Resonant Excitation of a Composite Beam

Phase–locked loops are widely used within communication technology for synchronizing signals by tracking their phases. Their functional principle can be applied for maintaining the resonant excitation of a beam whose eigenfrequency changes over time and, thus, representing an alternative excitation method for the adaptronic strut for machine tools shown in [1]. In this contribution an experimental set–up for resonant excitation of a composite beam is introduced. A particle is mounted on the beam, its position and mass are are adjustable to modify the eigenfrequency of the beam. The simulation results of the analytical examination of the test–rig are compared with the experimental results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Tracking of Mechanical System Parameters by Phase-Locked Loops

The eigenfrequencies of a mechanical system depend on several parameters such as mass-, damping-, stiffness-distribution and boundary conditions. Variation of these parameters results in variation of the Frequency Response Function (FRF). An eigenfrequency can be tracked by resonant excitation with a Phase-Locked Loop (PLL) in order to monitor a certain parameter. As an example, a cantilever beam with a time-variant mass at its end is analyzed. In this contribution, the FRF of this beam will be derived, which is the prerequisite for the following resonant excitation. The modified PLL-design for the resonant excitation of higher modes will be explained and finally tested at the state-space model of the beam. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

On the refined modeling of the force distribution in a bistable magnetoelastic energy harvesting system due to a magnetic field

Abstract: The problem of a bistable magnetoelastic beam under base excitation was discussed in [1] under the aspect of chaotic behaviour in mechanical systems. Three decades later the system was used in [2] to design an energy harvesting system which performs well under harmonic excitation for a broad range of excitation frequencies due to its bistability. The initial modeling was tailored to obtain a model with one degree of freedom based on the assumption that the magnetic force acts on the beam tip only. A more appropriate model can be found when considering a distributed force along the beam. The authors present the force distribution on a ferromagnetic beam due to the magnetic field of two permanent magnets. A semi-analytic method is used to compute the magnetic field. The force distribution can in future be used to derive a refined nonlinear dynamical model for the ferromagnetic elastic beam. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Active suppression of nonlinear composite beam vibrations by selected control algorithms

This paper is focused on application of different control algorithms for a flexible, geometrically nonlinear beam-like structure with Macro Fiber Composite (MFC) actuator. Based on the mathematical model of a geometrically nonlinear beam, analytical solutions for Nonlinear Saturation Controller (NSC) are obtained using Multiple Scale Method. Effectiveness of different control strategies is evaluated by numerical simulations in Matlab–Simulink software. Then, the Digital Signal Processing (DSP) controller and selected control algorithms are implemented to the physical system to compare numerical and experimental results. Detailed analysis for the NSC system is carried out, especially for high level of amplitude and wide range of frequencies of excitation. Finally, the efficiency of the considered controllers is tested experimentally for a more complex autoparametric “L-shape” beam system.     

The geometrically nonlinear dynamic responses of simply supported beams under moving loads

This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as an Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load–structure system is discussed, and Kelvin−Voigt material model is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by using the Newmark method and Newton−Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability.     

Nonlinear analysis of a parametrically excited beam with intermediate support by using Multi-dimensional incremental harmonic balance method

In this paper, a nonlinear Euler-Bernoulli beam under a concentrated harmonic excitation with intermediate nonlinear support is investigated. Continuous expression for the kinetic energy, potential energy and dissipation function are constructed. An energy method based on the Lagrange equation combined with the Galerkin truncation is used for discretizing the governing equation. The Multi-dimensional incremental harmonic balance method (MIHBM) is derived, and the comparisons between the numerical results and the approximate analytical solutions based on the MIHBM verify the excellent accuracy of the MIHBM. The steady state dynamic of the beam is investigated by MIHBM. In order to investigate the energy transmission and understand the vibration response of the Euler-Bernoulli beam, the effects of the key parameters on the dynamic behaviors are studied and discussed, individually. The results show that the amplitude-frequency curves exhibits softening nonlinear behavior in the super-harmonic resonance region, and near resonant region the hardening nonlinear behavior is observed depending on the different parameters. Nonlinear dynamic analysis, such as bifurcation, 3-D frequency spectrum, waveform, frequency spectrum, phase diagram and Poincaré map, are also presented in order to study the influences of the key parameters on the vibration behaviors for the beam in a more accurate manner. In addition, the path to chaotic motion is observed to be through a sequence of the periodic motion and quasi-periodic motion.     

Dynamics of a linear beam with an attached local nonlinear energy sink

We provide numerical evidence of passive and broadband targeted energy transfer from a linear flexible beam under shock excitation to a local essentially nonlinear lightweight attachment that acts, in essence, as nonlinear energy sink—NES. It is shown that the NES absorbs shock energy in a one-way, irreversible fashion and dissipates this energy locally, without ‘spreading’ it back to the linear beam. Moreover, we show numerically that an appropriately designed and placed NES can passively absorb and locally dissipate a major portion of the shock energy of the beam, up to an optimal value of 87%. The implementation of the NES concept to the shock isolation of practical engineering structures and to other applications is discussed.     

Nonlinear behavior and characterization of a piezoelectric laminated microbeam system

A methodology of analyzing and characterizing the responses of a piezoelectric laminated microbeam system actuated by AC and DC voltages is developed in this research. The present development is based on the piezoelectric theory, Euler–Bernoulli hypothesis, and a newly developed periodicity–ratio (P–R) approach. The electric excitation loading on the beam is considered to be generated by AC and DC interactions. The control voltage of the piezoelectric layer and the geometric nonlinearity of the beam are also taken into account. The analysis of the nonlinear motion trend of the beam system with multiple parameters is carried out with the employment of the P–R criterion. The findings of the research are significant for the design of microbeam systems and micro-structures.     

Structural vibration control by means of nonlinear pendulum dampers

This paper discusses the application of a nonlinear Pendulum Tuned Mass Damper (PTMD) for the reduction of structural vibrations. Pendulum dynamic absorbers are used extensively to reduce the vibration level of slender elastic structures such as towers. A PTMD is a device consisting of a suspended mass, and a damper that is attached to the tower in order to reduce its dynamic response. The primary eigenfrequency of the nonlinear damper is tuned to a particular structural frequency. Energy is dissipated by the damping force acting on the structure. Here, the PTMD is applied to a tower as a continuous system consisting of distributed mass and elasticity. The optimum values of PTMD parameters are found based on minimization of the response of the tower tip–point. Time history and frequency domain responses for the tower with PTMD in linear and nonlinear condition are compared. In addition, the equations of motion of active pendulum control are intruduced. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elimination of tall building vibration resulting from ground motions

In this paper, the vibration problems of tall buildings are considered. The focus is on vibration caused by earthquakes, semi–seismic phenomena and ground vibrations of other origins. The construction consists of the main system and a vibration eliminator (passive tuned mass damper – pendulum type) which is attuned to the first eigenfrequency of the main structure. The analysis focuses on elimination of structure vibration caused by horizontal components of ground motions, while the functioning of the eliminator is simultaneously influenced by the vertical component (parametric effect – the possibility of improper functioning of the device). The vertical periodic movement of the support point can cause changes of the vibration eliminator's stiffness. In such a case parametric excitation occurs in the system, which signifies that parametric resonance may appear. The numerical analysis of the problem was performed with the Newmark method in conjunction with FEM. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic analysis of dielectric elastomer actuators

Dielectric elastomer actuators (DEAs) with a typical sandwich structure of electrode-elastomer-electrode arrangement provide large deformation. DEAs outperform most large displacement actuators in terms of light weight, low cost, and high efficiency. The actuation mechanism of DEAs relies primarily on the electrostatic force or the Maxwell stress which is due to the reaction of material polarization in an electric field. By a dynamic nonlinear electromechanical continuum model implemented with the finite element method, the dynamic response of a homogeneous DEA is studied. Results show that the actuation magnitudes can be increased when the frequency of the applied electric field approaches to the eigenfrequency of the DEAs. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Residual saturated porous media – from oscillating water blobs to waves in ground earth

A model for wave propagation in residual saturated porous media is presented distinguishing enclosed fluid clusters with respect to their eigenfrequency and damping properties. The additional micro-structure information of cluster specific damping is preserved during the formal upscaling process and allows a stronger coupling between micro- and macro-scale than characterisation via eigenfrequencies alone. A numerical example of sandstone filled with air and liquid clusters of different eigenfrequency and damping distributions is given. If energy dissipation due to viscous damping dominates energy storage due to cluster oscillations, the damping distribution is more influential than the eigenfrequency distribution and vice versa. Spreading the damping distribution around a constant mean value supported the effect of capillary forces and spreading the eigenfrequency distribution around a constant mean value supported the effect of viscous damping in the investigated samples. For a wide distribution of the liquid clusters' damping properties, two damping mechanisms of propagating waves occur at the same time: damping due to viscous effects (for highly damped clusters) and energy storage by cluster oscillations (for underdamped clusters). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Active control of secondary resonances piezoelectric sandwich beams

Secondary resonances of piezoelectric/elastic/piezoelectric sandwich beams submitted to active control are studied in this paper. The proportional and derivative nonlinear potential feedback controls via piezoelectric sensor and actuator layers are used. The dynamics of the beam is modelled by a highly nonlinear ordinary-differential equation. The method of multiple scales is applied and approximate solutions are obtained for hard excitations. Analytical frequency and phase-amplitude relationships as well as the time response are explicitly given for various super- and subharmonic resonances. Static and dynamic stability criteria are elaborated and critical displacement and excitation amplitudes associated to the resulting unstable zones are analytically given. The feedback parameters effects on the subharmonic and superharmonic resonances and on their stability are investigated.     

Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects

The main objective of this study is to predict both the subharmonic and superharmonic resonances of the nonlinear oscillation of nanobeams in the presence of surface free energy effects. To this purpose, Gurtin–Murdoch elasticity theory is adopted to the classical beam theory in order to consider the surface Lame constants, surface mass density, and residual surface stress within the differential equations of motion. The Galerkin method together with the method of multiple scales is utilized to investigate the size-dependent response of nanobeams under hard excitations corresponding to various boundary conditions. A parametric analysis is carried out to indicate the influence of the surface elastic parameters on the frequency-response as well as amplitude-response of the nonlinear secondary resonance including multiple vibration modes and interactions between them. It is seen that for the superharmonic excitation, except for the clamped–free boundary condition, the jump phenomenon is along the hardening direction, while in the clamped–free end supports, it is along the softening direction. Moreover, it is revealed that for the subharmonic excitation, within a specific range of the excitation amplitude, the nanobeam is excited, and this range shifts to lower external force by incorporating the surface free energy effects. It is found that in the case of superharmonic excitation, the value of the excitation frequency associated with the bifurcation point at the peak of the frequency-response curve increases by taking the surface free energy effect into consideration.     

Application of CFRP as a rotor shaft material

Conclusions Theoretical analysis and tests performed on rotors with composite shaft show that there is a sufficiently wide rotation stability region in the rotor parameter space despite comparatively high damping of a polymeric composite with respect to steel. Optimum parameters of the shaft (lay-up, thickness) and bearing (radial stiffness, damping) can be found within this region for each given rotor ensuring a low vibration level at critical frequencies.If rotor system parameters are far enough from the instability threshold, maximum vibration level is observed when rotor passes the first eigenfrequency zone. Further increase of rotation frequency leads to a rotor self-centering, and vibration level does not change passing the second eigenfrequency zone. The rotor response is not sensitive to small changes in rotor system parameters. If rotor system parameters are close to the instability threshold, vibration level at the second eigenfrequency dominates, and a small variation of bearing parameters causes significant changes in the vibration level.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 227–240, March–April, 1995.     

基于剪切效应纤维梁单元的结构非线性有限元数值模拟

基于Euler-Bernoulli梁理论的经典纤维模型忽略了剪切变形给截面带来的影响,为了得到更加精确的梁单元模型,该文基于考虑剪切效应的纤维梁单元,根据Timoshenko梁理论,推导了该纤维梁单元的刚度矩阵,并结合弹塑性增量理论,同时考虑了几何非线性和材料非线性的双重影响,建立了压弯剪复杂应力状态下结构非线性有限元...     

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part II: Theoretical analysis

In Part I of this work (Comm. Nonlin. Sci. Numer. Simulat. 18 (2013) 1710–1724), an experimental investigation on nonlinear low-frequency gravity water waves in a cylindrical shell subjected to high-frequency horizontal excitations was reported. To reveal the mechanism of this phenomenon, a theoretical analysis is now presented as Part II of the work. A set of nonlinear equations for two mode interactions is established based on variational principle of fluid-shell coupled system. Theory proofs that for high frequency mode of circumferential wave number m nonlinear interaction exits only with gravity wave modes of circumferential wave number zero or 2m. Multi-scale analysis reveals that appearance of such phenomenon is due to Hopf bifurcation of the dynamic system. Curves of critic excitation force with respect to excitation frequency are obtained by analysis. Theoretical results show good qualitative and quantitative agreement with experimental observations.     

Influence of internal parametric and kinematic excitation on nonlinear gearbox vibration

This paper deals with mathematical modelling of nonlinear vibration of large rotating shaft systems with gears and rollingelement bearings. Gearing and bearing couplings bring into the system nonlinear phenomena like impact motions due to the possibility of the mesh interruption. The motion of the system is influenced by the internal kinematic excitation in gearing and by the parametric excitation caused by periodic change of number of teeth in gear meshing. The influence of simultaneous internal kinematic and parametric excitation is investigated in dependence on revolutions of the driving shaft of a test-gearbox. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)