PERIODIC MOTION

In this section we investigate two cases of periodic motion which can be compared with exact solutions. A simple spring–mass system is considered first, and then we look at the large motion of a pendulum. The primary objective is to examine how well time domain problems can be solved. In particular, is appreciable numerical damping induced into the solution, and are the nonlinearities properly accounted for. The two examples also address the question of how well the flexible and rigid connections function in the time domain.

The first test consists of a spring, modeled by a vertical bar with a Young’s modulus of 10 kips/in², a length of 100 ft, and a constant circular cross section of 1 ft diameter. The block has a weight of 1000 kips as is shown in Figure 1. To initiate the motion, the mass was displaced by 10 ft in the positive $z$-direction and released.

Figure 1: Setup for Spring Vibration

For uniaxial deformation, the deflection of the bar under load $P$ is

$$ \delta = \frac{PL}{AE} \tag*{(1)} $$

and the spring constant is

$$ \begin{aligned} k &= \frac{P}{\delta} = \frac{AE}{L} \end{aligned} \tag*{(2)} $$

The quantities verified are the natural frequency, the force needed to stretch the spring 10 ft, and the stretched length of the spring with a 1000 kips block. The force required to change the length of the bar 10 ft can be determined by solving (1) for $P$,

$$ \begin{aligned} P = \frac{\delta EA}{L} \end{aligned} \tag*{(3)} $$

which for a $\delta=$ 10 ft and the bar particulars given above results in $P=$ 113.09 kips. The natural frequency and period of the system are given by

$$ \begin{aligned} \omega_n &= \sqrt{\frac{k}{m}} \end{aligned} \tag*{(4)} $$

and

$$ \begin{aligned} T = \frac{2\pi}{\omega_n} \end{aligned} \tag*{(5)} $$

Using these equations with the spring constant of 11.3 kips/ft given by (2) yields a natural period of 10.37 seconds. The stretched length of the spring corresponds to the average length of the spring during oscillation. In Figure 2 we show the mean oscillation being about $z=$ −89 ft. Recalling that the spring is 100 ft long at $z=$ 0 ft will conclude that the stretched length of the spring is 189 ft. Again, using (1) with the weight of the block gives the stretched length of the bar to be 88.5 ft from its $z=$ 0 ft position.

Below we present the comparison of hand calculated values to those from MOSES. As can be seen there is a good comparison between the force needed to move the block to $z=$ 10 ft, the natural period, and the stretched length. The force to move the block to $z=$ 10 ft was taken from the MOSES status report, the mean $z$-location was taken from the statistics of the block motion, and the natural period was taken from the plot of oscillatory motion for 100 seconds. Finally, notice that the numerical integration did not induce any perceptible error.

Comparison of Exact Solution and MOSES for Spring-Mass System

Next we consider the large angular motion of a pendulum. This is by far the more interesting of the two tests, since the governing equation is nonlinear. The system consists of a 100 ft bar of constant section, hinged at one end. The bar is initially rotated 90 degrees and released.

Figure 2: Vibration of Spring-Mass System

The equation of motion for the pendulum in terms of the angle, $\theta$, the mass, $m$, the length of bar, $l$, and the moment of inertia, $I_o$, is

$$ \begin{aligned} \ddot{\theta} + \omega^2 \sin \theta &= 0 \end{aligned} \tag*{(6)} $$

where

$$ \begin{aligned} \omega^2 &= \frac{3g}{2l} \end{aligned} \tag*{(7)} $$

and $g$ is the acceleration of gravity. The period for motion that satisfies this equation is an elliptic integral, and it can be represented as

$$ \begin{aligned} T &= \frac{2}{\pi} K(2\pi \omega) \end{aligned} \tag*{(8)} $$

where $K$ is dependent on the initial amplitude. For 90 degrees, $K=$ 1.854, which yields a period of $T=$ 10.66 seconds. As can be seen from Figure 3, the computed results have a period that is the same as the predicted value. It is also worth noting the plateaus in the acceleration versus time plot shown in Figure 4. The motion here is certainly not described by a cosine.

Figure 3: Oscillation of Pendulum Motion

Figure 4: Acceleration of Pendulum Motion