Vibration Equation Research Articles

Forced vibration assessment of moderately thick carbon nanotube-reinforced composite plates

Carbon nanotubes, distributed in composite sheets with various arrangements, can lead to the development of materials with functionally graded properties. In the present study, the method of spectral components was used to investigate the free vibration frequencies and dynamic responses of single-layer sheets reinforced with carbon nanotubes. The spectral component method offers significant advantages for high-frequency dynamic problems, as it requires fewer fine elements to solve boundary conditions. Based on the first-order shear theory, the governing equations of sheet vibration are derived. The discrete Fourier transform is applied to convert the differential equations from the time domain to the frequency domain. Using dynamic shape functions obtained from the exact solutions, the stiffness matrix of the spectral element is constructed. Dynamic frequency responses are then derived, and time-domain responses are obtained using the inverse Fourier transform. The study extracted the exact natural frequencies of functionally graded sheets and verified them with results from the literature, showing high accuracy with a minimal number of elements. The spectral finite element method of fundamental natural frequencies for different ratios of modulus of elasticity and width-to-thickness ratios obtained were investigated and compared with the results of research. The forced vibration was addressed, demonstrating that the method efficiently captures dynamic responses under different carbon nanotube distributions, with spectral displacements verified through numerical comparison. These findings demonstrate the capability and efficiency of the spectral finite element method for analyzing carbon nanotube-reinforced composite plates.

Nonlinear resonance of fractional order viscoelastic PET films under temperature loading

The effects of oven temperature during printing on nonlinear vibration for fractional-order PET films are considered in this paper. The effect of temperature, fractional order modelling and some other parameters are analysed with respect to the response of the resonance. Fractional order kelvin-Voigt ontological relationship is used to describe the characteristics of viscoelastic materials. The differential equations for nonlinear vibrations are inferred according to the second law of Newton and the theory of von Karman. Discretization for nonlinear equations on locomotion using the Bubnov–Galerkin method. Forced co-oscillatory amplitude-frequency response equations for thin-films systems under temperature loading were calculated using the multiple scales method. Results of numeral results show that temperature, and fractional-order visco-elastic modelling influence the membrane's response to resonance. These results provide a basis for studying fractional-order visco-elastic films vibrations and identifying regions of stable operation in moving systems to prevent divergent instabilities for flexible electronic device manufacturing.

Dynamic characteristics of shape-memory alloy plates immersed in liquid subjected to blast loading

As a critical structure in ship and marine engineering, axially moving shape-memory alloy (SMA) plates may exhibit complex dynamic behaviors. However, there are few studies investigating the dynamic response of SMA plates under underwater blast, let alone consider the effect of initial geometric imperfection and axial motion. Therefore, further studies on such topic are warranted. Based on this, the current paper dedicates to conducting the dynamic research of the geometrically imperfect SMA plate subjected to underwater blast. Firstly, adopting Falk's polynomial constitutive model, the stress strain constitutive relationship of SMA plate can be determined. Secondly, the Hamilton's principle is employed to derive the vibration equation. Thirdly, the assumed mode method and Runge-Kutta technique are applied for solving the dynamic system. After that, several comparative analyses are performed. Finally, the influences of immersed depth, axially moving velocity, initial geometric imperfection, elastic foundation, pulse load type, aspect ratio, length-to-thickness ratio, as well as blast loading parameter on the transient response path are discussed. These findings can contribute to understanding the dynamic behavior of the geometrically imperfect SMA plates immersed in water when subjected to blast loading, and thus providing valuable thinking for the manufacturing and design of submarine equipment.

Examination of parameters affecting the free vibration frequency of 3D orthogonal woven composite rectangular plates

Three-dimensional orthogonal woven composites (3DOWC) can be widely used in various industries, including aerospace, automotive and military. This research examines the linear free vibration behavior of 3DOWC, focusing on the effects of important parameters, such as thickness, dimensional ratio, and binder fiber volume fraction under different boundary conditions. To achieve this, Ritz theory is utilized to calculate the natural frequency of the system. First, the governing equations for the linear free vibrations of 3DOWC are derived. After forming the energy function of the system and minimizing it, the frequency equations of the system are obtained.

Coupled Thermo-Electric-Elastic Piezoelectric Vibration Energy Harvester With Axial Movement: Modeling, Verification, and Analysis

Abstract In the field of rail transport and aerospace field, vibration energy harvesting is inevitably subjected to coupled excitations, including train wheel–track interaction induced friction heat and forced vibration, periodic thermal radiation, and vibration excitation. This paper investigates a coupled thermo-electric-elastic piezoelectric vibration energy harvester with axial movement under external heat flux and mechanical force load. The coupled forced vibration equation, coupled electric equation, and coupled thermoelastic heat conduction equation are derived and solved by Green's function theory. To analyze the effect of excitations on the response characteristics, the decoupled method is utilized to solve the coupled multi-field equations and obtain the displacement, electric, and temperature distribution closed-form solutions. The displacement coupling effect induced temperature distribution and the thermo-electric coupling effect triggered displacement are respectively decoupled and analyzed. The obtained closed-form temperature distribution and displacement solutions are verified by the finite element method. To further verify the obtained solutions, a numerical method is conducted by decoupling the coupled multi-field equations and comparing them with prior solutions. Additionally, the different height-to-length ratios, axially moving speeds, and external force load are analyzed in detail. The results indicate that the displacement, temperature distribution, and output voltage vary with external conditions due to the coupled multi-field effect. Overall, this work investigates the thermo-electric-elastic coupling effect on the axially moving piezoelectric energy harvesting, which is beneficial to promote theoretical investigations of the coupled multi-field energy harvesting system and accelerate the practical applications in the aerospace field.

LONGITUDINAL-RADIAL VIBRATIONS OF A VISCOELASTIC CYLINDRICAL THREE-LAYER STRUCTURE

The paper considers a cylindrical three-layer structure of arbitrary thickness made of viscoelastic material. It consists of two external bearing layers and a middle layer, the materials of which are generally different. The problem of nonstationary longitudinal-radial vibrations of such a structure is formulated. Based on the exact solutions in transformations of the three-dimensional problem of the linear theory of viscoelasticity for a circular cylindrical three-layer body, a mathematical model of its nonstationary longitudinal-radial vibrations is developed. Equations are derived that allow, based on the results of solving the vibration equations, to determine the stress-strain state of a cylindrical structure and its layers in arbitrary sections. The results obtained allow for special cases of transition into cylindrical viscoelastic and elastic two-layer structures, as well as into homogeneous single-layer cylindrical structures and round rods.

Magneto-Thermoelastic Principal Parameter Resonance of a Functionally Graded Cylindrical Shell with Axial Tension

The principal parameter resonance of a ferromagnetic functionally graded (FG) cylindrical shell under the action of axial time-varying tension in magnetic and temperature fields is investigated. The temperature dependence of physical parameters for functionally graded materials (FGMs) is considered. Meanwhile, the tension bending coupling effect is eliminated by introducing the physical neutral surface. The kinetic and strain energies are gained with the Kirchhoff–Love shell theory. Based on the nonlinear magnetization characteristics of ferromagnetic materials, the electromagnetic force acting on the shell is calculated. The nonlinear vibration equations are obtained through Hamilton’s principle. Afterward, the vibration equations are discretized and solved by Galerkin and multiscale methods, respectively. The stability criterion for steady-state motion is established utilizing Lyapunov stability theory. After example analysis, the effects of magnetic field intensity, temperature and power law index on the static deflection are elucidated. Subsequently, the impacts of these parameters, as well as axial tension, on the amplitude-frequency characteristics, resonance amplitude, and multiple solution regions are discussed explicitly. Results indicate that the stiffness can be enhanced due to the generation of static deflection. The amplitude decreases with increasing magnetic field intensity, temperature, and power law index. When the magnetic field intensity surpasses a threshold, the resonance phenomenon disappears.

Transmission and Dissipation of Vibration in a Dynamic Vibration Absorber-Roller System Based on Particle Damping Technology

The research of rolling mill vibration theory has always been a scientific problem in the field of rolling forming, which is very important to the quality of sheet metal and the stable operation of equipment. The essence of rolling mill vibration is the transfer of energy, which is generated from inside and outside. Based on particle damping technology, a dynamic vibration absorber (DVA) is proposed to control the vertical vibration of roll in the rolling process from the point of energy transfer and dissipation. A nonlinear vibration equation for the DVA-roller system is solved by the incremental harmonic balance method. Based on the obtained solutions, the effects of the basic parameters of the DVA on the properties of vibration transmission are investigated by using the power flow method, which provides theoretical guidance for the selection of the basic parameters of the DVA. Furthermore, the influence of the parameters of the particles on the overall dissipation of energy of the particle group is analyzed in a more systematic way, which provides a reference for the selection of the material and diameter and other parameters of the particles in the practical application of the DVA. The effect of particle parameters on roll amplitude inhibition is studied by experiments. The experimental results agree with the theoretical analysis, which proves the correctness of the theoretical analysis and the feasibility of the particle damping absorber. This research proposes a particle damping absorber to absorb and dissipate the energy transfer in rolling process, which provides a new idea for nonlinear dynamic analysis and stability control of rolling mills, and has important guiding significance for practical production.

Free vibration of doubly-curved panels reinforced by carbon nanotubes: New analytic solutions under non-Lévy-type boundary conditions

Existing analytic solutions for the free vibration of functionally graded carbon nanotube reinforced doubly-curved panels primarily address the cases with two parallel simply supported boundaries, known as Lévy-type boundary conditions (BCs). However, doubly-curved panels with non-Lévy-type BCs are more commonly encountered in practical engineering applications, yet their analytic solutions are rarely available due to significant mathematical challenges. This gap motivates us to develop new analytic free vibration solutions under these more complex BCs. The nanocomposites’ material properties are first computed according to the rule of mixture. The Hamiltonian-system governing equation for the free vibration of doubly-curved panels is then formulated from the Donnell-Mushtari theory, and is solved by adopting the analytic symplectic superposition method. The obtained analytic solutions are derived without requiring predefined solution forms, and have been thoroughly validated by comparison with the results from the finite element method. By utilizing the accurate analytic solutions, the effects of aspect ratios, BCs, types of CNT distributions, and volume fractions of CNT on the free vibration behaviors are further analyzed. The present solution procedure and the resulting analytic solutions are expected to be useful for dynamic modeling of composite shell panels, supporting both future research and practical applications.

Simulation modeling of wafer grinding surface roughness considering grinding vibration

When using the workpiece rotation method to grind wafers, the grinding end vibration will deteriorate the surface roughness of the wafers. To study the impact law of vibration on the surface roughness of wafers during the grinding procedure, this paper presents a new approach to simulate and model the surface roughness of wafer grinding considering the grinding vibration. Firstly, the dynamics model under the consideration of grinding force was established for the grinding end of the grinding wheel and workpiece turntable. Secondly, using the iterative method to solve the dynamic equations that have been established, the vibration equation is obtained by fitting the displacement vibration curve of the end. Then, by reconstructing the surface grain of the gear teeth, a simulation model of wafer grinding surface roughness was established considering material removal, grain motion and grinding vibration. And then the grinding comparison test was conducted to compare the simulation and test surface roughness measurement results. The maximum deviation of the surface roughness Sz and Sa was 7.7 % and 5.4 %, respectively. The results indicate the accuracy of the modeling. Finally, based on the established wafer roughness model, explore the impact of vibration on wafer roughness during the grinding procedure. This model provides a reference for the research of wafer precision grinding technology.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

Experimental and theoretical evaluation of axial forces in short steel ropes

AbstractAssessing rope force in bridge structural health monitoring, particularly for shorter lengths, poses challenges. The vibration method, commonly utilized for taut strings, yields inaccurate results for short ropes due to neglecting bending stiffness. To address this, the differential equation of lateral vibration of a prismatic beam possessing bending stiffness EI, evenly distributed mass m under the tension force N is solved approximately and numerically using FEM for greater accuracy. Nonlinear fitting via the Gauss‐Newton aids in refining results. Laboratory experiments, varying axial forces and rope characteristics, validated these methods, offering recommendations for improved accuracy.

Multi-modal control and position optimization of solar panels

This paper investigates the impact of the position of piezoelectric actuators on the vibration control effect of the membrane plane composed of solar sail materials with a given unit area. Firstly, the nonlinear vibration equation of the membrane structure is perfected based on the system equation established by Yifan Lu et al. Then, the modal coupling effect pointed out by Liu Xiang is considered, and the fourth-order modal displacement generated by the piezoelectric actuator in vibration control is derived. Subsequently, the sliding mode controller used in vibration control is designed based on Lyapunov stability theory, and its effectiveness is verified through Simulink simulation. The sliding mode controller is used to calculate the required excitation signal and apply it to the piezoelectric patches. The reverse force is generated through the inverse piezoelectric effect of the patches to actively suppress the vibration of the solar sail. By changing the position and size of each piezoelectric actuator on the solar sail plane, this paper compares the impact of different layout positions on vibration control, and proposes an optimal layout scheme for the position of the piezoelectric patches in terms of control effectiveness.

Numerical Simulation Study on Vibration Characteristics and Influencing Factors of Coal Containing Geological Structure

Accurately determining the natural frequency of coal-containing geological structures is crucial for preventing mine dynamic disasters and utilizing vibration waves to break coal and enhance its permeability. Based on the modal theory of rock, vibration models of coal-containing geological structures, including layering and fractures are established. By analysis, the undamped vibration equation and its characteristic equation for both the layered coal system and the fractured coal system are derived. Subsequently, the Lanczos method is employed to solve the system’s vibration modes using ABAQUS. The effects of the layering position, layering thickness, layering physical properties, crack width, and crack length on the natural frequency and vibration response of coal-containing geological structures are investigated. The results indicate that when a single influencing factor is altered, the displacement response distribution of the coal body vibration system with geological structures remains essentially the same, and these single influencing factors have a minimal impact on the vibration displacement of the coal-containing geological structure. The natural frequency of the system decreases exponentially as the distance between the layering and the geometric center of the coal system with geological structures increases. The presence of layering in the coal system with geological structures significantly reduces the system’s natural frequency. The natural frequency of the coal system with geological structures increases in a power function manner as the layering elastic modulus increases. Conversely, the natural frequency decreases with an increase in crack length. When the change ranges of crack width and bedding thickness are the same, the natural frequency of the fractured coal body system exhibits more significant changes. The natural frequency of the coal system with geological structures initially decreases and then increases as bedding thickness and crack width increase. The trend in the natural frequency changes and the position of the extreme point are related to the ratio of the elastic modulus and density of the geological structure.

Analytical method for free vibration of steel pipe piles partially submerged in water and penetrated in layered soil

This paper presents a novel analytical method for dynamic characteristics of steel pipe piles submerged in water based on the thin shell theory by using the state space method. The state equation with respect to the length direction is first derived by idealizing the steel pipe pile in water as a thin cylindrical shell, where the soil around the pile is modelled as an elastic foundation with stiffnesses in three directions and the surrounding seawater is considered through added equivalent mass. The governing equations for the free vibration of steel pipe piles are derived after the expansion of Fourier series for the state vectors along the circumferential direction. The frequency equation for free vibration and its corresponding mode shapes are then obtained, which is used to analyze the dynamic characteristics of steel pipe piles in water. Due to the advantage of the state space method, arbitrary boundary conditions of the steel pipe pile are conveniently achieved as well as the effects of seawater, soil, the piling hammer and guiding frame. Finally, the present analytical method is verified and demonstrated in detail by several numerical examples, and results show its simplicity and efficiency.

A nonlinear vibration isolator with an essentially nonlinear converter

This paper introduces nonlinearity into bilinear springs and proposes an efficient asymmetrical nonlinear frequency converter (NC) that has a non-reciprocal transmission, which cascades a nonlinear spring and a rope. To solve the vibration equations of both NC and the conventional NES, the Runge-Kutta method is utilized. A comparison is conducted between the amplitude attenuation rates of the NC and the conventional NES. The rate of energy dissipation is indicated by analyzing the remaining energy. According to numerical analysis results, NC has a better performance in suppressing vibration than the conventional NES within a certain range. The energy dissipation rate is higher under a wide initial condition. In addition, NC can cause energy to shift from low-frequency regions to high-frequency regions with wide frequency bands. NC has a broader frequency range and a wider range of applications. This research offers a new approach and strategy for exploring effective devices for damping vibrations.

Random thermal-vibration mechanisms of sandwich ventral fin-type plate-shell systems with porous functionally graded core

The stochastic thermal-vibration mechanisms within a sandwich ventral fin-type plate-shell system, featuring a porous functionally graded (FG) core, are exhaustively analyzed under various random loading conditions employing an innovative node-based, meshless computational approach. The studied structure is decoupled into several plates and open cylindrical panel according to the geometric characteristics, and the mechanical relationships at the structural boundaries or connection interfaces are equivalently simulated by using penalty parameters. Following the general Hamilton's principle, the meshless approach combined with the first-order shear deformation theory (FSDT) incorporating thermal effects is employed to derive the vibration equations of the sandwich ventral fin-type plate-shell systems. Also, the pseudo excitation method (PEM) is introduced to calculate stationary and nonstationary random responses. In order to verify the accuracy of the meshless algorithm in this study, the convergence and correctness are studied comprehensively. And then, the effects of some parameters such as temperature variation, porosity parameters, power-law index and random excitations on the thermal vibration characteristics of the sandwich ventral fin-type plate-shell systems with porous FG core are presented. The results show that the power-law index and temperature can increase the frequency parameter of the structure. The smooth power spectral density (PSD) excitation only affects the amplitude of the response curve, and does not affect the frequency corresponding to the peak. In the analysis of non-stationary random vibration, the influence of modulation parameters on response is very significant.

Thermomechanical buckling and vibration of composite sandwich doubly curved shells with carbon nanotube‐reinforced face layers

AbstractIn this paper, the thermomechanical buckling and vibration analysis are investigated for carbon nanotube‐reinforced composite sandwich doubly curved shells (CNCSDS) under the action of both thermal loads and in‐plane compressive loads. Based on the von‐Karman non‐linear kinematic relations, First Order Shear Theory, and Hamilton's principle, the vibration and stability equations for CNCSDS under thermomechanical loads are deduced. For obtaining the critical thermomechanical buckling load and vibration frequency, an exact method is adopted. Subsequently, the results are validated by comparing with the finding in published literature and simulation results. At last, the influence of system parameters on critical thermomechanical buckling load and vibration frequency is studied and displayed graphically. Meanwhile, some new results about thermomechanical buckling and vibration analysis of CNCSDS are proposed for the first time.Highlights Thermomechanical buckling and vibration of carbon nanotube‐reinforced composite sandwich doubly curved shells was studied. Vibration and stability equations under thermomechanical loads was deduced. The change rules of thermomechanical buckling load and frequency were obtained.

Study of Resonance between Bogie Hunting and Carbody Mode via Field Measurements and Dynamic Simulation.

By addressing the phenomenon of carbody abnormal vibrations in the field, the acceleration of the carbody and bogie was measured using accelerometers, and the diamond mode of the carbody was identified. The equivalent conicity of the wheelset and the acceleration at the frame end indicated that the shaking of the carbody was caused by bogie hunting. In the SIMPACK simulation, the acceleration frequency and amplitude at the frame end and midsection of the side beam were calculated. The lateral deformation amplitude of the side beam in the finite element model was extracted, and a modal shape function for the diamond-shaped mode was established. By utilizing the modal vibration equation, the modal generalized forces of the carbody were computed, revealing that, during carbody shaking, the yaw damper force contributed significantly among the forces of the secondary suspension, with the phase difference between the front and rear bogies approaching 180°. This insight offers a novel perspective for subsequent active control strategies. Subsequently, these modal generalized forces were applied as external excitation to a coupled vibration model encompassing both the carbody and transformer. Aiming to reduce the acceleration amplitude at the side beam, the transformer was treated as a dynamic vibration absorber, allowing for the optimization of its lateral suspension parameters. As a result, the lateral and vertical acceleration amplitudes at the side beam were concurrently reduced, with the maximum decrease reaching 58.5%, significantly enhancing the ride comfort.

Size-dependent nonlinear free vibration of magneto-electro-elastic nanobeams by incorporating modified couple stress and nonlocal elasticity theory

Magneto-Electro-Elastic (MEE) Composites, as an innovative functional material blend, are composed of multiple materials, boasting exceptional strength, rigidity, and an extraordinary magneto-electric interaction effect. This paper establishes a nonlocal modified couple stress (NL-MCS) magneto-electro-elastic nanobeam dynamic model. To accurately capture the intricate influences of scale effects on nanostructures, This model meticulously examines scale effects from two distinct perspectives: leveraging nonlocal elasticity theory to elucidate the softening phenomena in nanostructures stemming from long-range particle interactions, and employing modified couple stress theory to reveal the hardening effects attributed to the rotational behavior of particles within the structure. By incorporating Von Karman geometric nonlinearity, Reddy’s third-order shear deformation theory and Maxwell’s equations, the governing equations for the nonlinear free vibration of MEE nanobeams are derived using Hamilton’s principle. Finally, a two-step perturbation method is employed to solve these equations. Two-step perturbation method disintegrates the solution process into two stages, iteratively approximating and refining the solution, thereby progressively unraveling the intricate details and enhancing the precision of the solution in a systematic manner. Finally, the nonlinear free vibration behavior of MEE nanobeams is explored under the coupled magnetic-electric-elastic fields, with a focus on the effects of various factors that including length scale parameters, nonlocal parameters, Winkler-Pasternak coefficients, span-to-thickness ratios, applied voltages and magnetic potentials.