Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 2, pp. 601-612, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.601

INFLUENCE OF UNCERTAINTY IN AERODYNAMIC PERFORMANCE ON

THE DYNAMIC RESPONSE OF A TWO STAGE GEAR SYSTEM

Manel Tounsi, Moez Beyaoui, Kamel Abboudi, Nabih Feki,

Lassaad Walha, Mohamed Haddar

Laboratory of Mechanics, Modelling and Mansufacturing (LA2MP), National School of Engineers of Sfax, Sfax, Tunisia

e-mail: manelt@yahoo.fr; moez.beyaoui@yahoo.fr; kamalo1982@yahoo.fr; fekinabih@gmail.com; walhalassaad@yahoo.fr;

mohamed.haddar@enis.rnu.tn

In this paper, the nonlinear dynamic response in a wind turbine system is considered and
the quantification of uncertainty effects on the variability of this nonlinear response is inve-
stigated.Under dynamic conditions, a lumpedmodelwith 12 degrees of freedom is proposed
taking into account the uncertainty associated to the power coefficient of the input aerody-
namic torque. The dynamic response of the two-stage spur gear system is obtained using
ODE45 solver ofMatlab. The Polynomial Chaos (PC)method is used to introduce the un-
certainties on the proposedmodel. A comparison between the two dynamic responses given
by the proposed lumped dynamic model takes into account the uncertainty. It is perfor-
med on the existed model without uncertainty. Thus, the efficiency and robustness of the
proposed newmethodology is evaluated.

Keywords: gearbox, uncertainty, power coefficient, random parameter, polynomial chaos

1. Introduction

Recently, due to increasing demand for energy, there has been a rapid development of wind
turbines all over the world. This constant growth in energy consumption and polluting effects
associatedare in theheartof the issueof the environmental care, so thatan increasingattention is
being paid towind energy.Generally, wind turbines are one of themachines that take advantage
of wind energy to generate electrical power.

During preliminary design of dynamic systems,many physical parameters can have a signifi-
cant effect on the vibration response of the system. Indeed, some features can generate nonlinear
responses need to be taken into account. The aerodynamic complexities are involved in optimi-
sation of wind turbine systems in an attempt to maximise its performance. Their aerodynamic
and dynamic properties have a decisive influence on the entire system. These properties are
responsible of rotor capability to convert wind energy intomechanical energy. Thus, the overall
efficiency of the energy conversion in the wind turbine is determined.

Several studies have beendeveloped to study the dynamicbehaviour ofwind turbines (Abbo-
udi et al., 2011; Helsen et al., 2011; Zhu et al., 2014). However, themodelling of thesemechanical
systemsadmits strongdispersionsanduncertainties. In this context, designparametersmayvary
in an uncertain way during themanufacturingmonitoring or operation. Thus, the responsemay
change in some uncertain way. Therefore, the formulation of dynamical systems requires intro-
ducing uncertainties into input parameters. In this field,Wei et al. (2015) studied the dynamic
response of a geared transmission system of a wind turbine with uncertainty.

To take into account the uncertainties, differentmethods are reported in the literature, such
as Monte Carlo simulations (Rubinstein, 1981; Kalos and Whitlock, 1986), Polynomial Chaos
Expansion (Wiener, 1938; Ghanem and Spanos, 1991; Fisher and Bhattacharya, 2008).



602 M. Tounsi et al.

The main idea of the polynomial chaos methods is to transform the stochastic differential
equations by means of an intrusive Galerkin projection (Ghanem and Spanos, 1991; Jakerman
and Roberts, 2009) into a deterministic set of differential equations. Moreover, mechanical sys-
tems operate under parametric and external excitation uncertainties. Such as reported in the
literature, thePolynomial Chaos approach is the efficientmethod comparing to theMonteCarlo
approach for quantifying the effects of such uncertainties on the system response.

The capabilities of polynomial chaos have been illustrated in numerous fields, such as envi-
ronmental and biological problems (Isukapalli et al., 1998a,b), fluid dynamics (Pettersson et al.,
2009; Chantrasmi et al., 2066), multibody dynamic systems (Sandu et al., 2006a,b).

In this study, the main originality is that the treatment of uncertainties in the dynamic
analysis of a wind turbine system is proposed. The dynamic behaviour of nonlinear systems
is investigated in order to analyse the robustness and reliability. For that, a dynamic lumped
model of a two-stage gear system is developed in this paper. Three-bladed horizontal-axis wind
turbines are considered with 12 degrees of freedom in the presence of the aerodynamic torque
that is highlighted by an uncertain coefficient of performance belonging to a well-defined inte-
rval. Finally, the main goal of this work is to determine the dynamic behaviour of the gearbox
transmission system of the wind turbine generated by uncertainty parameters.

2. Dynamic modelling

Thestudied system is awind turbine.The increased speedmechanism is a two-stage gear system.
It is composed by two trains of gearings supposed without manufacturing defects. In order to
make this systemmore reliable, resistant and sustainable, a numerical analysis of themechanical
system is developed to study the dynamic response.

Figure 1 shows the dynamic model of the two stages gear system. The power transmission
of the wind turbine is composed of the two-stage spur gear system. It is presented by three
main blocks. The first block (j =1) includes wheel 11 representing the turbine, main shaft and
gear 12. The second block (j = 2) includes gear 21, intermediate flexible shaft and gear 22.
Finally, the third block (j =3) is composed by gear 31, intermediate shaft and wheel 32 which
is the representative wheel of the electrical generator.

Every block j is supported by a flexible bearing having two stiffnesses: the bending stiff-
ness kxj and the traction-compression stiffness kyj. Each intermediate flexible shaft has a ne-
gligible mass compared to the turbine and the generator. It admits some torsional stiffness kθj.
Wheels (11) and (32) characterise respectively themotor side (inertia I11) and the receiving side
(inertia I32). Angular displacements of each wheel about their rotation axes are denoted by θji.
The indices j=1 to 3 designates the number of the block and the indices i=1 to 2 designates
the two wheels of each block.

Besides, the linear displacements of the bearing denoted by xj and yj are measured in the
plane which is orthogonal to the axes of rotation of the wheels. Each pair of wheels is linked
throughflexible teeth.Thisflexibility causesdisplacements.Thegear-meshcontacts aremodelled
by a linear time varying stiffness k(t) along the lines of action in the spur gear stage.

The gear mesh stiffness can bemodelled by a sinusoid wave or by a square wave depending
on the type of gear employed (for spur gear the stiffness function is a square wave, for helical
gear it is a sinusoid wave function). So the periodic square wave form is themost representative
for description of operation of gear systems (Fig. 2).

The terms εα are the contacts ratio corresponding to the two gear mesh contacts and Te is
the mesh period.

The teeth deflection, denoted by δi(t), is projected along the line of action because the gear
mesh stiffness is defined along this direction.



Influence of uncertainty in aerodynamic performance... 603

Fig. 1. Components of the wind turbine system

Fig. 2. Modelling of the mesh stiffness fluctuation

The first deflection δ1(t) along the first gear-mesh contact is given by

δ1(t)= (x1−x2)sinα1+(y1−y2)cosα1+rb12θ12+rb21θ21 (2.1)

while the deflection δ2(t) can be written by

δ2(t)= (x2−x3)sinα2+(−y2+y3)cosα2+rb22θ22+rb31θ31 (2.2)

whileαn represents the pressure angle (generally equal to 20
◦) and rbji are the base radii of the

wheels.

3. Aerodynamic torque

The maximisation of the power coefficient presents a fundamental role in the wind turbine
design to optimise the extraction of energy and to increase the efficiency (Beltran et al., 2011;
Buckspan, 2012). The power coefficient is defined by the ratio of power available on the primary
shaft and the power of wind. The optimum design of the aerodynamic unit of a wind turbine
can be achieved from considering uncertainty of the power coefficient.

For the wind turbine system studied in this paper, we consider that the rotor is composed
of three blades removed by an angle of 120◦ (Gebreslassie et al., 2013) and connected by a hub,
which houses the system for regulating the angular speed.



604 M. Tounsi et al.

The rotor is presented by wheel (11) rotating with some angular velocity and have an input
aerodynamic torque to the power transmission system such as shown in Fig. 1.
Sloth et al. (2011) considered that the power in the wind depends on the wind speed, air

density, and the swept area.Here, the aerodynamic torque is expressed by the following equation
(Lei et al., 2013)

Caero = ρairAR
3Ω2Cp (3.1)

where ρair represents the air density, A andR are the area and the radius of the rotor, respec-
tively, Ω is the angular velocity andCp is the power coefficient.
The power coefficient for the existingmodel is assumed deterministic by the following empi-

rical expression (Abboudi et al., 2011)

Cp =0.44
(125

λ
−6.94

)

e
16.5

λ (3.2)

where λ=ΩR/V (t) and V (t) is the wind velocity.

4. Formulation of equations of motion

The Lagrange formalism leads to the set of differential equations governing the system motion

MẌ+(Ks+K(t))X=F(Cp) (4.1)

The generalised vector of coordinates X is defined by

X(t)= [x1,y1,x2,y2,x3,y3,θ11,θ12,θ21,θ22,θ31,θ32]
T (4.2)

Thematrix M representing the global mass matrix is expressed by

M=

[

ML 0

0 MA

]

ML = diag(m1,m1,m2,m2,m3,m3) MA = diag(I11,I12,I21,I22,I31,I32)

(4.3)

wheremj is the mass of the block j and Iji is the inertia.
Thematrix Ks is the average stiffness matrix of the structure defined by

Ks =

[

Kp 0

0 Kθ

]

(4.4)

where Kp represents the bearing stiffness and Kθ represents the torsional stiffness matrix of
shafts

Kp = diag(kx1,ky1,kx2,ky2,kx3,ky3)

Kθ =



















kθ1 −kθ1 0 0 0 0
−kθ1 kθ1 0 0 0 0
0 0 kθ2 −kθ2 0 0
0 0 −kθ2 kθ2 0 0
0 0 0 0 kθ3 −kθ3
0 0 0 0 −kθ3 kθ3



















(4.5)

Thematrix K(t) is the gear mesh stiffness matrix

K(t)=Km+Kv(t) (4.6)



Influence of uncertainty in aerodynamic performance... 605

Each gear mesh stiffness variation is approximately modelled by the function K(t). It is
composed of an average componentKm and a variable componentKv(t)

Kv(t)=

{

Kmin if t< ta

Kmax else
where ta =(2−εα)Te (4.7)

The external force vector F can be written by

F= [0,0,0,0,0,0,Caero,0,0,0,0,−Cr]
T (4.8)

where Cr presents the receiving torque. It is defined by the aerodynamic torque Caero divided
by the gear ratioGR expressed by

GR=
(Z12Z22
Z21Z31

)

(4.9)

5. Modal analysis and dynamic response

The technological and dimensional parameters of the two-stage gear system (Abboudi et al.,
2011) are summarised in Table 1.

Table 1. System parameters

Description Symbol Value Units

Gear material density (42CrMo4) ρ 7860 Kg/m3

Rotor diameter D 12 m

Bending stiffness kxj 7 ·10
8 N/m

Traction stiffness – compression kyj 6 ·10
8 N/m

Average mesh stiffness km 2 ·10
8 N/m

Torsional stiffness of the shaft kθj 5 ·10
6 Nm/rad

Number of teeth Z(12), Z(21) 72, 18 –
Z(22), Z(31) 54, 18 –

Module of teeth m 0.016 m

Contact ratio εα1-εα2 1.67-1.64 –

In this contribution, the modal analysis focuses on the dynamic properties of system under
vibrational excitation is considered. The goal of the modal analysis is to determine the natural
modevibration and frequencies of a structure.Thus, the stiffnessmatrix of themodel is assumed
to be the averagematrix in order to determine the eigenvalue andmodal vibration of the system.
The dynamic system response is different at each natural frequency. A null eigenvalue indicates
rigid bodymotion. The dashed lines indicate the initial wheel positions.
Figure 3 represents the referenceposition and someeigenmodes of the two-stage gear system.

The fifth mode (mode of pure translation) is relative to the fifth eigen value wp5 =4600rad/s.
The first mode (mode of pure rotation) characterises the rigid body motion. Finally, the tenth
mode is relative to the tenth eigen value wp10 = 40700rad/s, in fact this mode is a combined
mode of translation and rotation.
In order to compare the two models with and without uncertainty, the power coefficient of

the aerodynamic torque is consideredwithout uncertainty in this Section.TheNewmarkmethod
is employed to resolve the equations of motion obtained by the Lagrange formalism.
Figure 4 presents evolution of the displacements of the first (input) and the third (output)

bearings. The figures show that the bearing dynamic behaviour is symmetric according to the
y direction as a function of the x direction: y= f(x).



606 M. Tounsi et al.

Fig. 3. Mode shapes of the gear system; (a) reference position, (b) pure rotationmode (rigid body
motion), (c) pure translationmode (f5 =740Hz), (d) combinedmode (f10 =6460Hz)

Fig. 4. Displacements of the first and third bearing; (a) first bearing, (b) third bearing

Figure 5presents thefluctuation of deflections of thefirst and second tooth.Thesedeflections
are due to teeth flexibility. The deflection has an amplitude in the order of 10−5m.

Fig. 5. Fluctuation of deflections in the first and second stage; (a) First stage; (b) second stage

6. Application of the polynomial chaos method

The fundamental idea of this approach is to establish a separation between the stochastic com-
ponents of a random function and its deterministic components. The randomprocess of interest
is approximated by sums of orthogonal polynomial chaos of random independent variables. In
this context, any uncertain parameter can be viewed as a second order random process. The-
refore, the second order random process z can be expanded in terms of orthogonal polynomial
chaos as (Nechak et al., 2011)



Influence of uncertainty in aerodynamic performance... 607

z=
∞
∑

j=0

zjφj(ξ) (6.1)

where ξ is a vector of standard normal random variables with the known joint density func-
tion W(ξ), zj are stochastic modes of the random process z and φj are orthogonal polynomial
functions satisfying the orthogonally relation

〈φiφj〉=

∫

φi(ξ)φj(ξ)W(ξ) dξ (6.2)

where 〈·〉 means the internal product operator and W(ξ) is the probability density function
(PDF)of randomvariables thatmakeupthevector ξ.ThePDF(XiuandKarniadakis, 2002) acts
as a weighting function in the orthogonally relation for φj(ξ). Therefore, the type of orthogonal
expansion polynomials depends on the nature of the stochastic process through the PDF of the
random variables that describe the probability space. In practice, the generalised polynomial
chaos expansion is truncated to a finite number of termsP. The truncation of the infinite series
is necessary to keep the problem computationally feasible. In this work, we will truncate the
series in such a way that all expansion polynomials up to a certain maximum degree, denoted
by p, are included.Thenumber of terms (P+1) in the expansionnow follows fromthismaximum
degree r and the dimensionality n of the random vector according to

P =
(r+n)!

r!n!
(6.3)

Then, the computing of z is transformed into the problem of finding the coefficients zj of its
truncated expansion. The intrusive and non-intrusive approaches are defined to calculate these
coefficients called stochastic modes. The non-intrusive approach is shown to be more efficient
than the intrusive approach. This approach requires simulations that correspond to particular
samples of the random variables and needs no modifications of the stochastic model, contrary
to the instructive approach.
The system in this work is equivalently expressed as follows

MQ̈+K(t)Q=F(Cp) (6.4)

A representation in the state space can reduce the order of the system to get a first order
system, and it can be written as follows

q̇(t)=Aq(t)+f(q(t),Cp) (6.5)

The robust analysis is based on the system representation in the phase space defined by the
displacements and velocities

q(t)= [θ11, θ̇11,θ12, θ̇12,θ21, θ̇21,θ22, θ̇22,θ31, θ̇31,θ32, θ̇32,

x1, ẋ1,x2, ẋ2,x3, ẋ3,y1, ẏ1,y2, ẏ2,y3, ẏ3]
T

(6.6)

The coefficient of performance of the aerodynamic torque is supposed a random variable
according to a uniform distribution law defined as follows

Cp(ξ)=
b+a

2
+
b−a

2
ξ (6.7)

According to the state of the art, the Legendre polynomials are the best suited to deal with
uniform uncertainties. Here ξ is distributed uniformly within the orthogonally interval [−1,1]
of the Legendre polynomials. It models the uncertainty of the parameter Cp in the interval
[a,b] = [0.35,0.45].



608 M. Tounsi et al.

The Legendre polynomials calculated using the recurrence relation are as follows

(n+1)Ln+1(x)= (2n+1)xLn(x)−nLn−1(x) L0(x)= 1 L1(x)=x (6.8)

The decomposing of the random in the Legendre polynomial basis using the Galerkin pro-
jection allows generating a non-linear deterministic differential equation system

θ̈11,l =−
kθ1
Im
(θ11,l−θ12,l)+

1

〈L2
l
(ξ)〉

ρAR3

Im

p
∑

j=0

p
∑

k=0

θ̇11,jθ̇11,k〈Cp(ξ),Lj(ξ),Lk(ξ),Ll(ξ)〉

θ̈12,l =
kθ1
I1
(θ11,l−θ12,l)−

Rb1
I1
K1(t)δ1,l

θ̈21,l =
kθ2
I2
(−θ21,l+θ22,l)−

Rb2
I2
K1(t)δ1,l

θ̈22,l =
kθ2
I3
(θ21,l−θ22,l)+

Rb3
I3
K2(t)δ2,l

θ̈31,l =
kθ3
I4
(−θ31,l+θ32,l)+

Rb4
I4
K2(t)δ2,l

θ̈32,l =
kθ3
Ir
(θ31,l−θ32,l)−

1

〈L2
l
(ξ)〉

ρAR3

Ir
p
∑

j=0

p
∑

k=0

θ̇11,jθ̇11,k〈Cp(ξ)Lj(ξ)Lk(ξ)Ll(ξ)〉
1

GR

(6.9)

and

ẍ1 =−
kx1
M1
x1,l+

sinϕ1
M1
K1(t)δ1,l

ẍ2 =−
kx2
M2
x2,l−

sinϕ1
M2
K1(t)δ1,l−K2(t)

sinϕ2
M2
δ2,l

ẍ3 =−
kx3
M3
x3,l+K2(t)

sinϕ2
M3
δ2,l

ÿ1 =−
ky1
M1
y1,l−

cosϕ1
M1
K1(t)δ1,l

ÿ2 =−
ky2
M2
y2,l+

cosϕ1
M2
K1(t)δ1,l+

cosϕ2
M2
K2(t)δ2,l

ÿ3 =−
ky3
M3
y3,l−

cosϕ2
M3
K2(t)δ2,l

(6.10)

7. Uncertainty in the dynamic response of the two stages gear system

In this Section, the dynamic behaviour of the two stage gear transmission system is investigated.
The sesults are presentedusing the polynomial chaosmethod.ThePCresults are comparedwith
the results of the deterministic system derived in Section 5.

Figures 6 and 7 represent the mean value and standard deviation of the input angular
displacement θ11(t) and the linear displacement x1(t), respectively. The signal is random and it
fluctuates aroundtheboundaryconditions (zero value).Thestandarddeviation allowspredicting
thevariation domain around the average value of the response.Themean value and the standard
deviation of the dynamic displacement have the same order of amplitude.



Influence of uncertainty in aerodynamic performance... 609

Fig. 6. Instantaneousmean value and standard deviation of θ11(t); (a) mean value,
(b) standard deviation

Fig. 7. Instantaneous mean value and standard deviation of x1(t); (a) mean value,
(b) standard deviation

Figure 8 represents deflection of the first and the second tooth. The signal fluctuates around
the zero value with an amplitude of the in order to 10−3. The signal is sinusoidal and have the
same form of the deterministic model (Fig. 5), therefore, the polynomial chaos results provide a
very good accuracy. In the case of uncertainty, at each time t, the performance coefficient varies
randomly in the range of [0.35,0.45]. By contrast, it is constant in the case without uncertainty
(deterministicmodel). So, there aremany curves of teeth deflection relative to each performance
coefficient. Here, Fig. 8, including Fig. 5, presents a more accurate range.

Fig. 8. Fluctuation of teeth deflection; (a) first bearing, (b) second bearing



610 M. Tounsi et al.

Thefluctuation of the aerodynamic torquewith consideration of theuncertainty according to
the power coefficient is plotted in Fig. 9. The signal is sinusoidal and the amplitude is increasing
between 0 and 60Nm.

Fig. 9. Fluctuation of the aerodynamic torque

The results presented in Fig. 10 are found through orbits of the shaft. The orbits are con-
structed by using displacements in the x- and y-directions. Figure 10 shows the evolution of
orbits for the first and third bearingwith a set of the randomparameter definedpreviously. The
bearings behave in an arbitrary way, which is not observed in the case in the model without
uncertainty (Fig. 4).

Fig. 10. Evolutions of bearings displacements y= f(x); (a) first bearing, (b) third bearing

8. Conclusions

The probabilistic dynamic response of a wind turbine system witha two-stage gearbox trans-
mission system generated by an unceartain input aerodynamic torque has been incestigated. A
new application of the polynomial chaos (PC) method is derived to study the influence of the
input uncertainty parameter. The system structural dynamic response is presented using the
polynomial chaos theory. Therefore, a set of mathematical equations is developed in order to
predict the dynamic behavior of the two-stage spur gear system. Results of the uncertainmodel
using the PCmethod are compared with the deterministic model.

The results suggest that the polynomial chaos method takes into account the uncertainty
with a good efficiency. So, the PC approach can be considered as an efficient tool to take into
account unceartianties in the study of dynamic behaviour of gearbox systems.



Influence of uncertainty in aerodynamic performance... 611

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Manuscript received February 3, 2015; accepted for print October 15, 2015