A Linearised Input-Output Representation for Control Synthesis in - PowerPoint PPT Presentation

§ 12.3 & paper ES A Linearised Input-Output Representation for Control Synthesis in Flexible Multibody System A Linearised Input-Output Representation for Dynamics Control Synthesis in Flexible Multibody System Layout Dynamics • Finite element representation of flexible multibody systems • Equations of motion and reaction • Linearised equations of motion and reaction J.B. Jonker, J. van Dijk and R.G.K.M. Aarts • Linearised state-space equations Department of Mechanical Automation and Mechatronics • Stationary and equilibrium solutions University of Twente • From state-space equations to transfer function(s) The Netherlands • Illustrative examples • Conclusions Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 1 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 2 § 2 Finite element representation of multibody systems Planar flexible beam element y R q n y � � cos φ p − sin φ p R p ≡ R q n x sin φ p cos φ p φ q � � q cos φ q − sin φ q R q ≡ β ( k ) sin φ q cos φ q R p n y R p n x n y l ( k ) ≡ x q − x p φ p p = [ x q − x p , y q − y p ] T x n x Physical description of a flexible multibody system � ( x q − x p ) 2 + ( y q − y p ) 2 � 1 / 2 − l ( k ) Elongation: ε ( k ) = D ( k ) ( x ( k ) ) = 1 1 0 ε ( k ) = D ( k ) ( x ( k ) ) = − ( R p n y , l ( k ) ) Bending: Element k with set of nodal coordinates x ( k ) (Cartesian and rotational) in a 2 2 fixed inertial coordinate system and deformation modes specified by a vector of ε ( k ) = D ( k ) ( x ( k ) ) = ( R q n y , l ( k ) ) deformation parameters e ( k ) . 3 3 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 3 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 4

§ 3 Kinematic analysis Geometric transfer functions q : generalised coordinates x ( m ) and e ( m ) x = F ( x ) ( q ) Deformation equations e = F ( e ) ( q ) e = D ( x ) x : nodal coordinates e = ∂ D Velocities x = D x D ˙ ˙ x e : deformation mode coordinates ∂ ˙ x = D q F ( x ) ˙ D q F : first-order geometric transfer function ˙ q Partitioning: e = D q F ( e ) ˙ D 2 ˙ q q F : second-order geometric transfer function   x ( o ) fixed coordinates Accelerations    x ( c )  x = dependent nodal coordinates   q F ( x ) ˙ x = D 2 q + D q F ( x ) ¨ x ( m ) ¨ q ˙ q absolute generalized / independent coordinates q F ( e ) ˙ e = D 2 q + D q F ( e ) ¨ ¨ q ˙ q   e ( o ) rigid / zero deformations    e ( m )  e = relative generalized /independent coordinates   e ( c ) dependent deformations Generalised coordinates x ( m ) , e ( m ) collected in vector q with ndof kinematic degrees of freedom. Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 5 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 6 § 6 § 4 Equations of reaction for unknown stress resultants and reaction forces Equations of motion expressed the kinematic degrees of freedom q :     f ( o ) σ ( o ) q = D F ( x ) T ( f − M D 2 F ( x,c ) ˙ q ) − D F ( e ) T σ     ¯ M ( q )¨ q ˙ ( D x D ) T σ = f − M ¨ f ( c )    σ ( m )  with partitioning f =  and σ = x    f ( m ) σ ( c ) M = D F ( x ) T M D F ( x ) ¯ system mass matrix       f ( o ) − M ( o,c ) ¨ x ( c ) − M ( o,m ) ¨ ( D ( o ) D ( o ) ) T ( D ( o ) D ( m ) ) T ( D ( o ) D ( c ) ) T x ( m ) σ ( o )       − M ( c,c ) ¨ x ( c ) − M ( c,m ) ¨ ( D ( c ) D ( o ) ) T ( D ( c ) D ( m ) ) T ( D ( c ) D ( c ) ) T D F ( x ) T f = D F ( x,c ) T f ( c ) + D F ( x,m ) T f ( m ) σ ( m ) f ( c ) x ( m )  =      nodal forces f ( m ) − M ( m,c ) ¨ x ( c ) − M ( m,m ) ¨ ( D ( m ) D ( o ) ) T ( D ( m ) D ( m ) ) T ( D ( m ) D ( c ) ) T σ ( c ) x ( m ) D F ( e ) T σ = D F ( e,m ) T σ ( m ) + D F ( e,c ) T σ ( c ) stress resultants If the square matrix [( D ( c ) D ( o ) ) T , ( D ( c ) D ( m ) ) T ] is non-singular, then     � � � �� � � � � �  σ ( m ) S ( m,m ) S ( m,c )  S ( m,m ) S ( m,c ) σ ( m ) e ( m ) e ( m ) σ ( o ) � x ( m ) − ( D ( c ) D ( c ) ) T σ ( c ) � ˙ f ( c ) − M ( c,c ) ¨ x ( c ) − M ( c,m ) ¨ a = ˜  + d d  = + D 1 , σ ( c ) σ ( c ) S ( c,m ) S ( c,c ) e ( c ) S ( c,m ) S ( c,c ) e ( c ) σ ( m ) ˙ a d d � ( D ( c ) D ( o ) ) T , ( D ( c ) D ( m ) ) T � − 1 . Elastic coefficients S ( m,m ) , S ( m,c ) and S ( c,c ) (symmetric matrices) with ˜ D 1 = Viscous damping coefficients S ( m,m ) , S ( m,c ) and D ( c,c ) (symmetric matrices) d d d Vector σ ( c ) is known from the previous slide, so the reaction forces f ( o ) and Driving forces and torques σ ( m ) and σ ( c ) a . a the driving forces f ( m ) are then determined as well. Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 7 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 8

§ 7 & § 12.3 § 5 State equations Linearised equations: prefix δ indicates small variations � � q d : q d � � � � � � dynamic degrees of freedom (to be computed) q d q d δ q d q = q r : q r 0 x = x 0 + δ x q = q 0 + δ q so q = = + rheonomic degrees of freedom (known) q r q r δ q r 0     � ¯ � � � � � � � � �  D q d F ( x ) T  D q d F ( e ) T ¯ q d q d q d q d M dd M dr ¨ ˙ ˙ δ ˙  ( f − M D 2 F ( x ) ˙  σ 0 = q ˙ q ) − x = ˙ ˙ x 0 + δ ˙ x q = ˙ ˙ q 0 + δ ˙ q so ˙ q = = + ¯ ¯ q r q r q r q r D q r F ( x ) T D q r F ( e ) T M rd M rr ¨ ˙ ˙ δ ˙ 0 � � � � � � q d q d q d ¨ ¨ δ ¨ q r ¯ q d = ¯ q , t ) − ¯ 0 M dd ( q )¨ f d ( q , ˙ M dr ¨ f : nodal forces x = ¨ ¨ x 0 + δ ¨ x q = ¨ ¨ q 0 + δ ¨ q so ¨ q = = + q r q r q r ¨ ¨ δ ¨ 0 M dd = D q d F ( x ) T M D q d F ( x ) ¯ σ : stress resultants M dr = D q d F ( x ) T M D q r F ( x ) ¯ M : mass matrix Stresses σ = σ 0 + δ σ a and forces f = f 0 + δ f . f d = D q d F ( x ) T ( f − M D 2 F ( x ) ˙ q ) − D q d F ( e ) T σ ¯ q ˙ Linearised equations of kinematics δ x = D F ( x ) δ q , Non-linear state-space equations q + ( D 2 F ( x ) ˙ x = D F ( x ) δ ˙ δ ˙ q ) δ q , � � � � � � q d q d q d ˙ q + 2( D 2 F ( x ) ˙ q + D 3 F ( x ) ˙ d x = D F ( x ) δ ¨ q + ( D 2 F ( x ) ¨ δ ¨ q ) δ ˙ q ˙ q ) δ q = with state vector z = M − 1 q d q d ¯ dd (¯ f d − ¯ q r ) ˙ ˙ dt M dr ¨ with third-order geometric transfer function D 3 F ( x ) . Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 9 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 10 § 7.2 § 7.3 Linearised equations of motion Linearised equations of reaction � � � � δ q = D F ( x ) T δ f − D F ( e ) T δ σ a ¯ C + ¯ ¯ K + ¯ ¯ N + ¯ M δ ¨ q + D δ ˙ q + G First order terms in Taylor series expansion: with D F ( x ) T δ f = D F ( x,c ) T δ f ( c ) + D F ( x,m ) T δ f ( m ) ( D x D ) T δ σ + (( D 2 x D ) T σ ) δ x = δ f + ( D x f in ) δ x + ( D ˙ x f in ) δ ˙ x and D F ( e ) T δ σ a = D F ( e,m ) T δ σ ( m ) + D F ( e,c ) T δ σ ( c ) a . − D x ( M ¨ x ) δ x − M δ ¨ a x or ( D x D ) T δ σ = δ f + M ( x ) δ ¨ q − ( N ( x ) + G ( x ) ) δ q q − C ( x ) δ ˙ M = D F ( x ) T M D F ( x ) ¯ C = D F ( x ) T � � x f in ) D F ( x ) + 2 M D 2 F ( x ) ˙ ( D ˙ ¯ q M ( x ) = M D F ( x ) D = D F ( e ) T S d D F ( e ) ¯ C ( x ) = ( D ˙ x f in ) D F ( x ) + 2 M D 2 F ( x ) ˙ q K = D F ( e ) T S D F ( e ) ¯ N ( x ) = D x ( M ¨ x − f in ) D F ( x ) + ( D ˙ x f in ) D 2 F ( x ) ˙ q N = D F ( x ) T � x − f in ) D F ( x ) + ( D ˙ x f in ) D 2 F ( x ) ˙ q + D 3 F ( x ) ˙ D x ( M ¨ + M ( D 2 F ( x ) ¨ ¯ q q ˙ q ) � �� q + ( D 3 F ( x ) ˙ + D F ( e ) T S d D 2 F ( e ) ˙ D 2 F ( x ) ¨ G ( x ) = (( D 2 x D ) T σ ) D F ( x ) + M q )˙ q q G = − D 2 F ( x ) T [ f − M ¨ x ] − D 2 F ( e ) T σ ¯ Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 11 Jonker/van Dijk/Aarts FMSA4CP / Input-Output / 12