Vibration Equation Research Articles

Structural dynamic properties of stiffened composite plates with embedded multi-layered viscoelastic damping membranes

Aiming at optimizing stiffened composite structures with embedded multi-layered viscoelastic damping membranes, we developed a dynamic analytical model based on the first-order shear deformation theory, in which the vibration equations were derived from both the modulus theory and the composite material mechanics theory. The theoretical solution that satisfied the displacement boundary conditions was obtained by employing the Navier method. Results were simulated by the finite element method to verify correctness of the dynamic analytical model. In addition, the effects of structural parameters on dynamic performance (including first-order natural frequency and modal loss factor) were studied using the developed model.

Free Vibration Analysis of a Spinning Composite Laminated Truncated Conical Shell under Hygrothermal Environment

This paper is concerned with free vibration characteristics of a spinning composite laminated truncated conical shell subjected to hygrothermal environment. Hygrothermal strains are introduced into the constitutive law of single-layer material, and fiber orientation lies symmetrically with respect to the midplane of the composite shell. Considering the spin-induced Coriolis and centrifugal forces, as well as initial hoop tension, the governing equations of free vibration of the composite conical shell with hygrothermal effects are derived on the basis of Love’s thin-shell theory and Hamilton’s principle. The solution of the equations is derived using the Galerkin approach. Then, a detailed parametric study on natural frequencies and critical spinning speeds is numerically performed. Results indicate that the Coriolis force induces an asymmetric influence on natural frequencies of forward and backward traveling waves, while the centrifugal force enhances the frequencies of both traveling waves symmetrically. Initial hoop tension plays a major role in the increase of critical spinning angular speed. Temperature, moisture concentration, and design parameters show the significant influence on the free vibration characteristics of the conical shell, and thermal expansion deformation is nonnegligible in the free vibration analysis.

Response adjustable performance of a visco-elastomer sandwich plate with harmonic parameters and distributed supported masses under random loading

Vibration control of composite structures with distributed masses under random loadings is a significant issue. Adjustability of dynamic characteristics including response spectrum peaks and valleys is important for structural vibration control. The vibration control design in space contains structure and conformation designs which combination results in periodic composite structures. In the present paper, spatial periodicity control design is proposed. Stochastic response adjustable performance of a visco-elastomer sandwich plate with harmonic distribution of geometrical and physical parameters and distributed supported masses under random base motion loading is studied. Both facial layer thickness and core layer modulus of the sandwich plate are considered as harmonic distribution in length and width directions as well as periodically distributed masses. Partial differential equations of coupling motions of the sandwich plate system are derived and converted into ordinary differential equations for multi-mode coupling vibration. Generalized stiffness, damping, and mass coefficients are functions of the harmonic distribution parameters. An analysis solution with frequency response function and response spectral density expressions of the sandwich plate system is obtained. Numerical results are given to show the response adjustable performance through the harmonic geometrical and physical parameters and distributed masses. The results have a potential application to stochastic vibration control or dynamic optimization design of smart composite structure systems.

Study of nonlinear vibration of resonant gyroscope based on uncertainty analysis method

To study the nonlinear vibration characteristics of resonant gyroscope, the vibration model of the resonant beam is established primarily, aiming for the analytical formulas of the nonlinear stiffness coefficients and linear intrinsic frequencies of the gyroscope. With that foundation, the influence of the structural parameters of the resonant beam on the frequency tracking error caused by nonlinear vibration is analyzed thoroughly at first. Secondly, by random sampling, an uncertainty analysis model is established to evaluate the impact of each structural parameter of the resonant beam on the critical performance indexes of the gyroscope such as linear intrinsic frequency, nonlinear frequency offset and scale factor of the resonant beam. Finally, an equivalent circuit is constructed corresponding to the nonlinear vibration equations of the resonant gyroscope based on the electromechanical analogy method. By studying the nonlinearity of the circuit for equivalence, it turns out that the vibration characteristics of the equivalent circuit are consistent with that of the resonant gyroscope obtained by using the uncertainty analysis method. It effectively verifies the feasibility of the nonlinear vibration research method for the resonant gyroscope based on uncertainty analysis method.

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To study the nonlinear vibration characteristics of resonant gyroscope, the vibration model of the resonant beam is established primarily, aiming for the analytical formulas of the nonlinear stiffness coefficients and linear intrinsic frequencies of the gyroscope. With that foundation, the influence of the structural parameters of the resonant beam on the frequency tracking error caused by nonlinear vibration is analyzed thoroughly at first. Secondly, by random sampling, an uncertainty analysis model is established to evaluate the impact of each structural parameter of the resonant beam on the critical performance indexes of the gyroscope such as linear intrinsic frequency, nonlinear frequency offset and scale factor of the resonant beam. Finally, an equivalent circuit is constructed corresponding to the nonlinear vibration equations of the resonant gyroscope based on the electromechanical analogy method. By studying the nonlinearity of the circuit for equivalence, it turns out that the vibration characteristics of the equivalent circuit are consistent with that of the resonant gyroscope obtained by using the uncertainty analysis method. It effectively verifies the feasibility of the nonlinear vibration research method for the resonant gyroscope based on uncertainty analysis method.

An indirect method to determine nonlinearly elastic shear stress-strain constitutive relationships for nonlinear torsional vibration of nonlinearly elastic shafts

PurposeIn this paper, the authors aim to propose an effective method to indirectly determine nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials, and then study the nonlinear free torsional vibration of Al–1%Si shaft.Design/methodology/approachIn this study the authors use BoxLucas1 model to fit the determined-experimentally nonlinear elastic normal stress–strain constitutive relationship curve of Al–1%Si, a typical case of isotropic nonlinear elasticity materials, and then derive its nonlinear shear stress-strain constitutive relationships based on the fitting constitutive relationships and general equations of plane-stress and plane-strain transformation. Hamilton’s principle is utilized to gain nonlinear governing equation and boundary conditions for free torsional vibration of Al–1%Si shaft. Differential quadrature method and an iterative algorithm are employed to numerically solve the gained equations of motion.Findings The effect of four variables, namely dimensionless fundamental vibration amplitude , radius and length , and nonlinear-elasticity intensity factor , on frequencies and mode shapes of the shafts is obtained. Numerical results are in good agreement with reference solutions, and show that compared with linearly elastic shear stress-strain constitutive relationships of the shafts made of the nonlinear elasticity materials, its actual nonlinearly elastic shear stress-strain constitutive relationships have smaller torsion frequencies. In addition, but having opposite hardening effect, the rest of the four variables have softening effect on nonlinearly elastic torsion frequencies. Eventually, taking into account nonlinearly elastic shear stress-strain constitutive relationships, changes of the four factors, i.e. , , and , cause inflation and deflation behaviors of mode shapes in nonlinear free torsional vibration.Originality/valueThe study could provide a reference for indirectly determining nonlinear elastic shear stress-strain constitutive relationships for nonlinear elasticity materials and for structure design of torsional shaft made of nonlinear elasticity materials.

Vibrational frequencies and modes of a kinked inflatable beam

Inflated beams are under increasing application in various structures. Many inflatable structures have kinks during use due to folding or initial material defects. A specific emphasis is often placed on determining the static characteristics of inflated beams in bending and wrinkling. From the perspective of mobility, a kink is needed to realize the movement of the structure, namely the vibration-containing kink. There is a lack of knowledge in dynamic conditions, especially for simple and accurate solutions for kinked inflatable beams. This paper studies the vibrational frequencies and modes of kinked inflatable beams and deals with the theoretical analysis of the natural frequencies of kinked inflatable beams. The first-order natural frequency of the kinked inflatable beam is predicted by introducing the kink parameters, and the inflated pressure into the vibration equation of the beam. The predictions on the natural frequency agree well with the vibration test. Results reveal that the kink decreases the first-order natural frequency of the inflated beam greatly, and the beam has no global vibration mode but only a higher mode. The present results can provide meaningful guidance for innovative inflated structure design.

Vibration control of an articulated tower with a tuned mass damper subjected to the inertial force of ground acceleration

ABSTRACT This research proposes a new approach, in which a tuned mass damper (TMD) is installed in an articulated tower subjected to moving loads caused by ground. Then, the vibration equations of the system are formulated for analyzing and selecting the optimal parameters of the tuned mass damper to reduce the harmful vibrations. The maximization of equivalent viscous resistance method is applied to determine the optimum parameters of the tuned mass damper. The purpose of designing the tuned mass damper is to suppress the vibrations of the articulated tower. The results show that the vibration of the articulated tower is effectively eliminated by using the optimum parameters of the tuned mass damper in this paper. In particular, the optimum expressions are given in analytical solutions, which involve the parameters in well-understood forms to derive the exact solution of the optimum parameter. These optimum expressions, helping the researchers easily design the tuned mass damper on controlling vibrations of the articulated tower.

Dynamic characteristics analysis of complex aircraft pipeline system using MDSMA algorithm

PurposeThe dynamic characteristics prediction and frequency-modulation of pipeline was an important work for the design of aircraft hydraulic structure.Design/methodology/approachA complex pipeline was deemed as a combination of several segments of straight-pipe-element (SPE). The 3D vibration equations of each SPE were established in their local coordinate system based on Timoshenko-beam model and Euler-beam model, respectively. The dynamic-stiffness-matrixes were deduced from the dispersion relation of these equations. According to the complex pipeline layout in the global coordinate system, a multi dynamic stiffness matrixes assembling (MDSMA) algorithm was carried out to establish the characteristic equations of the whole complex pipeline. The MDSMA solutions were verified to be consistent with experimental results.FindingsThe MDSMA method based on Timoshenko-Beam model was more suitable for the short span aviation pipeline and the vibration at high frequency stage (>350 Hz). The layout affected the pipeline's in-plane stiffness and out-plane stiffness, for the Z-shaped pipe, each order natural mode took place on the ZP and NP alternately. Reasonable designs of bending position and bending radius were effective means for complex pipeline frequency-modulation.Originality/valueA new dynamic modeling method of aircraft complex pipeline was proposed to obtain the influence of pipeline layout parameters on dynamic characteristics.

An improved dynamics modeling during milling of the thin-walled parts based on magnetorheological damping fixture

Aerospace thin-walled parts vibrates during the process of milling due to their characteristics of low rigidity, which influences the machining quality of the workpiece. Based on the characteristics of magnetorheological fluid excitation solidification, a magnetorheological damping fixture is designed to semi-actively suppress the vibration generated during the machining process. This paper simplifies the multi-characteristic structure of aerospace thin-walled parts into rectangular thin plates. In accordance with Kirchhoff G.’s small deflection bending theory of elastic thin-plates and Daramberg’s principle, the differential equations for the transverse vibration of the thin-walled parts are established, which aims to obtain the natural frequency as well as vibration mode function of the thin-walled parts. In consideration of damping characteristics and the external dynamic milling force after the magnetorheological fluid excitation solidification, this paper uses the mode superposition method and an improved dynamic response mathematical model of the magnetorheological damping fixture thin-walled parts system of the linear multiple degrees of freedom, which has different volumes of magnetorheological fluids under forced vibration, was established. The maximum error between the predicted and measured values ​​of the fixture-workpiece system dynamic response of displacement and acceleration is 16.2% and 15.5%, respectively. Finally, this paper verifies the feasibility as well as the effectiveness of the model through dynamics experiment and milling machining experiment.

An Analytical, Numerical, and Experimental Investigation on Transverse Vibrations of a Finite Locally Resonant Beam

An analytical, numerical, and experimental investigation on the transverse vibrations of a finite beam with periodically arrayed beam-like resonators was carried out. A continuous-discrete model of the finite locally resonant beam was established by employing the “mass-spring- mass” subsystem. The analytical solution of the coupling vibration equations was derived based on the modal superposition method, and the analytical expression of average velocity response and vibration transmissibility were given. Then, the minimum periodic number of different units which could result in a bandgap was determined. Finally, the bandgap of a finite locally resonant beam was confirmed by a vibration experiment on a simply supported beam with twelve uniformly distributed beam-like resonators. The numerical and experimental results show that finite locally resonant beams have low-frequency bandgaps like infinite locally resonant beams, and the bandgap position is close to the resonance frequency of resonators. In addition, for a beam with a different type of locally resonant units, the minimum number of units that can generate the bandgap is nearly the same. Within considered frequency ranges, the experimental results are consistent with the theoretical results, meaning that the transverse vibration in locally resonant beams could be substantially attenuated. The conclusions may be supported to the application of locally resonant theory to control low-frequency vibration and radiation noise.

Stability of axially functionally graded pipe conveying fluid: Generalized integral transform solution

The stability of axially functionally graded pipe conveying fluid is analyzed based on the Euler–Bernoulli beam theory. The governing equation for transverse vibration is transformed into an infinite system of second-order ordinary differential equations by the generalized integral transform technique, which is truncated at finite terms to find the complex frequency as a function of flow and structural parameters. The effects of the elastic modulus and density distributions on the linear stability of axially functionally graded pipe are investigated by parametric studies. It is shown that the complex frequency, critical velocity of buckling instability and coupled flutter of the axially functionally graded pipe were significantly influenced by the elastic modulus gradient. The density gradient distribution significantly affects the complex frequency of the axially functionally graded pipe.

The influence of wind load on the stress characteristics of dropper for a high-speed railway

In this paper, the influence of wind load on the stress characteristics of dropper for a high-speed railway is studied. The section method is used to obtain the vibration equation of the dropper, and the finite difference method is used to convert the partial differential equation of the dropper vibration into a finite difference equation. Through the force analysis of the contact line section upwind, the vibration equation of the contact line with air damping is obtained. Thus, the partial differential equation of the motion can determine the initial and boundary conditions of each dropper. The MATLAB program is written to obtain the stress change diagram of each dropper, and the influence of wind load on the stress characteristics of the dropper is analyzed. The results show that the maximum tensile and compressive stresses of each dropper are different due to the difference of wind load and dropper location, and the dropper in the middle location is most sensitive to the change of wind load. At a low wind speed, there are immediate rebound and obvious attenuation vibration stages during the stress change of dropper. In addition, with the increasing speed of wind load, the stress amplitude and the maximum tensile stress of dropper do not almost change or decrease except the middle dropper.

Whirl dynamics of an axially functionally graded liquid-filled rotor considering shear deformation and rotary inertia

In this study, whirl characteristics and stability of an axially functionally graded (AFG) liquid-filled rotor are investigated. The rotor is modeled based on the spinning Timoshenko beam theory. The governing equations for flexural vibration are derived via Hamilton’s principle. For pinned–pinned AFG liquid-filled rotor, the analytical solutions are derived for both the exact whirl frequency equation and the stability model. To validate the present formulations, comparative studies by numerical solutions available in the literature are conducted. Some numerical examples are performed to investigate the effects of gradient parameter, mass ratio, cavity ratio, rotary inertia, and shear deformation on the whirl speed, the critical spinning speed, and the stability of the AFG liquid-filled rotor system. The results show that these parameters have noticeable influences on dynamic behavior and stability of the rotor system. In particular, the rotary inertia and shear deformation play an important role in the stability analysis for different length rotors.

Flow induced bifurcation and phase-plane stability analysis of branched nanotubes resting on two parameter foundation in a magnetic environment

This paper explores the nonlinear partial differential equations (PDEs) for transverse and longitudinal vibrations of slightly curved fluid-conveying embedded branched nanotubes operating in a magnetic environment. Using Bernoulli–Euler, Nonlocal and Hamilton theories, equations of motion governing the vibrations of the nanotubes are developed. Thereafter, the developed inimitable equations are solved using the numerical PDE solver and PDE-tool in MATLAB. With the numerical solutions, visualizations and parametric studies were carried out. The results indicate that increasing the branch angle at the downstream decreases quantitative stability. Furthermore, the stability results acquired via simulation point out that the magnetic field has a damping or attenuating influence of about 20%. The solutions as presented in this study match with existing results from previous studies, hence the verifications and validations of this work. It is envisioned that the results obtained will provide better physical insights into the stability criteria for carbon nanotubes and will enhance the designs of nanomechanical carbon nanotubes-based devices that convey fluid and operate in elastic and magnetic environments.

Vibration suppression of collaborative robot based on modified trajectory planning

PurposeThe paper aims to reduce the low-frequency resonance and residual vibration of the robot during the operation, improve the working accuracy and efficiency. A reduced weight and large load-to-weight ratio can improve the practical application of a collaborative robot. However, flexibility caused by the reduced weight and large load-to-weight ratio leads to low-frequency resonance and residual vibration during the operation of the robot, which reduces the working accuracy and efficiency. The vibrations of the collaborative robot are suppressed using a modified trajectory-planning method.Design/methodology/approachA rigid-flexible coupling dynamics model of the collaborative robot is established using the finite element and Lagrange methods, and the vibration equation of the robot is derived. Trajectory planning is performed with the excitation force as the optimization objective, and the trajectory planning method is modified to reduce the vibration of the collaborative robot and ensure the precision of the robot terminal.FindingsThe vibration amplitude is reduced by 80%. The maximum torque amplitude of the joint before the vibration suppression reaches 50 N·m. After vibration suppression, the maximum torque amplitude of the joint is 10 N·m, and the resonance phenomenon is eliminated during the operation process. Consequently, the effectiveness of the modified trajectory planning method is verified, where the vibration and residual vibration in the movement of the collaborative robot are significantly reduced, and the positioning accuracy and working efficiency of the robot are improved.Originality/valueThis method can greatly reduce the vibration and residual vibration of the collaborative robot, improve the positioning accuracy and work efficiency and promote the rapid application and development of collaborative robots in the industrial and service fields.

A three-machine equivalent system frequency response model and its closed-form solution

The frequency response (FR) of the large-scale power system shows significant frequency spatial distribution characteristics (FSDC), still using uniform frequency models may cause large errors in the FR solution, and the FSDC should be preserved in the modeling. In this paper, a three-machine equivalent system frequency response (TM-SFR) model is proposed to preserve the FSDC by the parameter equivalent aggregation method, including load, network, and generator. The TM-SFR model is further simplified into a solvable forced vibration equation by loop-opening and approximation, and its closed-form solution (CFS) is obtained by the modal superposition method, to analytically solve the FR following a major disturbance. Case studies are introduced to verify the performance of the TM-SFR model and its CFS over the WSCC 9-bus system, the New England 39-bus system, and the IEEE 118-bus system. The results prove that the proposed model can preserve the intermachine frequency oscillation caused by the FSDC in various scenarios, and improve the FR solution accuracy at high solution speed, which can provide a more reliable basis for frequency stability emergency control.

Monotone iterative technique for nonlocal problems of damped elastic systems with delay

In this paper, based on the monotone iterative technique and the operator semigroup theory, a class of nonlocal problem of structural damped elastic systems with delay is studied in the case of noncompact semigroups in ordered Banach space. First, on the premise of the existence of upper and lower mild solutions, the existence and uniqueness of mild solutions for the elastic system are obtained. Then, an existence theorem of positive mild solutions for elastic system is obtained without assuming lower and upper mild solutions. Finally, as the application of abstract results, the existence and uniqueness of mild solutions and positive mild solutions for two classes of nonlocal damped beam vibration equations with delay are discussed.

On the coupled theory of thermoelastic double-porosity materials

In this paper, the linear model of thermoelastic double-porosity materials is proposed in which the coupled phenomenon of the concepts of Darcy’s law and the volume fractions is considered. The basic systems of governing equations of motion and steady vibrations are obtained. Then, the boundary value problems of steady vibrations are investigated by means of the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, on the basis of Green’s identities and the surface and volume potentials the existence and uniqueness theorems for the classical solutions of the internal and external basic boundary value problems of the considered theory are proved.

Comparative Study on Damping Ratio of Different Materials

The concept of vibrations has an eminent role in the field of Mechanical operations. The operation of a machine may introduce additional vibrations which is undesirable in many domains, primarily engineered systems and methods are being developed in order to prevent transfer of vibration to the concerned engineering system. This is achieved with the help of vibration isolators (Dampers). The vibrational aspect of analysis has a detrimental effect in order to understand the response of system. There are various software packages available in the market to track down the frequency response of the specimen for analysis purpose. The article includes the comparison of the dynamic properties of Steel, Aluminum, Granite and Mineral Casting. The main aim of this comparison is to study the natural frequencies and mode shapes and showcase that the Mineral Casting has a better damping than the latter specimens under observation. This would help to figure out the scope of Mineral Casting for machine tool application. Experimentation is carried out using FFT analyzer and the specimens were mounted in Free-Free Configuration. In short, the experiment is a case of Damped Free Vibration and the results are validated based on general equations of free vibration, etc. Key Words: Composite, Damped Free Vibration, Damping, FFT Analyzer, Frequency Response, Mineral Casting, Vibration Isolators.