Conceptual Physics

damping force always opposes (resists) the motion of an object, which means

sometimes it acts in the same direction as the restoring force (when the object moves

away from equilibrium), and sometimes in the opposite direction (when the object

moves toward equilibrium). At all times, however, it is opposing the motion.

Instead of moving in SHM, the system moves back to its equilibrium point and stops, or

it may oscillate a few times with smaller and smaller amplitude before resting at its

equilibrium point. The fluid “dampens” the motion, reducing the amplitude of the

oscillations. The result is a relatively fast yet smooth return to the equilibrium position.

The resistive force of the fluid in a system like this is often proportional to the velocity,

and opposite in direction. In Equation 1, you see the equation for the damping force. It

equals the negative of b (the damping coefficient) times the velocity. (You may note that

this is similar to the formula for air resistance, where the drag force depends on the

square of the velocity.) The negative sign indicates that the damping force opposes the

motion that causes it.

In Equation 2, you also see the equation for the net force FN. The net force is the sum

of the restoring forces and the damping force. (If you look at the equation, it may seem

that two negatives combine to make a larger number, but the sign of the velocity is the

opposite of the displacement as the system moves toward equilibrium.)

The graph in Equation 3 illustrates three types of damping. The blue line represents a

critically damped system. The damping force is such that the system returns to

equilibrium as quickly as possible and stops at that point.

The green line represents a system that is overdamped. The damping force is greater

than the minimum needed to prevent oscillations. The system returns to equilibrium

without oscillating, but it takes longer to do so than a critically damped system.

The red line is a system that is underdamped. It oscillates about the equilibrium point,

with ever diminishing amplitude.

With certain shock absorbers, the system can be adjusted, which means that the

damping coefficient can be tuned based on rider preferences. Beginners often prefer an

underdamped system. The bike bounces a bit but there is less of a “jolt” because the

shock absorber acts more slowly. Advanced riders sometimes prefer a critically damped

or overdamped “harder” ride, trading off a less smooth ride in exchange for regaining

control of the bicycle more quickly.

Damped oscillations

Damping causes oscillations to diminish

Damping force

Opposes motion

Often proportional to velocity

Damping force

Fd = íbv

Fd = damping force

b = damping coefficient

v = velocity

Net force

ȈF = íkxíbv

ȈF = net force

k = spring constant

x = displacement

b = damping coefficient

v = velocity

(^288) Copyright Kinetic Books Co. 2000-2007 Chapter 14