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Procedures
In contrast, operations such as
S _ {\alpha \beta} ^ {q} (f) = H _ {\bar {\alpha}} (f) S _ {\bar {\alpha} \bar {\beta}} ^ {f} (f) H _ {\bar {\beta}} ^ {*} (f)
are relatively inexpensive since they are done independently for each combination of modes.
Forming
S _ {\bar {N} \bar {N}} ^ {u} (f) = \phi_ {\alpha} ^ {\bar {N}} S _ {\alpha \beta} ^ {q} (f) \phi_ {\beta} ^ {\bar {N}}
may be expensive if we choose to compute the results for a large selection of physical variables (displacements, velocities, accelerations, stresses, etc.). ABAQUS/Standard will calculate the response only for the element and nodal variables requested. However, if *RESTART is requested with the *RANDOM RESPONSE procedure, all variables are computed at the requested restart frequency, which can add substantially to the computational cost. The user is advised to write the restart file for the last increment only. To reduce the computational cost of random response analysis,
ABAQUS/Standard assumes that the cross-spectral density matrix for the loading can be separated into a frequency-dependent scalar function (containing the units of the CSD) and a set of coupling terms that are independent of frequency as follows:
S _ {N M} ^ {F} (f) = \sum_ {J} P ^ {J} (f) \sum_ {I} \Psi_ {N M} ^ {I J} \qquad \mathrm{for} N < M,
S _ {N M} ^ {F} (f) = \left[ S _ {M N} ^ {F} (f) \right] ^ {*} \quad \text {for} N > M,
S _ {N N} ^ {F} (f) = \Re \left(\sum_ {J} P ^ {J} (f) \sum_ {I} \Psi_ {N N} ^ {I J}\right).
Here J is the number of *CORRELATION options included in the *RANDOM RESPONSE step. Each such option references an input complex frequency function, P ^ { J } ( f ) . The spatial cross-correlations are then defined by the complex set of values \Psi _ { N M } ^ { I J } .
With this approach the cross-spectral density function for the generalized loads, S _ { \alpha \beta } ^ { f } ( f ) , can be constructed as
S _ {\alpha \beta} ^ {f} (f) = \sum_ {J} P ^ {J} (f) \left(\phi_ {\alpha} ^ {N} \phi_ {\beta} ^ {M} \sum_ {I} \Psi_ {N M} ^ {I J}\right) \qquad \mathrm{for} N < M,
S _ {\alpha \beta} ^ {f} (f) = \left[ S _ {\beta \alpha} ^ {f} (f) \right] ^ {*} \quad \text { for } N > M,
S _ {\alpha \alpha} ^ {f} (f) = \Re \left[ \sum_ {J} P ^ {J} (f) \left(\phi_ {\alpha} ^ {N} \phi_ {\alpha} ^ {M} \sum_ {I} \Psi_ {N M} ^ {I J}\right) \right].
Since the \phi _ { \alpha } ^ { N } \phi _ { \beta } ^ { M } \sum _ { I } \Psi _ { N M } ^ { I J } are not functions of frequency, they can be computed once only, leaving the frequency-dependent operations to be done only in the space of the eigenmodes.
Although this procedure is not natural for typical correlated loadings (like road excitation or jet noise), loadings can always be defined this way by using enough *CORRELATION options. The approach then reduces the computational cost for models with many loaded physical degrees of freedom. The approach works well for uncorrelated and fully correlated loadings, which are quite common cases.
Decibel conversion
ABAQUS/Standard allows the user to provide an input PSD (say P ( f ) ) in decibel units rather than units of power/frequency. There are various ways to convert from decibel units to units of power/frequency, depending on how the frequencies in one octave band are related to the frequencies in the next. A general formula relates the center (midband) frequencies f _ { \mathrm { c } } between octaves as
\frac {f _ {\mathrm{c}} ^ {(i)}}{f _ {\mathrm{c}} ^ {(i - 1)}} = 2 ^ {x},
where the superscript (i) denotes ith octave band and x is a chosen value. For example, x = 1 for full octave band conversion, and x = 1 / 3 for one-third octave band conversion. ABAQUS/Standard uses full octave band conversion to convert from decibel units to units of power/frequency. For full octave band conversion, as shown by the above equation, the center frequency doubles from octave band to octave band.
Since decibel units are based upon log scales, the center frequency for an octave band bounded by lower frequency f _ { \mathrm { L } } and upper frequency f _ { \mathrm { U } } is given by
\log f _ {\mathrm{c}} = \frac {1}{2} \log f _ {\mathrm{U}} + \frac {1}{2} \log f _ {\mathrm{L}}.
Since f _ { \mathrm { U } } / f _ { \mathrm { L } } = 2 ^ { x } , we can easily show that
f _ {\mathrm{U}} = f _ {\mathrm{c}} 2 ^ {0. 5 x}
and
f _ {\mathrm{L}} = f _ {\mathrm{c}} 2 ^ {- 0. 5 x}.
Thus, the change in frequency within any given octave band is
\Delta f = f _ {\mathrm{U}} - f _ {\mathrm{L}} = f _ {\mathrm{c}} [ 2 ^ {0. 5 x} - 2 ^ {- 0. 5 x} ].
To convert from one type of conversion formula to the next, we need the following general decibel to power/frequency conversion equation:
d b (f) = 1 0 \log \frac {P (f)}{P _ {\mathrm{ref}} ^ {(x)} / \Delta f},
Procedures
where P _ { \mathrm { r e f } } ^ { ( x ) } is a reference power value. The subscript (x) means that the power reference is given for the type of conversion represented by x (e.g., full octave band conversion for x = 1 or one-third octave band conversion for x = 1 / 3 ) . When x = 1 , we will simply use the notation P _ { \mathrm { r e f } } . Thus, since ABAQUS/Standard uses a full octave band conversion, \Delta f = f _ { \mathrm { c } } / \sqrt { 2 } and
P (f) = \frac {\sqrt {2}}{f _ {\mathrm{c}}} P _ {\mathrm{ref}} 1 0 ^ {d b (f) / 1 0}.
The PSD data can be given with respect to some other type of octave band frequency scale. In that case we can convert the PSD data at those frequencies coinciding with the full octave band scale by computing an equivalent full octave band reference power based on the following ratio:
\frac {P _ {\mathrm{ref}}}{P _ {\mathrm{ref}} ^ {(x)}} = \frac {1}{\sqrt {2} [ 2 ^ {0 . 5 x} - 2 ^ {- 0 . 5 x} ]}.
For example, if we are given P (1=3)ref ( P _ { \mathrm { r e f } } ^ { ( 1 / 3 ) } i.e., one-third octave band frequency scale), the equivalent full octave band reference power value would be
P _ {\mathrm{ref}} = 3. 0 5 3 6 P _ {\mathrm{ref}} ^ {(1 / 3)}.
This conversion would be valid only at the one-third octave band center frequencies that coincide with the full octave band center frequencies. Thus, only every third data point should be considered.
2.5.9 Base motions in modal-based procedures
Structures subjected to ground motion by earthquakes or other excitations such as explosions or dynamic action of machinery are examples in which support motions may have to be considered in the analysis of dynamic response. For modal-based dynamic analyses using the *MODAL DYNAMIC, *STEADY STATE DYNAMICS, and *RANDOM RESPONSE procedures, the support motions are simulated by prescribed excitations called base motions that are applied to the suppressed degrees of freedom. The suppressed degrees of freedom are grouped into one or more bases by using the BASE NAME parameter on the *BOUNDARY option. Multiple bases are required if base motions cannot be described by a single set of rigid body motions. A common case is that of a bridge whose supports are subjected to the same earthquake record but with a time shift.
The degrees of freedom that are suppressed without being assigned to a named base make up the primary base, which typically is the only base if the motion can be described by a single set of rigid body motions. The suppressed degrees of freedom that are associated with named boundary conditions make up the secondary base or bases. ABAQUS/Standard uses different approaches to handle primary and secondary base motions. The modal participation method is used for primary base motions, and the "big mass" method is used for secondary base motions. Multiple bases can be used only in the *MODAL DYNAMIC and *STEADY STATE DYNAMICS procedures.
Primary base motions
Procedures
Let us consider structural motions relative to the base motion, u _ { b } . The total response, \left\{ { { u } _ { t } } \right\} , of the dynamic system will now consist of the relative response, \{ u \} , and the applied base motion excitation, \left\{ { { u } _ { b } } \right\} :
\{u _ {t} \} = \{u \} + \{u _ {b} \},
with similar expressions for velocities and accelerations. Substituting \left\{ { { u } _ { t } } \right\} in the linearized equation of motion gives
[ M ] \{\ddot {u} \} + [ C ] \{\dot {u} \} + [ K ] \{u \} = - [ M ] \{\ddot {u} _ {b} \}.
The base acceleration is converted into applied inertia loads - [ M ] \{ \ddot { u } _ { b } \} . Here it has been assumed that there is no damping on rigid body modes (i.e., Rayleigh damping with \alpha \neq 0 is not allowed). If the prescribed excitation is given in the form of a displacement or a velocity, ABAQUS/Standard differentiates it to obtain the acceleration. The base motion vector can be expressed in terms of the rigid body mode vectors, \{ T \} _ { j } , and time dependent base motion values, z _ { j } , j = 1 , 2 , . . . , 6 :
\{u _ {b} \} = \sum_ {j = 1} ^ {6} \{T \} _ {j} z _ {j}.
Projecting the equation of motion into the eigenspace we have
\ddot {q} _ {m} + \frac {c _ {m}}{m _ {m}} \dot {q} _ {m} + \frac {k _ {m}}{m _ {m}} q _ {m} = - \sum_ {j = 1} ^ {6} (\Gamma_ {m}) _ {j} \ddot {z} _ {j},
where q _ { m } and \{ \phi \} _ { m } denote the relative generalized coordinate and mode shape for the mode m ; k _ { m } , c _ { m } and m _ { m } are modal stiffness, modal damping, and modal mass, respectively; and
\left(\Gamma_ {m}\right) _ {j} = \frac {1}{m _ {m}} \{\phi \} _ {m} ^ {T} [ M ] \{T \} _ {j}
is the modal participation factor for mode m and degree of freedom j .
Kinematic conditions defined under the *BOUNDARY option when it is used without the BASE NAME parameter in a *FREQUENCY step cannot be changed in any of the subsequent modal-based procedures. The kinematic constraints are built into the eigenvectors and into the participation factors for each mode, which implies that all degrees of freedom in the primary base must be subjected to the same rigid body motion.
The participation factors are used to calculate the equivalent forcing function, and the equation of motion is solved for the relative quantities (such as relative displacements, relative velocities, and relative accelerations--output variables U, V, and A, respectively). To obtain total kinematic quantities (such as total displacements, total velocities, and total accelerations--output variables TU, TV, and TA, respectively), the primary base motions are added to the relative responses.
Secondary base motions
The base motion treatment described above cannot be applied to secondary bases. Instead, ABAQUS/Standard uses a "big mass" approach to simulate the motion of secondary bases. In this approach a big mass (much bigger than the total mass of the structure) is added to each degree of freedom in a secondary base during the *FREQUENCY step. This generates additional low frequency modes associated with the masses M _ { b i g } . As more big masses are applied, more low frequency modes will be extracted in the frequency analysis step. To keep the number of frequencies of interest the same, the number of eigenvalues extracted is automatically increased. Hence, the size of the subspace will grow proportionally to the number of degrees of freedom associated with secondary bases.
The desired base motion is obtained by applying a point force to each degree of freedom in the modal superposition step:
P _ {s} ^ {N} = M _ {b i g} \ddot {u _ {s}} ^ {N},
where M _ { b i g } is the big mass and \ddot { u _ { s } } ^ { N } is the applied acceleration prescribed for degree of freedom N associated with secondary supports.
Using the notation used in the equation of motion for primary base motions, the equation of motion for combined primary and secondary base motions is readily written as
[ M ] \{\ddot {u} \} + [ C ] \{\dot {u} \} + [ K ] \{u \} = - [ M ] \{\ddot {u} _ {b} \} + \{P _ {s} \},
with
\{P _ {s} \} = \sum_ {i} [ M _ {b i g} ^ {i} ] \{\ddot {u} _ {s} ^ {i} \},
where [ M _ { b i g } ^ { i } ] is the diagonal matrix containing the big masses for secondary base i and \{ \ddot { u _ { s } ^ { i } } \} is the base motion applied to this base. The mass matrix [M ] now contains the mass of the structure as well as the big masses associated with the secondary bases. Projecting the equation of motion into the eigenspace (expanded by the low frequency modes) we obtain
\ddot {q} _ {m} + \frac {c _ {m}}{m _ {m}} \dot {q} _ {m} + \frac {k _ {m}}{m _ {m}} q _ {m} = - \sum_ {j = 1} ^ {6} (\Gamma_ {m}) _ {j} \ddot {z} _ {j} + \frac {1}{m _ {m}} \{\phi \} _ {m} ^ {T} \{P _ {s} \}.
Again, the quantities solved for are relative to the primary support, including those obtained at the secondary supports.
The big masses should be chosen as large as possible to obtain accurate base motions but should not be so large as to cause excessive round-off errors or overflows. To provide six digits of numerical accuracy, ABAQUS/Standard chooses each big mass equal to 1 0 ^ { 6 } times the total mass of the structure and each big rotary inertia equal to 1 0 ^ { 6 } times the total moment of inertia of the structure.
The big masses, which are introduced in the *FREQUENCY step, are not included in the model for other steps in a multiple step analysis. Hence, the total mass of the structure and the printed messages about masses and inertia of the entire model are not affected. However, the presence of the masses will be noticeable in the output tables printed in the eigenvalue extraction step, as well as in the information for the generalized masses and effective masses. See ``Mode-based steady-state dynamic analysis,'' Section 6.3.6 of the ABAQUS/Standard User's Manual, and ``Double cantilever subjected to multiple base motions,'' Section 1.4.12 of the ABAQUS Benchmarks Manual, for further details about the use of the base motion feature.
2.6 Complex harmonic oscillations
2.6.1 Direct steady-state dynamic analysis
For structures subjected to continuous harmonic excitation, ABAQUS/Standard offers a "direct" steady-state dynamic analysis procedure in addition to the "modal" procedure described in ``Steady-state linear dynamic analysis,'' Section 2.5.7, and the "subspace" procedure described in ``Subspace-based steady-state dynamic analysis,'' Section 2.6.2. This procedure belongs to the perturbation procedures, where the perturbed solution is obtained by linearization about the current base state. For the calculation of the base state, the structure may exhibit material and geometrical nonlinear behavior as well as contact nonlinearities. Viscous damping can be included in the procedure, using the Rayleigh damping coefficients specified under the material definition. Discrete damping (such as dashpot elements) can be included. The procedure can also be used for coupled acoustic-structural medium analysis (as described in ``Coupled acoustic-structural medium analysis,'' Section 2.9.1), with piezoelectric medium (as described in ``Piezoelectric analysis,'' Section 2.10.1), and with viscoelastic material modeling (as described in ``Frequency domain viscoelasticity,'' Section 4.7.3). All properties can be frequency-dependent.
The formulation is based on the dynamic virtual work equation,
Equation 2.6.1-1
\int_ {V} \rho \delta \mathbf {u} \cdot \ddot {\mathbf {u}} d V + \int_ {V} \rho \alpha_ {c} \delta \mathbf {u} \cdot \dot {\mathbf {u}} d V + \int_ {V} \delta \pmb {\varepsilon}: \pmb {\sigma} d V - \int_ {S _ {\mathrm{t}}} \delta \mathbf {u} \cdot \mathbf {t} d S = 0,
where u_ and uÄ are the velocity and the acceleration, ½ is the density of the material, \alpha _ { c } is the mass proportional damping factor (part of the Rayleigh damping assumption), ¾ is the stress, t is the surface traction, and ±" is the strain variation that is compatible with the displacement variation ±u. The discretized form of this equation is
\delta u ^ {N} \bigg \{M ^ {N M} \ddot {u} ^ {M} + C _ {(m)} ^ {N M} \dot {u} ^ {M} + I ^ {N} - P ^ {N} \bigg \} = 0,
Equation 2.6.1-2
where the following definitions apply:
Equation 2.6.1-3
Procedures
M ^ {N M} = \int_ {V} \rho \mathbf {N} ^ {N} \cdot \mathbf {N} ^ {M} d V \qquad \mathrm{isthemassmatrix},
C _ {(m)} ^ {N M} = \int_ {V} \rho \alpha_ {c} \mathbf {N} ^ {N} \cdot \mathbf {N} ^ {M} d V \qquad \mathrm{isthemassdampingmatrix},
I ^ {N} = \int_ {V} \boldsymbol {\beta} ^ {N}: \boldsymbol {\sigma} d V \quad \text {is the internal load vector},
P ^ {N} = \int_ {S _ {\mathrm{t}}} \mathbf {N} ^ {N} \cdot \mathbf {t} d S \qquad \mathrm{istheexternalloadvector}.
For the steady-state harmonic response we assume that the structure undergoes small harmonic vibrations about a deformed, stressed state, defined by the subscript 0. Since steady-state dynamics belongs to the perturbation procedures, the load and response in the step define the change from the base state. The change in internal force vector follows by linearization:
\Delta I ^ {N} = \int_ {V} \left[ \Delta \pmb {\beta} ^ {N}: \pmb {\sigma} + \pmb {\beta} ^ {N}: \Delta \pmb {\sigma} \right] d V.
The change in stress can be written in the form
\Delta \pmb {\sigma} = \mathbf {D} ^ {e l}: (\Delta \pmb {\varepsilon} + \beta_ {c} \Delta \dot {\pmb {\varepsilon}}),
where \mathbf { D } ^ { e l } is the elasticity matrix for the material and \beta _ { c } is the stiffness proportional damping factor (the other part of the Rayleigh damping assumption). The strain and strain rate changes follow from the displacement and velocity changes:
\Delta \varepsilon = \boldsymbol {\beta} ^ {M} \Delta u ^ {M}, \quad \Delta \dot {\varepsilon} = \boldsymbol {\beta} ^ {M} \Delta \dot {u} ^ {M}.
This allows us to write Equation 2.6.1-2 as
Equation 2.6.1-4
\delta u ^ {N} \bigg \{M ^ {N M} \ddot {u} ^ {M} + (C _ {(m)} ^ {N M} + C _ {(k)} ^ {N M}) \dot {u} ^ {M} + K ^ {N M} u ^ {M} - P ^ {N} \bigg \} = 0,
where we have defined the stiffness matrix
K ^ {N M} = \int_ {V} \left[ \frac {\partial \pmb {\beta} ^ {N}}{\partial u ^ {M}}: \pmb {\sigma} _ {0} + \pmb {\beta} ^ {N}: \mathbf {D} ^ {e l}: \pmb {\beta} ^ {M} \right] d V
and the stiffness damping matrix
C _ {(k)} ^ {N M} = \int_ {V} \beta_ {c} \boldsymbol {\beta} ^ {N}: \mathbf {D} ^ {e l}: \boldsymbol {\beta} ^ {M} d V.
For harmonic excitation and response we can write
\Delta u ^ {M} = \left(\Re \left(u ^ {M}\right) + i \Im \left(u ^ {M}\right)\right) \exp i \Omega t
and
\Delta P ^ {N} = \left(\Re \left(P ^ {N}\right) + i \Im \left(P ^ {N}\right)\right) \exp i \Omega t,
where \Re \left( u ^ { M } \right) and \Im \left( u ^ { M } \right) are the real and imaginary parts of the amplitudes of the displacement, < \left( P ^ { N } \right) and \mathfrak { F } \left( P ^ { N } \right) are the real and imaginary parts of the amplitude of the force applied to the structure and − is the circular frequency. Substituting the expressions for harmonic excitation and response in Equation 2.6.1-4 and writing the result in matrix form yields
Equation 2.6.1-5
\left[ \begin{array}{c c} \Re \left[ A ^ {N M} \right] & \Im \left[ A ^ {N M} \right] \\ \Im \left[ A ^ {N M} \right] & - \Re \left[ A ^ {N M} \right] \end{array} \right] \left\{ \begin{array}{c} \Re \left(u ^ {M}\right) \\ \Im \left(u ^ {M}\right) \end{array} \right\} = \left\{ \begin{array}{c} \Re \left(P ^ {N}\right) \\ - \Im \left(P ^ {N}\right) \end{array} \right\},
where
\Re \left[ A ^ {N M} \right] = K ^ {N M} - \Omega^ {2} M ^ {N M}
\Im \left[ A ^ {N M} \right] = - \Omega (C _ {(m)} ^ {N M} + C _ {(k)} ^ {N M}).
Note that both the real and imaginary parts of A ^ { N M } are symmetric.
The procedure is activated by including the DIRECT parameter on the *STEADY STATE DYNAMICS option. Both real (LOAD CASE=1) and imaginary (LOAD CASE=2) loads can be defined.
As output ABAQUS/Standard provides amplitudes and phases for all element and nodal variables at the requested frequencies. For this procedure all *AMPLITUDE references must be given in the frequency domain.
2.6.2 Subspace-based steady-state dynamic analysis
For structures subjected to continuous harmonic excitation, ABAQUS/Standard offers a "subspace" steady-state dynamic analysis procedure in addition to the "modal" procedure described in ``Steady-state linear dynamic analysis,'' Section 2.5.7, and the "direct" procedure described in ``Direct steady-state dynamic analysis,'' Section 2.6.1. The procedure is activated by including the SUBSPACE PROJECTION parameter on the *STEADY STATE DYNAMICS option. This procedure is a perturbation procedure, where the perturbed solution is obtained by linearization about the current base state. For the calculation of the base state the structure may exhibit material and geometrical nonlinear behavior as well as contact nonlinearities. Viscous damping can be included in the procedure using the Rayleigh damping coefficients specified under the material definition. The procedure can also be used for viscoelastic material modeling (``Frequency domain viscoelasticity,'' Section 4.7.3). Discrete damping (such as dashpots) can be included. It cannot be used for coupled acoustic-structural medium analysis. The main advantage of this method is that it allows frequency-dependent behavior to be considered at a relatively small cost increase over the purely linear analysis via the "modal" procedure
Procedures
described in ``Steady-state linear dynamic analysis,'' Section 2.5.7.
The discretized form of the linearized dynamic virtual work equation can be written as
Equation 2.6.2-1
\delta u ^ {N} \left\{M ^ {N M} \ddot {u} ^ {M} + C ^ {N M} \dot {u} ^ {M} + K ^ {N M} u ^ {M} - P ^ {N} \right\} = 0,
where the following definitions apply:
M ^ {N M} = \int_ {V} \rho \mathbf {N} ^ {N} \cdot \mathbf {N} ^ {M} d V \qquad \mathrm{isthemassmatrix},
K ^ {N M} = \int_ {V} \left[ \frac {\partial \pmb {\beta} ^ {N}}{\partial u ^ {M}}: \pmb {\sigma} _ {0} + \pmb {\beta} ^ {N}: \mathbf {D} ^ {e l}: \pmb {\beta} ^ {M} \right] d V \quad \mathrm{isthestiffnessmatrix},
P ^ {N} = \int_ {S _ {\mathbf {t}}} \mathbf {N} ^ {N} \cdot \mathbf {t} d S \qquad \mathrm{istheexternalloadvector,}
C ^ { N M } is the damping matrix, u_ and uÄ are the velocity and the acceleration, \rho is the density of the material, \pmb { \sigma } _ { 0 } is the stress in the base state, t is the surface traction, and \mathbf { D } ^ { e l } is the elasticity matrix for the material. We assume that both the stiffness K ^ { N M } and the damping C ^ { N M } are frequency-dependent.
The *FREQUENCY step prior to the *STEADY STATE DYNAMICS, SUBSPACE PROJECTION analysis has extracted \scriptstyle n _ { e i g } eigenmodes of the undamped system using
(- \omega^ {2} M ^ {M N} + K ^ {M N}) \phi^ {N} = 0,
where \omega is the eigenfrequency in radians/time. The procedure assumes that the complex displacement changes for the damped system can be written in the form
Equation 2.6.2-2
u ^ {M} = \phi_ {\beta} ^ {M} a _ {\beta}, \quad \beta = 1, \dots , n _ {e i g},
where a _ { \beta } are the complex modal amplitudes. Using Equation 2.6.2-2 in Equation 2.6.2-1 and pre-multiplying with \phi _ { \alpha } , the equation of motion projected onto the subspace is provided
M _ {\alpha \beta} \ddot {a} _ {\beta} + C _ {\alpha \beta} (\Omega) \dot {a} _ {\beta} + K _ {\alpha \beta} (\Omega) a _ {\beta} = P _ {\alpha} (\Omega) \quad \alpha , \beta = 1, \dots , n _ {e i g},
Equation 2.6.2-3
where
M _ {\alpha \beta} = \phi_ {\alpha} ^ {N} M ^ {N M} \phi_ {\beta} ^ {M},
K _ {\alpha \beta} (\Omega) = \phi_ {\alpha} ^ {N} K ^ {N M} (\Omega) \phi_ {\beta} ^ {M},
C _ {\alpha \beta} (\Omega) = \phi_ {\alpha} ^ {N} C ^ {N M} (\Omega) \phi_ {\beta} ^ {M},
P _ {\alpha} (\Omega) = \phi_ {\alpha} ^ {N} P ^ {N} (\Omega),
and − is the excitation frequency. Since the eigenmodes are not orthogonal to the damping and stiffness matrices in the equilibrium equations (because of the frequency-dependent properties), the projected damping and stiffness matrices are not diagonal.
For harmonic excitation and response we can write
a _ {\beta} = \left(\Re \left(a _ {\beta}\right) + i \Im \left(a _ {\beta}\right)\right) \exp i \Omega t \quad \text {and} \quad P _ {\alpha} = \left(\Re \left(P _ {\alpha}\right) + i \Im \left(P _ {\alpha}\right)\right) \exp i \Omega t,
where \Re \left( a _ { \beta } \right) and \Im \left( a _ { \beta } \right) are the real and imaginary parts of the modal amplitudes and \Re \left( P _ { \alpha } \right) and \Im \left( P _ { \alpha } \right) are the real (LOAD CASE=1) and imaginary (LOAD CASE=2) parts of the amplitude of the force applied to the structure after projection onto the subspace. Substituting the expressions for harmonic excitation and response in Equation 2.6.2-3 and writing the result in matrix form yields
\left[ \begin{array}{c c} \Re \left[ A _ {\alpha \beta} \right] & \Im \left[ A _ {\alpha \beta} \right] \\ \Im \left[ A _ {\alpha \beta} \right] & - \Re \left[ A _ {\alpha \beta} \right] \end{array} \right] \left\{ \begin{array}{c} \Re \left(a _ {\beta}\right) \\ \Im \left(a _ {\beta}\right) \end{array} \right\} = \left\{ \begin{array}{c} \Re \left(P _ {\alpha}\right) \\ - \Im \left(P _ {\alpha}\right) \end{array} \right\},
where
\Re \left[ A _ {\alpha \beta} \right] = K _ {\alpha \beta} - \Omega^ {2} M _ {\alpha \beta},
\Im \left[ A _ {\alpha \beta} \right] = - \Omega C _ {\alpha \beta}.
The equation is solved for the real and imaginary part of the complex modal amplitudes a _ { \beta } , and Equation 2.6.2-2 can be used to compute the real and imaginary part of the nodal displacements. As output ABAQUS/Standard provides amplitudes and phases for all element and nodal variables at the requested frequencies. All *AMPLITUDE references must be given in the frequency domain.
2.7 Steady-state transport analysis
2.7.1 Steady-state transport analysis
ABAQUS/Standard provides a specialized analysis capability to model the steady-state behavior of a cylindrical deformable body rolling along a flat rigid surface. The capability uses a reference frame that removes the explicit time dependence from the problem so that a purely spatially dependent analysis can be performed. For an axisymmetric body traveling at a constant ground velocity and constant angular rolling velocity, a steady state is possible in a frame that moves at the speed of the ground velocity but does not spin with the body in the rolling motion. This choice of reference frame allows the finite element mesh to remain stationary so that only the part of the body in the contact zone requires fine meshing.
Kinematics of steady-state rolling
The kinematics of the rolling problem are described in terms of a coordinate frame that moves along with the ground motion of the body. In this moving frame the rigid body rotation is described in a spatial or Eulerian manner and the deformation in a material or Lagrangian manner. It is this kinematic