Alembert Virtual Displacement

The principle of virtual work is a statement of static equilibrium of mechanical systems which represents the first variational principle of mechanics and is a transition from Newtonian to Lagrangian mechanics.

Virtual displacement

Consider a system of N particles in which the position of i th particle in space is represented by r_i ( i = 1,2… N ).

Then the virtual displacement represents the imagined infinitesimal changes δ r_i in these position vectors that are consistent with the constraints of the system, but are otherwise arbitrary.

 
It may be noted that virtual displacements are not true displacements but are small variation in the system coordinates resulting from imagining the system in a slightly displaced position, a process that does not necessitate any corresponding change in time and hence the forces and constraints do not change during this process.

Hence the virtual displacements take place instantaneously. This is in direct contrast with the actual system, which requires certain amount of time to evolve, during which time the forces and constraint may change.

Lagrange introduces δ to emphasize the virtual character of the instantaneous variation in opposed to the symbol d designating actual differential of position taking place in time interval dt.

The virtual displacement being infinitesimal, obey the rules of differential calculus.

Considering the system of N particles (Figure 1) subject to a constraint in the form     

                                                              

Then the virtual displacement must be such that

              
It may be noted that both t and C are same in both equations

Expanding eqn.  in Taylor's series about the position x₁, y₁, z₁......, x_n, y_n, z_n we have

                     
Neglecting the terms of order δ² and higher from equation (3) and using equation (1) one may obtain

  .                                                                                      
So that only 3n-1 of the virtual displacements is arbitrary. In general, the number of arbitrary virtual displacements coincides with the number of degrees of freedom of the system. For a system in equilibrium, the resultant force on each particle should be equal to zero. Assuming that each of the n particles ( m_i ) is acted upon by a set of forces (i = 1, 2, ...n)  For a system in equilibrium, the resultant force on each particle should be
equal to zero.

So the virtual work performed by the resultant  is -

                                                                                             
For the entire system of particle the virtual work can be given by

                                                                                                 
Let    be the external applied force and be the constraint force acting on the i th particle.
 
Also ƒ_ij  is the internal force exerted by particle m_j on particle m_i .

So the resultant force on the i th particle can be given by

                                                                                      
Substituting equation

                                                                                      
Considering constraint forces that are normal to the virtual displacement , the work done by the constraint forces through virtual displacements compatible with the system constraints is zero.

Or                                                                                                                      
System obeying equation (9) is a reversible system. As the internal forces acts in pair, the work done by the internal force for the whole system is zero. So equation (8) reduces to

                                                                                                                     
Equation (10) states that, the work performed by the applied forces in infinitesimal reversible virtual displacements compatible with the system constraints is zero.

To derive the equilibrium equation for such cases one may write   using m independent generalized coordinates qk, (k = 1, 2, ....,m)  One may note that m ≤ n

r_i = r_i (q₁, q₂.....q_r)   i = 1,2........n
                                                                                        
The virtual displacement vector an be expressed in terms of the generalized virtual displacement δ qk (k =1,2....m) in the form

                                                                               
Substituting equation, one obtains

are known as generalized forces.

The Generalized Principle of d'Alembert

The Principle of Virtual Work provides a statement of the static equilibrium of a mechanical system. As using d'Alembert principle one may convert a dynamical system into an equivalent static system by incorporating the concept of inertial force, the Virtual Work principle now can be extended to dynamical systems. Considering a system of particles, the d‘Alembert Principle for a particle of mass mi can be written as

Here,   is the applied force,   is the constraint force, is the internal force and is the inertial force acting on the ith particle. Equation (1) can be regarded as a statement of dynamic equilibrium of particle mi.

Now, the Principle of Virtual work can be extended to dynamic system thus producing the first variational principle of dynamics which can be expressed by the following equation.

Considering a system of Considering a system of n particles and assuming that the virtual displacements (i = 1,2,…n) are reversible so that  , one can write for the system of particles

Equation (3), which utilizes both the principle of virtual work of statics and d'Alembert principle of dynamics, is referred to as generalized principle of d'Alembert.
The Sum of the applied force and the inertial force is sometime referred to as effective force.

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