Transverse Vibration Of Beam Research Articles

Non-linear vibration and bifurcation analysis of Euler-Bernoulli beam under parametric excitation

This paper presents an analysis of the non-linear vibrations of beams, which play a crucial role in various industrial and construction structures. Understanding the transverse vibrations of beams and accurately determining their frequency response is essential for achieving optimal design and structural performance. The novelty of this study lies in conducting a transverse non-linear vibration analysis of a three-dimensional beam while considering the effect of mid-plane elongation. By incorporating this aspect into the analysis, the study aims to provide deeper insights into the dynamic behavior of beams subjected to non-linear effects. A multiple-time scale approach has been adopted to conduct this research. To verify the accuracy of the method as well as the accuracy of the outcomes gained from this method, a contrast has been made with the 4th-order Runge-Kutta technique, which indicates that the results obtained are acceptable. The frequency response of the beam indicates the presence of a phenomenon of splitting into two non-linear branches during the three-dimensional vibrations of the beam, as well as a hardening state in the frequency response as a result of stretching the middle plane of the beam. Furthermore, a parametric study was conducted in which different parameters were examined to determine the starting point of non-linear bifurcation. As a result, the damping coefficient and resonance deviation parameter are two factors that affect the preference for critical bifurcation over safe bifurcation. Furthermore, the stretching of the middle plane results in a higher non-linear term coefficient in the vibration equations of the beam, which increases the oscillation frequency of the beam.

Active Vibration Damping Using Rotor Torque Control in Electric Aircraft

Vibration damping in aerospace structures can decrease the likelihood of failures and instabilities and improve passenger comfort. This paper introduces the novel idea of damping vibration using the electric proprotors on aircraft without compromising flight control. The equations of motion of a cantilevered beam with a propeller at the tip driven by an electric motor are obtained using Hamilton’s principle, solved analytically in the frequency domain, and approximately in the time domain. Feeding back the beam tip angular rate to the motor torque is shown to asymptotically stabilize all transverse beam vibration modes. The overall vibration control consists of the inner rate feedback damping loop with an outer rotor speed control loop. Experimental frequency response and step response validate the models and show that the closed-loop damping in the first mode is three times higher than open loop with less than 1% rotor speed change for a 3% initial tip displacement. Theoretical results give good agreement with experiments. Parametric studies based on a 12 kg quadcopter indicate that if the rotor inertia is sufficiently large the proposed control can provide 8% damping with minimal impact on rotor speed.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

Exact solutions of Euler–Bernoulli beams

In numerous real-world applications, transverse vibrations of beams are nonlinear in nature. It is a task to solve nonlinear beam systems due to their substantial dependence on the 4 variables of the system and the boundary conditions. To comprehend the nonlinear vibration characteristics, it is essential to do a precise parametric analysis. This research demonstrates an approximation solution for odd and even nonlinear transverse vibrating beams using the Laplace-based variation iteration method, and the formulation of the beams depends on the Galerkin approximation. For the solution of the nonlinear differential equation, this method is efficient as compared to the existing methods in the literature because the solutions exactly match with the numerical solutions. The Laplace-based variation iteration method has been used for the first time to obtain the solution to this important problem. To demonstrate the applicability and precision of the Laplace-based iteration method, several initial conditions are applied to the governing equation for nonlinearly vibrating transverse beams. The natural frequencies and periodic response curves are computed using Laplace-based VIM and compared with the Runge–Kutta RK4 method. In contrast to the RK4, the results demonstrate that the proposed method yields excellent consensus. The Lagrange multiplier is widely regarded as one of the most essential concepts in variational theory. The result obtained are displayed in the table form. Highlights The highlights of the solution of the Euler–Bernoulli beam equation with quintic nonlinearity using Lagrange multiplier are: 1. Introducing the constraint of the boundary conditions into the equation using Lagrange multipliers. 2. Formulating the equations for the Lagrange multipliers and the deflection of the beam. 3. Solving the resulting system of algebraic equations using numerical methods. 4. Obtaining the deflection of the beam as a function of its length and the applied load. 5. Analyzing the behavior of the beam under different loads and boundary conditions.

Analytical and Experimental Investigation of the Rotary Inertia Effects of Unequal End Masses on Transverse Vibration of Beams

In this study, the transverse vibration of free–free slender beams with two unequal end masses attached were studied. The effects of the rotary inertia of the end masses on the free vibration of the beam were investigated. An exact frequency equation and the boundary conditions were obtained by using the Euler–Bernoulli beam theory and Hamilton’s principle. Natural frequencies and mode shapes of the beams in transverse vibrations were calculated for various combinations of physical and geometrical parameters, such as mass ratios, the distances between the attachment point and the center of the masses, etc. The effects of an increase in the rotational inertia of the end masses and different mass ratios on the natural frequencies and mode shapes of the beam are presented. It is shown that the increase in the rotational inertia of the end masses had a greater effect at low frequencies of the beam. In addition, experimental tests were performed to validate the obtained analytical results. A good agreement was obtained between the analytical and experimental results. The main scope of this study was to reveal the effects of the rotary inertia of the end masses on the dynamic behavior of the beam. Thus, the aim is to contribute to the understanding of the properties of the end mass and the effect of rotary inertia on the dynamics of end-mass-attached structures. Furthermore, the results obtained from this research are helpful for designing end-mass-attached structures, such as micromechanical sensors, energy harvesters, and Stockbridge-type dynamic absorbers.

The 100th Anniversary of the Timoshenko–Ehrenfest Beam Model

The 100th Anniversary of the Timoshenko–Ehrenfest Beam Model

Supercritical nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading

Dynamics of a hyperelastic beam in a buckled state subjected to a harmonic axial load is investigated in this work. For the static case, the buckled configuration of the hyperelastic beam is first determined through an asymptotic method when the axial load is in excess of its critical load. Then, the governing equation of vibration of the buckled beam is derived, which is a complex nonlinear partial differential equation with varying coefficients. The first natural frequency of the buckled beam is obtained by linear analysis and effects of material and geometrical parameters on it are numerically investigated. By applying the Galerkin method, the governing equation for nonlinear transverse vibration of the beam in the buckled state is transformed into a series of strongly nonlinear ordinary differential equations. Dynamic characteristics of the hyperelastic beam are investigated by the Runge–Kutta method. Bifurcation diagrams, time traces, phase-plane portraits, and Poincare sections are obtained for different values of the external excitation frequency and amplitude of variation of the axial load. Results show that different dynamic behaviors of the hyperelastic buckled beam such as periodic, multi-periodic, quasiperiodic, and chaotic motions can be found when the amplitude of the axial load is varied. Amplitude–frequency responses of the hyperelastic buckled beam are obtained by the Runge–Kutta method and harmonic balance method. Results from the Runge–Kutta method are in good agreement with those from the harmonic balance method. Lastly, effects of the external mean axial load, amplitude of variation of the axial load, geometrical and material parameters on amplitude–frequency responses of the buckled beam are numerically investigated.

Study on Vibration Band Gap Characteristics of a Branched Shape Periodic Structure Using the GDQR

In this study, a new periodic structure with special vibration band gap properties is introduced. This structure consists of a main beam and several cantilever beam elements connected to this main beam in the branched shape. Two models with different number of beam elements and geometrical parameters are considered for this periodic structure. The transverse vibrations of beams are solved using the generalized differential quadrature rule (GDQR) method to calculate the first four band gaps of each model. Investigating the influences of geometrical parameters on the band gaps shows that some bands are close to each other for specific ranges of geometrical parameters values. Furthermore, as the number of beam elements increases, the number of close band gaps increases. Having more than two close band gaps means that this periodic structure has a relatively wide band gap in total. Furthermore, this wide band can move to low frequency ranges by changing the geometrical parameters. Absorbing vibrations over a wide band gap at low frequency ranges makes this periodic structure a good vibration absorber. Verification of the analytical method using ANSYS software shows that the GDQR method can be used for vibration analysis of beam-like structures with high accuracy.

Nonlinear Transverse Vibration of a Hyperelastic Beam Under Harmonically Varying Axial Loading

Abstract Nonlinear transverse vibration of a hyperelastic beam under a harmonically varying axial load is analyzed in this work. Equations of motion of the beam are derived via the extended Hamilton's principle, where transverse vibration is coupled with longitudinal vibration. The governing equation of nonlinear transverse vibration of the beam is obtained by decoupling the equations of motion. By applying the Galerkin method, the governing equation transforms to a series of nonlinear ordinary differential equations (ODEs). Response of the beam is obtained via three different methods: the Runge–Kutta method, multiple scales method, and harmonic balance method. Time histories, phase-plane portraits, fast Fourier transforms (FFTs), and amplitude–frequency responses of nonlinear transverse vibration of the beam are obtained. Comparison of results from the three methods is made. Results from the multiple scales method are in good agreement with those from the harmonic balance and Runge–Kutta methods when the amplitude of vibration is small. Effects of the material parameter and geometrical parameter of the beam on its amplitude–frequency responses are analyzed.

Nonlinear transverse vibration of a hyperelastic beam under harmonic axial loading in the subcritical buckling regime

Equations of motion of a hyperelastic beam under time-varying axial loading are derived via the extended Hamilton’s principle in this work, where the transverse vibration is coupled with the longitudinal vibration, and nonlinear vibrations of the beam in the subcritical buckling regime are investigated. Complex nonlinear boundary conditions of the beam are determined under some geometric constraints. The critical buckling load is first determined through linear bifurcation analysis. Effects of material and geometric parameters on the forced longitudinal vibration of the beam are numerically investigated. Steady harmonic shapes of the beam at different times under harmonic axial loading are determined. The beam is in the barreling deformation state even when the axial load is not in excess of the critical buckling load. The governing equation for the nonlinear transverse vibration of the beam is obtained by decoupling its equations of motion. Natural frequencies of the free linearized transverse vibration of the beam are studied. By applying the eigenfunction expansion method, the governing equation for the nonlinear transverse vibration of the beam transforms to a series of strongly nonlinear ordinary differential equations (ODEs). Two-to-one internal resonance of the beam is studied by the numerical integration method and its phase-plane portraits are obtained. The harmonic balance method and pseudo arc-length method are used to determine steady-state periodic solutions of the beam from the strongly nonlinear ODEs, and amplitude-frequency responses of the beam are determined. Effects of the external mean axial load, excitation amplitude, and damping coefficient on the amplitude-frequency response of the beam are numerically investigated. Combined effects of the external excitation amplitude and frequency on response amplitudes are also investigated.

A New Approach to Compute Natural Frequencies and Mode Shapes of One-Dimensional Continuous Structures With Arbitrary Nonuniformities

AbstractOne-dimensional continuous structures include longitudinal vibration of bars, torsional vibration of circular shafts, and transverse vibration of beams. Using the linear time-varying system theory, algorithms are developed in this paper to compute natural frequencies and mode shapes of these structures with nonuniform spatial parameters (mass distributions, material properties and cross-sectional areas) which can have jump discontinuities. A general numerical approach has been presented to include Dirac-delta functions and their spatial derivatives due to jump discontinuities. Numerical results are presented to illustrate the application of these techniques to the solution of different types of spatial variations of parameters and boundary conditions.

Analysis of complete vibration bandgaps in a new periodic lattice model using the differential quadrature method

In this study, a new periodic lattice model with special vibration-absorbing properties is introduced. This periodic structure consists of the connected beam elements with circular cross-sections. Four models with different sets of cross-sectional radii are considered for this periodic lattice. The theoretical equations of longitudinal, torsional, and transverse vibrations of beams are solved using the combination of generalized differential quadrature and generalized differential quadrature rule methods to calculate the first three complete bandgaps. Investigating the effects of geometrical parameters on the bandgaps shows that all bands are close to each other for specific values of the cross-sectional radii. Having close bandgaps means that this periodic structure has a relatively wide bandgap in total. Furthermore, this wide band can move to low-frequency ranges by changing the lattice thickness. Absorbing both in-plane and out-of-plane vibrations over a wide bandgap at low-frequency ranges makes this periodic lattice a good vibration absorber. Verification of the analytical method using ANSYS software shows that the combination of generalized differential quadrature and generalized differential quadrature rule methods can be used for vibration analysis of two- or three-dimensional structures such as frames and trusses with high accuracy.

Exact mathematical solution for nonlinear free transverse vibration of beams

In the present paper, an exact mathematical solution has been obtained for nonlinear free transverse vibration of beams, for the first time. The nonlinear governing partial differential equation in un-deformed coordinates system has been converted in two coupled partial differential equations in deformed coordinates system. A mathematical explanation is obtained for nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of beam. It is shown that as the th mode of transverse vibration excited, the mode 2 th of in-plane vibration will be developed. The result of present work is compared with those obtained from Galerkin method and the observed agreement confirms the exact mathematical solution. It is shown that the governing equation is linear in the time domain. As a parameter, the amplitude to length ratio has been proposed to show when the nonlinear terms become dominant in the behavior of structure.

An analytical-numerical approach to vibration analysis of periodic Timoshenko beams

The subject of this article is analysis of transverse vibrations of beams which geometric and material properties vary periodically along the longitudinal axis. The aim is to present averaged models that take into account the shear deformation and geometric non-linearity, and to analyse transverse vibrations of such beams in moderately large deflection range. As the theoretical foundations, we use Timoshenko beam theory with von Kármán-type non-linearity. This results in obtaining new differential equations with constant coefficients, some of which explicitly depend on the beam inhomogeneity period size. Then, a reasonably simplified model is proposed to describe the vibrations of the considered beams in the low frequency range. The differential equations are transformed into a system of algebraic equations according to the Galerkin method. The response of the beam to transverse harmonic load is investigated by means of a pseudo arc-length continuation scheme. Non-linear coupling between vibration modes and the possibility of superharmonic resonance occurrence are taken into account. As an example of application, few special cases of beam geometry and boundary conditions are examined and compared. The results have the potential application to structural vibration control.

Nonlinear isolation of transverse vibration of pre-pressure beams

Research on nonlinear isolation has always focused on the vibration of discrete systems. The elastic vibration of the main structure itself is always ignored. In order to study the influence of the multimodal elastic vibration of the main structure on the nonlinear isolation effect, nonlinear isolation of the transverse vibration of a pre-pressure elastic beam is studied in this paper. Three linear springs are utilized to build a nonlinear isolation system, in which horizontal springs are utilized to provide non-linearity and to achieve quasi-zero stiffness. The transverse vibration of the beam is isolated by the elastic support at the two ends. The Galerkin truncation method (GTM) is used to solve the response of the forced vibration. The isolation effect of the primary resonance of the pre-pressure beam is presented. The effects of the axial pre-pressure and nonlinear isolation system on vibration transmission are explored. Results of the GTM are confirmed by utilizing the finite difference method (FDM). It also illustrates the effectiveness of the proposed difference method for nonlinear support structures. The numerical results demonstrate that under certain conditions, the quasi-zero stiffness isolation system may increase the transmission of high-order modal vibration of the elastic continuum. Furthermore, this work finds that the initial axial pre-pressure could be beneficial to vibration isolation of elastic structures.

Free vibration analysis of axially functionally graded tapered, stepped, and continuously segmented rods and beams

Abstract In this paper a new non-iterative computational technique referred to as the symbolic-numeric method of initial parameters (SNMIP) is proposed. The SNMIP represents a modification of the iterative numeric method of initial parameters in differential form known in the literature. The SNMIP is applied to study free vibrations of Euler-Bernoulli axially functionally graded tapered, stepped, and continuously segmented rods and beams with elastically restrained end with attached masses. Both the longitudinal vibration of rods and transverse vibration of beams are considered. The influence of the attached masses and springs on the natural frequencies of vibration of axially functionally graded rods and beams is examined. The validity and accuracy of the method are proven through the comparison with the known results in the available literature.

Identification of equivalent mechanical properties for unreinforced masonry walls

Transverse vibrations of beams are an interesting topic for numerous engineering fields. The equations of motion develop under various assumptions and associate a frequency equation. This paper expresses the frequency equation for two specific models, using the Timoshenko beam theory. The first model is a typical cantilever beam with an additional mass attached to the free end. The second model considers a beam partially clamped at the base and with an elastic connection of the additional mass to the free end. The relevance of the used theory is discussed and the importance of each term of the equation is studied. These models find applications in unreinforced masonry. They are indeed fitted to identify, from white noise tests, equivalent mechanical properties of unreinforced load-bearing clay masonry walls including soundproofing devices in the perspective of modelling these elements with a macro-scale approach. Discrepancies with respect to current standards are underscored.

Identification of equivalent mechanical properties for unreinforced masonry walls

Transverse vibrations of beams are an interesting topic for numerous engineering fields. The equations of motion develop under various assumptions and associate a frequency equation. This paper expresses the frequency equation for two specific models, using the Timoshenko beam theory. The first model is a typical cantilever beam with an additional mass attached to the free end. The second model considers a beam partially clamped at the base and with an elastic connection of the additional mass to the free end. The relevance of the used theory is discussed and the importance of each term of the equation is studied. These models find applications in unreinforced masonry. They are indeed fitted to identify, from white noise tests, equivalent mechanical properties of unreinforced load-bearing clay masonry walls including soundproofing devices in the perspective of modelling these elements with a macro-scale approach. Discrepancies with respect to current standards are underscored.

3-node Basic Displacement Functions in Analysis of Non-Prismatic Beams

Purpose– Analysis of non-prismatic beams has been focused of attention due to wide use in complex structures such as aircraft, turbine blades and space vehicles. Apart from aesthetic aspect, optimization of strength and weight is achieved in use of this type of structures. The purpose of this paper is to present new shape functions, namely 3-node Basic Displacement Functions (BDFs) for derivation of structural matrices for general non-prismatic Euler-Bernoulli beam elements. Design/methodology/approach– Static analysis and free transverse vibration of non-prismatic beams are extracted studied from a mechanical point of view. Following structural/mechanical principles, new static shape functions are in terms of BDFs, which are obtained using unit-dummy-load method. All types of cross-sections and cross-sectional dimensions of the beam element could be considered in this method. Findings– According to the outcome of static analysis, it is verified that exact results are obtained by applying one or a few elements. Furthermore, it is observed that results from both static and free transverse vibration analysis are in good agreement with the previous published once in the literature. Research limitations/implications– The method can be extended to structural analysis of curved and Timoshenko beams as well as plates and shells. Furthermore, exact dynamic shape functions can be derived using BDFs by solving the governing equation for transverse vibration of beams. Originality/value– The present investigation introduces new shape functions, namely 3-node Basic Displacement Functions (BDFs) extended from 2-node functions, and then compares its performance with previous element.

Fluidic flexible matrix composite damping treatment for a cantilever beam

This paper presents a novel approach for damping the vibration of a cantilever beam by bonding multiple fluidic flexible matrix composite (F2MC) tubes to the beam and using the strain induced fluid pumping to dissipate energy. Transverse beam vibration strains the F2MC tube and generates fluid flow through an energy dissipating orifice. An optimally sized orifice maximizes energy dissipation, greatly reducing the resonant peaks and increasing modal damping. An analytical model is developed based on Euler–Bernoulli beam theory and Lekhnitskii׳s solution for anisotropic layered tubes. Using miniature tubes, a laboratory-scale F2MC-integrated beam prototype is constructed and experimentally tested. The experimental results agree well with the theoretical predictions, provided the fluid bulk modulus is reduced to reflect the entrained air in the fluidic circuit. Based on the validated model, a design space study calculates the modal damping for various tube attachment locations and the orifice sizes. The results show that damping ratios of 32percent and 16percent are achievable in the first and second modes of a cantilever beam, respectively, using an F2MC damping treatment.