Dynamics of Nonholonomic Systems

Transcription

1 Dynamics of Nonholonomic Systems 1

2 Example: When is the skate in equilibrium? x 2 Number of degrees of freedom n = 3 degrees of freedom m= 1 nonholonomic constraint Generalized coordinates (x 1, x 2, x 3 ) Generalized speeds (coordinates of C and angle formed with x axis) C l A v P = x 1 a 1 + x 2 a 2 + lsin x 3 a 1 + lcos x 3 a 2 P x 3 u x 1 ( ) = u 1 a 1 + u 2 ( lsin x 3 a 1 + lcos x 3 a 2 ) + u 3 a 2 F x 3 Static equilibrium for a holonomic system i.e., the force must be zero! Yet, a force passing through C (i.e., when l=0) perpendicular to u will keep the nonholonomic skate in equilibrium 2

3 Nonholonomic Systems Key Idea in Principle of Virtual Work Project forces along directions that are unconstrained (i.e., along directions of motion These directions are given by the partial velocities Key Idea for Extension to Nonholonomic systems Project forces along directions that are unconstrained (i.e., along directions of motion 3

4 Example x 2 Number of degrees of freedom for the nonholonomic system n m = 2 degrees of freedom l P u Generalized coordinates (x 1, x 2, x 3 ) C x 3 x 1 Generalized speeds 4

5 Nonholonomic Systems Key Idea in Principle of Virtual Work Project forces along directions that are unconstrained (i.e., along directions of motion) These directions are given by the partial velocities Key Idea for Extension to Nonholonomic systems Project forces along directions that are unconstrained (i.e., along directions of motion) These directions are given by nonholonomic partial velocities Partial Velocities do not account for directions that are unavailable because of nonholonomic constraints Introduce nonholonomic partial velocities = subset of directions available after incorporating the nonholonomic constraints 5

6 Principle of Virtual Work for Nonholonomic Systems Are Q j = 0? n generalized coordinates p speeds The speeds are not independent! k = p + 1,, n Q j δw = (a F ) i r P i q j δq j p N n N (a + F ) i j=1 i=1 i=1 k= p +1 p = N (a F ) P i v i j r Pi q k n [ ] + N (a F ) P [ i v i k ] j=1 i=1 δq j k= p +1 i=1 p N n (a = F ) P i v i P j + A kj v i k i=1 k= p +1 δq j j=1 p N (a = F ) P [ i v i j ] δq j j=1 i=1 δq k 6 p j=1 A kj δq j

7 Principle of Virtual Work for Nonholonomic Systems A nonholonomic system of N particles (P 1, P 2,, P N ) with n speeds (u 1, u 2,, u n ), p of which are independent is in static equilibrium if and only if the p nonholonomic generalized forces are all zero. The jth nonholonomic generalized force given by must equal zero! δw = Q j = p N (a F ) i v [ P i ] j δq j = 0 j=1 N i=1 i=1 (a F ) P [ i v i j ] 7

8 Example x 2 Static Equilibrium if and only if l P u C (F x, F y ) x 3 x 1 F x = F y = 0 Unless l =0 If l =0, equilibrium when 8

9 Nonholonomic system with m dependent speeds n 1 p 1 A = [A lj ] is a n p matrix whose rank is equal to p: 9

10 Example x 2 Number of degrees of freedom n m = 2 degrees of freedom l P u Generalized coordinates (x 1, x 2, x 3 ) C x 3 x 1 Generalized speeds 10

11 Generalized coordinates to independent speeds MEAM 535 Generalized coordinates Derivatives of Generalized coordinates Generalized speeds Independent Generalized speeds q u n 1 = [ Y] n n q n 1 + Z n 1 q n 1 = [ W] n n u n 1 + X n 1 [ W] = [ Y] 1 q n 1 = W ( n n A n p u p 1 + B ) n 1 + X n 1 = ( W n n A n p )u p 1 ( W n n B n 1 + X n 1 ) Γ n p Θ n 1 11

12 Nonholonomic system with m dependent speeds But We define the n p matrix Γ 12

13 D Alembert s Principle for Nonholonomic Systems Apply the Principle of Virtual Work to a system of particles F i - m i a i = 0 (-m i a i ), the inertial force, is an applied force acting on the ith particle Nonholonomic active generalized forces Nonholonomic generalized active inertial forces 13

14 Kane s equations for nonholonomic systems The motion of a nonholonomic system with N particles and n speeds, with p of them independent, is governed by p equations of motion given by: Main Advantage Kane s equations: Only p equations of motion As many equations as there are independent speeds (degrees of freedom) 14

15 Example 1 Single particle, mass m Coordinates x, y, and z Generalized speeds External force F x, F y, and F z Nonholonomic constraint Generalized active and inertial forces 15

16 The jth Generalized Inertial Force Kane s equations Kane - Lagrange equations for a holonomic system with n speeds u 1, u 2,, u n : The jth Generalized Inertial Force Kane s equations Kane - Lagrange equations for a nonholonomic system with n speeds u 1, u 2,, u p : 16

17 Summary Coordinate transformation (Jacobian) matrix W kj Generalized speeds: u n u 1 u 1, u 2,, u n Generalized active and inertial forces: u 2 Projection Γ kj Only p of the speeds are independent n Q ʹ + * ( Q ʹ k k )W kj = 0 k=1 u p n Q ʹ + * ( Q ʹ k k )Γ kj = 0 k=1 u 1, u 2,, u p 17

18 Lagrange s Equations for a Nonolonomic Conservative System Kane - Lagrange equations for a nonholonomic system with p speeds u 1, u 2,, u p : If we further assume that the forces acting on the system are conservative, we can find a potential function, V(q 1, q 2,..., q n, t) such that all (holonomic) generalized active forces can be expressed as partial derivatives of the potential function: 18

19 Example 2: Rolling Disk (Simplified) τ d θ C (x, y) τ s radius R φ 19

20 Example 2: Rolling Disk (Simplified) τ d θ C (x, y) τ s radius R φ 20