Numerical Analysis of a Three Element Microbeam Array Subject to Electrodynamical Parametric Excitation

2008 ◽  
Author(s):  
Stefanie Gutschmidt ◽  
Oded Gottlieb
Keyword(s):  
Dynamic Behavior ◽  
Parametric Excitation ◽  
Nearest Neighbor ◽  
Equations Of Motion ◽  
Nonlinear Effects ◽  
Theoretical Models ◽  
System Response ◽  
Lumped Mass ◽  
Internal Resonances ◽  
Temporal Interaction

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, documented theoretical models consist of lumped-mass models. While a lumped-mass approach is useful for a qualitative understanding of the system response it does not resolve the spatio-temporal interaction of the individual elements in the array. Thus, we employ a consistent nonlinear continuum model to investigate the nonlinear dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations focus on the behavior of a small size array in its 1:1:1 internal, parametric, and 3:1 internal resonances, which correspond to low, medium and high DC-voltage input, respectively. The dynamic equations of motion are solved numerically. The dynamic behavior of the three beam systems reveals coexisting periodic and aperiodic solutions. Similarities in the comprehensive bifurcation structures of the three beam systems provide insight to the nearest neighbor response of multi-element microbeam arrays subject to electrodynamic parametric excitation.

Internal Resonances in Microbeam Arrays Subject to Electrodynamical Parametric Excitation

2007 ◽  
Author(s):  
Stefanie Gutschmidt ◽  
Oded Gottlieb
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Nonlinear Effects ◽  
Theoretical Models ◽  
System Stability ◽  
Dynamic Features ◽  
Lumped Mass ◽  
Internal Resonances ◽  
Weakly Nonlinear ◽  
Temporal Interaction

The dynamic response of parametrically excited microbeam arrays is governed by nonlinear effects which directly influence their performance. To date, documented theoretical models consist of lumped-mass systems which do not resolve the spatio-temporal interaction of the individual elements and reproduce measured array response only qualitatively. A consistent nonlinear continuum model is derived using the extended Hamilton’s principle to capture the salient dynamic features of an array of N nonlinearly coupled microbeams. The nonlinear dynamic equations of motion are solved analytically using the asymptotic multiple-scales method for the weakly nonlinear system. Stability analysis of the resulting coexisting solutions enables construction of a comprehensive bifurcation structure for the system. Analytically obtained results for the weakly nonlinear limit of two coupled microbeams are verified numerically.

scholarly journals Dynamic behavior of the nonlinear planetary gear model in nonstationary conditions

2020 ◽  
pp. 095440622094104
Author(s):  
Ahmed Hammami ◽  
Ayoub Mbarek ◽  
Alfonso Fernández ◽  
Fakher Chaari ◽  
Fernando Viadero ◽  
...  
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Planetary Gear ◽  
Nonlinear Effects ◽  
Hertzian Contact ◽  
Mesh Stiffness ◽  
Planetary Gearbox ◽  
Time Frequency ◽  
Run Up ◽  
Nonlinear Equations Of Motion

The nonlinear effects in gearboxes are a key concern to describe accurately their dynamic behavior. This task is difficult for complex gear systems such as planetary gearboxes. The main aim of this work is to provide responses to overcome this difficulty especially in nonstationary operating regimes by investigating a back-to-back planetary gearbox in steady conditions and in the run-up regime. The nonlinear Hertzian contact of teeth pair is modeled in stationary and nonstationary run-up regime. Then it is incorporated in to a torsional model of the planetary gearbox through different mesh stiffness functions. In addition, motor torque and external load variation are taken into account. The nonlinear equations of motion of the back-to-back planetary gearbox are computed through the Newmark- β algorithm combined with the method of Newton–Raphson. An experimental validation of the proposed numerical model is done through a test bench for both stationary and run-up regimes. The vibration characteristics are extracted and correlated to speed and torque. Time–frequency analysis is implemented to characterize the transient regime during the run-up.

INTERNAL RESONANCES AND BIFURCATIONS OF AN ARRAY BELOW THE FIRST PULL-IN INSTABILITY

2010 ◽  
Vol 20 (03) ◽  
pp. 605-618 ◽  
Author(s):  
STEFANIE GUTSCHMIDT ◽  
ODED GOTTLIEB
Keyword(s):  
Nearest Neighbor ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Internal Resonances ◽  
Large Size ◽  
Element Array ◽  
Nonlinear Equations Of Motion ◽  
Chaotic Transients ◽  
Nonlinear Continuum ◽  
Continuation Technique

A nonlinear continuum model is used to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations concentrate on the region below the array's first pull-in instability, which is shown to be governed by several internal three-to-one and combination resonances. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weak nonlinear system. The analytically obtained periodic response of two coupled microbeams is numerically evaluated by a continuation technique and complemented by a numerical analysis of a three-element array which exhibits quasi-periodic responses and lengthy chaotic transients. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays.

scholarly journals Nonlinear Dynamic Behavior of a Flexible Structure to Combined External Acoustic and Parametric Excitation

2006 ◽  
Vol 13 (4-5) ◽  
pp. 233-254 ◽  
Author(s):  
Paulo S. Varoto ◽  
Demian G. Silva
Keyword(s):  
Dynamic Response ◽  
Dynamic Behavior ◽  
Parametric Excitation ◽  
Accurate Determination ◽  
Longitudinal Axis ◽  
Equation Of Motion ◽  
Driving Mechanism ◽  
Lumped Mass ◽  
Mass System ◽  
Tip Mass

Flexible structures are frequently subjected to multiple inputs when in the field environment. The accurate determination of the system dynamic response to multiple inputs depends on how much information is available from the excitation sources that act on the system under study. Detailed information include, but are not restricted to appropriate characterization of the excitation sources in terms of their variation in time and in space for the case of distributed loads. Another important aspect related to the excitation sources is how inputs of different nature contribute to the measured dynamic response. A particular and important driving mechanism that can occur in practical situations is the parametric resonance. Another important input that occurs frequently in practice is related to acoustic pressure distributions that is a distributed type of loading. In this paper, detailed theoretical and experimental investigations on the dynamic response of a flexible cantilever beam carrying a tip mass to simultaneously applied external acoustic and parametric excitation signals have been performed. A mathematical model for transverse nonlinear vibration is obtained by employing Lagrange’s equations where important nonlinear effects such as the beam’s curvature and quadratic viscous damping are accounted for in the equation of motion. The beam is driven by two excitation sources, a sinusoidal motion applied to the beam’s fixed end and parallel to its longitudinal axis and a distributed sinusoidal acoustic load applied orthogonally to the beam’s longitudinal axis. The major goal here is to investigate theoretically as well as experimentally the dynamic behavior of the beam-lumped mass system under the action of these two excitation sources. Results from an extensive experimental work show how these two excitation sources interacts for various testing conditions. These experimental results are validated through numerically simulated results obtained from the solution of the system’s nonlinear equation of motion.

scholarly journals Dynamics Modeling and Motion Simulation of USV/UUV with Linked Underwater Cable

2020 ◽  
Vol 8 (5) ◽  
pp. 318
Author(s):  
Sung Min Hong ◽  
Kyoung Nam Ha ◽  
Joon-Young Kim
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Underwater Vehicle ◽  
Open Loop ◽  
Motion Simulation ◽  
Dynamics Modeling ◽  
Lumped Mass ◽  
Unmanned Surface Vehicle ◽  
Unmanned Underwater Vehicle ◽  
Dynamic Simulator

This paper describes a study on the dynamic modeling and the motion simulation of an unmanned ocean platform to overcome the limitations of existing unmanned ocean platforms for ocean exploration. The proposed unmanned ocean vehicle combines an unmanned surface vehicle and unmanned underwater vehicle with an underwater cable. This platform is connected by underwater cable, and the forces generated in each platform can influence each other’s dynamic motion. Therefore, before developing and operating an unmanned ocean platform, it is necessary to derive a dynamic equation and analyze dynamic behavior using it. In this paper, Newton’s second law and lumped-mass method are used to derive the equations of motion of unmanned surface vehicle, unmanned underwater vehicle, and underwater cable. As the underwater cable among the components of the unmanned ocean platform is expected to affect the motion of unmanned surface vehicle and unmanned underwater vehicle, the similarity of modeling is described by comparing with the cable modeling results and the experimental data. Finally, we constructed a dynamic simulator using Matlab and Simulink, and analyzed the dynamic behavior of the unmanned ocean platform through open-loop simulation.

Dynamic analysis of a helical gear reduction by experimental and numerical methods

2020 ◽  
Vol 68 (1) ◽  
pp. 48-58
Author(s):  
Chao Liu ◽  
Zongde Fang ◽  
Fang Guo ◽  
Long Xiang ◽  
Yabin Guan ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Numerical Models ◽  
Helical Gear ◽  
Operating Conditions ◽  
Dynamic Responses ◽  
Lumped Mass ◽  
Gear Mesh

Presented in this study is investigation of dynamic behavior of a helical gear reduction by experimental and numerical methods. A closed-loop test rig is designed to measure vibrations of the example system, and the basic principle as well as relevant signal processing method is introduced. A hybrid user-defined element model is established to predict relative vibration acceleration at the gear mesh in a direction normal to contact surfaces. The other two numerical models are also constructed by lumped mass method and contact FEM to compare with the previous model in terms of dynamic responses of the system. First, the experiment data demonstrate that the loaded transmission error calculated by LTCA method is generally acceptable and that the assumption ignoring the tooth backlash is valid under the conditions of large loads. Second, under the common operating conditions, the system vibrations obtained by the experimental and numerical methods primarily occur at the first fourth-order meshing frequencies and that the maximum vibration amplitude, for each method, appears on the fourth-order meshing frequency. Moreover, root-mean-square (RMS) value of the acceleration increases with the increasing loads. Finally, according to the comparison of the simulation results, the variation tendencies of the RMS value along with input rotational speed agree well and that the frequencies where the resonances occur keep coincident generally. With summaries of merit and demerit, application of each numerical method is suggested for dynamic analysis of cylindrical gear system, which aids designers for desirable dynamic behavior of the system and better solutions to engineering problems.

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.

On the Parametric Excitation of Pendulum-Type Vibration Absorber

1961 ◽  
Vol 28 (3) ◽  
pp. 330-334 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Energy Transfer ◽  
Numerical Solution ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Vibration Absorber ◽  
Single Cycle ◽  
Linearized System ◽  
Nonlinear Equations Of Motion ◽  
Beam Oscillation ◽  
Energy Interchange

The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

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