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.

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.

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.

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.

A Method to Improve the Modal Convergence for Structures With External Forcing

1987 ◽  
Vol 54 (4) ◽  
pp. 904-909 ◽  
Author(s):  
Keqin Gu ◽  
Benson H. Tongue
Keyword(s):  
Spatial Distribution ◽  
Free Vibration ◽  
Convergence Rate ◽  
Traditional Approach ◽  
Vibration Modes ◽  
External Forcing ◽  
Slow Convergence ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Assumed Mode

The traditional approach of using free vibration modes in the assumed mode method often leads to an extremely slow convergence rate, especially when discete interactive forces are involved. By introducing a number of forced modes, significant improvements can be achieved. These forced modes are intrinsic to the structure and the spatial distribution of forces. The motion of the structure can be described exactly by these forced modes and a few free vibration modes provided that certain conditions are satisfied. The forced modes can be viewed as an extension of static modes. The development of a forced mode formulation is outlined and a numerical example is presented.

scholarly journals Nonlinear Controller Design for a Flexible Spacecraft

2020 ◽  
Vol 67 (4) ◽  
pp. 1500-1520
Author(s):  
Jose Luis Redondo Gutiérrez ◽  
Ansgar Heidecker
Keyword(s):  
Controller Design ◽  
Flexible Structures ◽  
Rigid Motion ◽  
Nonlinear Controller ◽  
Assumed Mode Method ◽  
Control Approach ◽  
Mode Method ◽  
Study Case ◽  
Nonlinear Controller Design ◽  
Assumed Mode

AbstractThis paper combines the nonlinear Udwadia-Kalaba control approach with the Assumed Mode Method to model flexible structures and derives an attitude controller for a spacecraft. The study case of this paper is a satellite with four flexible cantilever beams attached to a rigid central hub. Two main topics are covered in this paper. The first one is the formulation of the equation of motion and the second one is the nonlinear controller design. The combination of these two techniques is able to provide a controller that damps the vibration of a flexible structure while achieving the desired rigid-motion state.

scholarly journals One Dimensional Shock Refraction at a Plasma-cold Gas Interface

1973 ◽  
Vol 28 (10) ◽  
pp. 1668-1675
Author(s):  
J. Gerofi ◽  
H. K. Messerle
Keyword(s):  
Plasma Temperature ◽  
Region Of Interest ◽  
Initial Pressure ◽  
Step Function ◽  
Initial Plasma ◽  
Function Discontinuity ◽  
Definition Of ◽  
Shock Refraction ◽  
Weak Shocks ◽  
Cold Gas

Refraction and reflection of a shockwave at a plasma/cold gas interface has been studied using an R.F. preheated section in an electrothermal shock tube. The gas used in the experiment was Argon at initial pressures from 10 to 30 Torr, with initial temperature of 9000 K.A detailed numerical analysis of the refraction event has been undertaken using a method that does not require definition of an effective γ. Methods that do use such a γ are not accurate, except for very weak shocks. Because of short ionization times it has been possible to assume equilibrium behind the various shock waves. Calculations suggest that in the region of interest, reflected and refracted shock velocities depend primarily on initial shock velocity, slightly on initial plasma temperature and very weakly on initial pressure. The analysis covers initial temperatures of 6000 to 12 000 K and initial pressures of 10 to 50 Torr.Calculations and experimental results are presented. These show that a step function discontinuity is a good approximation to the nature of the plasma cold gas interface in this situation.

Squeal Analysis of a Disc Brake Considering Damping between a Disc and a Pad Lining

2012 ◽  
Vol 157-158 ◽  
pp. 1000-1003
Author(s):  
Ke Wei Zhou ◽  
Cheol Kim ◽  
Min Ok Yun ◽  
Ju Young Kim
Keyword(s):  
Finite Element ◽  
Equations Of Motion ◽  
Element Model ◽  
Disc Brake ◽  
Vibration Modes ◽  
Assumed Mode Method ◽  
Frictional Damping ◽  
System Components ◽  
Mode Method ◽  
Assumed Mode

The improved equations of motion for a friction-engaged brake system have been newly derived on the basis of the assumed mode method and frictional damping. The equations of motion with a finite element model were constructed by a set of vibration modes found from FE modal analysis on all system components. Consequently, the modal information of system components are combined with equations of motion derived from the analytical model. Numerical analysis showed the mode which was unstable in an undamped case became stable in a damped case.

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