Question 7.16: Constant energy curves in the phase plane. Use the energy pr...

The free body diagram is shown in Fig. 6.13a. The weight of the oscillator and the normal surface reaction do no work in any rectilinear motion along the smooth horizontal surface . The elastic potential energy of the linear spring force acting on m is given by (7.65):  V(x)=\frac{1}{2} k x^2,  wherein V(0) = 0 in the natural state x = 0. The system is conservative with kinetic energy   K=\frac{1}{2} m \dot{x}^2,  so the energy principle (7.73) yields

V_e=\frac{1}{2} k x^2               (7.65)

K+V=E, \text { a constant. }               (7.73)

\frac{1}{2} m \dot{x}^2  +  \frac{1}{2} k x^2=E                          (7.84a)

The equation of motion is obtained by differentiation of (7.84a) with respect to the path variable x or with respect to time; we find   m \ddot{x}  +  k x=0.  This agrees with (6.65a) in which  p=\sqrt{k / m}.

Now let us examine the curves in the xv-plane, called the phase plane. Because k > 0, the total energy E in (7.84a) is a positive constant determin ed from  assigned initial data. Suppose that x(0)=x_0 and   \dot{x}(0)=v_0  at t = 0, then   E=\frac{1}{2} m v_0^2  +  \frac{1}{2} k x_0^2.  We introduce

\varepsilon^2 \equiv \frac{2 E}{m},                 (7.84b )

and write  v=\dot{x}  to cast (7.84a) in the form

\left(\frac{x}{\varepsilon / p}\right)^2  +  \left(\frac{v}{\varepsilon}\right)^2=1.                  (7.84c)

For any given spring-mass pair (m, k), the phase plane curve described by (7.84c) is an ellipse whose axes are determined by the constant   \varepsilon.  For each  choice of initial data,  \varepsilon  has a different value; and hence (7.84c) describes a family of concentric ellipses each of which is traversed in the same time   \tau=2 \pi / p,  the period of the oscillation, and on each of which ε  is a constant fixed by the total energy E, In consequence, the phase plane curves for a conservative dynamical system are called energy curves . In physical terms, (7.84c) shows that e is equal to the maximum speed in the periodic motion, which occurs at the natural state x = 0, and   x_A \equiv \varepsilon / p  is the symmetric amplitude of the oscillation, the maximum displacement from the natural state—it marks the extreme states in the motion at which v = 0.