Dynamics of Flexible Multibody Systems: A Finite Element Approach

Mathematical notations (/3) Dynamics of Flexible Multibody Systems: A Finite Element Approach Partial derivatives q q,..., q n T x F(q) F (q),..., F nx (q) T Prof.dr.ir. J.B. Jonker and dr.ir. R.G.K.M. Aarts Department of Mechanical Automation and Mechatronics University of Twente The Netherlands D q F F q Fi F i,j q j nx n nx n D 2 qf 2 F 2 q 2 F i F i,jk q j q k nx n n nx n n DvM / DvM / Mathematical notations (2/3) Mathematical notations (3/3) Chain rule of differentiation D(x F(q)) D (x F(q)),..., D ne (x F(q)) T Duality and the scalar product P < f, ẋ > D q D D Di q F l D i,l F l,j F l q j ne n D xdd q F ne n D 2 qd 2 D 2 q 2 D i F l F m Di 2 F + l F l F m q j q k F ne n n l q j q k ne n n D i,lm F l,j F m,k ne n n + D i,l F l,jk ne n n (D 2 xdd q F)D q F + D x DD 2 qf < u, v > u T v < u, Av > u T Av (u T Av) T (Av) T u v T A T u (v T A T u) T (A T u) T v < A T u, v > DvM / 2 DvM / 3

Physical representation Finite element representation hinge 3 hinge 4 beam 2 beam 4 beam 3 hinge 2 hinge 5 beam end-effector hinge DvM / 4 DvM / 5 Multibody representation Multibody formulation { x, y, φ x 2, y 2, φ 2 x 3, y 3, φ 3 } joint 4 body 5 joint 5 body 6 body 4 joint 3 body 3 joint 2 body 2 joint body x and y constraints at point A: x 2 l() cos φ y 2 l() sin φ DvM / 6 DvM / 7

Multibody formulation { x, y, φ x 2, y 2, φ 2 x 3, y 3, φ 3 } Multibody formulation { x, y, φ x 2, y 2, φ 2 x 3, y 3, φ 3 } x and y constraints at point B: x + 2 l() cos φ x 2 + 2 l(2) cos φ 2 y + 2 l() sin φ y 2 + 2 l(2) sin φ 2 x and y constraints at point C: x 2 + 2 l(2) cos φ 2 x 3 2 l(3) cos φ 3 y 2 + 2 l(2) sin φ 2 y 3 2 l(3) sin φ 3 DvM / 7 DvM / 7 Multibody formulation { x, y, φ x 2, y 2, φ 2 x 3, y 3, φ 3 } Multibody formulation { x, y, φ x 2, y 2, φ 2 x 3, y 3, φ 3 } x and y constraints at point A: x 2 l() cos φ y 2 l() sin φ x and y constraints at point B: x + 2 l() cos φ x 2 + 2 l(2) cos φ 2 y + 2 l() sin φ y 2 + 2 l(2) sin φ 2 x and y constraints at point C: x 2 + 2 l(2) cos φ 2 x 3 2 l(3) cos φ 3 y 2 + 2 l(2) sin φ 2 y 3 2 l(3) sin φ 3 x and y constraints at point D: x 3 2 l(3) cos φ 3 l 4 y 3 2 l(3) sin φ 3 x and y constraints at point D: x 3 2 l(3) cos φ 3 l 4 y 3 2 l(3) sin φ 3 DvM / 7 DvM / 8

Finite element formulation { φ x 2, y 2, φ 2 x 3, y 3, φ 3 } Finite element formulation { φ x 2, y 2, φ 2 x 3, y 3, φ 3 } link element : (x 2 ) 2 + (y 2 ) 2 /2 l () ( ) φ arccos x 2 l () link element 2: (x 3 x 2 ) 2 + (y 3 y 2 ) 2 /2 l (2) ( ) φ 2 arccos x 3 x 2 l (2) DvM / 9 DvM / 9 Finite element formulation { φ x 2, y 2, φ 2 x 3, y 3, φ 3 } Finite element formulation { φ x 2, y 2, φ 2 x 3, y 3, φ 3 } link element : (x 2 ) 2 + (y 2 ) 2 /2 l () ( ) φ arccos x 2 l () link element 2: (x 3 x 2 ) 2 + (y 3 y 2 ) 2 /2 l (2) ( ) φ 2 arccos x 3 x 2 l (2) link element 3: (x 3 x 4 ) 2 + (y 3 ) 2 /2 l (3) ( φ 3 arccos x 3 l (4) ) l (3) DvM / 9 link element 3: (x 3 x 4 ) 2 + (y 3 ) 2 /2 l (3) ( φ 3 arccos x 3 l (4) ) l (3) DvM /

Finite element representation of mechanisms Finite element formulation H ST B B B ST B H ST B T ST B H Planar excavator mechanism T T T Truss Slider Truss Beam Hinge DvM / Deformation functions: e (k) D (k) (x (k) ) e (k) i D (k) i (x (k) j ) x (k) vector of nodal coordinates e (k) vector of generalised deformations The number of generalised deformations is equal to the number of nodal coordinates minus the number of degrees of freedom of the element as a rigid body. The deformations should be invariant under rigid body motions and are generally non-lineair functions of the nodal coordinates. The functions D i are chosen such that they have a clear physical meaning which facilitates the description of strength and stiffness. DvM / 2 Finite element formulation Planar (slider) truss element Derivative functions: velocities: ė (k) DD (k) ẋ (k) ė (k) i accelerations: D(k) i x (k) ẋ (k) j D (k) i,j ẋ(k) j j ë (k) (D 2 D (k) ẋ (k) )ẋ (k) + DD (k) ẍ (k) ë (k) i 2 D (k) i x (k) j x (k) l D (k) i,jl ẋ(k) j ẋ (k) l ẋ (k) j ẋ (k) l + D (k) i,j ẍ(k) j + D(k) i x (k) ẍ (k) j j DvM / 3 x (k) x truss p x q x p, y p x q, y q T l (k) x q x p ((x q x p ) 2 + (y q y p ) 2 ) /2 e (k) D (k) (x(k) ) l (k) l (k) DvM / 4

Planar (slider) truss element (cont.) D D (k) x pd(k) (xq x p ) (x q x p ) 2 + (y q y p ) 2 /2 cos β (x p ) (y p ) (x q ) (y q ) D (k),j cos β sin β cos β sin β cos β xq x p l (k) and sin β yq y p l (k) Planar (slider) truss element (cont.) D 2 D (k) 2 x p y pd(k) (x q y p x p ) (x q x p ) 2 + (y q y p ) 2 /2 (x q x p )(y q y p ) (x q x p ) 2 + (y q y p ) 2 3/2 D (k),jl l (k) cos β xq x p cos β sin β l (k) (x p ) (y p ) (x q ) (y q ) sin 2 β sin β cos β sin 2 β sin β cos β sin β cos β cos 2 β sin β cos β cos 2 β sin 2 β sin β cos β sin 2 β sin β cos β sin β cos β cos 2 β sin β cos β cos 2 β l (k) and sin β yq y p l (k) DvM / 5 DvM / 6 Sliding bar Sliding bar v ė DD ẋ 2 3, 2, 2 3, 2 ẏ q 2 3v + 2 ẏ q ẏ q v 3 ë ẋ T D 2 D ẋ + DD ẍ 4 4 3 4 4 3 v v,,, ẏ q 4 3 3 4 4 3 3 4 2 4 4 3 4 4 3 4 3 3 4 4 3 3 4 ẏ q ÿ q 4v 2 + 2 3, 2, 2 3, 2 ÿ q DvM / 7 DvM / 8

Planar beam element Planar beam element x (k) x beam p x q x p, y p, φ p x q, y q, φ q T elongation:ε (k) D (k) (x(k) ) l (k) l (k) bending: ε (k) 2 D(k) 2 (x(k) ) (R p n y, l (k) ) (x q x p )sin φ p (y q y p )cos φ p ε (k) 3 D(k) 3 (x(k) ) (R q n y, l (k) ) (x q x p )sin φ q + (y q y p )cos φ q R p cos φ p sin φ p sin φ p cos φ p and R q cos φ q sin φ q sin φ q cos φ q DvM / 9 DvM / 2 Planar beam element Planar beam element D (k) i,j (x p ) (y p ) (φ p ) (x q ) (y q ) (φ q ) cos β sin β cos β sin β sin φ p cos φ p l x (k) cos φ p + l y (k) sin φ p sin φ p cos φ p sin φ q cos φ q sin φ q cos φ q l x (k) cos φ q l y (k) sin φ q bending: ε (k) 2 D(k) 2 (x(k) ) (R p n y, l (k) ) (x q x p )sin φ p (y q y p )cos φ p ε (k) 3 D(k) 3 (x(k) ) (R q n y, l (k) ) (x q x p )sin φ q + (y q y p )cos φ q DvM / 2 D (k),jl l (k) (x p ) (y p ) (φ p ) (x q ) (y q ) (φ q ) sin 2 β sin β cos β sin 2 β sin β cos β sin β cos β cos 2 β sin β cos β cos 2 β sin 2 β sin β cos β sin 2 β sin β cos β sin β cos β cos 2 β sin β cos β cos 2 β cos β l(k) x l xq x p and sin β l(k) y (k) l (k) l yq y p (k) l (k) DvM / 22

D (k) 2,jl D (k) 3,jl Planar beam element (x p ) (y p ) (φ p ) (x q ) (y q ) (φ q ) cos φ p sin φ p cos φ p sin φ p ε (k) 2 cos φ p sin φ p cos φ p sin φ p cos φ q sin φ q cos φ q sin φ q cos φ q sin φ q cos φ q sin φ q ε (k) 3 Planar rigid beam element x p x (k) rigid beam φ x q xp, y p φ x q, y q T cos β l(k) x l xq x p and sin β l(k) y (k) l (k) l yq y p (k) l (k) elongation:ε (k) D (k) (x(k) ) l (k) l (k) DvM / 23 DvM / 24 Planar rigid beam element Planar rigid beam element (x p ) (y p ) (φ) (x q ) (y q ) D (k) i,j cos β sin φ sin β cos φ l (k) x cos φ + l y (k) sin φ cos β sin φ sin β cos φ bending: R (k) R (k) (φ) ε (k) 2 D(k) 2 (x(k) ) (R (k) n y, l (k) ) (x q x p )sin φ (y q y p )cos φ cos φ sin φ sin φ cos φ D (k),jl D (k) 2,jl l (k) sin 2 β sin β cos β sin 2 β sin β cos β cos φ sin β cos β cos 2 β sin β cos β cos 2 β sin φ cos φ sin φ ε (k) 2 cos φ sin φ sin 2 β sin β cos β sin 2 β sin β cos β cos φ sin β cos β cos 2 β sin β cos β cos 2 β sin φ DvM / 24 DvM / 25

Planar hinge element Simple pendulum x (k) hinge φp, φ q T e (k) D (k) (x(k) ) φ q φ p D (k),j (φ p ) (φ q ), (φ p ) (φ q ) D (k),jl ẋ q ε cos β sin β ε 2 sin φ cos φ l ẏ q φ ẋq cos β sin β sin φ cos φ ẏ q l φ ẋ q 2 lω, ẏq 2 3 lω DvM / 26 DvM / 27 ẍq cos β sin β sin φ cos φ ÿ q Simple pendulum (/l) (ẋ q ) 2 sin 2 β 2ẋ q ẏ q sin β cos β + (ẏ q ) 2 cos 2 β φ(ẋ q cos φ + ẏ q sin φ) l φ Assembly process Planar kinematic analysis The interconnections between the elements are accomplished by indicating common nodes between the elements. p translational node, p rotational node. φ 2 3 2 2 ẍq 2 3 ÿ q lω2 pin-joint hinge-joint rigid-joint ẍ q 2 3 lω 2, ÿ q 2 lω2 DvM / 28 DvM / 29

Planar kinematic analysis Various supports Define a vector x of global nodal coordinates x x,..., x nx T, Then for each element the deformation functions D i can be described in terms of the components of vector x, that is e D. (x). e ne D ne (x) or e D(x) Kinematic constraints can be introduced by putting conditions on the nodal coordinates x i as well as by imposing conditions on the deformations e i. support rigid joint roller pinned joint roller pinned joint diagrammatic form constraints const. const. const. const. const. const. const. const. clamped const. const. const. DvM / 3 DvM / 3 Absolute constraint conditions Support conditions x () i C () i Driving conditions x (m) i (t) C (m) i (t) Relative constraint conditions () Holonomic constraints e () i, D () (x) Driving conditions e (m) i (t) C (m) i (t), D (m) (x) e (m) (t) x () -------- x x (c) nxo -------- nxc x (m) nx fixed or support coordinates dependent or calculable coordinates absolute generalized coordinates e () -------- e e (m) neo -------- nem e (c) ne fixed prescribed deformation parameters relative generalized coordinates redundant deformation parameters DvM / 32 DvM / 33

Relative constraint conditions (2) Partitioning of vectors x and e Relative constraint conditions (3) If nxc nxo nem, where nem represents the number of holonomic constraints, then according to the implicit function theorem nxc nxo nem number of unknown or calculable coordinates x (c) i number of holonomic constraints If nxc nxonem then a solution for given (x (m), e (m) ) exists. DOF is number of kinematic degrees of freedom DOF nx nxc + nem neo nx nxo neo e () D () (x) neo ------- nem e (m) -------------- D (m) (x) can be solved for x (c) as function of the generalized coordinates (x (m), e (m) ), and next the redundant deformation mode coordinates can be calculated from nem e ne (c) D (c) (x) DvM / 34 DvM / 35 Four-bar mechanism () Four-bar mechanism (2) x () y, x 4, x 5, y 5 T, e () ε, ε 2, ε 4, ε 5 T x (c) x 2, y 2, x 3, y 3, y 4 T, e (m) ε 3 x (m) x, e (c) ε 6 DvM / 36 ε D (x) ((x 3 x ) 2 + (y 3 ) 2 ) /2 l () e () ε 2 D 2 (x) ((x 2 x ) 2 + (y 2 ) 2 ) /2 l (2) ------- e (m) ε 4 D 4 (x) ((x 4 x 3 ) 2 + (y 4 y 3 ) 2 ) /2 l (4) ε 5 D 5 (x) ((x 4 x 2 ) 2 + (y 4 y 2 ) 2 ) /2 l (5) ---- ---------- ----------------------------------------------------------- ε 3 D 3 (x) ((x 3 x 2 ) 2 + (y 3 y 2 ) 2 ) /2 l (3) e (c) ε 6 D 6 (x) ((x 3 ) 2 + (y 3 ) 2 ) /2 l (6) DvM / 37

Geometric transfer functions () Geometric transfer functions (2) solve e D(x) D q F (e) D x DD q F (x) D q F (e), D q F (x) for x (c) and e (c) as function of the generalized coordinates x (m) q e (m) The solutions are expressed by the geometric transfer functions differentiate D 2 qf (e) (D 2 xdd q F (x) )D q F (x) + D x DD 2 qf (x) D 2 qf (e), D 2 qf (x) e F (e) (q) x F (x) (q) F (e) D(F (x) ) for all q differentiate D 2 qf is calculated after D q F with the same linear equations. The components of D x D and D 2 xd are obtained from the element contributions D x D (k) and D 2 xd (k). D q F (e) D x DD q F (x) D q F (e), D q F (x) DvM / 38 DvM / 39 First order geometric transfer functions () First order geometric transfer functions (2) Dq F (e) D x D Dq F (x) DF (e,) D DF (e,m) x D () -------------- D x D (m) DF (e,c) -------------- DF (x) D x D (c) nxo nxc nx DF (e,) neo DF (e,m) D () D () D (c) D () D (m) D () nem -------------- D () D (m) D (c) D (m) D (m) D (m) DF (x,) DF (x,c) ne DF (e,c) --------------------------------------------------------- D () D (c) D (c) D (c) D (m) D (c) DF (x,m) DF (e,) e() q DF (e,m) e(m) q DF (x,) x() q DF (x,m) x(m) q e() e(), neo x (m) e (m) O, O neo e(m) e(m), nem x (m) e (m) O, I x() x(), nxo x (m) e (m) O, O nxc x(m) x(m), nxm x (m) e (m) I, O DvM / 4 DvM / 4

First order geometric transfer functions (3) nxo nxc nx DF (e,) neo DF (e,m) D () D () D (c) D () D (m) D () nem -------------- D () D (m) D (c) D (m) D (m) D (m) DF (x,) DF (x,c) ne DF (e,c) --------------------------------------------------------- D () D (c) D (c) D (c) D (m) D (c) DF (x,m) nxo DF (x,c) D (c) D () DF (e,) D nxc D (c) D (m) DF (e,m) (m) D () D (m) D (m) DF (x,m) D (c) D () O, O D D (c) D (m) (m) D () O, I D (m) D (m) I, O D (c) D () D (c) D (m) D (m) D () O D (m) D (m) I nem DF (e,c) D nec (c) D (c) DF (x,c) + D (m) D (c) DF (x,m) DvM / 42 Second order geometric transfer functions () D ij F (e) (D 2 xd D i F (x) )D j F (x) + D x D D ij F (x) D ij F (e,) (D D ij F (e,m) 2 D () D i F (x) )D j F (x) ---------------- (D 2 D (m) D i F (x) )D j F (x) D ij F (e,c) ------------------------------------------ (D 2 D (c) D i F (x) )D j F (x) D () D () D (c) D () D (m) D () + D () D (m) D (c) D (m) D (m) D (m) D ij F (x,) --------------------------------------------------------- D ij F (x,c) D () D (c) D (c) D (c) D (m) D (c) D ij F (x,m) s DvM / 43 Second order geometric transfer functions (2) Velocities and accelerations D ij F (e,m) D ij F (x,m) D ij F (x,c) D (c) D () (D 2 D () D i F (x) )D j F (x) D (c) D (m) (D 2 D (m) D i F (x) )D j F (x) Velocities: ẋ F(x) x (m) ẋ(m) + F(x) e (m) ė(m), or ẋ DF (x) q ė F(e) x (m) ẋ(m) + F(e) e (m) ė(m), or ė DF (e) q D ij F (e,c) (D 2 D (c) D i F (x) )D j F (x) + D (c) D (c) D ij F (x,c) D ij D (k) D ji D (k) D ij F (x) D ji F (x) and D ij F (e) D ji F (e) Accelerations: ẍ (D 2 F (x) q) q + DF (x) q ë (D 2 F (e) q) q + DF (e) q DvM / 44 DvM / 45

Analysis of accelerations () Analysis of accelerations (2) ë (D 2 Dẋ)ẋ + DDẍ ë () (D ë (m) 2 D () ẋ)ẋ ------- (D 2 D (m) ẋ)ẋ ë (c) (D 2 D (c) ẋ)ẋ D () D () D (c) D () D (m) D () + D () D (m) D (c) D (m) D (m) D (m) ẍ () --------------------------------------------------------- ẍ (c) D () D (c) D (c) D (c) D (m) D (c) ẍ (m) ẍ (c) D (c) D () D (c) D (m) D (m) D () O D (m) D (m) I ẍ(m) ë (m) ẍ (c) DF (x,c) ẍ (m) D ë (m) (c) D () (D 2 D () ẋ)ẋ D (c) D (m) (D 2 D (m) ẋ)ẋ ẍ (c) (D 2 F (x,c) q) q + DF (x,c) q (D 2 F (x,c) q) q D (c) D () D (c) D (m) (D 2 D () ẋ)ẋ (D 2 D (m) ẋ)ẋ (D 2 D () ẋ)ẋ (D 2 D (m) ẋ)ẋ DvM / 46 DvM / 47 Analysis of accelerations (3) Solution to the kinematic problem ë (c) DF (e,c) ẍ (m) +(D 2 D (c) ẋ)ẋ ë (m) D (c) D (c) D (c) D () D (c) D (m) (D 2 D () ẋ)ẋ (D 2 D (m) ẋ)ẋ ë (c) (D 2 F (e,c) q) q + DF (e,c) q (D 2 F (e,c) q) q D (c) D (c) D (c) D () +(D 2 D (c) ẋ)ẋ D (c) D (m) (D 2 D () ẋ)ẋ (D 2 D (m) ẋ)ẋ DvM / 48 DvM / 49

Solution to the kinematic problem (a) Solution to the kinematic problem (b) DvM / 5 DvM / 5 Example crank connecting rod ė () ẋ () D () D () D (c) D () D (m) D () ẋ (c) ẋ (m) x (c) x(c) + DF(x,c) q Di F (x,c) D (c) D () D (m) D () x, x 2, x3 4, l() 2, y, y 2 3, y 3, l(2) 2 3 x () x x (c) x (m) x, y, y 3 x 2, y 2 x 3 T DF (x,c) 2 2 3 2 3 2 2 3 3 4 3 4 DvM / 52 DvM / 53

x (c) D () (x ) x2 y 2 3 + 4 3 4.7945.359 3 x (c).8296.5964.92857.375.75.2994.7945.359.539.22955 x (c).7.5367 D () (x ).6.3 D () (x ).6 < δ x (c).75.539.2994.22955.696.6949 D () (x ).582.782 D () (x ).975 x (c).84599.5339.9537.384.582.782.39.583 DvM / 54 DvM / 55 Planar dynamic analysis Mechanics of materials Basic Laws: Newton: f p d dt (mp ẋ p ) m p ẍ p Euler: T p d dt (Jp ω p ) J p φ p f p T p M p J p ẍp φ p f p : nodal forces T p : nodal torque (moment) m p : mass J p : rotational inertia with, M p respect to center of mass p m p m p Prismatic bar Equilibrium: f p f q Imaginary cut at section mn Normal stress resultant: σ f q f p Hook s Law: σ S ε EA l ε ε l longitudinal elongation S longitudinal stiffness E modulus of elasticity A cross-section area DvM / 56 DvM / 57

Principle of virtual work () Principle of virtual work (2) Used to solve problems of static equilibrium. Virtual displacements are imaginary, infinitesimal changes in the nodal coordinates x i and deformation coordinates e i that are consistent with the system constraints, but are otherwise arbitrary; they are not true displacements but small variations in the coordinates. The symbol δ was introduced by Lagrange to emphasize the virtual character of instantaneous variations, as opposed to the symbol d which designates actual differentials of coordinates taking place in the time interval dt, during which interval forces and constraints may change. Work done by the real forces and stress resultants during a so-called virtual displacement is called virtual work. If a (deformable) structure in equilibrium under the action of a system of loads is given a small virtual deformation, then the virtual work done by the external forces (or loads) is equal to the virtual work done by the internal forces (or stress resultants). Prismatic bar (deformable) f p δx p + f q δx q σ δe f p δx p + f q δx q σ (δx q δx p ) δe δx q δx p δx q and δx p f q σ, f p σ DvM / 58 DvM / 59 D Alembert s principle Principle of virtual power () Newton s second law can be rewritten as f p M p ẍ p This equation enables us to extend the principle of virtual work to the dynamical case < (f p M p ẍ p ), δx p > δx p The virtual power of the external forces, inclusive of the inertial forces, acting on the element, must be equal to zero for all virtual velocity distributions that are free of deformation. < (f (k) f (k) in (ẋ(k) ) M (k) ẍ (k) ), δẋ (k) > for all virtual velocities δẋ (k) satisfying DD (k) δẋ (k) With the vector of Lagrange multipliers σ (k), we then obtain the equations of motion for element k < (f (k) f (k) in M(k) ẍ (k) ), δẋ (k) > < σ (k),dd (k) δẋ (k) > DvM / 6 for all δẋ (k) DvM / 6

Principle of virtual power (2) Stress resultants of the planar truss element < (f (k) f (k) in M(k) ẍ (k) ), δẋ (k) > < σ (k),dd (k) δẋ (k) > for all δẋ (k) With the transpose of DD (k) we obtain DD (k)t σ (k) f (k) f (k) in M(k) ẍ (k) Static case: ẋ (k) and ẍ (k) DD (k)t σ (k) f (k) (equilibrium equations) DvM / 62 DD (k)t σ (k) f (k) f p σ (k) cos β f q sin β cos β f p x sin β σ (k) f p f y x p + fx q cos β fx q sin β fy q fy p + fy q DvM / 63 Stress resultants of the planar beam element Stress resultants of the planar hinge element l (k) l (k) σ (k) σ (k) 2 σ (k) 3 f p x f p y T p fx q f q y T q f p σ (k), σ(k) 2 σ (k) T 3 T p σ (k) 2 l(k) f q σ (k), σ(k) 2 + σ (k) T 3 T q σ (k) 3 l(k) DvM / 64 σ (k) T p σ (k) T q σ (k) T p T q DvM / 65

Stiffness properties of planar elements () Stiffness properties of planar elements (2) Stresses of flexible elements are characterized by Hooke s law: σ (k) S (k) ε (k) Axial deformation: σ (k) E (k) A (k) ε x (Hooke s law) ε x du dx (axial strain) l (k) ε x dx l (k) l (k) ε (k) S (k) E(k) A (k) l (k) DvM / 66 DvM / 67 Elastic line concept () If the deformations ε (k) 2 and ε (k) 3 remain sufficiently small (ε (k) i l (k) ), then in the elastic range they are linearly related to bending moments σ (k) 2 l(k) and σ (k) 3 l(k). Slender beam (d (k) /l (k) ) subjected to bending moments σ (k) 2 l(k) and σ (k) 3 l(k) at its end points. E (k) I (k)d4 w dx 4 Integrating four times, Elastic line concept (2) for x l(k) w(x) 6 c x 3 + 2 c 2x 2 + c 3 x + c 4 w(), dw(x) dx x ε (k) 2 /l(k) w(l (k) ), dw(x) dx xl (k) ε (k) 3 /l(k) DvM / 68 DvM / 69

w(x) E (k) I (k) d2 w(x) dx 2 E (k) I (k) d2 w(x) dx 2 Elastic line concept (3) ( ) x 3 ( ) x 2 ( ) ( x 2 + l (k) l (k) l (k) ε (k) ) x 3 ( ) x 2 2 l (k) l (k) ε (k) 3 x xl (k) σ (k) 2 l(k) σ (k) 3 l(k) σ (k) 2 E(k) I (k) ( (l (k) ) 3 4ε (k) ) 2 2ε(k) 3 σ (k) 3 E(k) I (k) ( (l (k) ) 3 2ε (k) ) 2 + 4ε(k) 3 Stiffness properties of planar elements (3) Beam element: σ (k) S (k) ε (k) σ (k) σ (k) 2 σ (k) 3 E (k) A (k) l (k) 4 E(k) I (k) (l (k) ) 3 2 E(k) I (k) (l (k) ) 3 2 E(k) I (k) (l (k) ) 3 4 E(k) I (k) (l (k) ) 3 ε (k) ε (k) 2 ε (k) 3 DvM / 7 DvM / 7 Elastic line beam w(x) ε (k) 2 L (x) ε (k) 3 L 2(x) ( ) x 3 ( ) x 2 x L (x) 2 + l (k) l (k) l (k) ( ) x 3 ( ) x 2 L 2 (x) l (k) l (k) Inclusion of second-order geometric effects ε x du(x) dx + 2 ( ) 2 dw(x) dx ε (k) l (k) l (k) + 2 l (k) ( ) 2 dw(x) dx dx l (k) ( ) 2 dw(x) 2 dx dx 6l (k) ε (k) 2, ε (k) 4 3 4 ε(k) 2 ε (k) 3 ε (k) l (k) l (k) + ( 3l (k) 2 ε (k) ) 2 ( (k) 2 + ε 2 ε(k) 3 + 2 ε (k) ) 2 3 DvM / 72 DvM / 73

Inertia properties of finite elements Lumped mass formulation Lumped mass formulation Consistent mass formulation In this idealisation rigid bodies with equivalent mass and rotational inertia are attached to the end nodes of the element. The lumped masses and rotational inertias are calculated by assuming that the element behaves like a rigid body. Conditions for dynamical equivalence:. The mass of the element should be equal to the total mass of the lumped system. 2. The center of mass of the element and of the discrete mass model should coincide. 3. The rotational inertia of the element and that of the discrete model should be equal. DvM / 74 DvM / 75 Lumped mass formulation Lumped mass formulation Symmetrically shaped elements Asymmetrically shaped elements. m m p + m q 2. Center of gravity located at x l/2 m p m q m/2 3. J c J p + J q + 2 m/2 (l/2) 2 J p + J q + 4 ml2 DvM / 76 DvM / 77

Lumped mass formulation Lumped mass formulation x c x p + s p x p + R p s p R p cos φ p sin φ p sin φ p cos φ p ẋ c ẋ p + φ p R p s p R p dr p dφ p, φ c φ p, φ c φ p sin φ p cos φ p cos φ p sin φ p DvM / 78 f c in mp ẍ c, T c in Jc φ c < f p, δẋ p > + < T p, δ φ p > < m p ẍ c, δẋ c > < J c φ c, δ φ c > for all δẋ c, δ φ c satisfying δẋ c δẋ p + δ φ p R p s p, δ φ c δ φ p DvM / 79 Lumped mass formulation < (f p m p ẍ c ), δẋ p > + < (T p J c φ c ), δ φ p > < m p ẍ c, R p s p δ φ p > < m p s p T R p Tẍc, δ φ p > for all δẋ p, δ φ p < (f p m p ẍ c ), δẋ p > + < (T p m p s p T R p Tẍc J c φ c ), δ φ p > for all δẋ p, δ φ p f p m p ẍ c, T p m p s p T R p Tẍc J c φ c ẍ c ẍ p + φ p R p s p ( φ p ) 2 R p s p, φ c φ p m p I m p R p ẍp s p f p + m m p s p T R p T J p φ p p R p s p ( φ p ) 2 T p + m p s p 2 ( φ p ) 2 m p I m p m p If s p s p, then m p ẍp I f p J p φ p T p, J p J c + m p s p 2 DvM / 8 DvM / 8

Consistent mass formulation Consistent mass matrix of planar truss element Basic assumptions: Position of every point on the element described by polynomial functions whose parameters depend on the nodal positions and orientations. Distributed inertia forces are replaced by equivalent concentrated forces in the element nodes using the virtual power principle. Interpolation functions are used which are consistent with the interpolations used in the calculation of the stiffness matrix (elastic line concept). r(ξ) ( ξ)x p + ξx q ξ s l (k) Virtual power of distributed inertia forces rdm: m (k) l (k) < δṙ, r > dξ DvM / 82 DvM / 83 m (k) l (k) < δṙ, r > dξ ṙ ( ξ)ẋ p + ξẋ q, m (k) l (k) r ( ξ)ẍ p + ξẍ q < δṙ, r > dξ δẋ (k)t M c (k) ẍ (k) M c (k) m (k) l (k) ( ξ 2 )I ξ( ξ)i ξ( ξ)i ξ 2 dξ I (x p ) (y p ) (x q ) (y q ) M (k) c m(k) l (k) 6 2 2 ------------------------------------ 2 2 m(k) l (k) 6 2I I I 2I DvM / 84 Consistent mass matrix of planar beam element ξ s l (k) Hermite interpolation: two points and two slope conditions at the end points r(ξ)( 3ξ 2 + 2ξ 3 )x p + (ξ 2ξ 2 + ξ 3 )(l (k) + e (k) )Rp n x +(3ξ 2 2ξ 3 )x q + ( ξ 2 + ξ 3 )(l (k) + e (k) )Rq n x DvM / 85

Consistent mass matrix of planar beam element ṙ(ξ)( 3ξ 2 + 2ξ 3 )ẋ p + (ξ 2ξ 2 + ξ 3 )l (k) φ p R p n x +(3ξ 2 2ξ 3 )ẋ q + ( ξ 2 + ξ 3 )l (k) φ q R q n x r(ξ)( 3ξ 2 + 2ξ 3 )ẍ p + (ξ 2ξ 2 + ξ 3 )l (k) φ p R p n x +(3ξ 2 2ξ 3 )ẍ q + ( ξ 2 + ξ 3 )l (k) φ q R q n x ξ s l (k) (ξ 2ξ 2 + ξ 3 )l (k) ( φ p ) 2 R p n x ( ξ 2 + ξ 3 )l (k) ( φ q ) 2 R q n x DvM / 86 m (k) l (k) < δṙ, r > dξ δẋ (k)t M (k) c ẍ (k) f (k) in M (k) c m(k) l (k) 42 f (k) in m(k) l (k) 42 (x p ) (φ p ) (x q ) (φ q ) 56I 22l (k) R p n x 54I 3l (k) R q n x 4(l (k) )2 3l (k) nt xr p 3(l (k) )2 n T xr p R q n x 56I 22l (k) R q n x 4(l (k) )2 22l (k) Rp n x 3l (k) Rq n x 4(l (k) )2 n T xr pt R p n x 3(l (k) )2 n T xr p R q n x 3l (k) Rp n x 22l (k) Rq n x 3(l (k) )2 n T xr qt R p n x 4(l (k) )2 n T xr qt R q n x ( φ p ) 2 ( φ q ) 2 DvM / 87 Equations of motion with multipliers Global mass matrix M: M k (M (k) l + M (k) c ) Virtual power equation: < (f (c) M (c,c) ẍ (c) ), δẋ (c) >< σ (c), δė (c) > for all virtual velocities δẋ (c) and δė (c) satisfying D (c) D () δẋ (c) δė (c) D (c) D (c) δẋ (c) < (f (c) M (c,c) ẍ (c) ), δẋ (c) > < σ (),D (c) D () δẋ (c) > for all δẋ (c). < σ (c),d (c) D (c) δẋ (c) > With the transpose transformations D (c) D ()T and D (c) D (c)t M (c,c) ẍ (c) + D (c) D ()T σ () f (c) D (c) D (c)t σ (c) D () (x) System of differential algebraical equations (DAE s) M (c,c) D (c) D ()T D (c) D () ẍ(c) f (c) D σ () (c) D (c)t σ (c) (D (c) D (c) D () ẋ (c) )ẋ (c) DvM / 88 DvM / 89

Lagrange s form of Jourdain s principle Combines the computational advantages of Newton-Euler approach and Lagrange s method. objective I geometric transfer functions objective II D Alembert s forces principle of virtual power (Jourdain s principle) Equations of motion in terms of independent coordinates () Global mass matrix M: M k (M (k) l + M (k) c ) Vector of kinematic degrees of freedom q: q d q q d : vector of dynamic degrees of freedom q r q r : vector of rheonomic coordinates equations of motion in terms of independent coordinates DvM / 9 DvM / 9 Equations of motion in terms of independent coordinates (2) Equations of motion in terms of independent coordinates (3) Virtual power equation: < (f Mẍ), δẋ >< σ, δė > With the transpose transformations D q df (x)t and D q df (e)t : < D q df (x)t (f Mẍ), δ q d >< D q df (e)t σ, δ q d > for all for all δ q d δẋ D q df (x) δ q d and δė D q df (e) δ q d D q df (x)t (f Mẍ) D q df (e)t σ ẋ D q df (x) q d + D q rf (x) q r ẍ D q df (x) q d + D q rf (x) q r + (D 2 qf (x) q) q System of ordinary differential equations (ODE s) M q d D q df (x)t f M (( D 2 qf (x) q ) q + D q rf (x) q r) D q df (e)t σ where M D q df (x)t MD q df (x) DvM / 92 DvM / 93

Virtual power equation: Kinetostatic analysis < (f(x, ẋ, t) M(x)ẍ), δẋ >< σ, δė > for all δė DD(x)δẋ (DD) T σ f Mẍ (Equations of reaction) f () σ () M (,) M (,c) M (,m) f f (c) f (m), σ σ (m) σ (c), M M (c,) M (c,c) M (c,m) M (m,) M (m,c) M (m,m) neo nem nec (D () D () ) T (D () D (m) ) T (D () D (c) ) T nxo --------------------------------------------------- σ (D (c) D () ) T (D (c) D (m) ) T (D (c) D (c) ) T () σ (m) nxc --------------------------------------------------- (D nxm (m) D () ) T (D (m) D (m) ) T (D (m) D (c) ) T σ (c) f () M (,c) ẍ (c) M (,m) ẍ (m) f (c) M (c,c) ẍ (c) M (c,m) ẍ (m) f (m) M (m,c) ẍ (c) M (m,m) ẍ (m) σ () σ (m) (D (c) D () ) T,(D (c) D (m) ) T f (c) M (c,c) ẍ (c) M (c,m) ẍ (m) D (c) D (c) T σ (c) σ (c) Sε (c) + S d ε (c) (Kelvin-Voigt model) DvM / 94 DvM / 95 Planar mechanism with four truss elements () x () x x (c) x (m) x, y, y 4 x 2, y 2, x 3, y 3 x 4 T e e () e, e 2, e 3, e 4 T x, x 2, x 3, x 4 l (3) 4 y, y 2 l () 6, y 3 l () + l (2) 9, y 4 l () + l (2) 9 ė () ẋ () D () D () D (c) D () D (m) D () ẋ (c) ẋ (m) DvM / 96 DvM / 97

ė () ẋ x y y 4 x 2 y 2 x 3 y 3 x 4 ẏ ẏ 4 ---- ẋ 2 ẏ 2 3 5 4 5 3 5 4 5 ẋ 3 ẏ 3 ---- ẋ 4 ẋ 2 ẋ (c) ẏ 2 ẋ 3 ẋ4 ẏ 3 4 ẋ4 5 3 5 4 5 DF (x),,,,, T DF (x)t f Mẍ x y y 4 x 2 y 2 x 3 y 3 x 4 f x 2 fy f 4 2 y ------- ----------------------------------------- ---- f Mẍ 4 2 ẍ 4 2 ẍ 4 ------- fx 4 ----------------------------------------- ---- ẍ 4 ẍ,, 2,, 2, 2 T 8 2 + f 4 x, or f 4 x N DvM / 98 DvM / 99 σ σ () σ 2 (D (c) D () ) T f (c) σ 3 σ 4 4 5 8 6 3 5 2 2 Equations of motion: DF (x)t D 2 D () DF(x) 6 DF (x)t D 2 D () 2 DF(x) 3 DF (x)t D 2 D () 3 DF(x) 4,,, 6,,,,,, DvM / DvM /

DF (x)t D 2 D () 4 DF(x) 5 25 9 2 9 2,,, 2 6 2 6 9 2 9 2 2 6 2 6 D q df (x) MD q df (x) 2 + 3 D q df (x) MD 2 q d F (x) (ẋ 4 ) 2 4 (ẋ4 ) 2 D 2 F (x,c) D (c) D () (D 2 D () DF (x) )DF (x) DF (x)t f f 4 x + f 2 x f 4 x 4 6 8 D 2 F (x,c) 6 4 5 3 5 6 DF (e)t σ 3ẍ 4 f 4 x 4 4 (ẋ4 ) 2 DvM / 2 DvM / 3 Planar mechanism with two truss elements x, x 2 4, x 3, l () 5, y 3, y 2, y 3, q d y T, e 2 x y x x 2 y 2, DF (x) 3 4 x 3 y 3 3 4 q d y T, e 2 ----- ----- a, (D 2 qf (x) q d ) q d ẏ ẏ, ė 2 ----- ė 2 ----- a ----- DvM / 4 DvM / 5

e e, DF (e) e 2 (D 2 qf (e) q d ) q d ẏ ẏ, ė 2 ----- ė 2 a 25 64 (ẏ ) 2 M diagm, m, m 2, m 2, m 3, m 3 D q df (x)t MD q df (x) m + 6 9 (m 2 + m 3 ) 3 4 m 3 3 4 m 3 m 3 D q df (x)t f f y 3 4 (f2 x + fx) 3 f 3 x D q df (x)t M(D 2 qf (x) q) q σ 2 Se 2, DF (e)t σ 3 4 (m 2 + m 3 )a m 3 a m + 6 9 (m 2 + m 3 ) 3 4 m ÿ 3 f 4 3 m y 3 4 (f2 x + f 3 x) 3 m 3 ë 2 fx 3 75 256 (m 2 + m 3 )(ẏ ) 2 25 64 m 3(ẏ ) 2 Se 2 σ 2 DvM / 6 DvM / 7 Numerical integration of equations of motion Numerical integration of equations of motion (a) DvM / 8 DvM / 9

Numerical integration of equations of motion (b) Rotation of body coordinates r r x, r y, r z T, r T r x, r y, r z r Rr R R T r R T r DvM / DvM / Rotation of body coordinates Finite rotations Matrix R is defined by nine direction cosines. R R 2 R 3 R R 2 R 22 R 23 R 3 R 32 R 33 R e x e x cos( e x, e x ) R 2 e x e y cos( e x, e y ) R 3 e x e z cos( e x, e z ) R 2 e y e x cos( e y, e x ) R 22 e y e y cos( e y, e y ) R 23 e y e z cos( e y, e z ) R 3 e z e x cos( e z, e x ) R 32 e z e y cos( e z, e y ) R 33 e z e z cos( e z, e z ) Only three of these components are independent which can be described by 3 angles, called Euler angles, involving 3 successive rotations. DvM / 2 (a) (b) (c) The positions of a block: (a) initially, (b) after 9 rotations about the x- and y-axes, (c) after 9 rotations about the y- and x-axes. DvM / 3

Euler angles Euler angles Cannot be summed as vector quantities. Rotation matrices contain trigoniometric functions. Rotation matrices contain singularities. Euler s theorem on the motion of a body The general displacement of a body with one point fixed is equivalent to a single rotation about some axis through that point. The theorem dictates that the orientation of the body-fixed axes at any time t can be obtained by single rotation about some axis. (orientational axis of rotation) DvM / 4 DvM / 5 Euler parameters λ cos φ/2 λ e x sin φ/2 λ 2 e y sin φ/2 λ 3 e z sin φ/2 Euler parameters Euler parameters (normed quaternions) Describe rotational matrix in a natural way. Rotation matrices contain algebraic functions. No singularities of rotation matrix. λ 2 + λ2 λ2 2 λ2 3 2(λ λ 2 λ λ 3 ) 2(λ λ 3 + λ λ 2 ) R 2(λ λ 2 + λ λ 3 ) λ 2 λ2 + λ2 2 λ2 3 2(λ 2λ 3 λ λ ) 2(λ λ 3 λ λ 2 ) 2(λ 2 λ 3 + λ λ ) λ 2 λ2 λ2 2 + λ2 3 λ T λ λ 2 + λ2 + λ2 2 + λ2 3 R T R DvM / 6 DvM / 7

r Rr Angular velocities and Euler parameters ṙ Ṙr + Rṙ ṙ Ṙr r ω r or ṙ Ωr ω z ω y Ω ω z ω x ω y ω x R T r r ṙ ṘR T r Ω ṘR T Angular velocities and Euler parameters (continued) Ṙ ij R ij,k λ k k k k 2 k 3 2λ 2λ 3 2λ 2 2λ 2λ 2 2λ 3 2λ 2 2λ 2λ 2λ 3 2λ 2λ Rij,k 2λ 3 2λ 2λ 2λ 2 2λ 2λ 2λ 2λ 2 2λ 3 2λ 2λ 3 2λ 2 2λ 2 2λ 2λ 2λ 3 2λ 2λ 2λ 2λ 3 2λ 2 2λ 2λ 2 2λ 3 Ω ij R ik,l R kj λ l DvM / 8 DvM / 9 Angular velocities and Euler parameters (continued) λ ω x λ ω y 2 λ λ 3 λ 2 λ 2 λ 3 λ λ λ λ ω z λ 3 λ 2 λ λ 2 λ 3 Identities with Euler parameters Λλ Λ λ ΛΛ T Λ Λ T I ω 2Λ λ λ ω x λ ω y 2 λ λ 3 λ 2 λ 2 λ 3 λ λ λ λ ω z λ 3 λ 2 λ λ 2 λ 3 ω 2Λ λ R ΛΛ T Λ T Λ Λ T Λ I λλ T R ΛΛ T Λ λ Λ λ DvM / 2 DvM / 2

Identities with Euler parameters Manipulator with six degrees of freedom Λ λ Λ λ B H H H Λ λ Λ λ H B B wrist Λ Λ T ΛΛ T Ṙ 2Λ Λ T 2 ΛΛ T ST B H H ST B H Slider Truss Beam Hinge R ij R ij,k λ k + R ij,kl λ k λ l support DvM / 22 DvM / 23 Spatial (slider) truss element Spatial (slider) truss element (cont.) First-order partial derivatives: D (p) i D (k) l i l (k), D (q) i D (k) l i l (k) l x q x p, l 2 y q y p, l 3 z q z p x (k) x truss p x q x p, y p, z p x q, y q, z q T e (k) D (k) (x(k) ) l (k) l (k) l (k) x q x p ((x q x p ) 2 + (y q y p ) 2 + (z q z p ) 2 ) /2 Second-order partial derivatives: D (p) ij D(k) D (q) ij D(k) l (k) δ ij (l (k) ) 2 l il j D (p) i D (q) j D (k) D (q) i D (p) j D (k) l (k) δ ij if i j if i j δ ij (l (k) ) 2 l il j DvM / 24 DvM / 25

Spatial (slider) truss element (cont.) Spatial beam element Third-order partial derivatives: D (p) ijk D(k) D(q) ij D(p) k D(k) D (q) i (l (k) ) 3 D (p) j D (q) k D(k) D (p) i D (q) jk D(k) δ ij l k + δ ik l j + δ jk l i 3 (l (k) ) 2 l il j l k D (q) ijk D(k) D(p) ij D(q) k D(k) D (q) i (l (k) ) 3 D (p) jk D(k) D (p) i D (q) j D (p) k D(k) δ ij l k + δ ik l j + δ jk l i 3 (l (k) ) 2 l il j l k δ ij if i j if i j DvM / 26 x (k) beam x p, y p, z p λ p, λp, λp 2, λp 3 xq, y q, z q λ q, λq, λq 2, T λq 3 DvM / 27 λ λ, λ, λ 2, λ 3 T Euler parameters λ 2 + λ2 λ2 2 λ2 3 2(λ λ 2 λ λ 3 ) 2(λ λ 3 + λ λ 2 ) R 2(λ λ 2 + λ λ 3 ) λ 2 λ2 + λ2 2 λ2 3 2(λ 2λ 3 λ λ ) 2(λ λ 3 λ λ 2 ) 2(λ 2 λ 3 + λ λ ) λ 2 λ2 λ2 2 + λ2 3 Constraint equation λ T λ λ 2 + λ2 + λ2 2 + λ2 3 R T R I elongation: ε (k) D (k) l (k) l (k) torsion: ε (k) 2 D (k) 2 2 l(k) (R p n z, R q n y ) (R p n y, R q n z ) bending: ε (k) 3 D (k) 3 (R p n z, l (k) ) ε (k) 4 D (k) 4 (R q n z, l (k) ) ε (k) 5 D (k) 5 (R p n y, l (k) ) ε (k) 6 D (k) 6 (R q n y, l (k) ) l (k) x q x p x q x p, y q y p, z q z p T DvM / 28 DvM / 29

Visualization of bending deformations Torsional deformations ε (k) 2 2 l(k) (R p n z, R q n y ) (R p n y, R q n z ) ε (k) 3 (R p n z, l (k) ) ε (k) 5 (R p n y, l (k) ) ε (k) 4 (R q n z, l (k) ) ε (k) 6 (R q n y, l (k) ) DvM / 3 DvM / 3 Second-order geometric effects (Jaap Meijaard, 996): Spatial hinge element ε (k) ε(k) + 3l (k) 2(ε (k) 3 )2 + ε (k) 3 ε(k) 4 + 2(ε(k) 4 )2 +2(ε (k) 5 )2 + ε (k) 5 ε(k) 6 + 2(ε(k) 6 )2 ε (k) 2 ε (k) 2 + ( ε(k) 3 ε(k) 6 + ε(k) 4 ε(k) 5 )/l(k) ε (k) 3 ε (k) 3 + ε(k) 2 ( ε(k) 5 + ε(k) 6 )/(6l(k) ) ε (k) 4 ε (k) 4 ε(k) 2 ( ε(k) 5 + ε(k) 6 )/(6l(k) ) ε (k) 5 ε (k) 5 ε(k) 2 ( ε(k) 3 + ε(k) 4 )/(6l(k) ) ε (k) 6 ε (k) 6 + ε(k) 2 ( ε(k) 3 + ε(k) 4 )/(6l(k) ) x (k) λ p hinge λ q λ p, λp, λp 2, λp 3 λq, λq, λq 2, T λq 3 relative rotation: e (k) D (k) ATAN2 (Rp n y, R q n z ) (R p n y, R q n y ) bending: ε (k) 2 D (k) 2 (R p n y, R q n x ) ε (k) 3 D (k) 3 (R p n z, R q n x ) DvM / 32 DvM / 33

Dynamics of a rigid body Constraint equation Euler parameters λ T λ The λ-element e (λ) D (λ) λ T λ x body x p λ p x p, y p, z p λ p, λp, λp 2, T λp 3 DvM / 34 DvM / 35 Dynamics of a gyrostatic body f (x,p) d dt (Mp ẋ p ) f (ω,p) d dt (Jp ω p + h p ) ω p 2Λ p λ p λ p Λ p λ p λp 3 λ p 2 λ p 2 λ p 3 λ p λp λ p 3 λp 2 λ p λ p ė (λ,p) 2λ p,t λ p DvM / 36 DvM / 37

Principle of virtual power Principle of virtual power < (f (ω) d (Jω + h)), δω > dt for all δω 2Λδ λ, associated with the virtual motion of the body < (f (ω) d dt (Jω + h)),2λδ λ > < σ (λ),2λ T δ λ > for all δ λ < (f (λ) 2Λ T d dt (JΛ λ + h) 2σ (λ) λ), δ λ > for all δ λ f (λ) 2Λ T f (ω) λ λ 2 λ 3 f (λ) 2 λ λ 3 λ 2 λ 3 λ λ f(ω) λ 2 λ λ J RJ R T, h Rh λ R ΛΛ T, Λ λ λ 3 λ 2 λ 2 λ 3 λ λ λ 3 λ 2 λ λ f (λ) 4Λ T J Λ λ + 8 Λ T J Λ λ + 4 Λ T h + 2σ (λ) λ DvM / 38 DvM / 39 ẍ M 4Λ T J Λ 2λ λ ë (λ) 2 λ T λ + 2λ T λ Principle of virtual power σ (λ) f(x) f (λ) 8 Λ T J Λ λ 4 Λ T h M ẍ 4Λ T J Λ f (x) 2λ 2λ T λ σ (λ) f (λ) 8 Λ T J Λ λ 4 Λ T h 2 λ T λ Spatial beam element M (k) l f (k) in Lumped mass formulation x p λ p x q λ q M p 4Λ p T J p Λ p M q 4Λ q T J q Λ q 8Λ p T J p Λ p λ p 8Λ q T J q Λ q λ q DvM / 4 DvM / 4

Lumped mass formulation Spatial (slider) truss element M (k) l x p x q M p O O M q Consistent mass formulation of the spatial truss element Spatial hinge element λ p λ q M (k) 4Λ p T J p Λ p O l O 4Λ q T J q Λ q f (k) 8Λ p T in J p Λ p λ p 8Λ q T J q Λ q λ q r(ξ) ( ξ)x p + ξx q m (k) l (k) M (k) c m(k) l (k) 6 < δṙ, r > dξ δẋ (k)t M (k) c ẍ (k) 2I I I 2I, I DvM / 42 DvM / 43 Consistent mass formulation of the spatial beam element Consistent mass formulation of the spatial beam element r(ξ)( 3ξ 2 + 2ξ 3 )x p + (ξ 2ξ 2 + ξ 3 )(l (k) + e (k) )Rp n x +(3ξ 2 2ξ 3 )x q + ( ξ 2 + ξ 3 )(l (k) + e (k) )Rq n x M (k) c m(k) l (k) 42 (x p ) (λ p ) (x q ) (λ q ) 56I 22l (k) A 54I 3l(k) B 4(l (k) )2 A T A 3l (k) AT 3(l (k) )2 A T B )2 B T B 56I 22l (k) B 4(l (k) f (k) in m(k) l (k) 42 (l (k) A Rp n x λ p, B Rq n x λ q l (k) (22A λ p λ p 3B λ q λ) )2 ( 3B T A λ p λ p + 4B T B λ q λ q ) (l (k) )2 (4A T A λ p λ p 3A T B λ q λ q ) l (k) (3A λ p λ p 22B λ q λ q ), A A λ p, B B λ q A DvM / 44 DvM / 45

Linearized equations (of motion) checking stability of motion computation of stationary and equilibrium solutions natural frequencies buckling loads linearized state-space formulations (semi-)implicit numerical integration methods perturbation methods Coefficient matrices obtained by: numerical differentiation analytically using the geometric transfer function formalism The latter approach leads to a system of second order linearized equations in which the matrix coefficients possess all physical and mathematical properties of the system. DvM / 46 Linearized equations of kinematics Consider small preturbations δq, δ q, δ q around nominal trajectory (q, q. q ) q q + δq q q + δ q q q + δ q e + δe F (e) (q + δq) F (e) + DF(e) δq +... δe DF (e) δq ė + δė DF (e) (q + δq)( q + δ q) e F (e) (q) ė DF (e) q (DF (e) + D2 F (e) δq +...)( q + δ q) DF (e) q + DF (e) δ q + (D2 F (e) q )δq δė DF (e) δ q + (D2 F (e) q )δq DvM / 47 q q + δq q q + δ q q q + δ q Linearized equations of kinematics x + δx F (x) (q + δq) F (x) + DF (x) δq +... δx DF (x) δq ẋ + δẋ DF (x) (q + δq)( q + δ q) x F (x) (q) ẋ DF (x) q ẍ DF (x) q + (D 2 F (x) q) q (DF (x) + D 2 F (x) δq +...)( q + δ q) DF (x) q + DF (x) δ q + (D2 F (x) q )δq δẋ DF (x) δ q + (D2 F (x) q )δq ẍ + δẍ DF (x) (q + δq)( q + δ q) + (D 2 F (x) (q + δq)( q + δ q))( q + δ q) (DF (x) + D 2 F (x) δq +...)( q + δ q) + ((D 2 F (x) + D 3 F (x) δq +...)( q + δ q))( q + δ q) δẍ DF (x) δ q + 2(D2 F (x) q )δ q + (D 2 F (x) q + (D 3 F (x) q ) q )δq Linearized equations of motion M q d D q df f (x)t M((D 2 q df(x) q) q + D q rf (x) q r ) D q df (e)t σ M D q df (x)t MD q df (x) D q df (x) D q df (x) + D 2 q df(x) δq +... D 2 q df(x) D 2 q df(x) + D 3 q df(x) δq +... M q d M(q d + δqd )( q d + δ qd ) (D q df (x)t + D 2 q df(x)t δq d )(M + D x M D q df (x) δqd ) (D q df (x) + D 2 q df(x) δqd )( q d + δ qd ) M q d + M δ q d + (D q df (x)t M D 2 q df(x) q )δq +(D q df (x)t D x M DF (x) q D q df (x) )δq +(D 2 q df(x)t q M D q df (x) )δq DvM / 48 DvM / 49

Linearized equations of motion Forces acting on the system are conservative: M δ q d + (C + D )δ q d + (K + N + G )δq d M, D, K and G are symmetric matrices, C and N need not. () Reduced mass matrix: M D q df (x)t M D q df (x), M (M (k) l + M (k) c ) k (2) Velocity sensitivity matrix: C D q df (x)t Dẋfin D q df (x) + 2M D q dd q F (x) q (3) Damping matrix: D D q df (e)t S d D q df (e), S d k S (k) d, S (k) d σ(k) ė (k) Linearized equations of motion (4) Structural stiffness matrix: K D q df (e)t SD q df (e), S k (5) Dynamic stiffness matrix: S (k), S (k) σ(k) ε (k) N D q df (x)t D x (f in M ẍ )D q df (x) ( + M D q dd q F (x) q + (D q d D 2 qf (x) ) q ) q + Dẋfin D q dd q F (x) q +D q df (e)t S d D q dd q F (e) q (6) Geometric stiffness matrix: G D 2 q d F (x)t f M ẍ D 2 q d F (e)t σ DvM / 5 DvM / 5 Linearized state equations Stationary and equilibrium solutions δq d δz δ q d δż A(t)δz M q d D q df (x)t f M((D 2 q d F (x) q) q + D q rf (x) q r ) D q df (e)t Stationary or equilibrium solutions are solutions for which q d is constant, i.e. q d, q d A(t) O I M K + N + G M C + D Stationary solutions can be obtained for q r by solving the algebraic equation f D q df (x)t (f M(D 2 q d F (x) q) q) D q df (e)t σ Newton-Raphson method with Jacobian matrix D q d f D q d f (K + N + G ) DvM / 52 DvM / 53

Stability of stationary solution M δ q d + (C + D )δ q d + (K + N + G )δq d δz δq dt, δ q dt T The symmetric eigenvalue problem (N is symmetric) M δ q d + (K + N + G ) δq d Natural frequencies ω i : δż Az δz(t) e λt δz det( ω 2 i M + K + N + G ) The real part of the eigenvalues controls the stability characteristics of the system. If all eigenvalues have negative real parts, the solution is stable. If some eigenvalue has a positive real part the solution is instable. If some eigenvalue is purely imaginary or zero we are in a bifurcation point. Buckling loads: critical loading parameters λ i det(k + λ i G ) λ i f i /f f i buckling load f reference load DvM / 54 DvM / 55 Four-bar mechanism e 3 mg/k e σ e () e (m) ε, ε 2, ε 4, ε 5 e 3 T σ () σ (m) σ, σ 2, σ 4, σ 5 σ 3 T q d e 3 Nominal configuration: x, x 2 2 2, x 3 2 2, x 4 y, y 2 2 2, y 3 2 2, y 4 2 DF (e),,, T S diag,,, k DvM / 56 DvM / 57

x f x () x (c) x, y, x 4 x 2, y 2, x 3, y 3, y 4 T f () f (c) fx, fy, fx 4,,,, mg T DF (x),, 2, 2, 2, 2, T D 2 F (x),,, 2 2,, T 2 2, 2 D 3 F (x),,, 3 2,, 3 2, 3 T M diag,, m,,,, m T DvM / 58 Equation of motion for nominal configuration: më 3 + 2m(ė 3 ) 2 + ke 3 mg DvM / 59 Coefficient matrices of the linearized equations of motion: M DF (x)t M DF (x) m C 2DF (x)t M D 2 F (x) ė3 2 2mė 3 K DF (e)t SDF (e) k G D 2 F (x)t f + D 2 F (x)t M DF (x) ë3+ + D 2 F (x)t M D 2 F (x) (ė 3 ) 2 2mg + 2më 3 + 2mė 2 3 N DF (x)t M D 2 F (x) ë3 + DF (x)t M D 3 F (x) (ė 3) 2 Linearized equation of motion: mδë 3 + 2 2mė 3 δė 3 + (k 2mg + 2 2më 3 + 5m(ė 3 ) 2 )δe 3 Small vibration about a stable equilibrium position (ë 3 ė 3, e 3 mg/k): mδë 3 + (k 2mg)δe 3 When k mg, it reduces to the well-known form mδë 3 + kδe 3 2më 3 + 3m(ė 3 ) 2 DvM / 6 DvM / 6

Rotating beam m (e,e) m (e,ε) m (ε,e) M (ε,ε) δë (m) δ ε (m) + K (ε,ε) + (N + G ) (ε,ε) δe (m) δε (m) det( ω 2 i M + K + N + G ) (N is symmetric) Zero bending stiffness, K (ε,ε) (cord condition) - Within the plane of rotation: (ω c,i ) 2 2i 2 i, i,2,..., Ω 2 - Perpendicular to the plane of rotation: (ω c,i ) 2 2i 2 i, i,2,..., Ω 2 First and second in-plane bending frequencies ω b and ω c of the beam and the cord respectively as functions of the angular rate Ω DvM / 62 DvM / 63 Rotating mass spring system Stationary solution (r, r 2 ): r r 2 r r 2 k + k 2 m φ 2 k 2 r k l k2 k 2 m 2 φ 2 k 2 l 2 ) r 2 k 2 l 2 (m r 2 k l + m 2 k 2 l 2 ) φ 2 + k k 2 l m m 2 φ 4 (k 2 m 2 + k 2 m + k m 2 ) φ 2 + k k 2 Non-linear equations of motion: m r m φ 2 r k (r l ) k 2 (r 2 r l 2 ) m 2 r 2 m 2 φ 2 r 2 k 2 (r 2 r l 2 ) m r 2 φ 2 k 2 l 2 + k k 2 (l + l 2 ) m m 2 φ 4 (k 2 m 2 + k 2 m + k m 2 ) φ 2 + k k 2 DvM / 64 DvM / 65

Rotating mass spring system Cantilever beam (vibration modes).5.5 2 3 4 5 6 7 8 9 ω.875 EI ml 4.5 2 3 4 5 6 7 8 9 Linearized equations of motion (N is symmetric): m δ r k + k + 2 m φ 2 k 2 δr m 2 δ r 2 k 2 k 2 m 2 φ 2 δr 2 ( 2 m k + k det ω i + 2 m φ 2 ) k 2 m 2 k 2 k 2 m 2 φ 2 DvM / 66 ω 2 4.694 EI ml 4 ω 3 7.856 EI ml 4.5.5.5.5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 DvM / 67 Cantilever beam (buckling modes) Cantilever beam (lateral buckling).5 F.5 2 3 4 5 6 7 8 9 F th π2 EI l 2.5 2 3 4 5 6 7 8 9 F.5 2 3 4 5 6 7 8 9.5 2 3 4 5 6 7 8 9 F cr (4) /F th.459 EIS F th 4.3 t l 4 DvM / 68 DvM / 69

Cantilever beam subject to concentrated end force 2 3 4 5 l F DvM / 7