Constant & Linear Accel

Constant & Linear Accel - CE 573: Structural...

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CE 573: Structural Dynamics Numerical Evaluation of Dynamic Response – Newmark Methods Example Given: Mass of tank , stiffness k = 10 kips/in, undamped natural period ( 2 0.2533 kip-sec /in m = 1.0 sec T = 6.283 rad/sec ω = ), and 0.05 ξ = . Tank starts at rest and is subjected to half-cycle sine pulse F(t) as shown. Required: Determine the response x(t) from 0.0 sec t = to 1.0 sec t = using Newmark’s constant acceleration method and linear acceleration method with 0.1 sec t ∆ = . Solution: 1. Need to check time step for stability – we will assume that stability is OK. 2. Preliminary Calculations – 2 0.0 in; 0.0 in/sec; 0.0 kips. 0.0 in/sec . ooo oo o Fc uk u o m uu F u −− == = ⇒= = ± ± ±± 3. Constant Acceleration Method: () ( ) ( )( )( ) 2 22 kips 42 in 2 4 0.2533 kip-sec /in 2 2 0.2533 kip-sec /in 0.05 6.283 rad/sec 10 0.1 sec 0.1 sec 114.50 kips/in. mc ii t t kk =+ + = + + = 4. Linear Acceleration Method: ( ) ( )( )( ) 2 kips 63 in 2 6 0.2533 kip-sec /in 3 2 0.2533 kip-sec /in 0.05 6.283 rad/sec 10 0.1 sec 0.1 sec 166.75 kips/in. t t + + =
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5. Numerical solution (via spreadsheet) Constant Acceleration Method Mass Damping Stiffness
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Constant & Linear Accel - CE 573: Structural...

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