analysis of the local response from which the contact force can be theoretically derived. Yang J and Chun [4], Sun and Yang S [5], Suemasu et al [6], Liu and Swaddiwudhipong [7], Olsson [8], Zheng and Binienda [9], employed the Hertzian contact law in their formulations for understanding the impact events. Clerence Zener (1941) proposed an analytical solution for an isotropic plate impacted by spherical impactor based on infinite Hertzian and Kirchhoff-Love theory [10]. Olsson (2000) showed that small mass and large mass impactors of identical impact energy initiate different plate responses [11]. Lee et al. [3, 12] studied the transmission of energy flow in the plate from a structural intensity approach. However, we took a different approach and implement two feature extraction models that analyze the transmitted AUSs. We redefined the analytical solutions using constructed indices and designed an algorithm to predict the force history generated from an impact more accurately. 11.2 Analysis For a small displacement in a flat plate, we use the unified particle motion equation given by μ∇2uþ λþμ ð Þ∇∇ u ð Þþf ¼ρ p€uþF, ð11:1Þ where, u is the particle displacement vector, λ and μ are the Lame parameters, ρp is the density of the plate, f is the body force, and F is the externally transmitted force [13, 14]. The homogenous solution of the Eq. 11.1 (i.e. F ¼f ¼0) yields the solution of shear horizontal wave and coupled Lamb waves. However, in this analysis it is important to note that the externally applied load is non-uniform and is transmitted with a natural intensity factor e that is governed by the unique coupled properties of the impactor and the plate, F¼ e E1,2; ρ1,2; σ1,2; h; R ð □ v ρ2€u X; Y; Z; t ð ÞdV, ð11:2Þ where, Ep,i is the elastic modulus of the plate (p) and the impactor (i), likewiseρis the density, σ is the Poisson ratio, wis the plate thickness andRis the radius of the impactor. Our objective is twofold (1) accurately estimate the material properties of the impactors and (2) experimentally investigate Fand correlate with the TSSC sensor signals. Here we use the strength of the signal at a distance dfrom the impact, after degeneration of the displacement due to damped wave propagation and the geometric spreading of the energy using the following equation, where b is the function of coupled material properties. €ud ¼ 1 ffifffifid p e bd€u: ð11:3Þ The displacement and force history at the point of impact can be analytically approximated using Clarence Zener’s (CZ) theory for hard material impactors, while for soft materials like Teflon, there is huge discrepancy. Consequently, data driven indices are constructed for more accurate results. Accurate results are needed for damage prediction which depends on the force transmitted and dissipated in the structure as wave propagates particularly due to the coupled plate-impactor properties. The theory accords with the fact that force history of the impact event is dependent on coupled physical properties of the impactor and the plate, from which a dimensionless parameter λcan be obtained. We express all parameters in their respective domain, keeping the plate parameters constant in order to isolate impactor parameters. The theoretical solution is applicable to the impact events, whose effects are concluded before the arrival of the reflected waves from the plate boundaries. It is safe to assume that the impact events are elastic and complete after the impactor to plate collision, pressure exertion and retraction. However, the displacement on the plate at the point of impact remains perfectly inelastic until the boundary reflections return to the origin [10]. The first equation is the acceleration of the impactor modeled as a spherical body and is given by m d2s dt2 ¼ F, ð11:4Þ where s is taken as the displacement of the center of the sphere in contact with the plate, mis its mass, and Fis the plate reaction. Displacement of the mid-plane of the plate at the point of impact is said to be directly proportional to impulse and is expressed as 80 C. Agbasi and S. Banerjee
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