Steady-state linear dynamic analysis

This section describes steady-state linear dynamic analysis in Abaqus/Standard using a set of eigenmodes extracted in a previous eigenfrequency step to calculate the steady-state solution as a function of the frequency of the applied excitation.

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In Other Guides
Mode-Based Steady-State Dynamic Analysis

ProductsAbaqus/Standard

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 eigenfrequency 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 αth mode gives

(1)q¨α+cαq˙α+ωα2qα=1mα(f1α+if2α)exp(iΩt),

where qα is the amplitude of mode α (the αth “generalized coordinate”), cα is the damping associated with this mode (see below), ωα is the undamped frequency of the mode, mα is the generalized mass associated with the mode, and (f1α+if2α)exp(iΩt) is the forcing associated with this mode. The forcing is defined by the frequency, Ω, and the real and imaginary parts of the nodal equivalent forces, F1N and F2N, projected onto the eigenmode ϕαN:

f1α+if2α=ϕαN(F1N+iF2N).

In this equation summation is implied by the repeat of the superscript N indicating a degree of 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 α. The load vector is written in terms of its real and imaginary parts, F1N and F2N, since this is the manner in which the loading is defined in Abaqus/Standard. It is equivalently possible to write the loading in terms of its magnitude, F0N, and phase, Ψ, as FN=F0Nexpi(Ωt+Ψ), where F1N=F0cosΨ and F2N=F0sinΨ.

Several representations of modal damping are provided. Modal damping defines cα=2ξαωα, where ξα is the fraction of critical damping in the mode. Structural damping gives a damping force proportional to the modal amplitude:

cαq˙α=isαωα2qα,

where sα is the structural damping coefficient for the mode. Rayleigh damping is defined by cα=βα+γαωα2; βα and γα are the Rayleigh coefficients damping low and high frequency modes, respectively. Rayleigh damping can be reproduced exactly by modal damping as

ξα=βα2ωα+γαωα2.

Introducing all of these damping definitions into Equation 1 gives

(2)q¨α+2ξαωαq˙α+(βα+γαωα2)q˙α+isαωα2qα+ωα2qα=1mα(f1α+if2α)exp(iΩt).

The solution to this equation is

(3)qα=H0αf0αexpi(Ωt+Ψα),

where f0α=(f1α)2+(f2α)2 is the amplitude of the projected load vector and H0α(Ω) is the amplitude of the complex “transfer function” for mode α. H0αf0α defines the response in mode α from the force projection onto that mode and is defined by its real and imaginary parts as

(Hα)=1mα[f1α(ωα2-Ω2)(ωα2-Ω2)2+(ηαΩ)2+f2αηαΩ(ωα2-Ω2)2+(ηαΩ)2](Hα)=1mα[-f1αηαΩ(ωα2-Ω2)2+(ηαΩ)2+f2α(ωα2-Ω2)(ωα2-Ω2)2+(ηαΩ)2],

where ηα denotes

ηα=2ξαωα+βα+γαωα2+sαωα2Ω.

The amplitude of the response is

H0αf0α=(Hα)2+(Hα)2=1mα1(ωα2-Ω2)2+(ηαΩ)2f0α,

and the phase angle of the response is

Ψα=arctan((Hα)/(Hα)).

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

f1α=-1mαϕαNMNMe^jMa1jexp(iΩt),
f2α=-1mαϕαNMNMe^jMa2jexp(iΩt),

where MNM is the structure's mass matrix and e^jM is a vector that has unit magnitude in the direction of the base acceleration at any grounded node and is otherwise zero; a1j and a2j 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 a1=-Ωv1 and a2=-Ωv2, where v1 and v2 are the real and imaginary parts of the velocity, or a1=-Ω2u1 and a2=-Ω2u2, where u1 and u2 are the real and imaginary parts of the displacement.

The peak amplitude of any physical variable, uN, is available from the modal amplitudes as

uN=αϕαNqα.

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.