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The free vibration responses are divergent based on a complex time-domain damping model. The traditional step by step integration method cannot be used to calculate the time-domain responses. Based on the theoretic solution that eliminates the divergent term for the complex damping model, the equivalent viscous damping model is proposed and the corresponding average acceleration method is realized in this paper. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified, and the computational efficiency is high.

The role of the spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates is investigated in this study. The key achievement of this work is that the classical nonlocal elasticity theory is modified to take into account the dependence of nonlocal parameters on the varying of materials through the thickness of the functionally graded sandwich nanoplates. Hamilton’s principle is adopted to establish the governing equations of motion using a new inverse hyperbolic shear deformation theory. Numerical results are carried out via Navier’s solution for the fully simply supported rectangular functionally graded sandwich nanoplates, and they are compared with the available results to confirm the accuracy and efficiency of the proposed algorithm. Besides, the effects of some parameters such as the spatial variation of the nonlocal parameters, the aspect ratio, the side-to-thickness ratio as well as the power-law index on the free vibration of the nanoplates are also investigated cautiously. The results show that the variation of the nonlocal parameters plays a significant role in the free vibration of the functionally graded sandwich nanoplates, which is never investigated in the literature. The present methodology could be applied to the design and application of the micro/nanostructures.

This study investigates the application of physics-informed neural networks (PINN) for bending and free vibration analysis of three-dimensional functionally graded (TDFG) porous beams. The beam material properties are assumed to vary continuously in three dimensions according to an arbitrary function. The governing equations of motion are obtained using Hamilton's principle and solved by a PINN computational approach. The beam deflection is approximated with a deep feedforward neural network which its input is the spatial coordinate. The network parameters are trained by minimizing a loss function comprised of the governing differential equation and the boundary conditions. The beam natural frequency is considered as an unknown parameter in the governing equation; thus, it has to be obtained by solving an inverse problem. This procedure makes it possible to find higher modes’ natural frequencies, which is impossible according to the previous PINN methods. A systematic procedure for tuning the network's hyperparameters is done based on the Taguchi design of the experiment and the grey relational analysis. The PINN results are validated with analytical and numerical reference solutions. Effects of material distribution, elastic foundation and porosity factor, and porosity distribution type on the bending behavior and natural frequencies of TDFG beams are investigated.

Toroidal tuned liquid column dampers (TLCDs) are recently designed devices that extend the application of TLCDs to multidirectional vibration control. Toroidal TLCDs are promising in suppressing the horizontal vibration response of structures. This study further explores the potential and optimization scheme of toroidal TLCDs for multidirectional pitching vibration mitigation. Firstly, equations of motion for a toroidal TLCD-structure system in pitching motion are presented. A non-iterative analytical closed-form solution for calculating the dynamic response of the system under harmonic loading is developed. Subsequently, optimized frequency tuning ratio and flow resistance coefficient can be obtained. The optimization results theoretically confirm the direction-independent control performance of toroidal TLCDs. A design example of toroidal TLCDs is presented in detail as a reference. Finally, a pendulum rotational structure for TLCDs testing in pitching motion is constructed. The multidirectionality, effectiveness and robustness of toroidal TLCDs in mitigating pitching vibration are verified by free vibration tests.

Due to their large and increasing size and the corrosive nature of salt water and high wind speeds, offshore wind turbines are required to be more robust, more rugged and more reliable than their onshore counterparts. The dynamic characteristics of the blade and its response to applied forces may be influenced dramatically by rotor rotational speed, which may even threaten the stability of the wind turbine. An accurate and computationally efficient structural dynamics model is essential for offshore wind turbines. A comprehensive model that takes the centrifugal stiffening effect into consideration could make rapid and accurate decisions with live data sensed from the structure. Moreover, this can enhance both the performance and reliability of wind turbines. When a rotating blade deflects in its plane of rotation or perpendicular to it, the centrifugal force exerts an inertia force that increases the natural frequencies and changes the mode shapes, leading to changes in the dynamic response of the blade. However, in the previous literature, studies of centrifugal stiffening are rarely found. This study investigates the influence of centrifugal stiffening on the free vibrations and dynamic response of offshore wind turbine blades. The National Renewable Energy Laboratory (NREL) 5 MW blade benchmark was considered to study the effect of angular speed in the flap-wise and edge-wise directions. The results demonstrate that the angular speed directly affects the modal features, which directly impacts the dynamic response. The results also show that the angular velocity effect in the flap-wise direction is more significant than its effect in the edge-wise direction.

This paper investigates the free vibration and buckling behaviors of functionally graded graphene platelets (FG-GPLs) reinforced porous beam under axially variable loads. The internal pores and GPLs are either uniformly or non-uniformly distributed along the thickness direction. Halpin–Tsai micromechanics model is used to calculate the effective elastic modulus. The variation of Poisson’s ratio along the thickness and the relation between mass density and porosity coefficients are determined using mechanical properties of closed-cell solid under the Gaussian random scheme. The equilibrium equations are derived by Hamilton’s principles, and critical buckling load and dimensionless natural frequency are determined by Ritz formulation. Results revealed that buckling and free vibration behavior of the porous FG-GPL beam are influenced by the GPLs grading pattern and the type of axially varying load. Furthermore, the grading pattern of porosity has more influence on the buckling behavior compared to the free vibration behavior. It is also observed that buckling mode and the fundamental vibration mode of the porous FG-GPL are influenced by the loading conditions and remain unaffected by the grading pattern of porosities and GPLs.

Structures and their components experience substantially large vibration amplitudes at resonance, which can cause their failure. The scope of this study is the utilization of silicone-coated steel balls in concrete as damping aggregates to suppress the resonance vibration. The heavy steel cores oscillate with a frequency close to the resonance frequency of the structure. Due to the phase difference between the vibrations of the cores and the structure, the cores counteract the vibration of the structure. The core-coating inclusions are randomly distributed in concrete similar to standard aggregates. This mixture is referred to as metaconcrete. The main goal of this work is to validate the ability of the inclusions to suppress mechanical vibration through laboratory experiments. For this purpose, two small-scale metaconcrete beams were cast and tested. In a free vibration test, the metaconcrete beams exhibited a larger damping ratio compared to a similar beam cast from conventional concrete. The vibration amplitudes of the metaconcrete beams at resonance were measured with a frequency sweep test. In comparison with the conventional concrete beam, both metaconcrete beams demonstrated smaller vibration amplitudes. Both experiments verified an improvement in the dynamic response of the metaconcrete beams at resonance vibration.

Graphene platelets (GPLs)-reinforced metal foam structures enhance the mechanical properties while maintaining the lightweight characteristics of metal foams. Further bonding piezoelectric actuator and sensor layers on the surfaces of GPLs-reinforced metal foam beams enables active vibration control, greatly expanding their applications in the aerospace industry. For the first time, this paper investigates the vibration characteristics and active vibration control of GPLs-reinforced metal foam beams with surface-bonded piezoelectric layers. The constant velocity feedback scheme is used to design the closed-loop controller including piezoelectric actuators and sensors. The effects of the GPLs on the linear and nonlinear free vibrations of the beams are numerically studied. The Newmark- β method combined with Newton’s iteration technique is used to calculate the nonlinear responses of the beams under different load forms including harmonic loads, impact loads, and moving loads. Additionally, special attention is given to the vibration reduction performance of the velocity feedback control on the responses of the beam.

Purpose Flexoelectricity refers to the phenomenon of the link between electrical polarization and strain gradient fields in piezoelectric materials, particularly at the nanoscale. The goal of this study is to look into the bending and vibration properties of piezoelectric sandwich nanoplates in more detail, focusing on the flexoelectric effect and nonlocal strain gradient. Method The plate consists of three distinct layers, including two outside skin layers formed of piezoelectric smart material exhibiting the flexoelectric effect and an inner core layer consisting of functionally graded porous (FGP) material. The general equations of motion with improved accuracy are derived using Hamilton’s principle and the revised higher order shear deformation plate theory (RPT). The use of the Galerkin–Vlasov method allows for the determination of the static bending characteristics and particular vibration frequencies of plates with various boundary conditions. A computational program is implemented using the Matlab software platform. Then, the program’s correctness is assessed by doing a comparative analysis using previously published, dependable findings in specific instances of the model described in the article. Moreover, it is crucial to carry out a comprehensive analysis to evaluate the influence of various attributes on a system’s static bending response and free vibration. The parameters include the flexoelectric effect, nonlocal and strain gradient parameters, elastic foundation stiffness coefficient, porosity coefficient, and geometric conditions. Results The results of this research have practical consequences for the effective planning and management of similar systems, such as micro-electro-mechanical and nano-electromechanical devices. The results of this work are an important premise for developing more complex problems shortly, such as buckling analysis and dynamics problems. In addition, this is also a valuable reference for engineers designing models in engineering practice. Conclusion The article presents a set of numerical experiments that provide significant findings. The stiffness of the plate is increased by the parameter f 14 , and therefore, the maximum deflection decreases as f 14 rises. Nonlocal and strain gradient coefficients have contrary effects on the structure, so modulating these two coefficients is crucial for regulating the structure’s stiffness. The ratio between the thicknesses of the piezoelectric layer and the FGP core layer plays a crucial role in the study of increasing or decreasing the overall structure’s stiffness. An augmentation in the grading index will lead to a decrease in the overall stiffness of the structure, while an augmentation in the elastic base stiffness would result in an increase in the overall stiffness. Furthermore, the porosity coefficient, contingent upon the characteristics of the object, will induce either an augmentation or diminution in the magnitude of the inherent vibration frequency. However, it is quite probable that the displacement value of the plate will be augmented. Some new points in the article: The plate is composed of three layers with two skin layers made of piezoelectric smart material with flexoelectric effect and a core layer made of functionally graded porous material. The nonlocal and strain gradient coefficients change along the thickness direction and the porosity. Navier and Galerkin–Vlasov analytical methods are applied with many different boundary conditions.

A new type of functionally graded material (FGM) with material properties varying in two or three directions is needed to obtain materials with better mechanical properties and high-temperature resistance for use in the military, aerospace, automotive, and engineering structures. Considering the porosity that occurs during the production of these materials, it has become necessary to examine the free vibration and buckling behaviors. Therefore, this study investigated free vibration and buckling analysis of porous two-directional functionally graded (2D FG) beams subject to various boundary conditions. A high-order finite element based on parabolic shear deformation theory (PSDT) is proposed to solve this problem. Three types of porosity distributions were used in the study (FGP-1, FGP-2, and FGP-3). The governing equations are derived from Lagrange’s principle. The material change in the beam volume in both directions is defined by a power-law rule. The dimensionless fundamental frequencies and critical buckling loads are obtained for various boundary conditions, gradation exponents ( p x , p z ), porosity coefficient ( e ), porosity distribution, and slenderness ( L/h ). The numerical results obtained with the proposed higher-order finite element are compatible with the literature.