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Multiple Degree of Freedom Systems: Influence Coefficients

We have so far used Newton’s laws to obtain the equations of motion for multiple degree of freedom systems. Another way to obtain them is to use \emph{influence coefficients}. (These are based on Newton’s laws as well, but are often easier to apply.) Influence coefficients relate the effects of forces and displacements on one part of a structure to another and are used extensively in structural engineering. Essentially, one set of influence coefficients can be associated with each matrix appearing in the equations of motion. Just as with the equations of motion, the influence coefficients are directly related to the choice of coordinates in a problem. A different set of coordinates for the same problem will result in a completely different set of influence coefficients.

For the stiffness matrix, there are two (related) sets of influence coefficients that can be used

  • Stiffness influence coefficients, and
  • Flexibility influence coefficients.

For the mass matrix we can use

  • Inertia influence coefficients.

Flexibility Influence Coefficients
The flexibility influence coefficients \ensuremath{a_{ij}} are defined as

\ensuremath{a_{ij}} = the deflection of coordinate i in the positive direction due to the application of a unit load at coordinate j in the positive direction.

Note that for a single degree of freedom system this simplifies to our previous definition for the equivalent stiffness. For a multiple degree of freedom system, these coefficients are determined in practice by applying a unit load to the system in the positive j coordinate direction and determining the displacement of all the coordinates in the system. This is a statics problem. Then \ensuremath{a_{ij}} is the deflection of the system in the (positive) j coordinate direction in this case.

Flexibility influence coefficients can also be applied to torsional systems where

\ensuremath{a_{ij}} = the \emph{rotation} at coordinate i in the positive direction due to the application of a unit \emph{torque} at coordinate j in the positive direction.

We can also use flexibility influence coefficients in mixed systems with both linear and rotational coordinates. The units of \ensuremath{a_{ij}} will then change depending on whether i and j are linear or angular coordinates.
Stiffness Influence Coefficients
The stiffness influence coefficients \ensuremath{k_{ij}} are defined as

\ensuremath{k_{ij}} = the force required at coordinate i in the positive direction to maintain a unit deflection of coordinate j in the positive direction while maintaining zero displacement for all of the other coordinates.

Note that for a single degree of freedom system this simplifies to our previous definition for equivalent stiffness. For a multiple degree of freedom system, these coefficients are determined in practice by considering the system when the j^{th} coordinate has a positive unit displacement while all other coordinates are held fixed with zero displacement. The forces required to maintain the structure in this deformed configuration are then solved statically. \ensuremath{k_{ij}} is then the force required in the positive i coordinate direction to maintain this deformed configuration.

Stiffness influence coefficients can also be applied to torsional systems where

\ensuremath{k_{ij}} = the \emph{torque} required at coordinate i in the positive direction to maintain a static unit \emph{rotation} of coordinate j in the positive direction while maintaining zero rotation for all of the other coordinates.}

We can also use single degree of freedom in mixed systems with both linear and rotational coordinates. The units of \ensuremath{k_{ij}} will then change depending on whether i and j are linear or angular coordinates.
Inertia Influence Coefficients
The inertia influence coefficients \ensuremath{m_{ij}} are defined as

\ensuremath{m_{ij}} = \text{the impulse that must be applied at coordinate i in the positive direction to produce a unit velocity at coordinate j in the positive direction while maintaining zero velocity at all the other coordinates.

An alternative equivalent definition is

\ensuremath{m_{ij}} = the force that must be applied at coordinate i in the positive direction to produce a unit acceleration at coordinate j in the positive direction while maintaining zero acceleration at all the other coordinates.

inertia influence coefficients can also be applied to torsional systems where

\ensuremath{m_{ij}} = the \emph{torque} at coordinate i in the positive direction that must be applied to produce a unit \emph{angular acceleration} at coordinate j in the positive direction while maintaining zero angular acceleration at all the other coordinates.

We can also use inertia influence coefficients in mixed systems with both linear and rotational coordinates. The units of \ensuremath{m_{ij}} will then change depending on whether i and j are linear or angular coordinates.

Properties of Stiffness and Flexibility Influence Coefficients

  • The total force F_i acting in the i^{th} coordinate direction can be determined from all of the displacements (x_j, j=1,2,\dots N) in the system from

        \[ F_i = \sum_{j-1}^N \ensuremath{k_{ij}} x_j, \qquad i = 1,2 \dots N, \]

    or in matrix form

    (8.32)   \[ \ensuremath{\bigl\{F\bigr\}} = \ensuremath{\bigl[k\bigr]}\!\!\ensuremath{\bigl\{x\bigr\}}, \]

    where \ensuremath{\bigl\{F\bigr\}} is a column vector of the forces acting on the system in the positive coordinate directions and \ensuremath{\bigl[k\bigr]} is the matrix of single degree of freedom. (Note that if one of the coordinates is an angular displacement, the corresponding term in \ensuremath{\bigl\{F\bigr\}} would be a torque.)Similarly, the total displacement in the i^{th} coordinate direction can be determined from all of the forces acting in the positive coordinate directions (F_j, j=1,2,\dots N) for the system from

        \[ x_i = \sum_{j-1}^N \ensuremath{a_{ij}} F_j, \qquad i = 1,2 \dots N, \]

    or again in matrix form

    (8.33)   \[ \ensuremath{\bigl\{x\bigr\}} = \ensuremath{\bigl[a\bigr]}\!\!\ensuremath{\bigl\{F\bigr\}}, \]

    where \ensuremath{\bigl[a\bigr]} is the matrix of flexibility influence coefficients. (Note that if one of the coordinates is an angular displacement, the corresponding term in \ensuremath{\bigl\{x\bigr\}} would be a rotation.)Substituting (8.33) into (8.32) shows that

        \[ \ensuremath{\bigl\{F\bigr\}} = \ensuremath{\bigl[k\bigr]}\ensuremath{\bigl\{x\bigr\}} = \ensuremath{\bigl[k\bigr]} \Bigl[ \ensuremath{\bigl[a\bigr]}\ensuremath{\bigl\{F\bigr\}} \Bigr] = \Bigl[\ensuremath{\bigl[k\bigr]}\ensuremath{\bigl[a\bigr]}\Bigr] \ensuremath{\bigl\{F\bigr\}}. \]

    Similarly,

        \[ \ensuremath{\bigl\{x\bigr\}} = \ensuremath{\bigl[a\bigr]}\ensuremath{\bigl\{F\bigr\}} = \ensuremath{\bigl[a\bigr]} \Bigl[ \ensuremath{\bigl[k\bigr]}\ensuremath{\bigl\{x\bigr\}} \Bigr] = \Bigl[\ensuremath{\bigl[a\bigr]}\ensuremath{\bigl[k\bigr]}\Bigr] \ensuremath{\bigl\{x\bigr\}}. \]

    These results show that

        \[ \ensuremath{\bigl[k\bigr]}\!\!\ensuremath{\bigl[a\bigr]} = \ensuremath{\bigl[a\bigr]}\!\!\ensuremath{\bigl[k\bigr]} = \ensuremath{\bigl[1\bigr]} \]

    which implies that

         \begin{align*} \ensuremath{\bigl[k\bigr]} &= \ensuremath{\bigl[a\bigr]}i, \\ \ensuremath{\bigl[a\bigr]} &= \ensuremath{\bigl[k\bigr]}i, \end{align*}

    or

    The matrices of stiffness and flexibility influence coefficients are \emph{inverses} of each other.
  • For linear systems, Maxwell’s Reciprocity Theorem tells us that the deflection at point i due to a unit load at point j in a structure is the same as the deflection at point j due to a unit load at point i (all in the positive directions). As a result we find that \ensuremath{a_{ij}} = \ensuremath{a_{ij}} which implies that \ensuremath{k_{ij}} = \ensuremath{k_{ji}}. These results lead to the conclusion that
    The matrices of stiffness and flexibility influence coefficients are \emph{symmetric}

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