A novel hybrid method, which simultaneously possesses the efficiency of Fourier spectral method (FSM) and the applicability of the finite element method (FEM), is presented for the vibration analysis of structures with elastic boundary conditions. The FSM, as one type of analytical approaches with excellent convergence and accuracy, is mainly limited to problems with relatively regular geometry. The purpose of the current study is to extend the FSM to problems with irregular geometry via the FEM and attempt to take full advantage of the FSM and the conventional FEM for structural vibration problems. The computational domain of general shape is divided into several subdomains firstly, some of which are represented by the FSM while the rest by the FEM. Then, fictitious springs are introduced for connecting these subdomains. Sufficient details are given to describe the development of such a hybrid method. Numerical examples of a one-dimensional Euler-Bernoulli beam and a two-dimensional rectangular plate show that the present method has good accuracy and efficiency. Further, one irregular-shaped plate which consists of one rectangular plate and one semi-circular plate also demonstrates the capability of the present method applied to irregular structures.
The needs for engineers to accurately predict performance of dynamic systems and to produce optimal designs as well as fast progress in hardware performance and constant decrease in the price of computers all lead to increasing popularity and application of the FEM in engineering. Though the FEM is applicable to problems of any geometric and material properties and boundary conditions, it can be very expensive and time consuming to solve large scale finite element models using a desktop computer, especially when the frequency involved is relatively high. When a parametric study is required in the analysis of some problems, such as for the rotor shaft-oil film bearing-flexible foundation system including several different components, remeshing and reanalysis are needed each time [
Besides simplifying a model to reduce computational workload, many attempts have been made within the last decades in order to replace or improve the conventional FEM with other methods [
For many problems such as ships and aeroplanes, the domain considered is geometrically complex and the application of the available spectral techniques to such structures would become difficult or even impossible. The main purpose of this paper is to combine the advantages of the FSM and FEM to develop a new hybrid method for problems with partly regular-shaped boundaries. Firstly, the structure is divided into two zones: one with regular boundaries described by the FSM and the other with irregular boundaries discretised by the FEM. At their interface, the coupling between these two zones is implemented by introducing several fictitious springs on the common edges. These two subsystems are coupled together using Hamilton’s equation. After these two methods are combined, it can be found that the computational efficiency is enhanced compared with the standard FEM, and limitations that the FSM can hardly be applied to complex structures are also overcome. Sections
Consider an Euler-Bernoulli beam elastically supported at its both ends. The equations of motion as well as the corresponding boundary conditions can be derived based on the energy principle, in which the total potential energy
For a beam with elastic boundary conditions, it is not easy to determine its eigenfunctions which can be used as displacement functions. In this study, the displacement function for FSM will be sought in the form of a simple series expansion as
Actually, the choice of these auxiliary functions is not unique; however, the appropriate construction of these functions can simplify the subsequent mathematical formulation significantly. These functions can be chosen as long as the following conditions are satisfied:
After briefly reviewing the main principles of FEM and FSM, development of a hybrid method from these two will be described.
As illustrated in Figure
Schematic of elastically restrained beam structure divided into FEM zone and FSM zone, with two types of fictitious springs introduced at zone interface.
The potential energy stored in these two springs can now be calculated as
When the spring constants of the fictitious springs involved are assigned very big values, the transverse displacements and rotations of the two subdomains at the interface will then be effectively equal.
Substituting (
In this section, several numerical examples are presented to show the accuracy of this hybrid method. Firstly, a straight beam with the following parameters is used in the calculation: beam length
Effect of the division ratio of the two computational domains on the modal frequencies of the beams structure with clamped-clamped boundary condition.
Mode order |
|
||||||
---|---|---|---|---|---|---|---|
|
Analytical results | 1.0 | 0.8 | 0.6 | 0.4 | 0.2 | 0.0 |
|
0.0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | |
1 | 1.5056 | 1.5056 | 1.5056 | 1.5056 | 1.5056 | 1.5056 | 1.5055 |
2 | 2.4998 | 2.4998 | 2.4997 | 2.4998 | 2.4997 | 2.4998 | 2.4996 |
3 | 3.5000 | 3.5000 | 3.5000 | 3.5000 | 3.5000 | 3.5000 | 3.5000 |
4 | 4.5000 | 4.5000 | 4.5000 | 4.5000 | 4.5000 | 4.5000 | 4.4999 |
5 | 5.5000 | 5.5000 | 5.4999 | 5.4999 | 5.4999 | 5.4999 | 5.5000 |
6 | 6.5000 | 6.5000 | 6.4998 | 6.5000 | 6.5000 | 6.4998 | 6.5001 |
7 | 7.5000 | 7.5000 | 7.4999 | 7.4998 | 7.4999 | 7.4999 | 7.5002 |
8 | 8.5000 | 8.5000 | 8.5000 | 8.4996 | 8.4997 | 8.5001 | 8.5003 |
9 | 9.5000 | 9.5000 | 9.4999 | 9.5000 | 9.5000 | 9.5000 | 9.5004 |
10 | 10.5000 | 10.4999 | 10.4993 | 10.4992 | 10.4992 | 10.4995 | 10.5007 |
Table
Comparison of computing time of the hybrid method with pure FEM and FEM with different truncated Fourier terms and/or FEM element numbers, in which a completely free-free beam structure is used.
Method | FSM truncated |
FEM element |
The 20th modal frequency | Degrees of freedom | Computing time (s) |
---|---|---|---|---|---|
Analytical | — | — | 19.50000000 | — | — |
|
|||||
FSM | 80 | — | 19.50000044 | 85 | 0.016 |
FEM | — | 600 | 19.50000073 | 1202 | 9.235 |
Hybrid | 40 | 300 | 19.49999961 | 647 | 7.66 |
FSM | 60 | — | 19.5000032 | 65 | 0.016 |
FEM | — | 500 | 19.50000153 | 1002 | 5.133 |
Hybrid | 30 | 250 | 19.50000005 | 537 | 3.947 |
FSM | 50 | — | 19.5000114 | 55 | 0.015 |
FEM | — | 250 | 19.50002436 | 502 | 0.64 |
Hybrid | 25 | 125 | 19.50001615 | 282 | 0.499 |
FSM | 30 | — | 19.50046727 | 35 | <0.01 |
FEM | — | 100 | 19.50094026 | 202 | 0.047 |
Hybrid | 15 | 50 | 19.50069514 | 122 | 0.031 |
In the first three lines of Table
Similar to the framework for vibration analysis of beam structures, vibrating plate structures with elastic edge supports can also be dealt with by the energy principle. For a rectangular plate with width
In the FEM description of a plate structure, each node has a lateral displacement and two rotations. The equation for vibration analysis of a plate structure can also be obtained in the form of (
In this study, the displacement function of a rectangular plate for the FSM will be sought in the form of series expansions as follows [
Compared with the hybrid beam model, the coefficients which couple these two methods are a little different. As illustrated in Figure
Schematic of elastically restrained plate structure divided into FEM zone and FSM zone, with two types of fictitious springs introduced at zone interface.
For the
And then the potential energy stored in linear spring can now be calculated as
The type of element used here is a 4-node incompatible plate element, in which the normal derivative at the edge is discontinuous. As a result, the rotation displacement at the edge cannot be obtained directly from derivatives of (
Substituting (
In this section, the accuracy and effectiveness of the proposed hybrid method will be demonstrated for predicting the modal behaviour of a 2-D plate structure with various boundary conditions and/or different shapes. As the first example, a rectangular plate with simply supported boundary condition at all its edges is considered. For the FSM, a simply supported edge can be treated as a special case while the stiffness coefficients for the translational springs become infinitely large and the rotational spring becomes zero. The plate dimension is 5 m wide and 10 m long with thickness
Comparison of the first ten natural frequencies of the rectangular plate.
Mode order | Analytical | FEAa | Hybridb | Difference |
---|---|---|---|---|
DOF = 2583 | DOF = 1357 | — | ||
1 | 0.9866 | 0.98624 | 0.9868 | −0.06% |
2 | 1.5785 | 1.5774 | 1.5796 | −0.14% |
3 | 2.5651 | 2.5624 | 2.5645 | −0.08% |
4 | 3.3544 | 3.3522 | 3.354 | −0.05% |
5 | 3.9463 | 3.9411 | 3.9447 | −0.09% |
6 | 3.9463 | 3.9411 | 3.9458 | −0.12% |
7 | 4.9329 | 4.9223 | 4.9275 | −0.11% |
8 | 5.7221 | 5.7131 | 5.7177 | −0.08% |
9 | 6.3141 | 6.2958 | 6.3057 | −0.16% |
10 | 7.3007 | 7.2930 | 7.2989 | −0.08% |
Note:
bResults with
Plots of the mode shape (meshed line denotes the FEM zone, while the continuous part denotes the FSM zone) for the rectangular. The (a) first, (b) second, (c) third, (d) fourth, (e) sixth, and (f) tenth mode shapes.
It is possible that the matrix
Influence of coupled springs on the former 6 mode frequencies of the rectangular plate.
In order to verify the accuracy and stability of the proposed hybrid method thoroughly, frequencies of higher modes are provided in Table
Comparison of higher natural frequencies of the rectangular plate.
Mode order | Analytical | FEAa | ERROR | Hybridb | ERROR |
---|---|---|---|---|---|
10 | 7.3007 | 7.2930 | 0.10% | 7.2989 | 0.02% |
20 | 12.8255 | 12.8055 | 0.16% | 12.8225 | 0.02% |
30 | 19.1396 | 19.0353 | 0.54% | 19.0681 | 0.37% |
40 | 23.0859 | 22.8829 | 0.88% | 22.9219 | 0.71% |
50 | 29.4000 | 29.0709 | 1.12% | 29.1143 | 0.97% |
60 | 35.5168 | 35.1455 | 1.05% | 35.1161 | 1.13% |
70 | 40.4496 | 40.0103 | 1.09% | 40.1129 | 0.83% |
80 | 45.7772 | 45.2644 | 1.12% | 45.3688 | 0.89% |
90 | 51.3020 | 51.0894 | 0.41% | 51.2793 | 0.04% |
100 | 56.8268 | 55.60429 | 2.15% | 55.7703 | 1.86% |
200 | 109.9047 | 106.3049 | 3.28% | 106.4953 | 3.10% |
Note:
bResults with
Another example of a rectangular plate (identical to the previous FSM rectangular plate with its width and length equal to 5 m) plus a matching semicircular plate (with its diameter equal to the width of the rectangular plate) is also solved and the results are presented in Table
Comparison of the first ten natural frequencies of the rectangular-half circular plate.
Mode order | FEAa | Hybridb | Difference |
---|---|---|---|
DOF = 2472 | DOF = 781 | — | |
1 | 0.8266 | 0.8265 | 0.01% |
2 | 0.8309 | 0.8311 | −0.02% |
3 | 1.876 | 1.876 | 0.00% |
4 | 1.9316 | 1.9308 | 0.04% |
5 | 2.1311 | 2.1299 | 0.06% |
6 | 2.8449 | 2.8444 | 0.02% |
7 | 3.5087 | 3.5053 | 0.10% |
8 | 3.9185 | 3.9167 | 0.05% |
9 | 4.7067 | 4.7043 | 0.05% |
10 | 4.9564 | 4.9553 | 0.02% |
Note:
bResults with
Plots of the mode shape (meshed line denotes the FEM zone, while the continuous part denotes the FSM zone) for the rectangular-circular plate. The (a) first, (b) second, (c) third, (d) fourth, (e) sixth, and (f) twelfth mode shapes.
It can also be seen from this example that the present method can be applied to structures with geometrically irregular boundary conditions. In these two 2-D examples, the validity and accuracy of the presented hybrid method are also confirmed by comparison with FEM results.
In this paper, a hybrid FEM-FSM modelling method for the vibration analysis of structures with elastic boundary conditions is proposed. The key point of this method is to divide the whole structure into FEM and FSM zones, and then fictitious springs are introduced to couple the two zones at their interface within the framework of energy principle. Comparative studies of the hybrid method with the FSM and FEM are carried out for modal characteristics, which confirm both the validity of the approach and its efficiency. The number of the degrees of freedom can also be significantly reduced compared with the conventional FEM. Although the concept of such a hybrid method is demonstrated only for a 1-D structure (Euler-Bernoulli beam) and two 2-D structures (plate) in this paper, the proposed approach can easily be extended to more complicated 3-D structures such as box-type structures, as well as structures with holes or notches.
Total potential energy
Total kinetic energy
Flexural rigidity
Young’s modulus
Second moment of inertia of the cross-section
Poisson’s ratio
Flexural displacement
Rotational displacement
Linear spring constants at
Rotational spring constants at
Total beam length, beam length of the FSM zone, and FEM zone, respectively
Angular frequency
Mass density
Cross-sectional area
Fourier series coefficients for beam
Index of Fourier series and supplemental terms
Fourier series truncation number in
Fictitious linear spring constants for beam
Fictitious rotational spring constants for beam
Plate dimension in
Plate dimension in
Double Fourier series coefficients for plate
Single Fourier series coefficients for plate
Number of elements
Four nodes number of element
Coordinates of the nodes
Index of nodes on the interface line
Total number of nodes on the interface line
Frequency parameter
Translational types of stiffness at
Translational types of stiffness at
Rotational types of stiffness at
Rotational types of stiffness at
Fictitious linear spring constants for plate
Fictitious rotational spring constants for plate
Stiffness matrix
Mass matrix.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the financial support from National Natural Science Foundation of China (Grant nos. 51375104 and 11202056) and Heilongjiang Province Funds for Distinguished Young Scientists (Grant no. JC201405). The authors would also like to acknowledge the helpful discussions with Professor W. L. Li of the Wayne State University concerning portions of Fourier spectral method.