6 Shock Response of an Antenna Structure Considering Geometric Nonlinearity 49 pyroshock which is the response of a system to high-frequency stress waves one example of mechanical shock. Generally, pyroshock is generated as a result of explosive charges in order to separate two stages of a rocket [8]. Moreover, gunfire shock is a repetitive wave originated from artillery shooting of military vehicle. Consequently, the antenna structure should withstand these mechanical shocks originating from various environmental effects. In literature, studies on the nonlinear dynamic characteristics of antenna structures exposed to mechanical shocks are limited since most researchers have investigated modal and random vibrations analyses of the antenna structure. Concerning the linear dynamic analyses of antenna structure, the simulations were performed by commercial finite element programs such as ABAQUS® and NASTRAN®. Static and dynamic analyses of dipoloop antenna radome were simulated by a linear finite element analysis by Reddy and Hussain [9]. Mechanical shock analysis was conducted on ABAQUS®. In this study, only stresses were evaluated on the dipoloop antenna radome. Lopatin and Morozov [10] studied the free vibration of thinwalled composite spoke of an umbrella-type deployable space antenna. The composite spoke of the deployable space antenna was modeled as a cantilever beam via including effects of transverse shear. On the other hand, the nonlinear dynamic response of the antenna structure under dynamic loads is characterized by limited number of researchers. Random and modal analyses of a gimbaled antenna including gap nonlinearity resulting from small clearances in the joints are studied by Su [11] where the nonlinearity is linearized and then the resulting linear systems is solved commercial finite element software. Moreover, Sreekantamurthy et al. [12] investigated static and dynamic loads such as inflation pressure, gravity and pretension loads on a parabolic reflector antenna by using commercial finite element software. In their work, geometric nonlinearity was included into the model, since the deformation of parabolic reflector antenna was large. Inherently, antenna structures have a larger dimension in longitudinal direction. When they receive high mechanical shock such as ballistic and pyrotechnical shocks, nonlinear effects play an important role on shock response of antenna structures. Although the nonlinear dynamic characteristics, under mechanical shock are different from the linear ones, there appears almost no study on this specific topic. However, especially in micro and nanoscale areas, many researchers investigated the dynamics response of micro electro mechanical systems, micro beams, micro switches and so forth which are under mechanical shock even by considering the nonlinear effects. As an initial attempt, some authors used single degree of freedom assumption to get a rough estimation of the dynamic response of micro systems. For example, Younis et al. [13] studied the performance of capacitive switches modeled as a single degree of freedom (SDOF) system under mechanical shock through including the effects of squeeze-film damping and electrostatic forces. Moreover, Li and Shemansky [14] treated the micro-machined structure as a single degree of freedom system as well as a distributed parameter model. For more accurate analysis, many authors used continuous beam models to simulate the response of micro systems to a mechanical shock. As an example, Younis et al. [15] investigated the simultaneous effects of mechanical shock and electrostatic forces on microstructures simulated as cantilever and clamped-clamped beams. In this particular study, reduced order model results based on Galerkin’s Method were compared with the ones obtained from commercial finite element software. Due to the large deformation of micro systems resulting from the applied mechanical shock, some researchers included nonlinearity to the models to predict the dynamic behavior in real life. For instance, Younis and Arafat included both geometric and inertia nonlinearities into their studies while analyzing the response of the cantilever microbeam activated by mechanical shock and electrostatic forces [16]. In their work, they analyzed the effects of cubic geometric and inertia nonlinearities on the cantilever microbeam by using reduced order model which is based upon Galerkin’s Method. In another study of Younis et al. [17], the response of the clamped-clamped microbeam was investigated through using four modes in the Galerkin based reduced order model including geometric nonlinearity. Moreover, Younis et al. [17] studied the effects of shape of shock pulse and package on the response of microbeam and validated the results via commercial finite element software. Furthermore, some researchers employed approximate solutions to the response of systems under mechanical shock through frequency domain approaches rather than time domain approach which is computationally expensive. As an example, Liang et al. [18] estimated shock response of the mast in ships using frequency domain method such as square root of the sum of squares (SRSS), complete quadratic combination method (CQC), naval research laboratory method (NRL) and absolute summation method (ABS). Alexander [19] mentioned the frequency domain methods which are applied to the nonlinear systems as well. Younis and Pitarresi [20] emphasized synthetic methods utilizing the static response and shock spectrum based on maximum responses of many single degree of freedom systems. In this study [20], linear and nonlinear response of microbeam found in synthetic method and Galerkin-based reduced order method employing six modes were compared in terms of different values of shock amplitude. Mechanical shock excitation is inherently applied to the base of structures. In literature, mechanical shock was simulated for continuous systems by base excitation which is either applied to the fixed boundary condition [21, 22] or distributed force applied through the structure [15–17, 20].
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