Modal superposition

The natural frequencies and mode shapes of a structure can be used to characterize its dynamic response to loads in the linear regime. The deformation of the structure can be calculated from a combination of the mode shapes of the structure using the modal superposition technique. Each mode shape is multiplied by a scale factor. The vector of displacements in the model, u, is defined as

u=∑i=1∞αi⁢ϕi,
where αi is the modal displacement and ϕi is the generalized coordinate of mode i. This technique is valid only for simulations with small displacements, linear elastic materials, and no contact conditions—in other words, linear problems.

In structural dynamic problems the response of a structure usually is dominated by a relatively small number of modes, making modal superposition a particularly efficient method for calculating the response of such systems. Consider a model containing 10,000 degrees of freedom. Direct integration of the dynamic equations of motion would require the solution of 10,000 simultaneous equations at each point in time. If the structural response is characterized by 100 modes, only 100 equations need to be solved every time increment. Moreover, the modal equations are uncoupled, whereas the original equations of motion are coupled. There is an initial cost in calculating the modes and frequencies, but the savings obtained in the calculation of the response greatly outweigh the cost.

If nonlinearities are present in the simulation, the natural frequencies may change significantly during the analysis, and modal superposition cannot be employed. In this case direct integration of the dynamic equation of equilibrium is required, which is much more expensive than modal analysis.

A problem should have the following characteristics for it to be suitable for linear transient dynamic analysis:

  • The system should be linear: linear material behavior, no contact conditions, and no nonlinear geometric effects.

  • The response should be dominated by relatively few frequencies. As the frequency content of the response increases, such as is the case in shock and impact problems, the modal superposition technique becomes less effective.

  • The dominant loading frequencies should be in the range of the extracted frequencies to ensure that the loads can be described accurately.

  • The initial accelerations generated by any suddenly applied loads should be described accurately by the eigenmodes.

  • The system should not be heavily damped.