Vibration Equation Research Articles

Dynamic Modeling and Analysis of a Temperature- Dependent Axially Translating Functionally Graded Cantilever Beam

In this paper, the dynamics of an axially translating functionally graded cantilever beam (FGCB) with time-varying length are studied under environmental temperature variations. Firstly, the governing partial differential equation for the FGCB under different temperatures is established based on the Euler–Bernoulli beam theory. Secondly, the Galerkin discretization and the assumed mode method (AMM) are employed to derive the vibration equations for each mode. Then, the coupling effects of the axial translation motion and the bending deformation on the vibration response of the FGCB are analyzed through numerical simulation. The results showed that different initial lengths, velocities and accelerations can influence the dynamic response greatly. Moreover, the increase in temperature will increase the response amplitude and decrease the frequency of FGCB, but the increase in functionally gradient parameter will decrease both the amplitude and the frequency of the FGCB. Finally, the wavelet transform (WT) is utilized to perform a time–frequency analysis. It is found that the time-dependent frequency obtained by using WT is consistent with the first-order static frequency obtained theoretically. The variation of frequency with time can be obtained quickly and accurately by using WT. The results of this research are of great significance for the application of composite axially translating beams in practical engineering.

Nonlinear vibrations of variable speed rotating graphene platelets reinforced blades subjected to combined parametric and forced excitation

It is generally acknowledged that the disturbance of rotating speed will cause parametric resonance for blades, and the presence of rotor displacement also can make the blade vibrate transversely. However, the coupled resonance mechanism of blades under the combined action of rotating speed disturbance and rotor displacement is not yet clear. To answer this question, this article investigates the nonlinear coupled resonance behavior of rotating blades in two cases: typical tuned (the frequency ratio of parametric and forced excitations equals 2:1) and frequency ratio detuned. A nonlinear vibration equation for rotating blades is established based on Euler beam theory in conjunction with geometric nonlinearity. The mixing rule and modified Halpin-Tsai model are used to calculate the effective properties of GPLRMF materials. Subsequently, using the method of varying amplitudes (MVA), an approximate analytical solution of the coupled resonance response is derived, the Jacobian matrix is used to determine the stability of non-trivial solutions, and the accuracy of the analytical results is verified with the aid of numerical solutions. Finally, the steady-state response of typical tuned and detuned coupled resonance is analyzed, and the influence mechanism of factors such as excitation amplitude, excitation phase angle and other parameters on coupled resonance is studied in detail. The results indicate that the supercritical coupled resonance curve exhibits double resonance peaks and jumps. Meanwhile, the steady-state response of coupled resonance highly depends on the excitation phase angle.

Vibration analysis of a partially covered beam with a shear thickening fluid core

The vibration responses of a sandwich beam with partially covered by shear thickening fluid (STF) layer under an impact load are investigated. The nonlinear governing equations of the flexural vibration are derived based on extended Hamilton’s principle and are solved by the finite difference method. The model is then validated and used to develop a complete parametric study of partially covered beams with the STF-filled core to properly design and place the STF patch. It is found that, for the first vibration mode, maximum damping, and the smallest change in the natural frequency are achieved when the coverage length of the partial STF patch exceeds 50% and the center of the patch is positioned at 56.25% from the left edge. For the second vibration mode, the coverage length is 37.5% and the center of the patch is located at 75% from the left edge of the beam. Additionally, it has been observed that maintaining a thickness ratio of 0.75 between the constraining layer and the base beam leads to increased damping, while simultaneously minimizing alterations in the natural frequency of the original beam. The results can be used for the structural design of sandwich beams partially covered by STF.

Nonlinear vibration characteristics of thermo-viscoelastic coupling of moving PET films with temperature-dependent elastic modulus

In order to improve the transport stability of roll-to-roll produced polyethylene terephthalate films (PET films) in high temperature environments, this paper investigates the nonlinear vibrational properties of moving PET films under a uniform temperature field by considering the temperature dependence of the elastic modulus. Firstly, constant tensile tests were conducted to test the elastic modulus values of PET films at different hot air temperatures, and the Wachtman model was improved to establish a temperature-dependent elastic modulus model for PET films. Secondly, based on the Kelvin-Voigt model, the nonlinear vibrational equilibrium differential equations of the system were established by applying the Bubnov-Galerkin method of discretization according to D’Alembert’s principle. Finally, numerical simulations based on the 4th-order Runge-Kutta method were carried out to analyze the effects of films width, external excitation frequency, external excitation amplitude, transfer velocity, ambient temperature, damping coefficient and viscoelasticity coefficient on the nonlinear vibration of the system. This provides a theoretical basis for the commissioning of stabilized transfer parameters for roll-to-roll production of PET films.

Harvesting vibration energy by quad-stable piezoelectric cantilever beam: Modeling, fabrication and testing

The combination of beams with piezoelectric patches has become a prevalent energy harvesting tool due to its ease of use. Typical energy harvesting systems are usually linear, and their efficiency is not satisfying due to low-frequency bandwidth. In this paper, a quad-stable piezoelectric vibration energy harvester is analyzed both numerically and experimentally. The primary purpose of this investigation is to analyze the static and dynamic characteristics of a proposed quad-stable system to consider its potential for application in broadband energy harvesting comprehensively. The harvester system consists of a slotted cantilever beam, a piezoelectric patch, a pair of tip-mass blocks, and a double-sided clip. The cantilever beam is subjected to pre-displacement constraints made by a mutual self-constraint at the free end of it. The nonlinear behaviors of the harvester system, including snap-through and softening phenomena, are analyzed using the assumed modes and finite element method (FEM). The harvester's vibration equation is solved numerically and through an FE model which is made by an in-house finite element software. A prototype is designed and fabricated to validate the mathematical model and FE simulation. The experimental force-displacement diagram of the harvester displays distinct discontinuities, reflecting abrupt transitions occurring while switching its stable states. The prototype dynamics are analyzed by harmonic base excitation with different amplitude levels in two conditions, including the presence and absence of the piezoelectric patch. The results obtained from the mathematical and FEM model demonstrate a satisfactory correlation with the experimental data. Furthermore, the experimental data reveal the occurrence of the snap-through phenomenon, accompanied by a significant widening of the frequency bandwidth at relatively high amplitude levels. The system has the ability to provide an average output electrical power of 0.288 mW for an electrical resistance of 3.262 kΩ at the excitation frequency of 12.695 Hz and base acceleration amplitude of 3g.

Dynamic simulation of a simply supported beam subjected by multiple constant moving forces

Abstract Dynamic simulation models, which are conducted with Simulink, are introduced to study behaviors of a simply supported beam under constant moving forces based on the vibration mechanics model established by Bernoulli-Euler beam theory. By utilizing the orthogonality of vibration, a set of decoupled forced vibration equations can be obtained, which can be constructed as a subsystem of the Simulink model. The generalized loads in each decoupled vibration equation are also separately formed into submodules to facilitate the extension of the action of a single moving load to multiple moving loads. Every partial and general simulation model in Matlab/Simulink are made and the simulation results indicate that dynamic responses are related to the distance between moving loads and beam span.

Sound transmission loss analysis of novel sandwich plate with star-shaped auxetic cellular cores

A star-shaped auxetic cellular metamaterial with a negative Poisson’s ratio (NPR) was employed in this study to model lightweight sandwich panels, and an analysis of its sound transmission characteristics was conducted. Utilizing the first-order shear deformation theory (FSDT) and the Hamiltonian principle, the free vibration equation for the sandwich plate is derived. Subsequently, leveraging the fluid-structure coupling condition and incorporating the Navier method, the sound transmission loss (STL) is analytically described. The efficacy of the proposed method is validated through numerical analysis and an acoustic insulation experiment using an impedance tube. Based on the theoretical modeling, it is calculated that the sound insulation effect of the star-shaped cellular core exceeds that of the traditional reentrant cellular with NPR. A systematic analysis is subsequently conducted to explore the impact of star-shaped cellular structure parameters on the STL of the sandwich panel. It can be obtained that the increase in core layer thickness leads to a reduction in the first-order resonance frequency of the sandwich panel, but concurrently diminishing its sound insulation performance. The change of the connecting rib length exerts a more considerable influence on the structure’s STL compared to the side rib length. While as the cell wall thickness reduces from 2.0 to 1.0 mm, the average STL of the sandwich plate across the analysis band decreases by 8.43%, from 46.3 to 42.7 dB.

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

$$ H^1$$-Galerkin mixed finite element method for the vibration equation of beam with structural damping

Resonance frequency band structure in heterojunction of one-dimension string-ball chain system

Using the concept of heterostructures in semiconductor physics and the similarity between the Schrödinger equation and the vibration equation of the one-dimension string-ball chain system, we create a heterojunction in our string-ball chain system by assembling a string-ball chain with two kinds of mass in a lattice-matched and an unmatched system and study the two systems through first-principle calculation, theoretical analysis, and experiment perspectives. A discontinuity in the frequency dependence is observed in the matched system, while a phase transition is discovered in the unmatched system. A proper theoretical analysis based on the wave function is also given to explain these phenomena.

Self-Similar Solutions for the Membrane Transverse Vibration Equation

Self-Similar Solutions for the Membrane Transverse Vibration Equation

On the Method for Solving the Problem of Ice Cover Deformation under an Arbitrary Moving Load

Introduction. The development of the polar areas of the World Ocean and the need to solve various problems associated with a large number of freezing inland water bodies issue new challenges for science. These challenges include the problem of studying the behavior of ice cover when exposed to various types of loads. Of great interest is the consideration of problems about the action of a moving load on the ice cover. A moving load simulates the effect of moving vehicles on ice. However, in papers devoted to the above problems, cases of load movement along a straight-line trajectory are considered. The objective of this research is to develop a method for studying the behavior of ice cover under the action of a load moving arbitrarily.Materials and Methods. The article proposes a method for solving the problem of the action of a force moving along an arbitrary trajectory on the ice cover of a reservoir of finite depth. The problem amounts to solving a system of two differential equations. The first of them models the behavior of the ice cover, and it is the equation of vibrations of a viscoelastic plate. The second equation simulates the behavior of fluid in a state of potential flow, and it is Laplace's equation. To solve the system of differential equations, integral transformations in time, space and variables were used. The resulting solution was expressed through an iterated integral, which was calculated using numerical methods.Results. The development and implementation of the method resulted in solving the problem of the movement of a concentrated force along an ice cover according to an arbitrary law. At the same time, studies were carried out on the behavior of displacements and stresses in the ice cover depending on the speed and acceleration of the movement of the vertical load, on the depth of the reservoir, and on the viscoelastic properties of ice. In addition, the distribution of the velocity vector of fluid particles along the depth of the reservoir was calculated.Discussion and Conclusion. The proposed method is very effective for solving problems of moving loads acting on the ice cover of a reservoir of finite depth. It provides solving problems about the action of a load moving along an ice cover along a complex trajectory. The results obtained can be used to calculate the stress and displacement of the ice cover during the laying of ice roads or the construction of airfields on the ice.

ОБЩИЙ ПОДХОД К РЕШЕНИЮ ЗАДАЧИ ИЗГИБА ПРЯМОУГОЛЬНОЙ ПЛАСТИНКИ С ЛЮБЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИ

In the work under consideration, an approach is proposed that makes it possible to calculate rectangular plates with any continuous types of boundary conditions from a unified position. The solution for plate deflections is constructed as a product of two functions, each of which represents its own function of the differential equation of bending vibrations of the rod with boundary conditions corresponding to the support of the rectangular plate. The problem reduces to an infinite system of linear algebraic equations for the coefficients of the eigenfunctions, which can be solved by the truncation method. Two examples of the calculation of rectangular plates with rigidly clamped and hingedly supported sides under the action of a concentrated force and with two clamped and two hingedly supported sides under the action of a uniformly distributed load are given.

Research on Electro-Hydraulic Servo Resonance Technology.

In this paper, an electro-hydraulic servo resonance technology is proposed to meet the loading requirements of a high-frequency sound fatigue test for large tonnage. First of all, we analyze the static and dynamic loading structure of electro-hydraulic servo vibration and establish the vibration equation of the system. Additionally, the modal and vibration characteristics of the system are analyzed by simulation, which verified the feasibility of the proposed electro-hydraulic servo resonant loading technology. Finally, the influence of various factors such as sample stiffness, lead screw stiffness, class II spring stiffness, class II weight mass, lower beam mass, and upper beam mass on the natural frequency and amplification coefficient of the system is analyzed. In this paper, a new technology is proposed to provide theoretical support for the research and development of large-tonnage high-frequency noise fatigue testing equipment.

A comparative study on the torsional vibration suppression of rotor system using variable steady state nonlinear energy sink

Current studies have shown that the introduction of bistable nonlinear energy sink (B-NES) and tristable nonlinear energy sink (T-NES) can reduce the effective threshold and alleviate the separation resonance curve compared with traditional NES under the low and higher external excitation respectively. However, there is a lack of comparative studies on the performance of B-NES, T-NES and monostable nonlinear energy sink (M-NES) in the field of torsional vibration engineering application. Aimed at this limitation, a variable steady state nonlinear energy sink (V-NES) is proposed to suppress the torsional vibration of shafting. Negative stiffness is generated by the pre-compression of the spring. The proposed V-NES can be easily switched between monostable, bistable and tristable cases by the moving position of the spring bases. Based on this, the governing equation of torsional vibration coupled with V-NES after proper orthogonal decomposition (POD) reduction is derived. The approximate analytical solution of the coupled system is obtained by using complex averaging method and verified numerically. Then, the vibration reduction effects of M-NES, B-NES and T-NES on the torsional vibration of shafting are compared. Throughout the study, the motion states of strongly modulated response (SMR), interwell oscillation, and chaotic strongly modulated response (CSMR) are observed in the V-NES. Excessively high external excitation amplitudes can make V-NES exhibit CSMR, diminishing the effectiveness of vibration reduction in comparison to inter-well oscillations.

Theoretical and experimental study of rigid rapier weft insertion vibration

In this study, vibration equations and models were established utilizing D’Alembert’s principle to address the resonance challenge of a rigid rapier loom operating in high-speed reciprocating motion, and the equations were subsequently solved using Galyokin’s methodology. A laboratory apparatus was constructed to evaluate the findings of the first-order vibration frequency under various rapier extension lengths to validate the precision of the theoretical model, which necessitated recalibration. The accuracy of the theoretical model is confirmed by comparing its predicted results with those of the experiments. In this report, the rigid rapier is further examined in order to correct the trapezoidal acceleration within the perspective of the vibration of the rigid rapier influenced by varied rapier head masses. The results demonstrate that as the mass of the rapier head increases, the maximum apex deviation of the rigid rapier during the weft-induced process also escalates. To enhance the stability of the rigid rapier gravitating process and to provide theoretical guidance and design implications in engineering, the theoretical model of this research can also examine the influence of gravity, rapier material, rapier dimension, and other variables on the oscillation of the rigid rapier gravitation process.

Girder quasi-static cumulative displacement-based suspension bridge nonlinear damper leakage diagnosis by eliminating stochastic traffic effects

Fluid viscous dampers (FVDs) are the key energy dissipation devices for long-span suspension bridges, and their leakage poses a threat to the stable and reliable operation of bridge restraint systems. Therefore, this paper proposed an FVD fluid leakage diagnosis method based on girder longitudinal displacement monitoring data. First, the longitudinal vibration equation of the main girder under traffic loads was derived to reveal the displacement characteristic of girder-end motion. On this basis, considering that different displacement components under traffic loads are coupled and transient analysis directly utilizing traffic loads as the excitation is complicated and inefficient, an analysis framework focusing on the influence mechanism of FVD leakage on quasi-static displacement was established. Second, an equivalent mechanical model for simulating the hysteresis relationship of leaked FVD was proposed and verified. Then, the influence of variations in the location, number, leakage level and parameters of the leaked FVD on the quasi-static displacement was studied. Third, the equivalent traffic load (ETL) indicator was constructed to characterize the traffic volume. Furthermore, a quasi-static cumulative displacement and ETL correlation model, which can effectively eliminate the interference of traffic flow variations, was proposed for the leakage diagnosis of FVD. Finally, the proposed method was verified by real suspension bridge monitoring data. The results demonstrated that the proposed method can effectively detect a mild increase in the girder-end cumulative displacement due to FVD fluid leakage.

Study of self-oscillations of a distributed rotor from a material with hereditarily elastic properties

Self-oscillatory solutions of dynamic equations of transverse vibrations of a distributed rotating ideally balanced rotor (shaft) on an isotropic support made of a material with nonlinear hereditary properties are studied. The derivation of dynamic equations (mathematical model), which are a system of two nonlinear partial differential equations with an infinite (integral) delay of the time argument, is given. A mathematical formulation of the initial–boundary value problem is formulated, its solvability and correctness are proven. The stability of the zero solution (stable rotation), the mechanisms of loss of stability, and the self-oscillatory solutions bifurcating in this case are studied. The possibility of bifurcation of periodic solutions (direct asynchronous precessions) and invariant two-dimensional tori (beat modes) is shown. The stability of self-oscillating solutions is determined by the parameters of the problem under consideration. The method of central manifolds of distributed systems and the theory of bifurcations were used as a research method. Asymptotic formulas are constructed for self-oscillating solutions.

A generalized method for in-plane eigenproblem of a cable network attached with multiple rigid cross-ties

Considering the cable sag as well as the impacts of cables’ dynamic axial forces on the transverse forces, this paper makes an effort to propose a generalized method for studying in-plane free vibration of the cable network attached with multiple rigid cross-ties. First, the vibration equations of the suspended cables are established and solved by separation-of-variables method. Then, the system is divided into several substructures, which are reassembled according to transfer matrix method (TMM). Using the boundary conditions, the characteristic equation of the system is yielded. In this way, the basic theory for solving the eigenproblem of a general cable network attached with rigid cross-ties is derived. Four examples are given to illustrate the applicability of the current method, and their frequencies and mode shapes are acquired by a developed calculating program. Meanwhile, a finite element model (FEM) is also established. It is found that the results are in a good agreement with those obtained by FEM, which verifies the validity and correctness of the method in this paper.

Electro-Mechanical Coupling Analysis of L-Shaped Three-Dimensional Braided Piezoelectric Composites Vibration Energy Harvester.

In this article, an L-shaped three-dimensional (3D) braided piezoelectric composite energy harvester (BPCEH) is established, which consists of an elastic layer composed of a 3D braided composite, flanked by upper and lower layers of piezoelectric material and two tuning mass blocks. Glass fiber and epoxy resin are used to produce a 3D braided composite. This L-shaped 3D BPCEH is mechanically designable and can be adapted to different work requirements by varying the braided angle of the 3D braided composite layer. The material parameters of 3D braided composites are predicted for different braided angles by means of a representative volume element (RVE). Electro-mechanical coupled vibration equations for the L-shaped 3D BPCEH are established. The impact of braided angles on voltage and power output is discussed in this article. Simulations using finite element method are conducted to analyze the voltage and power output responses at various braided angles. In addition, the effects of the mass of mass block B and the length of the beam on the output performance of the L-shaped 3D BPCEH are analyzed.

Theoretical and experimental transverse vibration analysis of a non-uniform composite helical tidal turbine foil

Recently, tidal energy has gained attention as an attractive renewable source of power generation. In this context, tidal turbine foils must operate in harsh conditions and extremely variable dynamic loads. Considering the expensive dynamic tests required for these large structures, developing validated dynamic models could be valuable for failure prediction and design purposes. In this paper, the free and forced vibration behaviour of the cantilevered part of a non-uniform helical tidal turbine foil made from the composite material is investigated theoretically and experimentally. In this regard, governing vibration equations of the foil with related boundary conditions are presented. The free vibration analysis of the foil is performed by solving a boundary-value problem, where a closed-form solution, based on the mode summation method, for the forced vibration of the foil is employed. Finally, the obtained numerical results (natural frequency and dynamic displacements) are compared with the corresponding results from experiments to validate the vibration equations and the employed solution. Acceptable agreement between numerical and experimental data confirms that the vibration model presented in this study can be employed to predict the dynamic response, parametric study, optimization, and design of the tidal foil without performing costly experiments.