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

Basic structural dynamics ITopics :

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 1981J.D. Holmes Wind Loading of Structures 2001Multi-degree of freedom structures Lect. 11

Response to random excitation

Basic structural dynamics IEquation of free vibration :Example : mass-spring-damper system :kcmass acceleration = spring force + damper forceequation of motionSingle degree of freedom system :

Basic structural dynamics ISingle degree of freedom system :

Equation of free vibration :Example : mass-spring-damper system :Ratio of damping to critical c/cc : often expressed as a percentage

Basic structural dynamics ISingle degree of freedom system :

Damper removed :Undamped natural frequency :Period of vibration, T :

Basic structural dynamics ISingle degree of freedom system :

Initial displacement = XoFree vibration following an initial displacement :

Basic structural dynamics ISingle degree of freedom system :

Free vibration following an initial displacement :

Basic structural dynamics ISingle degree of freedom system :

Free vibration following an initial displacement :

Basic structural dynamics ISingle degree of freedom system :

Response to sinusoidal excitation :Equation of motion :Steady state solution : = 2n

Basic structural dynamics ISingle degree of freedom system :

Critical damping ratio damping controls amplitude at resonanceAt n/n1 =1.0, H(n1) = 1/2Dynamic amplification factor, H(n)

Basic structural dynamics IIResponse to random excitation :

Consider an applied force with spectral density SF(n) :Spectral density of displacement :|H(n)|2 is the square of the dynamic amplification factor (mechanical admittance)Variance of displacement :see Lecture 5

Basic structural dynamics IIResponse 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(n1)

End of Lecture John Holmes225-405-3789 JHolmes@lsu.edu