Lagrange s Equations of Motion with Constraint Forces Kane s equations do not incorporate constraint forces 1
Ax0 A is m n of rank r MEAM 535 Review: Linear Algebra The row space, Col (A T ), dimension r Col (A T ) is spanned by: r 1 a 11 a 12 a 1n, r 2 a 21 a 22 a 2n,, r m a m1 a m2 The null space, N (A), dimension n r a mn. x N( A) r 1 r 2 N (A) r m R n orthogonal Col (A T ) http://ocw.mit.edu/ocwweb/mathematics/18-06spring-2005/videolectures/detail/lecture14.htm 2
y T A 0 A is m n of rank r MEAM 535 Review: Linear Algebra The column space, Col (A), dimension r Col (A) is spanned by columns c 1 a 11 a 21 a m1, c 2 a 12 a 22 a m2,, c n a 1n a 2n a mn. y N( A T ) c 1 c 2 c n The left null space, N (A T ), dimension m r dim Col(A) r dim Col(A T ) r dim N(A) n r dim N(A T ) m r orthogonal N (A T ) Col(A) R m 3
C m n q n 1 0 C is m n of rank m MEAM 535 Application to Velocity Constraints The row space, Col (C T ), dimension m Col (C T ) is spanned by: r 1 c 11 c 12 c 1n, r 2 c 21 c 22 c 2n,, r m c m1 c m2 The null space, N (C), dimension n m Physical Interpretation of N(C) N (C) set of admissible velocities (that don t violate constraints) c mn. Col (Γ) q N( C) r 1 r 2 N (C) orthogonal r m Col (C T ) R n Col (C T ) is the set of constraint forces orthogonal to admissible velocities! 4
Example 2: Rolling Disk (Simplified) τ d θ C (x, y) τ s radius R φ 5
Example: Rolling Disk Simplified τ d θ C (x, y) τ s radius R φ Two equations of motion Nonholonomic constraints provide two additional equations q n 1 Γ n p u p 1 C m n q n 1 0 C Γ 2 4 4 2 C m n Γ n p 0 6
Example 2: Rolling Disk (Simplified) τ d C θ τ s radius R φ Q * m q 1 0 m q 2, Q 0 I a q 3 τ d I t q 4 τ s (x, y) P T Γ 7
P T Γ 0 Γ is n p of rank p MEAM 535 Application to Generalized Forces The column space, Col (Γ), dimension p The left null space, N (Γ T ), dimension n p m c 1 Γ 11 Γ 21 Γ n1, c 2 Γ 12 Γ 22 Γ n2,, c p Γ 1p Γ 2p Γ np. P N( Γ T ) c 1 c 2 c p Physical Interpretation of N(Γ T ) set of admissible velocities (that don t violate constraints) set of constraint forces Col (Γ) N(C) N(Γ Τ ) Col (C T ) orthogonal N (Γ T ) Col(Γ) R n 8
Lagrange s Equations with Multipliers P k p Equations of Motion m Constraints P lies in the null space of Γ T m Columns of C T span the null space of Γ T There exist a vector of m constants (multipliers) λ, such that 9
Example: Rolling Disk Simplified τ d θ C (x, y) τ s radius R φ 10
Rolling Disk Simplified: Comparison τ d C (x, y) C θ m n τ s radius R φ Γ n p p+m equations in p+m unknowns p+m+m equations in p+m+m unknowns p equations n equations m equations 11
Multipliers are Constraint Forces! m Columns of C T span the null space of Γ T 1. C T λ are generalized forces (associated with the derivatives of generalized coordinates) 2. The m constants (multipliers) are coefficients for vectors that are orthogonal to the allowable directions of motion τ d θ C (x, y) τ s radius R φ 12
Constraint Forces for Holonomic Systems What if you choose more generalized coordinates than necessary? n is the number of generalized coordinates (more than necessary) p is the number of degrees of freedom (i.e., only p gen. coords. necessary) n > p Notice the parallel with nonholomic systems! n speeds, but only p independent speeds q n 1 [ W] n p u p 1 + X n 1 n Q + * ( Q k k )W kj 0 k1 p Equations of Motion m Constraints 13
y MEAM 535 Example: Particle in circular hoop R θ x LHS of Lagrange s equations of motion for unconstrained problem P r d dt P θ d dt ( ) L ṙ ( ) L θ L r L θ m r mr θ 2 + mg sin θ mr 2 θ + mgr cos θ. 14
Generalized speed: udθ/dt Velocities MEAM 535 Example B Q P Generalized Active Forces -Fa 1 x τa 3 Generalized Active Inertial Forces Find the constraint forces at the pin joint Q -m A a P -m x2dot a 1 15
Example (continued) p1 θ d e φ q n 1 [ W] n p u p 1 + X n 1 n3 16
Constraint Forces for Holonomic Systems q n 1 [ W] n p u p 1 + X n 1 p Equations of Motion m Constraints There exist a vector of m constants (multipliers) λ, such that 17
Example: Normal Force at P Generalized speed: udθ/dt Velocities B P x Generalized Active Forces -Fa 1 τa 3 Generalized Active Inertial Forces What if we relax the constraint that keeps the piston moving horizontally? 18
Example (continued) n1 r θ φ l y q n 1 [ W] n p u p 1 + X n 1 C p1, n2 19
Example: Normal Force at P Generalized speeds B Partial Velocities P Generalized Active Forces r θ φ l y -Fa 1 τa 3 x Generalized Active Inertial Forces 20