Forces of Constraint and Virtual work

Constraints are related to forces which restrict the motion of the system. Act in a direction ⟂ to the surface of constraints while the motion of the object is parallel to the surface.

Work done by forces of constraint = 0

Difficulties due to constraints and their removal #

1. In a system of N interacting particles the N system of equations are not independent and the equations of motion are to be written taking into consideration the equations of the constraints
2. Constraint forces are not known initially and are obtained from the solution of the problem that we are seeking.

To remove these difficulties, we formulate the mechanics in such a way that forces of constraint disappear.

Generalized Coordinates #

Set of independent coordinates sufficient in number to describe completely the state of configuration of a dynamical system. $q_1,q_2,q_3,…q_n$

Principle of Virtual Work #

When a system is in equilibrium $$ \begin{align} \delta W_i=\vec{F_i}\cdot \delta \vec{r_i}=0\\ \delta W=\sum_{i=1}^N \vec{F_i}\cdot \delta \vec{r_i}=0\\ \Rightarrow \sum_{i=1}^N \vec{F_i^a}\cdot \delta \vec{r_i}+\sum_{i=1}^N \vec{f_i}\cdot \delta \vec{r_i}=0\\ \sum_{i=1}^N \vec{F_i^a}\cdot \delta \vec{r_i}=0 \end{align} $$

• $F_i^a$: applied force
• $f_i$: force of constraint

We have proved that for equilibrium of a system, the virtual work of applied forces $F_i^a$ is zero.