time τ over which amplitude of oscillations decreases in e = 2.7 times.

Date: 2015-12-24; view: 859