53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 1 - Lecture 17: Response Spectra Reading materials: Sections 6.1, 6.2, and 6.3 1. Concepts In practical dynamic analysis situations we are interested in the maximum response. The graph showing the variation of the maximum response (maximum displacement, velocity, acceleration, or any other quantity) with the natural frequency (or natural period) of a single degree of freedom system to a specified forcing function is known as the response spectrum. A response spectrum is a plot of maximum response of a single degree of freedom system subject to a specific input, such as step loading and triangular pulse versus period of vibration or another suitable quantity. Example: Response spectra for a rectangular pulse loading T: fundamental period of the structure umax: maxium deflection over time ustatic: deflection if load F is treated as a static load Гmax: maximum dynamic load magnification factor53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 2 - 2. Response Spectrum of Sinusoidal Pulse Find the response spectrum for the sinusoidal pulse force using the initial conditions x(0)=v(0)=053/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 3 -53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 4 - 3. Usage of Response Spectrum SDOF systems Period of vibration: Maximum displacement MDOF systems Equations of motion Undamped free vibration mode shapes and frequencies Modal coordinates Damped modal equations53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 5 - Solution We know Here, 4. Response spectra using Duhamel’s integral In the above examples, the input force is simple and hence a closed form solution has been obtained for the response spectrum. If the input force is arbitrary, we can find the response spectrum only numerically. The peak displacement response of an undamped SDOF system subjected to a given load F(t) can be expressed via Duhamel’s integral Loading phase: t < td53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 6 - where F0 is the load magnitude. Free vibration phase: t >= td where utd and vtd are displacement and velocity at the end of the forced vibration phase. Rectangular Pulse53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 7 -53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 8 - Step force with ramp: maximum DLF occurs at the constant load phase.53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 9 - 5. Response spectra using Numerical Integration It is difficult to determine simple analytical expressions for maximum DLF for complicated loading. Equation of motion for a SDOF system Let then53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 10 - Half Sine Pulse Equation of motion53/58:153 Lecture 17 Fundamental of Vibration ______________________________________________________________________________ - 11