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 aij 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 aij = aji
Proof: Consider a linear system and now applying force ƒi the work done ½ force X displacement =
Then applying force ƒi, the work done =
However due to application of force ƒj i undergoes further displacement, aij ƒi and the additional work done by ƒi becomes (aij ƒi)ƒi.
So, total work done
Now if one reverses the order application of forces, i.e, first a force ƒj acts at j followed by a force ƒi acting at i , the work done will be
Since the work done in the two cases must be equal
hence, aij = aji (3)
Stiffness matrix :
For a three dof system, the force and displacements are related by stiffness matrix as
The stiffness kij 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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