## 2 Conceptual Model  
To introduce the proposed amplification mechanism, consider a simple transducer model, represented by an actuated point mass, impacting a rigid beam supported by a multi-axis spring, shown schematically in Figure 1. All reference displacements are measured from their nominal rest position under gravity. The dynamics of the point mass represent an idealized mass-spring-damper approximation of a piezoelectric transducer,  
\[m_{p}\bar{x}_{p}+c_{p}\hat{x}_{p}+k_{p}x_{p}=F_{eq}(t) \tag{1}\]  
where \(x_{p}\) is the displacement of the tip of the piezoelectric element (i.e., a piezoelectric bender or stack), \(m_{p}\) represents its effective mass, \(k_{p}\) is its effective stiffness and \(c_{p}\) is a damping constant. \(F_{eq}(t)\) represents the input forcing generated by the piezoelectric material as a function of time, \(t\), which depends on an applied voltage, \(V\), one or more piezoelectric coefficients, \(d_{ij}\), and the geometry of the element. For example, for a piezoelectric bimorph bender, shown schematically in Figure 2, input force may be approximated from the blocking force for the bender as a function of voltage, or  
\[F_{eq}(t)\approx 2d_{31}\frac{V(t)}{t_{p}}\big{(}w_{p}t_{p}E_{p}\big{)}\frac{g}{ L_{p}/2} \tag{2}\]  
where electromechanical coupling is primarily through the \(d_{31}\) piezoelectric constant, \(t_{p}\), \(w_{p}\), and \(L_{p}\) are the thickness, width, and length of the piezoelectric layers, \(E_{p}\) is the elastic modulus of the piezoelectric layer, and \(\bar{y}\) is the distance from the midline of each piezoelectric layer to the neutral axis of the bender (assuming opposite polarities for the top and bottom piezoelectric layers). The piezoelectric element interacts with a compliant structure capable of elastic deformation in translation, \(x_{L}\), and rotation, \(\theta_{L}\), with dynamics:  
\[\left[ \begin{array}{cc}m_{L}&0\\ 0&I_{L}\end{array} \right]\left[ \begin{array}{c}\bar{x}_{L}\\ \bar{\theta}_{L}\end{array} \right]+\left[ \begin{array}{cc}c_{x}&0\\ 0&c_{\theta}\end{array} \right]\left[ \begin{array}{c}\hat{x}_{L}\\ \dot{\theta}_{L}\end{array} \right]\] \[+\left[ \begin{array}{cc}k_{xx}&k_{x\theta}\\ k_{\theta x}&k_{\theta\theta}\end{array} \right]\left[ \begin{array}{c}x_{L}\\ \theta_{L}\end{array} \right]=\mathbf{0} \tag{3}\]  
where \(m_{L}\) and \(J_{L}\) are the equivalent mass and rotational inertia of the mechanism, \(c_{x}\) and \(c_{\theta}\) are damping constants in each axis, and \(k_{xx}\) and \(k_{\theta\theta}\) are spring constants between translational force and translational displacement, respectively, and \(k_{\theta x}\) and \(k_{x\theta}\) are stiffness matrix cross-axis terms. During simulation, the output mechanisms is treated as being under negligible external load except during impact, which is modeled by discrete changes in velocity states. Impact is assumed to occur whenever the piezoelectric actuator displacement would pass below the underlying load beam, taken to be a distance \(r\) from the compliant spring connection with nominal gap \(d\) (i.e., impact occurs if \(x_{p}>d+x_{L}-r\theta_{L}\)). For a very simple model of impact, the two structures are taken to be approximated by interaction according to a simple coefficient of restitution (CoR) model while obeying conservation of linear and angular momentum, or  
\[\hat{x}_{p}^{+}-\left(\hat{x}_{L}^{+}-r\hat{\theta}_{L}^{+}\right) = -\alpha\left[\hat{x}_{p}^{-}-\left(\hat{x}_{L}^{-}-r\hat{\theta}_ {L}^{-}\right)\right] \tag{4}\] \[m_{p}\hat{x}_{p}^{+}+m_{L}\hat{x}_{L}^{+} = m_{p}\hat{x}_{p}^{-}+m_{L}\hat{x}_{L}-\] (5) \[-rm_{p}\hat{x}_{p}^{+}+J_{L}\hat{\theta}_{L}^{+} = -rm_{p}\hat{x}_{p}^{-}+J_{L}\hat{\theta}_{L}^{-} \tag{6}\]  
where superscripts '+' and '-' denote state velocities after and before impact, respectively. Equation (4) is the CoR relationship, (5) the linear momentum balance, and (6) the angular momentum balance, measured in reference to the output spring location. Figure 1: Schematic model of transducer impacting elastic amplification mechanisms.