Acta of Bioengineering and Biomechanics Original paperVol. 16, No. 1, 2014 DOI: 10.5277/abb140111

Time dependent human hip joint lubrication for periodic motionwith stochastic asymmetric density function

KRZYSZTOF WIERZCHOLSKI*

Technical University of Koszalin, Institute of Technology and Education, Koszalin, Poland.

The present paper is concerned with the calculation of the human hip joint parameters for periodic, stochastic unsteady, motion withasymmetric probability density function for gap height. The asymmetric density function indicates that the stochastic probabilities of gapheight decreasing are different in comparison with the probabilities of the gap height increasing. The models of asymmetric densityfunctions are considered on the grounds of experimental observations. Some methods are proposed for calculation of pressure distribu-tions and load carrying capacities for unsteady stochastic conditions in a super thin layer of biological synovial fluid inside the slide bio-bearing gap limited by a spherical bone acetabulum. Numerical calculations are performed in Mathcad 12 Professional Program, by usingthe method of finite differences. This method assures stability of numerical solutions of partial differential equations and gives propervalues of pressure and load carrying capacity forces occurring in human hip joints.

Key words: Hip joint, stochastic effects, asymmetric density function, periodic motion

1. Introduction

The problem of hip joint or endoprosthesis lubri-cation for unsteady stochastic periodic motion hasalready been considered by Knoll, Cwanek, Mow,Pawlak [1]−[5] and additionally in the author’s papers[6]−[10]. Up to now the random considerations andsolution methods have been based on the probabilitysymmetric density functions of gap height changes.For example, the Gauss and pseudo-Gauss probabilitydensity functions of gap height changes are consid-ered. Such conditions assumed denote that in eacharbitrarily chosen time period, the probabilities of gapheight decreasing have the same rank as the probabil-ity values of gap height increasing. Random changesof the gap height of the natural normal and pathologi-cal human hip joint or gap height between head andacetabulum of endoprosthesis are caused mainly byvibrations in unsteady motion and the roughness ofthe joint surfaces.

From the many experimental observations [1],[11]−[17] it follows that during the arbitrarily chosentime period of unsteady periodic motion of the hipjoint the probabilities of the joint gap height decreas-ing are not equal to the probabilities of the gap in-creasing. Therefore, in the present paper, the asym-metric density function for probability gap heightchanges is taken into account.

Moreover, contrary to the foregoing papers [7],[18], the present paper presents a calculation algo-rithm which satisfies stability conditions of numer-ous numerical solutions of periodic lubricationproblems in the form of partial differential equationsand gives real values of fluid velocity componentsand load carrying capacities occurring in human hipjoints and finally real values of friction forces duringthe forward and backward locomotion of humanlimbs.

The present research aimed at the following:• To show a new unified calculation algorithm

for stochastic periodic load carrying capacity determi-

______________________________

* Corresponding author: Krzysztof Wierzcholski, Technical University of Koszalin, Institute of Technology and Education,ul. Śniadeckich 2, 75-453 Koszalin, Poland. Tel: +48 94 3478344, fax: +48 94 3426753, e-mail: krzysztof.wierzcholski@wp.pl

Received: March 17th, 2013Accepted for publication: August 8th, 2013

K. WIERZCHOLSKI84

nation of spherical, bio-bearing human hip joint sur-faces in the case of asymmetric density function.

• To indicate the time depended hydrodynamicpressure and load carrying capacity changes causedby the periodic lubrication for various standard de-viations with asymmetric probability density func-tion.

• To show the application of the semi-numericalmethod for stochastic periodic pressure calculationimplemented by Mathcad 12 Professional Program.

2. Materials and methods

2.1. Measurements of asymmetricprobability density function

for gap height

The probability density function of the gap heightwas assumed by virtue of experimental measurementsof cartilage sample roughness and its standard devia-tions. Random changes of cartilage surface are de-scribed using the probability asymmetrical densityfunctions on the basis of comparison between theresults of Cwanek and Wierzcholski’s experiments [1]and investigations of Dowson and Mow [2], [11], seeFig. 1a, Fig. 1b.

The measurements of samples of natural hip jointsurfaces are performed by means of a mechanical orlaser sensor, where normal (non-used) and pathologi-cal (used) cartilage samples are taken into account.During the measurements performed by means of themechanical sensor the samples of cartilages havingdimensions 2 mm × 2 mm are used, and for the meas-urements by using laser sensor the samples havingdimensions of about 10 mm × 10 mm are applied. Realgap height changes depend on the variations of carti-lage surface, environmental conditions, age of the jointconsidered, localization of collagen fibers, phospho-lipid solid lubricant particles [4].

Measurements of the values of changes on thesample surface (2 mm × 2 mm) of a normal cartilageresting on the spherical bone head of human hip joint,see Fig. 1, have been performed by using the micro-sensor laser installed in the Rank-Taylor-Hobson-Talyscan-150 Apparatus, and elaborated by means ofthe Talymap Expert and Microsoft Excel ComputerProgram [1], [10]. From 29 measured samples thefollowing parameters have been calculated: differ-ences between values of rises and deeps of bone headsurfaces in human hip joint (St), arithmetic mean be-tween values of 5 rises and 5 deeps of bone head sur-face (Sz), standard deviation of probability densityfunction of roughness distribution of cartilage surface(Sa) [1], [6]. The measured values of St oscillatewithin the interval from 9.79 to 24.7 micrometers. The

μm

20

15

10

5heig

ht o

f asp

eriti

es 1-1

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.82.0 mm

μm242220181614121086

1

1

a)

Fig. 1. Random changes of the joint gap height due to roughness and unsteady load: (a) measurement of roughness values on the samplesurface (2 mm × 2 mm) of normal joint cartilage taken from the bone head of human hip joint, (b) longitudinal section 1-1 of the surface

of normal hip joint cartilage of bone head measured by the mechanical sensor

a)

b)

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 85

measured Sz values fluctuate within the interval from8.52 to 14.7 micrometers. Finally, we find that thecalculated values of Sa are contained in the intervalfrom 0.78 to 1.96 micrometer.

For example, Fig. 2a, Fig. 2b show a hip jointFrankobal’s half-endoprosthesis in 3D and 2D geo-metrical structure where the head is seated onto thepin. Its head consists of the outer metal elementhaving spherical form and the inner element made ofpolyethylene.

In the geometrical structure presented in Fig. 2we have the following amplitude parameters: Sa =0.0604 μm, St = 0.632 μm and Sz = 0.538 μm [1],[6 ], [18].

The above-mentioned experimental results ob-tained complete the thesis about the asymmetric prob-ability density function.

2.2. Random gap height description

The unsteady time dependent gap height has peri-odic changes. The dimensionless gap height εT1 de-pends on the variable ϕ and ϑ1 and dimensionlesstime t1 and consists of two parts [6], [18], [19]

εT1 = εT /ε0 = εT1s(ϕ, ϑ1, t1) + δ1(ϕ, ϑ1, ξ),

εT1s = ε1 + ε1t , (1)

where εT1s denotes the total dimensionless height ofnominally smooth part of the thin fluid layer. This partof the gap height contains dimensionless correctionsof the gap height caused by the hyper elastic cartilagedeformations and is the sum of time independent ε1

and time dependent ε1t components. The symbol δ1denotes the dimensionless random part of changes ofthe gap height resulting from the vibrations, unsteadyloading and surface roughness asperities of cartilageor artificial surfaces measured from the nominal meanlevel. The symbol ξ describes the random variable,which characterizes the roughness arrangement.

Figure 3 presents the periodic time-dependent gapheight with periodic perturbations.

The time-independent value of the smooth part ofthe gap height has the following dimensional form[1], [18]

ε(ϕ, ϑ1) = ε0ε1(ϕ, ϑ1)

≡ Δεxcosϕ sinϑ1 + Δεy sinϕ sinϑ1 − Δεz cosϑ1 − R

+[(Δεxcosϕ sinϑ1 + Δεy sinϕ sinϑ1 − Δεz cosϑ1)2

+ (R + εmin)(R + 2D + εmin)]0.5, (2)

where ε − characteristic dimensional value of gapheight. We assume the centre of spherical bone headto be at the point O(0, 0, 0) and the centre of sphericalacetabulum at the point O1(x − Δεx, y − Δεy, z + Δεz).The eccentricity has the value D (see Fig. 3).

0.632 μm

1.0 mm

0.98 mm

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 mm

μm0.300.250.200.150.100.05

μm0.600.550.500.450.400.350.300.250.200.150.10

a)

b)

1

1

1-1

Fig. 2. Sample of a new head of endoprosthesis FRANKOBAL GL46MM:(a) measured roughness of surface, (b) vertical section of surface

a)

b)

K. WIERZCHOLSKI86

3. Results

3.1. Probability gapheight density function

and its cumulative values

At first we define the stochastic parameters.Stochastic changes of the joint gap height are de-

termined by virtue of stochastic parameters such asstandard deviation, average standard deviation, expec-tancy value and distributive function. The parametersmentioned are now defined. Expectancy stochasticoperator E imposed on the product of gap height (*)and probability density function f denotes average ran-dom changes of gap height defined by [19], [20]

.)((*)(*) 11 δδ dfE ×= ∫+∞

∞−

(3)

Standard deviation σ, average standard deviation℘ and distributive function F have the following form

height.gapaverage/

,(*)(*) 22

σ

σ

≡℘

−≡ EE(4)

.)()( 111

1

δδδδ

∫∞<

∞−

= dfF (5)

Pseudo-Gaussian symmetrical probability densityfunction

After the first part of measurements of the jointcartilage deformations we observe the case where

probabilities of the gap height random changes de-creasing are the same as probabilities of the gapheight random variations increasing. This result en-ables us to create a probability symmetrical densityfunction in the pseudo-Gaussian and triangle form.

• The pseudo-Gaussian symmetrical probabilitydensity function has the form [6], [19]

⎪⎪⎩

⎪⎪⎨

⎧

=>

=≤⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−

≡

,09375.1for0

,09375.13235for

35321

)(

131

1

321

113

c

fg

δ

δδδ

364583.013 =gσ . (6)

Symbol c13 = 1.093755 denotes the half total rangeof random variable of the thin layer thickness fornormal hip joint. The symbol σg13 = 0.364583 denotesthe dimensionless standard deviation. To obtain di-mensional value of the standard deviation σg we mustmultiply σg13 by the characteristic value of gap heightε0 = 10⋅10−6 m. In this case, the dimensional standarddeviation equals 3.7 micrometers. From the measure-ments we have obtained the standard deviation valueof about 3.5 micrometers for normal cartilage. Thevarious Gaussian dimensionless functions fg1 are pre-sented in Fig. 4.

• The classical triangle symmetrical probabilitydensity function has the form

⎪⎩

⎪⎨

⎧

>

≤≤−≤≤−+

≡Δ

.1for0,10for1,01for1

)(

1

11

11

1

δδδδδ

δf (7)

Triangle function fΔ described by formula (7) ispresented in Fig. 5a in graphical form.

Fig. 3. The gap height time-changes of human hip joint:(a) time-changes of the gap and limit velocity values, (b) stochastic changes of the joint gap,(c) the centre of the spherical bone head and acetabulum, (d) gap height changes with time

a)

b)

c)

d)

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 87

a)

b)

Fig. 5. Probability values of the gap height changesif probability of the gap height decreasing has the same rank

as probabilities of the gap height increasing:(a) symmetric probability density function,

(b) cumulative function

We put equation function fΔ from (7) into formula(5). Hence distributive function FΔ for symmetrictriangle density function is presented in Fig. 5b.

We put equation (7) into formula (4). Hence stan-dard deviation for symmetric triangle density functionhas the following form [19], [20]

.408248.0)()( 221 =−= ΔΔΔ fEfEσ (8)

To obtain dimensional value of the standard de-viation σΔ we must multiply σΔ1 by the characteristicvalue of the gap height ε0 = 10⋅10−6 m. In this case,the dimensional standard deviation equals 4.08 mi-crometers for normal cartilage.

3.2. Asymmetrical probabilitydensity function

for larger increase

From the next part of measurements of joint carti-lage deformations we observe the case where prob-abilities of the gap height random changes increasingare larger than probabilities of the gap height randomvariations decreasing. The experimental results ob-tained enable us to create the probability symmetricaldensity function in the following form

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

>

+≤≤+−

≤≤+−

≤≤−+

−≤≤−+

≡

.1for0

,143for33

,430for1

31

,041for13

,411for

31

31

)(

1

11

11

11

11

1

δ

δδ

δδ

δδ

δδ

δAf (9)

Asymmetric function fA described by formula (9)is presented in Fig. 6 in graphical form.

Fig. 4. Symmetric Gaussian probability density dimensionless functions for values of gap height changesif probabilities of gap height decreases and increases have the same order

K. WIERZCHOLSKI88

Fig. 6. Asymmetric density function in the casewhere probabilities of the gap height decreasing are less

than probabilities of the gap height increasing

We put equation (9) into formula (5). Hence dis-tributive function for asymmetric density function isdescribed by the following formula (10) and is pre-sented in Fig. 7

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+≤≤−+−

≤≤++−

≤≤−++

−≤≤−++

≡

= ∫∞<

∞−

.143for

213

23

,430for

41

61

,041for

41

23

,411for

61

31

61

)()(

112

1

112

1

112

1

112

1

111

1

δδδ

δδδ

δδδ

δδδ

δδδδ

dfF AA

(10)

Fig. 7. Distribution function for asymmetrical density functionelaborated by virtue of probability values presented in Fig. 6

We put equation (9) into formula (4). Hence stan-dard deviation for asymmetric density function has thefollowing form [19], [20]

.381837.0)()( 221 =−= AAA fEfEσ (11)

To obtain dimensional value of the standard de-viation σA we must multiply σA1 by the characteristicvalue of gap height ε0 = 10⋅10−6 m. In this case, thedimensional standard deviation equals 3.81837 mi-crometers. From the measurements we have obtainedthe standard deviation value of about 4.0 micrometersfor normal cartilage.

3.3. Asymmetrical probabilitydensity function

for smaller increase

The third part of measurements of joint cartilagedeformations shows the case where the probabilitiesof the gap height random changes increasing are lessthan the probability of the gap height random varia-tions decreasing. The experimental results obtainedenable us to create the probability symmetrical densityfunction in the following form

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

>

+≤≤+−

≤≤+−

≤≤−+

−≤≤−+

≡

.1for0

,141for

31

31

,410for13

,043for1

31

,431for33

)(

1

11

11

11

11

1

δ

δδ

δδ

δδ

δδ

δaf (12)

Asymmetric function fa described by formula (12)is presented in Fig. 8 in graphical form.

Fig. 8. Asymmetric density function in the case where probabilitiesof gap height decreasing are significantly larger

than probabilities of gap height increasing

We put equation (12) into formula (5). Hence dis-tributive function for asymmetric density function is

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 89

described by the following formula (13) and is pre-sented in Fig. 9

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

+≤≤++−

≤≤++−

≤≤−++

−≤≤−++

≡

= ∫∞<

∞−

.141for

65

31

61

,410for

43

23

,043for3

61

,31for233

23

)()(

112

1

112

1

112

1

112

1

111

1

δδδ

δδδ

δδδ

δδδ

δδδδ

dfF aa

(13)

Fig. 9. Distribution function for asymmetrical density functionelaborated by virtue of probability values presented in Fig. 8

We put equation (12) into formula (4). Hencestandard deviation for asymmetric density functionhas the following form [19], [20]

.381837.0)()( 221 =−= aaa fEfEσ (14)

To obtain dimensional value of the standard de-viation σa we must multiply σa1 by the characteristicvalue of gap height ε1 = 10⋅10−6 m. In this case, thedimensional standard deviation equals 3.81837 mi-crometers. From the measurements we have obtainedthe standard deviation value of about 4.0 micrometersfor normal cartilage.

3.4. Basic equationsand method of solutions

The synovial fluid velocity vector in spherical co-ordinates has the following components: νϕ, νr, νϑ inϕ-, r-, ϑ-directions, respectively. Spherical bone head

moves in the circumferential direction ϕ and meridiandirection ϑ contrarily to the acetabulum. Acetabulumand bone head surfaces vibrate in ϕ and ϑ directionswith various amplitudes and frequencies. Addition-ally, the acetabulum vibrates in the direction of thegap height which changes with time. The motion andvibrations of joint surfaces cause the synovial fluidflow in the hip joint gap. We assume rotational motionof human bone head in the circumferential directionwith the angular velocity ω1, and the meridian direc-tion with the angular velocity ω3. We have an un-symmetrical unsteady non isothermal synovial fluidflow in the gap, viscoelastic and unsteady propertiesof synovial fluid. Centrifugal forces are neglected. Wedenote: the peripheral velocity U = ω1R, constantvalue of the dimensional synovial fluid density ρ ≡ ρ0,changeable synovial fluid viscosity η = η0η1, charac-teristic dimensional value of dynamic viscosity η0,time dependent dimensional total gap height εT of hipjoint, R – radius of bone head, t0 – characteristic di-mensional time value, ε0 – characteristic dimensionalvalue of gap height. It is assumed that the product ofthe Deborah, Dβ ≡ βω1/η0, and Strouhal, Str = R/Ut0,numbers, i.e, DβStr and the product of the Reynolds,Re = ρUε0/η0, number, dimensionless clearance ψ,and Strouhal number, i.e., ReψStr have the same orderof magnitude and that DβStr >> Dβ . We perform anestimation of basic equation for boundary thin layerflow in a spherical joint gap. Neglecting the terms ofradial clearance ψ ≡ ε0/R ≈ 10−3, and centrifugalforces, in the governing equations in the sphericalcoordinates: ϕ, r, ϑ, and taking into account the abovementioned assumptions, we have [10]

),(

sin

1

2

3

10

αϕϕ

ϕϕ

ρβ

ηρ

ηϕρ

DOrtv

rv

rp

RRt

v

+∂∂

∂+

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛ ∂

−=∂

∂

(15.1)

,)2(22

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

+=rv

rv

rrp zϕβα

∂∂ (15.2)

),(

1

2

3

1

αϑϑ

ϑϑ

ρβ

ηρ

ηϑρ

DOrtv

rv

rp

tv o

+∂∂

∂+

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

+∂∂

−=∂

∂

(15.3)

0sinsin =⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

⎟⎠⎞

⎜⎝⎛+

∂

∂

RRv

rv

RR

v r ϑϑ

ϑϕ ϑϕ , (15.4)

K. WIERZCHOLSKI90

where 0 < ϕ ≤ 2πθ1, 0 < θ1 < 1, bm ≡ πR/8 ≤ ϑ ≤ πR/2≡ bs, 0 ≤ r ≤ εT, εT = ε0εT1, εT1 – total dimensionlessgap height, t = t0t1, t − dimensional time, t1 − dimen-sionless time, Oϕ, Oϑ – terms describing the one ordersmaller viscoelastic oil influences on the flow.

The terms multiplied by the pseudo-viscosity fac-tor β in Pas2 describe influences of viscoelastic prop-erties of the synovial fluid on the lubrication effects.The convection terms have been neglected. Only timederivatives of velocity component have been retainedin the left hand sides of equations (15.1) and (15.3).The joint bone head with hip joint gap and the rangeof the lubrication region of Ω : 0 ≤ ϕ ≤ π, πR/8 ≤ ϑ ≤πR/2, are presented in Fig. 10.

a)

b)

Fig. 10. Hip joint geometry: (a) bone head, hip joint gapand acetabulum, (b) the range of the lubrication region

resting on the spherical bone head

It is visible from the measurements performed thatradial velocity component has zero value on the bonehead and on the acetabulum surface attains a value ofthe first derivative of the total gap height with respectto time. Acetabulum and bone head move in circum-ferential, ϕ, and meridian, ϑ, direction. Moreover, wetake into account tangential time-dependent accelera-tion of bone head surface.

3.5. Stochastic Reynolds equations

Now we show the general form of stochastic Rey-nolds equation.

Imposing the boundary condition (19.1) on thesynovial fluid velocity component (18.2) in radial, i.e.,gap height direction we the obtain the following non-homogeneous stochastic partial differential Reynoldsequation of second order determining the unknownfunction of hydrodynamic pressure [6], [10]

),,,,(),(

)(sin

sin)()sin(6

)( sin6

sin)()(

)()(cosec1

3010

10

102

3

101

1

31

30

1

31

30

ωω

εεϑ

ϑεεϑ

ϕω

ϕεεϑω

ϑϑηη

εεϑ

ϕηηεε

ϕϑ

βα VUODDOt

ER

R

RER

ER

R

RpEER

pEERR

ppD

T

T

T

o

T

o

T

++∂

∂⎟⎠⎞

⎜⎝⎛−

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂

+

∂∂

⎟⎠⎞

⎜⎝⎛=

⎭⎬⎫

⎩⎨⎧

∂∂

∂∂

+

⎭⎬⎫

⎩⎨⎧

∂∂

∂∂

⎟⎠⎞

⎜⎝⎛

(16)

where symbol E(εT1) denotes expectancy operator ofdimensionless gap height.

The values of pressure distribution p are deter-mined in the lubrication region Ω. Total pressure hasvalues of the atmospheric pressure pat on the boundaryof the region Ω which is indicated in Fig. 10, and de-fined – on the basis of medical information – by thefollowing inequalities: Ω: 0 ≤ ϕ ≤ π, πR/8 ≤ α3 ≡ ϑ ≤πR/2. It is a section of the bowl of the sphere. TermsOpD, Op describe the pressure corrections caused byviscoelastic properties and periodic motion with vari-ous frequencies and amplitudes [6], [21].

At first, we determine the stochastic coefficientsfor the pseudo-Gaussian symmetric probability den-sity function.

By virtue of equations (3) and (6) we have [19],[20], [21]

,3)(

)()(

,)(

)()(

121

3111

311

31

111

111

sTgsTg

sTT

sTg

sTT

df

E

df

E

εσεδδ

δεε

εδδ

δεε

+=×

+=

=×

+=

∫

∫

∞+

∞−

+∞

∞−

(17)

where standard deviation equals σg1 = 0.364583.Dimensionless average deviation for the Gaussian

probability density function is defined in the form:℘g1 = σg1/avgεT1s < 1/3 where avgεT1s denotes averagedimensionless gap height and ℘g1 > 0.

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 91

Now, we derive the stochastic coefficients for clas-sical triangle symmetrical probability density func-tion.

By virtue of equations (3) and (7) we have [19],[20], [21]

,3)(

)()(

,)(

)()(

121

3111

311

31

111

111

sTsT

sTT

sT

sTT

df

E

df

E

εσεδδ

δεε

εδδ

δεε

ΔΔ

∞+

∞−

Δ

+∞

∞−

+=×

+=

=×

+=

∫

∫

(18)

where standard deviation equals σΔ1 = 0.408248.Dimensionless average deviation for triangle ran-

dom function is defined in the form: ℘Δ1 = σΔ1/avgεT1s

where ℘Δ1 > ℘g1.Now, we examine the stochastic coefficients for

asymmetrical density function for larger increase.By virtue of equations (3) and (9) we have [19],

[20], [21]

),25.0(3)25.0(

)()()(

,25.0)()()(

12

13

1

113

1131

111111

+++

×+=

+=×+=

∫

∫∞+

∞−

+∞

∞−

sTAsT

AsTT

sTAsTT

dfE

dfE

εσε

δδδεε

εδδδεε

(19)

where σA1 = 0.3818 81 30 68.Dimensionless average deviation for random

asymmetric function fA is defined in the form: :℘A1 =σA1/avg(εT1s + 0.25), where ℘g1 > ℘A1 > 0.

Now, we conclude with the stochastic coefficientsfor asymmetrical density function for smaller in-crease.

By virtue of equations (3) and (12) we have [19],[20], [21]

),25.0(3)25.0(

)()()(

,25.0)()()(

121

31

113

1131

111111

−+−+=

×+=

−=×+=

∫

∫∞+

∞−

+∞

∞−

sTasT

asTT

sTasTT

dfE

dfE

εσε

δδδεε

εδδδεε

(20)

where σa1 = 0.3818 81 30 68.Dimensionless average deviation for random

asymmetric function fa is defined in the form: ℘a1 =σa1/avg(εT1s − 0.25), where ℘a1 > ℘Δ1 > ℘g1 > ℘A1> 0.

3.6. Numerical calculations

The values of pressure distribution p are deter-mined in the lubrication region Ω. Total pressure hasvalues of the atmospheric pressure pat on the bound-ary of the region Ω which is indicated in Fig. 10, anddefined – on the basis of medical information – bythe following inequalities: Ω: 0 ≤ ϕ ≤ π, πR/8 ≤ α3 ≡ϑ ≤ πR/2. Numerical calculations are performed inMatlab 7.2 and Mathcad 12 Professional Program byvirtue of (16) using the finite difference method [22]for the region Ω, and for the following data:

– radius of the bone head in hip joint R =0.0265 [m],

– angular velocity in circumference direction onthe bone head surface ω1 = 1.10 [s–1],

– angular velocity in meridian direction on thebone head surface ω3 = −0.25 [s−1],

– angular velocity describing the periodical per-turbations in synovial fluid ω0 = 400 [s−1],

– angular velocity circumference perturbations onthe bone head surface ω10 = 0.10 [s−1],

– angular velocity meridian perturbations on thebone head surface ω30 = 0.02 [s−1],

– components of acetabulum centre eccentricities:Δεx = 2.5 [μm], Δεy = 0.5 [μm], Δεz = 2.0 [μm],

– characteristic value of synovial fluid viscosityη0 = 0.25 [Pas],

– characteristic value of synovial fluid densityρ0 = 1010 [kg/m3].

The minimum value of gap height equals εmin =5.8 μm, the maximum value of gap height equalsεmax = 11.50 μm. The measured roughness of jointsurfaces is taken into account. Pressure values arecalculated in the following instants within the timeperiods: t = 0 [s], t = π /ω0 [s], t = 2π /ω0 [s], ... In thefirst place, we perform numerical calculations forsynovial fluid without visco-elastic properties, i.e., forβ0 = 0.00000 Pas2 and in the next place we assumeviscoelastic properties for β0 = 0.00070 Pas2. Duringthe calculations, for the acetabulum, there are takeninto account the following time-independent dimen-sional amplitudes of measured vibration in the form oftangential velocity changes Vϕ0 = 0.001 m/s, Vϑ0 =0.0005 m/s. The time-independent dimensionlessscale of gap height perturbation changes has the valueBe = +0.0001 [10], [18]. The random effects for thepseudo-Gaussian density function ℘g1 = 1/3 is takeninto account.

Numerical calculations of time-varying pressuredistributions are performed by virtue of equation (16)for the gap height (1), (2). Their results are presented

K. WIERZCHOLSKI92

in Fig. 11, Fig. 12, Fig. 13 and Fig. 14. The time pe-riod of perturbations equals t = 2π /ωo.

Figure 11 shows the time-varying dimensionalpressure distribution in human hip joint where sto-chastic description of cartilage surface roughness isnot taken into account, hence ℘g1 = ℘Δ1 = ℘A1 = ℘a1= 0, and viscoelastic properties of the synovial fluid

are neglected for β0 = 0.0000 Pas2. For the time in-stants: t = 0, t = π /ω0, t = 2π /ω0, we obtain the fol-lowing values of joint capacity 1080.8 [N], 955.4 [N],1080.8 [N], respectively.

Figure 12 shows the time-varying dimensionalpressure distribution in human hip joint where sto-chastic description of roughness of cartilage surface is

0

1.50 [MPa]

0.75 [MPa]

R = 0.0265 [m], ηο = 0.25 [Pas], ωο = 400 [s−1], ω1 = 1.10 [1/s], ω3 = −0.25 [s−1]ω1ο = 0.15 [ s−1], ω3ο = −0.02 [ s−1], βο = 0, ℘g1 =℘Δ1 =℘A1 =℘a1 =0,

Lubrication surface = 20.38 [cm2]

t=0 and t=2π/ωo [s]pmax=1.503·106 [Pa]Ctot=1080.8 [N]

t=π/ωo [s]pmax=1.336·106 [Pa]Ctot=955.4 [N]

ω3 ω1

Fig. 11. The dimensional pressure distributions in human hip joint gap, caused by the rotation of the spherical bone headin circumferential direction ϕ and the meridian direction ϑ, simultaneously, where random effects and visco-elastic propertiesof synovial fluid are neglected for ℘g1 = ℘Δ1 = ℘A1 = ℘a1 = 0, β0 = 0. Calculations were performed for the non-zero valuesof the angular velocity: ω1 = 1.10 s−1, ω3 = −0.25 s−1 and non-zero angular velocity perturbations: ω10, ω30 in unsteady flow

and for the angular velocity of gap height perturbations, ω0 = 400 s−1

R = 0.0265 [m], ηο = 0.25 [Pas], ωο = 400 [s−1], ω1 = 1.10 [1/s], ω3 = −0.25 [s−1]ω1ο = 0.15 [ s−1], ω3ο = −0.02 [ s−1], βο = 0.0000 [Pas2], ℘g1 = 1/3,

Lubrication surface = 20.38 [cm2]

t=0 and t=2π/ωo [s]pmax=1.485·106 [Pa]Ctot=1059.5 [N]

t=π/ωo [s]pmax=1.321·106 [Pa]Ctot=934.9 [N]

ω3

ω1 ω1ω3

0

1.50 [MPa]

0.75 [MPa]

Fig. 12. The dimensional pressure distributions in human hip joint gap, caused by the rotation of the spherical bone headin the circumferential direction ϕ and the meridian direction ϑ, simultaneously, where only random effects accordingly

with the pseudo-Gaussian probability function are taken into account for ℘g1 = 1/3, and visco-elastic properties of synovial fluidare neglected for β0 = 0.00000 Pas2. Calculations were performed for non-zero values of the angular velocity:

ω1 = 1.10 s−1, ω3 = −0.25 s−1 and the non-zero angular velocity perturbations ω10, ω30,in unsteady flow and for the angular velocity of gap height perturbations, ω0 = 400 s−1

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 93

taken into account for ℘g1 = 1/3 and viscoelasticproperties of the synovial fluid are neglected, hencefor β0 = 0.00000 Pas2.

For the time instants: t = 0, t = π /ω0, t = 2π /ω0, weobtain the following values of joint capacity: 1059.5 [N],934.9 [N], 1059.5 [N], respectively [6]–[8].

R = 0.0265 [m], ηο = 0.25 [Pas], ωο = 400 [s−1], ω1 = 1.10 [1/s], ω3 = −0.25 [s−1]ω1ο = 0.15 [ s−1], ω3ο = −0.02 [ s−1], βο = 0.0007 [Pas2], ℘g1 =℘Δ1 =℘A1 =℘a1 =0,

Lubrication surface = 20.38 [cm2]

t=0 and t=2π/ωo [s]pmax=1.628·106 [Pa]Ctot=1174.8 [N]

t=π/ωo [s]pmax=1.243·106 [Pa]Ctot=884.8 [N]

ω3 ω1ω3 ω1

0

1.50 [MPa]

0.75 [MPa]

Fig. 13. The dimensional pressure distributions in human hip joint gap, caused by the rotation of the spherical bone headin the circumferential direction ϕ and the meridian direction ϑ, simultaneously, where all random effects are neglectedfor ℘g1 = ℘Δ1 = ℘A1 = ℘a1 = 0 and visco-elastic properties of synovial fluid taken into account for β0 = 0.00070 Pas2.

Calculations were performed for non-zero values of the angular velocity: ω1 = 1.10 s−1, ω3 = −0.25 s−1 and non-zero angularvelocity perturbations ω10, ω30 in unsteady flow and for the angular velocity of gap height perturbations, ω0 = 400 s−1

R = 0.0265 [m], ηο = 0.25 [Pas], ωο = 400 [s−1], ω1 = 1.10 [1/s], ω3 = −0.25 [s−1]ω1ο = 0.15 [ s−1], ω3ο = −0.02 [ s−1], βο = 0.0007 [Pas2], ℘g1 = 1/3,

Lubrication surface = 20.38 [cm2]

t=0 and t=2π/ωo [s]pmax=1.609·106 [Pa]Ctot=1152.9 [N]

t=π/ωo [s]pmax=1.228·106 [Pa]Ctot=864.8 [N]

ω3ω1

ω3ω1

0

1.50 [MPa]

0.75 [MPa]

Fig. 14. The dimensional pressure distributions in human hip joint gap, caused by the rotation of the spherical bone headin the circumferential direction ϕ and the meridian direction ϑ, simultaneously, where only random effects accordinglywith the pseudo-Gaussian density function are considered for ℘g1 = 1/3, and visco-elastic properties of synovial fluid

are taken into account for β0 = 0.00070 Pas2. Calculations were performed for the non-zero values of the angular velocity:ω1 = 1.10 s−1, ω3 = −0.25 s−1 and the nonzero angular velocity perturbations ω10, ω30 in unsteady flow

and for the angular velocity of gap height perturbations (frequencies), ω0 = 400 s−1

K. WIERZCHOLSKI94

Figure 13 shows the time-varying dimensionalpressure distribution in human hip joint where sto-chastic effects of cartilage surface roughness are ne-glected, hence ℘g1 = 0, and viscoelastic properties ofthe synovial fluid are taken into account for β0 =0.00070 Pas2. For the time instants: t = 0, t = π /ω0,t = 2π /ω0, we obtain the following values of jointcapacity: 1174.8 [N], 884.8 [N], 1174.8 [N], respec-tively.

Figure 14 shows the time-varying dimensional pres-sure distribution in the gap of spherical human hip jointfor ℘g1 = 1/3 and β0 = 0.00070 Pas2. Stochastic descrip-tion of cartilage surface roughness is taken into accountfor ℘g1 = 1/3. Viscoelastic properties of the synovialfluid are taken into account for β0 = 0.00070 Pas2. Forthe time instants: t = 0, t = π/ω0, t = 2π/ω0, we obtainthe following values of joint capacity: 1152.9 [N],864.8 [N], 1152.9 [N], respectively [10].

500

700

900

1100

1300

1500

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

capa

city

Cto

t [N

]

time [s]

βο=0.0000 [Pas2]; ℘g1=0ωo=400 [s−1]; Be=0,0001

βο=0.0000 [Pas2];℘g1=1/3βο=0.0007 [Pas2];℘g1=0βο=0.0007[Pas2]; ℘g1=1/3

1

3

2

4

1234

Fig. 15. Dimensional load carrying capacity distributions versus time within the time period where random effects accordinglywith the pseudo-Gaussian density function with ℘g1 = 1/3 are taken into account for four assumptions corresponding to

the following four cases presented in Fig. 11, Fig. 12, Fig. 13, Fig. 14, respectively:(1).℘g1 = 0, β0 = 0; (2). ℘g1 = 1/3, β0 = 0.0000 Pas2; (3). ℘g1 = 0, β0 = 0.0007; (4). ℘g1 = 1/3, β0 = 0.0007 [Pas2]

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

βο=0.0010 [Pas2]βο=0.0008 [Pas2]βο=0.0006 [Pas2]βο=0.0004 [Pas2]βο=0.0002 [Pas2]

βο=0.0009 [Pas2]βο=0.0007 [Pas2]βο=0.0005 [Pas2]βο=0.0003 [Pas2]βο=0.0001 [Pas2]

ωo=400[s−1]; Be=0.0001; ℘g1=1/3

time [s]

capa

city

Cto

t [N

]

9864

3

12

13579

246810

5 710

Fig. 16. Load carrying capacity distributions versus time within the time period of perturbations where random effectsaccordingly with the pseudo-Gaussian density function with ℘g1 = 1/3 are taken into account for the constant value

of periodicity of perturbations ω0 = 400 s−1 and various values of viscoelastic properties of synovial fluidrepresented by the pseudo-viscosity values inside interval: 0.00001 [Pas2] ≤ β0 ≤ 0.00040 [Pas2]

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 95

It is easy to see that the pressure distributions andcapacities presented in Fig. 11, Fig. 12, Fig. 13 andFig. 14 for the time instants: t = 0 s, t = 2π /ω0 s, i.e.,for t = 2 kπ /ω0 s and k = 0, 1, 2, ..., have the samevalues. The first upper illustrations on the left sides inFig. 11, Fig. 12, Fig. 13 and Fig. 14 show the pressuredistributions for the origin and end time of the periodof perturbations of the motion of human joint. Theillustrations for t = π /ω0 in Fig. 11, Fig. 12, Fig. 13and Fig. 14 show the pressure distributions for themiddle-time point of the period of perturbations of the

motion of human joint. Afterwards the pressure distri-butions return to the distributions which had beenshown in the first illustration occurring in each of thefigures 11, 12, 13, 14.

• The influences of the random changes of gapheight of human joint and visco-elastic synovial fluidproperties on the dimensional load carrying capacitydistributions in human hip, are presented in Fig. 15and considered below.

Figure 15 shows the capacity distributions versustime within the time period for constant periodicity of

Fig. 17. The influence of the shapes of symmetrical and asymmetrical probability density functions of gap height changesconnected with the probability average standard deviation on the hydrodynamic pressure distribution in human hip joint

Fig. 18. The graphical illustration of dependences between average standard deviations of human hip joint gap changesand load carrying capacities for unsteady lubrication process

K. WIERZCHOLSKI96

perturbation effects (frequencies), ω0 = 400 s−1, andfor four assumptions corresponding to the followingfour cases presented in Fig. 11, Fig. 12, Fig. 13 andFig. 14, respectively:

(1) ℘g1 = 0, β0 = 0 (random effects and viscoelas-tic properties are neglected),

(2) ℘g1 = 1/3, β0 = 0 (random effects are consid-ered, and viscoelastic properties are neglected),

(3) ℘g1 = 0, β0 = 0.0007 Pas2 (random effects areneglected, viscoelastic properties considered),

(4) ℘g1 = 1/3, β0 = 0.0007 Pas2 (random effectsand viscoelastic fluid properties are considered).

The symbol ℘g1 = 0 shows that the random effectsare neglected. The case for β0 = 0 represents theNewtonian properties of synovial fluid. In the casesmentioned we assume: ℘Δ1 = ℘A1 = ℘a1 = 0, i.e.,random effects of triangular and asymmetrical densityfunctions are neglected.

Figure 16 shows the load dimensional carrying ca-pacity distributions versus time within the time periodof constant value of its periodicity (frequencies) ω0 =400 [s−1] and for various viscoelastic properties ofsynovial fluid determined by the following dynamicpseudo-viscosity coefficients: β0 = 0.0001 [Pas2], β0 =0.0002 [Pas2], βo = 0.0003 [Pas2], β0 = 0.0004 [Pas2],β0 = 0.0005 [Pas2], β0 = 0.0006 [Pas2], β0 = 0.0007[Pas2], β0 = 0.0008 [Pas2], β0 = 0.0009 [Pas2], β0 =0.0010 [Pas2].

For unsteady lubrication process occurring in hu-man hip joint, the shapes of probability density func-tions of the hip joint gap height, have significant influ-ence on hydrodynamic pressure, friction forces andwear. The results obtained after numerous numericalcalculations by virtue of equations (16)–(20) are in-cluded in illustrations presented in Fig. 17 and Fig. 18.

The above-mentioned results are obtained fromthe author’s own and other authors and co-authorsnumerous experimental or analytical studies per-formed under unsteady, stochastic lubrication proc-esses for symmetric and asymmetric density func-tions occurring in deformed bonehead of human hipjoints [1]–[3], [9], [10], [18], [23], [24].

4. Discussion

The equation obtained in this paper for pressuredistribution in unsteady periodic conditions, stochasti-cally variable joint gap, visco-elastic properties of syno-vial fluid, and various frequencies and amplitudes ofvibrations of the bone head and acetabulum, tends – in

particular case – to the well known form of Reynoldsequation for steady motion, contained in the previousresults. From the numerical calculations it follows thatviscoelastic properties of synovial fluid increase hu-man hip joint capacities by at least 15%, and – insome cases – even by 60%. From numerical calcula-tions it follows that stochastic description of rough-ness of bone surfaces and stochastic description of thefilm thickness of synovial fluid changes the hip jointcapacity by about 11%. In this paper, two cases areassumed of probability density functions describingthe random part of gap height changes.

We take into account the symmetric probabilitydensity function, where random decreases of the gapheight changes are the same as the probabilities ofincreases. Such functions comprise pseudo-Gaussianand triangle function. The results obtained are validnot only for human hip joint but also for other humanjoints such as foot joint, elbow joint, knee joint,shoulder joint. On the grounds of experiments wepresent an asymmetric probability density function,where probabilities of the gap height increases arelarger (smaller) than probabilities of the gap heightrandom decreases.

To compare the pressure distribution we defineaverage deviation coefficient ℘ as a ratio of the stan-dard deviation and dimensionless average gap height.

If average standard deviation is equal zero, thenthe stochastic influences are neglected.

If average standard deviation increases, then thejoint load carrying capacity decreases.

It was found that for the pseudo-Gaussian prob-ability density function, the average standard devia-tion coefficient attains approximately the value 1/3.

For asymmetric probability density function theaverage standard deviation coefficient is smaller andlarger than 1/3 and has always positive values.

Initial research indicates that:• For unsteady lubricant conditions, if average

standard deviation increases (decreases), vibrationamplitude, i.e., maximum value of the gap heightchange decreases (increases). Hence the pressure de-creases (increases) and friction force and wear de-creases (increases). These phenomena are illustratedin Fig. 17 and Fig. 18.

• For steady lubricant conditions, average stan-dard deviation increases (decreases) imply increases(decreases) of the gap height changes. Hence the pres-sure increases (decreases) and friction force and wearincreases (decreases).

In the author’s opinion, the real human hip jointlubrication is always unsteady with random effectsand almost in each case the probability density func-

Time dependent human hip joint lubrication for periodic motion with stochastic asymmetric density function 97

tion of the values of joint gap height, load carryingcapacity, and friction forces is asymmetric.

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