An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

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REGULAR ARTICLE

An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash

Youssef Hilali, Bouazza Braikat^*, Hassane Lahmam, and Noureddine Damil

Laboratoire d’Ingénierie et Matériaux LIMAT, Faculté des Sciences Ben M’Sik, Université Hassan II de Casablanca, Sidi Othman, Casablanca, Morocco

Received: 22 February 2016 / Accepted: 17 January 2017

Abstract.In this work, we propose some regularization techniques to adapt the implicit high order algorithm based on the coupling of the Asymptotic Numerical Methods (ANM) (Cochelin et al., Méthode Asymptotique Numérique, Hermès-Lavoisier, Paris, 2007; Mottaqui et al., Comput. Methods Appl. Mech. Eng. 199 (2010) 1701–1709; Mottaqui et al., Math. Model. Nat. Phenom. 5 (2010) 16–22) and the implicit Newmark scheme for solving the non-linear problem of dynamic model of a two-stage spur gear system with backlash. The regularization technique is used to overcome the numerical difﬁculties of singularities existing in the considered problem as in the contact problems (Abichou et al., Comput. Methods Appl. Mech. Eng. 191 (2002) 5795–5810;

Aggoune et al., J. Comput. Appl. Math. 168 (2004) 1–9). This algorithm combines a time discretization technique, a homotopy method, a Taylor series expansions technique and a continuation method. The performance and effectiveness of this algorithm will be illustrated on two examples of one-stage and two-stage gears with spur teeth. The obtained results are compared with those obtained by the Newton–Raphson method coupled with the implicit Newmark scheme.

Keywords:spur gear system / high order algorithm / homotopy / vibration / backlash

1 Introduction

Gear transmission system is widely used in several mechanical engineering applications for the purpose of power and motion transmission. Practically, noise and vibration reduction of gear systems remains one of the main goals of researchers. Many excitation sources are responsible for the dynamic behaviour of such transmission. They can be divided into internal or external ones.

Concerning internal excitations, time varying mesh stiffness is considered as the main cause since there will a periodic ﬂuctuation of the number of teeth pairs in contact, many authors considered that this variation is the main source of observed noise and vibration [1–3]. For external excitations, variability of loading conditions and rotational speed are one of the main causes leading to both amplitude and frequency modulations [4,5]. Understand- ing the dynamic behaviour of gear transmission becomes more complicated when nonlinearity occurs [6]. They are induced basically by backlash leading to contact loss between mating gears or in bearings. Spur gear backlash is

considered as the amount of tooth space allowing smooth meshing and reduces friction. Dynamic behaviour of gear transmissions in presence of backlash was extensively studied from decades. Several studies [7–9] showed that gear dynamics cannot be predicted with linear model.

Munro et al. [10] developed a test rig to measure the dynamic transmission error of a spur gear pair and showed that the tooth separation takes place when the mean load is less than the design load. Kubo et al. [11] observed a jump in the frequency response of the gear pair with backlash. Authors modelled backlash between mating teeth by a discontinuous and non-differentiable function giving rise to a quite complicate nonlinear term in the equation of motion. Numerical techniques should be used in this case. Kahraman and Blankenship [12] used a digital simulation technique and the method of harmonic balance to solve the equation of motion. Wang [13] used lumped parameters models including backlash and solved the differential equation of motion using the piece-wise linear technique which gives only solutions for the equivalent linear systems. Cai and Hayashi [14] presented a nonlinear model in which backlash is considered as a time varying function, resulting in dynamic forces equal to zero when there is no contact between tooth pairs.

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©AFM,EDP Sciences2018

https://doi.org/10.1051/meca/2017006

M ^echanics

& I ^ndustry

Available online at:

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Yang et al. [15] developed a nonlinear time-varying dynamic model for right-angle gear pair systems, taking into account both backlash and asymmetric mesh effects. The resolution is done using an enhanced multi-term harmonic balance method with a modiﬁed discrete Fourier transform process and the numerical continuation method. They studied the stability of the steady-state harmonic balance solutions and the effects of the variation and asymmetry in mesh stiffness and directional rotation radius on the gear dynamic responses.

Walha et al. [16] investigated the nonlinear dynamics of a two-stage gear system including backlash and time varying mesh stiffness. The nonlinear dynamic response of the system is computed using a linearization technique. He showed clearly that the discontinuity of the motion induced contact loss.

Recently, Moradi and Salarieh [9] studied the nonlinear oscillations of spur gear pairs including the backlash nonlinearity is studied by using classical single degree of freedom model in terms of dynamic transmission. Backlash is considered as a displacement type nonlinear function approximated by a cubic function.

They implemented multiple scale method in order to compute forced vibration responses of the gear system including primary, super-harmonic and sub-harmonic resonances.

In the work [17], we have applied an implicit algorithm based on an implicit high order technique and a homotopy transformation. This transformation is performed by introducing a variable change from the initial conditions.

In this paper, there are two original ideas that we will discussed. One is the regularization of expression of the mesh force and the other is the application of the algorithm used in [18–20] for the nonlinear problem of dynamic model of one-stage and two-stage spur gear systems with backlash. This algorithm is based on the association of a high order implicit technique and a homotopy transformation. This homotopy transformation is performed by introducing a variable change from the solution evaluated at a previous time. The proposed algorithm is adapted to strong nonlinearity and is numerically efﬁcient since it requires a reduced computation time. The main idea of the proposed algorithm is to apply the homotopy transformation by introducing an arbitrary invertible matrix [K*] and an homotopy parameterevarying from zero to one [18–20].

When this parameter is zero, we obtain a simpliﬁed problem easy to solve and when the homotopy parameter is equal to one, we recover the initial problem without transformation. By applying the Taylor series technique, unknowns of the nonlinear problem are expanded in power series with respect to the homotopy parameter [21]. This technique allows one to transform the nonlinear problem into a sequence of linear ones which have the same tangent matrix [K*]. Consequently only one tangent matrix triangulation is needed to compute all the terms of the series for many time steps. A key point of the proposed algorithm is the possibility of choosing the tangent matrix to be decomposed and the possibility of computing many time steps with single tangent matrix decomposition [18–20]. To illustrate the performance

of the proposed algorithm, we give numerical comparisons with the classical implicit iterative algorithm of Newton– Raphson type [22].

2 Modelling of gear transmission including backlash

In this section, two dynamic models of gear transmission including tooth backlash, treated in [16], are presented.

The objective is to derive the equation of motion of the two models in order to implement the proposed algorithm.

2.1 Dynamic model of a one-stage spur gear system with backlash

A lumped parameters model of a single stage spur gear transmission is presented inFigure 1. The transmission is divided into two parts. The ﬁrst one is composed of a driving motor, a connecting shaft and the pinion and has a massm₁. The second one is composed of the wheel and the loading machine connected by the output shaft and has a massm₂. Motor, pinion, wheel and load are considered as rigid bodies, their respective inertias areI_m,I₁,I₂andI_L. The two parts are supported by bearings modelled by linear springsk_x₁,k_y₁, k_x₂ andk_y₂. Connecting shafts are modelled by torsional stiffnessku₁ andku₂. Each part has four degrees of freedom: two translations according tox andyand two rotations. The degrees of freedom vector can be written as:

fqg¼^t〈x₁;y₁;um;u₁;x₂;y₂;u₂;uL〉 ð1Þ Fig. 1. Single stage spur gear system model.

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Mesh stiffness is modelled by a functionkm_iðtÞspring acting along the line of action (seeFig. 2). This model has been used by Fakhfakh et al. [23]. This mesh stiffness is ﬂuctuating in rectangular form depending on the number of teeth in contact. In this case, the mesh stiffnesskmiðtÞis given as a function of an average component denoted byk_m_i and of aﬂuctuating component denoted byk_v_iðtÞ.

Thisﬂuctuating component is given by:

see equation(2)below

where n is an integer which designates the number of the period,idesignates the stage number, the termseai is the contact ratio at stage i,Te_i is the period and the average component k_m_i¼k_maxðea_i1Þ þk_minð2ea_iÞ, T_e₂ ¼^Z_Z²

3T_e₁ in the case of two-stage; with Z₂ and Z₃ are the teeth numbers. The reliability of the physical results is dependent on the validity of the model and that deﬁnite validation is still to come. An input torqueC_mis applied by the motor and a resisting torqueC_Lis opposed by the load. The expression of the transmission error describing the displacement along the line of action of mating teeth can be expressed as [24]:

dðtÞ ¼ ðx₁x₂ÞsinðaÞ þ ðy₁y₂ÞcosðaÞ

þu₁r_b₁þu₂r_b₂¼^Tfxgfqg ð3Þ

whereais the pressure angle,rb₁ andrb₂ are respectively the base radius of pinion and wheel, {x} is deﬁned by:

Tfxg ¼〈sinðaÞ;cosðaÞ;0;rb₁;sinðaÞ;cosðaÞ;rb₂;0〉 ð4Þ and, here, the degrees of freedom u1and u2 are algebraic quantities. A backlash between teeth is considered. To take into account of this backlash, one can write the expression of the mesh forceF_sas following:

Fs¼

k_m_iðtÞðdðtÞ þb_sÞ; if dðtÞ<bs

0; if bsdðtÞ b_s

k_m_iðtÞðdðtÞ b_sÞ; if dðtÞ>b_s 8>

>>

><

>>

:

ð5Þ

where 2b_sis the total backlash as shown inFigure 3.

Writing Lagrange formulation allows obtaining the equation of motion expressed by:

½Mf€qg þ ½Cf_qg þ ½K_bfqg þF_sðdðtÞÞfxg ¼ fF_extg ð6Þ where [M] is the inertia matrix given by:

½M ¼

m₁ 0 0 0 0 0 0 0

0 m₁ 0 0 0 0 0 0

0 0 I_m 0 0 0 0 0

0 0 0 I₁ 0 0 0 0

0 0 0 0 m₂ 0 0 0

0 0 0 0 0 m₂ 0 0

0 0 0 0 0 0 I₂ 0

0 0 0 0 0 0 0 I_L

2 66 66 66 66 66 4

3 77 77 77 77 77 5

ð7Þ

[K_b] is the bearings and shafts matrix expressed by:

½Kb ¼

kx₁ 0 0 0 0 0 0 0

0 ky₁ 0 0 0 0 0 0

0 0 ku₁ ku₁ 0 0 0 0

0 0 0 0 kx₂ 0 0 0

0 0 0 0 0 k_y₂ 0 0

0 0 0 0 0 0 k_u₂ ku₂

0 0 0 0 0 0 ku₂ k_u₂

2 66 66 66 66 66 4

3 77 77 77 77 77 5 ð8Þ

k

v₁

k

m_i

Gear mesh stiffness (N/m)

Time (s)

Fig. 2. Spring acting along the line of action.

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The vector of external applied forces {F_ext} is expressed by:

fFextg¼^t〈0;0;C_m;0;0;0;0;C_Z〉 ð9Þ The damping matrix [C] in this model is proportional both to the mass and stiffness matrices of the system given by:

½C ¼c₁½M þc₂½Kmoy ð10Þ wherec₁andc₂are the damping coefﬁcients which are expressed respectively in s¹ and s which can be determined from two damping ratios speciﬁed as in [25]

and [K_moy] is the average stiffness matrix of the system [16].

The initial conditions used for this problem are: {q} = {0}

andḟqg ¼ f0g.

2.2 Dynamic model of a two-stage spur gear system with backlash

A model of a two-stage spur gear transmission is presented in Figure 4. It can be divided into three parts. Part 1 is composed of driving unit, input shaft and pinion 1. Part 2 includes wheel 1 and pinion 2 connected by the intermediate shaft. Part 3 is composed of wheel 2, output shaft and load.

The system has 12 degrees of freedom (DOF) which can be detailed as follows:

– Translations of part 1 (having the mass m₁) along x (horizontal) andy(vertical) directions. The corresponding DOF arex₁andy₁.

– Translations of part 2 (having the mass m₂) x and y directions. The corresponding DOF arex₂andy₂. – Translations of part 3 (having the massm3) alongxandy

directions. The corresponding DOF arex₃andy₃. – Rotations of motoru_m, pinion 1u1, wheel 1u2, pinion 2u3,

wheel 2u₄and loadu_L.

The following inertia are considered:I_mfor the motor, I₁for the pinion 1, I₂for wheel 1, I₃ for pinion 2, I₄for wheel 2 andI_L for load. Input, output and intermediate shafts elasticity is modelled by torsional stiffness respectively k_u₁, k_u₂ and k_u₃. Bearings supporting input shaft are modelled by linear springskx₁ andky₁ acting alongxandyaxis. Bearings supporting intermediate shaft are modelled by linear springs kx₂ and ky₂. Bearings supporting intermediate shaft are modelled by linear springkx₃ andky₃.

Two mesh stiffness functionsk_m₁ðtÞandk_m₂ðtÞﬂuctu- ating in a rectangular shape are introduced between pinion 1 and wheel 1 and between pinion 2 and wheel 2 (seeFig. 5).

If we consider backlash for the two-stage, two mesh forces should be consideredFs₁ and Fs₂. One should be aware

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4

−0.2

−0.15

−0.1

−0.05 0 0.05 0.1 0.15 0.2

δ5(rad/s)

˜fsc(N) −0.22 −0.21 −0.2 −0.19 −0.18−0.02

−0.01 0 0.01

Non smoothed function Regularized function with η

sc=0.2 Regularized function with η

sc=0.1 Regularized function with η_sc=0.01 Regularized function with η

sc=0.001

Fig. 3. Contact loss modelling.

Fig. 4. Two-stage spur gear system model.

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here for phasing that can occur between the two mesh functions depending on the position of stage 1 with respect to stage 2.

The vector deﬁning the degrees of freedom {q} is expressed by:

fqg¼^t〈x₁;y₁;u_m;u₁;x₂;y₂;u₂;u₃;x₃;y₃;u₄;u_L〉 ð11Þ Two transmission errors d₁(t) and d₂(t) for the two- stage are expressed by [16]:

d₁ðtÞ ¼ ðx₁x₂Þsinða₁Þ þ ðy₁y₂Þcosða₁Þ þrb₁u₁rb₂u₂¼^Tfx₁gfqg

d₂ðtÞ ¼ ðx₂x₃Þsinða₂Þ ðy₂y₃Þcosða₂Þ þrb₃u₃rb₄u₄¼^Tfx₂gfqg

8>

>>

><

>>

:

ð12Þ

wherea₁anda₂are the pressure angles, {x₁} and {x₂} are given by:

Tfx₁g ¼< sinða₁Þ;cosða₁Þ;0;rb₁;sinða₁Þ;

cosða₁Þ;rb₂;0;0;0;0;0>

Tfx₂g ¼<0;0;0;0;sinða₂Þ;cosða₂Þ;0;rb₃;sinða₂Þ;

cosða₂Þ;rb₄;0>

8>

>>

><

>>

:

ð13Þ

and, here, the degrees of freedom u₁, u₂, u₃ and u₄ are positive quantities. Taking into account of the backlashes in the two mesh zones, the expression of the mesh forcesF_s₁ andFs₂respectively measured on the meshes in stage 1 and stage 2 are as following:

F_s₁ ¼

km₁ðtÞðd₁ðtÞ þb₁Þ; if d₁ðtÞ<b₁

0; if b₁d₁ðtÞ b₁

k_m₁ðtÞðd₁ðtÞ b₁Þ; if d₁ðtÞ>b₁ 8>

>>

><

>>

:

ð14Þ

F_s₂ ¼

km₂ðtÞðd₂ðtÞ þb₂Þ; if d₂ðtÞ<b₂

0; if b₂d₂ðtÞ b₂

km₂ðtÞðd₂ðtÞ b₂Þ; if d₂ðtÞ>b₂ 8>

>>

><

>>

:

ð15Þ

By the same Lagrange formulation it is possible to obtain the equation of motion of the two-stage gear system by:

½Mf€qg þ ½Cfqg þ ½K_ _bfqg þFs1ðd1Þfx1g þFs2ðd2ðtÞÞfx2g

¼ fFextg ð16Þ

where [M] is the inertia matrix given by:

0 0.5 1 1.5 2 2.5

x 10⁻³ 2

2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8x 10⁷

Gear mesh stiffness (N/m)

Time (s)

L

2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

ð17Þ

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[K_b] is the bearing and shafts matrix expressed by:

The vector of external applied forces{F_ext} is expressed by:

fFextg¼^tf0;0;C_m;0;0;0;0;0;0;0;0;C_Lg ð19Þ The initial conditions used for this problem are: {q} = {0}

and ḟqg ¼ f0g. So, the initial conditions of transmission errors deduced from equations(3)and(12)are given by:

dð0Þ ¼0

f g for one-stage d₁ð0Þ ¼0

d₂ð0Þ ¼0

( )

for two-stage ð20Þ

3 The proposed algorithm

The proposed algorithm for solving the problem(6)or(16) is based on four steps: a regularization technique, a time discretization, the introduction of a perturbation from the

solution at a previous time, a homotopy transformation and a Taylor series technique. The presented algorithm includes a generic procedure and it is a generic solver that can be applied to a wide class of nonlinear unsteady equations as equations (6) and (16). The Taylor series technique allows us to calculate all terms of the series with a single inversion of the tangent matrix.

3.1 Regularization technique

In the case of problems(6)and(16), non-smooth functions appearF_s(d(t)) deﬁned by(5),Fs₁

d₁ðtÞ

deﬁned by(14) andF_s₂

d₂ðtÞ

deﬁned by (15). The idea is to replace these non smooth functions by a smooth function denoted by:

F~s

dðtÞ

¼km_idðtÞ þ1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_m_i

dðtÞ bs

₂

þ4h²b²_s r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_m_i

dðtÞ þbs

₂

þ4h²b²_s r

Þ ð21Þ The function defined here involves a regularization parameterh. This parameter is chosen significantly small in order that the modified function is close to the non smooth function defined by (5) (see Fig. 6). To make easy the management of the regularization, we have defined dimensionless regularization parameter.

To obtain a quadratic problem, we introduce the new variablesh(d(t)) andg(d(t)) deﬁned by:

hðdðtÞÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_m_iðdðtÞ b_sÞ²þ4h²b²_s q

gðdðtÞÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k²_m_iðdðtÞ þb_sÞ²þ4h²b²_s q

8>

><

>>

: ð22Þ

Taking into account the equation (22), the modiﬁed function(21)is set in the following form:

F~_s dðtÞ

¼k_m_idðtÞ þ1 2 h

dðtÞ g

dðtÞ

ð23Þ

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

x 10⁻⁵

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

x 10⁻⁵

Deflection (m)

Meshing force (N)

non smooth function smooth function for η=0.01 smooth function for η=0.1 smooth function for η=0.2 smooth function for η=0.4

Fig. 6. The approximated of theðF~sdðtÞÞbased on a regularization technique for different values ofh.

½K

b

¼

k

x₁

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0

0 0 0 0 0 0 0 0 0 k

y₃

0 0

3.2 Time discretization

For solving the nonlinear unsteady problem(16), we use here the implicit Newmark scheme widely used in the resolution of dynamic problems [26]. This very popular implicit scheme requires less, but more expensive time steps to follow the equilibrium path of a dynamic system [26]. This scheme gives the expression of speed ḟq^nþ1g and accelerationfq^nþ1g at time (n + 1)Dt as follows:

fq_^nþ1g ¼a₀ðfq^nþ1g fqⁿgÞ fcⁿg

f€q^nþ1g ¼Dtba₀ðfq^nþ1g fqⁿgÞ þ fwⁿg; ð24Þ where fvⁿg ¼a1fq_ⁿg þa2f€qⁿg;fcⁿg ¼ð1a1DtbÞfq_ⁿgþ

Dtð1bÞ a2Dtb

ð Þfq€ⁿg,a₀¼_g_D¹_t2,a₁¼_g¹_D_t,a₂¼₂¹_g1,n denotes the time level,Dtis the time step, the constantsband g have the range 0b1,0 g ¹₂ with typical values beingb¼¹₂,g ¼¹₄. Using equation(24), the problem(6)at time (n+ 1)Dtis written as:

ða₀½M þba₀Dt½CÞðfq^nþ1g fqⁿgÞ þ ½Kbfq^nþ1g þF~sðd^nþ1Þfxg ¼ fFextg þ ½Mfcⁿg ½Cfvⁿg

ð25Þ where {qⁿ⁺¹},dⁿ⁺¹,hⁿ⁺¹andgⁿ⁺¹are the unknowns at time t= (n+ 1)Dt.

3.3 Introduction of a perturbation from the previous time Instead of searching the unknowns {qⁿ⁺¹},dⁿ⁺¹,hⁿ⁺¹and gⁿ⁺¹ one seeks the perturbations {Dq}, DD, DH and DG from the solutions {qⁿ},dⁿ,hⁿandgⁿat timet=nDtin the following manner.

fq^nþ1g ¼ fqⁿg þ fDQg d^nþ1 ¼dⁿþDD h^nþ1¼hⁿþDH g^nþ1¼gⁿþDG 8>

>>

><

>>

:

ð26Þ

The expressions(26)are introduced into equations(3), (22), and(25), we obtain a nonlinear problem:

½Kⁿ_TfDQg þ fFQg ¼ fSⁿg DD¼^TfxgfDQg

2hⁿDH¼k^nþ1_m_i ²ðdⁿbsÞ²þ4h²b²_shⁿ² þ2k^nþ1_m_i ² ðdⁿbsÞDDþk^nþ1_m_i ²DD²DH²

2gⁿDG¼k^nþ1_m_i ²ðdⁿþb_sÞ²þ4h²b²_sgⁿ² þ2k^nþ1_m_i ²ðdⁿþb_sÞDDþk^nþ1_m_i ²DD²DG² 8>

>>

<

>>

>:

ð27Þ where½Kⁿ_Tis the tangent matrix given by

½Kⁿ_T ¼a₀½M þDtba₀½C þ ½Kb þ ½Kⁿ_en ð28Þ with

½Kⁿ_en ¼ k^nþ1_m_i þk^nþ1_m

i 2

2

ðdⁿbsÞ

hⁿ ðdⁿþbsÞ gⁿ

!

fxg^Tfxg ð29Þ {FQ} is a quadratic form deﬁned by:

fFQg ¼1

4 k^nþ1_m_i ² 1 hⁿ 1

gⁿ

DD²DH² hⁿ þDG²

gⁿ

fxg ð30Þ and {Sⁿ} is the second hand side expressed by:

fSⁿg ¼ fFextg þ ½Mfcⁿg ½Cfvⁿg ½Kbfqⁿg ðk^nþ1_m_i dⁿþ1

2ðhⁿgⁿÞÞfxg 1

4

k^nþ1_m_i ²

ðdⁿbsÞ²

hⁿ ðdⁿþbsÞ² gⁿ

þ4h²b²_s

1

hⁿ 1 gⁿ

þgⁿhⁿ

fxg ð31Þ

3.4 Homotopy transformation

It is well known that the homotopy transformations have been developed to overcome the local convergence of the Newton-like methods and may provide a reliable way to solve the nonlinear equations [21]. In this work, a homotopy transformation is used while introducing an invertible matrix [K*] in the problem(27)and the parametereof this transformation is considered as an expanding parameter [18–20]:

½KfDWg þeðð½KⁿT ½KÞfDWg þ fFQgÞ ¼efSⁿg ð32Þ By this way, the solutions DW(e) of the nonlinear equation (32) passe continuously from 0 for e= 0 to the solution of the nonlinear problem(27)fore= 1.

3.5 Taylor series technique

The basic idea of this technique consists in searching the solution branches of the nonlinear problem(32)in the form of a truncated Taylor expansion:

fDWðeÞg ¼X^i¼p

i¼0

eⁱfDWig

DDðeÞ ¼X^i¼p

i¼0

eⁱDDi

DHðeÞ ¼X^i¼p

i¼0

eⁱDH_i

DGðeÞ ¼X^i¼p

i¼0

eⁱDGi

8>

>>

><

>>

:

ð33Þ

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Table 1. Parameters of the one-stage spur gear transmission.

Parameters Value

Pinion Wheel

Teeth numbers Z1= 20 Z2= 40

Damping coefﬁcients (1/s), (s) c₁= 510² c₂= 10⁵

Rotation speed (rpm) v1= 1500 v2= 750

Mass (kg) m₁= 1.8 m₂= 2.5

Mass moment of inertia (kg m²) I₁= 8.110⁴ I₂= 0.0045

Base circle (m) rb₁¼0:028 rb₂ ¼0:056

Primitive diameter (m) D₁= 0.06 D₂= 0.12

Motor torque (N m) C_m= 10

Mesh frequency (Hz) f_e₁ ¼500

Receiving torque (N m) C_L= 20

Bearings stiffness (N/m) kx₁¼ky₁¼kx₂¼ky₂¼10⁸

Torsional shaftﬂexibilities (N m/rad) k_u₁¼k_u₂¼10⁵

Pressure angle a= 20°

Modulus (m) M₁= 310³

Mean gear mesh stiffness (N/m) km₁ ¼10⁸

Contact ratio ea₁¼1:63

Teeth width (m) W₁= 3010³

Backlash (m) b_s= 10⁵

Material 42CrMo4

Density (kg/m³) r= 7860

0 0.02 0.04 0.06 0.08 0.1

−18

−16

−14

−12

−10

−8

−6

−4

−2

Time (s)

|| log10 (residual vector) ||

Newton−Raphson ANM order 3 ANM order 4 MAN order 5 MAN order 6

Fig. 7. Evolution of the solution quality log₁₀||{R}|| versus the timet.

0 0.02 0.04 0.06 0.08 0.1

0 0.2 0.4 0.6 0.8 1 1.2x 10⁻⁵

Time (s)

Transmission error (m) 2 4 6

x 10⁻³ 0.95

1 1.05 1.1 1.15

x 10⁻⁵

0.069 0.07 0.071 1

1.05 1.1x 10⁻⁵

ANM order 3 with η=10⁻² ANM order 3 with η=10⁻³ ANM order 3 with η=10⁻⁴ ANM order 3 with η=10⁻⁵ ANM order 3 with η=10⁻⁶

Fig. 8. Inﬂuence of regularization parameter on the evolution of the transmission errordversus the timetforh= 10²,h= 10³, h= 10⁴,h= 10⁵andh= 10⁶.

(9)

0 0.02 0.04 0.06 0.08 0.1 0

0.2 0.4 0.6 0.8 1 1.2x 10⁻⁵

Time (s)

Transmission error (m)

2 4 6 8 10

x 10⁻³ 0.95

1 1.05 1.1 1.15

x 10⁻⁵

ANM order 3 Newton−Raphson

Fig. 9. Time evolution of transmission error, Proposed algorithm truncated at orderp= 3 andk= 10⁴(ANM order 3), Newton– Raphson method with tolerancek= 10⁴(Newton–Raphson).

Table 2. Comparison between the number of triangulation tangent matrices for the both methods.

Number of triangulations

ofK* Number of triangulations

ofKⁿ_T

Proposed algorithm Newton–Raphson method

1 73 709

Table 3. Parameters of the two-stage spur gear transmission.

Parameters Value

Pinion 1 Wheel 1 Pinion 2 Wheel 2

Teeth number Z1= 30 Z2= 45 Z3= 30 Z4= 45

Mass (kg) m₁= 0.8889 m₂= 2.0001 m₃= 0.8889 m₄= 2.0001

Mass moment of inertia (kg m²) I₁= 0.0016 I₂= 0.0081 I₃= 0.0016 I₄= 0.0081

Rotation speed (rpm) v₁= 3000 v₂= 2000 v₃= 2000 v₄= 1000

Motor torque (N m) C_m= 10

Receiving torque (N m) C_L= 0

Bearings stiffnesses (N/m) k_x₁ ¼k_y₁ ¼k_x₂¼k_y₂¼k_x₃¼k_y₃¼10⁷ Torsional shaftﬂexibilities (N m/rad) ku₁¼ku₂ ¼ku₃¼10⁵

Pressure angle a₁=a₂= 20°

Damping coefﬁcients (1/s), (s) c₁= 10²,c₂= 10⁶

Mesh frequencies (Hz) f_e₁¼1500,f_e₂¼1000

Modulus (m) M₁=M₂= 410³

Mean gear mesh stiffness (N/m) km₁ ¼km₂¼3 10⁷

Contact ratio ea₁¼ea₂ ¼1;592

Teeth width (m) W1=W2= 3010³

Backlash (m) b= 210⁴

Material 42CrMo4

Density (kg/m³) r= 7860

(10)

By introducing the expansions(33)into equation(32) and by equating like powers ofe, we obtain a set of linear problems satisﬁed by the terms of the series(33)given by:

p¼1:

½KfDW₁g ¼ ð½K^nþ1_T ½KÞfDW₀g fFQðDW₀;DW₀Þg þ fS^nþ1g DD₁¼^TfxgfDW₁g

DH₁¼k^nþ1_m_i ²ðdⁿþDD₀bsÞ hⁿþDH₀ DD₁ DG₁¼k^nþ1_m_i ²ðdⁿþDD₀þbsÞ

gⁿþDG₀ DD₁ 8>

>>

<

>>

>:

ð34Þ

The solution is obtained step by step. Each step is represented by the power series (33). The length of each step is computed by requiring that the relative difference between the solutions with two consecutive truncation orders must remain small enough by comparison to a given accuracy parameterk. The step length is then computed by

0 0.02 0.04 0.06 0.08 0.1

−10

−8

−6

−4

−2 0 2 4x 10⁻⁷

Time(s)

Amplitude (m)

0.028 0.03 0.032 0.034 0.036 0.038

−2.5

−2

−1.5 x 10⁻⁷

0 0.02 0.04 0.06 0.08 0.1

−6

−4

−2 0 2 4 6 8x 10⁻⁷

Time (s)

Amplitude (m)

0.0260.0280.030.0320.0340.036 1

1.5 2 2.5

x 10⁻⁷

Fig. 10. Temporary evolution of the linear displacement of theﬁrst and second bearing, Proposed algorithm truncated at orderp= 3 andk= 10⁴(ANM order 3), Newton–Raphson method withk= 10⁴(Newton–Raphson).

0 0.005 0.01 0.015 0.02

−16

−14

−12

−10

−8

−6

−4

−2

Time (s) log10 || R ||

NR ANM order 3 ANM order 4 ANM order 5 ANM order 6 ANM order 7 ANM order 8

Fig. 11. Decimal logarithm of the norm the residual vector versus time, Proposed algorithm with different truncation orders p= 3, 4, 5, 6, 7, 8 (ANM orderp) andk= 10⁴, Newton–Raphson algorithm withk= 10⁴(NR).

p ≥ 2 :

½K

fW

p

g ¼ ð½K

^nþ1_T

½K

ÞfW

p1

g X

^p1

r¼0

fFQðW

r

; W

p1r

Þg D

p

¼

^T

fgfW

p

g

H

p

¼ 2k

^nþ1_m_i ²

ð

ⁿ

þ D

₀

b

s

ÞD

p

þ k

^nþ1_m_i ²

P

_p1

r¼1

D

r

D

_pr

P

_p1

r¼1

H

r

H

_pr

2ðh

ⁿ

þ H

₀

Þ

G

p

¼ 2k

^nþ1_m_i ²

ð

ⁿ

þ D

₀

þ b

s

ÞD

p

þ k

^nþ1_m_i ²

P

_p1

r¼1

D

r

D

_pr

P

_p1

r¼1

G

r

G

_pr

2ðg

ⁿ

þ G

0

Þ

8 >

> >

<

> >

> :

ð35Þ

(11)

deﬁning a maximal value of the parameteremax[18–20] as follows:

emax¼ kkfDW₁gk kfDWpgk

1

p1 ð36Þ

The solution of problem (25) is obtained while the parameteremax≥1.

4 Numerical applications

In this work, we present two typical examples of the gear system with one-stage and two-stage. The two nonlinear problems are solved by the detailed algorithm above. In both examples, the matrix [K*] has the same form as½Kⁿ_T

deﬁned at the initial conditions. IfK¼Kⁿ_T, our algorithm requires the inversion of this matrix each time step which demands a considerable computation time. This way of doing is similar to the iterative and incremental method type Newton–Raphson.

4.1 Example 1: one-stage gear system

Let us consider a spur gear system driven by a squirrel cage motor. Its characteristics are given inTable 1.

In this application, we have chosen a pre-conditioner [K*] which is taken equal to the tangent matrix½Kⁿ_T evaluated at the initial time, a time stepDt= 3.710⁶s, a tolerance parameter k= 10⁴ and various values for the truncation order for our proposed algorithm, a tolerance parameterk= 10⁴for Newton–Raphson algorithm and a regularization parameter h= 10⁶. The time interval for this study is [0, 0.09 s]. In a first step, we studied the influence of the truncation order on the solution quality which is defined from the residual vector:

fRg ¼ ½Kⁿ_TfDQg þ fFQðDQ;DQÞg fSⁿg ð37Þ The solution quality is measured by the decimal logarithm of the residual vector R. In Figure 7, we represent the evolution of the solution quality versus the time. We remark that this quality increases with the truncation order of the series forp= 3, 4, 5, 6. The curve of Figure 7is obtained by the used algorithm in a single step of continuation for orders greater or equal than 3. So the stage of the continuation of procedure is not used.

In this numerical test, we study the inﬂuence of several values of the regularization parameterhon the solution of the problem. In Figure 8, we represent the transmission error versus to time for h= 10², h= 10³, h= 10⁴, h= 10⁵andh= 10⁶. According to this test, we noticed that the solution of the problem is stabilized starting from the valueh= 10³(seeFig. 8).

0 0.005 0.01 0.015 0.02

−2 0 2 4 6 8 10 12x 10⁻⁵

Time (s) δ1 (m)

NR (b₁=b₂=0) NR (b₁=10⁻⁴,b₂=0) ANM (b₁=b₂=0) ANM (b₁=10⁻⁴,b₂=0)

Half of backlash

Backlash duration Contact loss

Half of backlash

Backlash duration Contact loss

0 0.005 0.01 0.015 0.02

−4

−2 0 2 4 6 8 10 12 14 16x 10⁻⁶

Time (s) δ2 (m)

NR (b₁=b₂=0) NR (b₁=10⁻⁴,b₂=0) ANM (b₁=b₂=0) ANM (b₁=10⁻⁴,b₂=0)

Backlash duration

The phase shift due to backlash

Fig. 12. Temporal evolution of transmission errors, proposed algorithm with truncation orderp= 3 (ANM) andk= 10⁴, Newton– Raphson algorithm withk= 10⁷(NR) for (b1= 0,b2= 0) and (b1= 10⁴,b2= 0).

0 0.005 0.01 0.015 0.02

0 1 2 3 4 5 6 7 8 9 10

Time (s)

Angular velocity (rad/s)

NR (b 1=b

2=0) NR (b₁=10⁻⁴,b₂=0) ANM (b₁=b₂=0) ANM (b

1=10⁻⁴,b 2=0)

θ˙1

θ˙3

θ˙4

Contact loss

Fig. 13. Temporal evolution of linear and nonlinear angular velocitiesﬂuctuation.

(12)

0 0.005 0.01 0.015 0.02

−0.025

−0.02

−0.015

−0.01

−0.005 0 0.005 0.01 0.015

Time (s)

˙x1(m/s)

NR (b₁=b₂=0) NR (b

1=10⁻⁴,b 2=0) ANM (b₁=b₂=0) ANM (b

1=10⁻⁴,b 2=0)

Backlash duration The phase shift due to backlash

0 0.005 0.01 0.015 0.02

−0.015

−0.01

−0.005 0 0.005 0.01

Time (s)

˙x3(m/s)

NR (b₁=b₂=0) NR (b

1=10⁻⁴,b 2=0) ANM (b₁=b₂=0) ANM (b

1=10⁻⁴,b 2=0)

The phase shift due to backlash

Fig. 14. Temporal evolution of bearingsẋ1andẋ3.

0 0.005 0.01 0.015 0.02

−150

−100

−50 0 50 1v00

Time (s)

¨x1(m/s2)

NR (b 1=b

2=0) NR (b

1=10⁻⁴,b 2=0) ANM (b

1=b 2=0) ANM (b

1=10⁻⁴,b 2=0)

0 0.005 0.01 0.015 0.02

−60

−40

−20 0 20 40 60

Time (s)

¨x3(m/s2)

NR (b 1=b

2=0) NR (b

1=10⁻⁴,b 2=0) ANM (b₁=b₂=0) ANM (b

1=10⁻⁴,b 2=0)

Fig. 15. Temporal evolution of bearingsẍ1andẍ3.

−2 0 2 4 6 8 10 12

x 10⁻⁵

−0.1

−0.05 0 0.05 0.1 0.15

Relative Teeth displacement (m)

Relative Teeth velocity (m/s) Linear velocity amplitude Nolinear velocity amplitude

NR (b 1=b

2=0) NR (b

1=10⁻⁴,b 2=0) ANM (b

1=b 2=0) ANM (b

1=10⁻⁴,b 2=0)

Half of backlash

−4 −2 0 2 4 6 8 10 12 14

x 10⁻⁶

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06 0.08

Relative Teeth displacement (m)

Relative Teeth velocity (m/s) Linear velocity amplitude Nolinear velocity amplitude

NR (b 1=b

2=0) NR (b

1=10⁻⁴,b 2=0) ANM (b

1=b 2=0) ANM (b

1=10⁻⁴,b 2=0)

Fig. 16. Relative teeth velocityﬂuctuation following teeth displacement in the case of one backlash located on theﬁrst stage.

An implicit algorithm for the dynamic study of nonlinear vibration of spur gear system with backlash