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ho6 - 16.20 HANDOUT#6 Fall 2002(Introduction To Structural...

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16.20 HANDOUT #6 Fall, 2002 (Introduction To) Structural Dynamics REGIMES (Structural) Wave (Quasi) - Static Dynamics Propagation Static f(natural f(speed of waves frequency of in material) structure) – Structural stiffness speed = – Structural “characteristic length” E ρ = region of transition SPRING-MASS SYSTEMS [Force/length] k d Static equation: F = kq ( ) = d t Dynamic: mq ˙˙ + k q = F t ( ) (No damping) ( ) (With damping) mq ˙˙ + cq ˙ + k q = F t Inertial Load = – (mass) x (acceleration) General form: m = mass matrix ~ ˙˙ mq + kq = F ~ ~ ~ ~ ~ ~ k = stiffness matrix Paul A. Lagace © 2002

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q t q t or F = force vector m q ˙˙ j + k ij q j = F i ~ ij q = d. o. f. vector ~ d. o. f. = degree of freedom i, j = 1, 2…n n = number of degrees of freedom of system FREE VIBRATION ( ) = C 1 sin ω t + C 2 cos ω t genera l solution for single spring-mass system natural frequency: ω = k m FORCED VIBRATION dirac delta function: δ ( t τ ) = 0 @ t τ δ ( t τ ) → ∞ @ t = τ −∞ δ ( t τ ) dt = 1 ( ) δ ( t τ ) dt = g ( ) −∞ g t τ unit impulse response: 1 ( ) = sin ω ( t
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