Wind loading and structural response Lecture 10 Dr

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

Basic structural dynamics I • Topics : • Revision of single degree-of freedom vibration theory • Response to sinusoidal excitation • Response to random excitation • Multi-degree of freedom structures – Lect. 11 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

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

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

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

Basic structural dynamics I • Single degree of freedom system : Free vibration following an initial displacement : c x m k Initial displacement = Xo

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

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

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

Basic structural dynamics I • Single degree of freedom system : Dynamic amplification factor, H(n) =0. 01 Critical damping ratio – damping controls amplitude at resonance 100. 0 =0. 05 10. 0 =0. 1 H(n) =0. 2 1. 0 0 =0. 5 1 2 3 0. 1 n/n 1 At n/n 1 =1. 0, H(n 1) = 1/2 Then, 4

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

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

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