Movement Equations 4

Michel BOREL, Former mechanics lecturer at CNAM Center in Versailles (1971-1999); associated professor at Petroleum and Motors National high School - ENSPM (1978-2008); engineer at Bertin Society (1970-1980); engineer at General Armement Direction - DGA (1980-2009). Georges VÉNIZÉLOS, Professor, chair of mechanical systems designing, Conservatoire National des Arts et Métiers (CNAM).

• Introduction ixTable of Notations xiiiChapter 1 Equilibrium, Stationary Movement and Oscillation of a Free Solid 11.1 Expression of the fundamental principle of dynamics for a free solid 11.2 Canonical form of the fundamental principle 41.2.1 Dynamic resultant 51.2.2 Dynamic moment at O s 51.2.3 Fundamental principle of dynamics 71.3 Equilibrium of the free solid 71.3.1 Equations of equilibrium 71.3.2 Stability of equilibrium 91.4 General equations of small movements of a free solid 91.4.1 Reminder of developments limited to the first order 91.4.2 Equations of small movements of the free solid 121.4.3 Analytical mechanics of free solids 431.5 Matrix expression of small movements of a free solid 591.5.1 Using vector representation 601.5.2 Using analytical mechanics 631.5.3 Relative situation of frames at the equilibrium 651.6 Stationary movement 671.6.1 Cyclic parameters 671.6.2 Characterizing a stationary movement 681.6.3 Conditions of realization of a stationary movement 691.6.4 Neighboring motions and stability of a stationary movement 711.6.5 Applications 74Chapter 2 Solving Equations of Small Movements 752.1 Linear differential systems with constant coefficients 752.1.1 General periodic solution of the homogeneous system 772.1.2 Particular solution to the system 792.1.3 Exercise 1 812.2 Laplace transformation 872.2.1 Definition 872.2.2 Linearity of a Laplace transformation 872.2.3 Laplace transforms for common functions 882.2.4 Functional properties of the Laplace transformation 912.2.5 Examples of use of the Laplace transform 932.2.6 Applications 96Chapter 3 Oscillator Studies 1153.1 Physical nature of oscillatory motion 1153.2 The single oscillator 1163.2.1 Definitions 1173.2.2 Conditions of an oscillatory motion 1173.2.3 Study of free oscillatory motion 1183.2.4 Study of forced oscillations 1233.2.5 Study of a modulated oscillatory signal 1263.3 Motion of coupled oscillators 1293.3.1 Coupling of two oscillators 1293.3.2 Study of free oscillation 1293.3.3 Applications: problem 6 1383.4. Oscillatory device of k oscillators – equilibrium and stability 1543.4.1 Approaching the problem 1543.4.2 Routh criteria 155Chapter 4 Gyroscopic Motion 1634.1 Gyroscopic coupling 1634.1.1 Composition of the device 1634.1.2 Velocity-distributing torsor 1664.1.3 Kinetic energies of all three components 1664.1.4 Equations of dynamics 1674.1.5 Equations of analytical mechanics 1684.1.6 Situations of equilibrium of the gyroscopic device 1724.1.7 Stability of the stationary movement 1734.2 Gyroscopic pendulum 1774.2.1 Composition of the device 1774.2.2 Velocity-distributing torsors 1784.2.3 Kinetic energies 1794.2.4 Lagrange equations 1794.2.5 Equilibrium and stability 1814.3 The gyro-compass 1844.3.1 Composition of the device 1844.3.2 Fundamental principle of dynamics 1864.3.3 Equations of analytical mechanics 1864.3.4 Stationary movement and stability 1934.3.5 Note for establishing Lagrange equations 1994.4 Applications: problem 7 – motion stabilizer 199Bibliography 213Index 215