Multi Degree Freedom

Properties of Vibrating Systems:

Since the elastic behavior of motion may be expressed in terms of the stiffness or flexibility, the equations of motion may be formulated by either the stiffness matrix [k]  or the flexibility matrix [a] .

In stiffness formulation , the force ƒ is expressed in terms of displacement x by
      
{ƒ} = [k] {x}                                                                         (1)

Also one may write
      
{x} = [k]^-1 {ƒ} = [a] {ƒ}                                                    (2)

This leads to the flexibility approach.

The choice as to which approach one should adopt depends on the problem. Some problems are more easily pursued as the basis of stiffness, whereas for others the flexibility approach may be desirable.

Flexibility matrix:

For a three degree-of-freedom system, the displacement as forces are related by flexibility matrix as

 
The flexibility influence coefficient a_ij is defined as the displacement at i due to unit force applied at j with all other forces equal to zero. Thus the first column represent displacement corresponding to ƒ₁ = 1, ƒ₂ = ƒ₃ = 0.  

Similarly, second column represents the displacements for ƒ₂ = 1 and ƒ₁ = ƒ₃ = 0 and so on.

Reciprocity theorem : States that in a linear system a_ij = a_ji

Proof: Consider a linear system and now applying force ƒ_i the work done ½  force X displacement =

 
Then applying force ƒ_i, the work done = 

However due to application of force ƒ_j i undergoes further displacement, a_ij ƒ_i  and the additional work done by ƒ_i becomes (a_ij ƒ_i)ƒ_i.

So, total work done
 


Now if one reverses the order application of forces, i.e, first a force ƒ_j acts at j followed by a force ƒ_i acting at i , the work done will be



Since the work done in the two cases must be equal

hence, a_ij = a_ji                                                          (3)
                                                                                                       
Stiffness matrix :

For a three dof system, the force and displacements are related by stiffness matrix as

The stiffness k_ij is defined, as the force required at point i to have unit displacement at point j, displacement at other places being zero. So k₁₁, k₂₁ and k₃₁ are the forces required at points 1,2,3 respectively to have unit displacement at 1, i.e.,
    
x₁ = 1, x₂ = x₃ = 0    

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