By Richard H. Rand

This article used to be built through Dr. Rand over numerous years of graduate-level guide in complicated dynamics, nonlinear vibrations, and perturbation equipment. It provides numerous subject matters in nonlinear dynamics, easily defined and modernized by using the MACCSYMA computing device algebra method. The emphasis is on utilizing machine algebra as a device for the derivation and the answer of nonlinear differential equations and encourages in-depth knowing of the actual and mathematical ideas concerned. Algebraic rules are provided along with examples, references, workouts and MACSYMA machine courses.

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**Example text**

Obtain the equations of motion by Lagrange' s equations. c. y -1] 2 + y. 2 ) 2' and then obtain the equations of motion by writing 4. A u niform solid sphere B of mass m and radius R rolls without slipping on a plane horizontal surface which rotates with constant angular speed {1 about a vertical axis through a fixed point P. 7. A sphere on a turntable e2 29 .. 30 METHOD OF GENERALIZED SPEEDS a. How many coordinates do you need to specify the configuration of the sphere? b. Derive the rolling-without-slipping constraint equations.

37). Then differentiate the constraint eq s . ( 1 . 6 ) and substitute the resulting expressions for:ie ' and y ' into the three eq uations of motion, thereby eliminating x and y entirely. The resulting eq uations may be solved for Cp' ,0 and '¢' to give: ( 1 . 38) ( 1 . 39) 23 ME THOD OF GENERALIZED SPEEDS 24 ( 1 . (1. 8) into the results of the method of generalized speeds, eq s . ( 1 . 23)-( 1 . 7. 25), gives eq uations which are eq uivalent to ( 1 . 38)-( 1 . 40). 8 Computer Algebra Computer algebra is a welcome tool for handling the computations involved in deriving the eq uations of motion in a sufficiently complicated problem.

A u niform solid sphere B of mass m and radius R rolls without slipping on a plane horizontal surface which rotates with constant angular speed {1 about a vertical axis through a fixed point P. 7. A sphere on a turntable e2 29 .. 30 METHOD OF GENERALIZED SPEEDS a. How many coordinates do you need to specify the configuration of the sphere? b. Derive the rolling-without-slipping constraint equations. c. Derive the equations of motion by the method of generalized speeds using the generalized speeds where x,y refer to the position of the center of mass of the sphere relative to an inertial frame E, and where d.