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Procedures

S ^ {D} = \frac {1}{\omega} S ^ {V} = \frac {1}{\omega^ {2}} S ^ {A}.

In ABAQUS/Standard it is assumed that the damping is always small, so these relationships are used whenever a conversion is needed.

A response spectrum is defined in the *SPECTRUM option by giving a table of values of S at increasing values of frequency, \omega , , for increasing values of damping, ». Linear interpolation on a logarithmic scale is used to compute the response for any required frequency and damping factor. Any number of spectra can be defined.

The *RESPONSE SPECTRUM procedure allows up to three spectra, which we denote by k, k = 1 , 2 , 3 , to be applied to the model in orthogonal physical directions defined by their direction cosines, \mathbf { t } ^ { k } . These spectra can come from different excitations (with a certain level of correlation between them), or they can be components of a single base excitation acting in an arbitrary direction.

When modal methods are used to define a model's response, the value of any physical variable is defined from the amplitudes of the modal responses (the "generalized coordinates"), q _ { \alpha } . The first stage in the response spectrum procedure is to estimate the peak values of these modal responses. For mode ® and spectrum k this is

(q _ {\alpha} ^ {\mathrm{max}}) _ {k} = c _ {k} S _ {k} ^ {D} \sum_ {j} t _ {j} ^ {k} \Gamma_ {\alpha j},

where c _ { k } is the scaling parameter defined in the *RESPONSE SPECTRUM option, S _ { k } ^ { D } ( \omega , \xi ) is the kth displacement spectrum, t _ { j } ^ { k } is the jth direction cosine for the kth spectrum, and \Gamma _ { \alpha j } is the participation factor for mode ® in direction j (see ``Variables associated with the natural modes of a model, '' Section 2.5.2, for the definition of \Gamma _ { \alpha j } ) . Similar expressions for ( { \dot { q } } _ { \alpha } ^ { \operatorname* { m a x } } ) _ { k } and ( { \overbrace { q } } _ { \alpha } ^ { \operatorname* { m a x } } ) _ { k } are obtained by using velocity or acceleration spectra in the above formula.

We now have estimates of the peak responses of the "generalized coordinates"--the amplitudes of the responses of the natural modes of the system for excitation in each direction. If the input spectra in the different directions are components of a single base excitation acting in an arbitrary direction, for each mode we combine these peak responses into a single value by algebraic summation of the values for the different spatial directions (specified by using COMP=ALGEBRAIC on the *RESPONSE SPECTRUM option):

q _ {\alpha} ^ {\mathrm{max}} = \sum_ {k} (q _ {\alpha} ^ {\mathrm{max}}) _ {k}.

In this case the modal combinations discussed below still apply, but the subscript k is no longer relevant and should be ignored.

Let us denote by ¡ Ri®(!; ») ¢mak \left( R _ { \alpha } ^ { i } ( \omega , \xi ) \right) _ { k } ^ { \mathrm { { m a x } } } the peak response of some physical variable R ^ { i } (a component of displacement, stress, section force, reaction force, etc.) caused by motion in the natural mode ® excited by the response spectrum in excitation direction k at frequency ! and with damping ». Denote the

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component of the eigenvector ® associated with R _ { \alpha } ^ { i } by \Phi _ { \alpha } ^ { i } . Then

(R _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}} = \Phi_ {\alpha} ^ {i} (q _ {\alpha} ^ {\mathrm{max}}) _ {k},

(\dot {R} _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}} = \Phi_ {\alpha} ^ {i} (\dot {q} _ {\alpha} ^ {\mathrm{max}}) _ {k},

and

(\ddot {R} _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}} = \Phi_ {\alpha} ^ {i} (\ddot {q} _ {\alpha} ^ {\mathrm{max}}) _ {k}.

We need to combine these estimates of the peak physical responses in the individual modes into estimates of the total peak response of the particular physical variable to the given spectrum, \left( R ^ { i } ( \omega , \xi ) \right) _ { k } ^ { \mathrm { m a x } } . Since the peak responses in the different modes will not in general occur at the same time, this combination is only an estimate, so several formulæ are offered, as follows:

Summation of the absolute values of the modal peak responses ( SUM=ABS) estimates

(R ^ {i}) _ {k} ^ {\max} = \sum_ {\alpha} \big | (R _ {\alpha} ^ {i}) _ {k} ^ {\max} \big |.

This provides the most conservative estimate of the peak response, since it assumes that all modes provide peak response in phase at the same time.

Square root of the summation of the squares (SUM=SRSS) estimates

(R ^ {i}) _ {k} ^ {\mathrm{max}} = \sqrt {\sum_ {\alpha} \left((R _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}}\right) ^ {2}}.

This summation usually provides a reasonable estimate if the natural frequencies of the modes are well separated.

The Naval Research Laboratory Method (SUM=NRL) distinguishes the mode, \beta , in which the physical variable has its maximum response, and adds the square root of the sum of squares of the peak responses in all other modes to the absolute value of the peak response of that mode. This gives the estimate

(R ^ {i}) _ {k} ^ {\max} = \left| (R _ {\beta} ^ {i}) _ {k} ^ {\max} \right| + \sqrt {\sum_ {\alpha / = \beta} \left((R _ {\alpha} ^ {i}) _ {k} ^ {\max}\right) ^ {2}}.

Again, the modes must be reasonably well-spaced in the frequency domain to obtain an accurate estimate with this method.

A variety of methods are available that aim to improve the estimation for structures with closely

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spaced frequencies. ABAQUS/Standard provides two of them: the Ten Percent Method recommended by Regulatory Guide 1.92 (1976) of the U.S. Nuclear Regulatory Commission and the Complete Quadratic Combination Method, which was first introduced by Der Kiureghian (1981) and developed by Smeby and Der Kiureghian (1984). Both methods reduce to the SRSS method if the modes are well separated with no coupling among them.

The Ten Percent Method (SUM=TENP) described in Regulatory Guide 1.92 modifies the SRSS method by adding a contribution from all pairs of modes ® and \beta whose frequencies are within 10% of each other, giving the estimate

(R ^ {i}) _ {k} ^ {\mathrm{max}} = \sqrt {\sum_ {\alpha} \left((R _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}}\right) ^ {2} + 2 \sum_ {\alpha <   \beta} \left| (R _ {\alpha} ^ {i}) _ {k} ^ {\mathrm{max}} (R _ {\beta} ^ {i}) _ {k} ^ {\mathrm{max}} \right|}.

The frequencies of modes ® and \beta are considered to be within 10% whenever

\frac {\omega_ {\beta} - \omega_ {\alpha}}{\omega_ {\beta}} \leq 0. 1, \quad \alpha <   \beta .

The Complete Quadratic Combination Method (SUM=CQC) combines the modal response with the formula

(R ^ {i}) _ {k} ^ {\max} = \sqrt {\sum_ {\alpha} \sum_ {\beta} (R _ {\alpha} ^ {i}) _ {k} ^ {\max} \rho_ {\alpha \beta} (R _ {\beta} ^ {i}) _ {k} ^ {\max}},

where \rho _ { \alpha \beta } are cross-correlation coefficients between modes ® and \beta _ { ; } , which depend on the ratio of frequencies and modal damping between the two modes:

\rho_ {\alpha \beta} = \frac {8 \sqrt {\xi_ {\alpha} \xi_ {\beta}} \left(\xi_ {\alpha} + r _ {\beta \alpha} \xi_ {\beta}\right) r _ {\beta \alpha} ^ {\frac {3}{2}}}{\left(1 - r _ {\beta \alpha} ^ {2}\right) ^ {2} + 4 \xi_ {\alpha} \xi_ {\beta} r _ {\beta \alpha} \left(1 + r _ {\beta \alpha} ^ {2}\right) + 4 \left(\xi_ {\alpha} ^ {2} + \xi_ {\beta} ^ {2}\right) r _ {\beta \alpha} ^ {2}},

where

r _ {\beta \alpha} = \frac {\omega_ {\beta}}{\omega_ {\alpha}}.

If double eigenvalues occur with the same damping coefficient, their correlation coefficient will be \rho _ { \alpha \beta } = 1 . If modes are well-spaced, their cross-correlation coefficient will be small ( \rho _ { \alpha \beta } < < 1 ) and the method will give the same results as the SRSS method. This method is usually recommended for asymmetrical building systems, since, in such cases, other methods can underestimate the response in the direction of motion and overestimate the response in the transverse direction (see ``Response spectra of a three-dimensional frame building,'' Section 2.2.3 of the ABAQUS Example Problems Manual).

For the case of different base excitations acting along orthogonal directions, we still need to sum over the directions indicated by the subscript k. Regulatory Guide 1.92 specifies that this directional summation be based on the square root of the sum of the squares summation rule, which is chosen by specifying COMP=SRSS on the *RESPONSE SPECTRUM option:

(R ^ {i}) ^ {\mathrm{max}} = \sqrt {\sum_ {k} \left((R ^ {i}) _ {k} ^ {\mathrm{max}}\right) ^ {2}}.

This rule is appropriate when the base motions in different directions are statistically independent (uncorrelated)--when they are acting along "principal directions." The existence of a set of directions along which the ground motions can be considered uncorrelated is discussed by Penzien and Watabe (1975). Such considerations are especially important when the CQC modal combination method is used. In the ABAQUS implementation of the CQC method it is assumed that the horizontal components act along the principal directions and are of equal intensity. For details on the CQC method applied to more general cases, see Smeby and Der Kiureghian (1984).

2.5.7 Steady-state linear dynamic analysis

Steady-state linear dynamic analysis predicts the linear response of a structure subjected to continuous harmonic excitation. In many cases steady-state linear dynamic analysis in ABAQUS/Standard uses the set of eigenmodes extracted in a previous *FREQUENCY step to calculate the steady-state solution as a function of the frequency of the applied excitation. ABAQUS/Standard also has a "direct" steady-state linear dynamic analysis procedure, in which the equations of steady harmonic motion of the system are solved directly without using the eigenmodes, and a "subspace" steady-state linear dynamic analysis procedure, in which the equations are projected onto a subspace of selected eigenmodes of the undamped system. These options are intended for systems in which the behavior is dependent on frequency, for when the model includes damping, or for systems in which the governing equations are not symmetric.

This section describes the linear steady-state response procedure based on the eigenmodes.

The projection of the equations of motion of the system onto the ®th mode gives

Equation 2.5.7-1

\ddot {q} _ {\alpha} + c _ {\alpha} \dot {q} _ {\alpha} + \omega_ {\alpha} ^ {2} q _ {\alpha} = \frac {1}{m _ {\alpha}} (f _ {1 \alpha} + i f _ {2 \alpha}) \exp (i \Omega t),

where q _ { \alpha } is the amplitude of mode ® (the ®th "generalized coordinate"), c _ { \alpha } is the damping associated with this mode (see below), \omega _ { \alpha } is the undamped frequency of the mode, m _ { \alpha } is the generalized mass associated with the mode, and \left( f _ { 1 \alpha } + i f _ { 2 \alpha } \right) exp(i−t) is the forcing associated with this mode. The forcing is defined by the frequency, −, and the real and imaginary parts of the nodal equivalent forces, F _ { 1 } ^ { N } and F _ { 2 } ^ { N } , projected onto the eigenmode \phi _ { \alpha } ^ { N } :

f _ {1 \alpha} + i f _ {2 \alpha} = \phi_ {\alpha} ^ {N} (F _ {1} ^ {N} + i F _ {2} ^ {N}).

In this equation summation is implied by the repeat of the superscript N indicating a degree of

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freedom in the model; but throughout this section we are working with only a single modal equation, so no summation is implied by the repeat of the mode subscript ®. The load vector is written in terms of its real and imaginary parts, F _ { 1 } ^ { N } and F _ { 2 } ^ { N } , since this is the manner in which the loading is defined in ABAQUS/Standard (as load case 1 and load case 2). It is equivalently possible to write the loading in terms of its magnitude, F _ { 0 } ^ { N } and phase, ª, as F ^ { N } = F _ { 0 } ^ { N } \exp { i ( \Omega t + \Psi ) } , where F _ { 1 } ^ { N } = F _ { 0 } cos ª and F _ { 2 } ^ { N } = F _ { 0 } sin ª.

Several representations of modal damping are provided. Modal damping defines c _ { \alpha } = 2 \xi _ { \alpha } \omega _ { \alpha } , where \xi _ { \alpha } is the fraction of critical damping in the mode. Structural damping gives a damping force proportional to the modal amplitude:

c _ {\alpha} \dot {q} _ {\alpha} = i s _ {\alpha} \omega_ {\alpha} ^ {2} q _ {\alpha},

where s _ { \alpha } is the structural damping coefficient for the mode. Rayleigh damping is defined by c _ { \alpha } = \beta _ { \alpha } + \gamma _ { \alpha } \omega _ { \alpha } ^ { 2 } ; \beta _ { \alpha } and \gamma _ { \alpha } are the Rayleigh coefficients damping low and high frequency modes, respectively. Rayleigh damping can be reproduced exactly by modal damping as

\xi_ {\alpha} = \frac {\beta_ {\alpha}}{2 \omega_ {\alpha}} + \frac {\gamma_ {\alpha} \omega_ {\alpha}}{2}.

Introducing all of these damping definitions into Equation 2.5.7-1 gives

Equation 2.5.7-2

\ddot {q} _ {\alpha} + 2 \xi_ {\alpha} \omega_ {\alpha} \dot {q} _ {\alpha} + (\beta_ {\alpha} + \gamma_ {\alpha} \omega_ {\alpha} ^ {2}) \dot {q} _ {\alpha} + i s _ {\alpha} \omega_ {\alpha} ^ {2} q _ {\alpha} + \omega_ {\alpha} ^ {2} q _ {\alpha} = \frac {1}{m _ {\alpha}} (f _ {1 \alpha} + i f _ {2 \alpha}) \exp (i \Omega t).

The solution to this equation is

Equation 2.5.7-3

q _ {\alpha} = H _ {0 \alpha} f _ {0 \alpha} \exp i (\Omega t + \Psi),

where f _ { 0 \alpha } = \sqrt { ( f _ { 1 \alpha } ) ^ { 2 } + ( f _ { 2 \alpha } ) ^ { 2 } } is the amplitude of the projected load vector and H _ { 0 \alpha } ( \Omega ) is the amplitude of the complex "transfer function" for mode ® that defines the response in mode ® from the force projection onto that mode and is defined by its real and imaginary parts as

\begin{array}{l} \Re (H _ {\alpha}) = \frac {1}{m _ {\alpha}} \left[ \frac {f _ {1 \alpha} (\omega_ {\alpha} ^ {2} - \Omega^ {2})}{(\omega_ {\alpha} ^ {2} - \Omega^ {2}) ^ {2} + (\eta_ {\alpha} \Omega) ^ {2}} + \frac {f _ {2 \alpha} \eta_ {\alpha} \Omega}{(\omega_ {\alpha} ^ {2} - \Omega^ {2}) ^ {2} + (\eta_ {\alpha} \Omega) ^ {2}} \right] \\ \Im (H _ {\alpha}) = \frac {1}{m _ {\alpha}} \left[ - \frac {f _ {1 \alpha} \eta_ {\alpha} \Omega}{\left(\omega_ {\alpha} ^ {2} - \Omega^ {2}\right) ^ {2} + \left(\eta_ {\alpha} \Omega\right) ^ {2}} + \frac {f _ {2 \alpha} \left(\omega_ {\alpha} ^ {2} - \Omega^ {2}\right)}{\left(\omega_ {\alpha} ^ {2} - \Omega^ {2}\right) ^ {2} + \left(\eta_ {\alpha} \Omega\right) ^ {2}} \right], \\ \end{array}

where \eta _ { \alpha } denotes

\eta_ {\alpha} = 2 \xi_ {\alpha} \omega_ {\alpha} + \beta_ {\alpha} + \gamma_ {\alpha} \omega_ {\alpha} ^ {2} + \frac {s _ {\alpha} \omega_ {\alpha} ^ {2}}{\Omega}.

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The amplitude of the response is

H _ {0 \alpha} f _ {0 \alpha} = \sqrt {\Re (H _ {\alpha}) ^ {2} + \Im (H _ {\alpha}) ^ {2}} f _ {0 \alpha} = \frac {1}{m _ {\alpha}} \frac {1}{\sqrt {\left(\omega_ {\alpha} ^ {2} - \Omega\right) ^ {2} + \left(\eta_ {\alpha} \Omega\right) ^ {2}}} f _ {0 \alpha},

and the phase angle of the response is

\Psi_ {\alpha} = \arctan \left(\Im (H _ {\alpha}) / \Re (H _ {\alpha})\right).

If a harmonic base motion is applied, the real and imaginary parts of the modal loads are given as

f _ {1 \alpha} = - \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {N} M ^ {N M} \hat {e} _ {j} ^ {M} a _ {1 j} \exp (i \Omega t),

f _ {2 \alpha} = - \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {N} M ^ {N M} \hat {e} _ {j} ^ {M} a _ {2 j} \exp (i \Omega t),

where M ^ { N M } is the structure's mass matrix and \hat { e } _ { j } ^ { M } is a vector that has unit magnitude in the direction of the base acceleration at any grounded node and is otherwise zero; a _ { 1 j } and a _ { 2 j } are the real and imaginary parts of the base acceleration. If the base motion is given as a velocity or displacement, the corresponding accelerations are a _ { 1 } = - \Omega v _ { 1 } , a 2 = - \Omega v _ { 2 } , where v _ { 1 } and v _ { 2 } are the real and imaginary parts of the velocity, or a _ { 1 } = - \Omega ^ { 2 } u _ { 1 } , a _ { 2 } = - \Omega ^ { 2 } u _ { 2 } , where u _ { 1 } and u _ { 2 } are the real are imaginary parts of the displacement.

The peak amplitude of any physical variable, u ^ { N } , is available from the modal amplitudes as

u ^ {N} = \sum_ {\alpha} \phi_ {\alpha} ^ {N} q _ {\alpha}.

Steady-state response is given as a frequency sweep through a user-specified range of frequencies. Since the structural response peaks around the natural frequencies, a bias function is used to cluster the response points around the frequencies. The biasing is described in ``Mode-based steady-state dynamic analysis,'' Section 6.3.6 of the ABAQUS/Standard User's Manual.

2.5.8 Random response analysis

Random response linear dynamic analysis is used to predict the response of a structure subjected to a nondeterministic continuous excitation that is expressed in a statistical sense by a cross-spectral density (CSD) matrix. The *RANDOM RESPONSE procedure uses the set of eigenmodes extracted in a previous *FREQUENCY step to calculate the corresponding power spectral densities ( PSD) of response variables (stresses, strains, displacements, etc.) and, hence--if required--the variance and root mean square values of these same variables. This section provides brief definitions and explanations of the terms used in this type of analysis. Detailed discussion of the theory of random response analysis is

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provided in the books by Clough and Penzien (1975), Hurty and Rubinstein (1964), and Thompson (1988).

Examples of random response analysis are the study of the response of an airplane to turbulence; the response of a car to road surface imperfections; the response of a structure to noise, such as the "jet noise" emitted by a jet engine; and the response of a building to an earthquake.

Since the loading is nondeterministic, it can be characterized only in a statistical sense. We need some assumptions to make this characterization possible. Although the excitation varies in time, in some sense it must be stationary--its statistical properties must not vary with time. Thus, if x(t) is the variable being considered (such as the height of the road surface in the case of a car driving down a rough road), then any statistical function of x, f (x), must have the same value regardless of what time origin we use to compute f :

f (x (t)) = f (x (t + \tau)) \quad \text { for   any } \tau .

We also need the excitation to be ergodic. This term means that, if we take several samples of the excitation, the time average of each sample is the same.

These restrictions ensure that the excitation is, statistically, constant. In the following discussion we also assume that the random variables are real, which is the case for the variables that we need to consider.

Statistical measures

We define some measures of a variable that characterize it in a statistical sense.

The mean value of a random variable x(t) is

E (x) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x (t) d t.

Since the dynamic response is computed about a static equilibrium configuration, the mean value of any dynamic input or response variable will always be zero:

E (x) = 0.

The variance of a random variable measures the average square difference between the point value of the variable and its mean:

\sigma_ {r} ^ {2} (x) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \bigl (x (t) - E (x) \bigr) ^ {2} d t.

Since E(x) = 0 for our applications, the variance is the same as the mean square value:

E (x ^ {2}) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x ^ {2} (t) d t.

The units of variance are (amplitude)2 , so that--for example--the variance of a force has units of (force)2 . Generally we prefer to use the same units as the variable itself. Therefore, output variables in ABAQUS/Standard are given as root mean square ("RMS") values, \sigma _ { r } ( x ) = \sqrt { \sigma _ { r } ^ { 2 } } .

Correlation

Correlation measures the similarity between two variables. Thus, the cross-correlation between two random functions of time, x _ { 1 } ( t ) and x _ { 2 } ( t ) , is the integration of the product of the two variables, with one of them shifted in time by some fixed value ¿ to allow for the possibility that they are similar but shifted in time.

(Such a case would arise, for example, in studying a car driving along a rough road. If the separation of the axles is d and the car is moving at a steady speed v , the back axle sees the same road profile as the front axle, but delayed by a time \tau = d / v . Assume that the road profile moves each wheel which, in turn applies a force to the car frame through the suspension. If F _ { 2 } is the force applied to the rear axle (as a *CLOAD in ABAQUS) and F _ { 1 } is the force applied to the front axle, F _ { 2 } ( t + d / v ) = F _ { 1 } ( t ) . )

The cross-correlation function is, thus, defined as

R _ {x _ {1} x _ {2}} (\tau) = \lim _ {T \rightarrow \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x _ {1} (t) x _ {2} (t + \tau) d t.

Since the mean value of any variable is zero, on average each variable has equal positive and negative content. If the variables are quite similar, their cross-correlation (for some values of ¿ ) will be large; if they are not similar, the product x _ { 1 } x _ { 2 } will sometimes be negative and sometimes positive so that the integral over all time will provide a much smaller value, regardless of the choice of \tau . .

Figure 2.5.8-1 Two random records to be correlated.

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| t | x₁(t) | x₂(t) | | ---- | ----- | ----- | | 0 | 0 | 0 | | 1 | 1 | 0 | | 2 | 0 | 1 | | 3 | 2 | 0 | | 4 | 0 | 2 | | 5 | 1 | 0 | | 6 | 0 | 1 | | 7 | 2 | 0 | | 8 | 0 | 2 | | 9 | 1 | 0 | | 10 | 0 | 1 | | 11 | 2 | 0 | | 12 | 0 | 2 | | 13 | 1 | 0 | | 14 | 0 | 1 | | 15 | 2 | 0 | | 16 | 0 | 2 | | 17 | 1 | 0 | | 18 | 0 | 1 | | 19 | 2 | 0 | | 20 | 0 | 2 | | 21 | 1 | 0 | | 22 | 0 | 1 | | 23 | 2 | 0 | | 24 | 0 | 2 | | 25 | 1 | 0 | | 26 | 0 | 1 | | 27 | 2 | 0 | | 28 | 0 | 2 | | 29 | 1 | 0 | | 30 | 0 | 1 | | 31 | 2 | 0 | | 32 | 0 | 2 | | 33 | 1 | 0 | | 34 | 0 | 1 | | 35 | 2 | 0 | | 36 | 0 | 2 | | 37 | 1 | 0 | | 38 | 0 | 1 | | 39 | 2 | 0 | | 40 | 0 | 2 | | 41 | 1 | 0 | | 42 | 0 | 1 | | 43 | 2 | 0 | | 44 | 0 | 2 | | 45 | 1 | 0 | | 46 | 0 | 1 | | 47 | 2 | 0 | | 48 | 0 | 2 | | 49 | 1 | 0 | | 50 | 0 | 1 | | 51 | 2 | 0 | | 52 | 0 | 2 | | 53 | 1 | 0 | | 54 | 0 | 1 | | 55 | 2 | 0 | | 56 | 0 | 2 | | 57 | 1 | 0 | | 58 | 0 | 1 | | 59 | 2 | 0 | | 60 | 0 | 2 | | 61 | 1 | 0 | | 62 | 0 | 1 | | 63 | 2 | 0 | | 64 | 0 | 2 | | 65 | 1 | 0 | | 66 | 0 | 1 | | 67 | 2 | 0 | | 68 | 0 | 2 | | 69 | 1 | 0 | | 70 | 0 | 1 | | 71 | 2 | 0 | | 72 | 0 | 2 | | 73 | 1 | 0 | | 74 | 0 | 1 | | 75 | 2 | 0 | | 76 | 0 | 2 | | 77 | 1 | 0 | | 78 | 0 | 1 | | 79 | 2 | 0 | | 80 | 0 | 2 | | 81 | 1 | 0 | | 82 | 0 | 1 | | 83 | 2 | 0 | | 84 | 0 | 2 | | 85 | 1 | 0 | | 86 | 0 | 1 | | 87 | 2 | 0 | | 88 | 0 | 2 | | 89 | 1 | 0 | | 90 | 0 | 1 | | 91 | 2 | 0 | | 92 | 0 | 2 | | 93 | 1 | 0 | | 94 | 0 | 1 | | 95 | 2 | 0 | | 96 | 0 | 2 | | 97 | 1 | 0 | | 98 | 0 | 1 | | 99 | 2 | 0 | | 100 | 0 | 2 |

A simple result is

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\begin{array}{l} R _ {x _ {1} x _ {2}} (\tau) = \lim _ {T \rightarrow \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x _ {1} (t) x _ {2} (t + \tau) d t \\ = \lim _ {T \rightarrow \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x _ {1} (t - \tau) x _ {2} (t) d t \\ = \lim _ {T \rightarrow \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x _ {2} (t) x _ {1} (t - \tau) d t \\ = R _ {x _ {2} x _ {1}} (- \tau). \\ \end{array}

For convenience the cross-correlation can be normalized to define the nondimensional normalized cross-correlation:

r _ {x _ {1} x _ {2}} (\tau) = \frac {R _ {x _ {1} x _ {2}} (\tau)}{\sqrt {\sigma_ {r} ^ {2} (x _ {1}) \sigma_ {r} ^ {2} (x _ {2})}}.

Thus, if x _ { 1 } = x _ { 2 } , r ( 0 ) = 1 ; { \mathrm { i f } } x _ { 1 } = - x _ { 2 } , r ( 0 ) = - 1 . If x _ { 1 } and x _ { 2 } are entirely dissimilar, r \to 0 . (When r = 0 the variables are said to be orthogonal.) Clearly - 1 \leq r \leq 1 , with small values of r indicating that x _ { 1 } and x _ { 2 } have quite different time histories.

Now consider the cross-correlation of a variable with itself: the autocorrelation. Intuitively we can see that if the variable is "very random," its autocorrelation will be very small whenever \tau \neq 0 ; : there will be no time shift that allows the variable to correlate with itself. However, if the variable is not so random--if it is just a vibration at a fixed frequency--the autocorrelation will be close to §1 whenever \tau is chosen to be some integer multiple of half the period of the vibration. Thus, the autocorrelation provides a measure of how random a variable really is.

The autocorrelation of a variable x ( t ) is, therefore,

R (\tau) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x (t) x (t + \tau) d t.

Clearly, as \tau \to 0 , R ( \tau ) \to \sigma _ { r } ^ { 2 } : the autocorrelation equals the variance (the mean square value). We can, therefore, also use the normalized autocorrelation:

r (\tau) = \frac {R (\tau)}{\sigma_ {r} ^ {2}}.

Obviously R ( \tau ) is symmetric about \tau = 0 :

R (- \tau) = R (\tau);

and the value of R ( \tau ) never exceeds its value at \tau = 0

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R (\tau) \leq R (0) = \sigma_ {r} ^ {2}.

The autocorrelation function of records with very similar amplitude over the wide range of frequencies drops off rapidly as ¿ increases. This kind of function is known as a "wide band" random function.

Figure 2.5.8-2 Wide band noise record and its autocorrelation.

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| t | Wide band noise x(t) | | ---- | -------------------- | | 0 | 0 | | 1 | 1 | | 2 | 0 | | 3 | 1 | | 4 | 0 | | 5 | 1 | | 6 | 0 | | 7 | 1 | | 8 | 0 | | 9 | 1 | | 10 | 0 | | 11 | 1 | | 12 | 0 | | 13 | 1 | | 14 | 0 | | 15 | 1 | | 16 | 0 | | 17 | 1 | | 18 | 0 | | 19 | 1 | | 20 | 0 | | 21 | 1 | | 22 | 0 | | 23 | 1 | | 24 | 0 | | 25 | 1 | | 26 | 0 | | 27 | 1 | | 28 | 0 | | 29 | 1 | | 30 | 0 | | 31 | 1 | | 32 | 0 | | 33 | 1 | | 34 | 0 | | 35 | 1 | | 36 | 0 | | 37 | 1 | | 38 | 0 | | 39 | 1 | | 40 | 0 | | 41 | 1 | | 42 | 0 | | 43 | 1 | | 44 | 0 | | 45 | 1 | | 46 | 0 | | 47 | 1 | | 48 | 0 | | 49 | 1 | | 50 | 0 | | 51 | 1 | | 52 | 0 | | 53 | 1 | | 54 | 0 | | 55 | 1 | | 56 | 0 | | 57 | 1 | | 58 | 0 | | 59 | 1 | | 60 | 0 | | 61 | 1 | | 62 | 0 | | 63 | 1 | | 64 | 0 | | 65 | 1 | | 66 | 0 | | 67 | 1 | | 68 | 0 | | 69 | 1 | | 70 | 0 | | 71 | 1 | | 72 | 0 | | 73 | 1 | | 74 | 0 | | 75 | 1 | | 76 | 0 | | 77 | 1 | | 78 | 0 | | 79 | 1 | | 80 | 0 | | 81 | 1 | | 82 | 0 | | 83 | 1 | | 84 | 0 | | 85 | 1 | | 86 | 0 | | 87 | 1 | | 88 | 0 | | 89 | 1 | | 90 | 0 | | 91 | 1 | | 92 | 0 | | 93 | 1 | | 94 | 0 | | 95 | 1 | | 96 | 0 | | 97 | 1 | | 98 | 0 | | 99 | 1 | | 100 | 0 |

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τ	R(τ)
0	Peak
τ	0

The most extreme wide band random function would have an autocorrelation that is just a delta function:

R (\tau) = \delta (\tau) = \left\{ \begin{array}{l l} \sigma_ {r} ^ {2} & \mathrm{if} \tau = 0 \\ 0 & \mathrm{if} \tau \neq 0. \end{array} \right.

Such a function is called white noise. White noise has the same amplitude at all frequencies, and its autocorrelation is zero except at \tau = 0 .

Figure 2.5.8-3 Autocorrelation of white noise.

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R(τ) σ_z^2 τ

Let us now consider the opposite case, known as a "narrow band" function. The extreme case of such a function is a simple sinusoidal vibration at a single frequency: x = A sin \omega _ { 0 } t . Then R ( \tau ) must also be periodic, since R ( \tau ) must attain the same value, R ( 0 ) = \sigma _ { r } ^ { 2 } , each time the shift, \tau , corresponds to the period of vibration. Performing the integration through time,

\begin{array}{l} R (\tau) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} A ^ {2} \sin \omega_ {0} t \sin \omega_ {0} (t + \tau) d t \\ = A ^ {2} \lim _ {T \to \infty} \frac {1}{T} \left(\cos \omega_ {0} \tau \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \sin^ {2} \omega_ {0} t d t + \sin \omega_ {0} \tau \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \sin \omega_ {0} t \cos \omega_ {0} t d t\right) \\ \end{array}