Vibration Modes Of Beams Research Articles

Analysis of the effect of nonlocal factors on the vibration of nanobeams

Abstract Currently, the Eluer-Bernoulli beam nonlocal theory does not fully consider the effects of foundation deformation and axial force on the beams, and cannot accurately reflect the real mechanical properties of nanobeams. The primary objective of this study is to introduce a novel computational method designed for an enhanced characterization of the vibrational behavior of nano beams. Initially, this method incorporates the influence of foundation deformation on beam bending, accounts for the effects of axial forces, integrates Eringen's nonlocal theory, and establishes a modified Euler-Bernoulli beam theory model for the first time, accompanied by a degradation validation of the model. Subsequently, the Laplace transform and Hasselman's complex mode synthesis method are utilized to solve the model, providing the first derivation of the state-space transfer function for the nano beam vibration model based on the modified Euler-Bernoulli beam theory that includes axial force effects. Lastly, the study elucidates the impact of nonlocal factors and various parameters on the vibration characteristics of nanobeams. The results show that the order n increases, and the peak frequency value moves in the direction where the nonlocal factor tends to zero. When the order n is less than 6, the frequency peak is mainly concentrated in the interval where the nonlocal factor is greater than 0 and less than 0.1. At the same order, the beam length increases, and the peak frequency moves in the direction of increasing non-local factor. The modified geometric parameters and the foundation beam stiffness parameters have a greater effect on the peak of the beam's vibration mode in the higher order case and a lesser effect in the lower order case. The larger the nonlocal factor, the larger the peak of the vibration mode. The foundation beam stiffness parameter also has an effect on the direction of the peak vibration mode at the 1st and 2nd orders. The research results will help to predict the vibration behavior of nano beams under different conditions more accurately and provide an important reference for related applications.

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.

Modal analysis and harmonic response analysis of energy-absorbing and anti-scouring columns

Many field observations have found that damage to the support did not increase with the magnitude of the impact ground pressure. To enhance the impact protection performance of “support and surrounding rock” system. From the perspective of preventing the resonance of the “support and surrounding rock” system, the dynamic response of energy-absorbing and anti-scouring column is studied. Using the block theory, a dynamic model of the roadway enclosure-absorbing anti-scouring column under impact loading was established, and the dynamic response equation of the rock block at the support end was obtained. Based on the structural dynamics and Love shell theory, the characteristic equations of the energy-absorbing and anti-scouring column were derived, and the theoretical equations for the natural frequency and vibration mode function of the energy-absorbing and anti-scouring column were obtained. The ABAQUS numerical simulation method was used to carry out pre-stress modal analysis and harmonious response analysis of energy-absorbing and anti-scouring columns and conventional columns. We obtained the natural frequencies and vibration modes of the two columns and analyzed the effect of the setting load on the modalities of the columns. The column dynamic response law was determined by monitoring the displacement and velocity response curves at different positions. The results show that the natural frequency and vibration mode of the energy-absorbing and anti-scouring column are related to the density, elastic modulus, and length. The vibration of the energy-absorbing and anti-scouring column has two forms: beam vibration mode and cylindrical shell vibration mode, and the first four orders of vibration mode are transverse bend. The setting load has a small effect on the natural frequency and vibration mode of both types of columns, where the natural frequency decreases as the setting load increases. When the excitation frequency is close to the vibration frequency of the surrounding rock, the radial, axial, and circumferential response amplitudes of the primary and secondary columns of the energy-absorbing and anti-scouring column are the largest. Reinforcement and strengthening measures should be implemented to reduce the column deformation amplitude.

Method for efficient excitation of selective vibration modes in pulsed laser photothermal actuation

Photothermal excitation based on thermoelastic mechanisms is widely used in non-destructive testing, precision operations, and driving micro-resonators. The narrow drive bandwidth of the high vibration mode in photothermal excitation limits its application to multi-mode drives. Controlling the laser’s irradiation position is an effective solution. In this study, we build a theoretical model to achieve selective and efficient excitation of different flexural vibration modes of beams with different supports. The model can be extended to other thermal and physical boundaries, which is validated by numerical simulations and experimental results. The results show that higher modes with complex periodic shapes can be efficiently excited by focusing the laser at the peak of the absolute value of the second derivative of the flexural mode while focusing the laser at the inflection point of the mode shape will result in extremely small amplitudes. Our study indicates that the thermal gradient plays a vital role in the oscillation of the beam. The conventional view assumes that the resonance of the photo-thermal excitation beam is caused by the local expansion and contraction of the material, which cannot completely explain the dependence principle of the photothermal vibration on the laser irradiation position. To investigate the mechanism of beam resonance under laser excitation, three excitation modes, unidirectional excitation, bidirectional in-phase excitation, and bidirectional anti-phase excitation, were established, and the conversion process of optical energy to mechanical energy under laser excitation was analyzed. These results provide new options for optimal excitation and multi-mode energy flow control in photothermal driving.

Finite element mesh refinement for in-plane and out-of-plane vibration of variable geometrical Timoshenko beams based on superconvergent vibration modes

PurposeIn this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.Design/methodology/approachAn adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.FindingsThe results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.Originality/valueThe proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

Effects of the Variational Consistency and Variational Inconsistency on Free Flexural Vibration Frequencies of Simply-supported Rectangular Isotropic Shear Deformable Beams

In order to derive governing differential equations of a displacement-based shear deformation beam theory, one can either utilise the variationally consistent approach (VCA) or the variationally inconsistent approach (VIA). In the VCA, variational principles are utilised to derive beam governing equations. Whereas in the VIA, beam gross / integrated two-dimensional equilibrium equations are utilised to derive beam governing equations. Shimpi et al. [2] have proposed the single variable shear deformation beam theory (SVSDBT) which is based on the VIA. Whereas, Shimpi et al. [1] have proposed the two variable shear deformation beam theory (TVSDBT) which is based on the VCA. In the SVSDBT and TVSDBT, the beam axial displacement and beam transverse displacement consist of bending and shearing components. In both theories, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. Both theories utilise linear strain-displacement relations. Displacement functions of both theories give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. For the case of dynamics, the SVSDBT has only one governing equation, and the TVSDBT has two governing equations which are inertially coupled. The aim of this paper is to present the comparison study on free flexural vibration frequencies of simply-supported rectangular isotropic shear deformable beams obtained by utilising the SVSDBT and TVSDBT. These frequencies are obtained for various values of the beam thickness-to-length ratio and beam vibration mode parameter. This comparison shows that for both theories, the VCA and VIA have a minimal effect on resulting beam free flexural vibration frequencies.

重要文化財法華経寺五重塔の地震・風応答観測とその考察

Motion behaviors of the national heritage five story wooden pagoda at Hokekyouji temple are observed from 2007. Continuing part 1,modal and response analyses for the pagoda are conducted. The model is made by using the beam vibration modes. Response analyses are carried out for both 3/11 earthquake and strong wind. As results of linear numerical calculations to strong earthquake motion and strong wind, the damping about 5%-10% would be agreed with observed response data. Modal analysis results showed the possibility of resonance between a pagoda frame and center column. However, in actual pagoda, it would be difficult to expect dynamic damper effects because of irregularities of pagoda structure.

Vibration Analyses of Elastic Wheelset Model with Concentrated Mass and Moment of Inertia

With the continuous improvement of train speed, the effect of the mass, the moment of inertia and the elasticity of the wheelset in the wheel-track vibration system is increasing and can not be ignored any more. By using the assumed mode method, a mode function of axle with a concentrated mass and a concentrated moment of inertia is obtained, which is compared with the straight beam's vibration mode function. Based on which a vehicle system dynamic model is constructed, considering the rigid body motion and elastic modes of the axle. By numerical method we got the force, vertical acceleration and the acceleration power spectrum density(PSD) of the rail-wheel coupling system, concentrated mass and moment of inertia Comparing to former researches, this methodology can more accurately reflect the axle vibration, the impact of wheel-rail force and attenuation characteristics.

Genetic algorithm based wireless vibration control of multiple modal for a beam by using photostrictive actuators

The lanthanum-modified lead zirconate titanate (PLZT) actuator, which are capable of converting photonic energy to mechanical motion, have great potential in applications of remote structural vibration control of smart structures and machines. In this paper, a novel genetic algorithm based controlling algorithm for multi-modal vibration control of beam structures via photostrictive actuators is proposed. Two pairs of photostrictive actuators are laminated with the beams and the alternation of light irradiation is in accordance with the changing of the corresponding modal velocity direction. The modal force indexes for beams with different boundary conditions are derived and a binary-coded GA is used to optimize the locations and sizes of photostrictive actuators to maximize the modal force index and guarantee the overall modal force index induced by two pairs of photostrictive actuators is positive. The control effect of multiple vibration modes of beams under irradiation of set/variable light intensity is analyzed. Numerical results demonstrate that the method is robust and efficient, and the use of strategically positioned actuator patches can effectively control the first two bending modes that dominate the structural vibration.

Coupled vibration of a cantilever micro-beam submerged in a bounded incompressible fluid domain

This paper investigates the free vibrations of a cantilever micro-beam submerged in a bounded frictionless and incompressible fluid cavity. Based on the Fourier–Bessel series expansion and using linear potential theory, an analytical method is proposed to analyze the eigenvalue problem, where the fluid effect emerges as an added mass. Wet beam vibration mode shapes together with the sloshing modes of the oscillating liquid are depicted. Moreover, effects of geometrical configuration and fluid density on the natural frequencies of the coupled system are evaluated. Results show that in spite of the high added mass values related to lower modes, presence of the fluid changes the higher modes more effectively.

建物の水平加振時の上下振動成分データを利用した鉄骨梁端部の破断検知

A vibration test for a 9-story building shows that fractures at steel beam-to-column connections change the vertical vibration characteristics at the center of the beams. By use of the result, this study proposes a fracture detecting method, in which beam vibration modes are compared by MAC (modal assurance criterion) value each other, then, fractures are detected to locate at beams related with peculiar MAC values.Finite element method analysis is executed for the 9-story building to examine the proposed method. MCS (Monte Carlo Simulation) is also executed in order to consider the effect of variation in beam vibration, which is found in practical buildings. The MCS shows that even with 15% of the maximum displacement of beam as variation, which is around half of that measured in a test for a 3-story building, the proposed method can be informative for fracture detection. Future works are to develop the criteria to judge whether to be fractured as well as to accumulate the data on practical buildings.

Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler-Bernoulli, Timoshenko, and Sheremet’ev-Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation O(h2) and the Bubnov-Galerkin finite element method. It is shown that for a relative thickness λ ⩽ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.

Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks

In this study, exact closed-form expressions for the vibration modes of the Euler–Bernoulli beam in the presence of multiple concentrated cracks are presented. The proposed expressions are provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations. Besides the evaluation of the natural frequencies, neither computational work nor recurrence expressions for the vibration modes are required. The cracks, that are not subjected to the closing phenomenon, are modelled as a sequence of Dirac's delta generalised functions in the flexural stiffness. The Eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions, which are already accounted for in the adopted flexural stiffness model. The vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed-form expressions in order to show their efficiency.

Modeling of a hybrid passive damping system

The modeling of a hybrid passive damping system is presented for suppressing the multiple vibration modes of beams in this paper. This hybrid passive damping system consists of a constrained layer damping and a resonant shunt circuit. In a passive mechanical constrained layer damping, a viscoelastic layer, which is sandwiched between a host structure and a cover layer, is used to suppress vibration amplitudes in the high frequency range. A passive electrical damping is designed for targeting the vibration amplitude in the low frequency range. The governing equations of motion of the hybrid passive damping system are derived through the Hamilton's principle. The obtained mathematical model is validated experimentally. The theoretical and experimental techniques presented provide an invaluable tool in controlling the multiple vibration modes across a wide frequency band.

Large Amplitude Free Vibration Study of Square Plates under Different Boundary Conditions Through a Static Analysis

The free v ibration problem of thin isotropic plates incorporating the effect. of geometric non-linearity is studied by developing a specific numerical methodology. The problem is formulated by using a variational method. The large amplitude vibration problem is addressed by solving the corresponding static problem first through an iterative schemne using a relaxation parameter. Subsequently. with the resulting displacement field, the dynamic problem is solved as a standard eigenvalue problem. The assumed deflection field, required for the analysis, is constituted through linear combinations of beam vibration modes corresponding to the specific boundary conditions of the plate. Typical results for the square plates, in the form of backbone curves, have been furnished in the dimensionless amplitude-frequency plane. Two different combinations of the boundary conditions for the purpose are chosen following an earlier benchmark work. Much insight on large amplitude dynamic behavior of the plate is obtained through the vibration mode shapes. presented for each case. A comparison of the results of the reduced problem with those of earlier studies indicates excellent agreement.

Vibration Characteristics of Fatigue Damage of Beam-Type Structural Components

We consider free vibration of an elastic cantilever beam with a closing edge crack. The crack is simulated by an elastic pivot whose compliance is taken to be finite and is determined from the energy equivalence criterion. An algorithm has been constructed for sequential calculation of the beam vibration modes for each beam vibration cycle, whereas the number of cycles is limited. It is demonstrated that the beam vibration modes other than the initially preset one are induced during the crack opening and closing. The distinct features of such modes have been studied. We have analyzed the sensitivity of the vibration-damage characteristics which are determined from the assessment of nonlinear distortions of the displacement, acceleration, and strain waveform of the cross sections of the beam with a closing crack.

Transient vibration and sound radiation of a rectangular plate with viscoelastic boundary supports

AbstractThis work considers transient vibration and sound radiation from an impact‐excited rectangular plate with viscoelastic boundary supports. The approach used is based on the modal strain energy (MSE) method. Vibration of the plate is approximated by a double infinite series in the spatial co‐ordinates. Each term of the series is constructed with vibration modes of beams having the same boundary conditions as the plate, multiplied by a time‐dependent function. Modal loss factor of each mode is obtained by the MSE method. The sound radiation pressure in the time and frequency domain is obtained by numerical integration of the Rayleigh integral. Effects of the viscoelastic boundary supports on the vibration response and the radiated sound pressure of the vibrating plate are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Transient vibration and sound radiation of a rectangular plate with viscoelastic boundary supports

This work considers transient vibration and sound radiation from an impact-excited rectangular plate with viscoelastic boundary supports. The approach used is based on the modal strain energy (MSE) method. Vibration of the plate is approximated by a double infinite series in the spatial co-ordinates. Each term of the series is constructed with vibration modes of beams having the same boundary conditions as the plate, multiplied by a time-dependent function. Modal loss factor of each mode is obtained by the MSE method. The sound radiation pressure in the time and frequency domain is obtained by numerical integration of the Rayleigh integral. Effects of the viscoelastic boundary supports on the vibration response and the radiated sound pressure of the vibrating plate are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C 1 continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C 1 continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.

Physico-acoustical properties of emission affected resonance spruce wood

High-quality resonant spruce wood for violin top plate and wood from traditional Slovak resonant spruce locality with various degrees of emission damage were studied experimentally. The modal analysis was done on quarter-cut spruce plates of wedge shape typical for violin making. Acoustical resonance and holographic interferometry were used to measure the relevant physico-acoustical characteristics. These characteristics (density, Young’s modulus, and acoustical constant) were obtained by using the bending beam vibration mode (2,0). The obtained results showed that the sorting of resonant spruce wood for violin based on empirical experiences has a physico-acoustical reason, and actual industrial emissions have not significant influence on relevant physico-acoustical characteristics of investigated resonant spruce wood.