Truncation errors of assumed shape modeling for flexible-link manipulators

2012 ◽  
Vol 226 (11) ◽  
pp. 2627-2644 ◽  
Author(s):  
Fatemeh Heidari ◽  
Mohammad Vakil ◽  
Reza Fotouhi ◽  
Peter N Nikiforuk
Keyword(s):  
Mode Shape ◽  
Dynamic Models ◽  
Transfer Functions ◽  
Degree Of Freedom ◽  
Flexible Manipulator ◽  
Mode Shapes ◽  
Physical Parameters ◽  
Flexible Link ◽  
Dimensional Parameters ◽  
Assumed Mode

The assumed mode shape method has been widely used to derive finite degree-of-freedom dynamic models for flexible-link manipulators, which theoretically have infinite degree-of-freedom dynamics. For a single flexible manipulator, this approximation changes locations of the zeros of transfer functions between base torque and end-effector displacement. The change in locations of zeros considerably affects accuracy of the model and therefore the performance of model-based controllers. This article presents a comprehensive study on the change in locations of zeros due to the truncation associated with assumed mode shape method. It is shown that the locations of approximate zeros depend on four non-dimensional parameters, whereas the locations of analytical zeros depend on only two non-dimensional parameters. Approximate zeros are obtained from assumed mode shape method models, whereas analytical zeros are derived from infinite order models. A thorough study of the differences between approximate zeros and analytical zeros versus the number of mode shapes as well as all the physical parameters is performed. Moreover, guidelines are provided to select the numbers of mode shapes such that the approximate zeros become close to the analytical zeros. These guidelines can easily be used by control and modeling engineers, making them valuable for modeling and control of flexible robot manipulators.

On the Accuracy of Assumed Mode Modeling for Flexible Manipulators

2011 ◽  
Author(s):  
F. Heidari ◽  
M. Vakil ◽  
R. Fotouhi
Keyword(s):  
Flexible Manipulator ◽  
Mode Shapes ◽  
Physical Parameters ◽  
Flexible Robot ◽  
Modeling And Control ◽  
Dimensional Parameters ◽  
Location Of Zeros ◽  
And Control ◽  
Assumed Mode ◽  
Comprehensive Study

Assumed mode shape method (AMM) has been widely used to derive finite degree-of-freedom (DOF) dynamic model for flexible link manipulators, which theoretically have infinite DOF dynamics. For single flexible manipulator, this approximation changes locations of the zeros of transfer function, between base torque and end-effector displacement. The change in locations of zeros considerably affects accuracy of the model and hence the performance of model-based controllers. This paper presents a comprehensive study on the change in location of zeros due to the truncation associated with AMM. It is shown that the locations of zeros of AMM model depend on four non-dimensional parameters while the locations of the analytical model depend on only two non-dimensional parameters; AMM zeros are obtained from AMM model while analytical zeros derived from infinite order model. A thorough study on the differences between AMM zeros and analytical zeros versus number of mode shapes as well as all the physical parameters is performed. Moreover, guidelines are provided to select the numbers of mode shapes such that the AMM zeros become close to the analytical zeros. These guidelines can easily be used by control engineers and thus makes them valuable for modeling and control of flexible robot manipulators.

Theoretical investigation into the modelling of a flexible beam with drive-line compliance

2002 ◽  
Vol 216 (8) ◽  
pp. 813-829 ◽  
Author(s):  
D J Purdy
Keyword(s):  
Theoretical Investigation ◽  
Transfer Functions ◽  
Flexible Beam ◽  
Direct Drive ◽  
Flexible Link ◽  
Line Model ◽  
Flexible Joint ◽  
Rigid Link ◽  
Dimensional Parameters ◽  
Weapons Systems

A comparison is made between the dynamics of three possible models for a flexible link and drive-line as used in some robotic or weapons systems. The three models considered are: model 1, a flexible link with compliance in the drive-line; model 2, flexible link with direct drive; and model 3, rigid link with drive-line compliance (flexible joint). Non-dimensional parameters are suggested for the models and comparisons are made between them, by examining the transfer functions poles (resonances) and zeros (anti-resonances). From the study, recommendations are made as to the suitability of the three models for different applications.

Representing Flexible Supports by Polynomial Transfer Functions

1998 ◽  
Author(s):  
José A. Vázquez ◽  
Lloyd E. Barrett
Keyword(s):  
Transfer Functions ◽  
Great Influence ◽  
Forced Response ◽  
Degree Of Freedom ◽  
Discrete Models ◽  
Mode Shapes ◽  
Rotor Systems ◽  
Flexible Supports ◽  
Set Up ◽  
The One

Flexible bearing supports may have a great influence in the calculation of forced response and stability of rotor systems. However, this effect is not always included in rotor analyses since an accurate model of the foundation and pedestals may be difficult and costly to obtain. It is common practice to use either a one degree of freedom model or a full modal analysis to represent the bearing supports. While the one degree of freedom model is easy to set up for computer calculations, it often requires experience to determine values for the stiffness, mass and damping of the model that will accurately represent the support under study. This model, however, fails to capture the dynamics of the system for stability analyses when more than one mode of the support structure is in the range of interest. On the other hand, modal representation provides much more information and can be measured experimentally, but requires measurement of the mode shapes. Even though modal representation can include all the dynamics of the system in the frequency range of interest, it provides much more information than is required for calculation of the rotor response and it is more difficult to use in calculation programs. This paper presents a procedure to include the support characteristics using transfer functions. Transfer functions permit modeling of multi-degree of freedom systems while maintaining the size of a one degree of freedom system (2×2 matrix if rotation at the bearing is not considered). Another advantage of transfer functions is that they can be obtained from existing discrete models, from modal information or can be measured directly. The fixed size of the transfer function matrix permits the method to be easily incorporated into rotor dynamic stability and forced response programs. The method is applied to stability calculations of models of typical industrial machines.

scholarly journals An Updating Method for Finite Element Models of Flexible-Link Mechanisms Based on an Equivalent Rigid-Link System

2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
R. Belotti ◽  
R. Caracciolo ◽  
I. Palomba ◽  
D. Richiedei ◽  
A. Trevisani
Keyword(s):  
Dynamic Models ◽  
Model Updating ◽  
Body Motion ◽  
Transformation Model ◽  
Mode Shapes ◽  
Deformation Model ◽  
Flexible Link ◽  
Static Tests ◽  
State Observers ◽  
Stiffness And Damping

This paper proposes a comprehensive methodology to update dynamic models of flexible-link mechanisms (FLMs) modeled through ordinary differential equations. The aim is to correct mass, stiffness, and damping matrices of dynamic models, usually based on nominal and uncertain parameters, to accurately represent the main vibrational modes within the bandwidth of interest. Indeed, the availability of accurate models is a fundamental step for the synthesis of effective controllers, state observers, and optimized motion profiles, as those employed in modern control schemes. The method takes advantage of the system dynamic model formulated through finite elements and through the representation of the total motion as the sum of a large rigid-body motion and the elastic deformation. Model updating is not straightforward since the resulting model is nonlinear and its coordinates cannot be directly measured. Hence, the nonlinear model is linearized about an equilibrium point to compute the eigenstructure and to compare it with the results of experimental modal analysis. Once consistency between the model coordinates and the experimental data is obtained through a suitable transformation, model updating has been performed solving a constrained convex optimization problem. Constraints also include results from static tests. Some tools to improve the problem conditioning are also proposed in the formulation adopted, to handle large dimensional models and achieve reliable results. The method has been experimentally applied to a challenging system: a planar six-bar linkage manipulator. The results prove their capability to improve the model accuracy in terms of eigenfrequencies and mode shapes.

Closed-Loop Behavior of a Feedback-Controlled Flexible Arm: A Comparative Study

1991 ◽  
Vol 10 (3) ◽  
pp. 263-275 ◽  
Author(s):  
Sabri Cetinkunt ◽  
Wen-Lung Yu
Keyword(s):  
Mode Shape ◽  
Closed Loop ◽  
Transfer Functions ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Dimensional Model ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Shape Models ◽  
Free Mode

The dynamics of mechanical systems with distributed flexi bility are described by infinite-dimensional mathematical models. In order to design afinite-dimensional controller, a finite-dimensional model of the system is needed. The con trol problem of a flexible beam is a typical example. The general practice in obtaining a finite-dimensional model is to use modal approximation for distributed flexibility, retain a finite number of modes, and truncate the rest. In this approx imation, the appropriate selection of the mode shape func tions and the number of modes is not clearly known. Mostly standard pinned-free and clamped-free mode shapes are used for the flexible beam model, retaining only two or three modes and truncating the rest. The actual system, on the other hand, is infinite-dimensional, and the modes describing its flexible behavior under feedback control would be neither pinned-free nor clamped-free boundary condition modes. Rather, the mode shapes themselves are a function of the feedback control. The infinite-dimensional transcendental transfer functions for a flexible beam are formulated without any modal ap proximation. Finite-dimensional transfer functions with different shapes and numbers of modes are formulated. The closed-loop performance predictions of different models under the same colocated and noncolocated controllers, which attempt to achieve high closed-loop bandwidth, are compared. Results are surprisingly consistent in all cases; the predictions of clamped-free mode shape models are much more accurate than the predictions of the pinned-free mode shape models.

VIBRATION OF DOUBLE BELLOWS TYPE EXPANSION JOINT IN LATERAL AND ROCKING MODES

2006 ◽  
Vol 06 (04) ◽  
pp. 575-588
Author(s):  
M. RADHAKRISHNA ◽  
C. KAMESWARA RAO
Keyword(s):  
Mode Shape ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Expansion Joint ◽  
Modes Of Vibration ◽  
Dimensional Parameters

In this paper, the exact frequency and mode shape expressions are derived for universal bellows type of expansion joint in lateral and rocking modes of vibration. The effect of equivalent support stiffness and mass on the natural frequencies and mode shapes are studied in detail and the results for a range of non-dimensional parameters are presented in graphical forms which should be useful for piping and bellow designers.

Singular perturbation adaptive control and vibration suppression of free-flying flexible space manipulators

2014 ◽  
Vol 229 (11) ◽  
pp. 1989-1997 ◽  
Author(s):  
Xiao Yan Yu ◽  
Li Chen
Keyword(s):  
Adaptive Control ◽  
Singular Perturbation ◽  
Lagrange Equation ◽  
Momentum Conservation ◽  
Flexible Manipulator ◽  
Physical Parameters ◽  
Flexible Link ◽  
Linear Quadratic ◽  
Flexible Links ◽  
Space Manipulators

Singular perturbation adaptive control is designed for free-flying space manipulators with multiple flexible links and unknown physical parameters. The dynamical Lagrange equation was established based on assumed mode technique and linear momentum conservation theory. A singular perturbation model has been formulated and used for designing a reduced-order controller. This controller consisted of a slow control component and a fast control component. An adaptive control law was constructed for the slow counterpart of the flexible manipulator. The flexible-link fast subsystem controller would damp out the vibrations of flexible links by optimal linear quadratic regulator method. Numerical simulations by undertaking a computer simulation of a two-flexible-link space manipulator using the fourth-order Runge–Kutta integration method showed that the link vibrations had been stabilized effectively with good tracking performance.

scholarly journals Vibration Analysis of a New Type of Compliant Mechanism with Flexible-Link, Using Perturbation Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. S. Viliani ◽  
H. Zohoor ◽  
M. H. Kargarnovin
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Vibration Analysis ◽  
Compliant Mechanism ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Flexible Link ◽  
Assumed Mode Method ◽  
New Type ◽  
Assumed Mode

Vibration analysis of a new type of compliant parallel mechanism with flexible intermediate links is investigated. The application of the Timoshenko beam theory to the mathematical modeling of the intermediate flexible link is described, and the equations of motion of the flexible links are obtained by using Lagrange’s equation of motion. The equations of motion are obtained in the form of a set of ordinary differential equations by using assumed mode method theory. The governing differential equations of motion are solved using perturbation method. The assumed mode shapes and frequencies are to be obtained based on clamped-clamped boundary conditions. Comparing perturbation method with Runge-Kutta-Fehlberg 4, 5th leads to highly accurate solutions, and the results are performed and discussed.

Spatial Exact Actuation of Flexible Deep Double-Curvature Shells

2007 ◽  
Author(s):  
Hong-Hao Yue ◽  
Xiao-Ying Gao ◽  
Bing-Yin Ren ◽  
Horn-Sen Tzou
Keyword(s):  
Mode Shape ◽  
Mode Shapes ◽  
Shape Functions ◽  
Shell System ◽  
Control Effectiveness ◽  
Natural Modes ◽  
Double Curvature ◽  
Spatially Distributed ◽  
Control Forces ◽  
Assumed Mode

Deep double-curvature shells are commonly used as key components in many advanced aerospace structures and mechanical systems, e.g., nozzles, injectors, horns, rocket fairings. Spatially distributed micro-actuation of a laminated flexible deep double curvature shell is investigated and its control effectiveness is evaluated in this study. Dynamic equations of the smart double curvature shell system are presented and modal control forces of spatial segmented piezoelectric actuators are carried out based on a new set of assumed mode shape functions with free boundary condition. Using these assumed mode shape functions, mode shapes of a free-floating deep shell are illustrated. Finally, via numerical simulation, control effectiveness of distributed actuator patches with respect to various natural modes, actuator locations and other factors which influence precision control and active actuation behavior of flexible deep double curvature shell structronic systems is evaluated.

A Comparison of Two Sub-structuring Techniques for Representing the Modal Properties of Tires

2001 ◽  
Vol 29 (1) ◽  
pp. 23-43 ◽  
Author(s):  
D. Tsihlas ◽  
T. Lacroix ◽  
B. Clayton
Keyword(s):  
Automobile Industry ◽  
Dynamic Models ◽  
Transfer Functions ◽  
Component Mode Synthesis ◽  
Numerical Techniques ◽  
The Past ◽  
Custom Made ◽  
Modal Behavior ◽  
Noise Vibration ◽  
Reduced Dynamic

Abstract Different numerical sub-structuring techniques for the representation of tire modal behavior have been developed in the past 20 years. By using these numerical techniques reduced dynamic models are obtained which can not only be used for internal studies but also be provided to the automobile industry and linked to reduced dynamic vehicle models in order to optimize the coupled vehicle-tire response for noise vibration and harshness purposes. Two techniques that have been developed in a custom-made finite element code are presented: 1) the component mode synthesis type models for which the wheel center interface is free and 2) the Craig and Bampton type models for which the wheel center interface is fixed. For both techniques the interface between the tire and the ground is fixed. The choice of fixed or free wheel center boundary condition is arbitrary. In this paper we will compare the formulation of these two numerical methods, and we will show the equivalency of both methods by showing the results obtained in terms of frequency and transfer functions. We will show that the two methods are equivalent in principle and the reduced dynamic models can be converted from one to the other. The advantages-disadvantages of each method will be discussed along with a comparison with experimentally obtained results.

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