1. Basic structural dynamics I Wind loading and structural response - Lecture 10 Dr. J.D. Holmes

2. Basic structural dynamics I Topics : Revision of single degree-of freedom vibration theory Response to sinusoidal excitation Refs. : R.W. Clough and J. Penzien ‘Dynamics of Structures’ 1975 R.R. Craig ‘Structural Dynamics’ 1981 J.D. Holmes ‘Wind Loading of Structures’ 2001 Multi-degree of freedom structures – Lect. 11 Response to random excitation

3. Basic structural dynamics I Equation of free vibration : Example : mass-spring-damper system : k c m x mass × acceleration = spring force + damper force - kx - c(dx/dt) equation of motion Single degree of freedom system :

4. Basic structural dynamics I Single degree of freedom system : Equation of free vibration : Example : mass-spring-damper system : Ratio of damping to critical c/c c : k c m x often expressed as a percentage

5. Basic structural dynamics I Single degree of freedom system : Damper removed : k m x(t) Equation of motion : is an equivalent static force (‘inertial’ force) Undamped natural frequency : Period of vibration, T :

6. Basic structural dynamics I Single degree of freedom system : Initial displacement = X o Free vibration following an initial displacement : k c m x

7. Basic structural dynamics I Single degree of freedom system : Free vibration following an initial displacement :

8. Basic structural dynamics I Single degree of freedom system : Free vibration following an initial displacement :

9. Basic structural dynamics I Single degree of freedom system : Response to sinusoidal excitation : Equation of motion : Steady state solution : k c m F(t)  = 2  n H(n) = Frequency of excitation

10. Basic structural dynamics I Single degree of freedom system : Critical damping ratio – damping controls amplitude at resonance 0.1 1.0 10.0 100.0 01234 n/n 1 H(n)  =0.01  =0.05  =0.1  =0.2  =0.5 At n/n 1 =1.0, H(n 1 ) = 1/2  Then, Dynamic amplification factor, H(n)

11. Basic structural dynamics II Response to random excitation : Consider an applied force with spectral density S F (n) : k c m F(t) Spectral density of displacement : |H(n)| 2 is the square of the dynamic amplification factor (mechanical admittance) Variance of displacement : see Lecture 5

12. Basic structural dynamics II Response to random excitation : Special case - constant force spectral density S F (n) = S o for all n (‘white noise’): The above ‘white noise’ approximation is used widely in wind engineering to calculate resonant response - with S o taken as S F (n 1 )

13. End of Lecture John Holmes 225-405-3789 JHolmes@lsu.edu