Behavior of hydrodynamic lubrication films under non-steady state speeds

Author’s Accepted Manuscript Behaviour of hydrodynamic lubrication films under non-steady state speeds H.C. Liu, F. Guo, B.B. Zhang, P.L. Wong

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To appear in: Tribiology International Received date: 3 June 2015 Revised date: 14 August 2015 Accepted date: 13 September 2015 Cite this article as: H.C. Liu, F. Guo, B.B. Zhang and P.L. Wong, Behaviour of hydrodynamic lubrication films under non-steady state speeds, Tribiology International, http://dx.doi.org/10.1016/j.triboint.2015.09.026 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Title: Behaviour of hydrodynamic lubrication films under non-steady state speeds Authors: H.C. Liu1, 2), F. Guo1)*, B.B. Zhang1), and P.L. Wong2) 1

School of Mechanical Engineering, Qingdao Technological University, 11 Fushun Road

Qingdao 266033, China 2

Department of Mechanical and Biomedical Engineering, City University of Hong Kong,

83 Tat Chee Avenue, Kowloon, Hong Kong, China

*Correspondence author: F. Guo E-mail: [email protected]

1

Abstract: The transient characteristics of hydrodynamic lubricating film thickness and its change rate (or squeeze effect) under start-up/shut-down and acceleration/deceleration motions were analyzed using a newly developed optical slider-on-disc test system. For a start-up/shut-down process, the change rate of film thickness attains its maximum when the final steady speed is achieved. In accelerating/decelerating motions, it was shown that the film thickness varies with some time lag to the speed, but its change rate is in phase with the speed. Influences of loads, frequencies and the maximum speed on the time lag were investigated. The film thickness hysteresis during acceleration-deceleration was explained by the measured squeeze effect. Numerical calculation was also carried out and the results quantitatively agree with the experiments.

Keywords: Transient hydrodynamic lubrication; squeeze effect; interferometry; film thickness.

2

Nomenclature b

:

Length of the slider in x (sliding) direction, m

h

:

Film thickness, m

h1, h0

:

Film thickness at inlet and outlet, m

H

:

Dimensionless film thickness, h/(h1h0)

H0

:

Dimensionless outlet film thickness, h0/(h1h0)

l

:

Width of the slider in y direction, m

p

:

Hydrodynamic pressure, Pa

P

:

Dimensionless pressure, p(h1h0)2/6bu0

Q

:

Dynamic parameter, (|∂h0/∂t|/h0)/(u/b)

t

:

Time, s

T

:

Dimensionless time, t/(b/u0)

u

:

x-component of sliding velocity, m/s

u0

:

Reference velocity, m/s

U

:

Dimensionless x-component of sliding velocity, u/u0

w

:

Applied load, N

W

:

Dimensionless applied load, w(h1h0)2/(6u0lb2)

W*

:

Dimensionless load-carrying capacity, (w/l)h02/(6ub2)

x

:

Cartesian coordinate in sliding direction, m

X

:

Dimensionless coordinate in sliding direction, x/b

y

:

Cartesian coordinate perpendicular to sliding direction, m

Y

:

Dimensionless coordinate perpendicular to sliding direction, y/l



:

Inclination angle of the slider, rad



:

Viscosity of lubricant, Pa·s

3

1 Introduction Hydrodynamic lubrication (HL) exists widely in industrial machine elements with conformal contacts, such as sliding/journal bearings, squeeze film dampers and hydro-viscous drive speed-regulating start [1]. Lubricating film thickness is one of the key operating parameters, which directly indicates the effectiveness of fluid film lubrication. Breakdown of lubrication film eventually gives rise to wear or failure of the lubricated surfaces, and is easy to occur under non-steady state conditions. For example, lubricating films undergo intermittent motions in a mechanical system driven by a step-motor, and failure may occur if the lubrication film is not well understood and designed. Therefore, it is an essential issue to obtain the true oil film thickness under dynamic conditions in HL studies. Up to now, transient

behaviours

of

hydrodynamic

lubrication

are

mainly

studied

through

theoretical/numerical analyses based on the Reynolds equation or the well-known NavierStokes equation. For plane slider or journal bearings, models and advanced algorithms have been presented to examine the film thickness, pressure variation, load-carrying capacity, temperature field and rheological effect under transient conditions, and large amounts of reports have been presented and references [2-6] only list some of the work. Venkateswarlu and Rodkiewicz [3] showed that when the sliding speed is closing to its final value in a startup process, the transient load capacity and drag force asymptotically approach their steady state values. Kennedy et al. [4] displayed that for a step change of the slider speed, the transient temperature and pressure are dependent of the initial conditions and the final speed of the slider. Yang and Rodkiewicz [5] numerically studied time-dependent behaviours of a centrally supported tilting pad bearings subjected to harmonic vibration and obtained the pressure and temperature change. A dynamic parameter was introduced to describe the effects of the tangential and normal motions. However, it is not easy to experimentally study transient HL film thickness in details, for example, those generated by a fixed-inclination slider bearing which is a basic model in lubrication theory [1]. Therefore some theoretical work has not been validated yet. A number of test methods have already been developed [714] and efforts are being made to study the transient behaviours. However, most of those experimental approaches enable the measurement of average film thickness only or the profile of the film shape with a relatively low resolution [9-14] and they are more suitable for industrial applications for identifying the effectiveness of lubrication. On the other hand, optical interferometry proves to be very successful in the laboratory measurement of non-conformal EHL film thickness, and has been used in transient EHL 4

under variable speed conditions including start-up, shut-down, reciprocating motion, unidirectional speed variation and other cyclic acceleration-deceleration motion [15-21]. Nishikawa et al. [15] presented EHL film data under reciprocating motion in ball-on-disc rolling/sliding contact and some intrinsic features of film building were revealed. The EHL film breathes cyclically as the wedging and squeezing action are not in phase, and film formation changes with oil types indicating the non-Newtonian and thermal effects. Their experimental results were subsequently correlated to the numerical analyses by Vahid et al. [22]. More measurements have been found in the literature to deliberately explore the EHL film thickness variation with variable entrainment speeds [16-21]. The time lag of the central film thickness to the speed was measured under reciprocating entrainment conditions [16, 17] and attributed to the movement to the contact centre of the thinner film at the contact rim, which is originally generated by squeeze effect. The results were correlated to the numerical analyses by Venner and Hagmeijer [23]. The intermolecular force was also presented by AlSamieh and Rahnejat [24] to account for the film thickness hysteresis when ultrathin film thickness is reached. However, the minimum film thickness does not show obvious phase shift with the speed in Ref. [17]. With unidirectional cyclic motion, it has been shown that there is film thickness hysteresis between the acceleration and the deceleration for both the central film thickness and the minimum film thickness. The film thickness in deceleration is larger, at shown in the work by Sugimura et al. [16], Glovnea and Spikes [18] and Ciulli et al. [19, 20]. Furthermore this film thickness hysteresis is tentatively attributed to the different lubricant entrainment at different times [19]. Generally, it can be argued that the above inherent behaviours under transient conditions mostly comes from the squeeze effect which is absent in steady state fluid film lubrication. Thus the understanding of the individual squeeze effect is certainly beneficial to the area of lubrication. However, EHL is a complicated process. The squeeze behavior and other effects are interrelated and cannot be readily separated. An alternative approach is to measure the squeeze behavior under hydrodynamic conditions where the flow behavior is only dominated by Reynolds equation. Unfortunately, the optical EHL test system cannot be directly in practice extended to the film thickness measurement of conformal HL due to difficulties in the flat-to-flat contact alignment and the limited measurement range. Recently, an optical slider-on-disc test rig [25] has been developed by the authors for detecting hydrodynamic lubrication film thickness. With a parallel mechanism for accurate slider inclination setting and a dichromatic interference intensity modulation (DIIM) approach for a rapid and a large range of measurement [26, 27], the slider-on-disc test rig is capable for the 5

measurement of transient HL film thickness. The objective of the present paper is to revisit the hydrodynamic lubrication theory with this newly developed optical test rig. The main focus is on the film formation and the individual squeeze effect under non-steady state conditions. 2 Experimental Method A custom-made optical slider-on-disc test rig [25] was used in the present study. As schematically illustrated in Fig. 1, the lubricated contact pair consists of a fixed-incline steel slider and a transparent rotating BK7 glass disk, in which a thin lubrication film can be generated once the disk rotates. The glass disk surface in contact with the slider was coated with a semi-reflective Cr coating which was protected by an additional transparent layer of SiO2. The sliding surface of the steel slider was highly polished to the roughness Ra around 9 nm. The slider inclination angle α can be known precisely through the number of fringes formed in the contact. The prescribed inclination angle can be adjusted and locked by the adjusting bolts located on the load arm. The term “film thickness” used in this study is referred to as the minimum film thickness h0 at the outlet of the slider bearing, as depicted in Fig. 1.

Fig. 1 Schematic diagram of the slider-on-disc setup using dichromatic interferometry. To enable measurement of the rather thick and rapidly changing transient film, dichromatic laser lights were adopted instead of the conventional white light/monochromatic light. Two sets of interference fringes were formed using two independent laser beams (with wavelengths of 653 nm and 532 nm) and recorded by a 3CCD camera at the rate of 25 fps. The film thickness h0 was obtained with the newly established dichromatic interference intensity modulation (DIIM) approach [26, 27]. In this approach, a modulated intensity signal is obtained by the intensity subtraction between the two sets of fringes (red and green). The modulating signal or beat wave of this modulated intensity signal presents an equivalent 6

wavelength, which is much larger than the wavelengths of the red and green components. With a proposed criterion to distinguish the first 3 half-cycles of the beat wave the measuring range can be significantly enlarged without wavelength ambiguity. With the red and green lights of 653 nm and 532 nm wavelengths respectively, the measurement range can be up to 4 μm. Table 1 Test conditions and properties of lubricant used in the experiments Test conditions Inclination 

5.3410−4 rad

Load w

5N

Maximum speed u

6 mm/s

Temperature

21 ℃ ± 0.5 ℃

Steel slider Size b × l

4 mm × 4 mm

Lubricant - PAO400 Viscosity  (21 ℃)

1.08 Pa·s

Refractive index n

1.47

PAO400 was used in all the experiments. Test conditions and lubricant properties are listed in Table 1. Several types of non-steady motions were employed, including: (a) start-up; (b)

shut-down;

(c)

continuous

acceleration/deceleration;

and

(d)

intermittent

acceleration/deceleration. These motions are schematically shown in Figure 2. All the tests were carried out with sliders of the same size of 4 mm× 4 mm at a constant inclination of 5.34 × 10-4 rad. If it is not specified, the applied load and the maximum speed for all cases are

0

Time (a) Start up

Time (b) Shut down

Velocity

Velocity

0

Velocity

Velocity

5 N and 6 mm/s, respectively.

0

0 Time (c) Acceleration/deceleration

7

Time (d) Intermittent

Fig. 2 Illustration of the non-steady state motions used in the experiments. The credibility of the present experimental system was validated by the measured film thickness vs. speed curves under steady conditions. The test speeds ranged from 1 mm/s to 8 mm/s. Interferograms of the longitudinal mid-section of the contact which were captured in the tests of different speeds are shown in Fig. 3(a). The dotted lines in the figure depict the position of the interference fringe of same order (equal film thickness). It shows that the slider lifts up when the sliding speed increases. Both the experimental and theoretical film thicknesses are plotted against speed at a log-log scale in Fig. 3(b), showing good agreement with each other. The log-log plot of film thickness vs. speed is linear and the speed index, i.e., the slope of the curve, is 0.5. The load capacity of a lubricated contact is determined by the speed and the corresponding film thickness. Decreasing the speed, thus, leads to the reduction of film thickness to bear the constant load applied. In other words, for a slider bearing as shown in Fig. 1, low film thickness and high speeds can enhance the load capacity [1].

Film thickness h0, m

3

2

Experimental Theoretical

1 1

(a)

2 3 4 5 6 7 89 Speed u, mm/s

(b)

Fig. 3 (a) Measured interferograms and (b) film thickness vs. speed in a slider-on-disk contact (load w = 5 N). It should be noted that different lubrication mechanisms may appear as film thickness changes with speeds. At moderate speeds used in the experiments, hydrodynamic effect dominates the lubrication. When the disc speed decreases, the outlet film thickness h0 decreases and hydrodynamic pressure is mostly generated in the region near the outlet. When the speed is reduced further, elastic deformation can be found in the outlet contact region and 8

the lubrication goes to EHL. Intermolecular forces, such as the solvation force and the Van der Waals pressure, will take effect when nano-meter film thickness is reached, which has been shown by Al-Samieh and Rahnejat [24]. The present experiments are concentrated on hydrodynamic lubrication, and to measure the film thickness and its change rate (or squeeze effect) under hydrodynamic lubrication regime. In fact, the contact configuration in Fig. 1 is not suitable for the measurement of EHL and ultrathin film lubrication. 3 Numerical Method The governing equation for a two-dimensional slider-on-disc contact hydrodynamically lubricated with an incompressible isothermal Newtonian fluid is written as,   h3 p    h3 p  h h    6u  12    x   x  y   y  x t

(1)

The boundary conditions for the pressure distribution:

 p |x 0,b  0   p | y  l /2  0

(2)

x h  h0  (h1  h0 )(1  ) b

(3)

 p( x, y)dxdy  w

(4)

The film thickness equation:

The force balance equation:

In the calculation, the time step was 1/240 second. The pressure distribution was evaluated with the multi-grid method [28]. Dimensionless equations were solved and the dimensionless parameters are given in the nomenclature. Five levels of grids were used arranging 257 nodes on both the x- and y-directions on the finest level. Convergence criteria of pressure and load on the finest grid level at each instant:

9

  Pi ,Kj  Pi ,Kj -1   10-4 K  Pi , j     Pi ,Kj  W  104  W 

(5)

Additionally, for the periodic and time-dependent speed in Fig. 2(c) and Fig. 2(d), the variables need to satisfy not only the boundary conditions given above, but also the following periodical condition:

 H H H K 0

K 0

K 1 0

 104

(6)

where H0 is the dimensionless outlet film thickness at time instant of one period, and the superscript K is the period number. The present model only considers hydrodynamic effect and is not suitable for very thin lubricating films where elastic deformation and intermolecular forces occur. 4 Results and Discussion 4.1 Start-up In the experiments of start-up motion the load was fixed to 5 N and the disk was accelerated at a constant rate to the final steady speed of 6 mm/s. Figure 4(a) shows the film thickness evolution at the constant acceleration of 3 mm/s2. During the start-up process, the transient film thickness is always smaller than the corresponding steady state film thickness. At the time instant of 2 s, the speed reaches its final value while the film thickness is still increasing. The curve of film thickness vs. time follows a stretched sigmoid shape with one end at the start-up origin and the other asymptotically approaching the final steady state value. From the plot of the change rate of film thickness with time (∂h0/∂t) in Fig 4(a), it can be noted that the film formation under a constant acceleration can be generally divided into three parts. At the beginning of the motion, the change rate of film thickness with time is relatively small in magnitude, but it increases rapidly and reaches the maximum. At the end, it decreases as could be expected. Venkateswarlu and Rodkiewicz [3] obtained a similar sigmoid shape for the transient load capacity vs. time which was calculated based on a fixed film thickness. In the present study, the calculation of film thickness was based on a constant load. However, a dimensionless load capacity parameter W* can be obtained through the following definition [29], 10

W *  (w l ) h02 (6ub2 )

(7)

where w = load (N), l = length of the slider in the y-direction (m, and 0.004 m in the present test), h0 = oil thickness at the outlet (m),  = dynamic viscosity of the lubricating oil (Pa·s), u = sliding speed (m/s) and b = length of the slider in the x-direction (m). Therefore, the present results are presented in the form of dimensionless load capacity vs. time, as shown in Fig. 4(b). The sigmoid shape of the load capacity vs. time shown in Fig. 4(b) is correlated with the results in [2]. Furthermore, the load capacity vs. time curve calculated based on the steady state film thickness shows an overshot before it settles at a steady value. The start-up characteristics of HL film as presented in the current study show large differences from EHL contacts [21] which are dominated by the growth of a stepped film.

1.5

Experimental h Theoretical h Theoretical transient h0/t

1.5

Theoretical steady state h

1.0

0.5

1.0 0.5

0.0 0.0 0

1

2

3 4 Time t, s

5

6

7

Load capacity W

2.0

*

0.012

h0/t, m/s

Film thickness h0, m

2.5

0.010 0.008 0.006 Experimental Theoretical steady state Theoretical transient

0.004 0.002 0.000

(a)

0

1

2

3 4 Time t, s

5

6

7

(b)

Fig. 4 (a) Film thickness and (b) dimensionless load capacity during start up with acceleration of 3 mm/s2 in 2 seconds (final velocity u = 6 mm/s). Moreover, it is interesting to find theoretically in Fig. 4(a) that the change rate of film thickness with time, ∂h0/∂t, reaches its maximum nearly at the same time as the speed attaining the final steady state value. To get more insight into this phenomenon, experiments under different accelerations have been carried out and the results are shown in Fig. 5. Figure 5(a) presents the temporal change in film thickness under different accelerations, 3 mm/s2, 6 mm/s2 and 12 mm/s2, respectively. Figure 5(b) shows the corresponding change rate of film thickness. The increase in the change rate of film thickness is faster for a higher acceleration. For the two with higher accelerations of 6 mm/s2 and 12 mm/s2, the film thickness change rate attains the maximum values almost at the time when the final steady speed is reached.

11

2.0 1.5

1.5

Experimental Theoretical 2 12 mm/s 2 6.0 mm/s 2 3.0 mm/s

Experimental Theoretical 2 12 mm/s 2 6.0 mm/s 2 3.0 mm/s

1.2 h0/t, m/s

Film thickness h0, m

2.5

1.0 0.5

0.9 0.6 0.3

0.0 0.1

0.0

1

0

1

2 3 Time t, s

Time t, s

(a)

4

5

(b)

Fig. 5 (a) Film thickness formation and (b) its changing rate vs. time during start-up at different accelerations (final steady velocity u = 6 mm/s). 4.2 Shut-down Shut-down experiments were also carried out with constant decelerations from an initial steady state velocity of 6 mm/s. The decrease of the film thickness is shown in Fig. 6 together with numerical simulated results. During the shut-down process, the transient film thickness at any deceleration is larger than that calculated based on the steady-state speed. The same trend was previously noted in EHL experiments [20]. When the sliding speed becomes zero, the film thickness is still quite large. In fact, the film thickness is governed by both the geometrical wedge and the squeeze effects. Once the rotation has been completely ceased, the slider approaches the disk surface slowly under a pure squeeze effect.

Experimental 2 3.0 mm/s 2 6.0 mm/s 2 12 mm/s Theoretical 2 3.0 mm/s 2 6.0 mm/s 2 12 mm/s

Film thickness h0, m

2.5 2.0 1.5 1.0 0.5 0.0 -1

0

1

2

3 4 5 Time t, s

6

7

8

9

Fig. 6 Film thickness variation with time at three constant decelerations (initial velocity u = 6 mm/s). 12

0.0 h0/t, m/s

-0.2 -0.4 -0.6

2

12 mm/s 2 6.0 mm/s 2 3.0 mm/s

-0.8 -1.0 -1.2

0

1

2 3 Time t, s

4

5

Fig. 7 Film thickness change rate vs. time during shut down (initial velocity u = 6 mm/s). Figure 7 shows the measured film thickness change rate with time during the shutdown process. It can be easily noted that the absolute change rate of film thickness with time, ∂h0/∂t, reaches the maximum almost at the same time when the disc stops or the final speed is reached. In the process of film thickness decreasing, both the tangential and normal motions could generate positive hydrodynamic pressure to give load capacity. To appreciate the role of these two motions, a dynamic parameter Q was defined by Yang and Rodkiewicz [5] as,

Q

h0 / t / h0

(7)

u/b

When Q = 1, these two effects are balanced. Figure 8 shows the temporal Q for the three cases of different decelerations. For low decelerations, the film thickness decreasing process is dominated by the entrainment speed, while for high decelerations, the dominating factor is the squeeze effect.

8

2

12 mm/s 2 6 mm/s 2 3 mm/s

7 6

Q

5 Theoretical

4 3 2 1 0

0.0

0.5

1.0 Time t, s 13

1.5

2.0

Fig. 8 Measured dynamic parameter Q vs. time during shut-down (initial velocity u = 6 mm/s). 4.3 Acceleration/deceleration Experiments were conducted with speeds in the form of cyclic triangular waves with different frequencies of same amplitude, as given in Fig. 2(c). The variations of film thickness which were obtained with various triangular speed waves of frequencies 0.25 Hz, 0.5 Hz and 1 Hz, respectively, are plotted in Fig. 9(a). To facilitate comparison, the abscissa is given in a normalized form of t/T, where t is the time and T is the period of the cyclic speed. It is shown that the measured film thickness vs. time curves are periodic. Their amplitude is a function of the frequency of speed wave, and a high frequency generates a small amplitude. The measured data and the theoretical results are well correlated. In Fig. 9(a), there is an obvious time lag of the film thickness with respect to the cyclic speed, which is attributed to the squeeze term, ∂h/∂t, in Eq. (1). When the speed does not decrease anymore and starts to increase, the squeeze term, ∂h/∂t, is negative, which can be inferred from Fig. 9(a). However, for the film thickness to increase immediately with the increasing speed, a positive ∂h/∂t is needed. Thus some transition time is necessary for the change of the squeeze term, ∂h/∂t, from negative value to positive value in such an engineering case. Therefore, the film thickness and the speed cannot change synchronously, and some time lag is found. This postulation can be demonstrated in Fig. 9(b), where the calculated squeeze term, ∂h/∂t, is given, and moreover it can be seen that the speed and the squeeze term are completely in phase. Under present non-steady state conditions, the hydrodynamic behavior comes from the two terms on the right-hand side of Eq. (1), namely, wedge effect and squeeze effect [1]. The sum of the two terms, which is a linear function of the speed u (∂h/∂x is a negative constant) and ∂h/∂t has to be negative for load carrying. When speed increases, the wedge effect becomes strong, and as a response ∂h/∂t has to change to accommodate the constant load applied. The above indicates that the squeeze effect of the lubricant would present some resistance to the change of the lubricating film thickness with speed, in other words, viscous damping effect. To get more insight into the film generation, the variation of film thickness for a complete speed cycle of 0.25 Hz frequency is drawn in Fig.10. The steady state film thickness corresponding to the speed cycle, which is calculated from Eq. (1) neglecting the squeeze term, is also plotted in the figure. It is easily to see that for the same speed, the film 14

thickness achieved during the deceleration process is larger than that during acceleration, and the variation of transient film thickness forms a hysteresis loop in a speed cycle. Such film hysteresis has also been reported in some EHL studies [16, 19, 20]. The hysteresis loop on the right side of the steady film thickness shows transient film thickness characterized by ∂h/∂t > 0 (increasing h0). Increasing the film thickness h0 would result in negative pressure (or reduction in pressure), if considering the squeeze term alone. As stated in last paragraph, the total load capacity of a slider bearing is contributed by the wedge and the squeeze effects. While the wedge term is always negative, the magnitude of the sum of the two terms is reduced if the squeeze term, ∂h/∂t, is positive. The film thickness would thus become thinner under a constant load, as the present experimental condition. As shown in Fig. 10, the transient film thickness in the hysteresis loop on the right is smaller than the corresponding steady film thickness in magnitude. Similarly, the transient film thickness on the left-hand side of the steady film thickness curve in Fig. 10 is larger than the steady film thickness since h0 is reducing, i.e. ∂h/∂t < 0. At the two turning points of the transient film thickness curves, ∂h/∂u is equal to 0, and ∂h/∂t, the squeeze term in Eq. (1), is also lost due to the linear relationship between u and t. Therefore the transient film thickness at these two points should be the same as the corresponding steady values. The experimental results excellently describe this prediction, where the hysteresis loop and the steady state film thickness vs. speed curve intersects at the two turning points.

Speed

2

1.4 0.2

0.4 0.6 Time t/T

0.8

4

0 -1

2

Speed

-2

0 0.0

1

6

-3

1.0

(a)

Speed u, mm/s

4

T= 4s 2s 1s

2 h0/t, m/s

1.6

6

Speed u, mm/s

Film thickness h0, m

3 T Exp. Theoretical 2.0 1s 2s 1.8 4s

0 0.0

0.2

0.4 0.6 Time t/T

0.8

1.0

(b)

Fig. 9 (a) Variation of film thickness and (b) the squeeze term h/t vs. normalized time during one cycle of different frequencies (triangular speed range: 0 to 6 mm/s, load w = 5 N).

15

Film thickness h0, m

2.5 Deceleration 2.0 1.5 Acceleration 1.0

Experimental Theoretical Steady State

0.5 0.0 0

1

2 3 4 Speed u, mm/s

5

6

Fig. 10 Film thickness vs. speed in an accelerated/decelerated motion at the frequency rate of 0.25 Hz (triangle wave speed range: 0 to 6 mm/s). More calculations have been carried out as shown in Fig. 11. It is found that the higher the frequency, the larger the time lag and the smaller the amplitude of transient film thickness. A higher frequency means the stronger squeeze effect, i.e., higher damping effect, which leads to more resistance to the change of film thickness with speed.

0.20

2.25 tlag/ Tperiod

Film thickness h0, m

Relative time lag

0.25

0.15 0.10 0.05 0.00

2.00 1.75 1.50 hmax

1.25

hmin

1.00

havg

0.75 0.1 Frequency f, Hz

0.1

1

1 Frequency f, Hz

(a)

(b)

Fig. 11 (a) Relative time lag and (b) amplitude of film thickness vs. frequency of triangular wave speed (speed range: 0 to 6 mm/s). 4.4 Intermittent acceleration/deceleration A periodic intermittent motion consisting of a triangular wave speed followed by a zero speed interval, as illustrated in Fig. 2(d), was adopted. The zero speed spans over one third of the whole cycle time and the speed varies from 0 to 6 mm/s as shown in Fig. 12. The variation of film thickness with time is also periodic. Figure 12 illustrates only one cycle. It can be seen that a short cycle produces a small amplitude of the film thickness – time curve, 16

but a large time lag with respect to the speed, which is similar to the behavior under continuous acceleration/deceleration. During the zero speed, there is no abrupt film thickness drop due to the squeeze effect.

T Exp. Theory 6s 3s 1.5s

1.8 1.6

6

4

1.4 1.2

2

1.0

Speed u, mm/s

Film thickness h0, m

2.0

Speed

0.8

0 0.0

0.2

0.4 0.6 Time t/T

0.8

1.0

Fig. 12 Variation of film thickness vs. normalized time under three frequencies in intermittent acceleration/deceleration motions (intermittent triangle speed range: 0 to 6 mm/s). Parametric studies were carried out to investigate the influences of the maximum speed and the load under intermittent motion on HL films. The period was fixed for 3 seconds for all experiments. Results of the experiments are plotted in Fig. 13, which shows that the experimental and theoretical results are well correlated. The change in film thickness within a speed cycle due to the effect of speed and load are plotted in Figs. 13(a) and (b), respectively. The results, as depicted in Fig. 13, show that low loads and small maximum speeds would extend the time delay. Moreover, higher maximum speed produces larger amplitude of film thickness – time variation.

2.8

12 mm/s 9 mm/s 6 mm/s Theoretical

2.4

Film thickness h0, m

Film thickness h0, m

2.8

2.0 1.6 1.2

Speed

0.8

2N 5N 10 N

2.4

Theoretical

2.0 1.6 1.2 0.8

Speed

0.4 0.0

0.5

1.0

1.5 2.0 Time t, s

2.5

3.0

0.0

(a)

0.5

1.0

1.5 2.0 Time t, s

2.5

3.0

(b)

Fig. 13 Response of film thickness corresponding to different HL parameters during cycle time of 3 s in an intermittent acceleration/deceleration motion. (a) Influences of maximum speed. (b) Influences of load, speed range from 0 to 6 mm/s. 17

4

Conclusion Experiments and theoretical analyses have been conducted aiming to understand the

individual squeeze effect on the hydrodynamic lubricating film formation in a slider-on-disc contact under different types of non-steady motions. The results can be summarized as the follows:

(1)

For a start-up and shut-down process, the film thickness approaches its steady value

asymptotically. Under the specified working conditions, the change rate of film thickness or the squeeze effect attains its maximum at about the same time as the final steady speed is achieved, which is demonstrated by both experiments and numerical analyses. (2)

With cyclic triangular wave speeds (acceleration/deceleration), variations of the film

thickness have the same time period of but lag behind the speed variation. However, the film thickness change rate/squeeze is in phase with the speed. The squeeze effect can explain the film thickness hysteresis between the deceleration and acceleration processes, and the larger film thickness in the deceleration process. (3)

In an accelerating/decelerating motion, the amplitude of the film thickness variation

decreases and the phase delay increases with increasing the frequency of the motion. In addition, smaller load and lower speed maximum give longer time delay. (4)

The optical slider-on-disc test rig equipped with dichromatic interference intensity

modulation is capable of capturing detailed hydrodynamic lubricating film thickness data under steady and transient conditions. The experiment data presented in the present paper validated quantitatively the classical hydrodynamic lubrication theory.

Acknowledgements The work described in this paper was supported by the National Natural Science Foundation of China (No.51275252) and the Research Grants Council of Hong Kong (Project 18

No. CityU123813). The authors would like to thank Dr. Li Xin-ming of QTECH for his help and valuable discussion.

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capacitance tomography. Meas Sci Rev 2014;14(1):8-15. [10] Azushima A. In situ 3D measurement of lubrication behavior at interface between tool and workpiece by direct fluorescence observation technique. Wear 2006;260(3):243-8. [11] Dwyer-Joyce RS, Drinkwater BW, Reddyhoff T. Operating limits for acoustic measurement of rolling bearing oil film thickness. Tribol Trans 2004;47(3):366-57. [12] Dwyer-Joyce RS, Harper P, Drinkwater BW. A method for the measurement of hydrodynamic oil films using ultrasonic reflection. Tribol Lett 2004;17(2):337-48. [13] Li P, Zhu Y, Zhang Y, Chen Z, Yan Y. Experimental study of the transient thermal effect 19

and the oil film thickness of the equalizing thrust bearing in the process of start-stop with load. IMechE J Eng Tribol 2013;227(1):26-33. [14] Kasolang S, Ahmed DI, Dwyer-Joyce RS, Yousif BF. Performance analysis of journal bearings using ultrasonic reflection. Tribol Int 2013;64:78-84. [15] Nishikawa H, Handa K, Kaneta M. Behavior of EHL films in reciprocating motion. JSME Int J Ser C 1995;38(3):558-67. [16] Sugimura J, Jones WR, Spikes HA. EHD film thickness in non-steady state contacts. ASME J Tribol 1998;120(3):442-52. [17] Glovnea RP, Spikes HA. Behavior of EHD films during reversal of entrainment in cyclically accelerated/decelerated motion. Tribol Trans 2002;45(2):177-84. [18] Glovnea RP, Spikes HA. The influence of lubricant properties on EHD film thickness in variable speed conditions. In Proceedings of the 2003 Leeds-Lyon Symposium on Tribology, vol. 43, 2004, Elsevier Tribology Series, pp. 401-8. [19] Ciulli E, Stadler K, Draexl T. The influence of the slide-to-roll ratio on the friction coefficient and film thickness of EHD point contacts under steady state and transient conditions. Tribol Int 2009;42(4):526-34. [20] Ciulli E. Non-steady state non-conformal contacts: friction and film thickness studies. Meccanica 2009;44(4):409-25. [21] Glovnea RP, Spikes HA. Elastohydrodynamic film formation at the start-up of the motion. IMechE J Eng Tribol 2001;215(2):125-38. [22] Vahid DJ, Rahnejat H, Jin ZM. Dowson D. Transient analysis of isothermal elastohydrodynamic circular point contacts. IMechE J Mech Eng Sci 2001;215(10):1159-72. [23] Venner CH, Hagmeijer R. Film thickness variations in elasto-hydrodynamically lubricated circular contacts induced by oscillatory entrainment speed conditions. IMechE J Eng Tribol 2008;222(4):533-47. [24] Al-Samieh M, Rahnejat H. Ultra-thin lubricating films under transient conditions. J Phys D Appl Phys 2001;34(17):2610-21. [25] Guo F, Wong PL, Fu Z, Ma C. Interferometry measurement of lubricating films in slider-on-disc contacts. Tribol Lett 2010;39(1):71-9. [26] Guo L, Wong PL, Guo F, Liu HC. Determination of thin hydrodynamic lubricating film thickness using dichromatic interferometry. Appl Opt 2014;53(26):6066-72. [27] Liu HC, Guo F, Guo L, Wong PL. A dichromatic interference intensity modulation 20

approach to measurement of lubricating film thickness. Tribol Lett 2015;58(15):1-11. [28] Venner CH, Lubrecht AA. Multi-level methods in lubrication. Elsevier; 2000: p.37. [29] Li X, Guo F, Yang S, Wong PL. Measurement of load-carrying capacity of thin lubricating films. J Tribol 2012;134(3):044501.

Behaviour of hydrodynamic lubrication films under non-steady state speeds H.C. Liu1, 2), F. Guo1)*, B.B. Zhang1), and P.L. Wong2) 1 School of Mechanical Engineering, Qingdao Technological University, 11 Fushun Road Qingdao 266033, China 2 Department of Mechanical and Biomedical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

Highlights: The transient characteristics of hydrodynamic lubricating film thickness and its change rate (or squeeze effect) under start-up/shut-down and acceleration/deceleration motions were analyzed using a newly developed optical slider-on-disc test system. For cyclic acceleration/deceleration, the film thickness varies out of phase with the speed, while its change rate, in phase. During start-up and shut-down, the change rate of film thickness attains the maximum almost at the same time when the final steady speed is achieved. Good quantitative agreement between the experiments and numerical results has been achieved.

21

Captions Fig. 1 Schematic diagram of the slider-on-disc setup using dichromatic laser interferometry. Fig. 2 Illustration of the non-steady state motions used in the experiments. Fig. 3 (a) Measured interferograms and (b) film thickness vs. speed in a slider-on-disk contact (load w = 5 N). Fig. 4 (a) Film thickness and (b) dimensionless load capacity during start up with acceleration of 3 mm/s2 in 2 seconds (final velocity u = 6 mm/s). Fig. 5 (a) Film thickness formation and (b) its changing rate vs. time during start-up at different accelerations (final steady velocity u = 6 mm/s). Fig. 6 Film thickness variation with time at three constant decelerations (initial velocity u = 6 mm/s). Fig. 7 Film thickness change rate vs. time during shut down (initial velocity u = 6 mm/s). Fig. 8 Measured dynamic parameter Q vs. time during shut-down (initial velocity u = 6 mm/s). Fig. 9 (a) Variation of film thickness and (b) the squeeze term h/t vs. normalized time during one cycle of different frequencies (triangular speed range: 0 to 6 mm/s, load w = 5 N). Fig. 10 Film thickness vs. speed in an accelerated/decelerated motion at the frequency rate of 0.25 Hz (triangle wave speed range: 0 to 6 mm/s). Fig. 11 (a) Relative time lag and (b) amplitude of film thickness vs. frequency of triangular wave speed (speed range: 0 to 6 mm/s). Fig. 12 Variation of film thickness vs. normalized time under three frequencies in intermittent acceleration/deceleration motions (intermittent triangle speed range: 0 to 6 mm/s). Fig. 13 Response of film thickness corresponding to different HL parameters during cycle time of 3 s in an intermittent acceleration/deceleration motion. (a) Influences of maximum speed. (b) Influences of load, speed range from 0 to 6 mm/s. 22

Table 1 Test conditions and properties of lubricant used in the experiments

Test conditions Inclination 

5.3410−4 rad

Load w

5N

Maximum speed u

6 mm/s

Temperature

21 ℃ ± 0.5 ℃

Steel slider Size b × l

4 mm × 4 mm

Lubricant - PAO400 Viscosity  (21 ℃)

1.08 Pa·s

Refractive index n

1.47

23