Table 6.3.11–1 Standard octave bands.

Band number	Band center (frequency, Hz)
1	1.0
2	2.0
3	4.0
4	8.0
5	16.0
6	31.5
7	63.0
8	125.0
9	250.0
10	500.0
11	1000.0
12	2000.0
13	4000.0
14	8000.0
15	16000.0

If you have data in terms of an alternative frequency scale (e.g., one-third octave band), an equivalent full octave band power reference value can be obtained as described in “Random response analysis,” Section 2.5.8 of the Abaqus Theory Guide. in decibels must be specified as a function of the frequency band; the associated midband frequencies are given in Table 6.3.11–1. # Alternate methods for defining frequency functions You can define a frequency function in an external file or in a user subroutine. Defining the frequency function in an external file The data to define a frequency function can be contained in an external file. Input File Usage: \*PSD-DEFINITION, NAME=name, TYPE=type, INPUT=file name Abaqus/CAE Usage: Load module: Create Amplitude; Type: PSD Definition; Specification units: Power, Decibel, or Gravity; Real, Imaginary, Frequency Defining the frequency function in a user subroutine Complicated frequency functions can be more easily defined by user subroutine UPSD than by entering data directly. Input File Usage: \*PSD-DEFINITION, NAME=name, TYPE=type, USER Any data lines given will be ignored if the USER parameter is specified. Abaqus/CAE Usage: Load module: Create Amplitude; Type: PSD Definition; Specification units: Power or Gravity; toggle on Specify data in an external user subroutine # Defining the correlation You define the cross-correlation between the applied nodal loads or base motions. You can also assign scaling (weight) factors to the frequency functions through the cross-correlation definition. Distributed loads are converted to equivalent nodal loads, which are treated as individual point loads with respect to the cross-correlation. The cross-correlation is defined in the random response step and references a particular load case number and frequency function. Three types of correlation can be defined: correlated, uncorrelated, and moving noise. As many correlations as needed to define the random loading can be specified unless the moving noise type is chosen, in which case only one correlation can appear in the step definition. • For the correlated type all terms in the cross-spectral density matrix are considered, which implies that the loads on all degrees of freedom within the load case are fully correlated (statistically dependent on each other). • For the uncorrelated type only diagonal terms in the cross-spectral density matrix are considered, which implies that no correlation exists between the load on one degree of freedom and the load on another. You should exercise caution when choosing the uncorrelated type with distributed loads since the equivalent nodal forces would be uncorrelated with each other (statistically independent). • For the moving noise type the terms in the correlation matrix depend on the relative position of the points where the loads are applied. This type can be used only in conjunction with concentrated point loads and distributed loads. In addition, the moving noise formulation assumes that the frequency function referenced by the cross-correlation defines a reference power spectral density function of the noise source. (It is a reference power spectral density because it can later be scaled by the magnitude of the loadings specified as distributed, concentrated point, or connector element loads.) Since the power spectral density is real-valued for real-valued variables, the frequency function must not contain imaginary terms when used with the moving noise type of cross-correlation. Input File Usage: Use one of the following options to define the correlation: \*CORRELATION, TYPE=CORRELATED, PSD=name \*CORRELATION, TYPE=UNCORRELATED, PSD=name \*CORRELATION, TYPE=MOVING NOISE For the moving noise type the reference to the power spectral density function must be given on each data line. Abaqus/CAE Usage: Load module; Create Boundary Condition; Step: random\_response\_step; Category: Mechanical; Types for Selected Step: Displacement base motion or Velocity base motion or Acceleration base motion; Correlation tabbed page: toggle on Specify correlation; Approach: Correlated or Uncorrelated; PSD: psd\_amplitude\_name # Specifying whether the correlation matrix is complex For correlated or uncorrelated cross-correlations you can specify whether or not both real and imaginary terms will be included in the spatial correlation matrix. This specification does not affect the imaginary terms given for the power spectral density frequency function. Input File Usage: Use one of the following options: \*CORRELATION, TYPE=CORRELATED, COMPLEX=YES or NO, PSD=name \*CORRELATION, TYPE=UNCORRELATED, COMPLEX=YES or NO, PSD=name Abaqus/CAE Usage: Load module; Create Boundary Condition; Step: random\_response\_step; Category: Mechanical; Types for Selected Step: Displacement base motion or Velocity base motion or Acceleration base motion; Correlation tabbed page: toggle on Specify correlation; Approach: Correlated or Uncorrelated; PSD: psd\_amplitude\_name; Real; Imaginary # Alternate methods for defining a correlation You can define a correlation in an external input file or in a user subroutine. Defining the correlation in an external input file The data to define a correlation can be contained in an external input file. Input File Usage: \*CORRELATION, TYPE=type, PSD=name, INPUT=file\_name Abaqus/CAE Usage: You cannot define a correlation in an external file in Abaqus/CAE. Defining the correlation in a user subroutine Simple excitations, such as uncorrelated white noise, are easily defined. Excitations involving more complicated correlations, including cases where the elements of the CSD matrix have different frequency dependencies, can be defined through user subroutine UCORR. If the user subroutine is specified, only the load case number must be entered as part of the correlation definition. A user subroutine cannot be used to define a moving noise correlation. For uncorrelated cross-correlations only the diagonal terms of the correlation matrix specified in UCORR will be used. The combination of the cross-correlation with the various kinds of applied loads is discussed in more detail below.

Input File Usage:	Use one of the following options:*CORRELATION, TYPE=CORRELATED, USER, COMPLEX=YES or NO, PSD=name*CORRELATION, TYPE=UNCORRELATED, USER, PSD=name
Abaqus/CAE Usage:	Load module; Create Boundary Condition; Step: random_response_step; Category: Mechanical; Types for Selected Step: Displacement base motion or Velocity base motion or Acceleration base motion; Correlation tabbed page: toggle on Specify correlation; Approach: User

# Selecting the modes and specifying damping You can select the modes to be used in modal superposition and specify damping values for all selected modes. # Selecting the modes You can select modes by specifying the mode numbers individually, by requesting that Abaqus/Standard generate the mode numbers automatically, or by requesting the modes that belong to specified frequency ranges. If you do not select the modes, all modes extracted in the prior eigenfrequency extraction step, including residual modes if they were activated, are used in the modal superposition.

Input File Usage:	Use one of the following options to select the modes by specifying mode numbers: *SELECT EIGENMODES, DEFINITION=MODE NUMBERS *SELECT EIGENMODES, GENERATE, DEFINITION=MODE NUMBERS Use the following option to select the modes by specifying a frequency range: *SELECT EIGENMODES, DEFINITION=FREQUENCY RANGE
Abaqus/CAE Usage:	You cannot select the modes in Abaqus/CAE; all modes extracted are used in the modal superposition.

# Specifying damping Damping is almost always specified for a random response analysis (see “Material damping,” Section 26.1.1). If damping is absent, the response of a structure will be unbounded if the forcing frequency is equal to an eigenfrequency of the structure. To get quantitatively accurate results, especially near natural frequencies, accurate specification of damping properties is essential. The various damping options available are discussed in “Material damping,” Section 26.1.1. You can define a damping coefficient for all or some of the modes used in the response calculation. The damping coefficient can be given for a specified mode number or for a specified frequency range. When damping is defined by specifying a frequency range, the damping coefficient for a mode is interpolated linearly between the specified frequencies. The frequency range can be discontinuous; the average damping value will be applied for an eigenfrequency at a discontinuity. The damping coefficients are assumed to be constant outside the range of specified frequencies.

Input File Usage:

Use the following option to define damping by specifying mode numbers:*MODAL DAMPING, DEFINITION=MODE NUMBERSUse the following option to define damping by specifying a frequency range:*MODAL DAMPING, DEFINITION=FREQUENCY RANGE

Abaqus/CAE Usage:

Use the following input to define damping by specifying mode numbers:Step module:Create Step:Linear perturbation:Random response:DampingDefining damping by specifying frequency ranges is not supported in Abaqus/CAE.

# Example of specifying damping Figure 6.3.11–3 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input:

*MODAL DAMPING, DEFINITION=FREQUENCY RANGE

f1, d1

f2, d2

f2, d3

f3, d3

f4, d4


line

| frequency | damping values | | --------- | -------------- | | f₁ | d₁ | | λ₁ | d₂ | | f₂ | d₃ | | f₃ | d₄ | | f₄ | d₄ | | λ₃ | d₄ |

Figure 6.3.11–3 Damping values specified by frequency range. # Rules for selecting modes and specifying damping coefficients The following rules apply for selecting modes and specifying modal damping coefficients: • No modal damping is included by default. • Mode selection and modal damping must be specified in the same way, using either mode numbers or a frequency range. • If you do not select any modes, all modes extracted in the prior frequency analysis, including residual modes if they were activated, will be used in the superposition. • If you do not specify damping coefficients for modes that you have selected, zero damping values will be used for these modes. • Damping is applied only to the modes that are selected. • Damping coefficients for selected modes that are beyond the specified frequency range are constant and equal to the damping coefficient specified for the first or the last frequency (depending which one is closer). This is consistent with the way Abaqus interprets amplitude definitions. # Initial conditions It is not appropriate to specify initial conditions in a random response analysis. # Boundary conditions It is not possible to prescribe nonzero displacements and rotations directly as boundary conditions (“Boundary conditions in Abaqus/Standard and Abaqus/Explicit,” Section 34.3.1) in mode-based dynamic response procedures. Therefore, in a random response analysis the motion of nodes can be specified only as base motion; nonzero displacement, velocity, or acceleration history definitions given as boundary conditions are ignored, and any changes in the support conditions from the eigenfrequency extraction step are flagged as errors. In addition, any amplitude definitions are ignored in a random response analysis. The method for prescribing motion in modal superposition procedures is described in “Transient modal dynamic analysis,” Section 6.3.7. In random response analysis only a single (primary) base can be defined. # Defining multiple load cases The excitation defined by the base motion is assigned to numbered load cases. These load cases are then referenced in the cross-correlation definition. The load cases are associated with frequency functions through the reference in the cross-correlation definition. Any number of load cases can be defined, but load case number 1 cannot be used if distributed loads are defined in the same step. Input File Usage: \*BASE MOTION, LOAD CASE=n Abaqus/CAE Usage: Base motions with load cases are not supported in Abaqus/CAE. # Converting base motion excitation to a cross-spectral density matrix When the excitation is provided by a base motion, it is converted directly into a cross-spectral density matrix projected onto the eigenspace through the modal participation factors (see “Natural frequency extraction,” Section 6.3.5), giving $$ S _ {\alpha \beta} (f) = \left\{ \begin{array}{l l} \sum_ {J} g _ {J} ^ {2} P ^ {J} (f) \sum_ {i} \sum_ {j} \Gamma_ {\alpha} ^ {i} \Gamma_ {\beta} ^ {j} \sum_ {I} \Psi_ {i j} ^ {I J} (2 \pi f) ^ {2 \lambda_ {I}} & \mathrm{for} \alpha < \beta , \\ \left[ S _ {\beta \alpha} (f) \right] ^ {*} & \mathrm{for} \alpha > \beta , \\ \mathrm{Re} \left(\sum_ {J} g _ {J} ^ {2} P ^ {J} (f) \sum_ {i} \sum_ {j} \Gamma_ {\alpha} ^ {i} \Gamma_ {\beta} ^ {j} \sum_ {I} \Psi_ {i j} ^ {I J} (2 \pi f) ^ {2 \lambda_ {I}}\right) & \mathrm{for} \alpha = \beta , \end{array} \right. $$ where the superscript \* denotes complex conjugate and where $\Gamma _ { \alpha } ^ { i }$ is the modal participation factor for mode in excitation direction $i ( i \mathrm { = } 1 \mathrm { - } 6 )$ ; $P ^ { J } ( f )$ is the frequency function referenced by the Jth cross-correlation and defined as a function of the frequency f in g units; $\Psi _ { i j } ^ { I J }$ is a matrix of weight factors indicating the fraction of $P ^ { J }$ to be associated with the correlation between base motion in directions i and j for load case $I ,$ as described below; $\lambda _ { I }$ , 1, or 2, depending on whether the base motion corresponding to load case I is defined in terms of an acceleration spectrum, a velocity spectrum, or a displacement spectrum (see “Transient modal dynamic analysis,” Section 6.3.7); and $g _ { J }$ is the user-specified acceleration of gravity for the same power spectral density frequency function that defines $P ^ { J }$ . If the cross-correlation is defined in user subroutine UCORR, $\Psi _ { i j } ^ { I J }$ is defined in the user subroutine. Otherwise, $\begin{array} { r l } { \Psi _ { i j } ^ { I J } } & { { } \quad = a ^ { I J } } \end{array}$ for all if the excitation is correlated or $\mathit { \Delta } = \mathit { a } ^ { I J } \delta _ { i j }$ if the excitation is uncorrelated, where $a ^ { I J }$ is the (complex) value of the weight factor by which to scale the frequency function $P ^ { J }$ used in load case I. # Loads The loading for random response analysis is defined in general terms by the cross-spectral density matrix $S _ { ( N , i ) ( M , j ) } ( f )$ , where $f$ is frequency in cycles per time and the subscripts $( N , i )$ and $( M , j )$ refer to degree of freedom i at node $N$ and degree of freedom $j$ at node M, respectively. Distributed loads are converted to equivalent nodal loads, which—for the formulation of the correlation matrix—are treated in the same way as concentrated point loads. The units of $S _ { ( N , i ) ( M , j ) } ( f )$ are $( \mathrm { f o r c e } ) ^ { 2 }$ or (moment)2 per frequency. In addition, any amplitude references on the concentrated point, connector element, or distributed load definitions are ignored in a random response analysis. # Defining multiple load cases Distributed loads will be assigned automatically to load case number 1. You assign a concentrated point load or connector element load to a numbered load case. Any number of concentrated point and connector element load cases can be specified, but load case number 1 cannot be used for a concentrated point or connector element load if a distributed load is present in the same step. The concentrated point, connector element, and distributed load cases are associated with frequency functions through the cross-correlation definition. Input File Usage: Use one or more of the following options: \*CLOAD, LOAD CASE=n $\ast \mathrm { C O N N E C T O R \ L O A D } , \mathrm { L O A D \ C A S E } { = } m$ \*DLOAD # Correlated and uncorrelated loading For correlated or uncorrelated cross-correlations, the cross-spectral density matrix is defined as $$ S _ {(N, i) (M, j)} (f) = \left\{ \begin{array}{l l} \sum_ {J} P ^ {J} (f) \sum_ {I} C _ {(N, i) (M, j)} ^ {I J} F _ {(N, i)} ^ {I} F _ {(M, j)} ^ {I} & \text {for} (N, i) < (M, j), \\ \left[ S _ {(M, i) (N, j)} (f) \right] ^ {*} & \text {for} (N, i) > (M, j), \\ \operatorname{Re} \left(\sum_ {J} P ^ {J} (f) \sum_ {I} C _ {(N, i) (M, j)} ^ {I J} F _ {(N, i)} ^ {I} F _ {(M, j)} ^ {I}\right) & \text {for} (N, i) = (M, j), \end{array} \right. $$ where the superscript \* denotes complex conjugate and where $$ \begin{array}{l l} F _ {(N, i)} ^ {I} & \text {is the load magnitude applied to degree of freedom i at node N for load case I ;} \\ P ^ {J} (f) & \text {is the frequency function referenced by the Jth cross - correlation and defined as a function of the frequency f in power (force) or decibel units ; and} \end{array} $$ $$ C _ {(N, i) (M, j)} ^ {I J} \quad \text { is a matrix of weight factors indicating the fraction of } P ^ {J} \text { to be associated with the } (N, i) (M, j) \text { cross - correlation term for load case } I, \text { as described below. } $$ If the cross-correlation is defined in user subroutine UCORR, is defined in the user subroutine. $C _ { ( N , i ) ( M , j ) } ^ { I J }$ Otherwise, $$ C _ {(N, i) (M, j)} ^ {I J} = a ^ {I J} \text { for all } (N, M, i, j) \text { if the excitation is correlated or } $$ $$ = a ^ {I J} \delta_ {N M} \delta_ {i j} \text { if the excitation is uncorrelated, } $$ where $a ^ { I J }$ is the (complex) value of the weight factor by which to scale the frequency function $P ^ { J }$ used in load case I. # Moving noise loading For moving noise cross-correlations, the cross-spectral density matrix is defined as $$ S _ {(N, i) (M, j)} (f) = \sum_ {I} P ^ {I} (f) \exp \left(i 2 \pi f (\mathbf {x} _ {N} - \mathbf {x} _ {M}) \cdot \mathbf {v} ^ {I} \frac {1}{\mathbf {v} ^ {I} \cdot \mathbf {v} ^ {I}}\right) F _ {(N, i)} ^ {I} F _ {(M, j)} ^ {I}, $$ where $F _ { ( N , i ) } ^ { I }$ is the load magnitude applied to degree of freedom i at node N for load case I; $P ^ { I } ( f )$ is the reference power spectral density function associated with load case I and defined as a function of the frequency f in power (force) or decibel units; $\mathbf { v } ^ { I }$ is the velocity vector of noise propagation given for load case ${ \cal I } _ { , } ^ { , }$ and $\mathbf { x } _ { N }$ are the coordinates of node N. This definition of moving noise implies that the different noise sources have no cross-correlation. Therefore, it is most generally used with only one noise source $( I = 1 \ \mathrm { o n l y } )$ . In addition, since $P ^ { I } ( f )$ （2 is the actual power spectral density of the moving noise source, it must be defined as a real-valued function. # Predefined fields Predefined fields, including temperature, cannot be used in random response analysis. # Material options As in any dynamic analysis procedure, mass or density (“Density,” Section 21.2.1) must be assigned to some regions of any separate parts of the model where dynamic response is required. The following material properties are not active during a random response analysis: plasticity and other inelastic effects, rate-dependent properties, thermal properties, mass diffusion properties, electrical properties, and pore fluid flow properties (see “General and linear perturbation procedures,” Section 6.1.3). # Elements Other than generalized axisymmetric elements with twist, any of the stress/displacement elements in Abaqus/Standard can be used in a random response analysis (see “Choosing the appropriate element for an analysis type,” Section 27.1.3). # Output In random response analysis the value of a variable is its power spectral density; all of the output variables in Abaqus/Standard are listed in “Abaqus/Standard output variable identifiers,” Section 4.2.1. Power spectral density values are not available for concentrated and distributed loads and for SINV. Options are also provided in random response analysis to obtain root mean square values for certain variables, as listed below. Total values include base motion, while relative values are measured relative to the base motion. Element integration point variables: RS Root mean square of all stress components. RE Root mean square of all strain components. Element nodal point variables:

MISES	Mises equivalent stress..
RMISES	Root mean square of Mises equivalent stress.

For connector elements, the following element output variables are available:

RCTF	Root mean square of connector total forces.
RCEF	Root mean square of connector elastic forces.
RCVF	Root mean square of connector viscous forces.
RCRF	Root mean square of connector reaction forces.
RCSF	Root mean square of connector friction forces.
RCU	Root mean square of connector relative displacements.
RCCU	Root mean square of connector constitutive displacements.

Nodal variables:

RU	Root mean square values of all components of the relative displacement/rotation at a node.
RTU	Root mean square values of all components of the total displacement/rotation at a node.
RV	Root mean square values of all components of the relative velocity at a node.
RTV	Root mean square values of all components of the total velocity at a node.
RA	Root mean square values of all components of the relative acceleration at a node.
RTA	Root mean square values of all components of the total acceleration at a node.
RRF	Root mean square values of all components of reaction forces and reaction moments at a node.

No energy values are available for a random response analysis. To reduce the computational cost of random response analysis, you should request output only for selected element and node sets. Abaqus/Standard will calculate the response for only the element and nodal variables requested. When MISES or RMISES output is requested, Abaqus/Standard stores the needed data in the output database (.odb) file and Abaqus/Viewer does the actual computation of the responses. These computations require element stress output in the frequency step preceding the random response step. Note that specifying the name of the element set in the output request in the random response step has no effect on these two output variables. If MISES or RMISES output for a selected set of elements is desired, the name of that element set needs to be specified for the element stress output request in the preceding frequency step. Unlike in other procedures, MISES and RMISES output for random response analysis is computed at the element nodal points and not at the element integration points.