15.7 Damped Oscillations

A pendulum left to itself gradually slows down and stops. The sound of a ringing bell gradually dies away. All real oscillators do run down—some very slowly but others quite quickly—as friction or other dissipative forces transform their mechanical energy into the thermal energy of the oscillator and its environment. An oscillation that runs down and stops is called a damped oscillation.

A photograph of a car on a rocky terrain. — The shock absorbers in cars and trucks are heavily damped springs. The vehicle’s vertical motion, after hitting a rock or a pothole, is a damped oscillation.

There are many possible reasons for the dissipation of energy, such as air resistance, friction, and internal forces within a metal spring as it flexes. The forces involved in dissipation are complex, but a simple linear drag model gives a reasonably accurate description of most damped oscillations. That is, we’ll assume a drag force that depends linearly on the velocity as

\vec{F}_{drag} = - b \vec{v} (model of the drag force)

(15.52)

where the minus sign is the mathematical statement that the force is always opposite in direction to the velocity in order to slow the object.

The damping constant b depends in a complicated way on the shape of the object and on the viscosity of the medium in which the particle moves. The units of b need to be such that they will give units of force when multiplied by units of velocity. As you can confirm, these units are $kg / s$ . A value $b = 0 kg / s$ corresponds to the limiting case of no resistance, in which case the mechanical energy is conserved.

FIGURE 15.21 shows a mass oscillating on a spring in the presence of a drag force. With the drag included, Newton’s second law is

(F_{net})_{x} = (F_{Sp})_{x} + (F_{drag})_{x} = - k x - b v_{x} = m a_{x}

(15.53)

Using $v_{x} = d x / d t$ and $a_{x} = d^{2} x / d t^{2}$ , we can write Equation 15.53 as

\frac{d^{2} x}{d t^{2}} = - \frac{k}{m} x - \frac{b}{m} \frac{d x}{d t}

(15.54)

Equation 15.54 is the equation of motion of a damped oscillator. If you compare it to Equation 15.32, the equation of motion for a block on a frictionless surface, you’ll see that it differs by the inclusion of the term involving $d x / d t$ .

The figure illustrates a block oscillating due to a horizontal spring attached to a wall on the left.

FIGURE 15.21 Full Alternative Text

Equation 15.54 is another second-order differential equation. We will simply assert (and, as a homework problem, you can confirm) that the solution is

x (t) = A e^{- b t / 2 m} cos (ω t + ϕ_{0}) (damped oscillator)

(15.55)

where the angular frequency is given by

ω = \sqrt{\frac{k}{m} - \frac{b^{2}}{4 m^{2}}} = \sqrt{ω_{0}^{2} - \frac{b^{2}}{4 m^{2}}}

(15.56)

Here $ω_{0} = \sqrt{k / m}$ is the angular frequency of an undamped oscillator $(b = 0)$ . The constant e is the base of natural logarithms, so $e^{- b t / 2 m}$ is an exponential function. Because $e^{0} = 1$ , Equation 15.55 reduces to our previous $x (t) = A cos (ω t + ϕ_{0})$ when $b = 0$ . This makes sense and gives us confidence in Equation 15.55.