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The variable section structure could be the physical model of many vibration problems, and its analysis becomes more complicated either. It is very important to know how to obtain the exact solution of the modal function and the natural frequency effectively. In this paper, a general analytical method, based on segmentation view and iteration calculation, is proposed to obtain the modal function and natural frequency of the beam with an arbitrary variable section. In the calculation, the section function of the beam is considered as an arbitrary function directly, and then the result is obtained by the proposed method that could have high precision. In addition, the total amount of calculation caused by high-order Taylor expansion is reduced greatly by comparing with the original Adomian decomposition method (ADM). Several examples of the typical beam with different variable sections are calculated to show the excellent calculation accuracy and convergence of the proposed method. The correctness and effectiveness of the proposed method are verified also by comparing the results of the several kinds of the theoretical method, finite element simulation, and experimental method.

The vibration energy harvesters often use the cantilever beam for the basic oscillator because of its simple structure, low rigidity, high energy density of bending vibration, and so on. The variable section structure shows some special performance, so it has excellent prospects in the field of the vibration piezoelectric energy harvester and attracts wide attention [

It is necessary to solve the modal function and natural frequency for studying the vibration characteristics of the beam [

Adomian decomposition method (ADM) includes series fitting to carry out multiorder iteration, and thus obtains a high-precision result [

Transfer matrix method (TM) is a kind of theoretical calculation method with strong expansibility. This method is used to associate the components with continuous conditions in a system, so as to achieve the purpose of analyzing the whole system. Fakoukakis and Kyriacou [

In the process of solving, ADM could obtain high-precision results although there exists the problem of large computation caused by high-order Taylor expansion. If the idea of piecewise solution in TM can be applied, the problem of excessive iteration calculation in the original ADM could be solved effectively. Therefore, the ADM is improved in this paper, and the Adomian decomposition-transfer matrix method (ADTM) is proposed which can be used to solve the mode function and natural frequency of the more general type of the beam with a variable section. The analysis idea of this improved analytic method is that it divides the whole beam into finite sections and builds the mode functions of each section first; then each discrete section is solved by association. The ADTM can not only obtain high-precision results of mode function and natural frequency of the beam with an arbitrary variable section, but also shorten the time required for calculation greatly.

Based on Euler–Bernoulli beam theory, the force analysis of transverse vibration of beams is shown in Figure

Force analysis of the Euler–Bernoulli beam. (a) Beam structure. (b) Microsegment d

The force and moment balance equations could be established according to the D’Alembert’s principle. The modal function and natural frequency of the beam could be gained by analyzing free vibration [

The Galerkin method is used to discretize the deflection function:

Substituting equation (

By substituting equation (

By integrating both sides of equation (

Building the iterative modal function [

In equation (

As shown in equation (

The original ADM is applied to discuss the fitting error of beams with different sections under the same order expansion. Taylor fitting calculation has unavoidable defects, which is the fitting error becomes larger when it is away from the expansion base point. If the error needs to be reduced, it should be increased in the order of expansion and then the term of expansion is also increased. According to the iterative character of the ADM method, the expansion term of the fitting function is multiplied continuously in the iterative process, which will lead to huge calculation. In the finite order, there will be some fitting errors between the fitting function and the original function.

Taking

Structure model and fitting error _{0} = 0.2, _{L} = 0.08,

In Figures

In order to reveal more intuitively the influence of Taylor fitting on the calculation quantity of ADM, the calculation quantity is simply estimated now. Because the Taylor fitting order _{n}. It can be seen that the variation of the expansion order

In addition, it can be found also that in some cases even though the order of Taylor series is up to _{m}

Failure example of ADM. (a) Concave beam with width function

In order to solve the problems of excessive calculation amount and failure mentioned above, a new analytical calculation method, ADTM, is proposed in this paper. The proposed method could divide the whole structure into some segments, and then length of each segment is smaller than the overall length. In this way, it could solve the disadvantage of fitting functions

Firstly, divide the whole structure into

Whole structure with

Natural frequency is the inherent property of the whole structure. Then, the natural frequencies of the monolithic structure before and after the section are the same, which can be expressed as

Substituting equation (

The relationship between both sides of iteration (

Then, substituting equation (

Combining equations (

The left side of equation (

Parameter

In this section, the cantilever structure is taken as an example to verify the excellence of ADTM by comparing the results obtained by several different methods. What needs illustration is that both ADM and ADTM are used to approximate the modal function and natural frequency by series expansion, so there are two kinds of errors in both methods when they are used to solve vibration equations. One is the truncation error caused by the finite series expansion of the method itself; the other is the fitting error caused by the Taylor series introduced in equation (

The unit mass and stiffness of the uniform beam are constant, so there is no Taylor fitting error term, when ADM or ADTM is used to solve the vibration equation. Then, the solution error is only caused by the truncation of the series expansion of the method itself. Table ^{10}.

Comparison of results from ADTM and ADM in the uniform beam case.

Frequency | Order of iteration | ADM | ADTM | FEM result | Theoretical result | |
---|---|---|---|---|---|---|

( | ( | |||||

_{1} | 79.0341 | 78.2747 | 78.1178 | 77.7096 | 78.0726 | |

78.0743 | 78.0727 | 78.0726 | ||||

78.0726 | 78.0726 | 78.0726 | ||||

Ω_{2} | 417.6492 | 454.8542 | 492.8109 | 487.1444 | 489.2727 | |

456.9292 | 488.6935 | 489.2750 | ||||

488.3615 | 489.2715 | 489.2727 | ||||

Ω_{3} | — | 1290.2621 | 1309.2891 | 1362.8536 | 1369.9775 | |

— | 1350.1889 | 1365.6756 | ||||

1095.9032 | 1369.6495 | 1369.9640 |

In Table

By using the natural frequency obtained in Table

The first three modal functions obtained by ADTM with

In order to understand more clearly, the error can be expressed as

Generally speaking, errors from different methods are not in the same order of magnitude. Figure

The errors of modal functions from different methods. (a) First modal, (b) second modal, and (c) third modal.

In addition, the experimental method is also used to further verify the modal solved by ADTM. Figure

Comparison of ADTM and experimental results. (a) Vibration modal. (b) Amplitude frequency response.

The trapezoidal convex beam mentioned in Figure _{1} = 10, _{2} = 10, and _{3} = 11, respectively. The comparison of the fitting error before and after segmentation is shown in Figure

Comparison of results from ADTM and ADM in the trapezoid convex beam case.

ADM | ADTM _{1} _{2} _{3} | |||||
---|---|---|---|---|---|---|

Order of iteration | _{1} (rad | _{2} (rad | _{3} (rad | _{1} (rad | _{2} (rad | _{3} (rad |

101.1648 | 502.4120 | 1349.6159 | 99.2913 | 533.8116 | 1420.8928 | |

99.8467 | 523.7288 | 1527.6533 | 99.2814 | 533.8033 | 1420.9502 | |

99.4269 | 531.2438 | 1347.9822 | 99.2802 | 533.8025 | 1420.9597 | |

99.3134 | 533.2523 | 1455.0812 | 99.2801 | 533.8025 | 1420.9609 | |

99.2868 | 533.6983 | 1426.7344 | 99.2801 | 533.8024 | 1420.9610 | |

99.2813 | 533.7847 | 1421.9121 | 99.2801 | 533.8024 | 1420.9610 |

Comparison of results from various methods in four cases.

Beam model | Frequency | FEM | ADTM _{1} _{2} _{3} | ADM ( | Semianalytical method ( | |||
---|---|---|---|---|---|---|---|---|

Result (rad | Result (rad | Error (‰) | Result (rad | Error (‰) | Result (rad | Error (‰) | ||

Trapezoidal beam | _{1} | 102.1261 | 101.8169 | 3.0 | 101.8159 | 3.0 | 101.3213 | 7.9 |

_{2} | 535.4677 | 533.3845 | 3.9 | 533.3869 | 3.9 | 530.4472 | 9.4 | |

_{3} | 1423.3859 | 1415.5921 | 5.4 | 1415.5946 | 5.5 | 1407.6242 | 11.1 | |

Trapezoidal convex beam | _{1} | 99.5983 | 99.2801 | 3.1 | 99.2813 | 3.1 | 98.7987 | 8.1 |

_{2} | 536.2030 | 533.8025 | 4.5 | 533.7847 | 4.5 | 530.2409 | 11.1 | |

_{3} | 1428.8301 | 1420.9610 | 5.5 | 1421.9121 | 4.8 | 1410.5328 | 12.8 | |

Concave beam | _{1} | 73.9735 | 74.2340 | 3.5 | — | — | 74.5634 | 7.9 |

_{2} | 499.8834 | 497.9771 | 3.8 | — | — | 496.4176 | 6.9 | |

_{3} | 1385.9541 | 1378.6030 | 5.3 | — | — | 1374.8716 | 7.9 |

Analysis of the trapezoid convex beam. (a) Structure model. (b) Error comparison of ADTM and ADM.

In Table ^{−1} order of magnitude, and the calculation speed is 2.5 times faster than that of the traditional method. From the point of view of the convergence radius, if the result is converged to 10^{−2} order of magnitude, the iteration order of the traditional method needs to reach

The concave section beam cannot be calculated by ADM that is mentioned above and shown in Figure _{1} = 11, and _{2} = 11; case two, _{1} = 10, _{2} = 10, and _{3} = 11; case three, _{1} = 11, _{2} = 10, _{3} = 10, and _{4} = 11.

Comparison of results from ADTM and the semianalytical method in the trapezoid concave beam case.

Segments | Order of iteration | ADTM | Semianalytical method | ||||
---|---|---|---|---|---|---|---|

_{1} (rad | _{2} (rad | _{3} (rad | _{1} (rad | _{2} (rad | _{3} (rad | ||

73.5041 | 491.8213 | 1362.2632 | 78.0726 | 489.2727 | 1369.9777 | ||

74.1666 | 497.3769 | 1372.2632 | |||||

74.2328 | 497.3795 | 1376.9487 | |||||

74.7595 | 497.1698 | 1377.4637 | 76.4328 | 488.4799 | 1350.0823 | ||

74.2509 | 497.9426 | 1378.4681 | |||||

74.2340 | 497.9771 | 1378.6030 | |||||

74.6325 | 497.3214 | 1378.1042 | 75.5023 | 495.5865 | 1355.1130 | ||

74.2414 | 497.9611 | 1378.5935 | |||||

74.2337 | 497.9768 | 1378.6141 |

In Table

In order to illustrate the general applicability of the proposed method, three methods are used to solve the more common trapezoid beam, as shown in Figure

Comparison of results from ADTM and ADM in the trapezoid beam case.

ADM ( | ADTM (_{1} _{2} _{3} | |||||
---|---|---|---|---|---|---|

Order of iteration | _{1} ( | _{2} ( | _{3} ( | _{1} ( | _{2} ( | _{3} ( |

101.1324 | 536.5036 | 1598.5717 | 101.8157 | 533.3674 | 1415.5740 | |

101.6175 | 534.5213 | 1497.6120 | 101.8161 | 533.3827 | 1415.5901 | |

101.7673 | 533.6933 | 1424.1163 | 101.8162 | 533.3844 | 1415.5919 | |

101.8057 | 533.4538 | 1416.5559 | 101.8162 | 533.3846 | 1415.5921 | |

101.8142 | 533.3980 | 1415.6736 | 101.8162 | 533.3846 | 1415.5921 | |

101.8159 | 533.3869 | 1415.5946 | 101.8162 | 533.3846 | 1415.5921 |

Comparison of results from ADTM and the semianalytical method in the trapezoid beam case.

ADTM ( | Semianalytical method | |||||
---|---|---|---|---|---|---|

Segments | _{1} ( | _{2} ( | _{3} ( | _{1} ( | _{2} ( | _{3} ( |

101.8150 | 533.3786 | 1415.5773 | 94.4962 | 496.5471 | 1373.9867 | |

101.8162 | 533.3827 | 1415.5901 | 98.4109 | 514.6177 | 1370.6740 | |

101.8162 | 533.3828 | 1415.5902 | 99.8686 | 522.2463 | 1387.4444 |

In Table ^{−2} order of magnitude, and the calculation speed is twice than that of ADM. From the point of view of the convergence radius, if the result converges to the 10^{−2} order of magnitude, then the iterative order of ADM needs to be ^{−2} order of magnitude. When ^{−5} order of magnitude. Moreover, the result from the semianalytical method converges to that from ADTM with

The results of the four examples above are compared together and listed in Table

In this paper, a new analytical method ADTM is proposed to solve modal function and natural frequency of a beam with a variable section. The proposed method could effectively combine advantages of high accuracy and fast calculation. Firstly, the uniform beam is analyzed to verify the correctness of the proposed method. The results from ADTM, the complete solution, FEM, and experiment are compared to show the accuracy of ADTM. Then, the excellent calculation accuracy and convergence of the ADTM are proved by comparing the results from different analytical methods with those of FEM under several typical examples. Compared with the ADM, the calculation time is shorter and the solution accuracy is higher by using the proposed method. In addition, some special structure models with a variable section which cannot be solved by ADM could also be analyzed. It is also further proved that ADTM is more universal for the excellent results.

The ADTM method is not restricted by traditional modal analytical methods. Based on the same principle, the modal functions and natural frequencies of the arbitrary variable sectioned structure under longitudinal, torsional, and bending coupled torsional vibration could be derived, and the influence of geometric nonlinearity of axial force applying on the beam with a variable section could be analyzed effectively.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant nos. 11602169, 11702188, 11772218, 11702192, and 51605330), the Natural Science Foundation of Tianjin City (Grant nos. 17JCYBJC18900 and 18JCYBJC19900), and the Scientific Research Project of Tianjin Municipal Education Commission (Grant no. 2017KJ262).