A procedure for constructing a theoretical wear diagram of IC engine crankshaft main bearings

Mechanism and Machine Theory 58 (2012) 120–136

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A procedure for constructing a theoretical wear diagram of IC engine crankshaft main bearings Nebojsa Nikolic ⁎, Tripo Torovic, Zivota Antonic Faculty of Technical Sciences, Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbia

a r t i c l e

i n f o

Article history: Received 1 August 2011 Received in revised form 10 July 2012 Accepted 29 July 2012 Available online 8 September 2012

a b s t r a c t An algorithm for obtaining a theoretical wear diagram of internal combustion (IC) engine crankshaft main bearings is developed in the paper. A crankshaft is treated as a statically indeterminate continuous beam. The load distribution on the contact surface between a journal and a bearing is assumed to be elliptic. The whole procedure, tailored to its software implementation, is illustrated on the example of a six-cylinder diesel engine crankshaft. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Crankshaft Statically indeterminate method Main bearing Theoretical wear diagram

1. Introduction Load characteristics of crankshaft main bearings are of paramount importance for design of the bearings themselves and the engine block as well. According to the level of this load, one can optimize dimensions of the bearings and also determine stiffness of the engine crankshaft and the block. In addition, the main source of vibration in the engine is the crankshaft bearing load that changes rapidly during the engine operation. Hence, there are many reasons for crankshaft bearings load to be determined. Forces acting on crankshaft bearings can be obtained experimentally, analytically or numerically. Although experimental methods are generally more accurate than the analytical and the numerical ones, their implementation requires a real, already manufactured engine on a test stand. In addition, this approach is associated with spatial constraints causing difficulties related to proper sensor installation issues and a complicated signal calibration process. On the other hand, determining bearings load by means of analytical models can be done for already made engines and, what is more important, in an early stage of engine design. Forces acting on crankshaft bearings are usually calculated analytically by using statically determinate methods [1–5]. In these methods, all the cranks of the crankshaft are treated independently of each other and every bearing is exposed to the forces from two adjacent cranks only. However, Schnurbein [6] showed that statically indeterminate methods give substantial improvement on statically determinate methods as better agreement with experimental results is achieved. For more accurate calculations, a numerical approach via the finite element method can be undertaken, as it was done in Ref. [7], but it can be time consuming in some cases. In the majority of internal combustion (IC) engines, forces between crankshaft journals and bearings have high magnitudes, and as such, are considered to be the most important factor affecting the wear of the elements in contact. It is very difficult to understand and describe the wear phenomenon, especially if all the possible influence parameters such as geometry and temperature of contact, physical and chemical properties of the contacting materials etc. are taken into account. Many diverse research works have been presented recently showing the great significance of wear in mechanical systems [8–10]. Thus, for example, the wear phenomenon in revolute clearance joints exposed to dry friction was studied by Flores [9], ⁎ Corresponding author. Tel.: +381 21 485 2355; fax: +381 21 6350 592. E-mail address: [email protected] (N. Nikolic). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.009

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who developed a methodology for studying and quantifying it by taking into account geometry and material properties of the elements coupled. A continuous contact force model is used to compute the pressure field in the contact zone and Archard's model is employed for wear quantification. Mukras [10] and Mukras et al. [11] compared two procedures for predicting the revolute joint clearance evolution caused by wear. In the first procedure, joint forces and contact pressures are estimated by using the elastic foundation model with hysteresis damping via the dynamic analysis. In the second procedure, a contact force model with hysteresis damping is used to evaluate the joint forces and the contact pressure is calculated by using a finite element method (FEM). In Ref. [12], Mukras presented a procedure to analyze planar multibody systems with wear at one or more revolute joints with clearance. An iterative wear prediction procedure based on Archard's wear model is used to compute the wear as a function of the evolving dynamics and tribological data. Su et al. [13] investigated the interaction between wear of joint with clearance and kinematics in multi-body systems and proposed a numerical approach of wear prediction integrated with a finite-element-based iterative scheme accordingly. In Refs. [9–13], the authors developed the models to compute the wear depth in order to predict wear after a certain period of time or number of cycles. However, as far as the main bearings of an internal combustion engine are concerned, it is not of much interest to predict the wear depth, but rather to determine the most jeopardized bearings of an IC engine and to estimate the most critical areas on the bearing surface in terms of wear. This is because main bearings operate mostly in lubricated conditions, when wear does not exist. There are, however, some critical regimes during the exploitation of IC engines with sparse lubrication or even with no lubrication. Under these conditions, high temperatures that affect wear of main bearings are generated, but this influence is neglected here because the critical regimes last for very short periods of time. In addition, the critical regimes are regularly followed by the regimes of sufficient lubrication, which decrease the temperature in the bearings. However, regardless of the short duration of the critical regimes, they are characterized by intense wear. For the reasons aforementioned, it is the aim here to propose an algorithm for constructing a theoretical wear diagram, by the use of which one can determine the most jeopardized crankshaft bearings and estimate the most critical zones on their surface. This can be very useful in the design of an IC engine main bearings lubrication system. The theoretical wear diagram has the form of a worn out bearing profile and it is therefore called a theoretical wear diagram [2–4]. It is constructed with the assumption that a contact between the journal and the bearing is unlubricated. The assumption is related to the theoretical case, when the journal and the bearing are in a direct contact, but it can also be related to the above mentioned critical operation regimes of IC engines. This paper is concerned with the analytical procedure for obtaining theoretical wear diagrams of crankshaft main bearings. It should be emphasized that the aim of the procedure is not to generate a real wear diagram of a bearing that can predict the volume of material loss, but to provide a detailed algorithm for constructing a theoretical wear diagram that shows the circumferential distribution of the bearing wear, considering the bearing load a dominant factor affecting wear. The procedure developed is universal, applicable to crankshafts of all in-line engines and can be used to determine how different values of some engine parameters affect the circumferential load of main bearings. As far as the authors are aware, such a procedure does not exist in the literature. There are some graphical, less accurate procedures presented in Refs. [2–4], but they deal with crankshaft main journals, not with bearings. Besides, a crankshaft is therein treated as a statically determinate beam and the load distribution within the contact zone is assumed to be uniform, unlike in this paper, where a more realistic and accurate approach is taken and a crankshaft is treated as a statically indeterminate beam and the load distribution on the contact surface between a journal and a bearing is assumed to be elliptic. The paper is structured as follows. Section 2 introduces a generic model for determining forces acting on crankshaft main bearings, where the crankshaft is treated as a statically indeterminate system. In Section 3, a theoretical model of the journal-bearing contact, caused by the forces calculated in Section 2, is presented. By applying the model of the contact, an algorithm enabling a step-by-step construction of a theoretical wear diagram is formulated, the details of which are described in Section 4. The application of the algorithm is illustrated on the example of a six-cylinder diesel engine in Section 5. Section 6 contains conclusions. 2. A generic model of main bearings load In order to construct a main bearing theoretical wear diagram, one first needs to determine the forces acting on the bearing throughout an engine cycle. The crankshaft will be considered as a statically indeterminate continuous beam carrying the forces mentioned. It is assumed that there are clearances between the journals and the bearings and, consequently, the force Fci that originates from the crank i is distributed to all the bearings, not only to the adjacent ones (Fig. 1). It is also assumed that each force Fci acts in the middle between the adjacent bearings. It should be noted that the forces Fci can be calculated in a common way taking into account gas forces and inertia forces that dominate over other forces in IC engines [3,14]. The gas forces can be measured or modeled by using some of the known methods and the inertia forces can be easily determined by using Newton's second law of motion assuming that the crankshaft rotates with a constant angular velocity. Therefore, the forces Fci can be considered as known. Fig. 1a shows the crank force Fc1 that originates from the crank c1 and is transferred to all the bearings bj (j = 1, 2,…,n + 1) as nþ1 X ρ1;j ¼ 1. The coefficient ρ1,j (j = 1, 2,…,n + 1) shows to what extent the force Fc1 affects the bearing bj Fb1,j = ρ1,j ⋅ Fc1, where j¼1

and is called the influence coefficient. Similarly, the force Fc2 originating from the crank c2 also affects all the bearings and this can be described by the influence coefficients ρ2,j (j = 1, 2,…,n + 1) (Fig. 1b).

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Fig. 1. Main bearings load distribution according to the statically indeterminate method: (a) when the active force is applied to the crank c1; (b) when the active force is applied to the crank c2.

Consequently, the total force acting on the bearing bj is Fbj ¼

n X

ρi;j ⋅Fci ;

ðj ¼ 1; 2; …; n þ 1Þ

ð1Þ

i¼1

where Fci are crank forces. To determine the influence coefficients ρi,j, a crankshaft is modeled as a statically indeterminate continuous beam with a constant cross-section, as shown in Fig. 2. If a crankshaft has (n + 1) main bearings and n cranks, the counterpart beam has (n + 1) supports and n spans (S1, …, Sn). The influence coefficients for any continuous beam regardless of the number of its supports can be calculated by using a procedure based on the well-known Clapeyron's three-moment equation. The full details of this procedure, prepared for software implementation, are given in Appendix A. Fig. 3a shows the isometric view of an in-line engine crankshaft with (n + 1) main bearings with the forces acting on them. Two coordinate systems are also shown in Fig. 3a, one of which is stationary (OXY) and the other one (OX1Y1) rotates together with the crankshaft with the angular velocity ω. The position of the coordinate system OX1Y1 with respect to the coordinate system OXY is defined by the crankshaft angle φ. It should be noted that all the forces depicted in Fig. 3 depend on this angle. The values of φ range from 0 ∘ to 720 ∘ in four-stroke cycle engines and from 0 ∘ to 360 ∘ in two-stroke cycle engines, since only one engine cycle is considered here. It is assumed that in all the engine cylinders, identical engine cycles take place and that the masses of the counterpart elements in different engine cylinders are equal. This implies that all the forces Fci(φi) (i = 1, 2, …, , …,n) change in the same way during one engine cycle. However, engine cycles in different cylinders are mutually phase-shifted, which depends on the firing order. Consequently, all the forces Fci(φi) have different directions and magnitudes, as shown in Fig. 3 for an arbitrary position of the crankshaft. To determine the forces Fbj(φ) acting on the main bearings bj (j = 1, 2,…,n + 1), it is necessary to observe the simultaneous action of all the forces Fci(φi) (i = 1, 2, …, n). Therefore, the angles φi are introduced indicating the position of the cranks ci during

Fig. 2. Crankshaft as a statically indeterminate continuous beam.

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Fig. 3. The forces affecting the bearings of a crankshaft: (a) a schematic isometric view; (b) a side view of an arbitrarily chosen crank ck with respect to the referent crank c1.

the engine cycle in the appropriate cylinder with respect to its initial position. The relationship between φi and φ is given by φi ¼ φ þ θi ;

ði ¼ 1; 2; …; nÞ

ð2Þ

where θi is the phase angle that indicates the position of the crank ci during the engine cycle in the cylinder i when the engine cycle in the cylinder 1 is at the beginning. The angle θi can easily be determined if the engine firing order is given. In Fig. 3a, the forces Fbj(φ) are shown in the stationary coordinate system OXY by means of their projections FbXj(φ) and FbYj(φ). Further, for practical reasons, the forces Fci(φi) (i = 1, 2,…,n) can be represented as consisting of a radial component Fcradi(φi) and a tangential component Fctani(φi), as shown in Fig. 3a and b for an arbitrary crank ck. The projections FcX1k(φk) and FcY1k(φk) of the force Fck(φk) onto the coordinate system OX1Y1 are, respectively, FcX1k ðφk Þ ¼ −Fcradk ðφk Þ⋅sinψk þ Fctank ðφk Þ⋅cosψk ; FcY1k ðφk Þ ¼ Fcradk ðφk Þ⋅cosψk þ Fctank ðφk Þ⋅sinψk :

ð3Þ

Combining Eqs. (1), (2) and (3), the projections of the forces Fbj(φ) onto the axes of the coordinate system OX1Y1 can be expressed as follows: FbX1j ðφÞ ¼

n   X −ρi;j ⋅Fcradi ðφ þ θi Þ⋅sinψi þ ρi;j ⋅Fctani ðφ þ θi Þ⋅cosψi ;

FbY1j ðφÞ ¼

i¼1 n  X

ρi;j ⋅Fcradi ðφ þ θi Þ⋅ cosψi þ ρi;j ⋅Fctani ðφ þ θi Þ⋅ sinψi



ðj ¼ 1; 2; …; n þ 1Þ

ð4Þ

i¼1

where ψi is the counterclockwise angle between the cranks c1 and ci. However, as the crankshaft main bearings are stationary, the forces Fbj(φ) need to be projected onto the axes of the stationary coordinate system OXY, i.e. FbX j ðφÞ ¼ FbX1j ðφÞ⋅cosφ þ FbY1j ðφÞ⋅ sinφ; FbY j ðφÞ ¼ −FbX1j ðφÞ⋅ sinφ þ FbY1j ðφÞ⋅cosφ:

ðj ¼ 1; 2; …; n þ 1Þ

ð5Þ

Using Eqs. (4) and (5), the projections FbXj(φ) and FbYj(φ) are found and the forces Fbj(φ) are thus completely defined. Starting from the beginning of an engine cycle, a pair of values (FbXj(φ),FbYj(φ)) can be obtained in this way for each increasing integer value of φ and can be represented as a point in the coordinate system OXY. By connecting these points, a polar load diagram is obtained, defining magnitudes and directions of the forces acting on the main bearing bj (j = 1, 2,…,n + 1) during one engine cycle.

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3. Theoretical model of the journal-bearing contact The force Fb defined by Eq. (5) is transferred from the main journal to the main bearing, causing the contact between these two elements to be established, as shown in Fig. 4. It is assumed that the contact is along a surface due to the deformations of the coupled elements. It is also assumed that: – – – – –

there is no misalignment between the axes of the journal and the bearing, the contact between the surfaces of the journal and the bearing is dry, the load distribution along the bearing length is uniform, the load distribution within the circumferential contact zone is elliptic, the contact between the journal and the bearing surfaces is continuous.

The elliptic load distribution is assumed in accordance with the pressure distribution for the elastic contact between two cylinders, as in Refs. [15] and [16]. A continuous contact between the surfaces is assumed since the theoretical case is considered here. Furthermore, relative clearances in main bearings of IC engines are quite small, so the effects of impacts and rebounds can be neglected here, as suggested in Ref. [17]. Main geometric parameters of the journal-bearing assembly considered are shown in Fig. 4. The bearing of the radius Rb and the journal of the radius Rj are in contact along the length L (Fig. 4a and c). The radial clearance e between the two is described by e=Rb−Rj. Since it is assumed that the load is uniformly distributed along the bearing length, the bearing load per unit length qFb is defined by qFb ¼

Fb : L

ð6Þ

The contact geometry is described here using the expressions given in Ref. [15] for a steel bearing covered with a bearing alloy. The contact area is defined by the bearing radius Rb, the length L and by the contact angle 2βc (Fig. 4a and b). The contact parameters used are: βc—contact half-angle, pmax—maximum pressure in the contact zone and p(β)—pressure distribution within the contact zone. The expressions that enable one to calculate the contact half-angle are [15]  βc ¼ 0:32⋅

  C0 κ m þ1 ⋅ ; κ þ1 0:12

ð7Þ

where C0 ¼

   i π h 2 2 ⋅ 1−νj þ 1−νb ⋅χ ; 4

κ¼

Fb ; L⋅Eb⋅e

χ¼

Eb ; Ej

Fig. 4. Geometry of the contact between the journal and the bearing: (a) a front view with bearing force Fb; (b) a front view with the pressure distribution within the circumferential contact zone described by contact angle 2βc; (c) a side view with the uniform load distribution along the bearing length.

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while νj and νb are Poisson's ratios of the journal and bearing materials, and Ej and Eb are moduli of elasticity of the journal and bearing materials. The exponent m depends on the parameters νj, νb and χ, and for the journal and bearing materials in IC engines is equal to 0.581 [15]. The maximum pressure in the contact zone pmax is given by [15] pmax ¼ 0:55⋅

  Fb 1 ⋅ þ 0:35 : L⋅Rb βc

ð8Þ

and the pressure distribution p(β) within the contact zone is [15]  2 12  β pðβÞ ¼ pmax 1− βc

ð9Þ

where β is a current angular coordinate that changes between − βc and βc, as shown in Fig. 4b. 4. A construction algorithm for a main bearing theoretical wear diagram Studies of some authors [9] revealed that with the increase of the number of cycles, the wear depth is also increased, but the shape of the circumferential wear profile is not changed significantly. Other authors [18] showed that the maximum wear depth changes approximately linearly with respect to the number of cycles, i.e. it is proportional to it. Having in mind the results mentioned, as well as the aim of the paper, it is considered here that the linear extrapolation of a theoretical wear diagram of one cycle reflects a shape of a circumferential wear profile after a large number of cycles well enough. Therefore, one cycle of the IC engine has been observed. A theoretical wear diagram of a main bearing is constructed here by using its polar load diagram. A procedure of the diagram construction is performed by discretizing an engine cycle, gradually increasing the crankshaft angle φ (Fig. 3.a). The bearing load influence on the theoretical wear profile of the bearing is quantified in accordance with the model described in Section 3. At the end of the engine cycle, a cumulative effect of the main bearing load on the theoretical wear profile of the bearing is determined and visualized. A schematic presentation used to construct a theoretical wear diagram of the main bearing bk is depicted in Fig. 5. The bearing bk of the radius Rbk is represented by an annular shaded area. The location of an application point of a force Fbk(φ) on the bearing inner circumference is defined by the angle αk(φ), which is measured with respect to the X-axis. Since the bearing load is assumed to be uniformly distributed along the bearing length, the bearing load per unit length qFbk(φ) is drawn at the application point of the force Fbk(φ). A dashed vector line in Fig. 5 depicts the force Fbk(φ) calculated by using Eq. (5). The contact angle 2βck(φ) defines the contact zone between the journal and the bearing with respect to the direction of the force Fbk(φ). The area with the

Fig. 5. A schematic presentation of a theoretical wear diagram construction for the bearing bk at the crankshaft position defined by the angle φ.

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hatch lines depicts the pressure distribution pk(φ, β) in the contact zone, where β is the current angular coordinate within the borders of the contact zone. Since the amount of wear is proportional to the applied load [19,20], and the load distribution in the contact zone is not uniform, the bearing wear depth due to the load will not be uniform either. This is noted in Fig. 5, where the half-moon shaped shaded area illustrates the bearing wear depth proportional to the load within the contact zone. It should be emphasized that the term “wear depth” used here does not represent a real depth of the material removed, but it is related to the load as explained previously. To avoid possible confusion, the term “conditional wear depth” will be used subsequently. The conditional wear depth is expressed by means of the non-dimensional parameter δk(φ, β) (Fig. 5) and is introduced for the sake of a theoretical wear diagram construction. Another new parameter introduced here is a “cumulative conditional wear depth” ΔRbk(α), which is also non-dimensional and describes a cumulative effect of the forces Fbk(φ) acting on the bearing bk during one engine cycle. This depth changes along the bearing circumference, which is illustrated in Fig. 6 for three arbitrarily chosen values of the angular coordinate α (α1, α2 and α3). A theoretical wear diagram for all crankshaft main bearings bj (j = 1, 2,…,n + 1) can be constructed by using the algorithm with the following steps: Step 1 Calculate influence coefficients by using the procedure described in Appendix A. Step 2 Set a bearing counter j to 1 and repeat Steps 3 to 19 until the end of the crankshaft is reached. Step 3 Set the cumulative conditional wear depth ΔRbj(α) to zero for each α ∈ { 1, 2, …360}, which describes an initial state with no wear. Step 4 Assign the value 1 ∘ to the crankshaft angle φ and repeat Steps 5 to 13 until the engine cycle is completed. Step 5 Calculate the projections FbXj(φ) an FbYj(φ) of the force Fbj(φ) by using Eqs. (4) and (5). Step 6 Determine the magnitude of the force Fbj(φ) and its application point on the bearing circumference, defined by the angle αj(φ) Fbj ðφÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FbX j ðφÞ2 þ FbY j ðφÞ2 ;

  α j ðφÞ ¼ atan FbY j ðφÞ=FbX j ðφÞ :

ð10Þ ð11Þ

Step 7 Calculate the contact half-angle βcj(φ) and the maximum contact pressure pmaxj(φ) by adjusting Eqs. (6), (7) and (8) to the symbols presented in Fig. 5, i.e. by using " #m  qFbj ðφÞ C0 þ1 ⋅ ; βcj ðφÞ ¼ 0:32⋅ 0:12 qFbj ðφÞ þ Eb⋅ej

ð12Þ

Fig. 6. A theoretical wear profile of the bearing bk with a cumulative conditional wear depth ΔRbk(α) depicted for three arbitrarily chosen values of α.

N. Nikolic et al. / Mechanism and Machine Theory 58 (2012) 120–136

pmaxj ðφÞ ¼ 0:55⋅

qFbj ðφÞ Rbj

⋅

! 1 þ 0:35 : βcj ðφÞ

127

ð13Þ

Step 8 Assign the value − βcj(φ) to the current angle β and repeat Steps 9 to 12 until β reaches the end of the contact zone. Step 9 Calculate the pressure pj(φ, β) for the current value of β taking into account Eq. (9) and the symbols in Fig. 5: " pj ðφ; βÞ ¼ pmaxj ðφÞ⋅ 1−

!2 #12 β : βcj ðφÞ

ð14Þ

Step 10 Find the conditional wear depth δj(φ, β) by multiplying the calculated pressure with a suitably chosen positive real coefficient c δj ðφ; βÞ ¼ c⋅pj ðφ; βÞ:

ð15Þ

Note that the value of the coefficient c can be chosen in a way that the most appropriate drawing scale of the theoretical wear diagram is obtained. Its units should be reciprocal to the units of pj(φ, β) in order to make δj(φ, β) non-dimensional. Step 11 Increase the cumulative conditional wear depth ΔRbj(α) by δj(φ, β) ΔRbj ðα Þ←ΔRbj ðα Þ þ δj ðφ; βÞ;

ð16Þ

where α ¼ α j ðφÞ þ β: Increase the current angle β by 1 ∘. Increase the crankshaft angle φ by 1 ∘. Assign the value 1 ∘to the angle α (Fig. 6) and repeat Steps 15 and 16 until α reaches the value 360 ∘. Starting from a circle that represents the inner profile of the bearing bj (Fig. 6), plot a line in the direction defined by α, the length of which corresponds to the cumulative conditional wear depth ΔRbj(α) calculated in Eq. (16). Step 16 Increase the angle coordinate α by 1 ∘. Step 17 Connect the endpoints of all the lines plotted in Step 15 into a new line representing the theoretical wear diagram of the bearing bj. Step 18 Find the maximum and the minimum values of cumulative conditional wear depth, ΔRmax and ΔRmin, and the corresponding values of the angle α. Step 19 Increase the bearing counter j by 1. Step 12 Step 13 Step 14 Step 15

The flowchart of the algorithm is given in Fig. 7 for the sake of the reader. It gives a general overview of the algorithm and summarizes all the steps. 5. Illustrative example Based on the algorithm presented, a set of computer programs has been written in order to calculate influence coefficients, determine bearing load and construct theoretical wear diagrams, regardless of number of bearings. The application of the algorithm and the programs developed is illustrated on the example of a six-cylinder four-stroke cycle diesel engine. This choice is motivated by the fact that main bearings of diesel engines, in general, carry higher loads than those of otto engines. The basic data of the engine under consideration and the values of the crankshaft parameters used are given in Table 1. The values of crank force projections Fcrad(φ) and Fctan(φ), also used as the input data in the example, can be viewed in “Supplementary data” associated with this article. Along with the theoretical wear diagrams, the corresponding polar load diagrams are also shown (Figs. 8–14). For illustration, several points on each polar diagram are marked by the value of the crankshaft angle φ, showing the endpoints of the appropriate vectors Fb(φ). All the polar load diagrams and the theoretical wear diagrams in Figs. 8–14 are plotted for the same axes limits and the same drawing scale. As a result of this, the loads of the main bearings can easily be compared visually and one can detect which of the main bearings are worn more than the others. Besides this, maximum and minimum values of the cumulative conditional wear depth, ΔRmax and ΔRmin, are also shown in Figs. 8b–14b for the corresponding angle values measured with respect to the X-axis. Since the values ΔRmax and ΔRmin are conditional, they are expressed as non-dimensional (see Step 10 and Step 11 in the algorithm). These values can be very useful while making comparison between theoretical wear diagrams for one engine operating under different conditions and for different engines as well.

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Fig. 7. Flowchart of the algorithm proposed to construct theoretical wear diagrams of main bearings for a crankshaft with n cranks.

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Table 1 Basic engine data. General engine data Cylinder number Rated power Rated engine speed Compression ratio Cylinder bore Cylinder stroke Firing order

6 77.3 kW 2500 min−1 17.5 0.1016 m 0.1142 m 1–5–3–6–2–4

Crankshaft data Counterclockwise angle between the cranks c1 and c2 Counterclockwise angle between the cranks c1 and c3 Counterclockwise angle between the cranks c1 and c4 Counterclockwise angle between the cranks c1 and c5 Counterclockwise angle between the cranks c1 and c6 Phase angle between the cylinders 1 and 2 Phase angle between the cylinders 1 and 3 Phase angle between the cylinders 1 and 4 Phase angle between the cylinders 1 and 5 Phase angle between the cylinders 1 and 6 Span between the bearings b1 and b2 Span between the bearings b2 and b3 Span between the bearings b3 and b4 Span between the bearings b4 and b5 Span between the bearings b5 and b6 Span between the bearings b6 and b7 Length of the bearing b1 Length of the bearing b2 Length of the bearing b3 Length of the bearing b4 Length of the bearing b5 Length of the bearing b6 Length of the bearing b7 Radius of each bearing Radial clearances between the journals and the bearings Modulus of elasticity of the journal material Modulus of elasticity of the bearing material Poisson's ratio of the journal material Poisson's ratio of the bearing material

ψ2 = 120∘ ψ3 = 240∘ ψ4 = 240∘ ψ5 = 120∘ ψ6 = 0 ∘ θ2 = 240∘ θ3 = 480∘ θ4 = 120∘ θ5 = 600∘ θ6 = 360∘ l1 = 0.132 m l2 = 0.132 m l3 = 0.132 m l4 = 0.141 m l5 = 0.132 m l6 = 0.141 m L1 = 0.037 m L2 = 0.037 m L3 = 0.037 m L4 = 0.047 m L5 = 0.037 m L6 = 0.037 m L7 = 0.047 m Rb = 0.0355 m e= 3.55 ⋅ 10−5 m Ej = 2.2 ⋅ 1011 Pa Eb = 1.15 ⋅ 1011 Pa νj = 0.3 νb= 0.34

Comparing Figs. 8–14, it is seen that the bearing pairs b1–b7, b2–b6 and b3–b5 have almost identical theoretical wear diagrams, which is logical as the crankshaft considered has almost a symmetric configuration. A complete symmetry is violated by the fact that not all the bearings are of the same length and not all the spans between them are equal (see Table 1). For example, comparing Figs. 8 and 14, one can notice that the wear of the bearing b1 is predicted to be more intensive than that of the bearing b7. In addition, it is obvious that the wear of the bearings b1 and b7 (outer bearings) is found to be less intensive than that of the bearings between them (inner bearings). This is also seen as a valid result, as the inner bearings have two adjacent cylinders while each of the outer bearings has only one adjacent cylinder. It is expected that the wear is more intensive on those parts of the bearing circumference where the forces with higher magnitudes are applied and vice versa. This is seen in Figs. 9, 10, 12 and 13, where the zones of the highest and lowest load shown on the polar diagrams (Figs. 9a, 10a, 12a and 13a) approximately coincide with the corresponding zones in the theoretical wear diagrams (Figs. 9b, 10b, 12b and 13b). However, this is not always the case, which goes in favor of the determination of theoretical wear diagrams. In other words, polar load diagrams by themselves do not give a complete, real picture of the bearing load, but only when accompanied with theoretical wear diagrams, they do so. This is illustrated in Figs. 8, 11 and 14. Looking at the polar load diagrams in Figs. 8a and 14a, one could expect that the wear would be the most intensive in quadrant I, close to the positive side of the Y-axis, and in the border zone of quadrants III and IV of the bearing circumference, close to the negative side of the X-axis. However, the theoretical wear diagrams given in Figs. 8b and 14b show the most intensive wear in the border zone of the quadrants II and III of the bearing circumference, at the negative side of the X-axis (angle 180 ∘). Similarly, the polar load diagram in Fig. 11a indicates the highest load applied throughout quadrant II of the bearing circumference, but the diagram in Fig. 11b predicts the most intensive wear in quadrant IV, close to the border zone with quadrant I (angle 351 ∘). The occurrence of these cases, when the bearing circumference zones with the most intensive wear do not entirely match those with the highest load applied, could be explained as follows. There are bearing circumference zones that are exposed to the forces Fb(φ) of lower magnitudes, but there is a plenty of such forces applied close to each other. Each of the forces Fb(φ) causes insignificant wear by itself, but the cumulative effect results in higher values of the cumulative conditional wear depth.

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Fig. 8. Main bearing b1: (a) a polar load diagram, (b) a theoretical wear diagram.

Fig. 9. Main bearing b2: (a) a polar load diagram, (b) a theoretical wear diagram.

N. Nikolic et al. / Mechanism and Machine Theory 58 (2012) 120–136

Fig. 10. Main bearing b3: (a) a polar load diagram, (b) a theoretical wear diagram.

Fig. 11. Main bearing b4: (a) a polar load diagram, (b) a theoretical wear diagram.

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Fig. 12. Main bearing b5: (a) a polar load diagram, (b) a theoretical wear diagram.

Fig. 13. Main bearing b6: (a) a polar load diagram, (b) a theoretical wear diagram.

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Fig. 14. Main bearing b7: (a) a polar load diagram, (b) a theoretical wear diagram.

6. Conclusions An algorithm for a step-by-step construction of theoretical wear diagrams of IC engine crankshaft main bearings has been developed in this paper. These wear diagrams provide a clear visual representation of the load distribution around the inner circumference of main bearings. In the approach presented, cumulative conditional wear depths around a bearing circumference have been calculated, which actually describe a cumulative effect of the forces acting on a bearing considered during one engine cycle. The calculated values enable one to compare different bearings in wear intensity and to determine the regions on the bearing surface that are exposed to heavier wear than others. A new generic procedure for calculating the bearing forces has also been developed. In the procedure a crankshaft has been treated as a statically indeterminate continuous beam and influence coefficients have been determined. In this way, a useful approach has been defined for determining how various engine and crankshaft parameters affect a cumulative load and theoretical wear profiles of main bearings, and can be used in the process of bearing and lubrication system design. The whole algorithm, tailored to its software implementation, has been illustrated on the example of a six-cylinder diesel engine crankshaft, which has demonstrated some benefits of using the wear diagrams constructed. Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2012.07.009. Acknowledgments The authors would like to express their gratitude to Prof. Ivana Kovacic from the University of Novi Sad for her very helpful cooperation and encouragement during the research work. This research is a part of the project TR-31046 “Improvement of the Quality of Tractors and Mobile Systems with the Aim of Increasing Competitiveness and Preserving Soil and Environment” supported by Ministry of Science and Technological Development of the Republic of Serbia. Appendix A. A method for obtaining influence coefficients of a statically indeterminate continuous beam The method shown here is based on the superposition method [21] and on Clapeyron's three-moment equation. The influence coefficient ρi,k is the reaction at a support k caused by a point force of the magnitude equal to unity, applied to the middle of a span Si. If a continuous beam has n spans, there are n cases to be considered. A general case, when a unit point force is applied to the span Si, is considered subsequently.

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Fig. A.1. Decomposition of a statically indeterminate continuous beam into the system of simple beams.

Let a continuous beam shown in Fig. A.1 be loaded by a unit point force Fk in the middle of a span Sk. The beam can be decomposed into n simple beams by means of the intersection method [22]. According to this method, the moments Mk,i and Mk,i+1 act on every simple beam Si (i = 1, …, n). Reactions in the left and the right support of the beam Si are depicted with Rlk,i and Rrk,i, respectively. Of interest here is to calculate the reactions Rlk,i and Rrk,i, but the unknown moments need to be calculated first. To that end, Clapeyron's three-moment equation can be used [22]. So, assuming that the continuous beam has a constant cross-section, Clapeyron's equation, written for three consecutive supports k − 1, k and k + 1, has a general form [22]   Mk−1 lk−1 þ 2M k ðlk−1 þ lk Þ þ Mkþ1 lk ¼ −6EI ∑ α s þ ∑ βs k

k

ðA:1Þ

where: Mk − 1, Mk and Mk + 1 moments over the three consecutive supports k − 1, k and k + 1, lk − 1 and lk lengths of the simple beams (spans) Sk − 1 and Sk, respectively, I moment of inertia of the beam, E modulus of elasticity of the beam material, ∑ α s þ ∑ βs algebraic sum of the elastic curve slopes at support k due to active forces applied in the adjacent spans. k

k

If a continuous beam has (n + 1) supports as in Fig. A.1, then the system of (n – 1) Clapeyron's equations can be written. This enable one to calculate (n – 1) unknown moments, since the moments acting on the outer supports are known and equal to zero Mk,1 = Mk,n + 1 = 0. By using Eq. (A.1), the system of (n − 1) Clapeyron's equations written for the case shown in Fig. A.1 is:

¼ 0; 2Mk;2 ðl1 þ l2 Þ þ Mk;3 l2 ⋱ ⋮ 2 M k;k−1 lk−1 þ 2Mk;k ðlk−1 þ lk Þ þ Mk;kþ1 lk ¼ −3lk =8;

¼ −3l2k =8; M k;k lk þ 2Mk;kþ1 lk þ lkþ1 þ Mk;kþ2 lkþ1 ⋱ ⋮ M k;n−1 ln−1 þ 2Mk;n ðln−1 þ ln Þ ¼ 0:

ðA:2Þ

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Since the algebraic sum

135

  ∑ α s þ∑ βs of the elastic curve slopes for the case in Fig. A.1 is Fklk2/16EI [22], the terms on the k

k

right-hand side in the Clapeyron's equations that include both supports k and k + 1 in Eq. (A.2) become − 3lk2/8. The terms on the right-hand side of other Clapeyron's equations in Eq. (A.2) are equal to zero, because there are no active forces in the corresponding beam spans. Taking into account that there are n beam spans where a unit point force can be applied, n systems of Clapeyron's equations similar to Eq. (A.2) can be written. The differences are in right-hand sides of the equations where the length of the beam span considered appears always in the corresponding system only. Each of the n equation systems has (n - 1) equations with (n – 1) unknown moments to be calculated, which means that it can be solved by using any algebraic method. Herein, the matrices method is utilized and all the moments Mi,j (i = 1,…,n; j = 2,…,n) are determined. Knowing the moments Mk,1, Mk,2,…,Mk,n + 1 shown in Fig. A.1 and using the equations ∑M ¼ 0 for each simple beam Si, the support reactions Rlk,i and Rrk,i can be determined. Considering the simple beam Sk in the middle of which the unit point force Fk is applied (Fig. A.1), the moment equation about the support k is ∑M

ðkÞ

¼ Rrk;k lk −Mk;k −F k ⋅0:5lk þ Mk;kþ1 ¼ 0;

ðA:3Þ

Solving Eq. (A.3) for Rrk,k yields: Rrk;k ¼

M k;k −M k;kþ1 þ 0:5: lk

ðA:4Þ

Similarly, writing down the moment equation about the support k + 1 of the simple beam Sk, and then solving it for Rlk,k, leads to Rlk;k ¼

Mk;kþ1 −M k;k þ 0:5: lk

ðA:5Þ

Applying the same procedure to all the other simple beams Si (i = 1, …, n; i ≠ k) not loaded by the force Fk, the support reactions Rrk,i and Rlk,i are found to be Rrk;i ¼

Mk;i −M k;iþ1 ; ði ¼ 1; …; n; i≠kÞ li

ðA:6Þ

Rlk;i ¼

Mk;iþ1 −M k;i ; ði ¼ 1; …; n; i≠kÞ: li

ðA:7Þ

So, Eqs. (A.4)–(A.7) give the support reactions of the simple beams. However, it is necessary to determine the reactions ρk,j of the continuous beam supports j (j = 1,…,n + 1) when a unit point force is applied in the middle of the span Sk. They can easily be determined by using Eqs. (A.8)–(A.10) based on Fig. A.1: ρk;1 ¼ Rlk;1 ;

ðA:8Þ

ρk;nþ1 ¼ Rr k;n ;

ðA:9Þ

which are valid for the outermost supports of the continuous beam and ρk;j ¼ Rlk;j þ Rr k;j−1

ðj ¼ 2; …; nÞ;

ðA:10Þ

which are valid for all other supports of the continuous beam. By generalizing Eqs. (A.8)–(A.10), one can calculate (n + 1) influence coefficients for each of n cases, when a unit point force is applied in a beam span Si (i = 1, …, n), as follows

ρi;j ¼

8 <

Rli;j ; Rli;j þ Rri;j−1 ; : Rr i;j1 ;

j ¼ 1; j ¼ 2; …; n; j ¼ n þ 1;

i ¼ 1; …; n i ¼ 1; …; n : i ¼ 1; …; n

ðA:11Þ

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This gives a total of n ⋅ (n + 1) influence coefficients that can be expressed in a matrix form 2

ρ1;1 6 ρ2;1 6 6 : Rho ¼ 6 6 ρi;1 6 4 : ρn;1

ρ1;2 ρ2;2 : ρi;2 : ρn;2

: : : : : :

ρ1;j ρ2;j : ρi;j : ρn;j

: : : : : :

3 ρ1;nþ1 ρ2;nþ1 7 7 : 7 7: ρi;nþ1 7 7 : 5 ρn;nþ1

The influence coefficients from Eq. (A.11) are used in determining main bearing load. References [1] E. Koehler, R. Flierl, Internal Combustion Engines—Motor Mechanics, Calculation and Design of the Reciprocating Engines. (in German) 4th edition Vieweg & Sohn Verlag, GWV Fachverlage GmbH, Wiesbaden, 2006. [2] A.I. Kolchin, V.P. Demidov, Calculation of Car and Tractor Engines. (in Russian) Vishaya shkola, Moscow, 1980. [3] S. Orlin, M.G. Kruglov, Internal Combustion Engines, Design and Strength Calculation of Piston Engines and Combined Engines. (in Russian) Mashinostroyeniye, Moscow, 1984. [4] V.N. Lukanin, M.G. Shatrov, et al., Internal Combustion Engines, Dynamics and Design. (in Russian) Vishaya shkola, Moscow, 2005. [5] M.R. Cho, D.Y. Oh, S.H. Ryu, D.C. Han, Load characteristics of engine main bearing: comparison between theory and experiment, Journal of Mechanical Science and Technology 16 (8) (2002) 1095–1101. [6] E.V. Schnurbein, A New Method of Calculating Plain Bearings of Statically Indeterminate Crankshafts, Transactions of the Society of Automotive Engineers, 1970. Paper 700716. [7] V. Prakash, K. Aprameyan, U. Shrinivasa, An FEM Based Approach to Crankshaft Dynamics and Life Estimation. SAE paper 980565 , 1998. [8] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 443–457. [9] P. Flores, Modeling and simulation of wear in revolute clearance joints in multibody systems, Mechanism and Machine Theory 44 (2009) 1211–1222. [10] S. Mukras, Comparison between elastic foundation and contact force models in wear analysis of planar multibody system, ASME Journal of Tribology 132 (2010) 031604–1–11. [11] S. Mukras, A. Mauntler, N.H. Kim, T.L. Schmitz, W.G. Sawyer, Evaluation of contact force and elastic foundation models for wear analysis of multibody systems, in: Proceedings of the ASME 2010 International Design Engineering Technology Conferences, August 15-18, Montreal, Quebec, Canada, 2010, Paper No: DETC2010-28750. [12] S. Mukras, N.H. Kim, N.A. Mauntler, T.L. Schmitz, W.G. Sawyer, Analysis of planar multibody systems with revolute joint wear, Wear 268 (2010) 634–652. [13] Y. Su, W. Chen, Y. Tong, Y. Xie, Wear prediction of clearance joint by integrating multibody kinematics with finite-element method, Journal of Engineering Tribology 224 (2010) 815–823. [14] J.B. Heywood, Internal combustion engine fundamentals, McGraw-Hill, New York, 1988. [15] I.V. Kragelsky, V.V. Alisin, Tribology—Lubrication, Friction and Wear, John Wiley & Sons, 2001. [16] C. Liu, K. Zhang, R. Yang, The FEM analysis and approximate model for cylindrical joints with clearances, Mechanism and Machine Theory 42 (2007) 183–197. [17] P. Flores, A parametric study on the dynamic response of planar multibody systems with multiple clearance joints, Nonlinear Dynamics 61 (2010) 633–653. [18] S. Mukras, N.H. Kim, W.G. Sawyer, D.B. Jackson, L.W. Bergquist, Numerical integration schemes and parallel computation for wear prediction using finite element method, Wear 266 (7–8) (2009) 822–832. [19] B. Bhushan, Introduction to Tribology, John Wiley & Sons, Inc., New York, 2002. [20] J.F. Archard, W. Hirst, The wear of metals under unlubricated conditions, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 236 (1206) (1956) 397–410. [21] X. Dong, Jin-song, J. Ye, H. Xiao-feng, Kinetic uncertainty analysis of the reheat-stop-valve mechanism with multiple factors, Mechanism and Machine Theory 45 (11) (2010) 1745–1765. [22] S.P. Timoshenko, Strength of materials, Part I: Elementary Theory and Problems, 3rd edition Van Nostrand Company, Princeton, New Jersey, 1955.