Problem 53 - Concepts and Applications of Finite Element Analysis

Question

A particle of unit mass is supported by a spring of unit stiffness. There is no damping and no external load. Thus \(k=m=\omega=1\). At time \(t=0\), the particle has zero displacement, zero acceleration, but unit velocity. Use. the central-difference method, Eq. 13, \(10-5\), to compute displacement versus time over five time steps. Use a \(\Delta t\) of (a) \(1.0\), (b) \(\sqrt{2.0}\), (c) \(2.0\), and (d)

Step-by-Step Solution

Answer

Using the central-difference method with different time steps, a plot of displacement over time will show oscillatory behavior, with the frequency of the oscillation depending on the time step. For smaller time steps, the displacement is closer to the exact solution, but requires more computations. For larger time steps, the solution is less accurate, but requires fewer computations.

Step 1: Setup

Given \(k=m=\omega=1\), and \(u(0)=0\), \(\dot{u}(0)=0\) and \(\ddot{u}(0)=1\). We can compute the displacement \(u(t)\) after the first few time steps.

Step 2: Compute Displacement for (a)

For \(\Delta t = 1.0\), first, find \(\ddot{u}(1)\) from the equation of motion \(\ddot{u}=-u\). For \(t=0\), this gives \(\ddot{u}(1) = -\ddot{u}(0) = -1\). Use these values to compute \(u(1)\) using the central difference method formula, \(u(t+\Delta t) = \Delta t^2 \cdot \ddot{u}(t) / 2 + u(t) + \Delta t \cdot \ddot{u}(t)\). Repeat this for five time steps.

Step 3: Compute Displacement for (b)

For \(\Delta t = \sqrt{2.0}\), repeat the process from Step 2, but with the new value for \(\Delta t\). This is a bigger time step, so expect the oscillations to be less frequent.

Step 4: Compute Displacement for (c)

For \(\Delta t = 2.0\), repeat the process from Step 2, but with the new value for \(\Delta t\). This is a larger time step, which will result in fewer computations, but also less precision.

Step 5: Compare

Compare the results obtained for the different time steps. The computed displacement over time should have a periodic behavior, resembling a sine or cosine wave, due to the nature of the system. However, due to the nature of the central-difference method, there may be some errors that increase over time, especially for larger time steps.