4 Modal Testing with Piezoelectric Stack Actuators 37 where Qp and Vp are vectors containing the electrical charges and voltages of each actuator, respectively; Mp,AA,j and Mp,BB,j are the mass contributions of the jth actuator to the connection DOFs of Systems Aand B, respectively; KE p,AA,j, KE p,BB,j, KE p,AB,j, and KE p,BA,j are the stiffness contributions of the jth actuator to the connection DOFs of Systems A and B with the latter two quantities coupling these two systems together; A,j and B,j are the matrices containing the electromechanical coupling for the jth actuator; and CS p is a diagonal matrix containing the capacitance of each actuator. The mechanical portion of these equations shows that the mass of the stack actuators is added to both Systems A and B at the connection DOFs and that the actuator stiffness couples these two systems together. Furthermore, the applied voltage appears as a mechanical force acting on the combined system at these same connection DOFs and with a direction in line with the actuators. Finally, the electrical portion of these equations relates the current required to drive the stack actuators with the prescribed voltages. Next, transform the above equations into the modal domain using the transformation xA xB = Tot,A Tot,B η (4.3) where Tot are the mode shapes identified from solving the eigenvalue problem associated with the mechanical portion of Eq. 4.2. Next, apply this transformation to Eq. 4.2 to obtain the modal equations of motion: ˜M¨η+ ˜C˙η+ ˜KEη = − t Tot,A Np j=1 t A,j + t Tot,B Np j=1 t B,j ˜ Vp (4.4) where ˜Mand ˜KE are the modal mass and stiffness matrices, ˜Cis the modal damping matrix with quantities obtained from a proportional-damping model, and ˜ is a matrix of electromechanical coupling quantities corresponding to each mode (rows of the matrix) and each actuator (columns of the matrix). This electromechanical coupling also corresponds to the modal participation factors with respect to the applied voltages. For example, if the electromechanical coupling is large for a given mode and actuator, an applied voltage will significantly excite the mode. Finally, convert the modal equations to the frequency domain and perform some algebra to obtain the matrix of transfer functions, Hηv, relating the applied voltages to the modal responses: η = ˜KE −ω 2 ˜M+iω˜C −1 ˜ Hηv Vp (4.5) For a single mode, this transfer function is Hηv,r,s = ηr Vp,s = (1/mr)˜θr,s ω2 r −ω2 +i2ζrωrω (4.6) which is similar in form to the transfer functions for a mechanical input, but with the modal electromechanical coupling factors serving as the modal participation factors. As such, typical modal curve-fitting algorithms can estimate the modes of the combined system using the transfer functions relating the voltages to the vibration responses. In many instances, we will prescribe the same voltage, Vp,0, to each actuator, which reduces the voltage vector toVp =1Vp,0 and results in the transfer function for a single mode Hηv,r = ηr Vp,0 = (1/mr) Np s ˜θr,s ω2 r −ω2 +i2ζrωrω (4.7) In this scenario, the modal participation will reduce to zero if the motion at the actuator-connection DOFs for System A and System B are the same (e.g., a rigid-body mode) or if the motion at the actuator-connection DOFs for System A and System B are equal in magnitude and opposite in sign (e.g., a plate twisting mode with actuators located on each corner).
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