Writing Constraint Equations for Two-Point and Three-Point Mounts

We describe how to obtain the constraint equations for a two point pivot and three point pivot. Designing a mechanism which can obtain a desired set of constraints is often an important step in kinematic or exact constraint machine design.

We begin with the simple lever mechanism shown in the figure below constraining the motion of two points A and C using the pivot at O.

We treat the bar as rigid and are only interested in small amplitude rotation of the pivot about O. For a rotation of angle $\theta$ about point O, the displacement at point C is

$u_C=L_A \sin \theta \approx L_C \theta$

and the displacement at point A is

$u_A=-L_A \sin \theta \approx -L_A \theta$

where the approximation is due to the assumption of small angles. Eliminating $\theta$ from both equations, we obtain a single constraint equation relating $u_A$ and $u_C$

$\frac{u_A}{L_A}+\frac{u_C}{L_C}=0$

Another method of deriving this set of constraints is to fit a line to the points O and C and then derive the constraint equation A must satisfy to also fall on this line. Let the (x,y) coordinates of X be (0,0) and of C be ( $L_C,u_C$ ). The line connecting the two is given by

$y=\frac{u_C}{L_C} x$

Then, substitute the coordinates of point A ( $x=-L_A,y=u_A$ ) to obtain

$\frac{u_A}{L_A}=-\frac{u_C}{L_C}$

Now consider a constraint of three points A, B, and C at the vertices of a rigid, triangular plate with a pivot O in the middle of the plate shown in the figure below which allows displacement of A, B, and C in the $z$ direction but not in the $x$ and $y$ (there is no $\theta_z$ motion of the plate).

Again, there are multiple ways to obtain the constraint equations. One could define angles of rotation $\theta_x$ and $\theta_y$ about to derive the constraints, but here we focus on using the equation of a plane to derive the constraints similar to the method above of fitting a line to two points.

Let the $(x,y,z)$ coordinates of the pivot O be $(0,0,0)$ and the coordinates of A $(x_A,y_A,u_A)$ , B $(x_B,y_B,u_B)$ , and C $(x_C,y_C,u_C)$ . The equation of the plane can be written setting the determinant equal to zero (note: there are a number of equivalent matrix determinants which can be used to define the plane)