Design Synthesis Fundamental Problem – locate a point (or set of points) fixed in a moving body that will pass through a series of points in space that satisfy geometrical constraints imposed by specific types of the mechanical guiding link. Specified points impose constraints on velocity and acceleration Specified points called “precision points”

Design Synthesis 3 Categories of Synthesis Position Generation – design a linkage to guide a rigid body through N specific positions/orientations Path generation – design a linkage to guide a point on a body to N specified positions Function Generation – design a linkage to generate a desired relation between input and output parameters

Design Synthesis Sewing machine 1,2,3,4 –path generator (4 bar linkage) 1,2,5,6 – function generator slider crank

Design Synthesis 4 Bar Linkage –guidance of a rigid body B r3 A r4 Ao A B 4 Bar Linkage –guidance of a rigid body d crank crank Synthesis requires: Specify A0 and B0 First position of A and B

Design Synthesis Constant link length constraint Crank Displacement Equations 1)

Design Synthesis Velocity and acceleration Fixed length Fixed position 2) acceleration 3) Constant length links

Design Synthesis Constant Slope Equations –Plane Sliders

Design Synthesis Three Position Crank Synthesis (crank guiding plane rigid body thru 3 positions) n=3 positions p1,p2,p3 and θ12,θ13 defined (on rigid body) 2 constraint equations that relate positions a0,a1,a2,a3 3) 4)

Design Synthesis Relate other positions (of the crank) to the initial position by: 5) 6) [D] is 3X3 displacement matrices, precalculated in terms of p1,p2,p3, and θ12,θ13

Matrix Methods in Kinematics Finding the Displacement Matrix by Inversion y B1 Known points C1 C2 A1 A2 B2 x

Matrix Methods in Kinematics Displacement Matrix by Inversion

Matrix Methods in Kinematics q q1 d a x y 2D Planar motion z D=d*a-1

Matrix Methods in Kinematics Using MATLAB inv(a) ans = 0.5000 -0.5000 2.0000 0.5000 0.5000 -3.0000 -1.0000 0 2.0000 d = 5 7 6 1 1 2 1 1 1 a = 2 2 1 4 6 5 1 1 1 >> e=d*inv(a) e = 0 1 1 -1 0 3 0 0 1 Displacement matrix

Design Synthesis Substitute into constraint equations 7) 8)

Design Synthesis since Expand eqns 7 and 8 in the form –see next 2 slides 2 equations Assume a0x and a0y, now have 2 equations and 2 unknowns (a1x and a1y)

Design Synthesis

Design Synthesis

Design Synthesis Results in 2 linear equations The crank displacement design equations 9) A,B,C are f(displacement matrix elements and a0 only)

Design Synthesis For velocity design constraint equations Constant length links 2) 10)

Design Synthesis Since: 11) 12) For velocity design constraint Linear equations where D,E,F are f(d)

Design Synthesis For acceleration design constraint equations Linear equations P,Q,R = f(d)

Design Synthesis Example: Crank Synthesis – 3 finitely separated positions of a moving plane Where is initial position of a1? (assumed)

Design Synthesis Displacement matrices

Design Synthesis Initial position

Design Synthesis Assume 2nd fixed pivot point at c0=(5,0) Displacement matrices [D] stay the same

Design Synthesis Initial Position (0.994,3.237) Starting angles and link lengths (3.54,-1.65)

Design Synthesis Suh & Radcliffe Fig 6.3

Design Synthesis

Design Synthesis

Design Synthesis Slider Synthesis

Design Synthesis

Design Synthesis Substitute, expand and group to get: 13)

Design Synthesis *It works

Design Synthesis Fortunately if θ12 or θ13 =0, A=0 Linear equation, straight line slider 13)

Design Synthesis Slider synthesis example Replace with slider

Design Synthesis Displacement matrices are the same

Design Synthesis Points are same but “offset” is different From 2 crank design

Design Synthesis

Design Synthesis

Design Synthesis HW 5 Four bar linkage synthesis Use a0=(5.,0) Find a1 (Slides 18,19) P1 P2 x y

Design Synthesis Four Point Synthesis – Crank Constraint Constraint equations will be non-linear, require iterative method for solution Newton-Raphson Method, finding root of non linear equation (1 variable)

Design Synthesis In kinematic analysis often have n non linear equations with n unknowns Expand Taylor series

Design Synthesis Solve for δs and update

Design Synthesis Constant length for guiding cranks Length constraint equation 14) 3 non linear equations with 4 unknowns a0x,a0y,a1x,a1y Assume 1 value and use N-R to solve for other 3 Get a series of fixed and moving crank points in the initial position Any pair should work but need to check mobility All possible fixed points = center point curve All possible moving points = circle point curve

Design Synthesis Example - 4 bar linkage P4 Added 4 th point

Design Synthesis Solving 14)

Design Synthesis

Design Synthesis

Design Synthesis

Design Synthesis

Design Synthesis Process Summary Start with constraint equations Crank length or slider slope Calculate D matrix for defined point/angles of rigid body (1 to 2, 1 to 3, 1 to j) Substitute displacement equation constraint eqns Get linear equations for initial crank/slider position (e.g.a1x,a1y) Determine initial position Determine additional positions using D equations

Design Synthesis –Function Generation Previously: Considered position synthesis (points/angles) Now: Consider mechanisms where output motion is specified function of input motion. (Crank and sliders) Approach: Function generator synthesis problem converted to equivalent rigid body guidance problem (last section) using principle of inversion Motion of guided rigid body described by the relative motion of the input member with respect to the output member

Design Synthesis –Function Generation Function generation requires consideration of error (error curve) Δ between mechanical output φ and theoretical function is called structural error Actual error curve is dependent on precision point spacing. Optimum spacing results in equal error between successive pairs of precision Points (over range x0 to xf)

Design Synthesis –Function Generation Chebyshev Spacing – good first approximation

Design Synthesis –Function Generation

Design Synthesis –Function Generation Four Bar Linkage Function Generator Displacement Matrix for rotation of a1 about a0 (0,0)

Matrix Methods in Kinematics Rotate about Z Components in the fixed system x-y Rotation matrix

Design Synthesis –Function Generation Four Bar Linkage Function Generator Rigid body rotation about bo (angles are rigid) Coincident with b0=b1 Establishing relative rotation of input link (a0 a1) to output( bo b1) Coupler link (a1,b1) acts as a guide crank for the input crank (a0,a1)

Matrix Methods in Kinematics Rigid Body Displacement Matrix 2d Cartesian Rotation p1 q1 Fixed x-y Reference Displacement Matrix May know p1,p Find position q

Matrix Methods in Kinematics Reference Displacement Matrix Now a displacement matrix 3x3 planar (2D)

Design Synthesis –Function Generation Displacement matrix about b0=(1.0,0.0) Rigid body rotation about b0

Design Synthesis –Function Generation Total relative Displacement Rigid body rotation Input crank rotation using

Design Synthesis –Function Generation Use same solution approach as position synthesis Use Dr elements for dikj b1x and b1y replace a0x and a0y - Center of relative motion of a relative to bob1

Design Synthesis –Function Generation Example: Synthesis 4 bar linkage with y x

Design Synthesis –Function Generation

Design Synthesis –Function Generation

Design Synthesis –Function Generation For other points, use displacement matrix for crank rotation Rotation about a0 (0,0)

Design Synthesis –Function Generation For b2 and b3, determine displacement matrix for rotation about b0 (1,0) p2x=p1x, p2y=p1y

Design Synthesis –Function Generation

Design Synthesis –Function Generation Slider Function Synthesis Input crank θ1j ~ x Output slide displacement d1j ~ y Assume slide slope is α relative to x axis Assume a0=(0,0) and b1 =(1,0) 1st position of output slider y=f(x) aj a1 θ1j Inversion of input crank about b1 b1j α (0,0) b1 a0 d1j (1,0)

Design Synthesis –Function Generation

Design Synthesis –Function Generation In the displacement matrix D Θ =0 for constant slope of slider y bj d1j b1 -d1j x (1,0) α

Design Synthesis –Function Generation Solve for a1x,a1y with Aj a1x+Bj a1y=Cj

Design Synthesis –Function Generation Initial position of crank pivot

Design Synthesis –Function Generation For positions a2 and a3

Design Synthesis –Function Generation -5.0 a1 (-4.348,-6.71) a3 (+1.67,-7.81) a2 (-0.410,-7.984)

Design Synthesis –Path Generation Path generation mechanism designed to guide one point on a Moving rigid body through a specified sequence of points in space. Plane Four Bar Path Generation Linkage Requires 2 guiding links to guide rigid body.

Design Synthesis –Path Generation Results in 2 constant length constraint equations b1 aj a1 bj b0 a0

Design Synthesis –Path Generation Specified path points Rotation angles to be determined

Design Synthesis –Path Generation # of equations depend on # of path precision points Ex. 2 path points 2 equations 9 unknowns # unknowns > equations Freedom of choice, pick design parameters to get solvable equations

Design Synthesis –Path Generation Relationship of path points to variables Path points Design equations # unkwns Specify Example variables Example calculate 2 9 7 a0,a1,b1,θ12 b0 3 4 10 (+θ13) 6 a0,a1,b0 b1,θ12,θ13 11 (+θ14) 5 a0,a1,b0x b0y,θ12,θ13,θ14 8 12 (+θ15) a0,b0 a0,a1,θ12,θ13, Θ14,θ15 16

Design Synthesis –Path Generation Example: Synthesis a path generator with

Design Synthesis –Path Generation Solution approach

Design Synthesis –Path Generation

Design Synthesis –Path Generation

Design Synthesis –Path Generation 1 2

Design Synthesis –Path Generation 3 4

Design Synthesis –Path Generation 1 2 3 4

Design Synthesis –Path Generation b0= (1.5,4.2) a1= (0.607,-1.127) b1= (-0.586,0.997)

Design Synthesis –Path Generation -68◦ -327◦ (not exactly)

Design Synthesis –Path Generation 5 path point solution

Homework #5