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© 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. Education Community Dynamic Simulation: Constraint Equations Objective The objective of this module is to develop the equations for ground, revolute, prismatic, and motion constraints for a planar mechanism. These equations will be developed for a piston-crank assembly in a Boxer style engine. These constraint equations will be used in the next Module (Module 4) to show how position, velocity, and accelerations are computed. Although the equations developed for this module are for a planar (2D) mechanism, the methods can be generalized to 3D mechanisms.

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© 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. Education Community Boxer Style Engine Boxer style engines have a horizontally opposed piston configuration. This has several advantages Lower center of gravity Lower vertical height Lighter weight Less vibration Boxer style engines are used by Porsche and Subaru. Because of their low vertical profile they are often called pancake engines. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 2

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© 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. Education Community Cross Section View Cylinder Liner Piston Connecting Rod Crank Shaft Piston Pin Crank Bearing Counterweight Piston Pin Bearing Bottom Bearing Cap Rod Bolt This module will use the piston-crank portion of this engine to demonstrate how kinematic and motion constraints are developed. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 3

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© 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. Education Community Planar System The boxer engine rotating assembly contains four piston assemblies. Constraint equations will be written for one piston assembly to demonstrate the process. This single assembly can be represented as a planar mechanism. A Dynamic Simulation of the complete system will be presented in another module. Cylinder 1 Cylinder 2 Cylinder 3 Cylinder 4 The planar equations will be developed for Cylinder 3. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 4

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© 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. Education Community Global Coordinate System The constraint equations will be referenced to the stationary coordinate system shown in the figure. This reference coordinate system is called the global coordinate system. Capital letters are used to indicate that a coordinate or vector refers to this coordinate system. Lower case letters will be used to indicate a coordinate or vector is referred to a body fixed coordinate system associated with a part. Z X Y X Cylinder 3 Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 5

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© 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. Education Community Part ID’s A Cylinder Liner B Piston C Connecting Rod D Crank Shaft E Crank Bearing (Not Visible) The process of developing the constraint equations is facilitated by identifying each component by a letter. The five components shown with letters make up the basic system for which the constraint equations will be written. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 6

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© 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. Education Community Mobility Gruebler’s equation can be used to establish the mobility of the planar mechanism. Bodies (B) = 5 Grounded bodies (G) = 2 Revolute joints (R) = 3 Prismatic joints (P) = 1 A Cylinder Liner B Piston C Connecting Rod D Crank Shaft E Crank Bearing (Not Visible) Mobility Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 7 A mobility of one will require one motion constraint.

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© 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. Education Community List of DOF’s The DOF’s are associated with a set of generalized coordinates. Each body has 3 DOF and 3 generalized coordinates. The generalized coordinates for the planar mechanism are listed on the right. Fifteen constraint equations must be developed that will enable each of the fifteen generalized coordinates to be determined. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 8 List of Generalized Coordinates Format Capital letter indicates that variable is associated with the global coordinate system. Center of Gravity Body X-coordinate of the cg of Body A

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© 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. Education Community Ground Joints The cylinder liners are pressed into the engine block and do not move. The pistons move relative to the cylinder liners and the combination make a prismatic joint. The cylinder liners must be mathematically grounded or fixed in space. Cylinder 1 Cylinder 2 Cylinder 3 Cylinder 4 Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 9

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© 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. Education Community Cylinder Liner Ground Equations The location of the center of gravity and the orientation of the principal axes of inertia are shown in the figures. The ground constraint equations that fix the position of the c.g. and orientation of the principal axes can be written as Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 10 x y y z

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© 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. Education Community Crank Bearing Ground Joint The crank bearing is fixed in the engine block and does not move. The crank shaft rotation relative to the crank bearing can be represented by a revolute joint. All of the parts in the planar system must lie in the global X-Y plane. Therefore, a “virtual” crank bearing will be placed at the origin of the global coordinate system so that the planar equations can be developed. Crank Bearing Constraint Equations X Y Z Virtual Crank Bearing Located at the Origin Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 11

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© 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. Education Community Summary of Ground Joint Equations Cylinder Liner Virtual Crank Bearing Each of these equations fix one DOF for the respective part in space. None of the equations are a function of time. None of the equations involve more than one part. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 12

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© 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. Education Community 2D Coordinate Transformation Matrix In subsequent slides it will be necessary to transform the components of a vector from a body fixed coordinate system to the global coordinate system. This transformation is accomplished with the transformation matrix [T( )]. From the figure, X Y x y Matrix Form Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 13

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© 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. Education Community Revolute Joint There are three revolute joints in the piston-crank assembly Between the piston and connecting rod Between the connecting rod and crankshaft Between the crankshaft and crank bearing The constraint equations for a revolute joint will be developed using the two bodies shown in the figure. Body A and B have the same translational motion at the joint but can have relative rotation. X Y Body A Body B xAxA yAyA AA xBxB yByB BB Two bodies connected at a common point that allows relative rotational motion. Joint Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 14

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© 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. Education Community Revolute Joint X Y Body A xAxA yAyA AA Joint 1 1 The position of Joint 1 on Body A relative to the global coordinate system is given by the equation The components of are written with respect to the global coordinate system base vectors and the components of are written with respect to the body fixed coordinate system. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 15

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© 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. Education Community Revolute Joint The components of the body fixed position vector must be transformed to the global coordinate system before the components of the two vectors can be added. This is accomplished using the transformation matrix introduced earlier. Position Vector Equation Component Form Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 16

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© 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. Education Community Revolute Joint The coordinates of Joint 1 on Body A are Similarly, the coordinates of Joint 1 on Body B are In a Revolute Joint the coordinates of the joint must be same for each body. Thus, or Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 17 General Form of the Constraint Equations for a Planar Revolute Joint

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© 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. Education Community Revolute Joint The general form of the constraint equations for a planar revolute joint is The specific equations for the three revolute joints in the piston-crank mechanism will now be developed 2 nd Revolute Joint Joint 1 Joint 2 Joint 3 Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 18

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© 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. Education Community 1 st Revolute Joint The location of the joint relative to the c.g. is needed to define the parameters & For the piston, x y Joint 1 C.G. 28 mm Piston Body B Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 19

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© 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. Education Community 1 st Revolute Joint Joint 1 Connecting Rod Body C The location of the joint relative to the c.g. is needed to define the parameters & From the picture, x y Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 20

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© 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. Education Community 1 st Revolute Joint Using the geometry from the piston and connecting rod, the revolute joint constraint equation becomes Joint 1 1 st Revolute Joint Constraint Equations Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 21

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© 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. Education Community 2 nd Revolute Joint 41.3 mm The location of the joint relative to the c.g. is needed to define the parameters & From the picture, General Form of Constraint Equation Body C Joint 2 x y Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 22

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© 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. Education Community 2 nd Revolute Joint 43 mm The location of the joint relative to the c.g. is needed to define the parameters & From the picture, General Form of Constraint Equation Joint 2 x y Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 23

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© 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. Education Community 2 nd Revolute Joint Using the geometry from the connecting rod and crank shaft, the revolute joint constraint equation becomes Joint 2 2 nd Revolute Joint Constraint Equations Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 24

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© 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. Education Community 3 rd Revolute Joint The c.g.’s of both the crank and crank shaft lie at the origin of the global coordinate system. Therefore, the body fixed coordinates of the joint relative to the c.g. are zero. General Form of Constraint Equation Joint 3 3 rd Revolute Joint Constraint Equations Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 25

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© 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. Education Community Summary of Revolute Joint Equations 2 nd Revolute Joint Joint 1 Joint 2 Joint 3 Body B Body C Body D Body E Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 26

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© 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. Education Community Prismatic Joint In the planar system the cylindrical joint between the cylinder liner and the piston acts like a prismatic joint. A prismatic joint allows two bodies to translate relative to each other along a common axis. The two bodies cannot rotate independent of each other. The equations for a planar prismatic joint are based on the geometry shown in the figure. X Y Common Axis Body A Body B xAxA yAyA AA xAxA yAyA BB Two bodies A & B that translate relative to one another along a common axis. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 27

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© 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. Education Community Prismatic Joint Constraint Equations The points P and Q in Body A lie on the common axis and are connected by the vector PQ. The points R and S in Body B lie on the common axis and are connected by the vector RS. The vector PR also lies on the common axis and connects the points P and R. The three vectors must be parallel. Alternatively, vectors PR and RS must be perpendicular to. X Y Common Axis P Q Body A Body B xAxA yAyA AA xAxA yAyA BB R S Two bodies A & B that translate relative to one another along a common axis. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 28

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© 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. Education Community Prismatic Joint Constraint Equations The vector PQ with components written with respect to the body fixed coordinate system of Body A are The components of the vector PQ with respect to the global coordinate system are X Y P Q Body A Body B xAxA yAyA AA xAxA yAyA BB R S Two bodies A & B that translate relative to one another along a common axis. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 29

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© 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. Education Community Prismatic Joint Constraint Equations The vector RS with components written with respect to the body fixed coordinate system of Body B are The components of the vector RS with respect to the global coordinate system are X Y P Q Body A Body B xAxA yAyA AA xAxA yAyA BB R S Two bodies A & B that translate relative to one another along a common axis. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 30

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© 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. Education Community Prismatic Joint Constraint Equations The third vector is directed from point P to point R. Point P has the coordinates Point R has the coordinates The vector has components X Y P Q Body A Body B xAxA yAyA AA xAxA yAyA BB R S Two bodies A & B that translate relative to one another along a common axis. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 31

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© 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. Education Community Prismatic Joint Constraint Equations The vector perpendicular to PQ has components The dot product of two vectors that are perpendicular to each other is zero. X Y P Q Body A Body B xAxA yAyA AA xAxA yAyA BB R S Two bodies A & B that translate relative to one another along a common axis. First Constraint Eq. Second Constraint Eq. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 32

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© 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. Education Community Prismatic Joint Constraint Equations Substituting the vector components from the previous slides into the first constraint equation yields Substituting the vector components from the previous slides into the second constraint equation yields Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 33

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© 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. Education Community Summary of Prismatic Constraint Equations The two constraint equations for a planar prismatic joint are 1 st Constraint Equation 2 nd Constraint Equation The vector components at the beginning and end of each equation are based on the body fixed coordinate systems and are constant. The only variables are the generalized coordinates of Body A and B. These equations are easily evaluated in a computer program. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 34

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© 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. Education Community Prismatic Joint The prismatic joint formed by the cylinder liner and the piston lies along the global X - axis. Point P is chosen to lie at the c.g. of the cylinder liner. Point Q is chosen to lie 1 mm to the right on the x -axis. Point R is chosen to lie at the c.g. of the piston. Point S is chosen to lie 1 mm to the right on the x -axis. x y x y P Q R S Vector components of PQ Vector components of RS Point Coordinates Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 35

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© 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. Education Community Prismatic Joint Substitution of the vector components and point coordinates into the two prismatic joint equations yields 1 st Constraint Equation which reduces to 2 nd Constraint Equation Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 36

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© 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. Education Community Motion Constraint One motion constraint is required to make the mechanism stable. The rotation of the crankshaft (Body D) will be given an angular speed of 3,000 rpm. A 3,000 rpm engine speed is equal to 314 rad/sec. Although all fifteen generalized coordinates are a function of time, this is the only constraint equation that explicitly contains time as a variable. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 37 Motion Constraint

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© 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. Education Community Summary of Constraint Equations There are five planar bodies each having three DOF giving a total of fifteen DOF. Fifteen unknowns requires fifteen equations. Ground Constraint 1 Ground Constraint 2 Revolute Joint 1 Revolute Joint 2 Revolute Joint 3 Prismatic Joint Motion Constraint Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 38

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© 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. Education Community Summary of Constraint Equations Only one of the constraint equations is time dependent (Motion Constraint). Most of the constraint equations are non-linear. All of the constraints are algebraic equations and none are differential equations. Geometric quantities (dimensions and distances) contained in the constraint equations can be found from information in a 3D CAD model. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 39

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© 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. Education Community Module Summary The constraint equations for ground, revolute, and prismatic joints have been developed for a planar mechanism. The constraint equation for a rotational motion constraint has been developed for a planar mechanism. These equations were used to determine the fifteen equations necessary for a piston-crank assembly taken from a Boxer engine model. In some cases the constraint equations are very simple and in other cases they are complex. Only the motion constraint is an explicit function of time. All of the constraint equations are algebraic. These equations will be applied in the next module: Module 4. Section 4 – Dynamic Simulation Module 3 – Constraint Equations Page 40

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