scholarly journals Geometrical Non-Linear Periodic Vibration of Plates

2000 ◽  
Author(s):  
M. Petyt ◽  
P. Ribeiro
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Internal Resonances ◽  
Resonance Frequencies ◽  
Modal Coupling ◽  
Non Linear ◽  
The Stability

Abstract Periodic, geometrically non-linear free and steady-state forced vibrations of fully clamped plates are investigated. The hierarchical finite element method (HFEM) and the harmonic balance method are used to derive the equations of motion in the frequency domain, which are solved by a continuation method. It is demonstrated that the HFEM requires far fewer degrees of freedom than the h-version of the FEM. Internal resonances due to modal coupling between modes with resonance frequencies related by a rational number, are discovered. In free vibration, internal resonances cause a very significant variation of the mode shape during the period of vibration. A similar behaviour is observed in steady-state forced vibration. The stability of the steady-state solutions is studied by Floquet’s theory and it is shown that stable multi-modal solutions occur.

Non-Linear Aeroelastic Stability of Wind Turbines

2013 ◽  
Vol 569-570 ◽  
pp. 531-538 ◽  
Author(s):  
Z.L. Zhang ◽  
M.T. Sichani ◽  
Jie Li ◽  
J.B. Chen ◽  
S.R.K. Nielsen
Keyword(s):  
Wind Turbine ◽  
Wind Turbines ◽  
Degrees Of Freedom ◽  
Dynamic Instability ◽  
Maximum Lyapunov Exponent ◽  
Moving Frames ◽  
Internal Resonances ◽  
Wind Turbine System ◽  
Non Linear ◽  
The Stability

As wind turbines increase in magnitude without a proportional increase in stiffness, the risk of dynamic instability is believed to increase. Wind turbines are time dependent systems due to the coupling between degrees of freedom defined in the fixed and moving frames of reference, which may trigger off internal resonances. Further, the rotational speed of the rotor is not constant due to the stochastic turbulence, which may also influence the stability. In this paper, a robust measure of the dynamic stability of wind turbines is suggested, which takes the collective blade pitch control and non-linear aero-elasticity into consideration. The stability of the wind turbine is determined by the maximum Lyapunov exponent of the system, which is operated directly on the non-linear state vector differential equations. Numerical examples show that this approach is robust for stability identification of the wind turbine system.

scholarly journals Nonlinear Modes of Vibration and Internal Resonances in Nonlocal Beams

2017 ◽  
Vol 12 (3) ◽  
Author(s):  
Pedro Ribeiro ◽  
Olivier Thomas
Keyword(s):  
Harmonic Balance ◽  
Rectangular Cross Section ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Harmonic Balance Method ◽  
Published Data ◽  
Internal Resonances ◽  
Nonlocal Effects ◽  
Nonlinear Modes ◽  
Modes Of Vibration

A nonlocal Bernoulli–Euler p-version finite-element (p-FE) is developed to investigate nonlinear modes of vibration and to analyze internal resonances of beams with dimensions of a few nanometers. The time domain equations of motion are transformed to the frequency domain via the harmonic balance method (HBM), and then, the equations of motion are solved by an arc-length continuation method. After comparisons with published data on beams with rectangular cross section and on carbon nanotubes (CNTs), the study focuses on the nonlinear modes of vibration of CNTs. It is verified that the p-FE proposed, which keeps the advantageous flexibility of the FEM, leads to accurate discretizations with a small number of degrees-of-freedom. The first three nonlinear modes of vibration are studied and it is found that higher order modes are more influenced by nonlocal effects than the first mode. Several harmonics are considered in the harmonic balance procedure, allowing us to discover modal interactions due to internal resonances. It is shown that the nonlocal effects alter the characteristics of the internal resonances. Furthermore, it is demonstrated that, due to the internal resonances, the nonlocal effects are still noticeable at lengths that are longer than what has been previously found.

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.

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

2021 ◽  
pp. 107754632199358
Author(s):  
Ali Fasihi ◽  
Majid Shahgholi ◽  
Saeed Ghahremani
Keyword(s):  
Energy Harvester ◽  
Equations Of Motion ◽  
Continuation Method ◽  
Dynamical Behavior ◽  
Harmonic Balance Method ◽  
Nonlinear Energy Sink ◽  
Piezoelectric Energy ◽  
Energy Sink ◽  
Two Degree Of Freedom ◽  
Nonlinear Energy

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.

Effects of Suspension Design on the Attitudes of a Car during Braking and Acceleration

1973 ◽  
Vol 187 (1) ◽  
pp. 787-794
Author(s):  
J. R. Ellis
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Degrees Of Freedom ◽  
Coupled Systems ◽  
Two Degrees Of Freedom ◽  
Simulation Techniques ◽  
Non Linear ◽  
Non Linear Systems ◽  
Effort Distribution

Two degrees of freedom models of a car are employed to demonstrate the effects of the suspension derivative ∂ x/∂ z on the pitch and bounce attitudes during braking or accelerating. The work equation is employed to show that brake effort distribution between the axles has a significant effect on the attitudes when anti-dive suspension characteristics are utilized. The steady-state positions in both pitch and bounce are developed for linear systems of typical suspensions that may be either standard or coupled systems. Non-linear systems are considered using simulation techniques. A description of some simulation circuits is contained in an appendix.

Periodic Steady State Response and Fatigue Analysis of a Nonlinear City Bus Model

2009 ◽  
Author(s):  
George Valsamos ◽  
Christos Theodosiou ◽  
Sotirios Natsiavas
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
High Stress ◽  
Direct Determination ◽  
Accurate Information ◽  
Steady State Response ◽  
State Response ◽  
Periodic Steady State

Dynamic response related to fatigue prediction of an urban bus is investigated. First, a quite complete model subjected to road excitation is employed in order to extract sufficiently reliable and accurate information in a fast way. The bus model is set up by applying the finite element method, resulting to an excessive number of degrees of freedom. In addition, the bus suspension units involve nonlinear characterstics. A step towards alleviating this difficulty is the application of an appropriate coordinate transformation, causing a drastic reduction in the dimension of the final set of the equations of motion. This allows the application of a systematic numerical methodology leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. Next, typical results were obtained for excitation resulting from selected urban road profiles. These profiles have either a known form or known statistical properties, expressed by an appropriate spatial power spectral density function. In all cases examined, the emphasis was put on investigating ride response. The main attention was focused on identifying areas of the bus suspension and frame subsystems where high stress levels are developed. This information is based on the idea of a nonlinear transfer function and provides the basis for applying suitable criteria in order to perform analyses leading to prediction of fatigue failure.

scholarly journals Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers

Micromachines ◽  
2018 ◽  
Vol 9 (11) ◽  
pp. 575 ◽  
Author(s):  
Aymen Jallouli ◽  
Najib Kacem ◽  
Joseph Lardies
Keyword(s):  
Differential Equations ◽  
Dynamic Behavior ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Ultrasonic Transducers ◽  
Initial Deflection ◽  
Geometric Imperfections ◽  
Resonance Frequencies

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Non-Linear Transient Stability Analysis of a Rigid Rotor Supported on Journal Bearings With Rectangular Dimples

2015 ◽  
Author(s):  
Ram Turaga
Keyword(s):  
Stability Analysis ◽  
Surface Texture ◽  
Transient Stability ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Pressure Curve ◽  
Rigid Rotor ◽  
Non Linear ◽  
The Stability ◽  
Transient Stability Analysis

The influence of deterministic surface texture on the sub-synchronous whirl stability of a rigid rotor has been studied. Non-linear transient stability analysis has been performed to study the stability of a rigid rotor supported on two symmetric journal bearings with a rectangular dimple of large aspect ratio. The surface texture parameters considered are dimple depth to minimum film thickness ratio and the location of the dimple on the bearing surface. Journal bearings of different Length to diameter ratios have been studied. The governing Reynolds equation for finite journal bearings with incompressible fluid has been solved using the Finite Element Method under isothermal conditions. The trajectories of the journal center have been obtained by solving the equations of motion of the journal center by the fourth-order Runge-Kutta method. When the dimple is located in the raising part of the pressure curve the positive rectangular dimple is seen to decrease the stability whereas the negative rectangular dimple is seen to improve the stability of the rigid rotor.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

On The Steady State and Dynamic Performance Characteristics of Floating Ring Bearings

1981 ◽  
Vol 103 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Chin-Hsiu Li ◽  
S. M. Rohde
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
High Speed ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Journal Bearings ◽  
Transient Problem ◽  
The Stability ◽  
Bearing System

An analysis of the steady state and dynamic characteristics of floating ring journal bearings has been performed. The stability characteristics of the bearing, based on linear theory, are given. The transient problem, in which the equations of motion for the bearing system are integrated in real time was studied. The effect of using finite bearing theory rather than the short bearing assumption was examined. Among the significant findings of this study is the existence of limit cycles in the regions of instability predicted by linear theory. Such results explain the superior stability characteristics of the floating ring bearing in high speed applications. An understanding of this nonlinear behavior, serves as the basis for new and rational criteria for the design of floating ring bearings.

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