What Is a Response Spectrum?

In engineering engineering, it is often required to know the maximum response value of the system after the impact load, that is, the maximum value of the displacement or acceleration of the vibration. Because the action time is short, the damping of the system is usually ignored when calculating the maximum response value, making the calculation result more safe. The relationship curve between the maximum response value and a certain parameter of the excitation (such as the time of the excitation) is called the response spectrum. ^[1]

The response spectrum is a graph of the relationship between the maximum response value of a system under a given stimulus and a certain parameter of the system or stimulus. The maximum response value can be the system's maximum displacement, maximum speed, maximum stress, or the moment when the maximum value occurs; the parameters can be selected as the natural frequency of the system or the action time of the excitation. The relevant quantities in the response spectrum are reduced to

Response spectrum analysis is an approximate algorithm for calculating the peak response of a structure under transient excitation. It is an extension of modal analysis. The application of response spectrum analysis is very wide. The most typical application is seismic response spectrum analysis in the civil industry. Response spectrum analysis is a standard analysis method for seismic analysis. It is used in seismic analysis of various structures, such as dams, power stations, and bridges. , Building construction, etc. Any structure or component subjected to earthquake or other vibration load can be checked by response spectrum analysis.

The response latent analysis is a frequency domain analysis. The input load is the frequency spectrum of the vibration load, such as seismic frequency spectrum, acceleration frequency spectrum, velocity spectrum, displacement frequency spectrum, etc. The response spectrum analysis calculates the peak response of the structure from the frequency domain perspective.

Load spectrum is defined as the relationship between the response amplitude and frequency. The response spectrum analysis calculates the maximum response of each mode shape of a structure under a given load spectrum. This maximum response is the product of the response coefficient and the mode shape. These mode shapes The combination of the maximum responses gives the overall response of the structure. The response spectrum analysis needs to first calculate the natural frequency and mode shape of the structure, and it must be performed after the natural frequency (modal) analysis.

An alternative method of response spectrum analysis is transient analysis. Transient analysis can obtain changes in the structure response at any time. Transient analysis is more accurate, but it takes more time. Response spectrum analysis ignores some information (such as phase, time history, etc.), but can quickly find the maximum response of the structure, which meets the requirements of many dynamic designs. ^[3]

figure 1

Undamped-mass system, subject to the rectangular pulse shown in Figure 1 on the right,

.

The response of the system is

among them,

For static

Deform the spring. Derivating the above formula, we get

make

:

for

, when

When x gets the maximum value for the first time

;

for

, when

At x

.

From the analysis above, when

At

At x

;when

At

Time,

It is always positive, x (t) monotonically increases, and it is impossible to obtain an extreme value of displacement. Its extreme value appears in the range of t> t ₁ and is equal to the amplitude of the residual vibration, that is,

. Therefore, the response spectrum of an undamped system to a rectangular pulse can be summarized as

If the

,

Is the ordinate,

For the abscissa, displacement response spectrum curve and time response spectrum curve can be obtained from the above formula, as shown in Figure 2 and Figure 3, respectively. ^[2]

figure 2

image 3