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Structural Vibration and Dynamics 275

TABLE 8.1
Comparisons of the Methods
Direct Methods Modal Method
Small models Large models
More accurate (with small Δt) Higher modes ignored
Single loading Multiple loading
Shock loading Periodic loading
… …

Then, solve the uncoupled equations using an integration method. We can use, for exam-
ple, 10% of the total modes (m = n/10). The advantages of the modal method are as follows:

• Uncoupled system
• Fewer equations
• No inverse of matrices
• More efficient for large problems

However, the modal method is less accurate if higher modes are important, which is the
case for structures under impact or shock loading. Table 8.1 summarizes the advantages
and disadvantages of the direct and modal methods for transient response analysis.

8.6 Modeling Examples
8.6.1 Modal Analysis
Take a bumper system as an example. Figure 8.10 shows a front bumper and the supporting
brackets in a car. The model is applied to study the dynamic responses of the bumper. Shell
elements were used for this study to obtain the natural frequencies and vibration modes.
Figure 8.11 shows the first mode of the bumper when it is constrained at the bracket locations.

FIGURE 8.10
FEA model of a front bumper and supporting brackets.