Stability analysis definition

@STABILITY_ANALYSIS_DEFINITION {
@STABILITY_ANALYSIS_NAME {StabAnaName} {
@SIGNAL_LIST {SigName1, SigName2,... SigNameN}
@RANK_NUMBER {r}
@TIME_STEP_SIZE {Δt}
@NUMBER_OF_TIME_STEPS {N}
@PERIOD {T}
@STABILITY_ANALYSIS_TYPE {StabAnaType}
@SVD_SOLVER_METHOD {SvdMethod}
@COMMENTS {CommentText}
}
}

NOTES

  1. It is often useful to perform a stability analysis based on a number signals characterizing the dynamic response of the system. Various procedures are used to extract the stability characteristics of the system, i.e., the frequency and damping of the system's modes presenting the lowest damping rates.
  2. Plots of the stability analysis results will be generated, if requested by the plotting control parameters. Stability analysis is performed during the signal analysis phase of the analysis, as explained in Processing Signals.
  3. Stability analysis will be performed based on N signals SigName1, SigName2,... SigNameN. Note that the preconditioning and conditioning operations defined for the signal will be applied to the raw signal data before it if used for stability analysis. Furthermore, the sole time steps that were archived, as implied by the archival rate, will be used in the analysis.
  4. Since the data contained in the various signals is contaminated with noise, it will only be possible to extract a limited number of modes of the system. Typically, if the system rank number is r = 2m, m modes will be extracted, i.e., m pairs of frequency and damping will be evaluated.
  5. The stability analysis procedure operates on signals sampled at a constant time step size Δt. Since the dynamic analysis could be run with a variable time step size, see time adaptivity in section~\ref{StpCtrl: Time Adaptivity}, computed responses are linearly interpolated to create new signals with a constant time step size Δt. A time step size, Δt, or a number of time steps, N, must be defined. These two quantities are linked by the following relationship: N Δt = tf - ti, where ti and tf are the initial and final times of the signal.
  6. The stability analysis algorithm is different when dealing with constant coefficient systems or periodic coefficient systems. If T = 0, the system is assumed to be a constant coefficient system; if T > 0 the system is assumed to be a periodic coefficient system with a period T.
  7. Two approaches are available to compute the stability characteristics of the system, according to the value of the parameter StabAnaType. The default value is StabAnaType = PRONY.
    • If StabAnaType = PARTIAL_FLOQUET, the partial Floquet with projection in a subspace is used.
    • If StabAnaType = PRONY, Prony's method with projection in a subspace is used.
  8. The procedures used to evaluate the system's stability characteristics rely on the singular decomposition of the Hankel matrices H0 and H1. Two methods are available to perform this task, according to the value of the parameters SvdMethod. The default value is SvdMethod = LANCZOS.
    • If SvdMethod = LANCZOS, the singular values of the Hankel matrices are computed using the Lanczos algorithm. This algorithm only computes the largest singular values of these matrices, but these are the singular values of interest. This approach is able to deal with large size Hankel matrices.
    • If SvdMethod = BI_QR, the singular values of the Hankel matrices are computed using the bi-diagonal QR algorithm. While the QR algorithm can compute all the singular values of these matrices, its computational cost become very heavy as the size of the matrices increases. This approach should only be used when the Hankel matrices are of modest size.
  9. It is possible to attach comments to the definition of the object; these comments have no effect on its definition.

Examples

Example 1.

The following example defines a stability analysis of a wing based on two signals, SignalWingTipDis and SignalWingTipRot, that extract the tip displacement and rotation of a wing, obtained from a single sensor, SensorWingTipDispl. Both signal use the same preconditioning, PreCon, to extract a portion of the entire simulation, from 0.3 to 1.0 sec. The signals are resampled using 200 equally spaced time steps.

SIGNAL_PRECONDITIONING_DEFINITION {
@SIGNAL_PRECONDITIONING_NAME {PreCon} {
@INITIAL_TIME {0.3}
@NORMALIZED_TIME_RANGE {0.0,0.7}
}
}
SIGNAL_DEFINITION {
@SIGNAL_NAME {SignalWingTipDis} {
@SENSOR_NAME {SensorWingTipDispl}
@CHANNEL_NUMBER {3}
@SIGNAL_PRECONDITIONING_NAME {PreCon}
}
@SIGNAL_NAME {SignalWingTipRot} {
@SENSOR_NAME {SensorWingTipDispl}
@CHANNEL_NUMBER {4}
@SIGNAL_PRECONDITIONING_NAME {PreCon}
}
}
@STABILITY_ANALYSIS_DEFINITION {
@STABILITY_ANALYSIS_NAME {StabAnaPF} {
@SIGNAL_LIST {SignalWingTipDis, SignalWingTipRot}
@RANK_NUMBER {4}
@NUMBER_OF_TIME_STEPS {200}
@STABILITY_ANALYSIS_TYPE {PARTIAL_FLOQUET}
}
}