Construction and movement of the centrodes of a slider crank mechanism
dr.B.Laczik Budapest University of Technology and Economics, HUNGARY
laczik@goliat.eik.bme.hu
The movement of the general planar linkage mechanism with one degree of freedom matches exactly the rolling of two centrode curves on each other without slipping.The first centrode is fixed to the frame of the linkage and the second centrode is fixed to the moving element of the mechanism.The example illustrates the analytic calculation and geometric construction of the centrodes for a slider crank mechanism, see. Fig.1.
Fig. 1
Calculation of the Centrode 1:
The basic dimensions of the mechanism are r = CA and l = AB.
Let the position angle BCA = . The instantaneous velocity centrum of the rod AB is the point P, which is the intersection of line CA and the line BP which is perpendicular to line BC, see Fig. 2.
The cosine theorem applied to triangle CBA is:
and the right-angled triangle CBP yields
So the co-ordinates of the point P of Centrode1 are
Fig.2 Fig. 3
Calculation of the Centrode 2:
If the rod AB is fixed, the velocity centrum of the rod CB is the point Q at the intersection of line CA and the line BQ which is perpendicular line to BC, see Fig. 3.
Now let the position angle BAC = . Using the cosine theorem, from tirangle CBA:
In the right-angled triangle CAQ the angle and the corresponding sinus theorem is
and additionally
While
so the co-ordinates of the point Q of Centrode 2 are
The animations are designed using classic geometric constructions of centrodes.
Fig. 4 illustrates the front and back sides of the actual construction of the centrodes and slider crank mechanism.
Fig. 4
