Revolute Joint Constrained degrees of freedom: UX, UY, UZ, ROTX, ROTY Universal Joint Constrained degrees of freedom: UX, UY, UZ, ROTY Slot Joint Constrained degrees of freedom: UY, UZ Translational Joint Constrained degrees of freedom: UY, UZ, ROTX, ROTY, ROTZ Cylindrical Joint Constrained degrees of freedom: UX, UY, ROTX, ROTY Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 16 Chapter 2: Modeling in a Multibody Simulation Spherical Joint Constrained degrees of freedom: UX, UY, UZ Planar Joint Constrained degrees of freedom: UZ, ROTX, ROTY 2.3.1.1. Joint Element Connectivity Definition A joint element is typically defined by specifying two nodes, I and J. These nodes may be arbitrarily located in space. There are instances, however, when one of the nodes needs to be considered as a “grounded” node. In such cases, specify either node I or node J appropriately. In cases when the node is grounded, the location of the grounded node is taken to be that of the other specified node. Example If the first node of the joint element is a grounded node, then the element definition is: E,,J or EN,Element- Number,,J Similarly, if the second node is the grounded node, then the element definition is: E,I, or EN,ElementNumber,I 17 Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2.3.1. Joint Element Types 2.3.1.2. Joint Element Section Definition Each joint element must have an associated section definition. Use the SECTYPE command to define the section type and subtype. Example The universal joint section definition is: SECTYPE,JOINT,UNIV,UNIV-01 2.3.1.3. Local Coordinate System Specification for Joint Elements Local coordinate systems at the nodes are required to define the kinematic constraints of a joint element. Use the SECJOINT command to do so. The local coordinate systems and their required orientation vary from one joint element to another. Input data requirements for each joint element differ. Typically, the local coordinate system is always defined at the first node of a joint element. The local coordinate system at the second node may be optional. If it is not specified, then the local coordinate system at the first node is usually assumed. The rotational components of the relative motion between the two nodes of the joint elements are quantified in terms of Bryant (or Cardan) angles that are evaluated based on these coordinate systems. Example The following figure illustrates the specification of the local coordinate system for a universal joint element: Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 18 Chapter 2: Modeling in a Multibody Simulation Figure 2.8: MPC184 Universal Joint Geometry The local coordinate system specification is: LOCAL,11,0 . . . LOCAL,12,0 . . . SECJOINT,LSYS,11,12 2.3.1.4. Stops or Limits with Joint Elements Stops or limit constraints in joints are imposed on the available components of relative motion between the two nodes of a joint element. The Lagrange multiplier method is used to implement these constraints. For 19 Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2.3.1. Joint Element Types static analysis, the stop constraints are based on the relative displacements (or relative rotations) of the free degrees of freedom. Stops in Transient Dynamic Analysis In a transient dynamic analysis, if relative displacement-based (or rotation-based) stop constraints are used, then the relative velocities and relative accelerations become inconsistent (oscillatory velocity and/or accel- erations are observed in many cases), implying that the energy and momentum due to the impact-like nature of the stops is not conserved. These inconsistencies are reasonably suppressed by imposing a numerical damping. However, numerical damping does not work appropriately in some cases. Thus, for the transient dynamic case, an energy-momentum conservation scheme is adopted. By this method, the user specified relative DOF stop values are taken into account, and constraints based on the relative velocity are imposed in such a way that the overall energy and momentum balance is achieved in a finite element sense. Irrespective of the integration scheme specified for the transient dynamic analysis, the Newmark method is used for the joint element when stops are specified. The energy-momentum conservation scheme for stops is implemented for all joints except the screw joint. In the case of the screw joint, the stops are imposed based on the relative displacements (or rotations). Defining Stops for Joint Elements You can impose stops or limits on the available components of relative motion between the two nodes of a joint element. The stops or limits essentially constrain the values of the free DOFs within a certain range. To specify minimum and maximum values, issue the SECSTOP command. The following figure shows how stops can be imposed on a revolute joint such that the motion is constrained. The axis of the revolute is assumed to be perpendicular to the plane of paper and is along the e 3 direction. Figure 2.9: Stops Imposed on a Revolute Joint φ −φ The local coordinate system specified at node I is assumed to be fixed in its initial configuration. However, the local coordinate system specified at node J evolves with the rotation of that node. The relative angle of rotation is given by: ψ = − ⋅ ⋅ − tan 1 1 2 1 1 e e e e I J I J Let the link with node J rotate with respect to the link with node I. This characteristic implies that the local coordinate system at node J rotates with respect to the local coordinate system at node I. Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 20 Chapter 2: Modeling in a Multibody Simulation For the configuration shown, the initial relative angle of rotation is zero degrees. A counterclockwise motion results in positive angles of rotation. Clockwise motion results in negative angles of rotation. If stops limit the movement of the link with node J (as shown), the stop conditions are specified as follows: SECSTOP,6,PHI min ,PHI max The next figure shows how stops can be imposed in a slot joint which involves displacements in the local e I 1 axis of node I. The relative distance between node J and node I is given by: ℓ = ⋅ −e x x 1 I J I ( ) where x I and x J are the position vectors of nodes I and J. The initial distance between the nodes I and J is l 0 and is a positive value. Figure 2.10: Stops Imposed on a Slot Joint e 1 I e I 2 e I 3 I max ℓ min ℓ o ℓ J e 1 I e I 2 e I 3 I max ℓ min ℓ o ℓ J The stops are defined as: SECSTOP,1, ℓ min , ℓ max The stops are defined as: SECSTOP,1,- ℓ min , ℓ max where ℓ min and ℓ max are both positive. where ℓ min is negative and ℓ max is positive. 2.3.1.5. Joint Mechanism Locks Locks or locking limits may also be imposed on the available components of relative motion between the two nodes of a joint element. Locks are basically used in joint mechanisms to “freeze” the joint in a desired configuration during the course of deformation. When the locks are activated on a particular component of relative motion, that component remains locked for the rest of the analysis. Issue the SECLOCK command to define lock limits. 21 Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2.3.1. Joint Element Types Referring to Figure 2.9: Stops Imposed on a Revolute Joint (p. 20), the locks for a revolute joint are specified as SECLOCK,6, Phi_Min,Phi_Max Referring to Figure 2.10: Stops Imposed on a Slot Joint (p. 21), the locks for the slot joint are specified as SEC- LOCK,1,l_Min,l_Max 2.3.2. Material Behavior of Joint Elements The following topics related to the material behavior of joint elements in a multibody analysis are available: 2.3.2.1. Stiffness and Damping Behavior of Joint Elements 2.3.2.2. Frictional Behavior For more information, see MPC184 Joint Materials in the Element Reference. 2.3.2.1. Stiffness and Damping Behavior of Joint Elements Linear or nonlinear stiffness and damping behavior can be associated with the free or unrestrained compon- ents of relative motion of the joint elements. In the case of linear stiffness or linear damping, the values are specified as coefficients of a 6 x 6 elasticity matrix using the TB,JOIN command with TBOPT = STIF or TBOPT = DAMP. The stiffness and damping values can be temperature-dependent. Depending on the joint element in use, only the appropriate coefficients of the stiffness or damping matrix are used in the joint element constitutive calculations. The nonlinear stiffness and damping behavior is specified using the TB,JOIN command with an appropriate TBOPT label. In the case of nonlinear stiffness, relative displacement (rotation) versus force (moment) values are specified using the TBPT command. For nonlinear damping behavior, velocity versus force behavior is specified using the TBPT command. (See Figure 2.11: Nonlinear Stiffness and Damping Behavior for Joints (p. 23) for a representation of the nonlinear stiffness or damping curve.) In either case, these values may be tem- perature dependent; use the TBTEMP command to define the temperature for the data table. Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 22 Chapter 2: Modeling in a Multibody Simulation Figure 2.11: Nonlinear Stiffness and Damping Behavior for Joints force (moment) Assumed behavior if u < u 1 displacement, rotation, or velocity f 6 f 5 f 4 f 3 u 1 u 2 u 3 u 4 u 5 u 6 f 1 f 2 Assumed behavior if u > u 6 You can specify the linear or nonlinear stiffness or damping behavior independently for each component of relative motion. However, if you specify linear stiffness for an unrestrained component of relative motion, you cannot specify nonlinear stiffness behavior on the same component of relative motion. The damping behavior is similarly restricted. If a joint element has more than one free or unrestrained component of rel- ative motion for example, the universal joint has two free components of relative motion then you can independently specify the stiffness or damping behavior as linear or nonlinear for each of the unrestricted components of relative motion. 2.3.2.2. Frictional Behavior Frictional behavior along the unrestrained components of relative motion influences the overall behavior of the joints. You can model Coulomb friction for joint elements via the TB,JOIN command with an appro- priate TBOPT label. Frictional behavior can be specified only for the following joints: Revolute Joint (x-axis and z-axis) Slot Joint Translational Joint The laws governing the frictional behavior of the joint are described below. Coulomb’s Law The classical Coulomb friction model is implemented for joints using a penalty formulation. The Coulomb friction model for joints is defined as: F F nlim = µ 23 Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2.3.2. Material Behavior of Joint Elements F F s ≤ lim Where, F s is the equivalent tangential force (or moment), F n is the normal force (or moment) in the joint, and µ is the current value of the coefficient of friction. The calculation of the normal force depends on the joint under consideration. If the equivalent tangential force F s is less than F lim , the state is known as the sticking state. If F s exceeds F lim , sliding occurs and the state is known as the sliding state. The sticking/sliding calculations determine when a point transitions from sticking to sliding or vice versa. Figure 2.12: Coulomb's Law Sliding F lim F n F s µ Exponential Friction Law The exponential friction law is used to smooth the transition between the static coefficient of friction and the dynamic coefficient of friction according to the formula (Benson and Hallquist): µ µ µ µ = + − − d s d c V e rel ( ) where: V rel = the relative slip rate µ s = the coefficient of friction in the static regime (stiction) µ d = the coefficient of friction in the dynamic regime c = decay coefficient Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 24 Chapter 2: Modeling in a Multibody Simulation Figure 2.13: Exponential Friction Law 2.3.2.2.1. Geometry specifications for Coulomb friction in Joints The modeling of Coulomb friction in joints requires some geometry specifications, depending on the type of joint under consideration. These quantities are used in the computation of the normal force (or moment) for Coulomb friction calculations. The SECJOINT command is used to specify these quantities. The following table outlines the required geometric quantities: Table 2.2 Required Geometric Quantities Geometric QuantitiesJoint Type Outer radius, Inner radius, Effective LengthRevolute Joint (x-axis and z-axis) None requiredSlot Joint Effective Length, Effective RadiusTranslational Joint If appropriate geometric quantities are not specified, then the corresponding normal force contributions will not be considered. The following section explains the normal force calculations and the geometric quantities required. 2.3.2.2.2. Calculation of Normal Forces for Coulomb Frictional Behavior The normal force (or moment) that is used in the Coulomb frictional behavior is based on the following forces that arise in a joint: • Lagrange Multiplier forces (or moments) due to the constraints • Interference fit forces (or moments) 25 Release 12.0 - © 2009 SAS IP, Inc. All rights reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2.3.2. Material Behavior of Joint Elements . SEC- LOCK,1,l_Min,l_Max 2 .3. 2. Material Behavior of Joint Elements The following topics related to the material behavior of joint elements in a multibody analysis are available: 2 .3. 2.1. Stiffness and. confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 24 Chapter 2: Modeling in a Multibody Simulation Figure 2. 13: Exponential Friction Law 2 .3. 2.2.1. Geometry specifications. reserved. - Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates. 2 .3. 1. Joint Element Types 2 .3. 1.2. Joint Element Section Definition Each joint element