Differential Equations and Linear Algebra, 4th Edition

9.4 Nonlinear Mechanical Systems

Now we apply the qualitative methods of Sections 9.1 and 9.2 to the analysis of simple mechanical systems like the mass-on-a-spring system shown in Fig. 9.4.1. Let m denote the mass in a suitable system of units and let x(t) denote the displacement of the mass at time t from its equilibrium position (in which the spring is unstretched). Previously we have always assumed that the force F(x) exerted by the spring on the mass is a linear function of x: $F(x)=-kx$ (Hooke’s law). In reality, however, every spring in nature actually is nonlinear (even if only slightly so). Moreover, springs in some automobile suspension systems deliberately are designed to be nonlinear. Here, then, we are interested specifically in the ...