Dynamic Modeling of Stacked, Multi-Plate Flexible Systems

The vibration analysis and control of stacked-plate mechanical systems such as circuit board assemblies is an important technical problem that often requires a complete and accurate description of the open loop system dynamics prior to controller development. Often, a preliminary finite element model (FEM) of the assembly of interest is developed to estimate the dynamics of the system prior to the execution of a validation modal experiment. The results of this modal test must then be used to update the stiffness, mass and damping matrices to yield accurate transfer functions throughout the structure. This paper undertakes the mathematical development of a general, dynamic, multi-plate system. The work proceeds with the description of a three-plate system dynamic model and the physical test setup with test article used for validation and testing purposes. A model update is performed using differentiated velocity (in the frequency domain) data measured at discrete points on the test article with a laser Doppler vibrometer (LDV) and force measurements collected with an impedance head at a corner of the base plate. Using these data, accelerance frequency response functions (FRFs) were computed and the first eight flexible mode shapes were estimated and compared to the corresponding FEM shapes using both percent frequency difference and modal assurance criterion (MAC). A preliminary model update was performed with the addition/modification of discrete, rotary stiffness elements between plates and finished with a final update utilizing analytical model improvement (AMI) techniques employing a target modal matrix composed of both FEM modal vectors and expanded/smoothed experimental modal vectors. In addition, the experimental data was also used to estimate percent critical damping values for the first eight flexible modes and a Rayleigh damping model was developed for the system. The updated model was validated by comparing the magnitude/phase relationships of the base plate force input and several important response regions to that of the experimental results. The resonant frequencies of the first eight modes of the updated model differed from the physical system by no more than 0.3 % and the MAC values for the first eight modes all surpassed 0.97.