Free Modal Analysis for Spiral Bevel Gear Wheel Based on the Lanczos Method

In order to obtain the spiral bevel gear wheel natural frequencies and mode shapes in the unconstrained state for the purpose of dynamic characteristics study, the spiral bevel gear wheel three-dimensional solid model of a mini-bus main reducer is established in this paper. The finite element model of spiral bevel gear wheel which consists of 32351 nodes, 18436 solid187 tetrahedrons finite element method elements is established by using free grid meshing method in this paper. Extract the first 6 orders modals parameters such as natural frequencies and main vibration mode shapes by using the Lanczos method. The new 1 to 4 orders modals are formed by comparing and merging 2 orders repeated modals. In order to verify the effectiveness of the finite element analysis results, the experiment modal test based on the impulse force hammer percussion transient single-point excitation and multi-point response analysis method has been done. The maximum difference value of natural frequency between experimental modal test result and finite element modal analysis results is 29.86 Hz, the maximum error rate is 0.41%, which confirmed the result of finite element method is effective and reliable. The conclusions reflect the vibration response characteristics of spiral bevel gear wheel, and provide theoretical basis for dynamic response, structure design and optimization of spiral bevel gear wheel.


INTRODUCTION
Spiral bevel gear with the good advantages of stable transmission, low noise, high contact coincidence ratio, is suitable for application on the automobile main reducer.But the vibration, shock and noise problems [1] caused by spiral bevel gear work conditions of high speed and heavy load influence the ride comfort, manipulation stability and fuel economy of the automobile [2].
Modal analysis is a basis of transient dynamic analysis, load prediction, spectrum analysis, modal superposition method for response (to determine the fatigue life, dynamic strength etc.), vibration control, vibration acoustic characteristics estimation and control, fault vibration and prediction and dynamic optimization design [3][4][5].Obtained the natural frequency of the structure by modal analysis can avoid the occurrence of resonance phenomenon, and provide some measures to reduced vibration, shock and noise.It is difficult to establish a real constraint conditions and the incentives are very complex during the finite element constraint modal analysis for the spiral bevel gear wheel, because of the spiral bevel gear wheel with very complex constraint conditions, while the free modal itself reflects the spiral bevel gear wheel inherent dynamic characteristics [6][7][8][9], so the free modal analysis of a mini bus main reducer spiral bevel gear wheel has been done in this paper.

Vibration Differential Equation
According to the vibration theory, the vibration differential equation of n degrees of freedom elastic system can be described in the physical coordinates as follow [42].
where, M is the mass matrix, C is the damping matrix, K is the stiffness matrix, X is the displacement column vectors,  X is the speed column vector,  X is the acceleration column vector, f (t) is the excitation force column vector.
For the free vibration system of non damping or can negligible damping with no external excitation force, substitute the equation f (t) = 0 into Eq. ( 1 can obtain the free vibration differential equation of n degrees of freedom elastic system as follow. Generally, Eq. ( 1) and Eq. ( 2) described by the physical coordinates are coupled with each other, in order to obtain the independent equations described by modal coordinates and modal parameters, it's necessary to carry out the vibration differential equation as shown in Eq. ( 2) a series of coordinate transformation, which transform the physical coordinate into modal coordinates, and makes the vibration differential equations decoupled, the homogeneous equations can be obtained as following shown.
where, ω i is the i th order natural frequency of the system, ϕ i is the i th order main vibration vectors of the system, i=1, 2,...... n.
The process of solving ω i and ϕ i is the process of modal analysis.The non damping free vibration equation Eq. ( 3) described by the modal coordinates has n orders natural frequencies ω n (possible value) and the corresponding n orders main modal vectors ϕ n , for each group of natural frequency ω i and the corresponding main vibration vectors ϕ i are represents a freedom vibration shape (modal) of single-degree, the free vibration shape of multi-degree system can be decomposed into a linear superposition of single-degree free vibration shape, that is to say the n degrees free vibration system is composed by linear combinations of n number of inherent vibration modal, therefore, any vibration form of the system is a superposition of the n number of main vibration modal.The lower order vibration modal of the system has the more prominent contribution rate and determines the dynamic response characteristics of the system, so, we should more focus on the lower order modal while the higher order modals is often not easy to stimulate [2].Therefore, the 1 st to 12 th orders modals natural frequencies and vibration modals of the spiral bevel gear wheel are extracted in this paper.

Lanczos Numerical Calculation Method
The solution of the free vibration differential Eq. ( 2) can be written in the form of where, Φ = ϕ 1 ,ϕ 2 ,ϕ i ,ϕ n { }.
Eq. ( 4) is a generalized eigenvalue problem.For the multi-degree of freedom constraints system with non rigid motion, the stiffness matrix K is real symmetric positive definite matrix which can be carried out Cholesky factor decomposition by using the square root method [43][44][45][46], the Eq. ( 4) can be rewritten in the form of AY = µY (5) where K=LL T (L is a lower triangular matrix), μ=1/ω 2 , A=L -1 ML -T , Y=L T Ф.
And for the multi-degree of freedom system with rigid body motion, the stiffness matrix K is positive semi-definite matrix which can transform it into positive definite matrix by the shift method, and then carry out the Cholesky factor decomposition.K and μ are determined by the Eq. ( 6).
The characteristics solutions of Eq. ( 5) are characteristics solution of Eq. ( 4) through a series of matrix transformations.Matrix A shown in Eq. ( 5) can convert into tridiagonal matrix by the Eq. ( 7) through selecting the initial Lanczos iterative vector V i which should satisfy the condition V i T V i = I , where I is an unit matrix [17, 47-49].
V T AVQ = µV T VQ (7) where, VQ=Y，and: The matrix V T AV=T is a three diagonal matrix, V T AV can be written as follow: The following equation can been obtained by combining the Eq. ( 8) and Eq. ( 9).AV = VT (10) The vector V i+1 is orthogonal with the former two vectors V i , V i-1 , and the Eq. ( 10) can be expressed as follows [47][48][49]: (11) where, α i , β i and V i+1 can be obtained by the Eq. ( 12) respectively.
After the completion of iterative process can form m number of Lanczos vectors V i (i=1, 2,…, m)， number m is 2 times to the order of eigenvalues to be solved.Solve the Eq. ( 8) based on the Sturm dichotomy theory can get the characteristics value solutions of specified order [50,51].Generally, the Eq. ( 9) cannot get all the characteristics value (n orders), only need to be truncated to m orders (m<n, m is usually 2 times to the order of eigenvalues to be solved).Solving the eigenvalue problem can be expressed as the following Eq.( 13) shown: where, μ i is the characteristics value obtained by truncating the three diagonal matrix.
After obtained the characteristics value μ i by using the dichotomy method to solve the Eq. ( 13), the characteristics vector Z in the Eq. ( 13) can obtain by inverse iteration, and then can obtain the characteristic vector Y of characteristic equation Eq. ( 5) by equation Y=VZ, then by equation Φ=L -T Y can get the original characteristics vector Φ and characteristics values ω i 2 = 1 / µ i (i =1, 2,…, m) of the Eq. ( 4).
In order to improve the stability of the iterative algorithm, the Lanczos vector sequence orthogonal method [52][53][54] was introduced, and suppose the following equation is established firstly.
where, I is an unit matrix, the (i-1) th element of vector {q i } is zero, and q i = 1 .
For 1≤i≤m， the following formula can be established: where, {e i } is the i th column of the unit matrix, and γ i =β i .

FINITE ELEMENT MODAL
There are a pair of spiral bevel gears installed on a mini bus main reducer, some parameters are shown in Table 1.
The three-dimensional model of spiral bevel gear wheel was established in the UG NX software.The middle part of the spiral bevel gear wheel was hollowed for installation of a differential according to the need of the actual work condition.Design 8*M10 threaded hole with 18 mm deep on the semifinished product of spiral bevel gear wheel for power output connecting with the differential case.

FINITE ELEMENT FREE MODAL ANALYSIS
The free modal analysis of spiral bevel gear wheel which non constraint and load added on the spiral bevel gear wheel was done, and the free modal parameters of the spiral bevel gear wheel was calculated.Usually the resonance more easily occurs at the lower order frequencies.Therefore, solve spiral bevel gear wheel free modal, only expands to the first 12 orders harmonics, and obtain the corresponding frequencies and main vibration mode shapes in this paper.The 1 st to 12 th orders modal parameters of spiral bevel gear wheel were calculated by using the Lanczos method.The first 6 orders natural frequencies are close to zero, and the corresponding main vibration mode shapes do not appear in obvious deformations, that is to say, the first 6 orders modal are rigid body modals which are three modals along the three-coordinate axes translation and three modals along the three-coordinate axes rotation.The true modals are the 7 th to 12 th order modals.Remove the first 6 orders rigid modals, then sorted the 7 th to 12 th order modals accordance to the order of 1 st to 6 th sequence, the natural frequencies are shown in Table 3.

Order Finite Element Method Nature Frequency/Hz
The 1 st order 2795 The 2 nd order 2796.3 The 3 rd order 3536.1 The 4 th order 3536.8 The 5 th order 7127.5 The 6 th order 7230.9 Select the 1 st to 6 th order of natural frequencies, take the overall deformation value as output indicator, create the 1 st to 6 th order of main vibration mode shape as shown in Figs.(3,8).3) is the 1 st order main vibration free mode shape, mainly indicates the 4 nodes bending vibration, the maximum total deformation value is 38.187 units.4) is the 2 nd order main vibration free mode shape, mainly indicates the 4 nodes bending vibration, the maximum total deformation value is 38.19 units.From the different value of the 1 st order and 2 nd order natural frequency is small (only 1.3Hz), the main vibration mode shape animation is very similar and the maximum total deformation value is very close (only 0.003 unit) can be identified that the 1 st order and 2 nd order should be the same order natural frequency (the experimental results also verified this conclusion later).So, the 1 st order and 2 nd order natural frequencies should be merged to the new 1 st order natural frequency.5) is the 3 rd order main vibration free mode shape, mainly indicates the 4 nodes torsional vibration, the maximum deformation value is 28.724 units.6) is the 4 th order main vibration free mode shape, mainly indicates the 4 nodes torsional vibration, the maximum total deformation value is 28.729 units.From the difference value of the 3 rd order and 4 th order natural frequency is small (only 0.7 Hz), the main vibration mode shape animation is very similar and the maximum total deformation value is very close (only 0.003 unit) can be identified that the 3 rd order and 4 th order should be the same order natural frequency (the experimental results also verified this conclusion later).So, the 3 rd order and 4 th order natural frequencies should be merged to the new 2 nd order natural frequency.7) is the 5 th order main vibration free mode shape, mainly indicates the umbrella vibration, the main vibration mode shape is similar like an umbrella opening and closing.The maximum total deformation value is 42.501 units.Fig. (7) is the new 3 rd order natural frequency because the influence of the last merged mode shape.Fig. (8) is the 6 th order main vibration free mode shape, mainly indicates 6 nodes circumferential modals, the maximum total deformation value is 40.194units.

THE EXPERIMENT MODAL TEST AND VERIFICATION
In order to verify the accuracy and effectiveness of the finite element modal analysis results based on the Lanczos method, the impulse force hammer single-point percussion transient excitation and multi-point response free experimental modal test for spiral bevel gear wheel has been done in this paper.The geometric modeling was established by using the 8 points octagon in space to approximate   The experimental hammer use Modally Tuned hammer made by American piezoelectric company with US.Patent No.4.799.375.The eight number of three-direction acceleration sensors also made by American piezoelectric company are uniformly and symmetrical installed on the back of the spiral bevel gear wheel by 502# glue as shown in Fig. (9).
The support is elastic rope suspension strut and the data acquisition is the LMS SCADA III with 24 channels of acquisition, 2 dedicated channels of speed acquisition, and 2 channels of signal output.The modal analysis system use modal analysis module of the LMS Test lab 9A.The layout of experimental modal test is shown in Fig. (10).
The modal parameters of the 1 st to 4 th order within the range of 0 kHz to 16 kHz are obtained by the experiment.The comparisons of natural frequency between the finite element analysis results and the experimental modal test result are shown in Table 4.
The 1 st order natural frequency of experimental test is 2787.48Hz, the damping ratio is 1.32%, the relative error ratio is 0.27%.The 2 nd order natural frequency of experimental test is 3522.95Hz, the damping ratio is 1.56%, the relative error ratio is 0.37%.The 3 rd order natural frequency of experimental test is 7146.12Hz, the damping ratio is 2.67%, the relative error ratio is 0.26%.The 4 th order natural frequency of experimental test is 7260.76Hz, the damping ratio is 2.89%, and the relative error ratio is 0.41%, and all the relative error ratio are the range of the engineering permissible value which is less than 5%.

CONCLUSION (1)
The three-dimensional solid model of spiral bevel gear wheel was established in the UG NX environment, and the finite element model of spiral bevel gear wheel was built after the definition of material property parameters and grid meshing on the ANSYS Workbench platform. (2) The free modal analysis of spiral bevel gear wheel has been done by using the Lanczos method for iterative calculation, extract the first 6 orders non zero natural frequencies and the main vibration mode shapes respectively, merged the 2 repeated orders of natural frequencies and obtain the real 4 orders natural frequencies and main vibration mode shapes. ( Export the spiral bevel gear wheel model established in UG NX as Parasolid file format, then import the exported file to ANSYS Workbench platform.The geometric model is shown in Fig. (1).Define the material property of spiral bevel gear wheel and mesh the grid, the finite element model was established as shown in Fig. (2).

Fig. (
Fig. (3) is the 1 st order main vibration free mode shape, mainly indicates the 4 nodes bending vibration, the maximum total deformation value is 38.187 units.

Fig. (
Fig. (4) is the 2 nd order main vibration free mode shape, mainly indicates the 4 nodes bending vibration, the maximum total deformation value is 38.19 units.From the different value of the 1 st order and 2 nd order natural frequency is small (only 1.3Hz), the main vibration mode shape animation is very similar and the maximum total deformation value is very close (only 0.003 unit) can be identified that the 1 st order and 2 nd order should be the same order natural frequency (the experimental results also verified this conclusion later).So, the 1 st order and 2 nd order natural frequencies should be merged to the new 1 st order natural frequency.

Fig. (
Fig. (5) is the 3 rd order main vibration free mode shape, mainly indicates the 4 nodes torsional vibration, the maximum deformation value is 28.724 units.

Fig. (
Fig. (6) is the 4 th order main vibration free mode shape, mainly indicates the 4 nodes torsional vibration, the maximum total deformation value is 28.729 units.From the difference value of the 3 rd order and 4 th order natural frequency is small (only 0.7 Hz), the main vibration mode shape animation is very similar and the maximum total deformation value is very close (only 0.003 unit) can be identified that the 3 rd order and 4 th order should be the same order natural frequency (the experimental results also verified this conclusion later).So, the 3 rd order and 4 th order natural frequencies should be merged to the new 2 nd order natural frequency.

Fig. (
Fig.(7) is the 5 th order main vibration free mode shape, mainly indicates the umbrella vibration, the main vibration mode shape is similar like an umbrella opening and closing.The maximum total deformation value is 42.501 units.Fig.(7) is the new 3 rd order natural frequency because the

Fig. ( 8
Fig. (8) is the new 4 th order natural frequency because the influence of the last merged mode shape.

Table 4 . The comparisons of natural frequency. Order Nature Frequency of Experimental Test/Hz Nature Frequency of Finite Element Method/Hz Damping Ratio The Difference/ Hz The Relative Error Rate
)The real first 4 orders of natural frequencies of experimental test for spiral bevel gear wheel are 2787.48Hz, 3522.95Hz, 7146.12Hz, 7260.76Hz respectively by free modal experimental test.The relative error rate of extraction for natural frequencies by using finite element modal analysis based on the Lanczos method are all the range of the engineering permissible value.