Correlation Analysis

Suppose there are two time histories, x_i(t) and x_j(t). The x_i(t) is a forcing function taken from loca-tion i and x_j(t) represents the corresponding structural response taken from location j. Similar to the computation of mean value, for a general random process, the cross-correlation function of x_i(t) and x_j(t) is defined by the ensemble average, written as

Rx x t t n n x t x ti j

and the autocorrelation function is defined by the ensemble average

R t t

Similar to the case of mean value computation, to investigate the statistical properties of the ran-dom process, Equations 2.225 and 2.226 require quite a collection of time histories x_i and x_j. Thus, in many cases, the temporal average is used instead of an ensemble average.

TABLE 2.1A

0.74 0.80 0.28 14.04 63.47 86.48

TABLE 2.1B

0.45 0.06 0.36 45.65 45.05 46.38

Suppose two pieces of sampled time histories are measured from a random process and are denoted as x₁(t) and x₂(t). The method used to correlate these two time histories is correlated based on

R

Here, for simplicity, R12 is used instead of Rx x1 2, to denote the cross-correlation function.

Mathematically, Equation 2.225 can be rewritten as

R

Since the above equations describe the relationship between the time histories of x₁(t) and x₂(t), it is referred to as the cross-correlation function.

In this particular case, when the time history x₂(t) is chosen to be x₁(t), the result is the auto-correlation function, which can be written as

R

The more general form is

R11

( )

τ ⁼ p t x t x t

( ) ( )

τ 1 1

(

⁺τ

)

dt

−∞

∫

∞ ^{, (2.229b)}

Here, p(t, τ) is the probability of the occurrence of the production of x1 taken at time t and t + τ.

From both the cross- and autocorrelation functions, it is seen that the resulting quantities from the integrations are functions of τ. The variable τ in the correlation domain is similar to the tem-poral variable t in the real-world time domain. When τ is chosen from zero to a rational value, the resulting integrals R₁₂(τ) or R11(τ) represent a deterministic process.

Note that in the above equations, the time histories x₁(t) and x₂(t) can be selected from a random process. After they are chosen, x₁(t) and x₂(t) are deterministic. The time histories x₁(t) and x₂(t) can also be known as periodic or transient signals, which are, of course, deterministic. Thus, the cor-relation analysis is not limited to random processes. However, in the following discussion, it will be seen that using the correlation functions for random processes will have additional limitations.

Further, when the time average is used to define the correlation functions, care should be taken to check whether the signals are ergodic.

In Figure 2.15a, the autocorrelation function of sin(ωt) is plotted. Here, ω = 2π (rad), which is the natural frequency of the system mentioned above. In Figure 2.15b, the cross-correlation function of sin(ωt), which is used as harmonic excitation, and the response displacement is plotted for the

SDOF system mentioned above. Furthermore, Figure 2.15c presents the autocorrelation function of [sin(ωt/5) + cos(ωt/3)/2]. Additionally, Figure 2.15d presents the cross-correlation function of [sin(ωt/5) + cos(ωt/3)/2], which is used as harmonic excitation, and the corresponding response displacement of the same SDOF system.

In addition, in Figure 2.15e, the autocorrelation function of the El Centro earthquake record is plotted, while Figure 2.15f provides the cross-correlation function of the El Centro earthquake, which is used as ground excitation, and the corresponding response displacement of the SDOF system.

From Figure 2.15, all these correlation functions look like free-decay time histories. That is, when the temporal variable t becomes quite large, the correlation functions tend toward zero. It is seen that for harmonic functions, the correlation functions decay slowly. For random excitation, the correla-tion funccorrela-tion will decay rather fast. In fact, it can be proven that for any time histories x₁(t) and x₂(t),

R11 R11 E x t1 2 x

0 12

( )

τ ^<

( )

^{= ⎡}_⎣⎢

( )

^⎤_⎦⎥⁼ _(2.230)

In the following discussion, it is seen that R11(0) actually stands for the average power of the process xR(t). Furthermore,

R12 R R

2

11 0 22 0

( )

τ ^<

( ) ( )

(2.231)

Correlation functions are very helpful tools in analyzing the random processes. For convenience, these functions are denoted as

0.50.4

Cross-correlation function of El Centro earthquake and SDOF response 1.2

Cross-correlation function of sin(ωt) and SDOF response

(c)

Autocorrelation function of sin(ωt/5)+cos(ωt/3)/2

30τ ^{40 50 60} (d)

Amplitude

Cross-correlation function of sin(ωt/5)+cos(ωt/3)/2 and SDOF response

τ

Autocorrelation function of El Centro earthquake

τ

R11

( )

τ ⁼ x x1, 1 (2.232) and

R12

( )

τ ⁼ x x1, 1 (2.233)

It can also be proven that the autocorrelation function is an even function of τ. That is,

R11

( )

−^{τ =}R11

( )

^τ (2.234)

On the other hand, the cross-correlation function is neither an even nor an odd function. However,

R12

( )

−^{τ =}R21

( )

^τ (2.235)

Because of the nature of the correlation function described in Equations 2.234 and 2.235, in practical computations, only the values of the correlation functions are calculated when

τ ≥0 (2.236)

Suppose a signal contains several frequency components, which can be expressed as trigo-nometric functions, say, the frequencies are ω1, ω2, and ω3. Also, suppose the signal has some random terms. From the definition of the autocorrelation function in Equation 2.229, it is real-ized that by using integration, the frequency components ω1, ω2, and ω3 will remain in the cor-relation function, and the random terms will vanish due to the orthogonality of the trigonometric functions. Thus, by using the autocorrelation function, these particular frequency components can be emphasized and random terms can be eliminated. Suppose two signals share the same frequency components, which can be expressed as trigonometric terms, say, the frequencies are ω1, ω2, and ω3. Also, suppose the signals have different frequency components and some random terms. From the definition of the cross-correlation function in Equation 2.228, it can also be realized that by using integration, the frequency components ω1, ω2, and ω3 will remain in the cross-correlation function, and the frequency terms that belong to individual signals and the random terms will also vanish, due to the orthogonality of trigonometric functions. Thus, by using the cross-correlation function, these particular frequency components shared by both signals can be emphasized and the frequency terms of the individual signals and the random terms can be eliminated.

The above points can be realized by examining the plots of the earthquake responses given in Figure 2.15.

In document Structural Damping (Page 139-142)