36 G. K. Lopp et al. as a point of comparison. A companion paper provides an overview of the model updating and analysis of this same test article [10]. Finally, Sect. 4.4 discusses the relevant conclusions and potential paths forward. 4.2 Transfer Function Derivation Consider the system in Fig. 4.1a that represents a set of piezoelectric stack actuators mounted between the test article (System A) and the mounting assembly (System B). Both System A and System B consist of a displacement vector (xA, xB), amass matrix (MA, MB), and a stiffness matrix (KA, KB). To represent a piezoelectric stack actuator, Fig. 4.1b shows a lumpedparameter model that consists of two masses connected by a stack of piezoelectric wafers. Following a similar derivation for the stack actuator as supplied in [11], the electromechanical equations of motion that govern the actuator dynamics are ⎡ ⎣ mp,1 0 0 0 mp,2 0 0 0 0 ⎤ ⎦ Mp ⎧⎨ ⎩ ¨xp,1 ¨xp,2 ¨Qp ⎫⎬ ⎭ +⎡ ⎢⎣ kE p −k E p 0 −kE p k E p 0 θ −θ 1 ⎤ ⎥⎦ KE p ⎧⎨ ⎩ xp,1 xp,2 Qp ⎫⎬ ⎭ =⎧⎨ ⎩ −θ θ CS p ⎫⎬ ⎭ Vp +⎧⎨ ⎩ −gp,1 gp,2 0 ⎫⎬ ⎭ (4.1) These equations contain three degrees-of-freedom (DOFs): xp,1 and xp,2 are the displacements at each end of the actuator andQp is the electrical charge. Furthermore, Vp is the voltage applied to the stack, andgp,1 andgp,2 are the reaction forces at each end of the actuator; mp,1 andmp,2 are the lumped masses, kp is the combined stiffness of the piezoelectric stack and mechanical spring, θ is the electromechanical coupling, and Cp is the capacitance of the stack; and the subscripts Eand S denote quantities at a constant electric field and strain, respectively. Next, connect System A to System B using a total of Np actuators and apply compatibility and force equilibrium constraints, which results in the electromechanical equations of motion of the combined system: ⎡ ⎢⎣ MA+ Np j=1 Mp,AA,j 0 0 0 MB + Np j=1 Mp,BB,j 0 0 0 0 ⎤ ⎥⎦ ⎧⎨ ⎩ ¨xA ¨xB ¨Qp ⎫⎬ ⎭ +⎡ ⎢⎢ ⎣ KA+ Np j=1 KE p,AA,j − Np j=1 KE p,AB,j 0 − Np j=1 KE p,BA,j KB + Np j=1 KE p,BB,j 0 Np j=1 A,j − Np j=1 B,j I ⎤ ⎥⎥ ⎦ ⎧⎨ ⎩ xA xB Qp ⎫⎬ ⎭ =⎡ ⎢⎣ − Np j=1 t A,j Np j=1 t B,j CS p ⎤ ⎥⎦Vp (4.2) Fig. 4.1 Diagram of a test article excited with piezoelectric stack actuators. (a) Combined system. (b) Stack actuator
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