Nonlinear Finite Element-Based Path Following of Periodic Solutions

2011 ◽  
Author(s):  
Andrea Arena ◽  
Giovanni Formica ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Vector Field ◽  
Frequency Response ◽  
Periodic Solutions ◽  
Parametric Resonance ◽  
Primary Resonance ◽  
Path Following ◽  
State Variables ◽  
Response Curves ◽  
Space Dependence

A computational framework is proposed to path follow the periodic solutions of nonlinear spatially continuous systems and more general coupled multiphysics problems represented by systems of partial differential equations with time-dependent excitations. The set of PDEs is cast in first order differential form (in time) u˙ = f(u,s,t;c) where u(s,t) is the vector collecting all state variables including the velocities/time rates, s is a space coordinate (here, one-dimensional systems are considered without lack of generality for the space dependence) and t denotes time. The vector field f depends, in general, not only on the classical state variables (such as positions and velocities) but also on the space gradients of the leading unknowns. The space gradients are introduced as part of the state variables. This is justified by the mathematical and computational requirements on the continuity in space up to the proper differential order of the space gradients associated with the unknown position vector field. The path following procedure employs, for the computation of the periodic solutions, only the evaluation of the vector field f. This part of the path following procedure within the proposed combined scheme was formerly implemented by Dankowicz and coworkers in a MATLAB software package called COCO. The here proposed procedure seeks to discretize the space dependence of the variables using finite elements based on Lagrangian polynomials which leads to a discrete form of the vector field f. A concurrent bifurcation analysis is carried out by calculating the eigenvalues of the monodromy matrix. A hinged-hinged nonlinear beam subject to a primary-resonance harmonic transverse load or to a parametric-resonance horizontal end displacement is considered as a case study. Some primary-resonance frequency-response curves are calculated along with their stability to assess the convergence of the discretization scheme. The frequency-response curves are shown to be in close agreement with those calculated by direct integration of the PDEs through the FE software called COMSOL Multiphysics. Besides primary-resonance direct forcing conditions, also parametric forcing causing the principal parametric resonance of the lowest two bending modes is considered through construction of the associated transition curves. The proposed approach integrates algorithms from the finite element and bifurcation domains thus enabling an accurate and effective unfolding of the bifurcation and post-bifurcation scenarios of nonautonomous PDEs with the underlying structures.

Coupling FEM With Parameter Continuation for Analysis of Bifurcations of Periodic Responses in Nonlinear Structures

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Giovanni Formica ◽  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Harry Dankowicz
Keyword(s):  
Finite Element ◽  
Primary Resonance ◽  
Path Following ◽  
Parameter Analysis ◽  
Finite Element Discretization ◽  
Response Curves ◽  
Computational Framework ◽  
Element Discretization ◽  
Explicit Finite Element ◽  
Parameter Continuation

A computational framework is proposed to perform parameter continuation of periodic solutions of nonlinear, distributed-parameter systems represented by partial differential equations with time-dependent coefficients and excitations. The path-following procedure, encoded in the general-purpose Matlab-based computational continuation core (referred to below as coco), employs only the evaluation of the vector field of an appropriate spatial discretization; for example as formulated through an explicit finite-element discretization or through reliance on a black-box discretization. An original contribution of this paper is a systematic treatment of the coupling of coco with Comsolmultiphysics, demonstrating the great flexibility afforded by this computational framework. Comsolmultiphysics provides embedded discretization algorithms capable of accommodating a great variety of mechanical/physical assumptions and multiphysics interactions. Within this framework, it is shown that a concurrent bifurcation analysis may be carried out together with parameter continuation of the corresponding monodromy matrices. As a case study, we consider a nonlinear beam, subject to a harmonic, transverse direct excitation for two different sets of boundary conditions and demonstrate how the proposed approach may be able to generate results for a variety of structural models with great ease. The numerical results include primary-resonance, frequency-response curves together with their stability and two-parameter analysis of multistability regions bounded by the loci of fold bifurcations that occur along the resonance curves. In addition, the results of comsol are validated for the Mettler model of slender beams against an in-house constructed finite-element discretization scheme, the convergence of which is assessed for increasing number of finite elements.

NONLINEAR DYNAMIC BEHAVIOR OF A ROTOR ACTIVE MAGNETIC BEARING

2010 ◽  
Vol 20 (12) ◽  
pp. 3935-3968 ◽  
Author(s):  
HONGKUI CHEN ◽  
USAMA H. HEGAZY
Keyword(s):  
Periodic Solutions ◽  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Primary Resonance ◽  
Period Doubling ◽  
Path Following ◽  
Magnetic Bearing ◽  
Hopf Bifurcations ◽  
Nonlinear Dynamic Behavior ◽  
Path Following Algorithm

The nonlinear dynamic behavior of a rigid disc-rotor supported by active magnetic bearings (AMB) is investigated, without gyroscopic effects. The vibration of the rotor is modeled by a coupled second order nonlinear ordinary differential equations with quadratic and cubic nonlinearities. Their approximate solutions are sought applying the method of multiple scales in the case of primary resonance. The Newton–Raphson method and the pseudo-arclength path-following algorithm are used to obtain the frequency response curves. Choosing the Hopf bifurcations as the initial points and applying the shooting method and the pseudo-arclength path-following algorithm, the periodic solution branches are obtained. At the same time, the Floquet theory is used to determine the stability of the periodic solutions. A detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. The three types of primary Hopf bifurcations are found for the first time in the rotor-AMB system. It is shown that the limit cycles undergo cyclic fold, period doubling bifurcations, and intermittent chaotic attractor, whereas the chaotic attractors undergo explosive bifurcation and boundary crises. In the regime of multiple coexisting solutions, multiple stable equilibriums, periodic solutions and chaotic solutions are the most interesting phenomena observed.

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.

Dynamic Characteristics Analysis and Topology Optimization of Column Based on Finite Element Method

2013 ◽  
Vol 721 ◽  
pp. 541-544
Author(s):  
Jing Chen ◽  
Ze Long Yang ◽  
Xian Xuan Li
Keyword(s):  
Finite Element ◽  
Topology Optimization ◽  
Frequency Response ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Response Analysis ◽  
Response Curves ◽  
Element Analysis ◽  
Column Structure

Aiming to improve the dynamic and static characteristics of a type of machining center column, the finite element modal analysis and harmonic response analysis of the column are performed, and this paper analyzes the dynamic characteristics of the column based on the first five mode shapes and natural frequencies of the column and the displacement - frequency response curves of the column. Topology optimization analysis of the column is performed with ANSYS, and the finite element analysis is performed on the column again after the column structure is improved based on the optimal distribution of material of the column structure and the design experience of column. The result shows that the first five natural frequencies of the column increase, the peak of the displacement - frequency response of the column decrease, and the dynamic characteristics are improved significantly.

Subharmonic Excitation for Electrically Actuated Microbeams

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Rf Mems ◽  
Electrostatic Force ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Global Dynamics ◽  
Response Curves ◽  
Electrically Actuated ◽  
Subharmonic Excitation ◽  
Order Of Magnitude

We present analysis of the global dynamics of electrically actuated microbeams under subharmonic excitation. The microbeams are excited by a DC electrostatic force and an AC harmonic force with a frequency tuned near twice their fundamental natural frequencies. We show that the dynamic pull-in instability can occur in this case for an electric load much lower than that predicted with static analysis and the same order-of-magnitude as that predicted in the case of primary-resonance excitation. We show that, once the subharmonic resonance is activated, all frequency-response curves reach pull-in, regardless of the magnitude of the AC forcing. Our results show a limited influence of the quality factor on the frequency response. This result and the fact that the frequency-response curves have very steep passband-to-stopband transitions make the combination of a DC voltage and a subhormonic of order one-half a promising candidate for designing improved high-sensitive RF MEMS filters.

Detailed Investigation of Capacitive Micromachined Ultrasound Transducer Design Space for Optimal Operation

2011 ◽  
Author(s):  
Onursal Onen ◽  
Rasim Guldiken
Keyword(s):  
Finite Element ◽  
Frequency Response ◽  
Material Selection ◽  
Optimal Operation ◽  
Design Guidelines ◽  
Ultrasound Transducers ◽  
Response Curves ◽  
Design Studies ◽  
Strong Function ◽  
Transducer Design

This paper presents our detailed design studies on capacitive micromachined ultrasound transducers (CMUTs) via finite element models developed in ANSYS 12. In this study, we will discuss our experimentally verified coupled field finite element simulations for operation frequency and bandwidth. The effects of device geometry, type of coupling media, material selection and properties are investigated for the frequency response, and bandwidth using an analytical approach. A commercial design of experiment software, Minitab is utilized for sample runs for obtaining response curves. The design principles for CMUTs are given in detail for effective and quick design. The design guidelines illustrate that, the frequency response is a strong function of transducer geometry, material selection and coupling media.

scholarly journals On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Nizar R. Jaber ◽  
Karim M. Masri ◽  
Mohammad I. Younis
Keyword(s):  
Frequency Response ◽  
Primary Resonance ◽  
Single Frequency ◽  
Response Curves ◽  
Shooting Technique ◽  
Softening Behavior ◽  
Dc Bias ◽  
The Stability ◽  
Good Agreement ◽  
Structural Layer

This work aims to investigate theoretically and experimentally various nonlinear dynamic behaviors of a doubly clamped microbeam near its primary resonance. Mainly, we investigate the transition behavior from hardening, mixed, and then softening behavior. We show in a single frequency–response curve, under a constant voltage load, the transition from hardening to softening behavior demonstrating the dominance of the quadratic electrostatic nonlinearity over the cubic geometric nonlinearity of the beam as the motion amplitudes becomes large, which may lead eventually to dynamic pull-in. The microbeam is fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from the bottom. Frequency sweep tests are conducted for different values of direct current (DC) bias revealing hardening, mixed, and softening behavior of the microbeam. A multimode Galerkin model combined with a shooting technique are implemented to generate the frequency–response curves and to analyze the stability of the periodic motions using the Floquet theory. The simulated curves show a good agreement with the experimental data.

Resonance behavior in coupled Van der Pol harmonic oscillators with controllers and delayed feedback

2020 ◽  
pp. 107754632093818
Author(s):  
Ashraf T EL-Sayed
Keyword(s):  
Frequency Response ◽  
Time Delays ◽  
Degrees Of Freedom ◽  
Primary Resonance ◽  
Delayed Feedback ◽  
Harmonic Oscillators ◽  
Feedback Signal ◽  
Response Curves ◽  
Internal Resonances ◽  
Van Der Pol

It has been revealed in the proposed work that a pair of delay positive position feedback control can lessen the vibration response of double Van der Pol oscillators with external forces. We also studied the effects of both the control and the delayed feedback signal gains to illustrate the low vibration amplitudes. The averaging perturbation process has been used to consider the frequency-response equations of amplitudes and modulation phases at the primary resonance and one-to-one internal resonances. According to the perturbation solutions for the four-degrees-of-freedom system, we presented the frequency response curves that were periodic in the time delays. The stability analysis presented in this study has shown optimum stable ranges. If the time delays increase, the steady-state amplitudes of the oscillator’s system will periodically result in few stable regions and more unstable ones. The numerical simulation has been introduced to check the analytical approximation. It was also found to be almost identical after presenting the comparison of the results.

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.

Vector field path following of a full-wing solar-powered Unmanned Aerial Vehicle (UAV) landing based on Dubins path: a lesson from multiple landing failures

2021 ◽  
pp. 1-30
Author(s):  
A. Guo ◽  
Z. Zhou ◽  
R. Wang ◽  
X. Zhao ◽  
X. Zhu
Keyword(s):  
Vector Field ◽  
Path Following ◽  
Endurance Performance ◽  
Electrical Energy ◽  
Lateral Control ◽  
Special Configuration ◽  
Straight Line ◽  
Lateral Distance ◽  
Solar Powered ◽  
Line Path

Abstract The full-wing solar-powered UAV has a large aspect ratio, special configuration, and excellent aerodynamic performance. This UAV converts solar energy into electrical energy for level flight and storage to improve endurance performance. The UAV only uses a differential throttle for lateral control, and the insufficient control capability during crosswind landing results in a large lateral distance bias and leads to multiple landing failures. This paper analyzes 11 landing failures and finds that a large lateral distance bias at the beginning of the approach and the coupling of base and differential throttle control is the main reason for multiple landing failures. To improve the landing performance, a heading angle-based vector field (VF) method is applied to the straight-line and orbit paths following and two novel 3D Dubins landing paths are proposed to reduce the initial lateral control bias. The results show that the straight-line path simulation exhibits similar phenomenon with the practical failure; the single helical path has the highest lateral control accuracy; the left-arc to left-arc (L-L) path avoids the saturation of the differential throttle; and both paths effectively improve the probability of successful landing.

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