In order to validate the proposed method, three cases were studied and compared with the results published in Refs.6,16,20.
The cantilevered shaft carrying single rotary inertia, located at \(x_{{m,1}} /l = 1.0\) or 0.5 respectively, were studied first (Fig. 4).
Figure 4
The cantilevered shaft carrying single rotary inertia.
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The dimensions and physical properties of the circular shaft studied in case one are the same as those in Ref.20 (\(l = 40\;{\text{in}}\), \(D = 1\;{\text{in}}\), \(G = 1.2 \times 10^{7} \;{\text{psi}}\), \(\rho = 0.283\;{\text{lbm}}/{\text{in}}^{3}\)). For convenience, non-dimensional parameters were introduced: \(I_{{m,\;i}}^{*} = \rho I_{p} l/J_{{m,\;i}}\), \(\xi = x/l\), \(\xi _{i} = x_{{m,\;i}} /l\), \(\beta = \omega /l\).
Substituting \(I_{m,\;1}^{*}\), \(\xi\), \(\xi_{1}\) into Eq. (29). Then non-dimensional frequency coefficients can be obtained. When \(\xi_{1} = 1\), the Eigen equation is given as Eq. (31), which is the same as it in Ref.6.
$$\beta \tan \beta = I_{m,1}^{*} .$$
(31)
When \(\xi_{1} = 0.5\), the Eigen equation is given as
$$\beta \tan \beta = 2I_{m,1}^{*} .$$
(32)
In order to compare the results of AM with the corresponding ones obtained by Gorman16 and Chen20, the lowest five non-dimensional frequency coefficients, \(\beta_{j}\) (j = 1–5) were shown in Table 1. Table 1 shows that the results of AM, Gorman16, and Chen20 are in good agreement, which means that the proposed method is accurate and effective for a simple model.
Table 1 The lowest five non-dimensional frequency coefficients for a cantilever shaft carrying a single rotary inertia.
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Further, to verify the applicability of the proposed method in complex models, case two was adopted. Hereby, a shaft carrying five inertias, as shown in Fig. 5 was studied. The dimensions and physical properties of the shaft are the same as in case one. Locations of rotary inertia are \(\xi_{1} = 0.1\), \(\xi_{2} = 0.3\), \(\xi_{3} = 0.5\), \(\xi_{4} = 0.7\), and \(\xi_{5} = 0.9\). Magnitudes of rotary inertia are \(I_{m,\;i}^{*} = 1.0\) (\(i = 1{-}5\)). Meanwhile, boundary conditions of PP and FP were considered. The lowest five natural frequencies, \(\Omega_{j}\) (\(j = 1 - 5\)), of the shaft were compared with FEM and Chen20, as shown in Table 2, and the corresponding mode shapes of twisting angles were shown in Figs. 6 and 7.
Figure 5
The shaft carrying five rotary inertias.
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Table 2 The lowest five natural frequencies for the shaft carrying five rotary inertia.
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Figure 6
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the PP support condition obtained from Chen20 and AM (This figure was created via Matlab 2015, https://ww2.mathworks.cn/products.html?s_tid=gn_ps, sub-images in Figs. 6, 7, 13, 14, 16, 17, 18, 19, 20 were also created by Matlab 2015).
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Figure 7
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the FP support conditions obtained from Chen20 and AM.
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The differences between \(\Omega_{AM}\) and \(\Omega_{{{\text{Re}} {\text{ference}}}} \;(\Omega_{FEM} ,\Omega_{Chen} )\) shown in the parentheses ( ) of Table 2 were calculated with the formula: \(\varepsilon _{j} = (\Omega _{{\text{Re} {\text{ference, }}j}} - \Omega _{{AM{\text{, }}j}} )/\Omega _{{AM{{, }}j}}\), where \(\Omega_{FEM,j}\) and \(\Omega_{Chen,j}\) denote the j-th natural frequencies of the shafts carrying ‘‘five’’ torsional springs obtained from AM, Chen20 and FEM, respectively. From Table 2 one finds that values of \(\Omega_{{AM{, }j}}\) agreement with \(\Omega_{FEM,j}\) and \(\Omega_{Chen,j}\) (for PP and FP shafts), hence the accuracy of the AM is good. It also can be found that values of \(\varepsilon_{j}\) between \(\Omega_{Chen,j}\) and \(\Omega_{AM}\) are all smaller than those between \(\Omega_{FEM,j}\) and \(\Omega_{AM}\). This is because the FEM model is approximate, and the calculation accuracy of natural frequency can be improved by improving the degree of freedom of the FEM model.
Figures 6 and 7 show the lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the support conditions: PP and PF, obtained from AM and Chen20. Mode shapes’ variation characteristics with \(x/l\), obtained by AM, are almost identical to the ones calculated by Chen20. However, for the same order mode, the mode shape obtained via AM is slightly different from Chen’s at the points where rotary inertias exist, which should be blamed on the application of the conclusion from Chen20 and the way of mapping. In Ref.20 it was pointed out that mode shapes of the shafts carrying five inertias are almost coincident with the ones for the shaft without carrying any concentrated elements and the former was not shown. To compare the results of the mode shapes, the aforementioned conclusion was directly used in this paper. The mapping way of mode shape was combing limited discrete points with trend lines. As a result, if the number of discrete points is small, a similar conclusion to Ref.20 can be obtained via AM, as shown in Figs. 8 and 9. Once the number of discrete points is increased, the aforementioned conclusion is no longer accurate (shown in Figs. 6, 7). In fact, concentrated elements have a significant influence on mode shapes, which can be found in many pieces of literatures7,21,25,26,27.
Figure 8
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the PP support conditions obtained from Chen20 and AM (small number of discrete points).
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Figure 9
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the FP support conditions obtained from Chen20 and AM (small number of discrete points).
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To verify the accuracy of the mode shapes obtained via the presented method, FEM was employed (solved by Ansys 14.0 https://www.ansys.com/zh-cn/products/structures/ansys-mechanical). The comparison of the lowest five mode shapes obtained by FEM with 51-element uniform discrete and AM was given in Figs. 10 and 11. While modeling by FEM, a simplified model of a uniform discrete section should include a sufficient number of elements with higher-order interpolation functions. Here, beam188 with quadric shape functions for 3-D (3-node) line element was applied. From Figs. 10 and 11, it can be seen that AM results are in good agreement with FEM results, indicating that the function of the mode shape via the proposed method is accurate.
Figure 10
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the PP support conditions obtained from FEM and AM (small number of discrete points).
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Figure 11
The lowest five mode shapes of twisting angles for the shaft carrying five concentrated elements in the FP support conditions obtained from FEM and AM (small number of discrete points).
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The findings in Table 2 and Figs. 6, 7, 8, 9, 10 and 11 indicate that the proposed method is accurate and effective for complex models. The results also show that concentrated elements have a significant influence on the modal shape. This phenomenon has also been found in Ref.7,21,25,26,27, which is another evidence of the accuracy of AM. The reason for the phenomenon is given in “Explanation of Phenomenon 5” section.
To determine the engineering applicability of the method, case three was employed. A free boundary shaft with one moving concentrated inertia, as shown in Fig. 12, was studied. The value of parameters are: \(N_{m} = 1\), \(l = L = 1\;{\text{m}}\), \(D = 0.1\;{\text{m}}\), \(H_{m,1} = 0.1\;{\text{m}}\), \(R_{m,1} = 0.2\;{\text{m}}\), \(\rho = 7850\;{\text{kg}}/{\text{m}}^{3}\), \(G = 81.5 \times 10^{9} \;{\text{Pa}}\), \(x_{m,\;1} = x_{m}\), \(J_{m,1} = J_{m}\). The results were compared with the lumped mass method (LMM) shown in Fig. 13.
Figure 12
The FF shaft with one moving concentrated inertia.
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Figure 13
The lowest six natural frequencies of the FF shaft carrying one moving concentrated inertia.
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From Fig. 13, it is seen that the lowest six natural frequencies of the FF shaft carrying one concentrated inertia analyzed by AM are the same as those obtained by LMM, which suggests that the presented method is applicable in practice.
According to the aforementioned comparisons, it is believed that the AM is an effective way for the title problem.