# What is a bilateral constraint?

In the realm of mechanics/rigid body dynamics, can anyone tell me what a bilateral constraint is? Can't seem to find any information on the exact definition, just uses of it such as "considering only bilateral holonomic constraints..."

Is this just saying there will be equal and opposite reaction forces? What are some examples of bilateral and non-bilateral constraints?

## 2 Answers

Simply stated, in the area of contact dynamics, a unilateral (bilateral) constraint refers to a one-sided (two-sided) constraint, i.e. a constraint described$^1$ via an inequality (equality) of some constraint function $f(q,\dot{q},t)$, respectively.

Here we are assuming that the variables $q^i$ are real (as opposed to complex). Technically, one demands that the constraint function $f(q,\dot{q},t)$ is regular, which usually means that it is sufficiently differentiable and that the gradient of $f$ does not vanish at points where the function $f$ itself vanishes.$^1$

In practice, bilateral constraints are easier to work with than unilateral constraints. Sometimes one can get away with treating an actual unilateral constraint as an effective bilateral constraint. Example: If one slowly pushes a box around on a horizontal surface $z=0$ (and it is a reasonable assumption that the box will not loose contact with the surface because of downward gravity), then one can replace the actual unilateral constraint $z\geq 0$ with an effective description in terms of a bilateral constraint $z=0$ .

References:

1. Jan Awrejcewicz, Classical Mechanics: Dynamics, Adv. Mech. Math., Vol. 29 (2012).

--

$^1$ Example: The constraint function $f(q)=q^2$ is a non-regular constraint function and therefore excluded. The inequality $q^2\leq 0$ is course the equality $q=0$ in disguise, and hence bilateral. The two sides are here $q>0$ and $q<0$.

• So just to see if I understand correctly (in less technical terms), let me ask about some examples: a unilateral constraint could prevent a rod from penetrating a solid surface, but does not prevent the rod from lifting away from the surface into free space; on the other hand, a bilateral constraint would e.g. be a rod attached to a solid surface by a joint, so that it can move neither into the surface nor away from the surface. Is that correct? – thkemp Nov 16 '13 at 17:38
• @thkemp : Right. Note that in the bilateral case, the joint is allowed to move around on the surface as long as it stay on the surface. This may e.g. be realized via some rail device. – Qmechanic Nov 16 '13 at 17:45
• Great thanks, that clears it up. I guess a joint would be a more specialized case where there are additional constraints on the allowable motion that is tangent to the surface, whereas the more general bilateral constraint is just on motion normal to the surface. – thkemp Nov 16 '13 at 18:28

bilateral constraint take the form $f(q,\dot{q},t)=0$ such as internal constraints which give rise to the constant distances between the points of a perfectly rigid body
unilateral constraint take the form $f(q,\dot{q},t)\leq0$ such as the motion of a body suspended from a flexible thread