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Linear static response structural optimization has been developed fairly well by using the finite element method for linear static analysis. However, development is extremely slow for structural optimization where a non linear static analysis technique is required. Optimization methods using equivalent static loads (ESLs) have been proposed to solve various structural optimization disciplines. The disciplines include linear dynamic response optimization, structural optimization for multi-body dynamic systems, structural optimization for flexible multi-body dynamic systems, nonlinear static response optimization and nonlinear dynamic response optimization. The ESL is defined as the static load that generates the same displacement field by an analysis which is not linear static. An analysis that is not linear static is carried out to evaluate the displacement field. ESLs are evaluated from the displacement field, linear static response optimization is performed by using the ESLs, and the design is updated. This process proceeds in a cyclic manner. A variety of problems have been solved by the ESLs methods. In this paper, the methods are completely overviewed. Various case studies are demonstrated and future research of the methods is discussed.

Thin cylindrical shells are susceptible to cracking under long-term load and external impact, and it is of considerable scientific and technical value to investigate the nonlinear vibration response characteristics and monitor the health condition of the shell structure. Based on the Flügge shell theory, the nonlinear dynamic model for the thin cylindrical shell is established. By the partial Fourier transform combined with the residue theorem, the forced vibration generation and propagation mechanism of the thin cylindrical shell are investigated, and the analytical solution of forced vibration displacement in the space domain is obtained. Then, the local flexibility matrix is derived from the perspective of fracture mechanics, and the continuous coordination condition on both sides of the straight crack is constructed using the linear spring model. Combined with the wave superposition principle, the analytical approach for nonlinear vibration response is proposed to reveal the evolution law of vibration characteristics of the thin cylindrical shell with a straight crack, and then, a straight crack identification method based on natural frequency isolines and amplitude maximization methods is presented. Finally, the effect of various morphological information of the straight crack on the nonlinear vibration response characteristics of the thin cylindrical shell is studied in detail, and a numerical case is conducted to verify the effectiveness of the proposed straight crack identification method.

This paper explores the nonlinear dynamic responses and bifurcations of the truncated sandwich simply supported porous conical shell with varying thickness under 1:1 internal resonance. Two skins with carbon fiber and a core with porous aluminum foam, which has an exponentially variable thickness along the generator and various porosity distribution types along the core thickness, make up the sandwich shell structure with varying stiffness. The porous shell structure is affected by a combination of the in-plane load, transverse excitation, thermal stress and aerodynamic force, which is formulated by employing first-order piston theory with a modified term for curvature. By way of FSDT, von-Karman geometrical formulations, Hamilton’s principle and Galerkin procedure, the nonlinear dynamic formulations in ordinary differential form for the variable stiffness porous sandwich shell structure are identified. The averaged equations in polar and Cartesian coordinate forms for the sandwich structure under the combined circumstance of 1:1 internal resonance, first-order main resonance and 1/2 subharmonic resonance are determined by multiple-scale technique. The frequency-amplitude and force–amplitude characteristic curves, phase portraits, time history and bifurcation diagrams are exhibited by numerical simulation. The impacts of the damping coefficient, detuning parameters, temperature increment, transverse and in-plane excitations on the nonlinear dynamics and bifurcation behaviors of variable thickness sandwich porous conical shell are demonstrated.

Carbon (C) fluxes in semiarid grasslands subject to precipitation variability play a critical role in the terrestrial C cycle. However, how ecosystem C fluxes respond to variability in precipitation (both decreases and increases precipitation along a gradient) remains unclear. In this study, we conducted a three-year field experiment in a semiarid grassland, with six precipitation treatments (precipitation decreased by 70%, 50%, and 30% [P–70%, P–50%, and P–30%], natural precipitation [P+0%], and precipitation increased by 30% and 50% [P+30% and P+50%]) to examine how variations in precipitation influence ecosystem C fluxes, specifically focusing on gross ecosystem productivity (GEP), ecosystem respiration (ER), and net ecosystem CO 2 exchange (NEE). We found that both decreased and increased precipitation significantly altered the GEP (from –26% to 14%), but only decreased precipitation significantly reduced the ER and NEE (from 1% to 31%), relative to their values during natural precipitation. This suggests that ecosystem C fluxes are more sensitive to decreased precipitation, and respond nonlinearly to the precipitation gradient. Furthermore, structural equation modeling indicated that the soil water content was the primary controlling factor driving changes in ecosystem C fluxes. Our research underscores the nonlinear response of ecosystem C fluxes to changes in precipitation within semiarid ecosystems, particularly their sensitivity to extreme drought. Considering this nonlinear response, it is crucial to improve dynamic models of the C cycle and predict ecosystem responses to precipitation variability.

In this study, sacrificial components were incorporated into self-centering railway bridge piers to improve the lateral stiffness. The seismic response of this new detail was investigated. First, the method to compute the initial uplift moment of the self-centering pier is given. In addition, shaking table tests were conducted on a free-rocking pier without sacrificial components, which was used to validate a two-spring numerical model. Good agreement was obtained between the numerical results and experimental data. Furthermore, the validated model was employed to investigate the influence of sacrificial components on the seismic response of rocking piers. For this purpose, two models were developed, with and without sacrificial components. Nonlinear response history analysis was then performed on both models under three historical motions. The results showed that compared to the one without sacrificial components, the rocking pier with sacrificial components has comparable displacement at the top of the pier, and maximum uplift moment at high amplitude motion. Therefore, incorporating sacrificial components into the rocking pier can increase the lateral stiffness at service load and low amplitude frequent earthquakes but can produce comparable response at high seismic excitation. These results provide support for performance-based seismic design of self-centering rocking piers.

Purpose When an in-plane harmonic loading is applied to a plate, the linear dynamic instability region (DIR) formed, gives only the range of frequencies where the plate becomes unstable, however, it doesn’t give any information about the transverse deformations. Hence, a nonlinear time history analysis is required to capture the actual time-varying deformations along with the nonlinear frequency response analyses to capture the actual frequency-varying deformations in the dynamic instability zone. Method In this study, a nonlinear mathematical model is developed within the finite element framework to analyze both the nonlinear frequency response and the time-history behavior of laminated composite plates attached with composite laminated stiffeners. The plates are subjected to time-dependent (harmonic) in-plane edge loading. An isoparametric finite element formulation is used to model the plate and stiffener components. To gain a comprehensive understanding of the instability characteristics of the stiffened plates, the Newmark-β method is employed for solving the linear and nonlinear dynamic equilibrium equations. Furthermore, the Incremental Harmonic Balance (IHB) method is utilized to trace the nonlinear frequency response path of the stiffened system. Results and Conclusions The results show that the dynamic responses under linear theory become unbounded within the DIR due to the lack of stabilizing mechanisms. Nonlinear theory introduces displacement-dependent restoring forces and geometric nonlinearities, which produce stabilizing effects and result in bounded periodic responses. Increasing the stiffener count enhances plate stiffness, causing the dynamic instability region to shrink and shift toward higher frequencies. This leads to reduced vibration amplitudes and improved stability across a wider frequency range. Overall, the nonlinear dynamic responses remain bounded and stable, demonstrating the system’s increased resilience to dynamic excitations.

In this study, a nonlinear energy sink (NES) is used to suppress the nonlinear aeroelastic response of graphene platelet reinforced composite (GPLRC) lattice sandwich plates in a supersonic airflow for the first time. The face sheets and lattice core trusses of lattice sandwich plates were reinforced with graphene platelets (GPLs). The effective elastic modulus of the GPLRC was solved using the Halpin–Tsai micromechanical model, and Poisson's ratio, mass density, and coefficient of thermal expansion were calculated using the rule of mixtures. Kirchhoff plate and first-order shear deformation theories were used separately to model the face sheets and lattice core layer of the structure. The nonlinear strain–displacement relationship was derived using the von Karman large-deformation theory. The aerodynamic load was simulated using the piston theory. The motion equations of the supported GPLRC lattice sandwich plates with an NES under supersonic flow were derived using the Lagrange equation and the assumed mode method. The nonlinear aeroelastic responses of the GPLRC lattice sandwich plate system coupled with an NES were solved using Newmark direct integration combined with the Newton–Raphson iteration technique. Finally, a detailed study of the effects of the NES on the suppression of the flutter behavior of GPLRC lattice sandwich plates was carried out. The results showed that within specific mass, damping, and nonlinear stiffness ranges, the NES could effectively suppress the nonlinear aeroelastic response of the GPLRC lattice sandwich plates.

This paper presents a comprehensive study on the nonlinear resonance response of the high-speed spindle system supported by preloaded ball bearings. Firstly, the nonlinear stiffness characteristic of the ball bearing-rotor system is studied based on the analysis of the contact nonlinearity and geometric nonlinearity generated by supporting ball bearings. Secondly, a simplified rigid rotor system is built to discuss its nonlinear “soft-spring” and “hard-spring” stiffness effects across varying speeds and external loads. Thirdly, the nonlinear resonance response mechanism of the spindle system near its 1st-order equivalent linear resonance frequency is predicted, and the effect of “soft-spring” and “hard-spring” stiffness characteristics on its resonance response mechanism is detailed analyzed. Finally, a hybrid-supported experimental spindle system is built to verify the theoretical prediction results on the nonlinear resonance response of the preloaded ball bearing-rotor system. The results show that the resonance response curves of the rotor system exhibit the leftward “bending” characteristic corresponding to the “soft-spring” stiffness effect under the low speed and light load ranges, while exhibit the rightward “bending” characteristic corresponding to the “hard-spring” stiffness effect under the high speed and heavy load ranges.

The calculation of observable averages in non-equilibrium regimes is one of the most important problems in statistical physics. Using the Hamiltonian approach of nonlinear response theory, we obtain a series expansion of the average excess work and illustrate it with specific examples of thermally isolated systems. We report the emergence of non-vanishing contributions for large switching times when the system is subjected to strong driving. The problem is solved by using an adapted multiple-scale method to supress these secular terms. Our paradigmatic examples show how the method is implemented generating a truncated series that obeys the Second Law of Thermodynamics.

This paper presents the development of a novel nonlinear dynamic model for partially and fully submerged rod pendulums. The pendulum undergoes oscillations in a quiescent water medium, wherein it experiences nonlinear hydrostatic and hydrodynamic forces that are integrated into the dynamics of the system. The analysis of the fixed points of the equation of motion is conducted to investigate the nonlinear static stability of the system and determine the static equilibrium angle. The static equilibrium characteristics exhibit a Pitchfork bifurcation, wherein the control parameter is the rod density ratio. The investigation also encompasses the examination of the impact of the depth and diameter ratios on the nonlinear static behavior. The investigation of the nonlinear frequency response to sinusoidal external torque applied to the hinge is subsequently carried out for the various potential equilibrium configurations. The investigation focuses on the influence of the system parameters, specifically the density, diameter, and depth ratios, across all configurations. The findings reveal that these parameters have varying effects.