Pdf a comparative study between codes of spectrum for a. A response spectrum is a graphical relationship of maximum values of acceleration, velocity, andor displacement response of an infinite series of elastic single. Instructional material complementing fema p751, design. Equation b2 is generally known as the duhamel integral for an undamped system. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. Its origin probably dates from civil engineering and the problem of seismic excitations, i. In particular, the single ordinate of the spectrum, which represents the. Duhamel integral malab code matlab answers matlab central. If the input force is arbitrary, we can find the response spectrum only numerically. He defined the srs shock response spectrum as the maximum response motion from a set of single dof oscillators covering the frequency range.
Plotting response spectrum seismology engineering engtips. The duhamel s formula that i know of reads as follows. The response of a damped system expressed by the duhamels integral is ob. The shock response spectrum is a useful tool for estimating the damage potential of a shock pulse, as well as for test level specification. Aided structural engineering program managed by the information. But avoid asking for help, clarification, or responding to other answers. It may be used to evaluate the response of an undamped sdof system to any form of dynamic loading tp, although in the case of arbitrary loadings the evaluation will have to be performed numerically. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. Response spectra using duhamels integral in the above examples, the input force is simple and hence a closed form solution has been obtained for the response spectrum. Note that milstd1540c and milstd810e require this format for certain shock environments. Effect of damping on shock spectra of impulse loads. Development of shock spectra for different pulse shapes using.
Response spectrum analysis provides insight into dynamic behavior by measuring pseudospectral acceleration. A representation of the solution of the cauchy problem or of a mixed problem for an inhomogeneous linear partial differential equation with homogeneous boundary conditions by means of the solution of the corresponding problem for the homogeneous equation. An example of smooth spectra is shown in figure 4 based on a building code design response spectrum for a site of high seismicity. Multipoint response spectrum different response spectra excite different points in the model. Widely available spreadsheet software is typically equipped with sophisticated features, such as programmability and handling of complexvalued data even fourier analysis, rendering these productivity tools fully tenable for seismic site response.
How to solve the differential equation with duhamels integral. Introduction to the computation of response spectrum for. Response spectrum is a curve drawn for any response quantity of sdof. First off, a spectrum is a grouping of the maximum response of many analysis. Singlepoint response spectrum a single response spectrum excites all specified points in the model. A response spectrum is used to provide the most descriptive representation of the influence of a given earthquake on a structure or machine. For the love of physics walter lewin may 16, 2011 duration.
A response spectrum is a plot of the maximum response of a number of idealized single degreeoffreedom oscillators see figure 2. The accelerograph records three orthogonal components of ground acceleration. Response spectra file exchange matlab central mathworks. Mod02 lec05 steps of analysis using software contd 31. Duhamels method can also be used to obtain the response directly using convolution integral. This package includes a collection of routines for the creation and operation of nonuniform rational lagrange nurl function geometries and its applications to isogeometric analysis of inplane and flexural vibration of thin plates, based on the nurl toolbox by bo liu.
Mod01 lec17 numerical problems in single degree of. A response spectrum is a plot of the peak or steadystate response displacement, velocity or acceleration of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. May 29, 2018 duhamel integral contd, numerical integrations, shock spectrum. Although seismic site response analysis can be performed using readily available computer software, occasionally the person doing. Often, the spectral response for a range of periods is available for a local historical earthquake clough 1962 or, based on the seismic history of an area, an acceleration spectrum envelope is developed for use in structural design. Based on your location, we recommend that you select. Using an approximation to xt in the integral, we can integrate the ode to obtain a new approximation. The response spectra of in acceleration, velocity and or displacement allows to involve the notion of. Response spectrum analysis rsa is a lineardynamic statistical analysis method which measures the contribution from each natural mode of vibration to indicate the likely maximum seismic response of an essentially elastic structure. Thanks for contributing an answer to mathematica stack exchange. Duhamel s method can also be used to obtain the response directly using convolution integral. Response to an arbitrary excitation convolution integral duhamels integral draw on example 4.
Simple vibration problems with matlab and some help. Development of shock spectra for different pulse shapes. The equation is derived analytically by using the duhamel integral for the determination of the timehistory response of the oscillator. One can use picardtype iteration to get the solution. Responsespectrum analysis technical knowledge base. Choose a web site to get translated content where available and see local events and offers. Husid plot, significant duration and bracketed duration. Here we will show you an example of a computer program in fortran. However, for design purposes, calculation of the duhmel integral is not always necessary. The duhamels integral computer code was developed to analytically test instrumentation and structures for survivability and reliability in gun launch environments and to assess the damage to the human ear due to nonclassical pressure pulses and to determine the shock survivability of proposed structural designs. Introduction the most direct description of an earthquake motion in time domain is provided by accelerograms that are recorded by instruments called strong motion accelerographs. Lecture 23 on mechanical vibrationsstructural dynamicspm.
Nuclear power plant buildings and components, for seismic loading airborne electronic equipment for shock loading commercial buildings in earthquake zones types of response spectrum analysis. Stiffness matrix, free vibration shear building and its response. Spectrumtospectrum methods for the generation of elastic. The total solution dynamic load factor example ao determine the response of a 1dof system to the step function.
The purpose of this report is to explain the shock response spectrum and to give a. Duhamel s integral linear systems based on treating the periodic motion as a series of short impulses. An advantage of this formulation is the fact that a systems behavior can be described accurately without all the mode shapes. The response of the oscillator is derived analytically based on duhamel s integral. I found one good procedure for solving the simple system of two equations with reducing on duhamel s integral, but i have problem to apply the same procedure on system with four equations. How to solve the differential equation with duhamels. This paper deals with response spectrum method and its application to various types of the structures. It is also possible to calculate the response of a sdof to an arbitrary forcing function directly in the time domain using duhamels integral. My original thought was to step through the integration using the tools from tutorialndsolvestatedata to build an interpolation of xt at each step for use in the integration term. Duhamel structural dynamics and vibration control lab.
Sdof spectrum analysis perform sdof spectrum analysis procedure generates the response spectra of a single degree of freedom system via numerical evaluation of the duhamel integral. Spectral pseudovelocity sv expressed in terms of duhamels. For derivation of the response spectrum, a fortran computer program was prepared. To find the response of system they used duhamel s integral method. Once solved, simply transform back from to by using eq. Displacement, pseudovelocity and pseudoacceleration response spectra for any values of damping ratio. The computer program described in this section calculates the response of the. Structures subject to earthquake it is similar to a vehicle moving on the ground. Ei fu earthquake engineering northern architecture. Note that the response calculation is carried out via a convolution integral, as explained later in the text.
The equation of motion for singlydegreeoffreedom is solved by duhamel integral. Virtual bat algorithm for the computation of duhamels integral. Seismic response of a sdof oscillator mounted on a sdof primary structure is studied first. Free vibration of the system after the end of ground excitation is accounted for in the analysis. The latter is expressed as a function of the response of the mode and that of the oscillator to the ground motion excitation. 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. A displacement response spectrum is the plot of maximum displacement of a single degree of freedom system sdof to a particular ground motion as a function. In this study, a direct spectratospectra method is developed for accurate and efficient generation of frs.
They are linked to python by using the comfortable f2py tool. So, you construct this function, thetat, using the aforementioned initial conditions, and the solution to your ode is the sum of the solution to the associated homogeneous equation and the convolution of thetat and your forcing function i. Each sdof system has a unique time history response to a give base input. Posted to course website in pdf format for printing by students. Biots apparatus consisted of a set of torsion pendulum which simulated mechanical systems with one degree of freedom 3. Mod02 lec06 dynamic analysis of articulated towers. Response spectrum analysis for support excitation earthquake motions will be studied. A response spectrum is a plot of maximum response of a single degree of freedom system.
Go to response to general dynamic loading, impulse loading, duhamel s integral, constant force, rectangular force, and rectangular load. This gives response of the system due to any impulse force. Equation of motion is solved using the duhamel integral. Download duhamel integral matlab source codes, duhamel. The superposition process involves the evaluation of an integral called the convolution integral or duhamels integral. So if you have a spectrum of displacement for a particular modelloading, this means that for a model that has a period of 0. I want to calculate acceleration spectrum for a given record ground motion and specified period in matlab. The shock response spectrum is the peak absolute acceleration response of each sdof. The response of a linear, viscously damped singledegree of freedom sdof system to a timevarying mechanical excitation p t is given by. In theory of vibrations, duhamels integral is a way of calculating the response of linear systems and structures to arbitrary timevarying external perturbation. All faculty, staff and students of the university of florida are required and. Convolution integral classical mechanics applied mathematics.
It is nearly identical to the true total acceleration response spectrum for lightly damped structures. The peak displacement response of an undamped sdof system subjected to a. It is also possible to calculate the response of a sdof to an arbitrary forcing function directly in the time domain using duhamel s integral. Duhamels integral is a theoretical equation that allows calculation of the response of a linear system with a single. Why use spreadsheets for seismic site response analysis. Smallwood, see an improved recursive formula for calculating. Response spectra can also be used in assessing the response of linear systems with multiple modes of oscillation multidegree of freedom systems, although they are only accurate for low levels of damping.
Characteristics of spectral responses for a ground motion from. They have been plotted ten different time period for different frequency ratio. The superposition process involves the evaluation of an integral called the convolution integral or duhamel s integral. Consider the following equation of motion for a sdof system subjected to an arbitrary force pt. The book will also better equip structural engineering consultants and practicing engineers in the use of standard software for seismic analysis of buildings, bridges, dams, and towers. Generally the dynamic magnification factor dmf in the shock spectra is maximum for rectangular pulse followed by halfsine wave and kalpa publications in civil engineering.
From the definition of duhamel s integral, it is obvious that c t is the response of an oscillator with the circular frequency. The duhamel s integral method based upon linear force variation over time step has been explained in this paper, which is also the methodology to estimate the response of the systems in this study. Seismic analysis method the maximum displacement of the sdof system having. Duhamel integral convolution, dynamic analysis by modal superposition, spectral analysis in earthquake engineering, dynamic response. B 5 numerical evaluation of the duhamel integral for an. The shock response spectrum srs allows to estimate transient acceleration signals in terms of a maximum response of a dynamic vibration system. I would like to know, if some can help me, how do i plot the response spectrum for spectral acceleration v period, sa v t. In theory of vibrations, duhamel s integral is a way of calculating the response of linear systems and structures to arbitrary timevarying external perturbation. Mod02 lec10 response control of multilegged articulated towers using tuned mass. A response spectrum is a graphical relationship of maximum values of acceleration, velocity, andor displacement response of an infinite series of elastic single degree of freedom sdof systems subjected to.