김경종
23 KiB
\begin{array}{l} = A ^ {2} \lim _ {T \to \infty} \frac {1}{T} \left(\cos \omega_ {0} \tau \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \frac {1}{2} (1 - \cos 2 \omega_ {0} t) d t + \sin \omega_ {0} \tau \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \frac {1}{2} \sin 2 \omega_ {0} t d t\right) \\ = \frac {1}{2} A ^ {2} \cos \omega_ {0} \tau = \sigma_ {r} ^ {2} \cos \omega_ {0} \tau . \\ \end{array}
Figure 2.5.8-4 Sine wave and its autocorrelation.
text_image
Type of record Sine wave x(t) = A sin ( ω₀t + θ ) t
line
| τ | Autocorrelation | |-------|-----------------| | 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 |The autocorrelation, R ( \tau ) , thus tells us about the nature of the random variable. If R ( \tau ) drops off rapidly as the time shift ¿ moves away from \tau = 0 , the variable has a broad frequency content; if it drops off more slowly and exhibits a cosine profile, the variable has a narrow frequency content centered around the frequency corresponding to the periodicity of R ( \tau ) .
Figure 2.5.8-5 Narrow band record and its autocorrelation.
text_image
Narrow band response
text_image
R(τ) = ce^{-k1τ1} cos ω₀τ O τ
We can extend this concept to detect the frequency content of a random variable by cross-correlating the variable with a sine wave: sweeping the wave over a range of frequencies and examining the cross-correlation tells us whether the random variable is dominated by oscillation at particular frequencies. We begin to see that the nature of stationary, ergodic random processes is best understood by examining them in the frequency domain.
As an illustration, consider a variable, x ( t ) , which contains many discrete frequencies. We can write x ( t ) in terms of a Fourier series expanded in N steps of a fundamental frequency !0:
x (t) = \sum_ {n = 1} ^ {N} \left[ a _ {n} \cos (i n \omega_ {0} t) + b _ {n} \sin (i n \omega_ {0} t) \right].
Procedures
The series begins with the n = 1 term because the mean of the variable must be zero. We can write this series more compactly as a complex Fourier series (keeping in mind that we will be interested only in the real part):
x (t) = \sum_ {n = - N} ^ {N} A _ {n} \exp (i n \omega_ {0} t),
where A _ { n } = \Re { ( A _ { n } ) } + i \Im { ( A _ { n } ) } is the complex amplitude of the nth term, A _ { n } ^ { * } is the complex conjugate of A _ { n } :
A _ {n} ^ {*} = \Re (A _ {n}) - i \Im (A _ {n}),
and
A _ {n} = \left\{ \begin{array}{l l} (a _ {n} - i b _ {n}) / 2 & \text {for} n > 0, \\ (a _ {n} + i b _ {n}) / 2 & \text {for} n < 0, \\ 0 & \text {for} n = 0. \end{array} \right.
The variance of x(t) is
\sigma_ {r} ^ {2} (x) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \sum_ {n = - N} ^ {N} A _ {n} ^ {*} \exp (- i n \omega_ {0} t) A _ {n} \exp (i n \omega_ {0} t) d t,
using the orthogonality of Fourier terms. Continuing,
\begin{array}{l} \sigma_ {r} ^ {2} (x) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} \sum_ {n = - N} ^ {N} A _ {n} A _ {n} ^ {*} d t \\ = \sum_ {n = - N} ^ {N} A _ {n} A _ {n} ^ {*} = \sum_ {n = - N} ^ {N} \left| A _ {n} \right| ^ {2} = \sum_ {n = - N} ^ {N} \sigma_ {r} ^ {2} (x _ {n}), \\ \end{array}
where
x _ {n} = \Re \left(A _ {n} \exp (i n \omega_ {0} t)\right)
is the nth component of the Fourier series.
Thus, thanks to the orthogonality of Fourier terms, the variance (the mean square value) of the series is the sum of the variances (the mean square values) of its components. In particular, we see that A _ { n } A _ { n } ^ { * } is the variance, or mean square value, of the variable at the frequency n!0.
The contribution to the variance of x, \sigma _ { r } ^ { 2 } ( x ) , at the frequency f _ { n } = n \omega _ { 0 } / ( 2 \pi ) , per unit frequency, is thus
Procedures
S _ {x} (f _ {n}) = \frac {2 \pi}{\omega_ {0}} A _ {n} A _ {n} ^ {*},
since we are stepping up the frequency range in steps of \Delta f = \omega _ { 0 } / ( 2 \pi ) . The variance can, therefore, be written as
\sigma_ {r} ^ {2} (x) = \sum_ {n = - N} ^ {N} S _ {x} (f _ {n}) \Delta f.
As we examine x as a function of frequency, S _ { x } ( f _ { n } ) tells us the amount of "power" (in the sense of mean square value) contained in x, per unit frequency, at the frequency f _ { n } . As we consider smaller and smaller intervals, \Delta f 0 , S _ { x } is the power spectral density ( P S D ) of the variable x:
\sigma_ {r} ^ {2} (x) = \int_ {- \infty} ^ {\infty} S _ {x} (f) d f,
where f is the frequency in cycles per time (usually Hz).
Notice that S _ { x } has units of (variable)2/frequency, where (variable) is the unit of the variable (displacement, force, stress, etc.). In this case "frequency" is almost always given in \mathrm { H z , } although--since ABAQUS does not have any built-in units--the frequency could be expressed in any other units of cycles per time. However, S _ { x } should not be given per circular frequency (radians per time): ABAQUS assumes that S _ { x } ( f ) , not S _ { x } ( \omega ) .
Fourier transforms
Since the variables of interest in random response analysis are characterized as functions of frequency, the Fourier transform plays a major role in converting from the time domain to the frequency domain and vice versa. The Fourier transform of x ( t ) , which we write as \overline { { x } } ( f ) , is defined by
x (t) = \int_ {- \infty} ^ {\infty} \overline {{{x}}} (f) \exp (i 2 \pi f t) d f
or, in terms of the circular frequency \omega = 2 \pi f
x (t) = \frac {1}{2 \pi} \int_ {- \infty} ^ {\infty} \overline {{x}} (\omega) \exp (i \omega t) d \omega .
Simple manipulation provides
x (t) = \frac {1}{2 \pi} \int_ {0} ^ {\infty} \left[ \left(\overline {{{x}}} (\omega) + \overline {{{x}}} (- \omega)\right) \cos \omega t + i \left(\overline {{{x}}} (\omega) - \overline {{{x}}} (- \omega)\right) \sin \omega t \right] d \omega ,
so \left( \overline { { x } } ( \omega ) + \overline { { x } } ( - \omega ) \right) / ( 2 \pi ) is the amplitude of the cosine term in x ( t ) at the circular frequency \omega , , while
Procedures
\left( \overline { { x } } ( \omega ) - \overline { { x } } ( - \omega ) \right) / ( 2 \pi ) is the amplitude of the sine term. We, thus, see the physical meaning of the Fourier transform--it provides the complex magnitude (the amplitude and phase) of the content of x(t) at a particular frequency.
If x(t) is real only (which is the case for the variables we need to consider), this expression shows that we must have
\Im (\overline {{{x}}} (\omega)) = - \Im (\overline {{{x}}} (- \omega))
and
\Re \bigl (\overline {{{x}}} (\omega) \bigr) = \Re \bigl (\overline {{{x}}} (- \omega) \bigr),
which means that
\overline {{{x}}} (- \omega) = \overline {{{x}}} ^ {*} (\omega).
For completeness we also note the inverse transformation,
\overline {{{x}}} (f) = \int_ {- \infty} ^ {\infty} x (t) \exp (- i 2 \pi f t) d t,
which shows that the transformations between the time and frequency domains are rather symmetrical.
We now need Parseval's theorem:
\begin{array}{l} \int_ {- \infty} ^ {\infty} x _ {1} (t) x _ {2} ^ {*} (t) d t = \int_ {- \infty} ^ {\infty} x _ {2} ^ {*} (t) \int_ {- \infty} ^ {\infty} \overline {{x}} _ {1} (f) \exp (i 2 \pi f t) d f d t \\ = \int_ {- \infty} ^ {\infty} \overline {{x}} _ {1} (f) \int_ {- \infty} ^ {\infty} x _ {2} ^ {*} (t) \exp (i 2 \pi f t) d t d f \\ = \int_ {- \infty} ^ {\infty} \overline {{x}} _ {1} (f) \overline {{x}} _ {2} ^ {*} (f) d f. \\ \end{array}
Applying this theorem to the variance (the mean square value):
\begin{array}{l} \sigma_ {r} ^ {2} = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x ^ {2} (t) d t \\ = \int_ {- \infty} ^ {\infty} \lim _ {T \to \infty} \frac {1}{T} \overline {{x}} (f) \overline {{x}} ^ {*} (f) d f. \\ \end{array}
Since x(t) is real, we know that { \overline { { x } } } ( - f ) = { \overline { { x } } } ^ { * } ( f ) , and so
\overline {{{x}}} (- f) \overline {{{x}}} ^ {*} (- f) = \overline {{{x}}} ^ {*} (f) \overline {{{x}}} (f) = \overline {{{x}}} (f) \overline {{{x}}} ^ {*} (f).
Procedures
We have already shown that we can write the variance in terms of the power spectral density as
\sigma_ {r} ^ {2} (x) = \int_ {- \infty} ^ {\infty} S _ {x} (f) d f.
By comparison,
S _ {x} (f) = \lim _ {T \to \infty} \frac {1}{T} \overline {{x}} (f) \overline {{x}} ^ {*} (f).
We also see that
S _ {x} (- f) = S _ {x} (f).
To avoid integration over negative frequencies, we write the variance as
\begin{array}{l} \sigma_ {r} ^ {2} (x) = \int_ {0} ^ {\infty} \left[ S _ {x} (f) + S _ {x} (- f) \right] d f \\ = \int_ {0} ^ {\infty} \tilde {S} _ {x} (f) d f, \\ \end{array}
where \tilde { S } _ { x } ( f ) is the single-sided PSD defined as
\tilde {S} _ {x} (f) = S _ {x} (f) + S _ {x} (- f), \quad \text { for } f \geq 0.
Since S _ { x } ( - f ) = S _ { x } ( f ) , we see that
\tilde {S} _ {x} (f) = 2 S _ {x} (f).
Now consider the autocorrelation function:
\begin{array}{l} R (\tau) = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x (t) x (t + \tau) d t \\ = \lim _ {T \to \infty} \frac {1}{T} \int_ {- \frac {T}{2}} ^ {\frac {T}{2}} x (t) \left[ \int_ {- \infty} ^ {\infty} \overline {{{x}}} (f) \exp \bigl (i 2 \pi f (t + \tau) \bigr) d f \right] d t \\ = \int_ {- \infty} ^ {\infty} \lim _ {T \to \infty} \frac {1}{T} \left[ \int_ {- \infty} ^ {\infty} x (t) \exp (i 2 \pi f t) d t \right] \overline {{x}} (f) \exp (i 2 \pi f \tau) d f \\ = \int_ {- \infty} ^ {\infty} \left(\lim _ {T \rightarrow \infty} \frac {1}{T} \overline {{x}} ^ {*} (f) \overline {{x}} (f)\right) \exp (i 2 \pi f \tau) d f \\ = \int_ {- \infty} ^ {\infty} S _ {x} (f) \exp (i 2 \pi f \tau) d f, \\ \end{array}
Procedures
(using the result above). Thus, the power spectral density is the Fourier transform of the autocorrelation function. The inverse transform is
S _ {x} (f) = \int_ {- \infty} ^ {\infty} R (\tau) \exp (- i 2 \pi f \tau) d \tau .
Since R ( \tau ) is symmetric about \tau = 0 \left( R ( - \tau ) = R ( \tau ) \right) , we can also write this equation as
\begin{array}{l} S _ {x} (f) = \int_ {0} ^ {\infty} R (\tau) \left(\exp (- i 2 \pi f \tau) + \exp (i 2 \pi f \tau)\right) d \tau \\ = 2 \int_ {0} ^ {\infty} R (\tau) \cos (2 \pi f \tau) d \tau , \\ \end{array}
so that
\tilde {S} _ {x} (f) = 4 \int_ {0} ^ {\infty} R (\tau) \cos (2 \pi f \tau) d \tau .
Cross-spectral density
Following a similar argument to that used above to develop the idea of the power spectral density, we can define the cross-spectral density (CSD) function, S _ { x _ { 1 } x _ { 2 } } ( f ) , which gives the cross-correlation between two variables, R _ { x _ { 1 } x _ { 2 } } ( \tau ) , as
R _ {x _ {1} x _ {2}} (\tau) = \int_ {- \infty} ^ {\infty} S _ {x _ {1} x _ {2}} (f) \exp (i 2 \pi f \tau) d f,
with the inverse transformation
S _ {x _ {1} x _ {2}} (f) = \int_ {- \infty} ^ {\infty} R _ {x _ {1} x _ {2}} (\tau) \exp (- i 2 \pi f \tau) d \tau .
Transforming the original definition of R _ { x _ { 1 } x _ { 2 } } ( \tau ) to the frequency domain provides
\begin{array}{l} R _ {x _ {1} x _ {2}} (\tau) = \lim _ {T \to \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) \int_ {- \infty} ^ {\infty} \overline {{x}} _ {2} (f) \exp (i 2 \pi f (t + \tau)) d f d t \\ = \int_ {- \infty} ^ {\infty} \lim _ {T \to \infty} \frac {1}{T} \left[ \int_ {- \infty} ^ {\infty} x _ {1} (t) \exp (i 2 \pi f t) d t \right] \overline {{x}} _ {2} (f) \exp (i 2 \pi f \tau) d f \\ = \int_ {- \infty} ^ {\infty} \left(\lim _ {T \to \infty} \frac {1}{T} \overline {{x}} _ {1} ^ {*} (f) \overline {{x}} _ {2} (f)\right) \exp (i 2 \pi f \tau) d f. \\ \end{array}
By comparison,
S _ {x _ {1} x _ {2}} (f) = \lim _ {T \to \infty} \frac {1}{T} \overline {{x}} _ {1} ^ {*} (f) \overline {{x}} _ {2} (f).
We could also write
\begin{array}{l} S _ {x _ {1} x _ {2}} (f) = \left[ \lim _ {T \rightarrow \infty} \frac {1}{T} \overline {{{x}}} _ {2} ^ {*} (f) \overline {{{x}}} _ {1} (f) \right] ^ {*} \\ = S _ {x _ {2} x _ {1}} ^ {*} (f) = S _ {x _ {2} x _ {1}} (- f). \\ \end{array}
Random response analysis
The general concept of random response analysis is now clear. A system is excited by some random loads, which are characterized in the frequency domain by a matrix of cross-spectral density functions, S _ { N M } \left( f \right) . Here we think of N and M as two of the degrees of freedom of the finite element model that are exposed to the random loads or prescribed boundary conditions (through the *BASE MOTION option).
In typical applications the range of frequencies will be limited to those to which we know the structure will respond--we do not need to consider frequencies that are higher than the modes in which we expect the structure to respond.
The values of S _ { N M } \left( f \right) might be provided by Fourier transformation of the cross-correlation of time records or by the Fourier transformation of the autocorrelation of a single time record, together with known geometric data, as in the case of the car driving along a roughly grooved road, where the autocorrelation of the road surface profile, together with the speed of the car and the axle separation, allow S _ { 1 2 } ( f ) to be defined for the front (1) and rear (2) axles, as shown above. (If, in this case, the road profile seen by the wheels on the left side of the car is not similar to that seen by the wheels on the right side of the car, the cross-correlation of the left and right road surface profiles will also be required to define the excitation.)
The system will respond to this excitation. We are usually interested in looking at the power spectral densities of the usual response variables--stress, displacement, etc. The PSD history of any particular variable will tell us the frequencies at which the system is most excited by the random loading.
We might also compute the cross-spectral densities between variables. These are usually not of interest, and ABAQUS/Standard does not provide them. (They might be needed if the analysis involves obtaining results that, in turn, will define the loading for some other system. For example, the response of a building to seismic loading might be used to obtain the motions of the attachment points for a piping system in the building so that the piping system can then be analyzed. The only option would be to model the entire system together.)
An overall picture is provided by looking at the variance (the mean square value) of any variable; the RMS value is provided for this purpose. The RMS value is used instead of variance because it has the same units as the variable itself. ABAQUS/Standard computes it by integrating the single-sided power
Procedures
spectral density of the variable over the frequency range, since
\sigma_ {r} = \sqrt {R (0)} = \sqrt {\int_ {0} ^ {\infty} \tilde {S} _ {x} (f) d f}.
This integration is performed numerically by using the trapezoidal rule over the range of frequencies given in the *RANDOM RESPONSE step:
\sigma_ {r} \approx \sqrt {\frac {1}{2} \big (\tilde {S} _ {1} (f _ {2} - f _ {1}) + \sum_ {i = 2} ^ {N - 1} \tilde {S} _ {i} (f _ {i + 1} - f _ {i - 1}) + \tilde {S} _ {N} (f _ {N} - f _ {N - 1}) \big)},
where { \tilde { S } } _ { 1 } denotes \tilde { S } _ { x } at the user-defined lower frequency range f _ { 1 } , \tilde { S } _ { i } is the \tilde { S } _ { x } at the frequency f _ { i } , and N is the number of points at which the response was calculated. N will depend on the number of eigenmodes used in the superposition and on the number of points chosen by the user in between the eigenfrequencies and given under the *RANDOM RESPONSE option.
The user must ensure that enough frequency points are specified so that this approximate integration will be sufficiently accurate.
The transformation of the problem into the frequency domain inherently assumes that the system under study is responding linearly: the *RANDOM RESPONSE procedure is considered as a linear perturbation analysis step.
What remains, then, is for us to consider how ABAQUS/Standard finds the linear response to the random excitation.
The frequency response function
Random response is studied in the frequency domain. Therefore, we need the transformation from load to response as a function of frequency. Since the random response is treated as the integration of a series of sinusoidal vibrations, this transformation is based on the same steady-state response function used in the *STEADY STATE DYNAMICS loading option and described in ``Steady-state linear dynamic analysis,'' Section 2.5.7.
The discrete (finite element) linear dynamic system has the equilibrium equation
\delta u ^ {N} \left(M ^ {N M} \ddot {u} ^ {M} + C ^ {N M} \dot {u} ^ {M} + K ^ {N M} u ^ {M}\right) = \delta u ^ {N} F ^ {N},
where M ^ { N M } is the mass matrix, C ^ { N M } is the damping matrix, K ^ { N M } is the stiffness matrix, F ^ { N } are the external loads, u ^ { N } is the value of degree of freedom N of the finite element model (usually a displacement or rotation component, or an acoustic pressure), and \delta u ^ { N } is an arbitrary virtual variation.
We project the problem onto the eigenmodes of the system. To do this the modes are first extracted from the undamped system:
Procedures
\left(M ^ {N M} \omega_ {\bar {\alpha}} ^ {2} - K ^ {N M}\right) \phi_ {\bar {\alpha}} ^ {N} = 0.
(Here, and throughout the remainder of this section, repeated subscripts and superscripts are assumed to be summed over the appropriate range except when they are barred, like ®¹ above. Roman superscripts and subscripts indicate physical degrees of freedom; Greek superscripts and subscripts indicate modal variables.)
Typically the structural dynamic response is well represented by a small number of the lower modes of the model, so the number of modes is usually O ( 1 0 ^ { 1 } ) – O ( 1 0 ^ { 2 } ) , while the number of physical degrees of freedom might be O ( 1 0 ^ { 3 } ) – O ( 1 0 ^ { 5 } ) .
The eigenmodes are orthogonal across the mass and stiffness matrices:
\phi_ {\alpha} ^ {N} M ^ {N M} \phi_ {\beta} ^ {M} = \left\{ \begin{array}{l l} m _ {\alpha} & \mathrm{if} \alpha = \beta \\ 0 & \mathrm{if} \alpha \neq \beta ; \end{array} \right.
\phi_ {\alpha} ^ {N} K ^ {N M} \phi_ {\beta} ^ {M} = \left\{ \begin{array}{l l} k _ {\alpha} & \text {if} \alpha = \beta \\ 0 & \text {if} \alpha \neq \beta . \end{array} \right.
(The eigenmodes are normalized so that the largest entry in \phi _ { \alpha } ^ { N } is 1.0. Thus, m _ { \alpha } \neq 1 in general. In contrast, some codes normalize the eigenmodes so that m _ { \alpha } = 1 . )
We assume that any damping is in the general form of "Rayleigh damping":
C ^ {N M} = \alpha M ^ {N M} + \beta K ^ {N M}
so that C ^ { N M } will also project into a diagonal damping matrix c _ { \alpha }
The problem, thus, projects into a set of uncoupled modal response equations,
m _ {\bar {\alpha}} \ddot {q} _ {\bar {\alpha}} + c _ {\bar {\alpha}} \dot {q} _ {\bar {\alpha}} + k _ {\bar {\alpha}} q _ {\bar {\alpha}} = f _ {\alpha},
where
f _ {\alpha} = \phi_ {\alpha} ^ {N} F ^ {N}
is the generalized load for mode ® and q _ { \alpha } is the "generalized coordinate" (the modal amplitude) for mode ®.
Steady-state excitation is of the form
f _ {\alpha} = A \exp (i 2 \pi f t)
and creates response of similar form that we write as
q _ {\alpha} = H A \exp (i 2 \pi f t),
where H ( f ) is the complex frequency response function defined in ``Steady-state linear dynamic analysis,'' Section 2.5.7.
Response development
Random loading is defined by the cross-spectral density matrix S _ { N M } ^ { F } ( f ) , which links all loaded degrees of freedom (N and M ). Projecting this matrix onto the modes provides the cross-spectral density function for the generalized (modal) loads:
S _ {\alpha \beta} ^ {f} (f) = \phi_ {\alpha} ^ {N} S _ {N M} ^ {F} (f) \phi_ {\beta} ^ {M}.
The complex frequency response function then defines the response of the generalized coordinates as
S _ {\alpha \beta} ^ {q} (f) = H _ {\bar {\alpha}} S _ {\bar {\alpha} \bar {\beta}} ^ {f} (f) H _ {\bar {\beta}} ^ {*},
where H _ { \alpha } ^ { * } is the complex conjugate of H _ { \alpha } .
Finally, the response of the physical variables is recovered from the modal responses as
S _ {N M} ^ {u} (f) = \phi_ {\alpha} ^ {N} S _ {\alpha \beta} ^ {q} (f) \phi_ {\beta} ^ {M}
so that the power spectral density of degree of freedom u ^ { N } is
S _ {\bar {N} \bar {N}} ^ {u} (f) = \phi_ {\alpha} ^ {\bar {N}} S _ {\alpha \beta} ^ {q} (f) \phi_ {\beta} ^ {\bar {N}}.
The PSDs for the velocity and acceleration of the same variable are
S _ {\bar {N} \bar {N}} ^ {\dot {u}} (f) = (2 \pi f) ^ {2} S _ {\bar {N} \bar {N}} ^ {u} (f)
and
S _ {\bar {N} \bar {N}} ^ {\ddot {u}} (f) = (2 \pi f) ^ {4} S _ {\bar {N} \bar {N}} ^ {u} (f).
Recall that we may typically have O ( 1 0 ^ { 1 } ) – O ( 1 0 ^ { 2 } ) eigenmodes but many more ( O ( 1 0 ^ { 3 } ) – O ( 1 0 ^ { 5 } ) ) （ physical degrees of freedom. Therefore, if many of the physical degrees of freedom are loaded (as in the case of a shell structure exposed to random acoustic noise), it may be computationally expensive to perform operations such as
S _ {\alpha \beta} ^ {f} (f) = \phi_ {\alpha} ^ {N} S _ {N M} ^ {F} (f) \phi_ {\beta} ^ {M},
which involve products over all loaded physical degrees of freedom and must be done at each frequency in the range considered.