Turbocharger rotor dynamics with foundation excitation | SpringerLink

Arch Appl Mech (2009) 79: 287–299 DOI 10.1007/s00419-008-0228-3

O R I G I NA L

Guangchi Ying · Guang Meng · Jianping Jing

Turbocharger rotor dynamics with foundation excitation

Received: 22 August 2007 / Accepted: 28 February 2008 / Published online: 5 April 2008 © Springer-Verlag 2008

Abstract To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established which includes the engine’s foundation excitation and nonlinear lubricant force. The rotor vibration response of eccentricity is simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed. Keywords Turbocharger · Rotor dynamics · Foundation excitation

1 Introduction A turbocharger is a rotational machine with extremely high rotational speed and light load. In a turbocharger, the exhaust gases with high temperature and high pressure from the cylinders of internal combustion engine through manifold and drive the turbine wheel, which is connected with a compressor wheel through a shaft, thus providing additional air for the combustion process. With the increase of fuel prices and demand for higher speeds, more and more vehicles are equipped with turbochargers to improve the efficiency of internal combustion engine and provide additional power. The demand for higher rotational speed and stability performance, along with lower cost continues to motivate the investigation of turbocharger rotor dynamics. Until now, much effort has been made to investigate the nonlinear behaviors of rotor-bearing system dynamics. References [1–4] studied the nonlinear rotor dynamics on the bifurcation, chaos, oil whirl/whip and stability problems. Li and Rohde [5] and Li [6] presented a finite length bearing model for the prediction of the stability response of floating ring bearing. In these efforts, the effects of imbalance and subsynchronicity on the stability of a rotor supported on floating ring bearings were studied. Howard [7,8] investigated the feasibility of supporting a turbocharger rotor on air foil bearings based upon predicted rotor dynamic stability, load accommodations and stress considerations, and demonstrated that foil bearings offer a plausible replacement for oil-lubricated bearings in diesel truck turbochargers. Naranjo et al. [9] presented evidence of subcritical and supercritical bifurcations and return to absence of whirl in a test rotor. Walton et al. [10] designed a small rotor to simulate a miniature turbojet engine or a turbocharger rotor mounted on compliant foil bearings, and presented good correlation between measurement and analysis results of the rotor bearing system dynamics with natural frequencies, rotor displacements and thrust load carrying ability. Recent researches [11–16] aimed G. C. Ying · G. Meng (B) · J. P. Jing State Key Laboratory of Mechanical System and Vibration, Shanghai JiaoTong University, 800 Dongchuan Rd., 200240 Shanghai, People’s Republic of China E-mail: [email protected]

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to develop test validated rotor dynamic models coupled to realistic bearing force models. Holt et al. [11] presented a complete nonlinear rotor dynamics model and comprehensive measurements of casing acceleration in a high speed turbocharger, and developed a nonlinear rotor dynamics program predicts well the frequency content of the measured response. Gjika and Groves [12] were concerned with the progress of nonlinear dynamic behavior modeling of a turbocharger rotor-radial bearing system with fully floating bearing design, and showed good correlation with the respect of synchronous response and total motion. San Andrés et al. [13–16] dealt with linear and nonlinear rotor-bearing models for prediction of the dynamic shaft response of automotive turbochargers supported on floating ring bearings or semi-floating ring bearing. In reference [15], more successful correlations of nonlinear rotor dynamics predictions to test data for a turbocharger further validated these models and opened the path for the implementation of the software as a virtual design tool. However, as a turbocharger is always founded on the engine, low frequency, with respect to engine’s running speed, large deflection vibrations are transferred from the engine to the turbocharger rotor through the hydrodynamic bearings. Even though these low frequency vibrations are far below the rotor’s running speed, they do affect its operation in a nonlinear way through the journal bearing clearance variation [17]. Traditional rotor dynamics mainly apply to rotational machines that are mounted on the ground, and presume that the foundation is stationary and the supporting stiffness is sufficiently large [18]. This presumption is obviously unreasonable for the turbocharger rotor-bearing system. The influence of the engine’s foundation excitation must be taken into account in the turbocharger rotor dynamics. In this effort, the dynamic model of a turbocharger rotor-bearing system is established, including the engine’s foundation excitation and nonlinear oil film force, and the nonlinear dynamic behaviors of a turbocharger rotor-bearing system are investigated. The vibration responses are simulated by numerical calculation. The bifurcation and chaos behaviors of nonlinear rotor dynamics with various rotational speeds are studied. The results with foundation excitation are compared with those without foundation excitation, and the differences are presented and discussed. 2 Governing equations of motion for a turbocharger rotor-bearing system The structure of a turbocharger rotor is depicted in Fig. 1, where the left end is a compressor wheel, and the right end is a turbine wheel. The floating ring bearing is supported in the middle of rotor, and is simplified as two journal bearings supported in rotor necks A and B. Such as shown in Fig. 2, the rotor is divided into three shaft sections and four disc elements. The X and Y directions are set in the disc plane, and Z direction is along with the shaft axis. The force sketch is depicted in Fig. 3 (only the forces in Z –X plane are shown). In this modeling, due to the cantilever style of wheels, the gyroscopic effect is taken into consideration and the torsional vibration, axial vibration, and shearing effect are ignored. For the disc element, the governing equations of motion are ⎧ ⎪ m i X¨ i + c0 X˙ i = FxiL − FxiR − Fbxi + m i ei ωr2 cos ωr t ⎪ ⎪ ⎪ ⎪ ⎨ Jyi Ψ¨ i + c0 Ψ˙ i − ωr Jzi Φ˙ i = −M L + M R yi yi i = 1, 2, 3, 4 (1) L R 2 sin ω t ⎪ ¨ ˙ Y Y m + c = F − F − F − m g + m e ω ⎪ i i 0 i byi i i i r r yi yi ⎪ ⎪ ⎪ ⎩ L − MR Jxi Φ¨ i + c0 Φ˙ i + ωr Jzi Ψ˙ i = Mxi xi compressor wheel

turbine wheel

bearings

l1

Fig. 1 Schematic diagram of a turbocharger rotor

l2

d3

B

d2

d1

A

l3


289

disc 1

Y

disc 2

Xe

disc 3

disc 4

Ψ X

Φ O

Z

ωr

Ye shaft section 1

shaft section 3

shaft section 2

Fig. 2 Mechanical model of a turbocharger rotor Xi

L Fxi

M yiR

Ψ i+1

Ψi

Zi

Fbxi L M yi

X i+1

Xi

Ψi

Z i+1

Zi R M yi

M yL,i+1

R Fxi

EI i

mi

J J zi

FxL,i +1

li

R

F yi

(a)

(b)

Fig. 3 Force sketch of disc and shaft section in Z -X plane: a the disc, b the shaft section

For the shaft sections, the dynamic equation is ⎧ l2 R + ⎪ X i+1 = X i + li Ψi + 2Ei Ii M yi ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ R + li F R ⎪ Ψi+1 = Ψi + EliIi M yi ⎪ ⎪ 2E Ii xi ⎪ ⎪ ⎪ L R + FRl ⎪ M = M ⎪ y,i+1 yi xi i ⎪ ⎪ ⎪ ⎪ L R ⎨ Fx,i+1 = Fxi

li3 6E Ii

FxiR

3 l2 ⎪ R + li F R ⎪ Yi+1 = Yi − li Φi + 2Ei Ii Mxi ⎪ ⎪ 6E Ii yi ⎪ ⎪ ⎪ 2 ⎪ l R − i FR ⎪ ⎪ Φi+1 = Φi − EliIi Mxi ⎪ 2E Ii yi ⎪ ⎪ ⎪ L R + FRl ⎪ M = M ⎪ ⎪ x,i+1 xi yi i ⎪ ⎪ ⎩ L R Fy,i+1 = Fyi

i = 1, 2, 3

(2)

Herein, X and Y are the translation displacements along X and Y directions, Φ and Ψ are the angular displacements around X and Y axis, m is the mass of disc, Jx , Jy and Jz are the moment of inertia, E is Young’s modulus, I is second axial moment of area, l is the length of shaft section, c0 is the damping coefficient, ωr is the rotational speed of a turbocharger rotor, e is eccentricity of disc, g is the acceleration of gravity, t is time, F is force, M is moment, respectively. The subscripts i = 1, 2, 3, 4 denote the compressor wheel, compressor neck, turbine neck and turbine wheel (Fig. 2), the subscript b denotes bearing, and the superscripts L and R denote left and right, respectively. For the compressor disc (i = 1) and the turbine disc (i = 4), one also has L = F L = F R = F R = 0, M L = M L = M R = M R = 0, and F Fx1 bx1 = Fby1 = Fbx4 = Fby4 = 0. y1 x4 y4 x1 y1 x4 x4

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Uniting all the equations, one can eliminate the inner forces and moments and derive the rotor dynamic equations. To make the equations dimensionless, the following variables are introduced: τ = ωr t, xi =

Xi , c

yi =

Yi Φi Ψi , φi = , ψi = , c π 2 π 2

i = 1, 2, 3, 4

where c is the clearance of bearing radius, and denote d( ) d2 ( ) , ( ) = dτ dτ 2 The dimensionless rotor dynamic equations can be expressed as ⎧ πl1 πl1 e1 + 2ζ x + 3E I1 ⎪ x 4x + ψ − 4x + ψ ⎪ 1 1 2 2 = c cos τ 1 λ 1 c c ⎪ l13 m 1 ω12 λ2 ⎪ ⎪ ⎪ ⎪ 2E I1 ⎪ 6c 6c + 2ζ ψ − Jz1 ϕ + ⎪ ψ x + 2ψ − x + ψ 1 2 =0 ⎪ 1 λ 1 Jy1 1 πl1 2 ⎪ l1 Jy1 ω12 λ2 πl1 1 ⎪ ⎪ ⎪ ⎪ πl1 πl1 3E I2 πl2 πl2 + 2ζ x + 3E I1 ⎪ ⎪ x −4x + 4x = − cmσ fωbx2 − ψ +4x − ψ + ψ −4x + ψ 1 1 2 2 2 2 3 3 2 2 ⎪2 λ 2 l 3 m 2 ω2 λ2 c c c c l23 m 2 ω12 λ2 ⎪ 2 1λ 1 1 ⎪ ⎪ ⎪ ⎪ 2ζ 6c 6c 6c 6c z2 ⎪ ψ + λ ψ2 − JJy2 ϕ2 + l J2EωI12 λ2 πl x1 + ψ1 − πl x2 +2ψ2 + l J2EωI22 λ2 πl x2 +2ψ2 − πl x3 +ψ3 = 0 ⎪ ⎪ 1 1 2 2 1 y2 1 2 y2 1 ⎪ 2 ⎪ ⎪ ⎪ ⎪x + 2ζ x + 3E I2 ⎪ −4x2 − πlc 2 ψ2 + 4x3 − πlc 2 ψ3 + l 3 m3EωI32 λ2 4x3 + πlc 3 ψ3 −4x4 + πlc 3 ψ4 = − cmσ fωbx3 ⎪ 3 2 2 2 ⎪ 3 1λ ⎪ 3 λ 3 l2 m 3 ω1 λ2 3 3 1 ⎪ ⎪ ⎪ 6c 6c 6c 6c ⎪ψ + 2ζ ψ − Jz3 ϕ + 2E I2 ⎪ x +ψ2 − πl x3 +2ψ3 + l J2EωI32 λ2 πl x3 + 2ψ3 − πl x 4 + ψ4 = 0 ⎪ 3 λ 3 Jy3 3 2 3 3 l2 Jy3 ω12 λ2 πl2 2 ⎪ 3 y3 1 ⎪ ⎪ ⎪ ⎪ 3E I3 πl3 πl3 e4 ⎪ ⎪x4 + 2ζ ⎪ λ x 4 + l 3 m 4 ω2 λ2 −4x 3 − c ψ3 + 4x 4 − c ψ4 = c cos τ ⎪ 3 1 ⎪ ⎪ ⎪ ⎪ Jz4 2ζ 6c 6c ⎪ =0 ϕ4 + l J2EωI32 λ2 πl x + ψ − x + 2ψ ⎨ψ4 + λ ψ4 − Jy4 3 3 4 4 πl 3 3 3 y4 1 (3) g ⎪ πl1 πl1 e1 + 2ζ y + 3E I1 ⎪ y 4y = − ϕ − 4y − ϕ sin τ − ⎪ 1 1 2 2 3 2 2 2 2 1 1 λ c c c ⎪ l1 m 1 ω1 λ cω1 λ ⎪ ⎪ ⎪ ⎪ 2ζ Jz1 2E I 6c 6c ⎪ ⎪ ϕ + λ ϕ1 + Jy1 ψ1 + l J ω12 λ2 − πl1 y1 + 2ϕ1 + πl1 y2 + ϕ2 = 0 ⎪ ⎪1 1 y1 1 ⎪ ⎪ ⎪ ⎪ σ f by2 g 2ζ 3E I1 πl1 πl1 3E I2 πl2 πl2 ⎪ ⎪ y −4y + 4y + y + + ϕ +4y + ϕ − ϕ −4y − ϕ 1 2 2 3 ⎪ c 1 c 2 c 2 c 3 = − cm 2 ω2 λ2 − cω2 λ2 l23 m 2 ω12 λ2 ⎪ 2 λ 2 l13 m 2 ω12 λ2 1 ⎪ ⎪ 1 ⎪ ⎪ J 2ζ 2E I 2E I 6c 6c 6c 6c z2 1 2 ⎪ ϕ2 + λ ϕ2 + Jy2 ψ2 + l J ω2 λ2 − πl1 y1 + ϕ1 + πl1 y2 + 2ϕ2 + l J ω2 λ2 − πl2 y2 +2ϕ2 + πl2 y3 +ϕ3 = 0 ⎪ ⎪ 1 y2 1 2 y2 1 ⎪ ⎪ ⎪ ⎪ σf g 2ζ ⎪ 3E I πl πl 3E I πl3 πl3 2 2 2 3 ⎪y3 + y3 + 3 −4y + 4y = − cm ωby3 + ϕ +4y + ϕ − ϕ −4y − ϕ 2 2 3 3 3 3 4 4 ⎪ 2 2 − cω2 λ2 λ c c c c ⎪ l2 m 3 ω12 λ2 l33 m 3 ω12 λ2 3 1λ ⎪ ⎪ 1 ⎪ ⎪ Jz3 2ζ 2E I 2E I 6c 6c 6c 6c ⎪ 2 3 ⎪ ϕ3 + λ ϕ3 + Jy3 ψ3 + l J ω2 λ2 − πl2 y2 + ϕ2 + πl2 y3 + 2ϕ3 + l J ω2 λ2 − πl3 y3 +2ϕ3 + πl3 y4 +ϕ4 = 0 ⎪ ⎪ 2 y3 1 3 y3 1 ⎪ ⎪ ⎪ ⎪ g 2ζ 3E I3 πl2 πl2 e4 ⎪ ⎪ ⎪y4 + λ y4 + l33 m 4 ω12 λ2 −4y3 + c ϕ3 + 4y4 + c ϕ4 = c sin τ − cω12 λ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ϕ + 2ζ ϕ + Jz4 ψ + 2E I32 2 − 6c y3 + ϕ3 + 6c y4 + 2ϕ4 = 0 4 λ 4 Jy4 4 πl3 πl3 l J ω λ ( ) =

3 y4 1

where ζ is the damping ratio, λ is the rotational speed ratio and is defined as λ = ωr ω1 , ω1 is the first-order critical rotational speed of a turbocharger rotor. The Adiletta oil film force model [19] is applied in this paper. The expression of Adiletta oil film force is Fbx = σ f bx ,

Fby = σ f by

2 2 R L σ = µλω1 R L c 2R

f bx f by

(x − 2y )2 + (y + 2x )2 3x V (x, y, α) − sin αG (x, y, α) − 2 cos αS (x, y, α) = 3yV (x, y, α) + cos αG (x, y, α) − 2 sin αS (x, y, α) 1 − x 2 − y2

(4) (5)

(6)


V (x, y, α) =

291

2 + (y cos α − x sin α) G (x, y, α) 1 − x 2 − y2

S (x, y, α) =

x cos α + y sin α

1 − (x cos α + y sin α)2 2 π y cos α − x sin α G (x, y, α) = + arctan 1 − x 2 − y2 2 1 − x 2 − y2 α = arctan

y + 2x π π y + 2x − − sgn y + 2x sgn x − 2y 2 x − 2y 2

(7) (8)

(9)

(10)

where σ is the Sommerfeld parameter, and V, S, G, α are the lubricant force variants, µ is the viscosity of lubricant oil, L is the length of bearing, and R is the radius of bearing, respectively. 3 Numerical simulations and dynamic behavior analysis The numerical calculation method is commonly used to solve the nonlinear differential equation. In this paper, the standard four-order Runge–Kutta method is applied to solve the dynamic responses of Eq. (3) and 500 cycles of the rotor’s dynamic responses are calculated. In order to exclude the transient responses, the first 400 cycles are discarded and the responses during the 401—500th cycles are regarded as a steady solution which is used to analyze the dynamic behavior of rotor. In this study, the bifurcation diagrams, together with the wave diagrams, FFT spectrums, orbits of disc centers and Poincaré maps are submitted to investigate the rotor dynamic behavior under different rotational speeds. The physical and geometrical parameters of the rotor are listed in Table 1. In order to investigate the effect of foundation excitation on the rotor dynamic behaviors, simulations are carried out both systems without and with foundation excitation. The results and discussions are presented in the following subsections. 3.1 Without foundation excitation If the foundation excitation is ignored, the rotor is supposed to be mounted on a static foundation. Using the dimensionless rotational speed as a control parameter to construct bifurcation diagrams, the bifurcation diagrams are given in Fig. 4. The wave diagrams, FFT spectrums, orbits and Poincaré maps (T = 2π) at different rotational speeds are given in Figs. 5–8 to illustrate the dynamic behaviors advanced. From Figs. 4–8, one can find that the vibration responses of the rotor without foundation excitation exhibit the following dynamic phenomena. (i) When the rotor rotates at a low speed, for example, ωr = 2π × 900 rad/s(λ = 0.67), the rotor performs synchronous and period-1 motion with small amplitude (Fig. 5). The vibration has only one singular frequency component accordant to the rotational speed. It is caused by the inertia forces of eccentricity. Table 1 The physical and geometrical parameters of rotor Parameters Mass Moment of inertia(10−6 kg · m2 ) Eccentricity distance Young’s modulus Length of rotor section Diameter of rotor section Damping ratio Journal bearing parameters The first-order critical speed a Obtained from Eq. (3) by substituting linear lubricant force f bxi 1 × 108 N/m for nonlinear lubricant force

Values m 1 =0.060 kg, m 2 =0.015 kg, m 3 =0.020 kg, m 4 =0.070 kg Jx1 = Jy1 = 9.0, Jz1 = 16.0; Jx2 = Jy2 = 0.2, Jz2 = 0.1 Jx3 = Jy3 = 0.2, Jz3 = 0.1; Jx4 = Jy4 = 8.0, Jz4 = 14.0 e1 = e4 = 0.05 mm, e2 = e3 = 0 E = 205 GPa l1 = 25 mm, l2 = 30 mm, l3 = 25 mm d1 = 10 mm, d2 = 8 mm, d3 = 14 mm ζ = 0.01 c = 0.1 mm, µ = 0.01Pa · s, R = 5 mm, L = 5 mm ω1 = 2π×1,348 rad/sa = k x xi , f byi = k y yi (i = 2, 3) and stiffness k x = k y =

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Fig. 4 The bifurcation diagrams of compressor disc response inx direction without foundation excitation: a λ = 0–3, b partial enlarged detail of λ = 1.2—1.5

(a) 0.5

(b) 0.4

x1

x1

0.3 0

0.2 0.1

-0.5 2600 2700 2800 2900 3000 3100

0

0

0.5

τ

1

1.5

2

0.5

1

λ1/ λ

(c) 0.5

(d)

1

y1

y1

0.5 0

0 -0.5

-0.5 -0.5

0

x1

0.5

-1 -1

-0.5

0

x1

Fig. 5 Response of compressor disc when rotational speed ωr = 2π × 900 rad/s(λ = 0.67) without foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

(ii) When the dimensionless rotational speed, λ reaches 1.26, oil whirl occurs (Fig. 6, ωr = 2π ×1,740 rad/s, λ = 1.29). Under this circumstance, there are two frequency components in the FFT spectrum, and the forced synchronous vibration is accompanied by oil whirl. An oil whirl is a kind of self-excited vibration due to fluid dynamic forces generated in the oil-lubricated bearings. As an oil whirl occurs, the rotor vibrates in the x–y plane and rotates around the bearing center. The ratio of the frequency of oil whirl to the rotational speed of the rotor is a constant and generally, it is 1/2. The amplitude of oil whirl is much higher than that of synchronous vibration. There are three points in the Poincaré map, which means that the oil whirl is period-3 motion. It can be found from Fig. 4b that there also exist period-5 and period-7 motions when the oil whirl occurs. If the direction of the rotor vibration coincides with the rotating direction of the rotor is referred to as a forward precession and the vibration direction oppose to the rotating direction of rotor is referred to as a reverse precession vibration, oil whirl is a rotor lateral forward subharmonic vibration. (iii) As the rotational speed reaches or exceeds twice of the first critical speed, the frequency of oil whirl approaches to the first critical speed, and the rotor vibrates very strongly. This vibration is referred to as oil whip. When oil whip occurs, the vibration changes to quasi-periodic motion, and one circle comes into being in the Poincaré map (Fig. 7, ωr = 2π × 2,700 rad/s, λ = 2.00). The frequency associated


(a)

293

(b)

2

1.5

1

x1

x1

1 0

0.5 -1 -2 2600 2700 2800 2900 3000 3100

0

0

0.5

1

τ

(c)

1.5

2

λ

(d)

1

0.4 0.2

0.5

y1

y1

0 0

-0.2 -0.5 -1

-2

-0.4 -1

0

1

-0.6 -2

2

-1

0

1

x1

x1

Fig. 6 Response of compressor disc when rotational speed ωr = 2π × 1740 rad/s(λ = 1.29) without foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

(a) 4

(b)

1.5

x1

x1

2

2

0 -2

1 0.5

-4 2600 2700 2800 2900 3000 3100

0

0

0.5

1

τ

(c)

1.5

2

λ

(d)

4

2 1

2

y1

y1

0 0

-1 -2 -4 -4

-2 -2

0

x1

2

4

-3 -4

-2

0

2

x1

Fig. 7 Response of compressor disc when rotational speed ωr = 2π × 2, 700 rad/s(λ = 2.00) without foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

with oil whip closes to the first natural frequency of the rotor and is independent on the rotation speed. The frequency-locked phenomenon is the significant characteristic of oil whip. (iv) With the continual increase of rotational speed, the point distributes in the Poincaré map is irregularly (Fig. 8, ωr = 2π × 5,400 rad/s, λ = 4.00). The vibration motion of the rotor becomes chaotic, and the chaos status comes into being. However, the oil whip still exists, and the frequency and amplitude of the oil whip do not change with the change of rotational speed. This verifies the frequency-locked characteristic of oil whip.

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(a)

(b)

4

1.5

x1

x1

2

2

0 -2

1 0.5

-4 2600 2700 2800 2900 3000 3100

0

0

0.5

1

τ

(c)

(d)

4

2

4 2

y1

2

y1

1.5

λ

0 -2

0 -2

-4 -4

-2

0

2

-4 -4

4

-2

x1

0

2

x1

Fig. 8 Response of compressor disc when rotational speed ωr = 2π × 5,400 rad/s(λ = 4.00) without foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

acceleration /m/s 2

(a) 80 60 40 20 0

0

100

200

300

400

500

600

400

500

600

frequency /Hz acceleration /m/s 2

(b) 60 40

20

0

0

100

200

300

frequency /Hz

Fig. 9 The acceleration of center housing vibration when the engine is running at 3,600 r/min: a X response, b Y response

3.2 With foundation excitation Since the hydrodynamic bearings that supports the rotor is set in the center housing of a turbocharger, the effect of vibration of center housing should be considered and treated as a foundation excitation for the turbocharger rotor dynamics. This foundation excitation can be obtained from experiment test when engine is running on. The acceleration of center housing tested from experiment at engine’s rating work condition (rotational speed 3,600 r/min, i.e., ωe = 2π×60 rad/s, and power 87 kW) is depicted in Fig. 9. Integrating the acceleration of the center housing vibration, one can obtain the velocity and displacement. Hence, the foundation excitation


295

Fig. 10 The bifurcation diagrams of compressor disc response inx direction with foundation excitation: a λ = 0–3 and ωr is the integer multiples of ωe , b partial enlarged detail of λ =1.2–1.5 and ωr is arbitrary

can be expressed as

X e = (1.48 cos ωe t + 2.37 cos 2ωe t + 1.07 cos 3ωe t + 3.03 cos 4ωe t) × 10−5 m Ye = (0.280 sin ωe t + 7.46 sin 2ωe t + 0.551 sin 3ωe t + 0.171 sin 4ωe t) × 10−5 m

X˙ e = −(0.558 sin ωe t + 1.79 sin 2ωe t + 1.21 sin 3ωe t + 4.57 sin 4ωe t) × 10−2 m/s Y˙e = (0.106 cos ωe t + 5.62 cos 2ωe t + 0.622 cos 3ωe t + 0.256 cos 4ωe t) × 10−2 m/s

(11)

(12)

where the first four harmonic response components are considered and the fifth and higher harmonic components are ignored. Since the vibrations along Z direction (axial direction of the turbocharger rotor) are ignored, the foundation excitation along Z direction does not need to present in this study. Considering the foundation excitation, the dimensionless displacements x and y and the velocities x and y in Eqs. (6)–(10) should be replaced by the dimensionless relative displacements x − xe and y − ye and dimensionless relative velocity x − xe and y − ye , respectively. The variables xe , ye and xe , ye are the dimensionless displacements and dimensionless velocities of the center housing inner obtained from Eqs. (11) and (12), respectively. The vibration responses of the rotor with foundation excitation exhibit far different dynamic phenomena from the responses without foundation excitation. In order to investigate the dynamic behaviors of the turbocharger rotor at different rotational speed, the turbocharger rotor’s rotational frequency is used as a control parameter to construct the bifurcation diagram. The ratio of turbocharger rotor’s rotational frequency to the engine crank shaft’s rotational frequency (60 Hz) is not an integer commonly, which means that when the engine crank shaft rotates one cycle, the turbocharger rotor does not return the original position. Hence, the dimen sionless time interval of the Poincaré map should be set as T = 2πω1 ωe . When the rotor performs the synchronous and period-1 motion at low speed, the number of point shown in Poincaré map equals to the denominator of the simplest fraction of rotor’s rotational frequency to the engine’s rotational frequency ωr b =n+ ωe a

(13)

where, n is an integer and b a is the simplest fraction. For instance, ωr = 2π × 260 rad/s, ωr ωe = 4 13 , the Poincaré map has three points (Fig. 11). This will produce a pseudo multiply periodic or quasi-periodic motion due to the nonsynchronism of a turbocharger rotor and engine crank shaft rotational frequencies. Similarly, when the rotor performs a period-m motion in case the oil whirl occurs, the point number of Poincaré map equals to m × a. Hence, Poincaré map cannot indicate the rotor’s motion in a direct way, except that the rotor’s rotational frequency is the integer multiples of the engine’s rotational frequency. This can also be observed from the bifurcation diagrams (Fig. 10). In Fig. 10a, the rotational frequencies of a turbocharger rotor are selected as the integer multiples of engine rotational frequency 60 Hz and period-1 motion is presented by one point in the bifurcation diagram. However, the bifurcation diagrams will be full of disorderly points, such as shown in Fig. 10b, if the rotational frequency of a turbocharger is not an integer multiples of engine rotation frequency.

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1

(b) 0.4

0.5

0.3

x1

x1

(a)

0 -0.5

0.2 0.1

-1 2600 2700 2800 2900 3000 3100

0

0

0.5

τ

(c)

1

1.5

2

0.5

1

λ1/λ

(d)

2

0 -0.2

1

y1

y1

-0.4 0

-0.6 -1 -2 -1

-0.8 -0.5

0

0.5

1

-1 -1

1.5

-0.5

x1

0

x1

Fig. 11 Response of compressor disc at rotational speed ωr = 2π ×260 rad/s(λ = 0.19, ωr /ωe = 4 31 ) with foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

(a)

(b)

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Fig. 12 Response of compressor disc at rotational speed ωr = 2π ×900 rad/s(λ = 0.67, ωr /ωe = 15) with foundation excitation: a wave diagram of x response, b FFT spectrum of x response, c orbit of disc center, d Poincaré map

The wave diagrams, FFT spectrums, orbits and Poincaré maps (T = 2πωr ωe ) at different rotational speeds are given in Figs. 11–15. At a low rotational speed when the oil whirl does not occur, the rotor vibration response with foundation excitation turns more complex than that without foundation excitation. For example, when the rotor rotates at a very low speed ωr = 2π × 260 rad/s(λ = 0.19, ωr ωe = 4 13 ), the response spectrum indicates there are many 1/3 harmonic components of engine rotational frequency, apart from the four low frequency components of engine excitation (Fig. 11). Similarly, the response spectrum of ωr = 2π × 250 rad/s, 2π × 255 rad/s, 2π × 270 rad/s, etc indicates there are many 1/6, 1/4, 1/2, etc., harmonic components of engine rotational frequency,


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respectively. It is evidence that the engine’s foundation excitation affects the turbocharger rotor dynamic behavior through the hydrodynamic bearings in a nonlinear way. More evidence that the foundation excitation affects the turbocharger rotor dynamic behavior is that when ωr = 2π × 900 rad/s(λ = 0.67, ωr ωe = 15), the rotor vibration response comes into being the period-2 motion (Fig. 12). If the foundation excitation is ignored, the response of system should be period-1 motion, such as shown in Fig. 5. It is obvious that the foundation excitation results in a strange change of the rotor vibration response. By comparing the results presented in Fig. 13 and those presented in Fig. 5, one can say when the rotor performs synchronous motion at ωr = 2π × 1, 080 rad/s(λ = 0.80, ωr ωe = 18), the vibration response is

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similar to that without foundation excitation, except that the FFT spectrum contains the multiple harmonic components of engine rotational frequency. These multiple harmonic components of engine rotation frequency not only include the calculated four orders presented in Eqs. (11) and (12), but also include higher order components. Withthe rotational speed increasing continually, the oil whirl will occur at ωr = 2π × 1, 740 rad/s(λ = 1.29, ωr ωe = 29). The vibration response is a period-3 motion, such as shown in Fig. 14. It is similar to that without foundation excitation. Despite the frequency spectrum contains the low-order harmonic components of engine excitation, the frequency of oil whirl dominates the frequency spectrum. Similarly, oil whip will still take place at ωr = 2π × 2, 700 rad/s(λ = 2.00, ωr ωe = 45). However, the dense points distributes in the Poincaré map disorderly (Fig. 15, compared to Fig. 7), which indicates that the system has reached the chaos status. If without foundation excitation, the system ought to reach the chaos status at λ = 4.00 (Fig. 8). It proves that on the influence of foundation excitation, the chaos status of the rotor system will come into being at a much lower rotational speed than that without foundation exaction. 4 Conclusion In this paper, the dynamic model of a turbocharger rotor-bearing system is established, including the engine’s foundation excitation and nonlinear lubricant force. The nonlinear dynamic behavior (bifurcation and chaos) of eccentricity is investigated by numerical simulation. The vibration responses show that the rotor dynamic behavior with foundation excitation differs greatly from that without foundation excitation. With the foundation excitation, the rotor dynamic behavior becomes more complicated, and the rotor will reach the chaos state at a lower rotational speed. Ackowledgments Financial support by The National Natural Science Foundation of China (No. 10572087) and China “973” Project (No. 2005CB724101) are gratefully acknowledged.

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