Residual modes for modal response analysis

This example illustrates the use of the residual modes capability in Abaqus and verifies the solution accuracy.

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ProductsAbaqus/Standard

In many modal response analyses, simplifying a model by reducing the number of degrees of freedom or extracting only a small subset of eigenmodes is often a common practice. These assumptions are beneficial for cost reductions, but the accuracy of the modal solution may suffer. To improve solution accuracy, the method of residual modes (see Natural Frequency Extraction) can be employed. This method extracts an additional set of modes based on loading conditions to help correct for errors introduced by mode truncation. Residual modes are orthogonal to retained eigenmodes and to each other and are computed when specified in the frequency step.

Problem description

A simple multiple-degree-of-freedom spring-mass system is used to demonstrate the capability of using residual modes to obtain high solution accuracy. The model consists of 4 masses and 5 springs, as shown in Figure 1. The assembled mass and stiffness matrices are as follows:

M=(100001000010000.5),
K=10,000⁢(2-100-12-100-12-100-12).

The mass for node 4 is set to half the value of the other three nodes so as to have four distinct modes for the system. A spatial loading of unit force R is applied to node 3 in the y-direction, where

R=(0010).

The eigenfrequencies and corresponding eigenmodes are given in the following table:

Mode No.Frequency (Hz)Nodal Eigendisplacements
Node 1Node 2Node 3Node 4
1 10.155 0.39948 0.63631 0.61408 0.03418
2 20.222 0.68548 0.26428 –0.58359 –0.48927
3 28.258 0.60461 –0.69678 0.19839 0.46815
4 34.963 0.07056 –0.19939 0.49292 –1.19360

The spatial loading is applied harmonically with an excitation frequency of 3 Hz to verify the steady-state response of the system. The single residual mode corresponding to the excitation load is included in the projected basis. A modal damping factor of 0.02 is applied to all the modes including the residual modes.

Results and discussion

Only one eigenmode is extracted to demonstrate the capability of improving the solution accuracy by extracting residual modes. The residual mode (RM) obtained by Abaqus is identical to that given in the reference.

Mode No.Frequency (Hz)Nodal Eigendisplacements
Node 1Node 2Node 3Node 4
Published solutions
1 10.155 0.39948 0.63631 0.61408 0.03418
RM 21.865 0.68548 0.26428 –0.58359 –0.48927
Abaqus
1 10.155 0.39948 0.63631 0.61408 0.03418
RM 21.865 0.68548 0.26428 –0.58359 –0.48927

For the 3 Hz harmonic response analysis, displacements and accelerations of all the nodes are presented for two cases. The first case uses only the first eigenmode, while the second case uses both the first eigenmode and the residual mode. The percentage error shows very clearly how solution accuracy can be significantly improved by adding the residual modes to the original set of eigenvectors.

 Displacements
Node 1Node 2Node 3Node 4
Published results (all modes) 4.52E−5 8.89E−5 1.29E−4 6.53E−5
Abaqus solutions
Case 1 (mode 1 only) 6.60E−5 1.05E−4 1.01E−4 5.65E−5
Case 2 (mode 1 with RM) 4.53E−5 8.88E−5 1.29E−4 6.51E−5
Percentage error
Case 1 46.02 18.11 –21.71 –13.63
Case 2 0.22 –0.11 0.00 –0.31
 Accelerations
Node 1Node 2Node 3Node 4
Published results (all modes) 1.61E−2 3.16E−2 4.59E−2 2.32E−2
Abaqus solutions
Case 1 (mode 1 only) 2.34E−2 3.73E−2 3.60E−2 2.00E−2
Case 2 (mode 1 with RM) 1.61E−2 3.16E−2 4.60E−2 2.31E−2
Percentage error
Case 1 45.34 18.04 –21.57 –13.79
Case 2 0.00 0.00 0.22 –0.43

References

  1. Dickens,  J. M., J. M. Nakagawa, and M. M. Wittbrodt, “A Critique of Mode Acceleration and Modal Truncation Augmentation Methods for Modal Response Analysis,” Computers & Structures, vol. 62, no. 6, pp. 985–998, 1997.

Figures

Figure 1. A four-degree-of-freedom spring-mass model.