Effects of nonlinear interactions of flexural modes on the performance of a beam autoparametric vibration absorber

2019 ◽  
Vol 26 (7-8) ◽  
pp. 459-474
Author(s):  
Saeed Mahmoudkhani ◽  
Hodjat Soleymani Meymand
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Primary System ◽  
Lumped Mass ◽  
Response Curves ◽  
Internal Resonances ◽  
The One

The performance of the cantilever beam autoparametric vibration absorber with a lumped mass attached at an arbitrary point on the beam span is investigated. The absorber would have a distinct feature that in addition to the two-to-one internal resonance, the one-to-three and one-to-five internal resonances would also occur between flexural modes of the beam by tuning the mass and position of the lumped mass. Special attention is paid on studying the effect of these resonances on increasing the effectiveness and extending the range of excitation amplitudes at which the autoparametric vibration absorber remains effective. The problem is formulated based on the third-order nonlinear Euler–Bernoulli beam theory, where the assumed-mode method is used for deriving the discretized equations of motion. The numerical continuation method is then applied to obtain the frequency response curves and detect the bifurcation points. The harmonic balance method is also employed for detecting the type of internal resonances between flexural modes by inspecting the frequency response curves corresponding to different harmonics of the response. Parametric studies on the performance of the absorber are conducted by varying the position and mass of the lumped mass, while the frequency ratio of the primary system to the first mode of the beam is kept equal to two. Results indicated that the one-to-five internal resonance is especially responsible for the considerable enhancement of the performance.

Analysis and experimental study on dynamic characteristics of an integrated quasi-zero stiffness isolator

2021 ◽  
pp. 1-28
Author(s):  
Haiping Liu ◽  
Kaili Xiao ◽  
Qi Lv ◽  
Yunlong Ma
Keyword(s):  
Frequency Response ◽  
High Reliability ◽  
Dynamic Performance ◽  
Harmonic Balance Method ◽  
Harmonic Excitation ◽  
Engineering Practice ◽  
Lumped Mass ◽  
Response Curves ◽  
Force Transmissibility ◽  
Force Excitation

Abstract The dynamic performance of an integrated quasi-zero stiffness (IQZS) isolator which is constructed by a single elastic structure is investigated in this study. This prototype exhibits the characteristics of the best simplicity, high reliability and without friction by using the minimum number of elements. For completeness, the static properties of the IQZS isolator are provided at first. And then, the dynamic behavior is analyzed and the frequency response under harmonic excitation is derived by using an equivalent mechanical model. Frequency response curves (FRCs) are obtained by using the harmonic balance method (HBM) under force excitation condition. Moreover, the dynamic performance of the nonlinear isolator supporting a lumped mass is investigated by using force transmissibility, which are derived by modelling and compared with an equivalent linear system with the same design parameter values. The isolation performance of the nonlinear isolator outperforms the linear counterpart for providing a larger isolation range. The effects of system parameters on the transmissibility are also examined. At last, the comparison between the analytical and experimental results under force excitation shows that the analytical model of the IQZS isolator is accuracy in terms of force transmissibility. The calculation results discussed may provide a theoretical basis for designing this class of IQZS isolator in engineering practice.

scholarly journals Nonlinear Vibration Analysis of Damaged Microplate considering Size Effect

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Jihai Yuan ◽  
Xiangmin Zhang ◽  
Changping Chen
Keyword(s):  
Size Effect ◽  
Frequency Response ◽  
Nonlinear Vibration ◽  
Vibration Analysis ◽  
Couple Stress Theory ◽  
Harmonic Balance Method ◽  
Damage Effect ◽  
Physical Parameters ◽  
Response Curves ◽  
Electrically Actuated

Since microplates are extensively used in MEMS devices such as microbumps, micromirrors, and microphones, this work aims to study nonlinear vibration of an electrically actuated microplate whose four edges are clamped. Based on the modified couple stress theory (MCST) and strain equivalent assumption, size effect and damage are taken into consideration in the present model. The dynamic governing partial differential equations of the microplate system were obtained using Hamilton’s principle and solved using the harmonic balance method after they are transformed into ordinary differential equation with regard to time. Size effect and damage effect on nonlinear free vibration of the microplate under DC voltage are discussed using frequency-response curve. In the forced vibration analysis, the frequency-response curves were also employed for the purpose of highlighting the influence of different physical parameters such as external excitation, damping coefficient, material length scale parameter, and damage variable when the system is under AC voltage. The results presented in this study may be helpful and useful for the dynamic stability of a electrically actuated microplate system.

scholarly journals Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

2020 ◽  
Vol 26 (13-14) ◽  
pp. 1119-1132 ◽  
Author(s):  
Vinciane Guillot ◽  
Arthur Givois ◽  
Mathieu Colin ◽  
Olivier Thomas ◽  
Alireza Ture Savadkoohi ◽  
...  
Keyword(s):  
Harmonic Balance ◽  
Internal Resonance ◽  
Harmonic Balance Method ◽  
Mode Shapes ◽  
Governing Equations ◽  
Internal Resonances ◽  
Thin Beam ◽  
Beam Focusing ◽  
Theoretical Results ◽  
Theoretical Developments

Experimental and theoretical results on the nonlinear dynamics of a homogeneous thin beam equipped with piezoelectric patches, presenting internal resonances, are provided. Two configurations are considered: a unimorph configuration composed of a beam with a single piezoelectric patch and a bimorph configuration with two collocated piezoelectric patches symmetrically glued on the two faces of the beam. The natural frequencies and mode shapes are measured and compared with those obtained by theoretical developments. Ratios of frequencies highlight the realization of 1:2 and 1:3 internal resonances, for both configurations, depending on the position of the piezoelectric patches on the length of the beam. Focusing on the 1:3 internal resonance, the governing equations are solved via a numerical harmonic balance method to find the periodic solutions of the system under harmonic forcing. A homodyne detection method is used experimentally to extract the harmonics of the measured vibration signals, on both configurations, and exchanges of energy between the modes in the 1:3 internal resonance are observed. A qualitative agreement is obtained with the model.

Theoretical and Experimental Study on Large Amplitude Vibrations of Clamped Viscoelastic Plates

2016 ◽  
Author(s):  
Prabakaran Balasubramanian ◽  
Giovanni Ferrari ◽  
Zenon J. Guzman N. del Prado ◽  
Marco Amabili
Keyword(s):  
Boundary Condition ◽  
Experimental Study ◽  
Frequency Response ◽  
Large Amplitude ◽  
Continuation Method ◽  
Mode Shapes ◽  
Harmonic Load ◽  
Response Curves ◽  
Voigt Model ◽  
Large Amplitude Vibrations

In this paper, the large amplitude vibrations of clamped-clamped thin viscoelastic rectangular plates due to a concentrated transversal harmonic load are investigated both theoretically and experimentally. Clamped boundary condition on all edges and von Kármán nonlinear strain-displacement relationships are considered while rotary inertia, geometric imperfections, and shear deformation are neglected. In the theoretical study, the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model. In-plane loads applied during the assembly of the plate are taken into account and clamped boundary conditions are modelled using artificial rotational springs. The nonlinear ordinary differential equations for the considered Kelvin-Voigt model are obtained using the generalized energy approach. These equations contain quadratic and cubic nonlinear viscoelasticity terms in addition to quadratic and cubic stiffness terms. Non-dimensionalization of variables is carried out and each second order equation is converted into two first order equations. The resulting system of equations is solved using AUTO (software based on the arclength continuation method that allows bifurcation analysis), to get the frequency-response curves at various force levels. Moreover numerical time integration of equations was also performed using the fourth-order Runge-Kutta method to understand the time response of the structure. In the experimental study, two rubber plates with different material and thicknesses were considered; a silicone plate with 0.0015 m thickness and a neoprene plate with 0.003 m thickness. The plates were fixed on a heavy rectangular metal frame thereby ensuring the clamped boundary condition on all edges. Linear experimental modal analysis was carried out as a first step to estimate the mode shapes and natural frequencies. In the second step, the nonlinear vibration response of the plate around its first resonance was measured at various harmonic force levels. At each force level, the amplitude of the harmonic excitation was kept constant by LMS Data Acquisition System and Test.Lab Stepped Sine software module while slowly varying the frequency of excitation to get the frequency-response curves. Laser Doppler Vibrometry was used to measure the response from the plate as it eliminates the possible mass loading effect introduced by any contact type sensors. A maximum amplitude of more than three times the thickness of the plate was achieved. The nonlinear response curves showed a typical hardening type nonlinearity along with sudden jumps as expected for plates. Experimental frequency-response curves were compared with theoretical results and a good agreement was found. The influence of nonlinear viscoelastic damping terms was clearly noticed on the response curves of the plate. The retardation time, measured in seconds decreases with increasing excitation force and larger amplitude vibrations.

scholarly journals Evaluation of the Autoparametric Pendulum Vibration Absorber for a Duffing System

2008 ◽  
Vol 15 (3-4) ◽  
pp. 355-368 ◽  
Author(s):  
Benjamın Vazquez-Gonzalez ◽  
Gerardo Silva-Navarro
Keyword(s):  
Frequency Response ◽  
Fixed Points ◽  
Multiple Scales ◽  
Coupled System ◽  
Vibration Absorber ◽  
Primary System ◽  
Duffing System ◽  
Approximate Frequency ◽  
The Stability ◽  
Cubic Stiffness

In this work we study the frequency and dynamic response of a damped Duffing system attached to a parametrically excited pendulum vibration absorber. The multiple scales method is applied to get the autoparametric resonance conditions and the results are compared with a similar application of a pendulum absorber for a linear primary system. The approximate frequency analysis reveals that the nonlinear dynamics of the externally excited system are suppressed by the pendulum absorber and, under this condition, the primary Duffing system yields a time response almost equivalent to that obtained for a linear primary system, although the absorber frequency response is drastically modified and affected by the cubic stiffness, thus modifying the jumps defined by the fixed points. In the absorber frequency response can be appreciated a good absorption capability for certain ranges of nonlinear stiffness and the internal coupling is maintained by the existing damping between the pendulum and the primary system. Moreover, the stability of the coupled system is also affected by some extra fixed points introduced by the cubic stiffness, which is illustrated with several amplitude-force responses. Some numerical simulations of the approximate frequency responses and dynamic behavior are performed to show the steady-state and transient responses.

Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Passive Control ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
System Response ◽  
Response Curves ◽  
System Parameters ◽  
Theoretical Results

Passive control of vibrations in an elastic structure subjected to horizontal, harmonic excitation by utilizing a nearly square liquid tank is investigated. When the natural frequency ratio 1:1:1 is satisfied among the natural frequencies of the structure and the two predominant sloshing modes (1,0) and (0,1), the performance of a nearly square tank as a tuned liquid damper (TLD) is expected to be superior to rectangular TLDs due to internal resonance. In the theoretical analysis, Galerkin's method is used to determine the modal equations of motion for liquid sloshing considering the nonlinearity of sloshing. Then, van der Pol's method is used to obtain the expressions for the frequency response curves for the structure and sloshing modes. Frequency response curves and bifurcation set diagrams are shown to investigate the influences of the aspect ratio of the tank cross section and the tank installation angle on the system response. From the theoretical results, the optimal values of the system parameters can be determined in order to achieve maximum efficiency of vibration suppression for the structure. Hopf bifurcations occur and amplitude modulated motions (AMMs) may appear depending on the values of the system parameters. Experiments were also conducted, and the theoretical results agreed well with the experimental data.

Passive/Active Autoparametric Cantilever Beam Absorber With Piezoelectric Actuator for a Two-Story Building-Like Structure

2015 ◽  
Vol 137 (1) ◽  
Author(s):  
Gerardo Silva-Navarro ◽  
Hugo F. Abundis-Fong
Keyword(s):  
Cantilever Beam ◽  
Active Vibration Control ◽  
Parametric Uncertainty ◽  
Vibration Absorber ◽  
Primary System ◽  
Internal Resonances ◽  
Piezoelectric Patch ◽  
Active Vibration ◽  
Stiffness And Damping ◽  
Story Building

This work deals with the design and experimental evaluation of a passive/active cantilever beam autoparametric vibration absorber mounted on a two-story building-like structure (primary system), with two rigid floors connected by flexible columns. The autoparametric vibration absorber consists of a cantilever beam with a piezoelectric patch actuator, cemented to its base, mounted on the top of the structure and actively controlled through an acquisition system. The overall system is then a coupled nonlinear oscillator subjected to sinusoidal excitation in the neighborhood of its external and internal resonances. The addition of the piezoelectric patch actuator to the cantilever beam absorber makes active the passive vibration absorber, thus enabling the possibility to control its equivalent stiffness and damping and, as a consequence, the implementation of an active vibration control scheme able to preserve, as possible, the autoparametric interaction as well as to compensate varying excitation frequencies and parametric uncertainty.

scholarly journals Nonlinear Dynamics of a Planar Hinged-Supported Beam with One End Lumped Mass and Longitudinal Elastic Support

2018 ◽  
Vol 241 ◽  
pp. 01016 ◽  
Author(s):  
Lukasz Kloda ◽  
Stefano Lenci ◽  
Jerzy Warminski
Keyword(s):  
Nonlinear Dynamics ◽  
Finite Element Method ◽  
Frequency Response ◽  
Spring Stiffness ◽  
Lumped Mass ◽  
Response Curves ◽  
Simply Supported ◽  
Softening Behaviour ◽  
Damped Oscillations ◽  
Linear Spring

Nonlinear forced-damped oscillations of a non-slender hinged simply supported beam with mass and spring attached to one end are investigated by mean of a finite element method. The frequency response curves are constructed numerically and the variability of hardening/softening behaviour of frequency response curves due to the lumped mass and axial linear spring stiffness is investigated. Resonant and sub resonant motion of beam midpoint as well as jumps between solution branches are highlighted.

Numerical Study of a New Variable Friction TMD

2011 ◽  
Vol 243-249 ◽  
pp. 5450-5457 ◽  
Author(s):  
Li Qin ◽  
Wei Ming Yan ◽  
Sheng Bo Guo
Keyword(s):  
Frequency Response ◽  
Numerical Study ◽  
Damping Ratio ◽  
Harmonic Balance Method ◽  
Structural Response ◽  
Harmonic Excitation ◽  
Response Curves ◽  
Equivalent Damping ◽  
Response Characteristics ◽  
Variable Friction

The paper proposes a new variable friction system, of which the friction force can increase linearly with the displacement of system. This new system can be used in TMD to avoid the disadvantage of Coulomb friction TMD. Using first order harmonic balance method, the equivalent damping ratio and frequency of SDOF variable friction system is deduced and analyzed. The frequency response characteristics of SDOF variable friction system is discussed. The control effectiveness of variable friction TMD under harmonic excitation is analyzed theoretically. The results demonstrate that the frequency response curves of variable friction TMD and classically damped TMD are similar and both can effectively reduce structural response under harmonic excitation.

Nonlinear Two-to-One Internal Resonance for Broadband Energy Harvesting

2015 ◽  
Author(s):  
D. X. Cao ◽  
S. Leadenham ◽  
A. Erturk
Keyword(s):  
Energy Harvesting ◽  
Frequency Response ◽  
Internal Resonance ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Frame Structure ◽  
Response Curves ◽  
Current Effort ◽  
Bandwidth Enhancement ◽  
Piezoelectric Coupling

The transformation of waste vibration energy into low-power electricity has been heavily researched to enable self-sustained wireless electronic components. Monostable and bistable nonlinear oscillators have been explored by several researchers in an effort to enhance the frequency bandwidth of operation. Linear two degree of freedom (2-DOF) configurations as well as combination of a nonlinear single-DOF harvester with a linear oscillator to constitute a nonlinear 2-DOF harvester have also been explored to develop broadband energy harvesters. In the present work, the concept of nonlinear internal resonance in a continuous frame structure is explored for broadband energy harvesting. The L-shaped beam-mass structure with quadratic nonlinearity was formerly studied in the nonlinear dynamics literature to demonstrate modal energy exchange and the saturation phenomenon when carefully tuned for two-to-one internal resonance. In the current effort, piezoelectric coupling is introduced, and electromechanical equations of the L-shaped energy harvester are employed to explore the primary resonance behaviors around the first and the second linear natural frequencies for bandwidth enhancement. Simulations using approximate analytical frequency response equations as well as time-domain numerical solutions reveal that 2-DOF configuration with quadratic and two-to-one internal resonance could extend the bandwidth enhancement capability. Both electrical power and shunted vibration frequency response curves of steady-state solutions are explored in detail. Effects of various electromechanical system parameters, such as piezoelectric coupling and load resistance, on the overall dynamics of the internal resonance energy harvesting system are reported.

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