AbstractA three-dimensional piston ring model has been developed using finite element method with eight-node hexahedral elements. The model predicts the piston ring conformability with the cylinder wall as well as the separation gap between the interfaces if existing in the radial direction. In addition to the radial interaction between the ring front face and the cylinder wall, the model also predicts the contact between the ring and groove sides in the axial direction. This means, the ring axial lift, ring twist, contact forces with the groove sides along the circumferential direction are all calculated simultaneously with the radial conformability prediction. The ring/groove side contact can be found for scraper ring at static condition, which is widely used as the second compression ring in a ring pack.

Thermal load is believed having significant influence on the ring pack performance. This thermal load influence is included in the model with a typical piston ring groove and cylinder wall temperature boundary condition.

The modeling procedure and results are presented in this work. The model analyzes the piston ring performance with the consideration of the variation along the ring circumference, which has not been considered by the two-dimensional models widely used currently.

IntroductionOne of the IC engine ring pack's major functions is to seal the combustion chamber by controlling blowby gas, which is known as the gas leaking from the combustion chamber, through the piston and ring pack, into the engine crankcase, causing power loss [8]. Ideally, a uniformly distributed contact pressure/force between the ring front face and the cylinder liner is believed to be desirable for 4-stroke engine operations. This is because a uniformly distributed pressure results in a ring conforming to the cylinder bore perfectly, thus seals the gas flow path between the ring front face and the cylinder liner. Negative ovality rings are widely used. The ring ovality is defined as the difference between the mutually perpendicular diameters: one diameter through the ring end gap and ring back, and the other

diameter perpendicular to this one, when the ring is enclosed in a flexible band (ISO 6621-2). Negative ovality rings result in lower ring-cylinder liner contact pressure at the ring butt ends when the thermal load across the ring is small. However, in several operation conditions, the thermal load influence is significant even for the second compression ring, such that the ring grows outward and conforms to the cylinder liner at the ring end butt, resulting a more uniform contact pressure between the interface. The pressure/force distribution depends on the ring free shape. D. C. Sun provided an approach to analyze the non-uniform ring bore contact problem [14]. Tian, T. modeled the piston ring dynamics assuming it is uniform along the ring periphery [17,18]. Mikhail A. Ejakov, et al, modeled ring twist behavior predicting ring axial, radial displacements, bending and twisting angles along the ring periphery over an engine cycle [6]. Ruddy, B. L., et al, proposed a ring twist model based on piston ring geometric properties and material modulus of elasiticity [13]. V. Dunaevsky, et al, demonstrated the complex 3D ring deformation due to the ring tension with the assumption of a uniform distributed pressure along the ring circumference [5]. Jiubo Ma, et al, computed the piston ring contact force distribution with three different numerical constraint models: gap element, cable element and thermal liner element models [10]. Liang Liu and Tian Tian presented an analytical tool for the ring conformability calculation using space beam elements with consideration of the ring lapping process [9]. Eduardo Tomanik and Rafael Bruno developed a ring radial force distribution measurement method with pin gauge that shows the contact force variations at different circumferential locations [20]. Tomanik also proposed a new criterion for ring conformability [19]. Adilson Tejada, et al, showed different ring pack configurations' impact on oil consumption and blowby [16]. However, the ring-cylinder bore interaction, including the separation gaps for the non-contact nodes and constraint forces for the in-contact nodes, remains a less understood topic in IC engine study. And the ring/groove side contact, which can exist for ring with non-symmetric cross-sections are not considered.

This work describes a three-dimensional (3D hereafter) analytical ring model to address the pressure/force distribution between the ring-cylinder bore interface along the ring circumference direction. In

Modeling of Piston Ring-Cylinder Bore-Piston Groove Contact

2015-01-1724

Published 04/14/2015

Chao Cheng, Ali Kharazmi, and Harold SchockMichigan State University

CITATION: Cheng, C., Kharazmi, A., and Schock, H., "Modeling of Piston Ring-Cylinder Bore-Piston Groove Contact," SAE Technical Paper 2015-01-1724, 2015, doi:10.4271/2015-01-1724.

Copyright © 2015 SAE International

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addition to this radial interaction, the model also takes into account the ring-groove side contact in the axial direction along the circumference for non-symmetric cross-section rings. The model solves the problem from the minimum strain energy point of view with realistic boundary condition assumptions. The penalty method is used to find the contact pairs between the ring/cylinder liner and ring/groove side interfaces. Dynamic analysis including the influence of the piston secondary motions [15] is not the focus of the work.

Modeling ApproachIn this section, the mathematical description of the ring model is presented. Ring mesh is generated based on the ring outer diameter (OD) curvature and the cross-sectional geometries. The penalty method used to solve the ring-cylinder bore contact is also discussed with a force release approach.

Finite Element DescriptionThe ring at its free shape is meshed with eight-node hexahedral elements. For each node, there are 3 degrees of freedoms (DOF) as the displacements in the X, Y, and Z directions in the global coordinate system. Thus an element has a total of 24 DOFs as can be found in Figure 1.

Figure 1. Hexahedral element and its DOF

The details of the formulation can be found from references [4, 7]. For each element, there are also 24 equivalent nodal loads corresponding to the DOFs. The finite element formulation for each element can be expressed as:

(1)

where Ki is the stiffness matrix for the ith element, qi and fi are the DOFs and load vector for the ith element. Detailed formulation of the components for the Ki matrix and fi vector can also be found in [7]. For each element, Ki is a 24×24 matrix while qi and fi are both 24×1 vectors.

After assembling all the elements based on their connectivity relations, the finite element expression for the system that defines the ring can be obtained:

(2)

The force vector f should take into consideration all loads acting on the ring, which include the gas pressure load around the ring, temperature gradient induced thermal load, friction load, constraint load from the cylinder bore or groove sides, etc. As the objective of this work is to calculate the static contact loads from the cylinder bore and the groove sides, only ring tension, constraint load and thermal load are considered. However, the ring constraint load is still an unknown since the contact pattern between the ring and cylinder and groove is not clear. The method of searching for the in-contact nodes will be discussed in the following section. The free shape ring mesh is shown in Figure 2.

Figure 2. Mesh of the ring with hexahedral elements

Due to the symmetric property of the ring, the boundary condition for the ring is that the nodes at the ring back cross-section are fixed in the X-direction only in the ring plane. These nodes are allowed to move in the radial and axial directions due to constraint forces. Additionally, the nodes can also move in the axial direction (Z-direction) if the ring cross-section is non-symmetric.

Penalty MethodThe penalty method is widely used for solving contact problems [1]. Compared to the other popular method, the Lagrange multiplier method, the order of the system is not increased using the penalty approach.

The problem can be formulated using the principle of minimum potential energy. This principle is a fundamental concept used in structure analysis analyzing structure deformation. It states that a structure should deform to a stationary state that minimizes its total potential energy, including the elastic strain energy and potential energy from the applied force. This principle is used to formulate the ring-cylinder bore contact problem. The formulation can then be expressed as:

To find nodal displacement: q

that minimizes the ring potential energy defined as:

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(3)

and subject to the constraint:

(4)

where ri is the radius of the ith cross-section at the ring face and RB is the radius of the cylinder wall; hT and hB are Z-coordinates of the deformed ring nodes at top and bottom sides, and GT and GB are Z-coordinates of the corresponding groove nodes at groove top and bottom respectively. Here, the constraints from groove sides in the axial direction are introduced in addition to the radial direction constraint as described by Cheng, C., et al, [2].

This constraint states that the piston ring outer radius should stay in contact (for the “=” case) or within (for the “<” case) the cylinder wall surface. It is not realistic for the piston ring locating beyond the cylinder wall, which is the case when ri > RB. The similar constraint criterion is also applicable to the groove constraints: the ring top side should stay in contact (for the “=” case) or below (for the “<” case) the groove top flank; while the ring bottom side should stay in contact (for the “=” case) or above (for the “>” case) the groove bottom flank. This constraint relation between the ring and cylinder bore, piston groove is shown in Figure 3 for a general ring cross-section geometry.

Figure 3. Constraints between ring and cylinder bore, groove sides

The minimization of the potential energy requires the derivative of Π with respect to q vanishing, which is:

(5)

At the same time, the constraint implies that if the nodes on the ring front face are in contact with the cylinder bore, the constraint forces should be non-zero and along the radial direction pointing inward to

the center of the cylinder bore; on the other hand, if the node on the ring front face is not in contact with the cylinder bore, the constraint force should be zero. For the groove constraint case, the constraint force on the groove top side should be downward (pointing to the groove bottom side) for the contact force; while the groove bottom side constraint forces should be upward (pointing to the groove top side) when the ring bottom is in contact with the groove bottom side. It is obvious that high nonlinearity is involved in the present contact problem. The penalty method is used to solve q by instead solving a sequence of specially constructed unconstrained optimization problems. That is, with the penalty method, an additional term that accounts for constraints is introduced into the system potential energy found in (3) as:

(6)

where λ is the penalty number and g represents the node geometric constraint and can be expressed in the matrix form as:

(7)

Here, c is the gap between the node's initial free state position, and the final deformed state and the matrix A projects the nodal DOF to the gap. Now, the minimization of the modified potential energy accounting for the penalties requires the derivatives of ΠP with respect to q to vanish, which yields the following relation:

(8)

From equation (8), the penalty method converts the geometric nonlinearity into material nonlinearity by modifying the potential energy term.

The penalty method introduces a penalty term into the system potential energy formula as expressed in (6). If the penalty number λ is equal to zero, then equation (6) becomes identical to equation (3). The penalty number λ is typically a very large number, such that a large cost is added to the objective function when the solution points lie outside the original feasible region. Thus by minimizing the modified potential energy accounting for the penalty term, the ring displacement is found and the constraint is satisfied. The recommended range for the penalty number λ found in literature [21] is:

A flowchart for solving the ring-cylinder bore contact problem using the penalty method can be found in Figure 4.

With the penalty method, all the constraint forces from the cylinder bore and the groove sides are found simultaneously.

The numerical model has been verified with a production top compression ring using two methods: one is by comparing the contact force distribution pattern; the other is a “light-tightness” method [2].

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This method is also adopted in this simulation model in order to identify the contact pairs between the ring/cylinder liner and ring/groove side interfaces.

Figure 4. Flowchart of solving ring-bore contact

Piston rings play important roles in dissipating the piston heat [11]. On the other side, the ring thermal load has significant influence on ring performance, especially for the top compression ring, which is directly exposed to the compression chamber. The thermal load on a top compression ring under convective boundary condition has been described previously [2]. The results showed it can significantly affect the ring/cylinder liner conformability for a top compression ring with symmetric cross-section. The thermal load's influence on a second compression ring is examined in this work and the results are given in the following result section.

Results and DiscussionOne example given in this section is a scraper ring with a taper face and cut-off at the ring inner upper corner, which promotes positive twist when installing the ring into the piston groove. The cross-section of the scraper ring is shown in Figure 5.

Figure. 5. Constraints on ring cross-section

From Figure 5, four nodes of the cross-section at a given circumference location are considered for the ring-piston groove side interaction and are numbered as node 1, node 2, node 3 and node 4 as shown. These four nodes are constrained by the groove in the axial direction. This means node 1 and 2 should stay in contact or above the groove bottom side; while node 3 and 4 should stay in contact or below the groove top side. Two nodes on the ring front face are constrained by the cylinder bore in the radial direction, at the front face top and bottom edges respectively. The groove has zero uptilt angle at its top and bottom sides. The nominal clearance between the groove and ring axial thicknesses is 0.1 mm.

The main parameters describing the ring are listed in Table 1.

Table 1. Main parameters for the ring

The scraper ring used in this study is a ring with negative ovality. The scraper ring has positive static twist due to the cut-off at the inner top corner.

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The constraint locations along the ring circumference are equally spaced with about 30° from one butt end to the other. The number of constraint locations is found to be able to represent the ring/cylinder liner/groove side contact force/pressure distribution pattern and also save calculation time. Increasing constraint locations will increase computation time exponentially; while decreasing the constraint locations may result in the contact force/pressure pattern not being well represented.

The deformed ring shape is shown in Figure 6 after installed into the cylinder liner and piston groove. The displacement in the z-direction (axial direction) is amplified by 100 times in order to illustrate the ring deformation distinctly.

Figure 6. Deformed ring shape without thermal influence

From Figure 6, especially the side view, the ring lifts up from its back to about 60° away from the ring end gap, after that, the ring drops in the axial direction.

The magnitude of the constraint forces at the ring top and bottom sides are shown in Table 2. As the ring end gap is not fully closed, the location is slightly less than 180°. The other half of the ring has the same constraint forces due to the symmetric boundary condition. The up and low locations refer to the top and bottom edges of the ring front face shown in Figure 5.

Table 2. Constraint forces along ring circumference without thermal influence

It is found that the radial constraint forces are only found at the lower edge of the ring front face due to the taper profile. This is because the scraper ring has a taper face of 1.5°, which means the ring can only touch the cylinder liner at its upper edge when the ring twists more than 1.5° negatively. However, due to the fact that the ring is a positive twist ring, the result becomes the ring front fact does not contact the cylinder liner at its upper edge. The contact pressure pattern is shown in Figure 7 based on a linearly distribution assumption.

Figure 7. Linear pressure distribution along circumference at lower edge of ring face

The ring twist is also examined along the circumference for this positive static twisted second ring as designed. The ring twist angle is shown in Figure 8 for half of the ring, as the other half is symmetric at the current boundary condition. The ring twists positively from the ring back to the ring end as expected. However, the twist angle varies along the circumference. From the ring back, the ring twist angle first increases slightly to about 0.22°, and then decreases towards the ring end at about 0.08°.

Figure 8. Twist angle along ring circumference without thermal influence

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The constraint forces between the ring and the piston groove is important since it dictates the contact pattern which will affect the ring-groove side wear eventually. However for this scraper ring investigated, no contact forces are found between the ring and groove sides at this static condition. This implies that the groove is large enough that the deformed ring does not touch with in both top and bottom sides after twist and axial movements.

Thermal Load InfluenceIn order to obtain a more accurate analysis of the piston ring deformation and subsequent force distribution at engine operating conditions, the thermal load needs to be considered due to the fact that the ring temperature can increase significantly during operation. Although not directly exposed to the combustion chamber as for the top compression ring, the second compression ring is still under considerably high thermal load when dissipating piston heat [11]. Under the boundary condition shown in Figure 9, the temperature distribution for the ring cross-section is shown in Figure 10.

Figure 9. Ring thermal boundary condition

Figure 10. Ring cross-section temperature distribution

Under the current boundary condition, it can be found that the temperature is higher at the ring inner top corner and lower at outer bottom corner. The average temperature is found to be 180.3 °C. As the thermal boundary condition is axisymmetric, the temperature distribution is identical for each cross-section.

The ring-cylinder liner contact was also analyzed for the ring under temperature influence. Cylinder bore thermal deformation is not considered in this study, as the focus of this study is on thermal load influence on the ring only. Similarly to the condition without thermal influence, the magnitudes of the constraint forces at the upper and lower edges along the ring circumference with temperature influence can be found in Table 3.

Figure 11. Deformed ring shape with thermal influence

Table 3. Constraint forces along ring circumference with thermal influence

As found from Table 3, the constraint forces are still only found at the lower edge of the front face. As mentioned for the ring/cylinder liner contact without thermal influence case, the lack of contact at the ring upper front face is due to the taper face and positive twist of the ring. The twist angles along the ring circumference for one half of the ring are shown in Figure 12 compared with the case without thermal influence.

It can be found from Figure 12, the ring twist changes significantly when thermal influence is considered. From the back of the ring to about 130° away, the ring twist angles are higher for the case with thermal influence. From about 130° to the ring butt end, the ring twist angles are smaller for the case considering thermal influence. The change in ring twist angle results from the non-symmetric cross-section geometry as well as the temperature gradient across the ring.

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The contact pressure between the ring face and cylinder liner interface can be found in Figure 13. It is found the contact pressures are high at the ring back, 90° and 150° away from the ring back compared to other locations.

Figure 12. Comparison in ring twist angle

Figure 13. Linear pressure distribution along circumference at lower edge of ring face

Between the ring and groove sides in the axial direction, no contact is found either for this case with thermal load influence.

Another example is given in this section illustrating the piston ring-cylinder bore-piston groove interaction. Another scraper ring is investigated with a larger cut-off at its inner top corner as shown in Figure 14. Other dimensions, including the ring radial width, axial thickness, groove thickness, etc, are the same as the ring studied previously in this work.

The nominal axial clearance between the ring and groove is also 0.1 mm as the previous ring studied.

Figure 14. Constraints on ring cross-section

Figure 15. Deformed ring shape

Figure 15 shows the side view of the deformed ring shape after installed into the piston groove. The displacement in the z-direction (axial direction) is amplified by 100 times in order to illustrate the ring deformation distinctly. In this case, the ring back and ring butt end are in contact with the groove bottom side; while the ring touches the groove top side at about 60° from the end gap.

Table 4. Constraint forces between ring and groove side

The constraint forces between the ring and the piston groove sides are important since it dictates the contact pattern which will affect the ring-groove side wear eventually. Table 4 lists the constraint forces between the ring and piston groove for the scraper ring at four locations shown in Figure 14. The ring-cylinder bore constraint forces are not shown in the table since the interest here is the ring-groove side interaction.

Table 4 shows that for the cross-sections that are in contact with the groove bottom (at ring back and at ring end), the contact occurs at the lower inner node (node 1 in Figure 14). And for the cross-section that touches the groove top, the contact occurs at the upper outer node

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(node 4 in Figure 14). This contact pattern is due to the fact that the ring twists positively, which results in a two-point contact between the ring and the groove sides.

In addition to the axial displacement variance, the ring twist angle also varies along the circumference as shown in Figure 16 for one half of the ring.

Figure 16. Twist angle along ring circumference

The twist angle plot shows that the scraper ring twists positively and the magnitude of the twist decreases from the ring back to the ring end gap. The maximum positive twist occurs near the ring back at less than 0.8°. And the minimum positive twist is found at the ring ends with higher than 0.2°. Comparing to the first ring studied, the static twist for this ring is much higher.

ConclusionsInteractions between piston ring and cylinder bore, and between ring and piston groove are complex 3D phenomena. A finite element ring-cylinder liner-piston groove contact model was developed with eight-node hexahedral elements to simulate this complex interaction using penalty method. A positive twisted scraper second ring with negative ovality is used as an example showing the simulation result of the model. The variation of constraint force/pressure along the ring circumference is found. The ring twist angle and axial ring movement are also found varying along the circumference. However, the axial ring movement is small such that the ring does not contact with the groove side for this ring even with temperature influence consideration. Another scraper ring with larger inner top cut-off is then studied. The result shows the ring contacts with the groove side at variable circumferential locations at either inner lower corners or outer upper corners due to the positive twist.

Temperature influence is found to have a significant impact on this ring-cylinder liner interaction as the constraint forces change along the ring circumference. Besides the contact force, ring twist is also affected with thermal load. That is the positive twist increases from ring back to about 130° and decreases from 130° toward the ring butt end for the first ring configuration.

The ring-cylinder liner-groove side interaction model also provides the geometry for 3D gas dynamics analysis. In addition to the gas flow paths that the current two-dimensional models predict, including

through the ring end gap, through the ring-groove sides as ring flutters, through the ring front face when ring radial collapses, gas can flow through an additional path across the ring. This gas flow path is formed due to the variance of ring axial displacements along the circumference. This can be important for the second compression ring, whose dynamics has been found having significant influence on engine blowby [3]. The ring-piston groove side contact can also affect local wear between the interaction surfaces and influence ring pack durability.

The present study provides a basis to integrate the ring-cylinder bore-piston groove interaction in a 3D manner. Along with the 3D ring dynamics, the gas dynamics, lubrication, friction, and wear can all be modeled in a more accurate way analyzing the internal combustion engine power cylinder system.

AcknowledgmentsWe acknowledge the support of Mid-Michigan Research for supporting this effort.

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