# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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The Oscillating Pendulum. The concepts of asymptotic stability, stability, and instability can be easily visualized in terms of an oscillating pendulum. Consider the configuration shown in Figure 9.2.2, in which a mass m is attached to one end of a rigid, but weightless, rod of length L. The other end of the rod is supported at the origin O, and the rod is free to rotate in the plane of the paper. The position of the pendulum is described by the angle Q between the rod and the downward vertical direction, with the counterclockwise direction taken as positive. The gravitational force mg acts downward, while the damping force c|dQ/dt|, where c is positive, is always opposite to the direction of motion. We assume that Q and dQ / dt are both positive. The equation of motion can be quickly derived from the principle of angular momentum,

474

Chapter 9. Nonlinear Differential Equations and Stability

FIGURE 9.2.2 An oscillating pendulum.

which states that the time rate of change of angular momentum about any point is equal to the moment of the resultant force about that point. The angular momentum about the origin is mL2{dO/ dt), so the governing equation is

r 2 d2Q T d9

mL —T = —cL mgL sinQ. (10)

dt2 dt

The factors L and L sin Q on the right side of Eq. (10) are the moment arms of the resistive force and of the gravitational force, respectively, while the minus signs are due to the fact that the two forces tend to make the pendulum rotate in the clockwise (negative) direction. You should verify, as an exercise, that the same equation is obtained for the other three possible sign combinations of Q and dQ / dt.

By straightforward algebraic operations we can write Eq. (10) in the standard form

d2Q c dQ g

—j +------------------ + — sin Q — °> (11)

dt mL dt L

or

d2Q dQ 2

—5- + y-------------+ m sin Q — 0, (12)

dt2 dt

where y — c/mL and m2 — g/L. To convert Eq. (12) to a system of two first order equations we let x — Q and y — dQ/dt; then

dx dy 2

— = y, — = — m sin x— y y? (13)

dt dt

Since y and m2 are constants, the system (13) is an autonomous system of the form (1). The critical points of Eqs. (13) are found by solving the equations

y — 0, —m2sin x — y y — 0.

We obtain y — 0 and x — ±nn, where n is an integer. These points correspond to two physical equilibrium positions, one with the mass directly below the point of support (Q — 0) and the other with the mass directly above the point of support (Q — n). Our intuition suggests that the first is stable and the second is unstable.

9.2 Autonomous Systems and Stability

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More precisely, if the mass is slightly displaced from the lower equilibrium position, it will oscillate back and forth with gradually decreasing amplitude, eventually approaching the equilibrium position as the initial potential energy is dissipated by the damping force. This type of motion illustrates asymptotic stability.

On the other hand, if the mass is slightly displaced from the upper equilibrium position, it will rapidly fall, under the influence of gravity, and will ultimately approach the lower equilibrium position in this case also. This type of motion illustrates instability. In practice, it is impossible to maintain the pendulum in its upward equilibrium position for any length of time without an external constraint mechanism since the slightest perturbation will cause the mass to fall.

Finally, consider the ideal situation in which the damping coefficient c (or y) is zero. In this case, if the mass is displaced slightly from its lower equilibrium position, it will oscillate indefinitely with constant amplitude about the equilibrium position. Since there is no dissipation in the system, the mass will remain near the equilibrium position, but will not approach it asymptotically. This type of motion is stable, but not asymptotically stable. In general, this motion is impossible to achieve experimentally because the slightest degree of air resistance or friction at the point of support will eventually cause the pendulum to approach its rest position.

These three types of motion are illustrated schematically in Figure 9.2.3. Solutions of the pendulum equations are discussed in more detail in the next section.

(a) (b) (c)

FIGURE 9.2.3 Qualitative motion of a pendulum. (a) With air resistance. (b) With or without air resistance. (c) Without air resistance.

Determination of Trajectories The trajectories of a two-dimensional autonomous system can sometimes be found by solving a related first order differential equation. From Eqs. (1) we have

dy = dy/dt G (x, y) (14)

dx dx/dt F(x, y)’ ( )

which is a first order equation in the variables x and y. Observe that such a reduction

is not usually possible if F and G depend also on t. If Eq. (14) can be solved by any

of the methods of Chapter 2, and if we write solutions (implicitly) in the form

H(x, y) = C’ (15)

then Eq. (15) is an equation for the trajectories of the system (14). In other words, the trajectories lie on the level curves of H(x, y). Keep in mind that there is no general

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