SPRING-WEIGHT-DAMPER SYSTEM

Almost all dynamic systems an be represented by combinations of springs, weights, and dampers. It is natural, therefore, to use this system to illustrate basic dynamic principles, since it can easily be visualized and even set up experimentally.

Figure 9.2 shows a simple spring-weight-damper system consisting of one of each of the elements. The variable of interest, x, is the distance between the instantaneous position of the center of the weight, xh and its static position, x„ when there is no load in the spring. At any one time, the forces acting on the weight in the x direction are of three types: the spring force, Fs, the damper force, Fd, and the inertia force, F[. In the absence of any external forces, the summation of these three forces must be zero:

Fs + FD + Fi= 0

Referring to Figure 9.2, this equation can be written:

kx + cx + mx = 0

where k is the spring rate in lb/ft; c is the damper rate in lb/(ft/sec); m is the mass in lb/^t/sec2); x is the displacement from the static position in ft; x is the velocity in ft/sec; and x is the acceleration in ft/sec2.

This is a classical differential equation, which can be solved by using the substitution:

This complicated-looking solution to an apparently simple system represents the six types of time histories shown in Figure 9.1 through various combinations of relative magnitudes of the constants, k, c, and да.

Case 1

k

c = 0, — > 0

The exponential terms can be written in terms of trignometric terms using Euler’s equations:

e, z = cos z + / sin z

and

e, z = cos z. — і sin г

Thus

This is a steady, or neutrally stable, oscillation with a half amplitude of x0 and a frequency of yjk/m radians per second, or a period of 2n/J(k/m) seconds as shown in the sketch.

This is a damped, or stable, oscillation with a frequency of у/(k/m) — [{c/l)/m]2 radians per second whose envelope decays asymtotically. The rate of decay to half amplitude is given by the equation:

or the time to half amplitude is:

In.5 .693

‘m ~~ с/г ~ с/г

m m

Case 3

. k I c/2 c < 0, – > — m m

(Note: Physically, there are no simple negative dampers, but in some systems a source of external energy gives the same effect.)

This is a negatively damped, or unstable, oscillation, with a frequency of yj(k/m) — [(c/2)/mf radians per second, whose envelope expands to double its amplitude every 1.386 mjc seconds.

In this case, the exponents of both e’s are real and negative, so that the motion is a pure convergence with time as in the sketch.

Note that the special subcase of [(с/2)/т^ — k/m represents the dividing line between convergence with oscillation and convergence without oscillation. The value of damping that satisfies this condition is called critical damping. This concept allows a useful alternative form of the equilibrium equation to be written by defining a damping ratio,

The concept of critical damping is sometimes used as a criterion for dynamic systems that might go unstable under certain circumstances by specifying a damping in terms of the critical damping. For example, a value of £ of 0.06,’or 6% of critical damping, is often used as a goal.

Case 5a

k < о

(Note: A simple system with a negative spring is a toggle switch poised at dead center.) Schematically, this system looks like this:

Since at least one e has a positive real exponent, the resultant time history is a pure divergence:

Case 5b

This solution has two es with positive exponents, so it is also a pure divergence:

Case 6

k = o

or

1

*0

Л___________

Time

Note that this solution, based on zero initial velocity, is good whether the damping is positive or negative. If damping were negative, however, any initial velocity would result in a pure divergence.