Review of single and multi-degree of freedom (mdof) systems: Stiffness and Flexibility Influence Coefficients
We can use the Newtonian approach to find the stiffness matrix and equations of motion. However, we can also use influence coefficients for both the mass and stiffness matrices.
Consider the definition of stiffness and flexibility influence coefficients:
is the force (moment) required at coordinate to maintain a unit linear (angular) displacement at coordinate with all other coordinates held fixed at zero
is the linear (angular) displacement at coordinate due to a unit force (moment) applied at coordinate with all other coordinates free to move.
NOTE: All forces (moments) and displacements (rotations) are taken to be applied in the positive direction. The sign of the resulting forces (moments) or displacements (rotations) are determined to be positive or negative.
Also note that:
As an example of MDOF, consider a general planar situation which can be specialized to different industrial situations. Many of these problems can be initially modelled as SDOF ones, and then MDOF analysis will allow interpretations of the simplifying assumptions made.
NOTE: While the body shown is a rectangular shape, it can be any rigid body with a mass and moment of inertia about an axis to the plane through the center of gravity. The horizontal stiffness components, designated by are most often just the lateral stiffness of the vertically mounted spring. For real springs, the lateral spring stiffness may be hard to find in the literature. For coil springs, it is often in the range of times the longitudinal value.
In matrix form:
Now find the stiffness matrix using influence coefficients.
First consider a unit displacement in the direction only:
For the direction unit displacement:
Assuming all summations for direction, direction and moment equal to :
This yields the same stiffness matrix.
As long as we use the motions about the center of gravity then the inertia (mass) matrix is just the mass or the mass moment of inertia about the center of gravity. If we use coordinates to describe the motion that are not related to the absolute motion of the center of gravity then the stiffness matrix may not be symmetric and the mass matrix will not be diagonal.
Inertia Influence Coefficients
The elements of the mass matrix, can be determined through the use of inertia influence coefficients. They are defined as the set of impulses applied at the points (representing the coordinate directions) respectively to produce a unit velocity at point and zero velocity at every other point.
NOTE: If denotes an angular coordinate then represents an angular velocity
Thus for a multi-degree of freedom system, the total impulses at points is:
To find the
- Assume that a set of impulses are applied at all points as to produce a unit velocity at point only(with zero velocity in all the after directions (coordinates)). By definition the set of impulses denote the influence inertia coefficients
- Repeat the procedure for each point .
First apply unit velocity to only:
Apply unit velocity(angular) to only
Now find the stiffness matrix
Apply unit deflection to only:
Apply unit deflection to only: