Active Shock Absorbers Recent advances in materials science have created reliable, long...

Active Shock Absorbers

Recent advances in materials science have created reliable, long-lasting fluids, MR fluids, which change their properties when subjected to a magnetic field. If an MR fluid is placed in a shock absorber, a change in an applied magnetic field can alter the damping capabilities of the fluid, so the damping coefficient can be adjusted dynamically. These “active” shock absorber systems have found application in such diverse objects as washing machines, prosthetic limbs, and car suspensions.∗

One of the first applications of this technology is called the Motion Master Ride Management System, an active shock absorber system for truck and school bus seats. Schematically, we can think of a truck seat as being attached to the rest of the truck by a spring and a dashpot (see Figure 2.69). For the perfect ride, we would want the spring to have spring constant k = 0 and the dashpot to have damping coefficient b = 0. In this case, the seat would float above the truck. For obvious reasons, the seat does have to be connected to the truck, so at least one of the two constants must be nonzero. The springs are chosen so that k is large enough to hold the seat firmly to the truck, and the damping coefficient b is chosen with the comfort of the driver in mind.

If b is very large, the seat is rigidly attached to the truck, which makes the ride very uncomfortable. On the other hand, if b is too small, the seat may “bottom out” when the truck hits a large bump. That is, the spring compresses so much that the seat

violently strikes the base. This response is both dangerous and uncomfortable. In practice, designers compromise between having b small (a smooth ride that has danger from large bumps) and b large (protection from large bumps but a rough ride). Another constant, the mass of a typical driver, must also be considered as a factor in the choice of b. Active damping allows adjustment of the damping coefficient according to the state of the system. That is, the damping coefficient b can be replaced by a function of y and v = dy/dt. As a first step in studying the possibilities in such a system, we consider a modification of the harmonic oscillator of the form

where m is the mass of the driver. In this case, the damping coefficient b(v) is assumed to be a function of the velocity v. For this lab, we assume that the units of mass and distance are chosen so that k = m = 1, and we study the equation

When the vertical velocity of the seat is near zero, we want small damping so that small bumps are not transmitted to the seat. When the vertical velocity of the seat is large, we want b(v) to be large to protect from “bottoming out” (and “topping out”). These criteria leave considerable freedom in the choice of the function b(v).

In this lab, we consider the behavior of a truck seat for three possible choices of the damping function b(v):

Your report: Address all of the items above. Be sure to keep the application in mind when describing the behavior of solutions. Phase portraits and y(t)-graphs are useful, but illustrations alone are not enough. In Part 4, address your analysis to an audience having your mathematical background but who have not considered this problem.

Repeat Part 1 for b(v) = 1 − e^−10v2 .