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).
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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
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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)
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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.
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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 2 T 1 T δI = (δL(q,q) +δW +δ (−kλ Φ − pΦ Φ))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 system of equations, the joints will add the above mentioned relations in the form of internal forces vector {Gint} and tangent stiffness [S]. The expressions for the internal forces vector and tangent stiffness is given by Eq.(11) for holonomic constraints,
B( pΦ + kλ) pBBT kB {G } = , [S] = int Φ T (11) k kB 0 and by Eq.(12) for non-holonomic constraints, in which it is present a pseudo- damping matrix [C] instead of the tangent stiffness [S]:
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B ( pΨ + kλ) pB BT kB {G } = nh , [C] = nh nh nh int Ψ T (12) k kBnh 0
The two matrices [B] and [Bnh] are the gradients of the constraints equations (or more precisely, their Jacobian matrix), as they appear in the expression of the virtual variation of the constraints:
T T ∂Φ ∂Ψ {δΦ} = [B]T {δq} = i {δq } , {δΨ} = [B ]T {δq} = i {δq } ∂ j nh & ∂ & j (13) q j q& j
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4 Classification of kinematic pairs
A standard classification for kinematic pairs is to group them in two main categories: lower and higher pairs. Joints with surface contact are referred to as lower pairs; the relative motions of coincident points of the links are similar, and an exchange of elements from one link to the other does not alter the relative motion of the parts. Joints with point or line contact are called higher pairs; the relative motion of the elements of higher pairs is generally quite complicated. An infinite number of higher pairs exist, and usually they can be replaced by a combination of lower pairs.
4.1 Lower pairs
In order to classify lower pairs, the potential relative motion between two rigid bodies linked by a joint and the related imposed constraints will be considered.
Hinge joint
The hinge joint is modeled by introducing 5 constraints: three to impose the equality of the positions x at the nodes A and B, and two fixing the two rotations about the two axes which are orthogonal to the hinge axis ξ3.
ξ 1 ξ HING 3 ξ Φ = B − A = = µ 2 i xi xi 0 i 1,2,3 1 B µ Φ = µ Tξ = 0 3 4 1 3 µ Φ = µ Tξ = 0 A 2 5 2 3
Fig. 1 – Hinge joint
The corresponding five Lagrangian multipliers λj, represent the reaction forces and the moment at the hinge, which are applied in order to respect the constraints.
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Prismatic joint
The prismatic joint is modeled by introducing 5 constraints: three to impose the equality of the rotations ψ at the nodes A and B, and two fixing the two translations along the two axes which are orthogonal to the allowed direction ξ3.
ξ PRIS 1 ξ 3 ξ Φ =ψ B −ψ A = = µ 2 i i i 0 i 1,2,3 1 B µ Φ = µ T (x B − x A ) = 0 3 4 1 µ Φ = µ T (x B − x A ) = 0 A 2 5 2
Fig. 2 – Prismatic joint
The corresponding five Lagrangian multipliers λj, represent the reaction forces and the moment at the prismatic joint, which are applied in order to respect the constraints.
Cylindrical joint
The cylindrical joint is modeled by introducing 4 constraints: two to impose the equality of the two translations in the plane orthogonal to the joint axis ξ3, and two fixing the two rotations leaving free only the rotation about axis ξ3.
ξ CYLI 1 ξ 3 Φ = µ T B − A = ξ 1 1 (x x ) 0 µ 2 1 B Φ = µ T (x B − x A ) = 0 µ 2 2 3 Φ = µ Tξ = 0 µ 3 1 3 2 Φ = µ Tξ = A 4 2 3 0
Fig. 3 – Cylindrical joint
The corresponding four Lagrangian multipliers λj, represent the reaction forces and the moment at the joint, which are applied in order to respect the constraints.
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Screw joint
The screw joint is similar to a cylindrical joint, with an additional constraint relating the translation along the axis to the rotation about it, by means of the pitch p of the screw.
ξ SCRE 1 ξ Φ = µ T (x B − x A ) = 0 3 1 1 ξ Φ = µ T (x B − x A ) = 0 µ α 2 2 2 B T T 1 Φ = µ ξ = 0 Φ = µ ξ = 0 µ 3 1 3 4 2 3 3 Φ = ξ T (x B − x A ) − pα = 0 µ 5 3 A 2 Φ = sin(θ −α) = 0 6 Fig. 4 – Screw joint
The angle of rotation of the screw, called α, is treated as an additional internal degree of freedom. Therefore, the joint imposes six constraints out of seven degrees of freedom (six of the second link plus the additional internal d.o.f. α). The Lagrange’s multipliers, corresponding to the two additonal constraint with respect to the cylindrical joint, represent the internal longitudinal force and the torsion moment about the screw axis.
Spherical joint
The spherical joint is modeled by simply introducing the 3 constraints to impose the equality of the positions x at the nodes A and B.
µ 1 SPHE ξ µ 1 ξ 3 3 Φ = B − A = = ξ i xi xi 0 i 1,2,3 ≡ 2 A B µ 2 Fig. 5 – Spherical joint
The corresponding Lagrangian multipliers λj, represent the reaction forces at the joint, which are applied in order to respect the constraints.
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Plane joint
The plane joint leaves free the translations on a plane and the rotation about the normal to the plane
ξ PLAN 1 ξ 3 Φ = µ Tξ = ξ 1 1 3 0 µ 2 1 B µ Φ = µ Tξ = 0 3 2 2 3 µ A 2 Φ = µ T B − A = 3 3 (x x ) 0
Fig. 6 – Plane joint
Almost all the lower joints can be completely specified, in an assembled configuration, by giving the coordinates of one node and one axis.
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4.2 Higher pairs
Universal joint (Hook joint)
The universal joint allows the transmission of the motion between two intersecting but not collinear shafts. It can be seen as a combination of two hinges, each of the latters having its axis coincident with one of the two axis of the joint’s cross.
µ 1 µ UNIV A 3 Φ = B − A = = µ i xi xi 0 i 1,2,3 2 Φ = µ Tξ = 4 1 3 0 ξ 1 ξ B 2 ξ 3
Fig. 7 – Universal joint
Point on plane
This joint imposes the second node to remain on a plane firmly attached to the first node, leaving free the three rotations and the two rotations on the plane.
ξ 1 ξ 3 ξ µ B 2 1 Φ = µ T B − A = µ 1 3 (x x ) 0 3 µ A 2
Fig. 8 – Point on plane joint
This joint can also be obtained as a combination of a plane joint and a spherical joint. SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 14
Rigid slider joint
This joint allows to represent sliding motion of a point on a rigid curve. The two translations normally oriented to the sliding curve are not allowed.
SLID
Φ = B − A − τ = B 1 x x RAr( ) 0 x(τ ) A r(τ ) = y(τ ) τ z( ) Fig. 9 – Rigid slider joint
The curve is given in its parametric format r(τ), and the arc-length parameter τ is used an additional internal degree of freedom.
Pulley-cable joint
This joint allows to represent cables sliding along a pulley. The cable is represented by linear interpolations between the points tangent to the pulley and the end nodes of the section under consideration, plus a linear angular interpolation for the remaining section on the pulley. There is a restriction to the use of this element: in fact, the end nodes of the cable segment N1 and N2 cannot pass through the pulley.
µ 1 PULL N N2 5 Φ = (x − x )T (x − x ) = 0 1 N1 N4 N3 N4 (x − x )T (x − x ) Φ = x − x − R = 0 θ = N3 N4 N3 N5 R 2 N3 N4 cos N4 α µ x − x x − x 2 Φ = (x − x )T µ = 0 N3 N4 N3 N5 3 N3 N4 3 N3 µ Φ = − T − = − × − T µ 4 (xN xN ) (xN xN ) 0 [(x x ) (x x )] 3 2 5 3 5 sinθ = N3 N5 N3 N4 3 T x − x x − x Φ = (x − x ) µ = 0 N3 N4 N3 N5 5 N3 N5 3 Φ = θ −α = 6 sin( ) 0
N 1 Fig. 10 – Pulley-cable joint
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5 Mecano elements syntax
In BACON menu window environment, all the kinematic pairs are available in the Joint menu, as shown in Fig. 11:
Fig. 11 – BACON menu for joints
In the command language of BACON the Mecano elements are all introduced by the keyword .MCE, whose general syntax is shown in Fig. 12.
.MCE ‘Type’ ["name"] I- N... [GP-] [ATT - -]
Fig. 12 – .MCE syntax
It can be observed that this syntax is analogous to the .MAI syntax already seen for the structure element library, but the concept of type is used instead of the concept of hypothesis.
Fig. 13 – BACON menu for joints properties
The properties of the Mecano elements can be set using the BACON menu “Join Properties”, as shown in Fig. 13.
.MCC ‘Type’ GP- / ATT - / I - [J - K - ]/ GROUP - ‘Parameters’...
Fig. 14 – .MCC syntax
In the command language of BACON the Mecano elements properties are all introduced by the keyword .MCC, whose general syntax is shown in Fig. 14. The list of parameters following .MCC syntax depends therefore on the element to which they are referred.
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6 MECANO: non linear finite element analysis
Static kinematic or dynamic analysis
Library of elements
• Structure : MECANO / STRUCTURE
• Mechanism : MECANO / MOTION
Possible coupling between MECANO / STRUCTURE and MECANO / MOTION
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6.1 Structure library
3D elements • Rod • Beam • Shell • Membrane • Volume Fourier multiharmonic elements • Shell • Volume Axisymmetrical elements • Membrane • Shell • Volume Plain strain elements • Membrane • Volume Plain stress elements • Rod
• Membrane
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6.2 Material library
All the material laws are available in every structural elements
• Linear elastic
• Hyperelastic material
• Elastoplastic material
• Viscoplastic material
• Viscoelastic material
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6.3 Rigid Kinematic Joints
RIGID HINGE JOINT CURVILINEAR SLIDER
CYLINDRICAL JOINT CAM ELEMENT
PRISMATIC JOINT GEAR ELEMENT
SCREW JOINT RACK-AND-
PINION JOINT
UNIVERSAL JOINT WHEEL ELEMENT
SPHERICAL WHEEL-RAIL
JOINT CONTACT ELEMENT
BREAKABLE USER ELEMENT CONNECTION
Fig. 15 – Mecano rigid joints library
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6.4 Flexible kinematic joints - linear material finite elements
SPHERICAL CABLE PULLEY
FLEXIBLE SLIDER
CYLINDRICAL CABLE RING
FLEXIBLE SLIDER
PRISMATIC
FLEXIBLE SLIDER
Fig. 16 – Mecano flexible joints library
BEAM ELEMENT SUPERELEMENT
MEMBRANE USER ELEMENT
ELEMENT
CABLE ELEMENT
Fig. 17 – Mecano flexible joints library
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6.5 Stiffners, dampers and transfert functions
SPRING DAMPER BUSHING DOUBLE STOP
ELASTIC DOUBLE STOP SPRING LAW MAXWELL MODEL
LAW
NON LINEAR ELASTIC LOCKING SPRING LAW LAW
NON LINEAR RIGID LOCKING DAMPER LAW LAW
NON LINEAR UNILATERAL RIGID SPRING*DAMPER CONTACT LAW LAW
HYSTERETIC FRICTION LAW SPRING LAW
PLASTIC SPRING NON LINEAR LAW FORCES LAWS
Fig. 18 –Stiffners, dampers and transfert functions
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6.6 Special boundary conditions and multi-point constraint
VOLUME POINT – PLANE INDICATOR
POINT – NON LINEAR SURFACE CONSTRAINT
POINT - POINT RIGID BODY
POINT - FACET USER ELEMENT
Fig. 19 –Special boundary conditions and multi-point constraint
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6.7 Control elements and computation elements
DIFFERENTIAL INERTIA EQUATIONS PROPERTIES INTEGRATOR COMPUTATION
ENVELOPE DIGITAL SIMULINK COMPUTATION CONTROL MATRIXX
RESULTANT HYDRAULIC COMPUTATION ACTUATOR
USER ELEMENT
Fig. 20 –Control elements and computation elements
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6.8 Functions definition
• Functions of 1, 2, 3, … n variables
• Analytical functions
• Piecewise linear functions
• Connection with CAD entities (arc, straight line, splines, nurbs, coons, ruledsuface,…)
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6.9 Boundary conditions
Initial conditions
• Initial positions • Initial velocities
Boundary conditions
• Fixation • Boolean identification
• = =− qqAB or qqAB
Contact
• Rigid link • Point-point • Planarity conditions • Point-rigid surface • Point-facet • Mesh merging
6.10 Loading function of time
Point support SAMTECH SA 2000 April 2000 MECANO Motion/ Theory Page 26
• Prescribed displacement, position, velocity and acceleration • Prescribed force (structural or follower)
Element support
• Pressure • Force per unit of length, surface (structural axes) • Centrifugal forces
Rotational or translational acceleration
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6.11 Computation flowchart
Special computation at time 0
• Assembly/Positioning (mechanism analysis) • Computation of kinematic modes (useful for mechanism analysis) • Initial static computation (before a dynamic analysis) • Initial velocity computation
Static, kinematic or dynamic analysis
Restart
6.12 Computation parameters
• Automatic time step choice • Resolution method • Strategy and storage selections (by point, by element)
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6.13 Postprocessing
CURVES VISUALISATION
• Time evolutions of an element or a point result • One result function of another one (trajectories) • Processing (derivative, integral, FFT,…) • Compatibility with preprocessing functions • …
MAP OF RESULTS
• Configuration, strains, stresses for a given time • Plot of mechanism 3D trajectories • …
ANIMATIONS
• Translation, rotation and zoom by the cursor • Wire frame moving mechanism
SAMTECH SA 2000 April 2000