Frictional Impact-Contacts in Multiple Flexible Links

2021 ◽  
pp. 2150075
Author(s):  
M. Ahmadizadeh ◽  
A. M. Shafei ◽  
R. Jafari
Keyword(s):  
Differential Equations ◽  
Recursive Algorithm ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Open Loop ◽  
Mode Shapes ◽  
Special Treatment ◽  
Exact Moment ◽  
Impact Phase ◽  
The Impact

Multiple impacts of 2D (planar) open-loop robotic systems composed of [Formula: see text] elastic links and revolute joints are studied in this paper. The dynamic equations of motion for such systems are derived by the Gibbs-Appell recursive algorithm, while the regularized method is employed to model the impact-contact mechanism. The Timoshenko beam theory is used to model the transverse vibrations of the links. Also, both the structural damping and air damping are considered to enhance the modeling accuracy. The system joints are assumed to be frictionless and slack-free, but friction force is included for the links colliding with the ground. The [Formula: see text]-flexible-link system considered goes through a flight phase and an impact phase during its motion. In the impact phase, new equations of motion are derived by including the terms caused by the viscoelastic forces in the system’s differential equations. Owing to the extremely short acting time of the impact force, the related differential equations can be solved only via special treatment, i.e. by detecting the exact moment of impact. To this end, entering or leaving the impact phase is analyzed and controlled with high precision by a special computational algorithm presented in this work. To demonstrate the efficacy and precision of the algorithm developed, computer simulations are conducted to study the dynamic behavior of a 3-link robotic mechanism. To investigate the effect of mode shape on the elastic deformation of links, four different mode shapes are used in the simulations and their results are compared.

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.

Free vibration analysis and post-critical free vibrations of nanocomposite rotating beams reinforced with graphene platelet

2021 ◽  
pp. 107754632110511
Author(s):  
Arameh Eyvazian ◽  
Chunwei Zhang ◽  
Farayi Musharavati ◽  
Afrasyab Khan ◽  
Mohammad Alkhedher
Keyword(s):  
Natural Frequency ◽  
Thermal Environment ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Weight Fraction ◽  
Free Vibrations ◽  
Graphene Platelet ◽  
The Impact

Treatment of the first natural frequency of a rotating nanocomposite beam reinforced with graphene platelet is discussed here. In regard of the Timoshenko beam theory hypothesis, the motion equations are acquired. The effective elasticity modulus of the rotating nanocomposite beam is specified resorting to the Halpin–Tsai micro mechanical model. The Ritz technique is utilized for the sake of discretization of the nonlinear equations of motion. The first natural frequency of the rotating nanocomposite beam prior to the buckling instability and the associated post-critical natural frequency is computed by means of a powerful iteration scheme in reliance on the Newton–Raphson method alongside the iteration strategy. The impact of adding the graphene platelet to a rotating isotropic beam in thermal ambient is discussed in detail. The impression of support conditions, and the weight fraction and the dispersion type of the graphene platelet on the acquired outcomes are studied. It is elucidated that when a beam has not undergone a temperature increment, by reinforcing the beam with graphene platelet, the natural frequency is enhanced. However, when the beam is in a thermal environment, at low-to-medium range of rotational velocity, adding the graphene platelet diminishes the first natural frequency of a rotating O-GPL nanocomposite beam. Depending on the temperature, the post-critical natural frequency of a rotating X-GPL nanocomposite beam may be enhanced or reduced by the growth of the graphene platelet weight fraction.

A Unified Frequency Equation and the Motion Analysis of a Moving Flexible Beam Carrying a Concentrated Mass

1999 ◽  
Author(s):  
S. Park ◽  
J. W. Lee ◽  
Y. Youm ◽  
W. K. Chung
Keyword(s):  
Natural Frequencies ◽  
Equations Of Motion ◽  
Frequency Equation ◽  
Open Loop ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Lumped Mass ◽  
Concentrated Mass ◽  
Forcing Function ◽  
The Mathematical Model

Abstract In this paper, the mathematical model of a Bernoulli-Euler cantilever beam fixed on a moving cart and carrying an intermediate lumped mass is derived. The equations of motion of the beam-mass-cart system is analyzed utilizing unconstrained modal analysis, and a unified frequency equation which can be generally applied to this kind of system is obtained. The change of natural frequencies and mode shapes with respect to the change of the mass ratios of the beam, the lumped mass and the cart and to the position of the lumped mass is investigated. The open-loop responses of the system by arbitrary forcing function are also obtained through numerical simulations.

Nonlinear Dynamic Modeling of Elastic Beam Fixed on a Moving Cart and Carrying Lumped Tip Mass Subjected to External Periodic Force

2009 ◽  
Author(s):  
Fadi A. Ghaith
Keyword(s):  
Linear Model ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Open Loop ◽  
Beam Deflection ◽  
Mode Shapes ◽  
Euler Beam ◽  
Periodic Force ◽  
Assumed Mode Method ◽  
Tip Mass

In the present work, a Bernoulli – Euler beam fixed on a moving cart and carrying lumped tip mass subjected to external periodic force is considered. Such a model could describe the motion of structures like forklift vehicles or ladder cars that carry heavy loads and military airplane wings with storage loads on their span. The nonlinear equations of motion which describe the global motion as well as the vibration motion were derived using Lagrangian approach under the inextensibility condition. In order to investigate the influence of the axial movement of the cart on the response of the system, unconstrained modal analysis has been carried out, and accurate mode shapes of the beam deflection were obtained. The assumed mode method was utilized for approximating the beam elastic deformation based on the single unconstrained mode shapes. Numerical simulation has been carried out to estimate the open-loop response of the nonlinear beam-mass-cart model as well as for the simplified linear model under the influence of the periodic excitation force. Also a comparison study between the responses of the linear and nonlinear models was established. It was shown that the maximum values of the beam tip deflection estimated from the nonlinear model are lower than the corresponding values obtained via the linear model, which reveals the importance of considering nonlinear hardening term in formulating the equations of motion for such system in order to come with more accurate and reliable model.

Active Damping of Vibrations of a Cantilever Beam by Axial Force Control

2007 ◽  
Author(s):  
Thomas Pumho¨ssel ◽  
Horst Ecker
Keyword(s):  
Cantilever Beam ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Open Loop ◽  
Active Damping ◽  
Nonlinear Feedback ◽  
Euler Beam ◽  
Bending Vibrations ◽  
Loop Control ◽  
Damping Of Vibrations

In several fields, e.g. aerospace applications, robotics or the bladings of turbomachinery, the active damping of vibrations of slender beams which are subject to free bending vibrations becomes more and more important. In this contribution a slender cantilever beam loaded with a controlled force at its tip, which always points to the clamping point of the beam, is treated. The equations of motion are obtained using the Bernoulli-Euler beam theory and d’Alemberts principle. To introduce artificial damping to the lateral vibrations of the beam, the force at the tip of the beam has to be controlled in a proper way. Two different methods are compared. One concept is the closed-loop control of the force. In this case a nonlinear feedback control law is used, based on axial velocity feedback of the tip of the beam and a state-dependent amplification. By contrast, the concept of open-loop parametric control works without any feedback of the actual vibrations of the mechanical structure. This approach applies the force as harmonic function of time with constant amplitude and frequency. Numerical results are carried out to compare and to demonstrate the effectiveness of both methods.

scholarly journals Parametric instability analysis of the top-tensioned riser in consideration of complex pre-stress distribution

2018 ◽  
Vol 10 (1) ◽  
pp. 168781401775389 ◽  
Author(s):  
Leixin Li ◽  
Luyun Chen
Keyword(s):  
Stress Distribution ◽  
Differential Equations ◽  
Parametric Instability ◽  
Beam Theory ◽  
Complex Loading ◽  
Welding Residual Stress ◽  
Parametric Vibration ◽  
Mode Shapes ◽  
Stress Influences ◽  
Euler Bernoulli Beam

The parametric vibration instability of a riser is studied in consideration of a complex pre-stress distribution. Differential equations of the riser are derived according to Euler–Bernoulli beam theory, and a method to solve the differential equations is proposed. With the parametric vibration of a top-tensioned riser as an example, the effects of the amplitude and direction of complex pre-stress on frequency, mode shapes, and instability characteristics are investigated. Results show that welding residual stress influences the dynamic response of the riser structure. A new approach to eliminate the complex loading of the riser is obtained.

Mechanical Behavior of an Electrostatically Actuated Microplate

2003 ◽  
Author(s):  
Xiaopeng Zhao ◽  
Eihab M. Abdel-Rahman ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Natural Frequencies ◽  
Finite Dimension ◽  
Equations Of Motion ◽  
Galerkin Approximation ◽  
Mode Shapes ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Linear Eigenvalue Problem ◽  
Electrically Actuated

We present a nonlinear model of electrically actuated microplates. The model accounts for the nonlinearity in the electric forcing as well as mid-plane stretching of the plate. We use a Galerkin approximation to reduce the partial-differential equations of motion to a finite-dimension system of nonlinearly coupled second-order ordinary-differential equations. We find the deflection of the microplate under DC voltage and study the pull-in phenomenon. The natural frequencies and mode shapes are then obtained around the deflected position of the microplate by solving the linear eigenvalue problem. The effect of various design parameters on both the static response and the dynamic characteristics are studied.

scholarly journals Differential Equations of Motion for Naturally Curved and Twisted Composite Space Beams

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Ying Hao ◽  
Wei He ◽  
Yanke Shi
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Cross Sections ◽  
Design Principle ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Coupled System ◽  
Variable Coefficients ◽  
Curve Beam ◽  
Design Factor

The differential equations of motion for naturally curved and twisted elastic space beams made of anisotropic materials with noncircular cross sections, being a coupled system consisting of 14 second-order partial differential equations with variable coefficients, are derived theoretically. The warping deformation of beam’s cross section, as a new design factor, is incorporated into the differential equations in addition to the anisotropy of material, the curvatures of the rod axis, the initial twist of the cross section, the rotary inertia, and the shear and axial deformations. Numerical examples show that the effect of warping deformation on the natural frequencies of the beam is significant under certain geometric and boundary conditions. This study focuses on improving and consummating the traditional theories to build a general curve beam theory, thereby providing new scientific research reference and design principle for curve beam designers.

scholarly journals Nonlinear pre and post-buckled analysis of curved beams using differential quadrature element method

2019 ◽  
Vol 14 (1) ◽  
Author(s):  
M. Zare ◽  
A. Asnafi
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Equations Of Motion ◽  
Differential Quadrature ◽  
Mode Shapes ◽  
Curved Beams ◽  
Buckling Loads ◽  
Differential Quadrature Element Method ◽  
Quadrature Element Method ◽  
Post Buckling

AbstractThis paper studied the in-plane elastic stability including pre and post-buckling analysis of curved beams considering the effects of shear deformations, rotary inertia, and the geometric nonlinearity due to large deformations. Firstly, the governing nonlinear equations of motion were derived. The problem was solved performing both the static and dynamic analysis using the numerical method of differential quadrature element method (DQEM) which is a new and efficient numerical method for rapidly solving linear and nonlinear differential equations. Firstly, the method was applied to the equilibrium equations, leading to a nonlinear algebraic system of equations that would be solved utilizing an arc length strategy. Secondly, the results of the static part were employed to linearize the dynamic differential equations of motion and their corresponding boundary and continuity conditions. Without any loss of generality, a clamped-clamped curved beam under a concentrated load was considered to obtain the buckling loads, natural frequencies, and mode shapes of the beam throughout the method. To validate the proposed method, the beam was modeled using a finite element simulation. A great agreement between the results was seen that showed the accuracy of the proposed method in predicting the pre and post-buckling behavior of the beam. The investigation also included an examination of the curvature parameter influencing the dynamic behavior of the problem. It was shown that the values of buckling loads were completely influenced by the curvature of the beam; also, due to the sharp change of longitudinal stiffness after bucking, the symmetric mode shapes changed more than it was expected.

Analysis of Electromechanical Performance of Energy Harvesting Skin Based on the Kirchhoff Plate Theory

2014 ◽  
Author(s):  
Heonjun Yoon ◽  
Byeng D. Youn ◽  
Heung S. Kim
Keyword(s):  
Energy Harvesting ◽  
Differential Equations ◽  
Analytical Model ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Electrical Circuit ◽  
Kirchhoff Plate ◽  
Mode Shapes ◽  
Kirchhoff Plate Theory

As a compact and durable design concept, energy harvesting skin (EH skin), which consists of piezoelectric patches directly attached onto the surface of a vibrating structure as one embodiment, has been recently proposed. This study aims at developing an electromechanically-coupled analytical model of the EH skin so as to understand its electromechanical behavior and get physical insights about important design considerations. Based on the Kirchhoff plate theory, the Hamilton’s principle is used to derive the differential equations of motion. The Rayleigh-Ritz method is implemented to calculate the natural frequency and the corresponding mode shapes of the EH skin. The electrical circuit equation is derived by substituting the piezoelectric constitutive relation into Gauss’s law. Finally, the steady-state output voltage is obtained by solving the differential equations of motion and electrical circuit equation simultaneously. The results of the analytical model are verified by comparing those of the finite element analysis (FEA) in a hierarchical manner.

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