60 H. C. Denning et al. approach a daunting task. Finite element analysis (FEA) has been used to model shock tests and shock loading [6, 7, 8] and has been used in several recent works to model resonant plate tests [3, 4, 9, 10, 11]. Efforts have been made to use predict the SRS of a component in a resonant plate shock test, showing promising results [11, 5]. This work explores the feasibility of using finite element analysis and dynamic substructuring to predict the shock response of a component in a resonant plate shock test. Experimental dynamic substructuring has received increasing attention in recent years in order to accelerate test and analysis work [12]. Dynamic substructuring allows for accurate modeling of a large system by breaking it into component parts. Each subsystem may be modeled using traditional finite element models or a model may be derived experimentally, depending on the application of interest [13]. Subsystem models can then be assembled to accurately predict the response of the total system, avoiding the need to perform a test on the assembly. Jacobson et al. [5] appear to have been the first to use substructuring to predict the response of a component in a resonant plate shock test. They used LaGrange-Multiplier Frequency Based Substructuring (LM-FBS) to predict and optimize the response of a metal block in a multi-axis shock test. Substructuring allowed them to evaluate 110 possible configurations of their test and select the one that would theoretically produce the best multi-axis SRS. However, they also noted that the substructuring predictions under-predicted most of the resonances, as well as the knee frequency in the SRS, and showed that this was due to the way in which the interface was modeled. This work seeks to build on those prior works by exploring a few questions. First, how many modes must be accurately modeled to predict the SRS in a resonant plate test? Shock has the potential to excite many modes, and it is unclear how many modes of each subcomponent would need to be measured or modeled in order for substructuring predictions to have adequate accuracy. Second, how sensitive is the SRS of a component to the dynamics of the component itself, as opposed to the dynamics of the resonant plate? It is hoped that the case studies presented here can lay the foundation needed to use modal substructuring to predict the SRS of an arbitrary component in a resonant plate test. This paper will first review the construction of an accurate model of a resonant plate shock response using FEA. Abaqus will be used to create all FE models presented in this work. A model is constructed and the modes of the plate are computed. A study is performed to determine the mesh density is required to achieve convergence. A MATLAB mathematical model is then constructed to derive the response and hence the shock response spectrum (SRS) of the plate. The model is then interrogated to determine the appropriate number of modes to use in this analysis. This reveals that the SRS is dominated by the symmetric bending modes of the plate, as these are these are the modes most strongly excited by the shock pulse. Higher modes have a minor effect unless the damping is very light. We then explore the influence of the device under test (DUT) on the shock response spectrum. A single degree-of-freedom (SDOF) system is coupled to the resonant plate at its center using substructuring and the fixed-base resonant frequency of the DUT is varied to see how that affects the SRS. The mass of the DUT is also varied, while keeping the resonant frequency constant, to explore its effect. In general, the SRS on the plate (i.e. at the base of the DUT) changes very little as the parameters of the DUT are varied. The following section reviews the methods used, in particulars of the finite element model and the analytical solution for the response of a system to an impulsive load. Section presents the results of the various case studies. Methods Finite Element Analysis (FEA) software has been used increasingly in the last several years to aid in predicting the shock response of resonant plate and resonant bar tests [3, 9]. For our research, the Abaqus package was utilized to simulate the modes of a 1000Hz resonant plate and these modes were imported into MATLAB to compute the plate’s response. The model was then augmented by adding a device under test (DUT), attached to the plate at its center, and the response and SRS computed as various parameters were varied. Developing the Abaqus Model A resonant plate test is used in the testing and qualification of parts. Its has proved especially useful in replicating pyroshock environments for aerospace applications. The test is performed by striking a heavy metal plate with a projectile. The force pulse input by the projectile is shaped by placing a ”programmer” material between the projectile and plate. The modes of the plate also serve to accentuate certain frequencies in the response. This research is concerned only with the standard case where the impact and the DUT are centered on opposite sides of the plate. Several steps were taken to create an accurate model of this test. Our plate model was created first without any device under test attached to verify the expected behavior of the plate when subjected to the impact. The model was setup using a standard508×508×50.8mm plate made of 6061 aluminum [4]. No
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