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A response spectrum is a plot of the peak or steady-state 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. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. One such use is in assessing the peak response of buildings to earthquakes. The science of strong ground motion may use some values from the ground response spectrum (calculated from recordings of surface ground motion from seismographs) for correlation with seismic damage.

If the input used in calculating a response spectrum is steady-state periodic, then the steady-state result is recorded. Damping must be present, or else the response will be infinite. For transient input (such as seismic ground motion), the peak response is reported. Some level of damping is generally assumed, but a value will be obtained even with no damping.

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Response spectra can also be used in assessing the response of linear systems with multiple modes of oscillation (multi-degree of freedom systems), although they are only accurate for low levels of damping. Modal analysis is performed to identify the modes, and the response in that mode can be picked from the response spectrum. These peak responses are then combined to estimate a total response. A typical combination method is the square root of the sum of the squares (SRSS) if the modal frequencies are not close. The result is typically different from that which would be calculated directly from an input, since phase information is lost in the process of generating the response spectrum.

The main limitation of response spectra is that they are only universally applicable for linear systems. Response spectra can be generated for non-linear systems, but are only applicable to systems with the same non-linearity, although attempts have been made to develop non-linear seismic design spectra with wider structural application. The results of this cannot be directly combined for multi-mode response.

Response spectra are very useful tools of earthquake engineering for analyzing the performance of structures and equipment in earthquakes, since many behave principally as simple oscillators (also known as single degree of freedom systems). Thus, if you can find out the natural frequency of the structure, then the peak response of the building can be estimated by reading the value from the ground response spectrum for the appropriate frequency. In most building codes in seismic regions, this value forms the basis for calculating the forces that a structure must be designed to resist (seismic analysis).

As mentioned earlier, the ground response spectrum is the response plot done at the free surface of the earth. Significant seismic damage may occur if the building response is 'in tune' with components of the ground motion (resonance), which may be identified from the response spectrum. This was observed in the 1985 Mexico City Earthquake[1] where the oscillation of the deep-soil lake bed was similar to the natural frequency of mid-rise concrete buildings, causing significant damage. Shorter (stiffer) and taller (more flexible) buildings suffered less damage.

In 1941 at Caltech, George W. Housner began to publish calculations of response spectra from accelerographs.[1] In the 1982 EERI Monograph on "Earthquake Design and Spectra",[2] Newmark and Hall describe how they developed an "idealized" seismic response spectrum based on a range of response spectra generated for available earthquake records. This was then further developed into a design response spectrum for use in structural design, and this basic form (with some modifications) is now the basis for structural design in seismic regions throughout the world (typically plotted against structural "period", the inverse of frequency). A nominal level of damping is assumed (5% of critical damping).

For "regular" low-rise buildings, the structural response to earthquakes is characterized by the fundamental mode (a "waving" back-and-forth), and most building codes permit design forces to be calculated from the design spectrum on the basis of that frequency, but for more complex structures, combination of the results for many modes (calculated through modal analysis) is often required. In extreme cases, where structures are either too irregular, too tall or of significance to a community in disaster response, the response spectrum approach is no longer appropriate, and more complex analysis is required, such as non-linear static or dynamic analysis like in seismic performance analysis technique.

Response-spectrum analysis (RSA) is a linear-dynamic 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. Response-spectrum analysis provides insight into dynamic behavior by measuring pseudo-spectral acceleration, velocity, or displacement as a function of structural period for a given time history and level of damping. It is practical to envelope response spectra such that a smooth curve represents the peak response for each realization of structural period.

Response-spectrum analysis is useful for design decision-making because it relates structural type-selection to dynamic performance. Structures of shorter period experience greater acceleration, whereas those of longer period experience greater displacement. Structural performance objectives should be taken into account during preliminary design and response-spectrum analysis.

Response spectrum analysis is a method to estimate the structural response to short, nondeterministic, transient dynamic events. Examples of such events are earthquakes and shocks. Since the exact time history of the load is not known, it is difficult to perform a time-dependent analysis. Due to the short length of the event, it cannot be considered as an ergodic ("stationary") process, so a random response approach is not applicable either.

The response spectrum method is based on a special type of mode superposition. The idea is to provide an input that gives a limit to how much an eigenmode having a certain natural frequency and damping can be excited by an event of this type.

A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The response spectrum is a function of the natural frequency of the oscillator and of its damping. Thus, it is not a direct representation of the frequency content of the excitation (as in a Fourier transform), but rather of the effect that the signal has on a postulated system with a single degree of freedom (SDOF).

For given values of , , and , this equation can be solved for a sufficiently long time. The displacement, velocity, and acceleration response spectra are defined as the maximum values caused by the acceleration history .

For a system without damping, the pseudoacceleration spectrum based on the relative displacement is actually equal to the absolute acceleration spectrum. This can be seen from the undamped equation of motion,

The absolute acceleration has a maximum at frequencies similar to the frequency content of the input signal. In this case, the peak is at 74 Hz (T = 13.5 ms). For high frequencies, the acceleration spectrum tends toward 50 g. This is a general observation for any signal: At high frequencies, the oscillator will behave as a rigid body, so the mass just follows the base motion. As an effect, the asymptotic value of the absolute acceleration spectrum always equals the peak base acceleration during the event.

Next, the relative velocity spectrum and the pseudovelocity spectrum are compared. As can be seen, they are quite different. The pseudovelocity and pseudoacceleration spectra do not represent the true relative spectra. This is a general observation, and the pseudo spectra should be viewed as different representations of the displacement spectrum.

A convenient way to represent a response spectrum is in a tripartite, or four-axis plot. In such plot, the relative displacement, pseudovelocity, and pseudoacceleration are shown simultaneously. This is possible, since they are related by a factor of frequency and frequency squared, respectively, which in a logarithmic plot just gives lines with different slopes. The tripartite plot is essentially a pseudovelocity plot but with two extra sets of skewed grid lines that represent the displacement and acceleration, respectively.

The response spectrum for the half sine pulse is actually somewhat atypical. The reason is that this pulse only has positive acceleration. If it is integrated with respect to time, such pulse corresponds to a resulting nonzero velocity and an ever-increasing displacement. Most events, like earthquakes, have the property that both displacement and velocity are zero both before and after the event. If a complete sine pulse is used instead of a half sine pulse, the characteristic low-frequency decay is also obtained.

The response spectra exhibit some interesting general properties. Higher damping will give lower response values and a smoother spectrum. Both these properties are related to the fact that the frequency response of an oscillator will have lower but wider peaks at higher damping.

The response spectrum of a single time signal is seldom of interest for an analysis, since it would be better to perform a direct time domain analysis of the structure with the original signal as input. As seen in the El Centro example above, a certain earthquake may give a response spectrum with significant peaks at certain frequencies. The peaks for another similar earthquake may, however, be located at other frequencies.

Design response spectra are often provided in terms of the period, rather than the frequency. Since one is the inverse of the other, the two graphs are just mirrored when plotting on a logarithmic scale. 2351a5e196