Question

Elementary Differential Equation Unit Step Function Problem

Project 2 A Spring-Mass Event Problenm A mass of magnitude m is confined to one-dimensional motion between two springs on a frictionless horizontal surface, as shown in Figure 4.P.3. The mass, which is unattached to either spring, undergoes simple inertial motion whenever the distance from the origin of the center point of the mass, x, satisfies lxl < L. When x 2 L, the mass is in contact with the spring on the right and, following Hooke's law, experiences a force in the negative x direction (to the left) proportional tox - L. Similarly, when x 3 -L, the mass is in contact with the spring on the left and experiences a force in the positive x direction (to the right) proportional tox + L This problem is an example of an event problem in differential equations. The events of interest occur whenever the mass initiates or terminates contact with a spring. A typical way to describe an event is to associate an event function g(t, x), which may or may not depend on t, with the problem. Then an event is said to occur at time * if g(t*, x(t 0 For example, an event function for this problem could be g(x)-(x - L(r + L). One strategy for finding numerical approximations to solutions of an event problem is to lo- cate the occurrence of events in time and restart the integration there so as to deal with the changes in the differential equation. Most modern commercial software packages contain ODE solvers that are capable of solving event problems -L 0 L FIGURE 4.P.3 A mass bouncing back and forth between two springs Project 2 PROBLEMS Use the unit step function to express the differential equation in Problem 1 in a single line 1. Assuming that both springs have spring constant k and that there is a damping force proportional to velocity x' with damping constant ?, write down 3, Is the differential equation derived in Problems I the differential equation that describes the motion of the mass. Note that the expression for the force . In the function depends on which of the three different parts of its domain x lies within and 2 linear or nonlinear? Explain why case that the damping constant ? > 0, find the critical points of the differential equation in Prob- lem 2 and discuss their stability properties 2. The Heaviside, or unit step function, is defined by 5. Consider the c ase of an undamped problem (y -0) using the parameter values L = 1 , m = 1 , k = 1 and initial conditions x(0)-2, x'(0)0. Find the solu- tion x(t) of the initial value problem for 0stst 0, x <c.

Answer 1