This week’s blog covers a recent technical support question, in which a user was trying to achieve the same results from both a harmonic analysis and a transient analysis (for a linear problem). Since I already put the information together to answer their question, I thought this information would also be useful to share with others. This post also compares results using a base excitation load versus an acceleration applied to the structure, with the base fixed.
So, here’s the problem, demonstrated with a simple beam model, and some important points along the way:
The problem is shown in Figure 1, where a sinusoidal input load is either applied as a base excitation (the “fixed” end of the beam has a prescribed time varying displacement applied) or one end of the beam is held fixed while an acceleration load vs. time is applied. The displacement response of the beam due to either case looks like what is shown.
Figure 1. Problem Overview

The analysis approach:
1. Perform a modal analysis to determine a natural frequency of the beam that will be excited.
2. Excite this natural frequency through either a harmonic or transient analysis.
Just a few quick comments worth mentioning.
 These analyses include damping, and its effect is easily shown by performing a transient analysis. When the input loading matches the natural frequency, without damping, the tip response will continue to amplify each cycle. With damping, the tip response will reach a steady state oscillation, as shown in Figure 2.
 The transient solution for the damped case shown in Figure 2 also starts off at rest and then sees the applied sinusoidal loading. For this case, the problem was just run out long enough to let the initial condition dampen out over time and reach a steady state condition.
 The transient solution results that are obtained and compared to the harmonic results are from a point in time when the oscillations have reached a steady state.
Figure 2. Damped vs Undamped Transient Analyses

As expected, the short answer to the original question is, yes, and the proof is in Table 1 below. It’s certainly possible to predict the same results with either a harmonic analysis or transient analysis, and it doesn’t matter if the applied loading is either an acceleration or base excitation. However, here are some items to consider when choosing the approach:
 In a transient analysis, the time step should be small enough to accurately capture the period of oscillation. For example, 20 points per cycle sufficiently captures the accuracy with a relatively low error in the solution outputs (as shown in Table 1 for the transient analyses). A larger time step, and subsequently fewer calculated solution points per cycle (e.g. 10), starts to significantly add to the solution error.
 A transient solution will inherently take much longer to run than the harmonic analysis due to the number of solutions that are calculated. However, both methods will produce the same results for this problem if proper numerical techniques are employed.
 Each analysis type may have different forms of damping that are available for use. The harmonic analysis can use a constant damping ratio, whereas in a transient analysis the Rayleigh damping that will result in a certain damping ratio has to be calculated.
 A description of some different types of damping and conversions can be found here in our resource library.
 A very small amount of numerical damping may be needed for the base excitation applied displacement loading case due to the spurious accelerations that can occur when applying a time varying displacement load. This is a consequence of the numerical integration scheme which is used. A base acceleration loading is usually preferred to a base displacement loading.
 A transient analysis can include nonlinear effects. If these are important and necessary, a harmonic analysis cannot be used, since it assumes a linear system and harmonic (sinusoidal) motion. If these effects are not needed, then a harmonic analysis will be significantly more efficient than the transient analysis method, as already described.
Table 1. Results of Study

The points and observations made within this study are certainly not meant to be all encompassing. If you have any additional comments to contribute on this topic, please add them to this discussion. I’d like to hear them!