Virtual Work & d’Alembert’s Principle

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Euler Equations ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Rigid Bodies ◽  
Gibbs Function ◽  
Constraint Force ◽  
Appell Equations ◽  
Jourdain's Principle ◽  
D'alembert's Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

Ostwald´s Rule and the Principle of the Shortest Way

2010 ◽  
Vol 224 (06) ◽  
pp. 929-934 ◽  
Author(s):  
Herbert W. Zimmermann
Keyword(s):  
Melting Point ◽  
Equilibrium State ◽  
Entropy Production ◽  
Thermodynamic Stability ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Geodesic Line ◽  
Final Equilibrium State ◽  
Relevant Variables ◽  
Non Equilibrium

AbstractWe consider a substance X with two monotropic modifications 1 and 2 of different thermodynamic stability ΔH1 < ΔH2. Ostwald´s rule states that first of all the instable modification 1 crystallizes on cooling down liquid X, which subsequently turns into the stable modification 2. Numerous examples verify this rule, however what is its reason? Ostwald´s rule can be traced back to the principle of the shortest way. We start with Hamilton´s principle and the Euler-Lagrange equation of classical mechanics and adapt it to thermodynamics. Now the relevant variables are the entropy S, the entropy production P = dS/dt, and the time t. Application of the Lagrangian F(S, P, t) leads us to the geodesic line S(t). The system moves along the geodesic line on the shortest way I from its initial non-equilibrium state i of entropy Si to the final equilibrium state f of entropy Sf. The two modifications 1 and 2 take different ways I1 and I2. According to the principle of the shortest way, I1 < I2 is realized in the first step of crystallization only. Now we consider a supercooled sample of liquid X at a temperature T just below the melting point of 1 and 2. Then the change of entropy ΔS1 = Sf 1 - Si 1 on crystallizing 1 can be related to the corresponding chang of enthalpy by ΔS1 = ΔH1/T. Now it can be shown that the shortest way of crystallization I1 corresponds under special, well-defined conditions to the smallest change of entropy ΔS1 < ΔS2 and thus enthalpy ΔH1 < ΔH2. In other words, the shortest way of crystallization I1 really leads us to the instable modification 1. This is Ostwald´s rule.

An Observation on Least Action Principle in Classical Mechanics Oriented on the Evaluation of Relativistic Error Concerning the Measurements of Light Propagating in a Liquid

2011 ◽  
Vol 1 (3) ◽  
pp. 14-24 ◽  
Author(s):  
Alessandro Massaro ◽  
Piero Adriano Massaro
Keyword(s):  
Euler Equations ◽  
Classical Mechanics ◽  
First Integrals ◽  
Action Principle ◽  
Classical Approach ◽  
Limit Case ◽  
Least Action Principle ◽  
Calculus Of Variation ◽  
Least Action ◽  
Approximate First Integrals

The authors prove that the standard least action principle implies a more general form of the same principle by which they can state generalized motion equation including the classical Euler equation as a particular case. This form is based on an observation regarding the last action principle about the limit case in the classical approach using symmetry violations. Furthermore the well known first integrals of the classical Euler equations become only approximate first integrals. The authors also prove a generalization of the fundamental lemma of the calculus of variation and we consider the application in electromagnetism.

Cayley kinematics and the Cayley form of dynamic equations

2005 ◽  
Vol 461 (2055) ◽  
pp. 761-781 ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Euler Equations ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Rigid Bodies ◽  
Matrix Form ◽  
Dynamic Equations ◽  
Cayley Transform ◽  
Higher Dimensional ◽  
The Matrix ◽  
General Motion

The Cayley transform and the Cayley–transform kinematic relationships are an important and fascinating set of results that have relevance in N –dimensional orientations and rotations. In this paper these results are used in two significant ways. First, they are used in a new derivation of the matrix form of the generalized Euler equations of motion for N –dimensional rigid bodies. Second, they are used to intimately relate the motion of general mechanical systems to the motion of higher–dimensional rigid bodies. This approach can be used to describe an enormous variety of systems, one example being the representation of general motion of an N –dimensional body as pure rotations of an ( N + 1)–dimensional body.

The Geometry of Virtual Work Dynamics in Screw Space

1992 ◽  
Vol 59 (2) ◽  
pp. 411-417 ◽  
Author(s):  
Steven Peterson
Keyword(s):  
Tangent Space ◽  
Screw Theory ◽  
Virtual Work ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Coordinate Space ◽  
Dynamical Equations ◽  
Least Action ◽  
System Of Rigid Bodies ◽  
Configuration Manifold

In this paper, screw theory is employed to develop a method for generating the dynamic equations of a system of rigid bodies. Exterior algebra is used to derive the structure of screw space from projective three space (homogeneous coordinate space). The dynamic equation formulation method is derived from the parametric form of the principle of least action, and it is shown that a set of screws exist which serves as a basis for the tangent space of the configuration manifold. Equations generated using this technique are analogs of Hamilton’s dynamical equations. The freedom screws defining the manifold’s tangent space are determined from the contact geometry of the joint using the virtual coefficient, which is developed from the principle of virtual work. This results in a method that eliminates all differentiation operations required by other virtual work techniques, producing a formulation method based solely on the geometry of the system of rigid bodies. The procedure is applied to the derivation of the dynamic equations for the first three links of the Stanford manipulator.

scholarly journals Modeling and Control of Flexible Hydraulic Robotic Arm

2011 ◽  
Vol 2-3 ◽  
pp. 334-339 ◽  
Author(s):  
Li Dai ◽  
Yao Wu ◽  
Jian Wang ◽  
Yun Gong Li ◽  
Yu Liu
Keyword(s):  
Hydraulic System ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Hydraulic Cylinder ◽  
Simulation Software ◽  
Robotic Arm ◽  
Converse Problem ◽  
Loop Control ◽  
Multi Body ◽  
Close Loop Control

Flexible hydraulic robotic arm is a complicated system which coupled by mechanics and hydraulics. It is widely applied in all kinds of large engineering equipments, such as concrete pump truck, bridge monitor truck, arm frame of crane, etc. The arm system of the hydraulic robotic arm is a multi-body system with redundant freedom, strong nonlinear, coupled with rigid and flexible characters. So it is of great theoretic value and real engineering significance to study the arm system of the robotic arm. In this theme, the movement of flexible hydraulic robotic arm and hydraulic cylinders are seperately analyzed with flexible multi-body dynamics, and the mechanical hydraulic dynamic model of the driving system and the arm system is built with Lagrange Equation and Virtual Work Theory. And the dynamic differential equation is built with the driving force of the hydraulic cylinder as the main force. With the track programming and the optimization method, the dynamic converse problem of the arm end track is researched, so as to get the optimized rotation angle when the arm end reaches the expected point. By using the PD control theory, without decoupling and rank-decreasing, only with feed back from the hydraulic system to realize the close loop control of the arm end position, pose and movement, the relationship between the hydraulic system and the end position & pose is studied, so that the flexible distortion is reduced and the libration is restrained. What’s more, the simulation model of the mechanical arms is built by the dynamic simulation software. The simulation result prove that the movement equation built by this way can clearly describe each dynamic character of the mechanical arms.

Transition from the mechanics of material points to the mechanics of structured particles

2016 ◽  
Vol 30 (04) ◽  
pp. 1650018 ◽  
Author(s):  
V. M. Somsikov
Keyword(s):  
Energy Equation ◽  
Lagrange Equation ◽  
Classical Mechanics ◽  
Infinitely Divisible ◽  
Holonomic Constraints ◽  
Time Symmetry ◽  
The Difference ◽  
Material Points ◽  
Irreversible Dynamics ◽  
The Way

In this paper, necessity of creation of mechanics of structured particles is discussed. The way to create this mechanics within the laws of classical mechanics with the use of energy equation is shown. The occurrence of breaking of time symmetry within the mechanics of structured particles is shown, as well as the introduction of concept of entropy in the framework of classical mechanics. The way to create the mechanics of non-equilibrium systems in the thermodynamic approach is shown. It is also shown that the use of hypothesis of holonomic constraints while deriving the canonical Lagrange equation made it impossible to describe irreversible dynamics. The difference between the mechanics of structured particles and the mechanics of material points is discussed. It is also shown that the matter is infinitely divisible according to the laws of classical mechanics.

Mechanism analysis of steel frames considering moment–shear interaction

2018 ◽  
Vol 22 (1) ◽  
pp. 254-267 ◽  
Author(s):  
Mohammad T Kazemi ◽  
Mahdi Sharifi ◽  
Jian Yang
Keyword(s):  
Virtual Work ◽  
Load Capacity ◽  
Steel Frames ◽  
Rigid Bodies ◽  
Mechanism Analysis ◽  
Link Length ◽  
Element Analysis ◽  
Eccentrically Braced Frames ◽  
Analysis And Design ◽  
Collapse Mechanisms

Considering the interaction of flexural moment and shear force in the steel frames with haunch or intermediate beam length and eccentrically braced frames with intermediate link length is a major concern of the structural analysis and design. This article contains two stages. In the first stage, to investigate the moment–shear interaction for the highly ductile steel I-sections, a study is carried out using finite element analysis and a simple and practical relationship is developed. In the second stage, a simple approach based on virtual work method for assemblage of interconnected rigid bodies is employed to consider collapse mechanisms with mixed hinges. Using this approach, the applicability of the proposed relationship is demonstrated for a one-bay portal frame by considering all possible collapse mechanisms including those containing mixed hinges. Some numerical examples are presented using the proposed approach. Results indicate that, in general, the effect of moment–shear interaction on the load capacity of the structures cannot be ignored, and the capacity could be estimated, without using step-by-step analysis. Finally, by satisfying kinematic compatibility requirements and normality condition, simplified relations are derived to estimate post-mechanism deformations of plastic hinges for a prescribed lateral drift.

scholarly journals Modeling and Simulation for Multi-Rotor Fixed-Wing UAV Based on Multibody Dynamics

2019 ◽  
Vol 37 (5) ◽  
pp. 928-934 ◽  
Author(s):  
Han Wu ◽  
Zhengping Wang ◽  
Zhou Zhou ◽  
Rui Wang
Keyword(s):  
Dynamic Model ◽  
Multibody Dynamics ◽  
Dynamic Modeling ◽  
Virtual Work ◽  
Lagrange Equation ◽  
Dynamic Change ◽  
Lagrangian Multiplier ◽  
And Control ◽  
Control Surfaces ◽  
Generalized Coordinate

Accurate dynamic modeling lays foundation for design and control of UAV. The dynamic model for the multi-rotor fixed-wing UAV was looked into and it was divided into fuselage, air-body, multi-rotors, vertical fin, vertical tail and control surfaces, based on the multibody dynamics. The force and moment model for each body was established and derived into the Lagrange equation of the second king by virtual work. By electing quaternion as generalized coordinate and introducing Lagrangian multiplier, the dynamic modeling was deduced and established. Finally, the comparison between the simulation results and the experimental can be found that the present dynamic model accurately describes the process of dynamic change of this UAV and lay foundation for the control of UAV.

Inspiring Learning: Assessment of Friction in a Real-World Model Using the Moving Frame Method in Dynamics

2018 ◽  
Author(s):  
Thorstein R. Rykkje ◽  
Daniel Leinebø ◽  
Erlend Sande Bergaas ◽  
Andreas Skjelde ◽  
Thomas J. Impelluso
Keyword(s):  
Undergraduate Students ◽  
Virtual Work ◽  
3D Visualization ◽  
Mechanical Energy ◽  
Rigid Bodies ◽  
Moving Frame ◽  
Integration Scheme ◽  
Moving Frames ◽  
Moving Frame Method ◽  
Frame Method

This project conducts research in energy dissipation. It also demonstrates the power of the new Moving Frame Method (MFM) in dynamics to inspire undergraduate students to embark on research in engineering. The MFM is founded on Lie Group Theory to model rotations of objects, Cartan’s moving frames to model the change of a frame in terms of the frame, and a new notation from the discipline of geometrical physics. The MFM presents a consistent notation for single bodies, linked systems and robotics. This work demonstrates that this new method is accessible by undergraduate students. The MFM structures the equations of motion on the Special Euclidean Group and the Principle of Virtual work. A restriction on the virtual angular velocities to enable variational methods empowers the method. This work implements an explicit fourth order Runge-Kutta numerical integration scheme. It assesses the change in mechanical energy. In addition, this work researches the energy losses due to friction in a system of linked rigid bodies. This research also builds the physical hardware and compares the theory and experiment using 3D visualization. The authors built the structure to observe the actual motion and approximate the energy loss functions. This project demonstrates the power of WebGL to supplement analyses with visualization.

Differential Equations

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Equation ◽  
Shortest Paths ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Variational Problems ◽  
Functional Derivative ◽  
Mathematical Framework ◽  
Analytical Mechanics ◽  
Mathematical Technique

This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.

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