Seismic Qualification of Structures, Systems, and Components by Test

Abstract

The general methods of seismic qualification of structures, systems, and components (SSCs) are grouped into the following categories.

Appendices

Appendix 1: Numerical Computation of Fourier Coefficients

For every simple forms of function x(t), the integrals can be evaluated easily.

$$ a_{0} = \frac{2}{T}\mathop \smallint \limits_{0}^{T} x\left( t \right)dt $$

(12.25)

$$ a_{n} = \frac{2}{T}\mathop \smallint \limits_{0}^{T} x\left( t \right)\cos \,n\omega t\,dt $$

(12.26)

$$ b_{n} = \frac{2}{T}\mathop \smallint \limits_{0}^{T} x\left( t \right)\sin \,n\omega t\,dt $$

(12.27)

However, the integration becomes involved if x(t) does not have a simple form. In some practical applications as with the case of experimental determination of the amplitudes of vibration transducers, the function x(t) is not available in the form of a mathematical expression; only the values of x(t) at a number of points t₁, t₂, … t_n are available as shown in Fig. 12.40.

Fig. 12.40

Pressure fluctuations of water in a pipe

In these cases, the coefficients a₀, a_n, and b_n of Eqs. (12.25)–(12.27) can be evaluated using a numerical integration procedure like trapezoidal or Simpson’s rule.

If t₁, t₂, … t_n are assumed to be an even number of equidistant point over period τ (N is also even) with the corresponding values $x_{1} = x\left( {t_{1} } \right)$, $x_{2} = x\left( {t_{2} } \right), \ldots x_{n} = x\left( {t_{n} } \right)$, respectively. The application of trapezoidal rule gives the coefficients a₀, a_n, and b_n (by setting T = N ∆t)

$$ a_{0} = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} x_{i} $$

(12.28)

$$ a_{n} = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} x_{i} \cos \frac{{2n\pi t_{i} }}{\tau } $$

(12.29)

$$ b_{n} = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} x_{i} \sin \frac{{2n\pi t_{i} }}{\tau } $$

(12.30)

Example 12.6

The pressure fluctuations of water in a pipe measured at 0.01 s intervals are given in Table 12.4. These fluctuations are repetitive in nature. Make a harmonic analysis of the pressure fluctuations, and determine the first three harmonics of the Fourier series expansion.

Table 12.4 Pressure fluctuations

Solution:

Since the given fluctuations repeat every 0.12 s, the period is τ = 0.12 s and circular frequency of the first harmonics is 2π radians per 0.12 s or ω = 2π/0.12. As the observed values in each wave is 12, a₀ can be obtained from Eq. (12.3)

$$ a_{0} = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} p_{i} = \frac{1}{6}\mathop \sum \limits_{i = 1}^{12} p_{i} = 79.50 $$

The coefficients a_n and b_n can be determined from Eqs. (12.4) and (12.5), respectively.

$$ \begin{aligned} a_{n} & = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} p_{i} \cos \frac{{2n\pi t_{i} }}{\tau } = \frac{1}{6}\mathop \sum \limits_{i = 1}^{12} p_{i} \cos \frac{{2n\pi t_{i} }}{0.12} \\ b_{n} & = \frac{2}{N}\mathop \sum \limits_{i = 1}^{N} p_{i} \sin \frac{{2n\pi t_{i} }}{\tau } = \frac{1}{6}\mathop \sum \limits_{i = 1}^{12} p_{i} \sin \frac{{2n\pi t_{i} }}{0.12} \\ \end{aligned} $$

With these coefficients and using Table 12.5, the Fourier series expansion of the pressure fluctuations p(t) can be obtained as

Table 12.5 Evaluation of various terms

$$\begin{aligned} p\left( t \right) & = 79.50/2 - 9.80\cos 52.36t - 0.82 \, \sin \, 52.36t - 27.67\cos 104.72t \\ &\quad + 0.29{ \sin }104.72t - 0.50\cos 157.08t + 0.0\sin 157.08t + \ldots \\ \end{aligned}$$

Considering three harmonics, approximate p(t) can be obtained as follows (Table 12.6).

Table 12.6 Approximate pressure variation with time

Comparison of actual pressure and approximate pressure with three Fourier harmonics considered is shown in Fig. 12.41.

Fig. 12.41

Actual pressure and approximate pressure with three Fourier harmonics

Appendix 2: Experimental Modal Analysis

12.2.1 A12.2.1 Introduction

Modal testing is the form of vibration testing of an object whereby the natural (modal) frequencies, modal masses, modal damping ratios, and mode shapes of the object under test are determined. A modal test consists of an acquisition phase and an analysis phase. The complete process is often referred to as a modal analysis or experimental modal analysis . The modal analysis provides a set of modal parameters that characterize the dynamic behavior of a structure.

There are several ways to do modal testing, but impact hammer testing and shaker testing are most common. The fundamental problem of modal parameter estimation consists of adjusting (estimating) the parameters in the model, so that the data predicted by the model approximates the measured data as closely as possible.

12.2.2 A12.2.2 Curve Fitting Methods for Estimation of Modal Parameters

Curve fitting is a numerical process that is typically used to represent a set of experimentally measured data points by some assumed analytical function. The results of this curve fitting process are the coefficients, or parameters, that are used in defining the analytical function. With regard to the frequency response function, the parameters that are calculated are its modal parameters (i.e., modal frequency , damping , and residue).

Although there are several ways in which curve fitting methods can be categorized, the most straightforward is single-mode versus multiple-mode classification.

12.2.3 A12.2.3 Single-Mode Methods

The basic assumption for single-mode approximations is that in the vicinity of a resonance , the response is due primarily to that single mode. The resonant frequency can be estimated from the frequency response data by observing the frequency at which any of the following trends occur:

• The magnitude of the frequency response is a maximum.

• The imaginary part of the frequency response is a maximum or minimum.

• The real part of the frequency response is zero.

• The response lags the input by 90° phase.

Example 12.7

Estimate the natural frequency of piping loop shown in Fig. 12.42 by testing.

Fig. 12.42

Test setup for shake table testing of piping loop

Solution:

Resonance search tests were carried out using the sine sweep signal as shown in Fig. 12.43 to estimate the natural frequency . The sweep rate used is 1 octave per minute. Starting initial frequency is 1 Hz; the sweep was carried out till 20 Hz which is the range of interest. Acceleration time histories (responses) have been measured at different locations. Response of the piping loop at location 6 is shown in Fig. 12.44.

Fig. 12.43

Sine sweep signal

Fig. 12.44

Response of the piping loop at location 6

Single-mode methods have been used to estimate the natural frequency of the piping loop. The real part of frequency response function (FRF) is shown in Fig. 12.45. From Fig. 12.45, it can be seen that real part of FRF is zero at 3.65 Hz, so the natural frequency is 3.65 Hz.

Fig. 12.45

Real part of frequency response function (FRF)

Appendix 3: Experimental Study for Evaluating Coherency of Response and Input Signal

An example of piping system shown in Fig. 12.46 is considered for evaluating coherence of response signal and excitation signal. Response acceleration was measured at various points as shown in Fig. 12.46.

Fig. 12.46

Location of accelerometers on piping system and view of piping system at shake table

The input signal has a sine sweep nature with starting frequency of 1 Hz and ending at 20 Hz with sweep rate of 1 octave per minute. Correlations and coherence between input signals along X and Y directions for locations A7X, A8Y, A10X, and A11Y have been found out, and the results are plotted in Figs. 12.47, 12.48, 12.49 and 12.50 respectively.

Fig. 12.47

a Input acceleration along X-direction b Response acceleration along X-direction c Power spectrum of input acceleration along X-direction d Power spectrum of response acceleration along X-direction e Coherency between input and response acceleration along X-direction

Fig. 12.48

a Input acceleration along X-direction b Response acceleration along X-direction c Power spectrum of input acceleration along X-direction d Power spectrum of response acceleration along X-direction e Coherency between input and response acceleration along X-direction

Fig. 12.49

a Input acceleration along Y-direction b Response acceleration along Y-direction c Power spectrum of input acceleration along Y-direction d Power spectrum of response acceleration along Y-direction e Coherency between input and response acceleration along Y-direction

Fig. 12.50

a Input acceleration along Y-direction b Response acceleration along Y-direction c Power spectrum of input acceleration along Y-direction d Power spectrum of response acceleration along Y-direction e Coherency between input and response acceleration along Y-direction

A.3.1 Coherence Between Input and A7X

Time history of input is shown in Fig. 12.47a.

Time history of A7X is shown in Fig. 12.47b:

Power spectrum of input signal is shown in Fig. 12.47c.

Power spectrum of A7X is shown in Fig. 12.47d.

Observation: The predominant frequency of the system is around 15.0 Hz as identified from the plot of Fourier spectrum for the output accelerometer reading. This frequency is to be verified using “coherency” between the input and output signals as shown in Fig. 12.47e.

It is seen that the predominant frequency of the system is found to be 15.0 Hz (in which coherency value is close to unity) which is in line with the frequency obtained by the frequency domain plot of the output time history . Hence, the fundamental frequency of the system is validated by both the approaches, which confirms the correctness of the approach.

A.3.2 Coherence Between Input and A10X

Time history of input is shown in Fig. 12.48a.

Time history of A10X is shown in Fig. 12.48b.

Power spectrum of input signal is shown in Fig. 12.48c.

Power spectrum of A10X is shown in Fig. 12.48d.

Observation: The predominant frequency of the system is around 15.0 Hz as identified from the plot of power spectrum for the output accelerometer reading. This frequency is to be verified using “coherency” between the input and output signals as shown below. Coherence between input and A10X signals is shown in Fig. 12.48e.

It is seen that the predominant frequency of the system is found to be 15.0 Hz (in which coherency value is close to unity) which is in line with the frequency obtained by the frequency domain plot of the output time history . Hence, the fundamental frequency of the system is validated by both the approaches, which confirms the correctness of the approach.

A.3.3 Coherence Between Input and A11Y

Time history of input is shown in Fig. 12.49a.

Time history of A11Y is shown in Fig. 12.49b.

Power spectrum of input signal is shown in Fig. 12.49c.

Power spectrum of A11Y is shown in Fig. 12.49d.

Observation: The predominant frequency of the system is around 5.78 Hz as identified from the plot of power spectrum for the output accelerometer reading. This frequency is to be verified using “coherency” between the input and output signals as shown in Fig. 12.49e.

It is seen that the predominant frequency of the system is found to be 5.0 Hz (in which coherency value is close to unity) which is in line with the frequency obtained by the frequency domain plot of the output time history . Another peak is observed at 13.25 Hz which represents the second mode of the piping system in Y-direction which is also seen in the power spectrum plot.

A.3.4 Coherence Between Input and A8Y

Time history of input is shown in Fig. 12.50a.

Time history of A8Y is shown in Fig. 12.50b.

Power spectrum of input signal is shown in Fig. 12.50c.

Power spectrum of A8Y is shown in Fig. 12.50d.

Observation: The predominant frequency of the system is around 5.0 Hz as identified from the plot of power spectrum for the output accelerometer reading. This frequency is to be verified using “coherency” between the input and output signals as shown in Fig. 12.50e.

It is seen that the predominant frequency of the system is found to be 5.0 Hz (in which coherency value is close to unity) which is in line with the frequency obtained by the frequency domain plot of the output time history . Another peak is observed at 13 Hz which represents the second mode of the piping system in Y-direction which is also seen in the power spectrum plot. Cross-correlation (or coherency) between signals A8Y and A10X in the piping system is shown in Fig. 12.51.

Fig. 12.51

Coherency between response acceleration along X and Y directions

As it can be seen that there is no single frequency at which there is coherence greater than 0.9. It is because these modes are mutually perpendicular in direction; hence, there is little correlation between them as can be seen in the plot above as the modes are significantly uncoupled. The need of this study is to identify and validate the natural frequencies of the piping system which is given in Table 12.7.

Table 12.7 Natural frequencies of the piping system

Appendix 4: Program to compute covariance of two time signals

clear all

clc

x=xlsread('C:\Users\Admin\Desktop\Acc_time.xlsx',1,'BQ:BR');

x=x';

N=length(x);

Rxx=zeros(1,N);

Rxy=zeros(1,N);

Ryy=zeros(1,N);

for m=1: N+1

for n=1: N-m+1

Rxx(m)=Rxx(m)+x(n,1)*x(n+m-1,1);

Rxy(m)=Rxy(m)+x(n,2)*x(n+m-1,1);

Ryy(m)=Ryy(m)+x(n,2)*x(n+m-1,2);

end

end

figure (1);

plot(x(:,1))

figure (2);

plot(x(:,2))

figure (3);

plot(Rxx)

figure (4);

plot(Rxy)

figure (5);

plot(Ryy)

Sxx=(fft(Rxx));

Sxy=(fft(Rxy));

Syy=(fft(Ryy));

%--Computation of magnitude square coherence between two signals (x & y)--

Cxy = mscohere(x,y)