2.2.3.      Frequency Response Based Measures

Modal effective mass is a very useful means of selecting target modes, but it is a fairly blunt instrument, and is strictly applicable only to the situation where the input to a structure is a base acceleration, and the output of interest is the interface force. This is a good measure of how that structure will interact with other components in a system, as well as which modes will contribute most to the interface loads, but often other performance indices are also of interest. It is possible to develop modal weightings based on specific transfer functions [15, 16] that generalize the concept of modal effective mass. Consider a Multiple Input Multiple Output (MIMO) system. The transfer function matrix from inputs to outputs can be expressed as follows:

                                       (2.2.3-1a)

                                       (2.2.3-1b)

                                    (2.2.3-1c)

where

            TFd       - transfer function matrix for displacement outputs

            TFv       - transfer function matrix for velocity outputs

            TFa       - transfer function matrix for acceleration outputs

                   - excitation frequency (rad/sec)

              -mode shape coefficients at outputs for ith mode

              - mode shape coefficients at inputs for ith mode[3]

                  - natural frequency (rad/sec) for ith mode

                   - critical damping ratio for ith mode

For a MIMO system, the transfer function is a matrix at each frequency, and the “size” of that matrix is typically described as the maximum singular value. Under the assumption of lightly damped modes and sufficient separation among modal frequencies, both the peak value of the transfer function maximum singular value and the RMS value can be expressed in terms of the peak and RMS values of the individual modal transfer functions as follows:

 

                                                (2.2.3-2a)

                                              (2.2.3-2b)

This provides a means for weighting the modes in terms of their contribution to either the peak or the RMS value of the transfer function matrix. The modal weighting in terms of their contribution to the peak value can be expressed as an Approximate Balanced Singular Value (ABSV) as follows:

                                            (2.2.3-3a)

                                            (2.2.3-3b)

                                            (2.2.3-3c)

The modal weighting in terms of their contribution to the RMS value can be expressed as a Modal Cost (MC) as follows:

                                             (2.2.3-4a)

                                                  (2.2.3-4b)

The RMS acceleration is unbounded, so there is not Modal Cost for acceleration output. Equations (2.2.3-3) and (2.2.3-4) can be used to develop a modal weighting that reflects the contribution of each mode to either the peak or RMS value of a transfer function matrix. While TAMKIT does not currently provide any means for evaluating these functions, they are fairly simple to calculate given mode shape data and provide a useful tool for identifying target modes.