MECANO Motion/ Theory Page 1 MECANO Motion Theory SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 2 Index 1 VARIATIONAL FORMULATION OF THE DYNAMIC PROBLEM ............................................. 3 1.1 CONSTRAINTS IN MECHANISM ANALYSIS........................................................................................... 4 2 CONSTRAINED STATIONARY VALUE PROBLEM....................................................................... 6 2.1 AUGMENTED LAGRANGIAN APPROACH ............................................................................................. 6 3 FORMULATION OF THE CONSTRAINED DYNAMIC PROBLEM............................................. 7 3.1 INTERNAL FORCES AND TANGENT STIFFNESS..................................................................................... 7 4 CLASSIFICATION OF KINEMATIC PAIRS ..................................................................................... 9 4.1 LOWER PAIRS..................................................................................................................................... 9 4.2 HIGHER PAIRS .................................................................................................................................. 13 5 MECANO ELEMENTS SYNTAX ....................................................................................................... 15 6 MECANO: NON LINEAR FINITE ELEMENT ANALYSIS ........................................................... 16 6.1 STRUCTURE LIBRARY....................................................................................................................... 17 6.2 MATERIAL LIBRARY......................................................................................................................... 18 6.3 RIGID KINEMATIC JOINTS ................................................................................................................ 19 6.4 FLEXIBLE KINEMATIC JOINTS - LINEAR MATERIAL FINITE ELEMENTS.............................................. 20 6.5 STIFFNERS, DAMPERS AND TRANSFERT FUNCTIONS......................................................................... 21 6.6 SPECIAL BOUNDARY CONDITIONS AND MULTI-POINT CONSTRAINT ................................................. 22 6.7 CONTROL ELEMENTS AND COMPUTATION ELEMENTS...................................................................... 23 6.8 FUNCTIONS DEFINITION ................................................................................................................... 24 6.9 BOUNDARY CONDITIONS.................................................................................................................. 25 6.10 LOADING FUNCTION OF TIME ........................................................................................................... 25 6.11 COMPUTATION FLOWCHART ............................................................................................................ 27 6.12 COMPUTATION PARAMETERS........................................................................................................... 27 6.13 POSTPROCESSING............................................................................................................................. 28 SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 3 1 Variational formulation of the dynamic problem Dynamic equations of motion can be seen as derived from the Hamilton’s principle (or principle of the Stationary Action) which states that « the actual path in the configuration space makes stationary the Hamiltonian Action I with respect to all arbitrary synchronous variations of the path between two instants t1 and t2 » t2 δI = δ L(q,q)dt = 0 (1) ∫t & 1 Note that the definition of synchronous variation implies that the path variations vanish at the two end points t1 and t2. In Eq.(1) the following symbols have been used: L is the Lagrangian L = T-V T kinetic energy function V potential energy function q(t) generalized coordinates The necessary conditions for the Hamiltonian Action I to have a stationary point, are given by the well-known Euler-Lagrange differential equations: d ∂L ∂L − = 0 k = 1,2,...,n ∂ ∂ (2) dt q&k qk In Eq.(2) it is assumed that all the external forces are conservative and they are therefore included in the potential energy function V, being the gradient of the latter. In case of presence of non conservative forces, the Hamilton principle can be extended in the form given by Eq.(3): t2 δI = (δL(q,q) + δW )dt = 0 (3) ∫t & 1 T where δW = δq Q is the virtual work of the generalized non conservative forces Qk conjugated to the generalized coordinates qk(t). SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 4 1.1 Constraints in mechanism analysis The variational formulation presented in Section 1 is based on the assumption that the generalized coordinates qk(t) are independent. In case of constrained dynamic problem, as in mechanism analysis, the approach is to work with a surplus of generalized coordinates and then adding the equations of constraints (for generality reasons). Before going further some definitions are necessary: • a mechanism is a device to transform a kind of motion into another one. • a rigid body is a system of particles whose relative distances remain unchanged • kinematic analysis is the process of predicting the motion once the mechanism has been specified • kinematic synthesis is the process of finding the geometry of a mechanism that will yield a desired set of motion characteristics • kinematic of flexible body is consists in predicting the motion of every point of the body in its deformed state • the individual bodies that form a mechanism are called links • joint or kinematic pair is the connection between two links in contact • kinematic chain is an assemblage of interconnected links • kinematic modes are corresponding to rigid body motions (zero energy) of the kinematic chain • kinematic degrees of freedom are the independent generalized coordinates corresponding to kinematic modes Constraints are imposed on links to make their motion useful to a desired purpose, thus diminishing their number of degrees of freedom. A kinematic pair imposes certain conditions on the relative motion between the two bodies in the pair. When these conditions are expressed in analytical form, they are called equations of constraint. A constraint equation describing a condition on the vector of the generalized coordinates qk(t) can be expressed as SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 5 Φ(q,t) = 0 (4) Algebraic constraint equations as in Eq.(4), which are twice continuously differentiable, are called holonomic. In case it does not explicitly depend on time t, the constraint is stationary. If the constraint equations contain relations between velocity components, which are not integrable, they are called non-holonomic constraints, Ψ(q,q&,t) = 0 (5) SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 6 2 Constrained stationary value problem As already anticipated in Section 1.1, in mechanism analysis the dynamic system is described with a surplus of n generalized coordinates qk(t) with respect to the (n-m) degrees of freedom of the system, and adding the corresponding m equations of constraints. From the variational point of view expressed by the Hamilton’s principle, this approach leads therefore to the problem of finding the stationary value of the functional I, subjected to the m constraints Φj. Recalling from calculus the Lagrange Multipliers approach for constrained optimization problems and the Kuhn-Tucker theorem, the following equivalence can be stated: min(I(q,q&)) q(t) − λT Φ ⇒ min(I(q,q&) ) (6) Φ(q,t) = 0 q(t) Eq.(6) expresses the fact that a constrained optimization problem is equivalent to an unconstrained optimization problem, in which the constraints, weighted by means of m Lagrangian Multipliers λj, are added to the functional I. 2.1 Augmented Lagrangian approach In order to improve the mathematical properties of the problem of finding the stationary value, as stated in Eq.(6), the augmented Lagrangian approach is adopted, leading to Eq.(7): min(I(q,q&)) q(t) − λT Φ − 1 ΦT Φ ⇒ min(I(q,q&) k 2 p ) (7) Φ(q,t) = 0 q(t) Τ Adding the quadratic form ½pΦ Φ to the functional I, has just the effect to add “some positive curvature” to the problem, since in the computation of the Hessian matrix the second derivatives of Φ are ignored, and both the quadratic form and its gradient vanish at the stationary point. The two constants k and p are penalty factor to better “scale” the constraints with respect to the dynamic problem. SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 7 3 Formulation of the constrained dynamic problem Applying the theory presented in Section 2 to the extended Hamilton’s principle, the constrained dynamic problem can be formulated as follows, t δ = 2 δ +δ +δ − λT Φ − 1 ΦT Φ = I ( L(q,q) W ( k p ))dt 0 (8) ∫t & 2 1 After some analytical developments, Eq.(8) can be expressed in the form d ∂L ∂L ∂Φ − = Q + (kλ + pΦ ) i ∂ ∂ k i i ∂ dt q&k qk qk (9) Φ = k (qk ,t) 0 or as in Eq.(10) when non-holonomic constraints are present: d ∂L ∂L ∂Ψ − = Q + (kλ + pΨ ) i ∂ ∂ k i i ∂ dt q&k qk q&k (10) Ψ = k i (qk ,q&k ,t) 0 3.1 Internal forces and tangent stiffness In the MECANO approach, each joint is treated in a similar way as finite elements, that is computing the relations between its local degrees of freedom, which are added to the system by the presence of the joint and its equations of constraints. Since the Newton-Raphson method is employed to solve the non-linear