Report

Dynamic Simulation: Constraint Kinematics Objective The objective of this module is to show how constraint equations are used to compute the position, velocity, and acceleration of the generalized coordinates. These equations are kinematic in nature because they do not consider the forces required to cause the motion. The kinematic and motion constraints developed in the previous module (Module 3) for the piston-crank mechanism are used to demonstrate the mathematics. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Notation Module 4 – Constraint Kinematics Page 2 The total set of constraint equations needed to define a mechanism includes both kinematic constraints and drive constraints. k q q, t d 0 q, t There are 15 generalized coordinates and 15 nonlinear constraint equations for the piston-crank assembly used in Module 3. Since the piston-crank has a mobility of one, only one of the fifteen equations will be a motion constraint that is an explicit function of time. © 2011 Autodesk q q k Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. d q, t is the set of generalized coordinates is the set of kinematic constraint equations is the set of motion constraint equations www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Position Module 4 – Constraint Kinematics Page 3 Solving the constraint equations will yield the value of each generalized coordinate at a specific instance of time. The constraint equations are non-linear and the NewtonRaphson method is used as the solution method. The Newton-Raphson method is iterative and converges when the constraint equations are satisfied. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Constraint Equations k q q, t d 0 q, t Newton-Raphson Equations q, t qi 1 qi q 1 q, t where q, t q is the Jacobian matrix www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Velocity Module 4 – Constraint Kinematics Page 4 The time derivative of the constraint equations is used to determine the velocities of the generalized coordinates. Since the generalized coordinates are a function of time and the constraint equations are a function of the generalized coordinates and time, the chain rule for partial differentiation must be used. © 2011 Autodesk Constraint Equations k q q, t d 0 q, t Time Derivative q, t q 0 t q t t Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Velocities 1 q t q t www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Acceleration Module 4 – Constraint Kinematics Page 5 The second time derivative of the constraint equations is used to determine the accelerations of the generalized coordinates. Since the generalized coordinates are a function of time and the constraint equations are a function of the generalized coordinates and time, the chain rule for partial differentiation must be used. © 2011 Autodesk 1st Time Derivative of Constraint Equations q, t q 0 t q t t 2nd Time Derivative of Constraint Equations 2 q, t 2 q qq 2 2 t q q q t t 2 q 2 2 2 0 q t t Accelerations qq 1 2 q q q 2 2 2 t q q 2 qt t t 2 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Summary of Equations Module 4 – Constraint Kinematics Page 6 Constraint Equations k q q, t d 0 q, t Velocities of Generalized Coordinates 1 q t q t Newton-Raphson Equations Used to determine the position (values of the generalized coordinates) at an instant in time. q, t qi 1 qi q 1 q, t Accelerations of Generalized Coordinates q 2 t q 2 © 2011 Autodesk 1 2 q 2 qq 2 2 q q qt t t Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Jacobian Module 4 – Constraint Kinematics Page 7 The Jacobian and its inverse is needed to determine the position, velocity, and acceleration of the generalized coordinates. Each i,j (row,column) term in the Jacobian matrix is given by J i, j i q j ith constraint equation J q Jacobian matrix Error messages indicating that the Jacobian is singular are sometimes encountered when running multi-body dynamic programs. This occurs when there is not a physically realizable solution. jth generalized coordinate © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Piston-Crank Constraint Equations Module 4 – Constraint Kinematics Page 8 The fifteen constraint equations developed for the piston-crank mechanism in Module 3 are given on the right. Note that only the motion constraint is an explicit function of time. 1) X cgA156.8 0 4) X cgE 0 2)Y cgA 0 5)YcgE 0 3) 0 6) cgE 0 A cg 7) X cgC 41.3 cos C X cgD 43 sin D 0 8)YcgC 41.3 sin C YcgD 43 cos D 0 9) X cgB 28 cos B X cgC 102.6 sin C 0 10)YcgB 28 sin B YcgC 102.6 cos C 0 11) X cgD X cgE 0 12)YcgD YcgE 0 0 1 cos A 13) 1 0 A 1 0 sin T 0 1 cos 14) 1 0 A 1 0 sin T A sin A cos A T A T sin cos A B A X CG X CG B A 0 YCG YCG cos B sin B sin B 1 0 B cos 0 15) D 314 t 0 © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Piston-Crank Jacobian Module 4 – Constraint Kinematics Page 9 The Jacobian of the constraint equations is given below. Although there are many terms, there are a lot of zeros and the derivatives are easily computed. 1 0 0 0 0 0 0 0 i J 0 q j 0 0 0 sin A 0 0 © 2011 Autodesk 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 41.3 sin C 0 1 0 0 0 43 cos D 0 0 0 0 0 1 0 0 0 28 sin B 0 1 1 0 41.3 cos C 102.6 cos C 0 0 1 43 sin D 0 0 0 0 0 1 28 cos B 0 1 102.6 sin 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 cos A 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 cos 0 0 A 0 0 0 0 A B A X cg X cg cos sin A A B A Ycg Ycg sin sin A cos B sin 2 A 0 0 0 0 0 sin A sin B cos A cos B 0 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Education Community Section 4 – Dynamic Simulation Velocities Module 4 – Constraint Kinematics Page 10 The velocities of the generalized coordinates are computed from the equation 1 q t q t Since the Jacobian is known, this equation can be solved if the array containing the time derivatives of the constraint equations is found. Only the motion constraint, (15), is an explicit function of time. © 2011 Autodesk Motion Constraint 15) D 314 t 0 0 0 0 0 Required Array 0 0 0 0 t 0 0 0 0 0 0 314 Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Acceleration Module 4 – Constraint Kinematics Page 11 The accelerations can be computed if each term is found. q 2 t q 2 Inverse of the Jacobian © 2011 Autodesk 1 2 q 2 q q 2 2 q q qt t t This term is explained on the next slide. The time derivative of the Jacobian is zero. Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. 0 0 0 0 0 0 0 0 t 0 0 0 0 0 0 314 www.autodesk.com/edcommunity 0 0 0 0 0 0 0 2 2 0 t 0 0 0 0 0 0 0 Education Community Section 4 – Dynamic Simulation Acceleration Term Module 4 – Constraint Kinematics Page 12 This term is evaluated by breaking it down into a series of operations that are easily done on a computer. q q q q Step 1) Multiply the Jacobian by the velocities. This creates a column array. Step 2) Take the derivative of each row with respect to each generalized coordinate. This operation is similar to finding the Jacobian and results in a matrix. Step 3) Multiply the matrix by the velocities. This results in a column array. © 2011 Autodesk q q q q q q q q q Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Redundant Constraints Module 4 – Constraint Kinematics Page 13 The Jacobian matrix is an important quantity and enables the position, velocity, and acceleration of the generalized coordinates to be found. The rank of a matrix is equal to the number of independent rows or columns. Independent rows or columns can not be written as a linear combination of other rows or columns. If rows or columns of the Jacobian are not independent the Jacobian is singular and the problem does not have a solution. Application of the methods contained in this module requires that the Jacobian have an inverse. This requires that the determinant of the Jacobian be non-zero or that the rank be equal to the number of generalized coordinates. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Redundant Constraints: Detection Module 4 – Constraint Kinematics Page 14 The Dynamic Simulation environment within Autodesk Inventor software assembles the Jacobian and determines its rank as each constraint is added. The rank gives the number of independent constraints. The difference between the number of generalized coordinates and the number of independent constraints is equal to the degree of mobility. The difference between the number of constraints and the number of independent constraints is equal to the degree of redundancy. © 2011 Autodesk nq number of generalized coordinates nc number of constraints nic number of independent constraints Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Degree of Mobility dom= nq – nic Degree of Redundancy dor = nc-nic www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Redundant Constraints: Reaction Forces Module 4 – Constraint Kinematics Page 15 A redundant constraint occurs when the motion associated with a DOF is enforced by too many constraint specifications. One or more of the constraint specifications can be removed without affecting the mobility of the system. The joint reactions can not be independently determined when redundant constraints are present. Although solutions can be obtained they are based on assumptions by the program as to which constraints to use. Different assumptions will yield different answers. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Joints having friction are particularly effected by redundant constraints. Friction forces are based on the joint normal forces. Therefore, the friction forces are incorrect if the joint normal forces are incorrect due to redundant constraints. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Redundant Constraint: Example Module 4 – Constraint Kinematics Page 16 A simple four bar mechanism will have redundant constraints if revolute joints are used at all joints. The ground link shown in the figure is fixed. The revolute joint at 1 prevents the drive link from rotating about its long axis and moving normal to the joint plane. The revolute joint at 2 prevents the coupler from rotating about its long axis and moving normal to the joint plane. The revolute joint at 4 prevents the rocker from rotating about its long axis and moving normal to the joint plane. © 2011 Autodesk 3 Coupler Rocker 2 Drive 1 Ground 4 A revolute joint at 3 is redundant because neither the rocker or coupler can rotate about their long axis or move normal to the joint plane due to the other revolute joints. These degrees of freedom are already restrained. Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Redundant Constraints: Example Module 4 – Constraint Kinematics Page 17 A point-line joint must be used at joint 3. A point-line joint restricts a point (the center point of the hole at joint 3 on the coupler) to remain on a line (the centerline of the hole at joint 3 on the rocker). Redundant joints can be confusing and a detailed analysis of what each joint is doing is required to figure out how to remove them. Point - Line 3 Coupler Revolute 2 Rocker Drive 1 Ground Revolute 4 Revolute An example of how to remove redundant constraints is provided in the next module: Module 5. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Section 4 – Dynamic Simulation Module Summary Module 4 – Constraint Kinematics Page 18 This module showed how the constraint equations can be used to find the position, velocity, and acceleration of the generalized coordinates. Kinematic relationships were used during the derivation and no mention of the forces required to impose the motion constraints was made. The constraint equations for the piston-crank introduced in the previous module (Module 3) were used to demonstrate the mathematical steps. The Newton-Raphson method is generally used to solve the constraint equations. The Jacobian is a key component of the overall solution process and the rank of the Jacobian is used to detect redundant constraints. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community