Front suspensions have become standard equipment on mountain bikes. Replacing the rigid fork that attaches the bicycle’s front tire to its frame, such suspensions absorb bump impact energy, shielding both frame and rider from jolts. Commonly used forks, however, use only one spring constant and treat bump impacts at high and low speeds—impacts that vary greatly in severity—essentially the same.

A suspension system with multiple settings that are adjustable while the bike is in motion would be attractive. One air and coil spring with an oil damper is available that permits an adjustment of the damping constant to the terrain as well as to the rider’s weight [17]. The suspension system model is shown in Figure AP3.4, where b is adjustable. Select the appropriate value for b so that the bike accommodates

(a) a large bump at high speeds and

(b) a small bump at low speeds. Assume that k2 = 1 and k1 = 2.

www f(t) Mass m k y(t) FIGURE AP3.4 Shock absorber. q(t)