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unit21 - MIT 16.20 Fall 2002 Unit 21 Influence Coefficients...

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MIT - 16.20 Fall, 2002 Unit 21 Influence Coefficients Readings : Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Paul A. Lagace © 2001
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MIT - 16.20 Fall, 2002 Have considered the vibrational behavior of a discrete system. How does one use this for a continuous structure? First need the concept of….. Influence Coefficients which tell how a force/displacement at a particular point “influences” a displacement/force at another point --> useful in matrix methods… finite element method lumped mass model (will use this in next unit) --> consider an arbitrary elastic body and define: Figure 21.1 Representation of general forces on an arbitrary elastic body Paul A. Lagace © 2001 Unit 21 - 2
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MIT - 16.20 Fall, 2002 q i = generalized displacement (linear or rotation) Q i = generalized force (force or moment/torque) Note that Q i and q i are: at the same point have the same sense (i.e. direction) of the same “type” (force displacement) (moment rotation) For a linear, elastic body, superposition applies, so can write: q 1 = C 11 Q 1 + C 12 Q 2 + C 13 Q 3 q i = C ij Q j q 2 = C 21 Q 1 + C 22 Q 2 + C 23 Q 3 q 3 = C 31 Q 1 + C 32 Q 2 + C 33 Q 3 or in Matrix Notation: q 1 C 11 C 12 C 13  Q 1 q 2 = C 21 C 22 C 23 Q 2 q 3 C 31 C 32 C 33 Q 3 Paul A. Lagace © 2001 Unit 21 - 3
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MIT - 16.20 Fall, 2002 Note : | | --> row { } --> column [ ] --> full matrix or { } = [ ] { } q i C ij Q j or q = C Q ~ ~ ~ C ij = Flexibility Influence Coefficient and it gives the deflection at i due to a unit load at j C 12
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