The differential equation of the damped harmonic motion is given by Eqn. (3.2):
where (–2mbx ) = (−β x ) is the damping force acting on the particle of mass m and ω is the natural frequency of the oscillator. Let x = exp (αt) be the trial solution of above equation. Then, we have
where A1 and A2 are two constants whose values can be determined from the initial conditions. Case I: b > ω (the damping force is large) The expression (3.8) for x represents a damped motion, the displacement x decreasing exponentially to zero (Fig. 3.1).
Equation (3.10) gives a damped oscillatory motion (Fig. 3.1). Its amplitude R exp (–bt) decreases exponentially with time. The time ** of damped oscillation is
Thus the time ** of damped oscillation is slightly greater than the undamped natural time ** when b ^ ω. In other words, the frequency of the damped oscillation
is less than the undamped natural frequency ω. Let us consider the simple case in which θ = 0 in Eqn. (3.10). Then cos ω′t = + 1 when
etc. Suppose that the values of x in both directions corresponding to these times are
Considering the absolute values of the displacements, we get
is called logarithmic decrement. Here ν is the frequency of the damped oscillatory motion. The logarithmic decrement is the logarithm of the ratio of two successive maxima in one direction = ln (xn/xn+2). Thus the damping coefficient b can be found from an experimental measurement of consecutive amplitudes. Since
The energy equation of the damped harmonic oscillator: We can regard equation (3.10) as a cosine function whose amplitude R exp (– bt) gradually decreases with time. For an undamped oscillator of amplitude R, the mechanical energy is constant and is given by E = 1 2 kR2. If the oscillator is damped, the mechanical energy is not constant but decreases with time. For a damped oscillator the amplitude is R exp (– bt) and the mechanical energy is
Like the amplitude the mechanical energy decreases exponentially with time. Case III: b = ω (critically damped motion)
When b = ω, we get only one root α = – b. One solution of Eqn. (3.2) is
The motion is non-oscillatory and the particle approaches origin slowly.
