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Item Nonlinear vibration and instability of a randomly distributed CNT-reinforced composite plate subjected to localized in-plane parametric excitation(Elsevier, 2022-01) Kumar, Rajesh; Patel, Shuvendu Narayan; Watts, GauravThis study presents a semi-analytical formulation for the nonlinear vibration and dynamic instability of a randomly distributed carbon nanotube-reinforced composite (RD-CNTRC) plate. Three cases of localized in-plane periodic loadings are studied. The analytical stress fields within the RD-CNTRC plate for all the in-plane stress components (σij, (i, j = x, y)) are developed by solving the in-plane elastic problem using Airy's stress approach. The effective mechanical properties of the RD-CNTRC plate are evaluated by the Eshelby-Mori-Tanaka technique. The plate is modeled based on higher-order shear deformation theory (HSDT) in conjunction with the von-Kármán nonlinearity. Using Hamilton's principle, the governing partial differential equations (PDEs) are derived, whose approximate solution is sought, referring to the Galerkin method. The resulting nonlinear ODEs are solved using the Incremental Harmonic Balance (IHB) Method to compute the nonlinear vibration response of the RD-CNTRC plate. Further dropping the nonlinear terms, these ODEs are solved by Bolotin's method to trace the instability region. The proposed semi-analytical method is an effective strategy for studying the influence of different parameters such as agglomeration models, CNT mass fraction, pre-loading, and boundary conditions on the nonlinear vibration and dynamic instability characteristics of the RD-CNTRC plates. The reduced computational effort allows the design phase to be supported in selecting parameters when designing RD-CNTRC plates with stability and vibration requirements.Item A semi-analytical approach for instability analysis of composite cylindrical shells subjected to harmonic axial loading(Elsevier, 2022-09) Kumar, RajeshIn this article, nonlinear vibration and dynamic stability analyses of simply supported laminated composite circular cylindrical shells subjected to periodic edge loading are carried out. A third-order shear deformation shell theory that considers all the nonlinear terms in all five kinematic parameters and rotary inertia is used to develop the present mathematical model so that the model is also valid for thick cylindrical shells. Hamilton’s principle, an energy-based approach, is used to obtain the governing partial differential equations (PDEs) of motion of the cylindrical shell. Further, these equations are reduced into ordinary differential equations by employing Galerkin’s method. The incremental harmonic balance (IHB) method in conjunction with the pseudo-arc-length method is used to obtain the frequency-amplitude response of the system. For obtaining the zone of instability regions, Bolotin’s method is adopted. For more practical significance, analysis of results is also extended by considering damping into account for the composite cylindrical shells. Time history response and phase portrait are plotted by adopting Newmark-beta method. The effects of the static load factor, dynamic load factor, modal damping coefficient, and stacking sequence on nonlinear vibration, instability regions and time history responses are also examined.Item Size-dependent nonlinear vibration and instability of a damped microplate subjected to in-plane parametric excitation(Elsevier, 2023-03) Kumar, RajeshThe semi-analytical framework for nonlinear vibration and dynamic instability of a damped microplate under periodic parametric excitation is presented. The microplate is modeled using the higher-order shear deformation theory (HSDT) in conjugation with the modified strain gradient theory (MSGT). The governing partial differential equations of motion are obtained using Hamilton’s principle and further solved using Galerkin’s method. The ordinary differential equations without the nonlinear terms are solved using Bolotin’s method to obtain the dynamic instability region. A combination of the incremental harmonic balance (IHB) and the arc-length continuation methods is used to obtain the nonlinear forced vibration response. The effect of initial displacements on the steady-state response of the microplate is discussed. The Newmark- method is used to obtain the time-history response plots. A comparison of results with those obtained from modified couple stress theory (MCST) and classical continuum theory (CCT) are examined. The effect of various parameters, such as the size of a plate, damping coefficient, static and dynamic loading factors, different boundary conditions, and different loading profiles, on the width of linear and nonlinear instability regions, are also studied.Item Nonlinear dynamic instability of laminated composite stiffened plates subjected to in-plane pulsating loading(Taylor & Francis, 2023-06) Patel, Shuvendu Narayan; Watts, Gaurav; Kumar, RajeshA nonlinear finite element dynamic instability analysis of laminated composite stiffened plates subjected to in-plane harmonic edge loading is presented in this article along with the linear and nonlinear dynamic response study. The eight-noded isoparametric degenerated shell element and a compatible three-noded curved beam element are used to model the stiffened plates. Bolotin method is applied to analyze the dynamic instability regions in linear case. Incremental Harmonic Balance (IHB) method is applied to solve the nonlinear frequency response equations and Newmark-β method is used to solve the linear and nonlinear time history response equations.Item Dynamic instability of composite skew plates using boundary characteristic orthogonal polynomials(World Scientific, 2014) Kumar, RajeshDynamic instability analysis of laminated composite skew plate for different skew angles subjected to different type of linearly varying in-plane loadings is investigated. The analysis also includes the instability of skew plate under uniform bi-axial in-plane loading. The skew plate structural model is based on higher order shear deformation theory (HSDT), which accurately predicts the numerical results for thick skew plate. The total energy functional is derived for the skew plates from total potential energy and kinetic energy of the plate. The strain energy which is the part of total potential energy contains membrane energy, bending energy, additional bending energy due to additional change in curvature and shear energy due to shear deformation, respectively. The total energy functional is mapped into a square plate over which a set of orthonormal polynomials satisfying the essential boundary conditions is generated by Gram–Schmidt orthogonalization process. Different boundary conditions of skew plate have been correctly incorporated by using Rayleigh–Ritz method in conjunction with Boundary Characteristics Orthonormal Polynomials (BCOPs). The boundaries of dynamic instability regions are traced by the periodic solution of governing differential equations (Mathieu type equations) with period T and 2T. The width of instability region for uniform loading is higher than various types of linearly varying loadings (keeping the same peak intensity). Effect of various parameters like skew angle, aspect ratio, span-to-thickness ratio, boundary conditions and static load factor on dynamic instability has been investigatedItem Dynamic instability of damped composite skew plates under non-uniform in-plane periodic loading(Elsevier, 2015-11) Kumar, RajeshIn this paper, the non-linear dynamic instability of damped composite skew plates under non-uniform in-plane periodic loadings is studied using analytical methods. The structural model for composite skew plate is based on first order shear deformation theory considering von-Kármán geometric nonlinearity. The analytical expressions for stress distributions within the composite skew plate subjected to three different types of non-uniform in-plane loadings are developed by solving, plane elasticity problem. Subsequently, using these stress distributions and via Hamilton principle, the equations governing the dynamic instability of composite skew plates are derived in terms of displacement (u–v–w) and rotation variables. The generalized differential quadrature method is used to reduce the governing partial differential equations into a set of ordinary differential (Mathieu type) equations. The dynamic instability regions are traced by the periodic solutions (with period T and 2T) to Mathieu-type differential equations. Numerical results are presented to demonstrate the influence of skew angles, boundary conditions, non-uniform loadings and damping on the dynamic instability regions. Furthermore, the characteristic features of linear and nonlinear response in stable and unstable regions are studied. This brings out various features of instability problem such as existence of beats, effect of non-linearity and damping on response and its dependence on forcing frequency.Item Analytical approach for dynamic instability analysis of functionally graded skew plate under periodic axial compression(Elsevier, 2017-09) Kumar, RajeshAnalytical studies on the dynamic instability analysis of a functionally graded (FG) skew plate subjected to uniform and linearly varying in-plane periodic loadings with four different types of boundary conditions are presented. The total energy functional of the FG skew plate is formulated based on Reddy's third order shear deformation theory (TSDT) and this functional is mapped from the physical domain to computational domain using transformation rule. The boundary characteristics orthonormal polynomials (BCOPs) are generated for different boundary conditions using Gram–Schmidt process, which satisfy the essential boundary conditions of skew plates in the computational domain. The energy functional is converted into a set of ordinary differential equations (Mathieu–Hill equations) using Rayleigh–Ritz method in conjunction with BCOPs. The solution of Mathieu–Hill equations describes the dynamic instability behavior of skew plate. The instability regions are traced using Bolotin method. The effect of skew angles, power-law distributions, span-to-thickness ratios, aspect ratios, boundary conditions and static load factors on the instability region of FG skew plates are presented. The result indicates that the width of instability region become narrow with the increase in skew angle. Moreover, the time history response and corresponding phase plot in the unstable and stable region is studied to identify the instability behavior such as existence of beats, bounded and unbounded response, and effect of forcing amplitude and its frequency on the response.Item Instability and Vibration Analyses of Functionally Graded Carbon Nanotube–Reinforced Laminated Composite Plate Subjected to Localized In-Plane Periodic Loading(ASCE, 2021-11) Kumar, Rajesh; Patel, Shuvendu NarayanCarbon nanotubes (CNTs) have attracted many researchers during the last three decades due to their versatile nature and excep-tional mechanical properties. In this study, a functionally graded CNT-reinforced laminated composite (FG-CNTRLC) plate subjected todifferent types of localized in-plane loadings was analyzed semianalytically to determine its dynamic instability and nonlinear vibrationcharacteristics. The effective mechanical properties of the FG-CNTRLC plate were estimated using the extended rule-of-mixture technique.The FG-CNTRLC plate was modeled based on higher-order shear deformation theory (HSDT) in conjunction with von Kármán nonlinearity.The distribution of prebuckling stresses within the plate due to localized in-plane loading was estimated by solving the in-plane elasticityproblem using Airy’s stress approach. The nonlinear governing partial differential equations (PDEs) of the FG-CNTRLC plate were derivedusing Hamilton’s principle. The Galerkin method was used to convert these nonlinear PDEs to the nonlinear ordinary differential equations(ODEs). The nonlinear ODEs were solved using the incremental harmonic balance (IHB) method to obtain the nonlinear vibration responseof the FG-CNTRLC plate. After dropping the nonlinear terms, the linear ODEs were solved by the Bolotin method to trace the dynamicinstability regions. The effect of different parameters such as volume fraction of CNTs, different types of localized in-plane loadings, typesof CNTs distribution, the static and dynamic load factor on the dynamic instability regions, and the nonlinear vibration characteristics of theFG-CNTRLC plate, were examinedItem Nonlinear vibration and instability of a randomly distributed CNT-reinforced composite plate subjected to localized in-plane parametric excitation(Elsevier, 2022-01) Patel, S. N.; Kumar, RajeshThis study presents a semi-analytical formulation for the nonlinear vibration and dynamic instability of a randomly distributed carbon nanotube-reinforced composite (RD-CNTRC) plate. Three cases of localized in-plane periodic loadings are studied. The analytical stress fields within the RD-CNTRC plate for all the in-plane stress components (σij, (i, j = x, y)) are developed by solving the in-plane elastic problem using Airy's stress approach. The effective mechanical properties of the RD-CNTRC plate are evaluated by the Eshelby-Mori-Tanaka technique. The plate is modeled based on higher-order shear deformation theory (HSDT) in conjunction with the von-Kármán nonlinearity. Using Hamilton's principle, the governing partial differential equations (PDEs) are derived, whose approximate solution is sought, referring to the Galerkin method. The resulting nonlinear ODEs are solved using the Incremental Harmonic Balance (IHB) Method to compute the nonlinear vibration response of the RD-CNTRC plate. Further dropping the nonlinear terms, these ODEs are solved by Bolotin's method to trace the instability region. The proposed semi-analytical method is an effective strategy for studying the influence of different parameters such as agglomeration models, CNT mass fraction, pre-loading, and boundary conditions on the nonlinear vibration and dynamic instability characteristics of the RD-CNTRC plates. The reduced computational effort allows the design phase to be supported in selecting parameters when designing RD-CNTRC plates with stability and vibration requirements.