Dynamic Behavior of Flexible Multiple Links Captured Inside a Closed Space

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
A. M. Shafei ◽  
H. R. Shafei
Keyword(s):  
Differential Equations ◽  
Beam Theory ◽  
Kinematic Chain ◽  
Algebraic Equations ◽  
Assumed Mode Method ◽  
Transformation Matrices ◽  
3D Space ◽  
Mode Method ◽  
Spatial Rotations ◽  
Euler Bernoulli Beam

This work presents a systematic method for the dynamic modeling of flexible multiple links that are confined within a closed environment. The behavior of such a system can be completely formulated by two different mathematical models. Highly coupled differential equations are employed to model the confined multilink system when it has no impact with the surrounding walls; and algebraic equations are exploited whenever this open kinematic chain system collides with the confining surfaces. Here, to avoid using the 4 × 4 transformation matrices, which suffers from high computational complexities for deriving the governing equations of flexible multiple links, 3 × 3 rotational matrices based on the recursive Gibbs-Appell formulation has been utilized. In fact, the main aspect of this paper is the automatic approach, which is used to switch from the differential equations to the algebraic equations when this multilink chain collides with the surrounding walls. In this study, the flexible links are modeled according to the Euler–Bernoulli beam theory (EBBT) and the assumed mode method. Moreover, in deriving the motion equations, the manipulators are not limited to have only planar motions. In fact, for systematic modeling of the motion of a multiflexible-link system in 3D space, two imaginary links are added to the n real links of a manipulator in order to model the spatial rotations of the system. Finally, two case studies are simulated to demonstrate the correctness of the proposed approach.

Oblique Impact of Multi-Flexible-Link Systems

2016 ◽  
Vol 24 (5) ◽  
pp. 904-923 ◽  
Author(s):  
AM Shafei ◽  
HR Shafei
Keyword(s):  
Beam Theory ◽  
Oblique Impact ◽  
Algebraic Equations ◽  
Flexible Link ◽  
Governing Equations ◽  
Assumed Mode Method ◽  
Virtual Links ◽  
Mode Method ◽  
Impact Phase ◽  
Curved Walls

The main goal of this paper is to present an automatic approach for the dynamic modeling of the oblique impact of a multi-flexible-link robotic manipulator. The behavior of a multi-flexible-link system confined inside a closed environment with curved walls can be completely expressed by two distinct mathematical models. A set of differential equations is employed to model the system when it has no contact with the curved walls (Flight phase); and a set of algebraic equations is used whenever it collides with the confining surfaces (Impact phase). In this article, in addition to the Assumed Mode Method (AMM), the Euler-Bernoulli Beam Theory (EBBT), and the Newton’s kinematic impact law, the Gibbs-Appell (G-A) formulation has been employed to derive the governing equations in both phases. Also, instead of using 3 × 3 rotational matrices, which involves lengthy kinematic and dynamic formulations for deriving the governing equations, 4 × 4 transformation matrices have been used. Moreover, for the systematic modeling of flexible multiple links through the space, two virtual links have been added to the n real links of a manipulator. Finally, two case studies have been simulated to demonstrate the validity of the proposed approach.

Exact solutions for the bending of Timoshenko beams using Eringen’s two-phase nonlocal model

2018 ◽  
Vol 24 (3) ◽  
pp. 559-572 ◽  
Author(s):  
Yuanbin Wang ◽  
Kai Huang ◽  
Xiaowu Zhu ◽  
Zhimei Lou
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Nonlocal Parameter ◽  
Beam Theory ◽  
Nonlocal Model ◽  
Bernoulli Beam ◽  
Two Phase ◽  
Differential Model ◽  
Euler Bernoulli Beam

Eringen’s nonlocal differential model has been widely used in the literature to predict the size effect in nanostructures. However, this model often gives rise to paradoxes, such as the cantilever beam under end-point loading. Recent studies of the nonlocal integral models based on Euler–Bernoulli beam theory overcome the aforementioned inconsistency. In this paper, we carry out an analytical study of the bending problem based on Eringen’s two-phase nonlocal model and Timoshenko beam theory, which accounts for a better representation of the bending behavior of short, stubby nanobeams where the nonlocal effect and transverse shear deformation are significant. The governing equations are established by the principal of virtual work, which turns out to be a system of integro-differential equations. With the help of a reduction method, the complicated system is reduced to a system of differential equations with mixed boundary conditions. After some detailed calculations, exact analytical solutions are obtained explicitly for four types of boundary conditions. Asymptotic analysis of the exact solutions reveals clearly that the nonlocal parameter has the effect of increasing the deflections. In addition, as compared with nonlocal Euler–Bernoulli beam, the shear effect is evident, and an additional scale effect is captured, indicating the importance of applying higher-order beam theories in the analysis of nanostructures.

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.

scholarly journals Vibrations of a Beam Moving Over Supports with Clearance

1994 ◽  
Vol 1 (6) ◽  
pp. 549-557
Author(s):  
H.P. Lee
Keyword(s):  
Piecewise Linear ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Euler Beam ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Euler Beam Theory ◽  
The Stability ◽  
Effect Of Support ◽  
Piecewise Linear Stiffness

The transverse vibration of a beam moving over two supports with clearance is analyzed using Euler beam theory. The equations of motion are formulated based on a Lagrangian approach and the assumed mode method. The supports with clearance are modeled as frictionless supports with piecewise-linear stiffness. A feature of the present formulation is that its complexity does not increase with increased number of supports. Results of numerical simulations are presented for various prescribed motions of the beam. The effect of support clearance on the stability of the beam is investigated.

scholarly journals Euler-Bernoulli Beam Theory: First-Order Analysis, Second-Order Analysis, Stability, and Vibration Analysis Using the Finite Difference Method

2021 ◽  
Author(s):  
Valentin Fogang
Keyword(s):  
Finite Difference Method ◽  
Differential Equations ◽  
Vibration Analysis ◽  
Beam Theory ◽  
Second Order ◽  
Bernoulli Beam ◽  
Order Analysis ◽  
First Order ◽  
The Finite Difference Method ◽  
Euler Bernoulli Beam

This paper presents an approach to the Euler-Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, the damping being considered. The efforts and displacements could be determined at any time.

Elastodynamic Modeling and Simulation of an Axially Accelerating Beam

2015 ◽  
Author(s):  
Fadi A. Ghaith ◽  
Ahmad Ayub
Keyword(s):  
Differential Equations ◽  
Plane Motion ◽  
Force Frequency ◽  
Major Focus ◽  
Assumed Mode Method ◽  
Vibrational Response ◽  
Runge Kutta Method ◽  
Mode Method ◽  
Elastodynamic Modeling ◽  
Assumed Mode

This paper aims to develop an accurate nonlinear mathematical model which may describe the coupled in-plane motion of an axially accelerating beam. The Extended Hamilton’s Principle was utilized to derive the partial differential equations governing the motion of a simply supported beam. The set of the ordinary differential equations were obtained by means of the assumed mode method. The derived elastodynamic model took into account the geometric non-linearity, the time-dependent axial velocity and the coupling between the transverse and longitudinal vibrations. The developed equations were solved numerically using the Runge-Kutta method and the obtained results were presented in terms of the vibrational response graphs and the system natural frequencies. The system dynamic characteristics were explored with a major focus on the influence of the velocity, acceleration and the excitation force frequency. The obtained results showed that the natural frequency decreased significantly at high axial velocities. Also it was found that the system may exhibit unstable behavior at high accelerations.

scholarly journals Vibration analysis and pull-in instability behavior in a multiwalled piezoelectric nanosensor with fluid flow conveyance

2020 ◽  
Vol 11 ◽  
pp. 1072-1081
Author(s):  
Sayyid H Hashemi Kachapi
Keyword(s):  
Differential Equations ◽  
Viscous Fluid ◽  
Natural Frequency ◽  
Mass Density ◽  
Fluid Velocity ◽  
Assumed Mode Method ◽  
Interface Theory ◽  
Mode Method ◽  
Assumed Mode ◽  
Piezoelectric Constants

In this work, surface/interface effects for pull-in voltage and viscous fluid velocity effects on the dimensionless natural frequency of fluid-conveying multiwalled piezoelectric nanosensors (FC-MWPENSs) based on cylindrical nanoshells is investigated using the Gurtin–Murdoch surface/interface theory. The nanosensor is embedded in a viscoelastic foundation and subjected to nonlinear van der Waals and electrostatic forces. Hamilton’s principle is used to derive the governing and boundary conditions and is also the assumed mode method used for changing the partial differential equations into ordinary differential equations. The influences of the surface/interface effect, such as Lame’s constants, residual stress, piezoelectric constants and mass density, are considered for analysis of the dimensionless natural frequency with respect to the viscous fluid velocity and pull-in voltage of the FC-MWPENSs.

Longitudinal Vibration Modeling and Control of a Flexible Transporter System with Arbitrarily Varying Cable Lengths

2005 ◽  
Vol 11 (3) ◽  
pp. 431-456 ◽  
Author(s):  
Yuhong Zhang ◽  
Sunil K. Agrawal ◽  
Peter Hagedorn
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Vibration ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Governing Equations ◽  
Partial Differential ◽  
Modeling And Control ◽  
Assumed Mode Method ◽  
Mode Method

We present a systematic procedure for deriving the model of a cable transporter system with arbitrarily varying cable lengths. The Hamilton principle is applied to derive the governing equations of motion. The derived governing equations are nonlinear partial differential equations. The results are verified using the Newton law. The assumed mode method is used to obtain an approximate numerical solution of the governing equations by transforming the infinite-dimensional partial differential equations into a finite-dimensional discretized system. We propose a Lyapunov controller, based directly on the governing partial differential equations, which can both dissipate the vibratory energy during the motion of the transporter and guarantee the attainment of the desired goal point. The validity of the proposed controller is verified by numerical simulation.

scholarly journals Passive Walking Biped Robot Model with Flexible Viscoelastic Legs

2021 ◽  
Author(s):  
Masoumeh Safartoobi ◽  
HamidReza Mohammadi Daniali ◽  
Morteza Dardel
Keyword(s):  
Stable Condition ◽  
Numerical Procedure ◽  
Beam Theory ◽  
Biped Robot ◽  
Step Function ◽  
Assumed Mode Method ◽  
Passive Walking ◽  
Walking Motion ◽  
Mode Method ◽  
Definition Of

Abstract To simulate the complex human walking motion accurately, a suitable biped model has to be proposed that can significantly translate the compliance of biological structures. In this way, the simplest passive walking model is often used as a standard benchmark for making the bipedal locomotion so natural and energy-efficient. This work is devoted to a presentation of the application of internal damping mechanism to the mathematical description of the simplest passive walking model with flexible legs. This feature can be taken into account by using the viscoelastic legs, which are constituted by the Kelvin–Voigt rheological model. Then, the update of the impulsive hybrid nonlinear dynamics of the simplest passive walker is obtained based on the Euler–Bernoulli’s beam theory and using a combination of Lagrange mechanics and the assumed mode method, along with the precise boundary conditions. The main goal of this study is to develop a numerical procedure based on the new definition of the step function for enforcing the biped start walking from stable condition and walking continuously. The study of the influence of various system parameters is carried out through bifurcation diagrams, highlighting the region of stable period-one gait cycles. Numerical simulations clearly prove that the overall effect of viscoelastic leg on the passive walking is efficient enough from the viewpoint of stability and energy dissipation.

Free Vibration Analysis of Frames Using the Transfer Dynamic Stiffness Matrix Method

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
Tsung-Hsien Tu ◽  
Jen-Fang Yu ◽  
Hsin-Chung Lien ◽  
Go-Long Tsai ◽  
B. P. Wang
Keyword(s):  
Free Vibration ◽  
Stiffness Matrix ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
System Matrix ◽  
Dynamic Stiffness Matrix ◽  
3D Space ◽  
Euler Bernoulli Beam ◽  
Good Agreement

A method for free vibration of 3D space frame structures employing transfer dynamic stiffness matrix (TDSM) method based on Euler–Bernoulli beam theory is developed in this paper. The exact TDSM of each member is assembled to obtain the system matrix that is frequency dependent. All free vibration eigensolutions including coincident roots for the characteristic equation can be obtained to any desired accuracy using the algorithm developed by Wittrick and Williams (1971, “A General Algorithm for Computing Natural Frequencies of Elastic Structures,” Q. J. Mech. Appl. Math., 24, pp. 263–284). Exact eigenfunction of structures can then be computed using the dynamic shape function and the corresponding eigenvector. The results showed good agreement with those computed by finite element method.

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