MT-1620 al.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
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
MT-1620 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 马 刁 Paul A Lagace @2001 Unit 21-2
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
MT-1620 al.2002 q= generalized displacement (linear or rotation Q generalized force(force or moment/torque Note that Q; and g;are at the same point have the same sense (i.e. direction of the same type force← displacement) ( moment e> rotation) For a linear, elastic body, superposition applies, so can write q1=C12+ C2 22+ C123 q= ci e Q C22 22+23 03 十C23 q C1Q1+(3 C32Q2+C3Q3 or in matrix notation q2 32 Paul A Lagace @2001 Unit 21-3
MIT - 16.20 Fall, 2002 qi = generalized displacement (linear or rotation) Qi = generalized force (force or moment/torque) Note that Qi and qi 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 = C11 Q1 + C12 Q2 + C13 Q3 qi = Cij Qj q 2 = C21 Q1 + C22 Q2 + C23 Q3 q 3 = C31 Q1 + C32 Q2 + C33 Q3 or in Matrix Notation: q 1 C11 C12 C13 Q1 q 2 = C21 C22 C23 Q2 q 3 C31 C32 C33 Q3 Paul A. Lagace © 2001 Unit 21 - 3
MT-1620 al.2002 Note:l」->roW 3-> column []--> full matrix q g=cQ Ci= Flexibility Influence Coefficient and it gives the deflection at i due to a unit load at M12=is deflection at 1 due to force at 2 Figure 21.2 Representation of deflection point 1 due to load at point 2 苏(Noe: Ci can mix types) Paul A Lagace @2001 Unit 21-4
MIT - 16.20 Fall, 2002 Note: | | --> row { } --> column [ ] --> full matrix or { } q i = [ ] Cij { } Qj or q = C Q ~ ~ ~ Cij = Flexibility Influence Coefficient and it gives the deflection at i due to a unit load at j C12 = is deflection at 1 due to force at 2 Figure 21.2 Representation of deflection point 1 due to load at point 2 (Note: Cij can mix types) Paul A. Lagace © 2001 Unit 21 - 4
MT-1620 al.2002 Very important theorem Maxwell's Theorem of Reciprocal Deflection (Maxwell's Reciprocity Theorem) Figure 21.3 Representation of loads and deflections at two points on an elastic body q1 due to unit load at 2 is equal to q 2 due to unit load at 1 Generally C:=C symmetric Paul A Lagace @2001 Unit 21-5
MIT - 16.20 Fall, 2002 Very important theorem: Maxwell’s Theorem of Reciprocal Deflection (Maxwell’s Reciprocity Theorem) Figure 21.3 Representation of loads and deflections at two points on an elastic body q1 due to unit load at 2 is equal to q2 due to unit load at 1 i.e. C12 = C21 Generally: Cij = Cji symmetric Paul A. Lagace © 2001 Unit 21 - 5