Constrained Dynamics

The Principle of Virtual Work

Principle of Virtual Work

Virtual Displacement

This refers to the change in configuration of the system as a result of any arbitrary infinitesimal change in the coordinates \(\delta r_i\) of the system.

Case I when there is no motion under constraints:

Imagine having a diamond hand and squeezing a diamond ball. The diamond hand will not deform and the diamond ball will not deform. The diamond hand will not penetrate the diamond ball and the diamond ball will not penetrate the diamond hand. This is the constraint imposed by the rigid body. The diamond hand and the diamond ball are rigid bodies. But because some force is being applied to do work on the diamond ball we can assoiciate this force with a displacement for the sake of analysis. This displacement is called the virtual displacement.

Case II when there is motion under constraints:

let us take another example of a simple pendulum. Let us freeze the frame when the bob is being released, meaning the cartesian position of the bob cannot be changed in this freezed frame but we can analyze the effect of the forces. Let the forces acting on the bob in this frozen frame produce a supposed virtual displacement. This virtual displacement is not real but it is useful for analysis.

\[\sum_{i}^{} \vec{F_i} \cdot \delta r_i= 0\]

where, \(\delta \vec{F_i}\) is the virtual force acting on the system at the particle \(r_i\).

The net work done by this virtual force to produce the virtual displacement is zero because the system is in equilibrium(remember the conditions of the fixed frame from the cases above).

decomposing this virtual force into applied and constraint forces, we get: \(\vec{F_i} = \vec{F_i}^{a} + \vec{F_i}^{c}\)

where, \(\vec{F_i}^{a}\) is the applied force and \(\vec{F_i}^{c}\) is the constraint force.

These forces are the direct result of the newton’s second law. They are called reaction forces. Imagine the case of a dexterous robotic hand holding a ball. The ball exerts a force on the finger which is the constraint force preventing the penetration of the finger into the surface of the ball and the finger exerts a force on the ball which is the applied force.

Then the above equation becomes:

\[\sum_{i}^{} \vec{F_i}^{a} \cdot \delta r_i + \sum_{i}^{} \vec{F_i}^{c} \cdot \delta r_i= 0\]

Let us look at these individual terms in detail.

Work done by constraint forces

let us consider 2 particles \(r_1\) and \(r_2\) in the pool of particles that represent the rigid body. Then the holonomic constraint imposed by the rigid body for those two particles is:

\[\|r_1 - r_2\|_{_2} = l_{12}\]

where \(l_{12}\) is the fixed distance between the two particles.

\[(r_1-r_2)^T(r_1-r_2) = l_{12}^2\]

and from the definition of virtual displacement, we have:

\[\| (r_1 + \delta r_1) - (r_2 + \delta r_2) \|_{_2} = l_{12}\]

by expanding and ignoring the higher order terms we get:

\[(r_1-r_2)^T(\delta r_1 - \delta r_2) = 0\]

by the definition of the constraint force acting from the particles \(r_1\) and \(r_2\), we have:

\[\vec{F_{12}}^{c} = \lambda_{12} (r_1-r_2)\]

where \(\lambda_{12}\) is the lagrange multiplier.

Similarly, from the particle \(r_2\) to \(r_3\) we have:

\[\vec{F_{21}}^{c} = \lambda_{21} (r_2-r_1)\]

where \(\lambda_{21}\) is the lagrange multiplier.

\(F_{12}^{c} = -F_{21}^{c}\) and \(\lambda_{12} = \lambda_{21} = \lambda\)

\[{\vec{F_{12}}^{c}} \delta r_1 = \lambda (r_1-r_2)^T \delta r_1\] \[{\vec{F_{21}}^{c}}\delta r_2 = -\lambda (r_1-r_2)^T \delta r_2\] \[\sum_{i}^{2} {\vec{F_i}^{c}} \delta r_i = \lambda (r_1-r_{2})^T \delta (r_1 - r_2)\]

or more generally,

\[\sum_{i}^{} {\vec{F_i}^{c}} \delta r_i = \lambda \sum_{i}^{} (r_i-r_{i+1})^T \delta (r_i-r_{i+1})\]

but because we already established that \((r_1-r_2)^T(\delta r_1 - \delta r_2) = 0\)

\[\sum_{i}^{} {\vec{F_i}^{c}} \delta r_i = 0\]

Work done by applied forces

which leads us to the equation:

\[\sum_{i}^{} {\vec{F_i}^{a}} \cdot \delta r_i = 0\]

This is called the principle of virtual work.