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ASE 365 - Lecture 3

ASE 365 - Lecture 3 - Discrete components •...

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Unformatted text preview: Discrete components • Elastic (spring) • Force is proportional to displacements : • Energy: (potential) x 1 x 2 FS FS k ) ( 1 2 x x k k s- = = δ F 2 1 2 2 ) ( 2 1 2 1 x x k k V- = = δ Discrete components (cont’d) • Dissipative (dashpot) • Force is proportional to velocities : • Energy: (not conserved!) 1 x 2 x c D F D F ) ( 1 2 x x c c d - = = δ F 2 1 2 2 ) ( x x c c E -- =- = δ • Inertial (mass) • Force is proportional to accelerations : • Energy: (kinetic) Discrete components (cont’d) x m Fm x m m = F 2 2 1 x m T = Equivalent springs, effective stiffness FBD: FS1 FS2 F k 1 x F k 2 : parallel in Springs eq eff k x F k = = : stiffness Effective k 1 k 2 2 1 2 1 2 1 2 2 1 1 ) ( k k x F k x k k F F F x k F x k F eq S S S S + = = + = + = = = , and , and Since k 1 x F k 2 : series in Springs . Again, x F k eq = : and , , , Eliminate 2 1 2 1 S S F F x x FS1 k 1 x1 F F x k 2 FS2 FS2 x 1 x 2 . or So 2 1 1 1 1 1 1 1 2 1 k k k x F k eq k k eq + = + = = 1 1 1 x k F S = 2 1 S S F F = ) ( 1 2 2 2 x x k F S- = F F S = 2 2 x x = x k F k F k F x x k F x = + = + = = 2 1 2 1 2 1 1 / / / / , : n deformatio Axial . Here, δ F k eq = : So , , L EA F k L E A F L E A F eq = = = ⇒ = = = δ δ δ ε ε σ σ F δ A E, L : materials of strength From L EA F F 1 1 L EA 2 2 L EA F 1 EA 2 EA 1 L 2 L 2 2 1 1 1 EA L EA L eq k + = Prototype SDOF system k c m x F FBD: fS f D F m : EOM Separating equilibrium from dynamics...
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ASE 365 - Lecture 3 - Discrete components •...

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