# Slider crank

by Roberto Lot

> restart: libname := libname, "C:/libs/MBSymba.mla": with(MBSymba_r6): ## Vector loop

definition of the vectors

> AC := make_VECTOR(ground,c,0,0): show(AC);

> AB := make_VECTOR(Rotate('Z',q),r,0,0): show(AB);

> BC := make_VECTOR(Rotate('Z',-theta),l,0,0): show(BC);

##NOTE: theta is clockwise, so it requires the minus (-) sign.   closing the vector loop: definition of a closure vector

> closure := AC - BC - AB: show(closure); this expression must be null

> {comp_X(closure) , comp_Y(closure)}; we find the solution with respect to the variables cand theta which depend on the generalized coordinate q(rotation of the crank)

> solution := solve(%,{theta,c}); we extract the solutions

> eval([theta, c], solution):

> theta := %;

> c := %%;  anothe trick for univocally identify variables from solution set

> c := subs(solution,c);

> theta := subs(solution,theta);  ## Numerical Solution

numerical solution for specified crank and rod lengths:
crank length r = 0.1 m
rod length l = 0.4 m

> dataset := [r=0.1, l=0.4]; > theta := subs(dataset, theta);

> c := subs(dataset,c);

> r := subs(dataset,r);

> l := subs(dataset,l);    Plot of the mechanism

> A := origin(ground): #show(A);

> B := A + AB: show(B): show(B);

> C := A + AC: show(C): show(C);  > vertex := [ [comp_X(A),comp_Y(A)] , [comp_X(B),comp_Y(B)] , [comp_X(C),comp_Y(C)] ]; > plot(theta, q=0..2*Pi, labels=[q, "theta"], title="theta vs. q"); > plot(c, q=0..2*Pi, labels=[q, "c"], title="location of the slider vs. q"); ## Animation of the Mechanism

##Substitute in values for problem

> with(plots):with(plottools):

##Generate frames for the animation

> N:=15; ##number of steps
for n from 0 to N-1 do
q := evalf(2*Pi*n/N);
##crank

> crank || n := plot({[ [comp_X(A)+0.01*cos(q),comp_Y(A)+0.01*sin(q)], [comp_X(B)-0.01*cos(q),comp_Y(B)-0.01*sin(q)] ] }, color=black, thickness=3 );

##connecting rod

> rod || n := plot( {[ [comp_X(B)+0.01*cos(theta),comp_Y(B)-0.01*sin(theta)], [comp_X(C)-0.01*cos(theta),comp_Y(C)+0.01*sin(theta)] ]}, color=black, thickness=3 );

## slider

> slider || n := rectangle([comp_X(C)-0.08/2, comp_Y(C)+0.02], [comp_X(C)+0.08/2, comp_Y(C)-0.02], color=grey );

##revolute joint A

> revj_A || n := disk( [comp_X(A),comp_Y(A)], 0.01, color=yellow );

##revolute joint B

> revj_B || n := disk( [comp_X(B),comp_Y(B)], 0.01, color=yellow );

##revolute joint C

> revj_C || n := disk( [comp_X(C),comp_Y(C)], 0.01, color=yellow );
##assembly the mechanis

> four_bar || n := display ( {crank||n, rod||n, slider||n, revj_A||n, revj_B||n, revj_C||n} ):

> end:

Display the animation

> display([four_bar||(0..N-1)], insequence=true, scaling=constrained);  