Movement Equations 5

Georges Vénizélos is a former Professor at the Conservatoire National des Arts et Métiers (CNAM), Paris, France, where he occupied the chair of mechanical systems design. Over his 35-year career at CNAM, he taught extensively and pursued research in the fields of solid mechanics and structural vibration.Abdelkhalak El Hami is Full Professor at INSA Rouen Normandy, France. He is the Director of the Department of Mechanics and responsible for the Chair of Mechanics at CNAM in Normandy, as well as several European educational projects.

• Preface ixTable of Notations xiChapter 1. Set of Solids with Neither Loops Nor Branches 11.1. Identifying a chain of solids with neither loops nor branches 11.2. Applying the fundamental principles of mechanics 21.2.1. Principle of effort generators 31.2.2. Principle of effort receivers 41.2.3. Applying the fundamental principle of dynamics 41.2.4. Theorem of mutual actions 71.2.5. Summary of equations obtained 81.3. Study of the movement of a chain of solids (case of three solids) 81.3.1. Applying the fundamental principle of dynamics 81.3.2. Solidifying parameters 91.3.3. Movement equations 111.3.4. Determining the link unknowns 181.4. Links between solids 181.4.1. Link associated with the point contact of two solids 181.4.2. Link torsor associated with the line contact of two solids 251.4.3. Link torsor associated with the surface contact of two solids 281.4.4. Fundamental links between two solids in contact 32Chapter 2. Vibration Mechanics of Systems of Solids 352.1. Movement equations of a set of solids 352.1.1. Configuring and situating a set of solids in a Galilean frame 352.1.2. Velocity distributors of n solids 372.1.3. Torsors associated with loads and efforts 382.1.4. General equation of dynamics derived from the fundamental principle 392.1.5. Applying analytical mechanics of movement 392.2. Linear oscillatory systems with n solids 422.2.1. Setting the problem as an equation 422.2.2. Equilibrium of a set of n solids 462.2.3. Oscillations of a set of n solids 472.2.4. Vibration eigen modes of a set of n solids 482.2.5. Influence of the initial conditions of the problem 532.3. Studying the vibrations of a continuous set by passing to the limit 542.3.1. Taking the boundary conditions into account at any instant 592.4. Exercises 622.4.1. Exercise 1: movement equations – equilibrium 622.4.2. Exercise 2: movement around an equilibrium position 752.4.3. Exercise 3: dynamics of an RTT robot (one rotation + two translations) 85Chapter 3. Vibrations with N Degrees of Freedom 973.1. Introduction 973.2. Homogeneous system – free vibrations (f1 = f2 = 0) 993.2.1. Without damping (cij = 0) 993.2.2. Solving the system (Σ) 1003.2.3. Damped free system 1083.3. Response on the time domain of an excited system 1123.4. Exercises 1133.4.1. Exercise 1: eigen modes of a system with 2 DOF 1133.4.2. Exercise 2: free and forced oscillations of a conservative 2-DOF system 1183.4.3. Exercise 3: calculation/test correlation 1253.4.4. Exercise 4: damped system with a single excited mode 1283.4.5. Exercise 5: system excited by the base 137Chapter 4. Modal Analysis of N Degrees of Freedom 1454.1. Introduction 1454.1.1. Normal modes 1454.2. Response in the frequency domain of a conservative structure subjected to a harmonic excitation 1464.3. Response of a structure with proportional viscous damping to a harmonic excitation 1504.4. Frequency response of a structure with proportional hysteretic damping 1534.5. Exercises 1554.5.1. Exercise 1: receptance matrix of a conservative structure 1554.5.2. Exercise 2: receptance matrix of a structure with proportional viscous damping 1644.5.3. Exercise 3: case of a non-diagonal mass matrix 170References 177Index 179