scholarly journals New form of the Euler-Bernoulli rod equation applied to robotic systems

2008 ◽  
Vol 35 (4) ◽  
pp. 381-406 ◽  
Author(s):  
Mirjana Filipovic
Keyword(s):  
Elastic Deformation ◽  
Bending Moment ◽  
Stationary Regime ◽  
Theoretical Background ◽  
Bernoulli Equation ◽  
Lumped Mass ◽  
Robot System ◽  
Elastic Line ◽  
Line Equation ◽  
Rod Equation

This paper presents a theoretical background and an example of extending the Euler-Bernoulli equation from several aspects. Euler-Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the bending moment of the considered mode. The stiffness matrix is a full matrix. Damping is an omnipresent elasticity characteristic of real systems, so that it is naturally included in the Euler-Bernoulli equation. It is shown that Daniel Bernoulli's particular integral is just one component of the total elastic deformation of the tip of any mode to which we have to add a component of the elastic deformation of a stationary regime in accordance with the complexity requirements of motion of an elastic robot system. The elastic line equation mode of link of a complex elastic robot system is defined based on the so-called 'Euler-Bernoulli Approach' (EBA). It is shown that the equation of equilibrium of all forces present at mode tip point ('Lumped-mass approach' (LMA)) follows directly from the elastic line equation for specified boundary conditions. This, in turn, proves the essential relationship between LMA and EBA approaches. In the defined mathematical model of a robotic system with multiple DOF (degree of freedom) in the presence of the second mode, the phenomenon of elasticity of both links and joints are considered simultaneously with the presence of the environment dynamics - all based on the previously presented theoretical premises. Simulation results are presented. .

Expansion of source equation of elastic line

Robotica ◽  
2008 ◽  
Vol 26 (6) ◽  
pp. 739-751 ◽  
Author(s):  
Mirjana Filipovic ◽  
Miomir Vukobratovic
Keyword(s):  
Bending Moment ◽  
Equation Of Motion ◽  
Robotic Systems ◽  
Lumped Mass ◽  
Elastic Line ◽  
Force Dynamics ◽  
Long Time ◽  
The Difference ◽  
Direct Outcome ◽  
The Relationship

SUMMARYThe paper is concerned with the relationship between the equation of elastic line motion, the “Euler-Bernoulli approach” (EBA), and equation of motion at the point of elastic line tip, the “Lumped-mass approach” (LMA). The Euler–Bernoulli equations (which have for a long time been used in the literature) should be expanded according to the requirements of the motion complexity of elastic robotic systems. The Euler–Bernoulli equation (based on the known laws of dynamics) should be supplemented with all the forces that are participating in the formation of the bending moment of the considered mode. This yields the difference in the structure of Euler–Bernoulli equations for each mode. The stiffness matrix is a full matrix. Mathematical model of the actuators also comprises coupling between elasticity forces. Particular integral of Daniel Bernoulli should be supplemented with the stationary character of elastic deformation of any point of the considered mode, caused by the present forces. General form of the elastic line is a direct outcome of the system motion dynamics, and can not be described by one scalar equation but by three equations for position and three equations for orientation of every point on that elastic line. Simulation results are shown for a selected robotic example involving the simultaneous presence of elasticity of the joint and of the link (two modes), as well the environment force dynamics.

scholarly journals Whole analogy between Daniel Bernoulli solution and direct kinematics solution

2010 ◽  
Vol 37 (1) ◽  
pp. 49-78 ◽  
Author(s):  
Mirjana Filipovic ◽  
Ana Djuric
Keyword(s):  
Mathematical Model ◽  
Elastic Deformation ◽  
Bernoulli Equation ◽  
Long Time ◽  
Rod Equation ◽  
The Relationship ◽  
Direct Kinematics ◽  
Motion Complexity

In this paper, the relationship between the original Euler-Bernoulli's rod equation and contemporary knowledge is established. The solution which Daniel Bernoulli defined for the simplest conditions is essentially the solution of 'direct kinematics'. For this reason, special attention is devoted to dynamics and kinematics of elastic mechanisms configuration. The Euler-Bernoulli equation and its solution (used in literature for a long time) should be expanded according to the requirements of the mechanisms motion complexity. The elastic deformation is a dynamic value that depends on the total mechanism movements dynamics. Mathematical model of the actuators comprises also elasticity forces.

scholarly journals Model of forming open-ended rings made of elastic rods

2019 ◽  
Vol 2019 (9) ◽  
pp. 3-6
Author(s):  
Альберт Королев ◽  
Albert Korolev ◽  
Александр Туренко ◽  
Alexandr Turenko
Keyword(s):  
Elastic Deformation ◽  
Structural Parameters ◽  
Bending Moment ◽  
Technological System ◽  
Elastic Rods

The problem of forming open-ended rings by a plastic bend is considered. The influence of technological system super-structural parameters upon a diameter of rings formed is analyzed. The dependences for the calculation of a bending moment of elastic deformation and a bending moment of a plastic one in different sections of a ring are offered.

Dynamics of Flexible Mechanical Systems Using Vibration and Static Correction Modes

1986 ◽  
Vol 108 (3) ◽  
pp. 315-322 ◽  
Author(s):  
W. S. Yoo ◽  
E. J. Haug
Keyword(s):  
Finite Element ◽  
Dynamic Analysis ◽  
Elastic Deformation ◽  
Large Scale ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Nonlinear Dynamic Analysis ◽  
Mode Analysis ◽  
Lumped Mass ◽  
Static Correction

A finite-element-based method is developed and applied for geometrically nonlinear dynamic analysis of spatial mechanical systems. Vibration and static correction modes are used to account for linear elastic deformation of components. Boundary conditions for vibration and static correction mode analysis are defined by kinematic constraints between components of a system. Constraint equations between flexible bodies are derived and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A standard, lumped mass finite-element structural analysis code is used to generate deformation modes and deformable body mass and stiffness information. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Two examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed.

Experimental Studies of the Longitudinal Flexibility of Balloon-Expandable Coronary Stents

2014 ◽  
Vol 538 ◽  
pp. 319-322
Author(s):  
Xiang Shen ◽  
Xiao Zhou
Keyword(s):  
Plastic Deformation ◽  
Elastic Deformation ◽  
Experimental Studies ◽  
Bending Moment ◽  
Coronary Stents ◽  
Bending Angle ◽  
Deformation Phase ◽  
Thin Walled Tube ◽  
Two Phases ◽  
Longitudinal Flexibility

Stents are medical devices used in cardiovascular intervention for unblocking the diseased arteries and restoring blood flow. A setup for the measurement of the longitudinal flexibility of a coronary stent was developed based on machine vision technology. A crimped stent made by medical stainless steel thin-walled tube was tested with the setup. The results show that the bending deformation of stent includes two phases: the elastic deformation phase and the plastic deformation phase. The bending angle changed very little with the increase of bending moment during the elastic deformation. However, the bending angle changed significantly when a small moment was applied during the plastic deformation. In conclusion, the experimental setup can be used to study and compare flexibility of different design kinds of coronary stents and provides a convenient tool for designers to improve bending characteristics of new stents.

scholarly journals Seismic Vibration Mitigation of Wind Turbine Tower Using Bi-Directional Tuned Mass Dampers

2020 ◽  
Vol 2020 ◽  
pp. 1-22
Author(s):  
Wanrun Li ◽  
Qing Zhang ◽  
Zhou Yang ◽  
Qingxin Zhu ◽  
Yongfeng Du
Keyword(s):  
Finite Element ◽  
Finite Element Model ◽  
Wind Turbine ◽  
Bending Moment ◽  
Element Model ◽  
Seismic Vibration ◽  
Lumped Mass ◽  
Tuned Mass Dampers ◽  
Vibration Mitigation ◽  
Wind Turbine Tower

Wind turbines have been increasingly erected in earthquake regions to harvest abundant wind energy. However, the wind turbine tower is slender and lightly damping, which exhibits high susceptibility to earthquake-induced vibration. It is challenging to mitigate the seismic vibration of the tower. In this study, a bi-directional tuned mass damper (BTMD) is proposed to mitigate the seismic vibration of the wind turbine tower. Meanwhile, a lumped-mass finite element model (LFEM) and a coupled blade tower finite element model (CBFEM) are used to investigate the vibration mitigation performance of the BTMD. First, the BTMD and corresponding dynamic equilibrium equations are systemically introduced. Accordingly, the optimum stiffness and damping of the BTMD at different mass ratios are investigated. Then, the dynamic prosperities of the LFEM and CBFEM are compared. Subsequently, the seismic responses of the wind turbine with the BTMD are conducted using the LFEM and CBFEM. Meanwhile, the mitigation performances of the BTMD under uni- and bi-directional earthquakes are investigated. The displacement, acceleration, and bending moment of the wind turbine tower are analyzed in time domain and frequency domain. Note that the influential factors, including mass ratio and structural frequency, on the vibration mitigation performance of the BTMD are investigated. Results show that the proposed BTMD can significantly mitigate the peak values of the top displacement and bottom bending moment. However, the blade tower coupling effect and frequency variation of the tower would have influences on the mitigation efficiency of the BTMD. The results enable a better understanding of the seismic vibration mitigation of the wind turbine tower using tuned mass dampers.

Dynamic Simulation of a Mooring Catenary Based on the Lumped-Mass Approach: OpenModelica and Python Implementations

2020 ◽  
Author(s):  
Savin Viswanathan ◽  
Christian Holden
Keyword(s):  
Initial Conditions ◽  
Constraint Equation ◽  
Equation Of Motion ◽  
Solution Procedure ◽  
Theoretical Background ◽  
Ocean Engineering ◽  
Lumped Mass ◽  
Engineering Software ◽  
Sea Bed ◽  
Drag Term

Abstract In this paper, the theoretical background behind the formulation and solution of a discretized lumped-mass mathematical-model of the physically continuous and inelastic mooring catenary is re-visited. The drag term of the Morison equation is used to determine the fluid loads, and the sea-bed interaction is prescribed as vertical spring loads on the interacting nodes. Numerical solution to the equation of motion is sought through a finite-difference method. The initial conditions are determined using the catenary theory, the instantaneous boundary conditions are prescribed, and the in-elasticity of the mooring segments is specified as the constraint equation. The solution procedure is then implemented as both Python and Modelica code. Results of the Modelica simulation are then compared with those generated using the popular ocean-engineering software, Orcaflex. Finally, conclusions are drawn based on the analysis of simulation results. The codes and results are made available for download.

Dynamic Modelling of an Elephant Trunk Like Flexible Bionic Manipulator

2019 ◽  
Author(s):  
Mrunal Kanti Mishra ◽  
Arun Kumar Samantaray ◽  
Goutam Chakraborty ◽  
Aditya Jain ◽  
Pushparaj Mani Pathak ◽  
...  
Keyword(s):  
Spatial Dynamics ◽  
Dynamic Modelling ◽  
Bending Moment ◽  
Flexible Manipulator ◽  
Lumped Mass ◽  
Elephant Trunk ◽  
End Effector ◽  
Body Dynamics ◽  
Section Model ◽  
Multi Body

Abstract In this paper, an attempt is made to model and study the planar and spatial dynamics of flexible elephant trunk-like manipulator by using multi-body dynamics software MSC-ADAMS. The flexible manipulator is modelled for bending with variable curvature. The entire manipulator length is divided into two sub-sections with associated lumped mass, damping and stiffness for the dynamic analysis. In this model, each section has three pressure actuated bellow tubes, which are modelled as simple spring-damper with the net mass distributed at the ends. Besides, a torsional spring-damper system is also incorporated in each section model to resist the bending about the transverse axes when the pressures in the bellow tubes are unequal. The manipulator is so designed that due to different actuation forces (corresponding to different bellows), the resultant action is finally a bending moment at the tip of each section. The effect of the gravitational force is also included. The change in behaviour of the end-effector position and orientation with respect to time is studied along with the elongation of bellow tubes. The nature of the velocity profile of the end-effector is also determined to study the behaviour of the manipulator.

Implementation and Verification of Cable Bending Stiffness in MoorDyn

2021 ◽  
Author(s):  
Matthew Hall ◽  
Senu Sirnivas ◽  
Yi-Hsiang Yu
Keyword(s):  
Wave Energy ◽  
Bending Moment ◽  
Bending Stiffness ◽  
Coupled Analysis ◽  
Axial Stiffness ◽  
Power Cable ◽  
Power Cables ◽  
Lumped Mass ◽  
Dynamic Power ◽  
Static Conditions

Abstract The relatively large motions experienced by floating wind turbines and wave energy converters pose a challenge for power cables, whose internal components provide significant bending resistance and are sensitive to deformation. The behavior and associated design considerations of power cables in these highly dynamic applications make coupled analysis relevant for design. Bending stiffness capabilities have recently been added to the lumped-mass mooring dynamics model MoorDyn to enable simulation of dynamic power cables. MoorDyn is a common modeling choice for floating wind energy simulation (often coupled with OpenFAST) and floating wave energy converter simulation (often coupled with WEC-Sim) but the model’s previous line elasticity formulation only considered axial stiffness. To properly capture the dynamics of power cables, a bending stiffness model has been added that approximates cable curvature based on the difference in tangent vectors of adjacent elements. The resulting bending moment is realized by applying forces on adjacent nodes, enabling cable modeling while leaving the underlying lumped-mass formulation unchanged. In this paper, the new bending stiffness implementation is verified in static conditions against analytical solutions and then in a dynamic power cable scenario in comparison with the commercial simulator OrcaFlex. The dynamic scenario uses prescribed motions and includes wave loadings on the cable. Results indicate correct implementation of bending stiffness and show close agreement with OrcaFlex.

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