- Email: [email protected]
Analytical wear model and its application for the wear simulation in automotive bush chain drive systems
Wear 446–447 (2020) 203193
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Analytical wear model and its application for the wear simulation in automotive bush chain drive systems Robert Tandler a ,∗, Niels Bohn a , Ulrich Gabbert b , Elmar Woschke b a b
BMW Group, Germany Otto von Guericke University Magdeburg, Germany
ARTICLE
INFO
Keywords: Wear Friction Wear modelling Finite element simulation Bush chain
ABSTRACT The wear of automotive chain drive systems after high mileages is numerically calculated based on Fleischer’s energetic wear equation. This equation is embedded in a FE-model, consisting of one single chain link only. Their time-variant positions and acting forces are taken from a multi-body simulation. A focus is on ensuring the quality of the FE-model and the contact between pin and bush, where here a penalty approach has provided a stable solution schema. The parameters of the wear model are derived from measurements. After each time increment the wear results in a changed surface geometry, which is used for the simulation of the next time increment. The enormous computation time is reduced by the development of a convenient extrapolation factor. The complex simulation approach is applied to the chain drive of a test vehicle after a mileage of about 50000 km. The comparison of the simulated and measured data demonstrates an agreeing correlation.
1. Introduction Modern combustion engines lead to the trend of high performances at low emission levels, which is especially present in the premium vehicle class. This is resulting in high moments on the crankshaft and further in high forces in the chain links of the chain drive. For this case it is important to calculate the wear of chain drive systems to receive a lifetime forecast. With the information of the produced wear in the chain drive system it is possible to optimise the bush and the pin of all chain links and decrease their wear volumes for different applications. An analysis of the recent literature for topics of friction and wear leads to the result that specific wear models are necessary for certain applications. Since the wear process of a bush chain in an automotive chain drive system seems to be dominated by abrasive wear, the focus of this section is set to abrasive wear models. As a first result the paper of Myshkin and Goryacheva [1] shows the problem of approaching different scaling factors in a microscopic or macroscopic view for the description of wear in a certain contact area. Many wear models with respect to a generated wear volume 𝑉 are based on the equations of Holm-Archard [2,3] 𝑠𝐹𝑁 𝐻 and Fleischer [4,5,6] 𝑉 =𝑘
𝑉 =
𝑊𝑅 , 𝑒
(1)
(2)
where 𝑉 is the wear volume, 𝑘 is a constant, 𝑠 is the sliding distance, 𝐹𝑁 is the contact normal force, 𝐻 is the hardness of the softer material, 𝑊𝑅 is the work of friction, and 𝑒 is the energy density of the material. These models are usually modified for any special applications. There is a large variety of wear equations available in the literature, which depend on numerous parameters. Some examples for this are given in the following. Beckmann und P. Dierich [7] developed a conceptional approach like Fleischer to determine a wear equation. For applications with mainly stationary periodic sliding, Páczelt and Mróz [8,9,10] have developed a wear model, where Coulombs friction law holds. Sappok und B. Sauer [11] built up an experimental set-up and measured the wear of a timing chain. They derived a time dependent wear equation depending on the half run-in period and a constant that is used during stationary wear. Further Antusch [12] investigated a correlation between wear and friction in his experiments. These measurements are done for fresh oil as for soot contaminated diesel oil. With respect to the wear models shown above, Popov [13] also stated an important influence of the surface roughness on the wear process in a system. For the numerical simulation of wear processes in engineering applications the finite element (FE) method is ideally suited. Põdra and Andersson [14] as well as Mukras et al. [15] describe FE-simulations with an underlying wear model. In these simulations the nodes of the mesh are moved if wear occurs. Further, there is an extrapolation factor described which has to be determined for a certain application.
∗ Corresponding author. E-mail address: [email protected] (R. Tandler).
https://doi.org/10.1016/j.wear.2020.203193 Received 2 December 2019; Received in revised form 7 January 2020; Accepted 14 January 2020 Available online 16 January 2020 0043-1648/© 2020 Elsevier B.V. All rights reserved.
Wear 446–447 (2020) 203193
R. Tandler et al.
̃ Podra and Andersson also perform an experimental investigation with a tribometer to verify their simulations. Schmidt et al. [16] describe a FE-model to calculate dry sliding wear on a tilted shaft-bushing bearing. In this model a ’Node-to-Surface’-Penalty contact formulation and a sectional exponential contact are used for the simulation. The exponential contact is aided by a linear contact at contact opening. The evaluation of the different wear equations in the literature has shown that in general the wear process is described by some basic parameters such as the contact forces, the hardness of the materials in contact, the sliding distance and the friction. Additionally some further parameters are included which are experimentally determined to ensure the agreement of the wear equation with measurements. These wear equations are mathematical models describing how the wear depends on external and internal influences. Consequently, such wear equations do not have the general validity of a physical law, meaning that for the same real problem different models could be applied. These models differ in their field of application, the considered dependencies, the quality and the range of validity. But, a wear equation should be consistent with considerations of physics. In order to this the general wear model of Fleischer is used to calculate the wear of timing chains numerically. For this purpose the wear equation is embedded in a finite element model (FE-model). This numerical approach and its experimental validation are presented in detail in Section 2. Finally, in Section 3 the results are discussed and an outlook on further investigations is given.
Fig. 1. Example for a nodal displacement of the FE-mesh.
2. Numerical embedding in a FE-simulation In this section the energy based wear equation of Fleischer is used for a numerical discretisation, since this model is related to physical aspects. Eq. (2) is an abstract extension of Eq. (1), since the model of Fleischer contains the correlation of wear to the work of friction in any system. As shown in the introduction a valid model for the work of friction should be determined for a special application. For an automotive bush chain drive system the experimental investigations of Tandler et al. [17] lead to the result that the work of friction can be approximated by Coulombs law of friction. In order to this the integrated wear volume 𝑉𝑉 is given by 𝑉𝑉 =
𝑠1 𝑊𝑅 = 𝑘 ⋅ 𝐹𝑁 ⋅ 𝜇 ⋅ 𝑑𝑠. ∫𝑠0 𝑒
Fig. 2. MKS-model of the automotive chain drive system.
The simulation of wear is done by the calculation of 𝑉𝑉 in Eq. (5) for each node (𝑙) in the FE-model with )(𝑙) (𝐼−1 ∑ 𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))𝛥𝑠(𝑖𝛥𝑡 ) . (6) 𝑉𝑉(𝑙) = 𝑖=0
Due to a belonging area 𝐴(𝑙) for each node (𝑙), a wear depth 𝑑 (𝑙) can be calculated trivially with 𝑑 (𝑙) = 𝑉𝑉(𝑙) ∕𝐴(𝑙) and the node coordinates are manipulated in each simulation routine. This is shown in Fig. 1 as schematic representation of a mesh segment. After a certain nodal displacement a new mesh is generated automatically with PYTHON in the ABAQUS CAE post processing and used as basis for the next simulation step. With respect to this and the elasticity of the model a new load distribution occurs on the contact surface in every step of the wear simulation. Consequently, after each simulation step a new geometry is calculated which results in the following step in a changed distribution of nodal forces and a new wear volume. From this again a new surface geometry is calculated. It is not necessary to transfer any other field parameters from one to the next simulation step. The parameters of Eq. (6) are defined in the following section.
(3)
for the materials reciprocal energy density 𝑘 = 1𝑒 , the sliding distance 𝑠 from 𝑠0 to 𝑠1 , the contact force 𝐹𝑁 and the friction coefficient 𝜇. In consideration of all measured dependencies [17] and the time 𝑡 of the dynamical system it follows 𝑡1
𝑉𝑉 =
∫𝑡0
𝑘𝐹𝑁 (𝑛, 𝑡)𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑡))
𝑑𝑠(𝑡) 𝑑𝑡, 𝑑𝑡
(4)
where 𝑛 is the engine speed, 𝜃 is the oil temperature, 𝜂 is the kinematic viscosity and 𝜓 is the parameter for soot content in the engine oil. The wear model of Eq. (4) is implemented in a finite element model within ABAQUS CAE 6.14-4. For this a specific working operation of the engine is chosen and verified in the finite element model. The ∑ 𝛥𝑠(𝑖𝛥𝑡 ) 𝑡 discretisation 𝑡 → 𝑖𝛥𝑡 , 𝑑𝑠(𝑡) → 𝛥𝑠(𝑖𝛥𝑡 ), ∫𝑡 1 𝑑𝑠(𝑡) 𝑑𝑡 → 𝐼−1 𝑖𝛥𝑡 , with 𝑖=0 𝑑𝑡 𝑖𝛥𝑡 0 the increment 𝑖 ∈ {0, … , 𝐼 − 1}, the total number of increments 𝐼 for discrete time values 𝛥𝑡 and the incremental sliding distance 𝛥𝑠(𝑖𝛥𝑡 ) ∶= 𝑠((𝑖 + 1)𝛥𝑡 ) − 𝑠(𝑖𝛥𝑡 ), 𝑠(0) ∶= 0 leads to the discretised wear equation for each solid body as 𝑉𝑉 = =
𝐼−1 ∑ 𝑖=0 𝐼−1 ∑
𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))
2.1. Definition of model parameters In this investigation a timing chain with coated pins and extruded hardened bushes is used. The detailed specification is shown in Bauer [18]. The contact normal force 𝐹𝑁 (𝑛, 𝑖𝛥𝑡 ) in Eq. (6) is given for every increment 𝑖 at a certain engine speed and part load operation from the FE-simulation and the multi body simulation (MBS) in comparison with the engine characteristics. The value of the load for the chain drive lies between 550 N and 700 N. The MBS-model for a simulation of the full chain drive system is shown in Fig. 2. The parameters for the stiffness, the mass and the damping factor of the certain components
𝛥𝑠(𝑖𝛥𝑡 ) 𝑖𝛥𝑡 𝑖𝛥𝑡
𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))𝛥𝑠(𝑖𝛥𝑡 ).
(5)
𝑖=0
2
Wear 446–447 (2020) 203193
R. Tandler et al.
are used from the BMW Group database and are embedded in the model. Every increment 𝑖 is related to one degree of shaft rotation of the engine and so defines a unique position of the chain link in the timing assembly. The assignment of a load 𝐹 (𝑛, 𝑖𝛥𝑡 ) and the contact sliding 𝑠(𝑖𝛥𝑡 ) on the chain link at two positions in the chain drive for the increments 𝑖 and 𝑖 + 𝑚 is schematically illustrated in Fig. 3. With this information the finite element model can be reduced to only one chain link and an amplitude information from the MBS. So for a specific position of the chain link the contact normal force at each node can be calculated in the finite element model with the implemented chain force. In the following a constant engine speed of 𝑛 = 5000 rpm, fresh engine oil 𝜓(𝑡) = 0 and a temperature 𝜃 = 100 ◦ C during engine operation are assumed. The corresponding coefficient of friction from the experimental investigations in Tandler et al. [17] is also implemented in the model. The factor 𝑘 gets a first approximation of 𝑘 = 1 ⋅ 10−10 mm2 /N from radionuclide wear measurements in a constant wear state with fresh oil. The assumption is, that this factor holds for bush and pin of a chain link to determine an extrapolation factor for the wear simulation. A description of the finite element model is given in the next chapter. 2.2. Description of the finite element model Due to the symmetry of the chain only one half of a bush chain link including symmetric boundary conditions is required. The developed finite element model is shown in Fig. 4. The contact area can be observed in more detail in Fig. 5.
Fig. 3. Assignment of the load and sliding distance on the chain link in a chain drive system.
Fig. 4. Symmetric finite element model of a bush chain link.
Fig. 5. Detail of the mesh in the contact area. 3
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 6. Scheme of a TIE-contact in ABAQUS CAE for two segmentation examples.
Fig. 7. Comparison between continuous meshed reference model and the TIE-model at the opened bush.
The angle 𝜑 of the chain link to its axis is given by the geometrical design of the chain drive system. During one chain rotation the chain link passes various contours with different curvatures like a chain wheel, which is resulting in the position dependent angle 𝜑. This engine specific angle and the chain force are implemented as amplitude in the dynamic ABAQUS CAE model. The symmetric model of one chain link with the nodal coordinate 𝑧 = 0 at the cutting plane is used to reduce the computational time of the FE-simulation. One chain link is enough for the simulation because all chain links of a bush chain statistically have the same load handling and sliding distance during the run time of the engine. A linear element type of C3D8R in ABAQUS CAE and a fine mesh in the contact region are applied to obtain sufficiently accurate values concerning the contact normal force and the contact sliding. This is required because the circular contour of the contact area has to be described numerically with a polygon. The smaller edges of the elements lead to a better approximation of a circle. Another
reason of the used fine mesh with element edge size ≈ 60 μm is that also topographically measured surface structures can be embedded into the model. A continuous meshed part with this element size would cause high computational times in the dynamic implicit simulation and consequently a TIE-contact is used at the bush and the pin to couple a coarse meshed part with a fine meshed part. The proper mesh density and the contact approach have been developed with help of several test simulations in order to find a compromise between computational time and a sufficient solution accuracy. To get a suitable computational time at a small relative error with respect to the full meshed model a TIE-coupling of 9 slave elements to 1 master element is important, as shown on the left side in Fig. 6. The TIE-coupling in ABAQUS CAE is defined as a slave node is bound to its nearest master node in the contact area. So it is important to have more nodes on the slave surface than on the master surface. Otherwise, many nodes are not bound by the TIE-coupling. Fig. 6 4
Wear 446–447 (2020) 203193
R. Tandler et al.
2.3. Determining an extrapolation factor for the wear simulation Since a common chain performs many millions of rotations during its lifetime it is important to determine an extrapolation factor 𝑒𝑥𝑝𝑓 for the simulation to get realistic computational times. To investigate this, high scaling factors for the wear volume are chosen at the beginning of this convergence study and gradually decreased until a reasonable converged numerical result occurs. The aim of this approach is to find a critical wear depth at the mesh nodes for each step of simulation such that numerical stability holds and the result is still physically reasonable. In order to Fig. 1 the schematic representation of the extrapolation factor is shown in Fig. 8, where the wear depth at each node gets scaled. In the following the wear process of a bush chain is analysed with the model parameters of chapter 3.1. The wear volume is calculated with an extrapolation factor and help of Eq. (5), which depends on the certain application and the different properties of the tribological system. If another chain or load case has to be simulated, the different properties of the tribological system have to be taken into account. In order to this the determined critical extrapolation factor leads to
Fig. 8. Schematic representation of the extrapolation factor.
illustrates that a segmentation of 3 slave elements to 1 master element per edge should be used to get a symmetric coupling of the slave nodes to the master nodes. A segmentation of 2 slave elements to 1 master element per edge would result in an asymmetric coupling, because the node in the middle of each edge will be coupled randomly, since the distances to the nearest master nodes are equal. The segmentation should be a small uneven natural number to get a minimal error with respect to a continuous meshed model. The investigation of the mesh quality is shown for the bush at a part load operation in Fig. 7. In Fig. 7 the pressure distribution at the surface of the bush is shown. At the picture on the left hand side the date are calculated with a continuous mesh without TIE-coupling. The results of the TIEcoupling are shown at the right hand picture of Fig. 7. The relative error of the peak pressure between both model is approximately 1,3%, the mean relative error over all circumferential nodes is approximately 2,8%. Due to the large reduction of the computing time in the TIE-coupling model this small deviations are acceptable. For the formulation of the sliding contact between pin and bush there are different methods available, such as the penalty method, the method of Lagrange multipliers and the method of augmented Lagrange multipliers. All methods have been investigated with respect to the pin and bush coupling. The method of Lagrange multipliers results in a large increase of the system of equations, since at each potential node of the very fine meshed contact area one additional degree of freedom – the unknown Lagrange multiplier – is introduced. This results in an huge increase of the computing time. The method of augmented Lagrange multipliers also increases the computing time because additional iterations are required to receive a converged solution. Finally, the penalty method has resulted in a sufficient accurate solution without an additional solution time. The constrained equations in the contact area can be written as ⃗ 𝑔(⃗ ⃗ 𝑢) ∶= 𝑍 𝑢⃗ − 𝑐⃗ = 0,
𝑒𝑥𝑝𝑓𝑛𝑒𝑤 = 𝑒𝑥𝑝𝑓𝑐𝑟𝑖𝑡 (𝑘, …)
(9)
with the critical wear volume 𝑉𝑐𝑟𝑖𝑡 corresponding to the critical extrapolation factor. Due to this a dynamical extrapolation factor is used to have a constant numerical quality at most efficient computational times. The value 𝑒𝑥𝑝𝑓𝑐𝑟𝑖𝑡 depends on the geometric model and its contact formulation, because both affect the numerical stability of the simulation. Convergence studies with different extrapolation factors for this application show that too high extrapolation factors lead to a reciprocally abrasion in the contact area. The reason for this is the simulated outlet contact pressure seen in Fig. 7, which is locally over scaled and causes to much wear depth at a certain position. An extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 still causes this effect slightly. To show this a chain link is simulated for the load case above and the wear surface evolving on the pin is calculated for different extrapolation factors at several steps of wear simulation. This means a wear depth at each node is calculated and in order to the nodal displacement a new mesh is generated after an extrapolated chain rotation. With this generated mesh the next step of simulation begins. The results of the simulation with an extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 are compared to those with the extrapolation factors 𝑒𝑥𝑝𝑓 = 200000, 𝑒𝑥𝑝𝑓 = 100000 and 𝑒𝑥𝑝𝑓 = 50000. The Fig. 9 illustrates these wear results after 600000 chain rotations. It can be observed that the extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 leads to a local over scaling of the wear depth at the contact edge. Since the contact surface is varying for the certain steps of a wear simulation, it is of interest to investigate the evolution of the wear surface for these extrapolation factors after another 600000 chain rotations. This is shown in Fig. 10. There is still a relative error between the extrapolation factors, but the tendency of deviation is decreasing. Further, Fig. 11 shows a comparison between the extrapolation factors after 1800000 chain rotations. The relative errors of the simulation results are shown in Table 1. With respect to this investigation and an acceptable computational time the critical extrapolation factor should be between 100000 and 200000, depending on the total number of wear routines. Since the relative error is decreasing for all cases with the duration of the wear analysis, the extrapolation factor of 200000 is chosen for further investigations. Regarding this, for the result of the maximal wear depth after 4000000 chain rotations a relative error of < 5% occurs between 𝑒𝑥𝑝𝑓 = 50000 and 𝑒𝑥𝑝𝑓 = 200000. The choice of 𝑒𝑥𝑝𝑓 = 200000 leads to the following contact pressure at the bush after several chain rotations as shown in Fig. 12. With this critical extrapolation factor a dynamical wear simulation with respect to the contact force, the friction coefficient and the relative contact sliding are performed.
(7) R𝑚×𝑛 ,
with the matrix of constraints 𝑍 ∈ 𝑚, 𝑛 ∈ N, the vector of all nodal translations 𝑢⃗ ∈ R𝑛 and a constant vector 𝑐⃗ ∈ R𝑚 . The penalty method, see Gabbert [19], results in an extension of the original system of the finite element stiffness equation as (𝐾 + 𝛼𝑍 𝑇 𝑍)⃗𝑢 = 𝑓⃗ + 𝛼𝑍 𝑇 𝑐⃗,
𝑉𝑐𝑟𝑖𝑡 , 𝑉𝑛𝑒𝑤
(8)
where 𝐾 ∈ R𝑛×𝑛 is the stiffness matrix of the system, 𝑓⃗ ∈ R𝑛 is the vector of external forces and 𝛼 ∈ R+ is the penalty factor. Here the 0 problem of a proper selection of the penalty factor 𝛼 arises. If the penalty factor is too small the constrained conditions are not sufficient accurate fulfilled. An increasing penalty factor results in an increasing loss of leading digits in the solution of the displacements 𝑢⃗, which is particularly critical if single precision real words are used during the calculation. A too large penalty number can finally result in a unsolvable singular system of equations. Several tests have demonstrated that in the given application a penalty number of 𝛼 ≈ 102 𝑘𝑖𝑖,𝑚𝑎𝑥 is a good compromise between stability of the solution process and the accuracy in the fulfilment of the constrained equations. Here 𝑘𝑖𝑖,𝑚𝑎𝑥 is the maximal entry of the main diagonal of the stiffness matrix 𝐾 in Eq. (8). 5
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 9. Comparison of the wear surface for different extrapolation factors after a total of 600000 chain rotations.
Fig. 10. Comparison of the wear surface for different extrapolation factors after a total of 1200000 chain rotations.
Fig. 11. Comparison of the wear surface for different extrapolation factors after a total of 1800000 chain rotations. 6
Wear 446–447 (2020) 203193
R. Tandler et al. Table 1 Comparison of different extrapolation factors after certain chain rotations. Comparison 𝑒𝑥𝑝𝑓
Rel. error 𝑉𝑚𝑎𝑥 after 600000 chain rotations
Rel. error 𝑉𝑚𝑎𝑥 after 1200000 chain rotations
Rel. error 𝑉𝑚𝑎𝑥 after 1800000 chain rotations
300000 vs. 200000 200000 vs. 100000 100000 vs. 50000
18.2% 11.8% 6.6%
13.8% 6.2% 3.2%
7.6% 4.0% 2.0%
With Fig. 14 it can be verified that a maximum wear depth occurs at the left and right end of the profile, because of an elastic deformation of the chain link during engine operation. The maximum delta of wear depth from the parts middle to an end of the polished area is approximately 6 μm. The results of the simulation described above after 20 routines of wear calculation with a critical extrapolation factor of 200000 at fresh oil is shown in Fig. 15. In Fig. 15 it can be observed that a nearly constant contact pressure occurs in the worn area within elastic effects due to the chain force. Further a polished area can be seen on the right side of Fig. 15. This coincides with the worn area of the pin after an engine run time of 48885 km as shown in Fig. 13. Another result is the correct tendency for this first approximated wear simulation. This can be seen in comparison of Fig. 15 with the result for the maximum wear depth of 5.028 μm to Fig. 14 where a delta wear depth of approximately 6 μm occurs at the measured pin. Finally, to show a comparison of the worn surface on the pin between simulation and real part the Fig. 16 should be taken into account. For this illustration the pin of the finite element model is symmetrised again. Fig. 12. Bushes contact pressure after several chain rotations.
3. Conclusions and further research With respect to Fleischer it is important to determine the work of friction to describe wear for any specific field of application. Based on extensive experimental investigations published in [17] the work of friction for the applied bush chains has been estimated. For the numerical analysis of the chain wear during operation it is appropriate to apply the finite element tool ABAQUS CAE. It is shown that the analysis of one chain link only is enough, since all chain links perform the same chain rotation during engine operation. The varying positions and chain forces of the chain link were estimated with help of multi body simulations and the results are then transferred to ABAQUS CAE. The quality of the FE-model is ensured by numerous convergence studies with respect to the mesh density and the contact formulation. Based on the convergence studies a critical extrapolation factor is estimated which leads to a maximum wear depth per wear simulation routine. With this critical extrapolation factor a dynamical wear simulation with respect to the contact force, the friction coefficient and the relative contact sliding is performed. The critical extrapolation factor also depends on the contact formulation with respect to the contacts geometry and the material specific wear factor k. The quality of the model is examined by a comparison of the simulation results with measurements of real engine chains of test vehicles, which demonstrates that the numerical wear simulation provides correct tendencies. In ongoing investigations the finite element model and the contact formulation are further improved to obtain lower computational costs and an increasing accuracy. Additionally the simulations are expanded to bush chains with winded bushes instead of extruded bushes, which are geometrically different at the outlet state. The objective is to apply the wear simulation approach to further chain types, such as oil pump chains and timing chains of certain engine types.
Fig. 13. Pin of a timing chain after 48885 km mileage of a test vehicle.
It can be observed that reasonable results occur, since the contact pressure is continuous distributed after multiple times of iterated wear simulation. Finally this should be verified in comparison to measurements of real engine parts in the next section. 2.4. Model comparison to measurements of chains from test vehicles The results of the simulation are compared to measurements of chain parts from test vehicles, where a similar load configuration is applied in the simulation. Fig. 13 illustrates a pin of an opened chain link after 48885 km mileage, where a polished area can be observed that depends on the intensity of wear processes in the system. The rough area around the worn area is still at output state, since there occurs no contact and also wear. The positive tensile load in the chain at all times is resulting in this wear surface. The simulation also confirms a contact opening on the unloaded surface of the pin and bush. It can be observed that the maximum wear depth must also occur at peak load in the simulation. The topographical profile of the pins worn area is measured with laser microscopy and is shown in Fig. 14.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 7
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 14. Topographical measurement of the pins loading surface with laser microscopy.
Fig. 15. Pins contact pressure (left) and wear depth (right) after 20 steps of wear simulation with a critical extrapolation factor of 200000.
Fig. 16. Comparison of the worn surface on the pin between simulation and real part.
References
[9] I. Páczelt, Z. Mróz, Variational approach to the analysis of steady state thermoelastic wear regimes, Internat. J. Numer. Methods Engrg. 81 (2010) 728–760, http://dx.doi.org/10.1002/nme.2709.
[1] N.K. Myshkin, I.G. Goryacheva, Tribology: Trends in the half-century development, J. Frict. Wear 37 (6) (2016) 513–516. [2] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (1953) 981–988, http://dx.doi.org/10.1063/1.1721448. [3] R. Holm, Electric Contacts, Gerbers, Uppsala, 1946. [4] G. Fleischer, Energetische Methode der Bestimmung des Verschleißes [Energetic methods to determine the wear of a system], Schmierungstechnik 9 (4) (1973) 269–274. [5] G. Fleischer, Zur Energetik der Reibung [The energetics of friction], Wiss. Z. Tech. Hochsch. Otto von Guericke 34 (8) (1990). [6] G. Fleischer, Zum energetischen Niveau von Reibpaarungen [Energetic level of frictional pairings], in: Schmierungstechnik VEB, Vol. 12, Verlag Technik, Berlin, 1985. [7] G. Beckmann und P. Dierich, A representation of the microhardness distribution and its consequences for wear prognoses, Wear 107 (1986) 195–212, http: //dx.doi.org/10.1016/0043-1648(86)90225-5. [8] I. Páczelt, Z. Mróz, On the analysis of steady sliding wear process, Tribol. Int. 42 (2009) 275–283, http://dx.doi.org/10.1016/j.triboint.2008.06.007.
[10] I. Páczelt, Z. Mróz, Analysis of thermo-mechanical wear problems for reciprocal punch sliding, Adv. Eng. Softw. 80 (2015) 139–155, http://dx.doi.org/10.1016/ j.advengsoft.2014.09.012. [11] D. Sappok und B. Sauer, Wear measurement on chain joint components using a roundness instrument, Period. Polytech. Mech. Eng. 59 (2015) 51–59, http: //dx.doi.org/10.3311/PPme.7780. [12] S. Antusch, Untersuchungen zum Einfluss von Rußim Öl auf den Motorenverschleiß- Betrachtung mechano-chemischer Reaktionen [Investigation of the influence of soot content in engine oil on wear of engines] Dissertation, Karlsruher Institut für Technologie, 2008. [13] V.L. Popov, Contact Mechanics and Friction - Physical Principles and Applications, Springer-Verlag, GmbH Germany, 2017, p. 2010, http://dx.doi.org/10. 1007/978-3-662-53081-8. [14] P. Põdra, S. Andersson, Simulating sliding wear with finite element method, Tribol. Int. 32 (1999) 71–81, http://dx.doi.org/10.1016/S0301-679X(99)000122. 8
Wear 446–447 (2020) 203193
R. Tandler et al.
[18] P. Bauer, Kettensteuertriebe - Stand der Technik, Anwendungen, Entwicklung und Herstellung [Chain drive systems - State of the art, application, development and production], in: Die Bibliothek Der Technik Band, Vol. 353, Süddeutscher Verlag onpact GmbH, 2013. [19] U. Gabbert, Berücksichtigung von Zwangsbedingungen in der FEM mittels der Penalty-Funktion-Methode [Consideration of constrains in the FEM due to the penalty method], in: Technische Mechanik 4 Heft 2, 1983.
[15] S. Mukras, N.H. Kim, W. Gregory Sawyer, D.B. Jackson, L.W. Bergquist, Numerical integration schemes and parallel computation for wear prediction using finite element method, Wear 266 (2009) 822–831, http://dx.doi.org/10.1016/j.wear. 2008.12.016. [16] A.A. Schmidt, T. Schmidt, O. Grabherr, D. Bartel, Transient wear simulation based on three-dimensional finite element analysis for a dry running tilted shaftbushing bearing, Wear 408–409 (2018) 171–179, http://dx.doi.org/10.1016/j. wear.2018.05.008. [17] R. Tandler, N. Bohn, U. Gabbert, E. Woschke, Experimental investigations of the internal friction in automotive bush chain drive systems, Tribol. Int. 140 (2019) http://dx.doi.org/10.1016/j.triboint.2019.105871.
9
Contents lists available at ScienceDirect
Wear journal homepage: www.elsevier.com/locate/wear
Analytical wear model and its application for the wear simulation in automotive bush chain drive systems Robert Tandler a ,∗, Niels Bohn a , Ulrich Gabbert b , Elmar Woschke b a b
BMW Group, Germany Otto von Guericke University Magdeburg, Germany
ARTICLE
INFO
Keywords: Wear Friction Wear modelling Finite element simulation Bush chain
ABSTRACT The wear of automotive chain drive systems after high mileages is numerically calculated based on Fleischer’s energetic wear equation. This equation is embedded in a FE-model, consisting of one single chain link only. Their time-variant positions and acting forces are taken from a multi-body simulation. A focus is on ensuring the quality of the FE-model and the contact between pin and bush, where here a penalty approach has provided a stable solution schema. The parameters of the wear model are derived from measurements. After each time increment the wear results in a changed surface geometry, which is used for the simulation of the next time increment. The enormous computation time is reduced by the development of a convenient extrapolation factor. The complex simulation approach is applied to the chain drive of a test vehicle after a mileage of about 50000 km. The comparison of the simulated and measured data demonstrates an agreeing correlation.
1. Introduction Modern combustion engines lead to the trend of high performances at low emission levels, which is especially present in the premium vehicle class. This is resulting in high moments on the crankshaft and further in high forces in the chain links of the chain drive. For this case it is important to calculate the wear of chain drive systems to receive a lifetime forecast. With the information of the produced wear in the chain drive system it is possible to optimise the bush and the pin of all chain links and decrease their wear volumes for different applications. An analysis of the recent literature for topics of friction and wear leads to the result that specific wear models are necessary for certain applications. Since the wear process of a bush chain in an automotive chain drive system seems to be dominated by abrasive wear, the focus of this section is set to abrasive wear models. As a first result the paper of Myshkin and Goryacheva [1] shows the problem of approaching different scaling factors in a microscopic or macroscopic view for the description of wear in a certain contact area. Many wear models with respect to a generated wear volume 𝑉 are based on the equations of Holm-Archard [2,3] 𝑠𝐹𝑁 𝐻 and Fleischer [4,5,6] 𝑉 =𝑘
𝑉 =
𝑊𝑅 , 𝑒
(1)
(2)
where 𝑉 is the wear volume, 𝑘 is a constant, 𝑠 is the sliding distance, 𝐹𝑁 is the contact normal force, 𝐻 is the hardness of the softer material, 𝑊𝑅 is the work of friction, and 𝑒 is the energy density of the material. These models are usually modified for any special applications. There is a large variety of wear equations available in the literature, which depend on numerous parameters. Some examples for this are given in the following. Beckmann und P. Dierich [7] developed a conceptional approach like Fleischer to determine a wear equation. For applications with mainly stationary periodic sliding, Páczelt and Mróz [8,9,10] have developed a wear model, where Coulombs friction law holds. Sappok und B. Sauer [11] built up an experimental set-up and measured the wear of a timing chain. They derived a time dependent wear equation depending on the half run-in period and a constant that is used during stationary wear. Further Antusch [12] investigated a correlation between wear and friction in his experiments. These measurements are done for fresh oil as for soot contaminated diesel oil. With respect to the wear models shown above, Popov [13] also stated an important influence of the surface roughness on the wear process in a system. For the numerical simulation of wear processes in engineering applications the finite element (FE) method is ideally suited. Põdra and Andersson [14] as well as Mukras et al. [15] describe FE-simulations with an underlying wear model. In these simulations the nodes of the mesh are moved if wear occurs. Further, there is an extrapolation factor described which has to be determined for a certain application.
∗ Corresponding author. E-mail address: [email protected] (R. Tandler).
https://doi.org/10.1016/j.wear.2020.203193 Received 2 December 2019; Received in revised form 7 January 2020; Accepted 14 January 2020 Available online 16 January 2020 0043-1648/© 2020 Elsevier B.V. All rights reserved.
Wear 446–447 (2020) 203193
R. Tandler et al.
̃ Podra and Andersson also perform an experimental investigation with a tribometer to verify their simulations. Schmidt et al. [16] describe a FE-model to calculate dry sliding wear on a tilted shaft-bushing bearing. In this model a ’Node-to-Surface’-Penalty contact formulation and a sectional exponential contact are used for the simulation. The exponential contact is aided by a linear contact at contact opening. The evaluation of the different wear equations in the literature has shown that in general the wear process is described by some basic parameters such as the contact forces, the hardness of the materials in contact, the sliding distance and the friction. Additionally some further parameters are included which are experimentally determined to ensure the agreement of the wear equation with measurements. These wear equations are mathematical models describing how the wear depends on external and internal influences. Consequently, such wear equations do not have the general validity of a physical law, meaning that for the same real problem different models could be applied. These models differ in their field of application, the considered dependencies, the quality and the range of validity. But, a wear equation should be consistent with considerations of physics. In order to this the general wear model of Fleischer is used to calculate the wear of timing chains numerically. For this purpose the wear equation is embedded in a finite element model (FE-model). This numerical approach and its experimental validation are presented in detail in Section 2. Finally, in Section 3 the results are discussed and an outlook on further investigations is given.
Fig. 1. Example for a nodal displacement of the FE-mesh.
2. Numerical embedding in a FE-simulation In this section the energy based wear equation of Fleischer is used for a numerical discretisation, since this model is related to physical aspects. Eq. (2) is an abstract extension of Eq. (1), since the model of Fleischer contains the correlation of wear to the work of friction in any system. As shown in the introduction a valid model for the work of friction should be determined for a special application. For an automotive bush chain drive system the experimental investigations of Tandler et al. [17] lead to the result that the work of friction can be approximated by Coulombs law of friction. In order to this the integrated wear volume 𝑉𝑉 is given by 𝑉𝑉 =
𝑠1 𝑊𝑅 = 𝑘 ⋅ 𝐹𝑁 ⋅ 𝜇 ⋅ 𝑑𝑠. ∫𝑠0 𝑒
Fig. 2. MKS-model of the automotive chain drive system.
The simulation of wear is done by the calculation of 𝑉𝑉 in Eq. (5) for each node (𝑙) in the FE-model with )(𝑙) (𝐼−1 ∑ 𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))𝛥𝑠(𝑖𝛥𝑡 ) . (6) 𝑉𝑉(𝑙) = 𝑖=0
Due to a belonging area 𝐴(𝑙) for each node (𝑙), a wear depth 𝑑 (𝑙) can be calculated trivially with 𝑑 (𝑙) = 𝑉𝑉(𝑙) ∕𝐴(𝑙) and the node coordinates are manipulated in each simulation routine. This is shown in Fig. 1 as schematic representation of a mesh segment. After a certain nodal displacement a new mesh is generated automatically with PYTHON in the ABAQUS CAE post processing and used as basis for the next simulation step. With respect to this and the elasticity of the model a new load distribution occurs on the contact surface in every step of the wear simulation. Consequently, after each simulation step a new geometry is calculated which results in the following step in a changed distribution of nodal forces and a new wear volume. From this again a new surface geometry is calculated. It is not necessary to transfer any other field parameters from one to the next simulation step. The parameters of Eq. (6) are defined in the following section.
(3)
for the materials reciprocal energy density 𝑘 = 1𝑒 , the sliding distance 𝑠 from 𝑠0 to 𝑠1 , the contact force 𝐹𝑁 and the friction coefficient 𝜇. In consideration of all measured dependencies [17] and the time 𝑡 of the dynamical system it follows 𝑡1
𝑉𝑉 =
∫𝑡0
𝑘𝐹𝑁 (𝑛, 𝑡)𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑡))
𝑑𝑠(𝑡) 𝑑𝑡, 𝑑𝑡
(4)
where 𝑛 is the engine speed, 𝜃 is the oil temperature, 𝜂 is the kinematic viscosity and 𝜓 is the parameter for soot content in the engine oil. The wear model of Eq. (4) is implemented in a finite element model within ABAQUS CAE 6.14-4. For this a specific working operation of the engine is chosen and verified in the finite element model. The ∑ 𝛥𝑠(𝑖𝛥𝑡 ) 𝑡 discretisation 𝑡 → 𝑖𝛥𝑡 , 𝑑𝑠(𝑡) → 𝛥𝑠(𝑖𝛥𝑡 ), ∫𝑡 1 𝑑𝑠(𝑡) 𝑑𝑡 → 𝐼−1 𝑖𝛥𝑡 , with 𝑖=0 𝑑𝑡 𝑖𝛥𝑡 0 the increment 𝑖 ∈ {0, … , 𝐼 − 1}, the total number of increments 𝐼 for discrete time values 𝛥𝑡 and the incremental sliding distance 𝛥𝑠(𝑖𝛥𝑡 ) ∶= 𝑠((𝑖 + 1)𝛥𝑡 ) − 𝑠(𝑖𝛥𝑡 ), 𝑠(0) ∶= 0 leads to the discretised wear equation for each solid body as 𝑉𝑉 = =
𝐼−1 ∑ 𝑖=0 𝐼−1 ∑
𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))
2.1. Definition of model parameters In this investigation a timing chain with coated pins and extruded hardened bushes is used. The detailed specification is shown in Bauer [18]. The contact normal force 𝐹𝑁 (𝑛, 𝑖𝛥𝑡 ) in Eq. (6) is given for every increment 𝑖 at a certain engine speed and part load operation from the FE-simulation and the multi body simulation (MBS) in comparison with the engine characteristics. The value of the load for the chain drive lies between 550 N and 700 N. The MBS-model for a simulation of the full chain drive system is shown in Fig. 2. The parameters for the stiffness, the mass and the damping factor of the certain components
𝛥𝑠(𝑖𝛥𝑡 ) 𝑖𝛥𝑡 𝑖𝛥𝑡
𝑘𝐹𝑁 (𝑛, 𝑖𝛥𝑡 )𝜇(𝑛, 𝜂(𝜃), … , 𝜓(𝑖𝛥𝑡 ))𝛥𝑠(𝑖𝛥𝑡 ).
(5)
𝑖=0
2
Wear 446–447 (2020) 203193
R. Tandler et al.
are used from the BMW Group database and are embedded in the model. Every increment 𝑖 is related to one degree of shaft rotation of the engine and so defines a unique position of the chain link in the timing assembly. The assignment of a load 𝐹 (𝑛, 𝑖𝛥𝑡 ) and the contact sliding 𝑠(𝑖𝛥𝑡 ) on the chain link at two positions in the chain drive for the increments 𝑖 and 𝑖 + 𝑚 is schematically illustrated in Fig. 3. With this information the finite element model can be reduced to only one chain link and an amplitude information from the MBS. So for a specific position of the chain link the contact normal force at each node can be calculated in the finite element model with the implemented chain force. In the following a constant engine speed of 𝑛 = 5000 rpm, fresh engine oil 𝜓(𝑡) = 0 and a temperature 𝜃 = 100 ◦ C during engine operation are assumed. The corresponding coefficient of friction from the experimental investigations in Tandler et al. [17] is also implemented in the model. The factor 𝑘 gets a first approximation of 𝑘 = 1 ⋅ 10−10 mm2 /N from radionuclide wear measurements in a constant wear state with fresh oil. The assumption is, that this factor holds for bush and pin of a chain link to determine an extrapolation factor for the wear simulation. A description of the finite element model is given in the next chapter. 2.2. Description of the finite element model Due to the symmetry of the chain only one half of a bush chain link including symmetric boundary conditions is required. The developed finite element model is shown in Fig. 4. The contact area can be observed in more detail in Fig. 5.
Fig. 3. Assignment of the load and sliding distance on the chain link in a chain drive system.
Fig. 4. Symmetric finite element model of a bush chain link.
Fig. 5. Detail of the mesh in the contact area. 3
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 6. Scheme of a TIE-contact in ABAQUS CAE for two segmentation examples.
Fig. 7. Comparison between continuous meshed reference model and the TIE-model at the opened bush.
The angle 𝜑 of the chain link to its axis is given by the geometrical design of the chain drive system. During one chain rotation the chain link passes various contours with different curvatures like a chain wheel, which is resulting in the position dependent angle 𝜑. This engine specific angle and the chain force are implemented as amplitude in the dynamic ABAQUS CAE model. The symmetric model of one chain link with the nodal coordinate 𝑧 = 0 at the cutting plane is used to reduce the computational time of the FE-simulation. One chain link is enough for the simulation because all chain links of a bush chain statistically have the same load handling and sliding distance during the run time of the engine. A linear element type of C3D8R in ABAQUS CAE and a fine mesh in the contact region are applied to obtain sufficiently accurate values concerning the contact normal force and the contact sliding. This is required because the circular contour of the contact area has to be described numerically with a polygon. The smaller edges of the elements lead to a better approximation of a circle. Another
reason of the used fine mesh with element edge size ≈ 60 μm is that also topographically measured surface structures can be embedded into the model. A continuous meshed part with this element size would cause high computational times in the dynamic implicit simulation and consequently a TIE-contact is used at the bush and the pin to couple a coarse meshed part with a fine meshed part. The proper mesh density and the contact approach have been developed with help of several test simulations in order to find a compromise between computational time and a sufficient solution accuracy. To get a suitable computational time at a small relative error with respect to the full meshed model a TIE-coupling of 9 slave elements to 1 master element is important, as shown on the left side in Fig. 6. The TIE-coupling in ABAQUS CAE is defined as a slave node is bound to its nearest master node in the contact area. So it is important to have more nodes on the slave surface than on the master surface. Otherwise, many nodes are not bound by the TIE-coupling. Fig. 6 4
Wear 446–447 (2020) 203193
R. Tandler et al.
2.3. Determining an extrapolation factor for the wear simulation Since a common chain performs many millions of rotations during its lifetime it is important to determine an extrapolation factor 𝑒𝑥𝑝𝑓 for the simulation to get realistic computational times. To investigate this, high scaling factors for the wear volume are chosen at the beginning of this convergence study and gradually decreased until a reasonable converged numerical result occurs. The aim of this approach is to find a critical wear depth at the mesh nodes for each step of simulation such that numerical stability holds and the result is still physically reasonable. In order to Fig. 1 the schematic representation of the extrapolation factor is shown in Fig. 8, where the wear depth at each node gets scaled. In the following the wear process of a bush chain is analysed with the model parameters of chapter 3.1. The wear volume is calculated with an extrapolation factor and help of Eq. (5), which depends on the certain application and the different properties of the tribological system. If another chain or load case has to be simulated, the different properties of the tribological system have to be taken into account. In order to this the determined critical extrapolation factor leads to
Fig. 8. Schematic representation of the extrapolation factor.
illustrates that a segmentation of 3 slave elements to 1 master element per edge should be used to get a symmetric coupling of the slave nodes to the master nodes. A segmentation of 2 slave elements to 1 master element per edge would result in an asymmetric coupling, because the node in the middle of each edge will be coupled randomly, since the distances to the nearest master nodes are equal. The segmentation should be a small uneven natural number to get a minimal error with respect to a continuous meshed model. The investigation of the mesh quality is shown for the bush at a part load operation in Fig. 7. In Fig. 7 the pressure distribution at the surface of the bush is shown. At the picture on the left hand side the date are calculated with a continuous mesh without TIE-coupling. The results of the TIEcoupling are shown at the right hand picture of Fig. 7. The relative error of the peak pressure between both model is approximately 1,3%, the mean relative error over all circumferential nodes is approximately 2,8%. Due to the large reduction of the computing time in the TIE-coupling model this small deviations are acceptable. For the formulation of the sliding contact between pin and bush there are different methods available, such as the penalty method, the method of Lagrange multipliers and the method of augmented Lagrange multipliers. All methods have been investigated with respect to the pin and bush coupling. The method of Lagrange multipliers results in a large increase of the system of equations, since at each potential node of the very fine meshed contact area one additional degree of freedom – the unknown Lagrange multiplier – is introduced. This results in an huge increase of the computing time. The method of augmented Lagrange multipliers also increases the computing time because additional iterations are required to receive a converged solution. Finally, the penalty method has resulted in a sufficient accurate solution without an additional solution time. The constrained equations in the contact area can be written as ⃗ 𝑔(⃗ ⃗ 𝑢) ∶= 𝑍 𝑢⃗ − 𝑐⃗ = 0,
𝑒𝑥𝑝𝑓𝑛𝑒𝑤 = 𝑒𝑥𝑝𝑓𝑐𝑟𝑖𝑡 (𝑘, …)
(9)
with the critical wear volume 𝑉𝑐𝑟𝑖𝑡 corresponding to the critical extrapolation factor. Due to this a dynamical extrapolation factor is used to have a constant numerical quality at most efficient computational times. The value 𝑒𝑥𝑝𝑓𝑐𝑟𝑖𝑡 depends on the geometric model and its contact formulation, because both affect the numerical stability of the simulation. Convergence studies with different extrapolation factors for this application show that too high extrapolation factors lead to a reciprocally abrasion in the contact area. The reason for this is the simulated outlet contact pressure seen in Fig. 7, which is locally over scaled and causes to much wear depth at a certain position. An extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 still causes this effect slightly. To show this a chain link is simulated for the load case above and the wear surface evolving on the pin is calculated for different extrapolation factors at several steps of wear simulation. This means a wear depth at each node is calculated and in order to the nodal displacement a new mesh is generated after an extrapolated chain rotation. With this generated mesh the next step of simulation begins. The results of the simulation with an extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 are compared to those with the extrapolation factors 𝑒𝑥𝑝𝑓 = 200000, 𝑒𝑥𝑝𝑓 = 100000 and 𝑒𝑥𝑝𝑓 = 50000. The Fig. 9 illustrates these wear results after 600000 chain rotations. It can be observed that the extrapolation factor of 𝑒𝑥𝑝𝑓 = 300000 leads to a local over scaling of the wear depth at the contact edge. Since the contact surface is varying for the certain steps of a wear simulation, it is of interest to investigate the evolution of the wear surface for these extrapolation factors after another 600000 chain rotations. This is shown in Fig. 10. There is still a relative error between the extrapolation factors, but the tendency of deviation is decreasing. Further, Fig. 11 shows a comparison between the extrapolation factors after 1800000 chain rotations. The relative errors of the simulation results are shown in Table 1. With respect to this investigation and an acceptable computational time the critical extrapolation factor should be between 100000 and 200000, depending on the total number of wear routines. Since the relative error is decreasing for all cases with the duration of the wear analysis, the extrapolation factor of 200000 is chosen for further investigations. Regarding this, for the result of the maximal wear depth after 4000000 chain rotations a relative error of < 5% occurs between 𝑒𝑥𝑝𝑓 = 50000 and 𝑒𝑥𝑝𝑓 = 200000. The choice of 𝑒𝑥𝑝𝑓 = 200000 leads to the following contact pressure at the bush after several chain rotations as shown in Fig. 12. With this critical extrapolation factor a dynamical wear simulation with respect to the contact force, the friction coefficient and the relative contact sliding are performed.
(7) R𝑚×𝑛 ,
with the matrix of constraints 𝑍 ∈ 𝑚, 𝑛 ∈ N, the vector of all nodal translations 𝑢⃗ ∈ R𝑛 and a constant vector 𝑐⃗ ∈ R𝑚 . The penalty method, see Gabbert [19], results in an extension of the original system of the finite element stiffness equation as (𝐾 + 𝛼𝑍 𝑇 𝑍)⃗𝑢 = 𝑓⃗ + 𝛼𝑍 𝑇 𝑐⃗,
𝑉𝑐𝑟𝑖𝑡 , 𝑉𝑛𝑒𝑤
(8)
where 𝐾 ∈ R𝑛×𝑛 is the stiffness matrix of the system, 𝑓⃗ ∈ R𝑛 is the vector of external forces and 𝛼 ∈ R+ is the penalty factor. Here the 0 problem of a proper selection of the penalty factor 𝛼 arises. If the penalty factor is too small the constrained conditions are not sufficient accurate fulfilled. An increasing penalty factor results in an increasing loss of leading digits in the solution of the displacements 𝑢⃗, which is particularly critical if single precision real words are used during the calculation. A too large penalty number can finally result in a unsolvable singular system of equations. Several tests have demonstrated that in the given application a penalty number of 𝛼 ≈ 102 𝑘𝑖𝑖,𝑚𝑎𝑥 is a good compromise between stability of the solution process and the accuracy in the fulfilment of the constrained equations. Here 𝑘𝑖𝑖,𝑚𝑎𝑥 is the maximal entry of the main diagonal of the stiffness matrix 𝐾 in Eq. (8). 5
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 9. Comparison of the wear surface for different extrapolation factors after a total of 600000 chain rotations.
Fig. 10. Comparison of the wear surface for different extrapolation factors after a total of 1200000 chain rotations.
Fig. 11. Comparison of the wear surface for different extrapolation factors after a total of 1800000 chain rotations. 6
Wear 446–447 (2020) 203193
R. Tandler et al. Table 1 Comparison of different extrapolation factors after certain chain rotations. Comparison 𝑒𝑥𝑝𝑓
Rel. error 𝑉𝑚𝑎𝑥 after 600000 chain rotations
Rel. error 𝑉𝑚𝑎𝑥 after 1200000 chain rotations
Rel. error 𝑉𝑚𝑎𝑥 after 1800000 chain rotations
300000 vs. 200000 200000 vs. 100000 100000 vs. 50000
18.2% 11.8% 6.6%
13.8% 6.2% 3.2%
7.6% 4.0% 2.0%
With Fig. 14 it can be verified that a maximum wear depth occurs at the left and right end of the profile, because of an elastic deformation of the chain link during engine operation. The maximum delta of wear depth from the parts middle to an end of the polished area is approximately 6 μm. The results of the simulation described above after 20 routines of wear calculation with a critical extrapolation factor of 200000 at fresh oil is shown in Fig. 15. In Fig. 15 it can be observed that a nearly constant contact pressure occurs in the worn area within elastic effects due to the chain force. Further a polished area can be seen on the right side of Fig. 15. This coincides with the worn area of the pin after an engine run time of 48885 km as shown in Fig. 13. Another result is the correct tendency for this first approximated wear simulation. This can be seen in comparison of Fig. 15 with the result for the maximum wear depth of 5.028 μm to Fig. 14 where a delta wear depth of approximately 6 μm occurs at the measured pin. Finally, to show a comparison of the worn surface on the pin between simulation and real part the Fig. 16 should be taken into account. For this illustration the pin of the finite element model is symmetrised again. Fig. 12. Bushes contact pressure after several chain rotations.
3. Conclusions and further research With respect to Fleischer it is important to determine the work of friction to describe wear for any specific field of application. Based on extensive experimental investigations published in [17] the work of friction for the applied bush chains has been estimated. For the numerical analysis of the chain wear during operation it is appropriate to apply the finite element tool ABAQUS CAE. It is shown that the analysis of one chain link only is enough, since all chain links perform the same chain rotation during engine operation. The varying positions and chain forces of the chain link were estimated with help of multi body simulations and the results are then transferred to ABAQUS CAE. The quality of the FE-model is ensured by numerous convergence studies with respect to the mesh density and the contact formulation. Based on the convergence studies a critical extrapolation factor is estimated which leads to a maximum wear depth per wear simulation routine. With this critical extrapolation factor a dynamical wear simulation with respect to the contact force, the friction coefficient and the relative contact sliding is performed. The critical extrapolation factor also depends on the contact formulation with respect to the contacts geometry and the material specific wear factor k. The quality of the model is examined by a comparison of the simulation results with measurements of real engine chains of test vehicles, which demonstrates that the numerical wear simulation provides correct tendencies. In ongoing investigations the finite element model and the contact formulation are further improved to obtain lower computational costs and an increasing accuracy. Additionally the simulations are expanded to bush chains with winded bushes instead of extruded bushes, which are geometrically different at the outlet state. The objective is to apply the wear simulation approach to further chain types, such as oil pump chains and timing chains of certain engine types.
Fig. 13. Pin of a timing chain after 48885 km mileage of a test vehicle.
It can be observed that reasonable results occur, since the contact pressure is continuous distributed after multiple times of iterated wear simulation. Finally this should be verified in comparison to measurements of real engine parts in the next section. 2.4. Model comparison to measurements of chains from test vehicles The results of the simulation are compared to measurements of chain parts from test vehicles, where a similar load configuration is applied in the simulation. Fig. 13 illustrates a pin of an opened chain link after 48885 km mileage, where a polished area can be observed that depends on the intensity of wear processes in the system. The rough area around the worn area is still at output state, since there occurs no contact and also wear. The positive tensile load in the chain at all times is resulting in this wear surface. The simulation also confirms a contact opening on the unloaded surface of the pin and bush. It can be observed that the maximum wear depth must also occur at peak load in the simulation. The topographical profile of the pins worn area is measured with laser microscopy and is shown in Fig. 14.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 7
Wear 446–447 (2020) 203193
R. Tandler et al.
Fig. 14. Topographical measurement of the pins loading surface with laser microscopy.
Fig. 15. Pins contact pressure (left) and wear depth (right) after 20 steps of wear simulation with a critical extrapolation factor of 200000.
Fig. 16. Comparison of the worn surface on the pin between simulation and real part.
References
[9] I. Páczelt, Z. Mróz, Variational approach to the analysis of steady state thermoelastic wear regimes, Internat. J. Numer. Methods Engrg. 81 (2010) 728–760, http://dx.doi.org/10.1002/nme.2709.
[1] N.K. Myshkin, I.G. Goryacheva, Tribology: Trends in the half-century development, J. Frict. Wear 37 (6) (2016) 513–516. [2] J.F. Archard, Contact and rubbing of flat surfaces, J. Appl. Phys. 24 (1953) 981–988, http://dx.doi.org/10.1063/1.1721448. [3] R. Holm, Electric Contacts, Gerbers, Uppsala, 1946. [4] G. Fleischer, Energetische Methode der Bestimmung des Verschleißes [Energetic methods to determine the wear of a system], Schmierungstechnik 9 (4) (1973) 269–274. [5] G. Fleischer, Zur Energetik der Reibung [The energetics of friction], Wiss. Z. Tech. Hochsch. Otto von Guericke 34 (8) (1990). [6] G. Fleischer, Zum energetischen Niveau von Reibpaarungen [Energetic level of frictional pairings], in: Schmierungstechnik VEB, Vol. 12, Verlag Technik, Berlin, 1985. [7] G. Beckmann und P. Dierich, A representation of the microhardness distribution and its consequences for wear prognoses, Wear 107 (1986) 195–212, http: //dx.doi.org/10.1016/0043-1648(86)90225-5. [8] I. Páczelt, Z. Mróz, On the analysis of steady sliding wear process, Tribol. Int. 42 (2009) 275–283, http://dx.doi.org/10.1016/j.triboint.2008.06.007.
[10] I. Páczelt, Z. Mróz, Analysis of thermo-mechanical wear problems for reciprocal punch sliding, Adv. Eng. Softw. 80 (2015) 139–155, http://dx.doi.org/10.1016/ j.advengsoft.2014.09.012. [11] D. Sappok und B. Sauer, Wear measurement on chain joint components using a roundness instrument, Period. Polytech. Mech. Eng. 59 (2015) 51–59, http: //dx.doi.org/10.3311/PPme.7780. [12] S. Antusch, Untersuchungen zum Einfluss von Rußim Öl auf den Motorenverschleiß- Betrachtung mechano-chemischer Reaktionen [Investigation of the influence of soot content in engine oil on wear of engines] Dissertation, Karlsruher Institut für Technologie, 2008. [13] V.L. Popov, Contact Mechanics and Friction - Physical Principles and Applications, Springer-Verlag, GmbH Germany, 2017, p. 2010, http://dx.doi.org/10. 1007/978-3-662-53081-8. [14] P. Põdra, S. Andersson, Simulating sliding wear with finite element method, Tribol. Int. 32 (1999) 71–81, http://dx.doi.org/10.1016/S0301-679X(99)000122. 8
Wear 446–447 (2020) 203193
R. Tandler et al.
[18] P. Bauer, Kettensteuertriebe - Stand der Technik, Anwendungen, Entwicklung und Herstellung [Chain drive systems - State of the art, application, development and production], in: Die Bibliothek Der Technik Band, Vol. 353, Süddeutscher Verlag onpact GmbH, 2013. [19] U. Gabbert, Berücksichtigung von Zwangsbedingungen in der FEM mittels der Penalty-Funktion-Methode [Consideration of constrains in the FEM due to the penalty method], in: Technische Mechanik 4 Heft 2, 1983.
[15] S. Mukras, N.H. Kim, W. Gregory Sawyer, D.B. Jackson, L.W. Bergquist, Numerical integration schemes and parallel computation for wear prediction using finite element method, Wear 266 (2009) 822–831, http://dx.doi.org/10.1016/j.wear. 2008.12.016. [16] A.A. Schmidt, T. Schmidt, O. Grabherr, D. Bartel, Transient wear simulation based on three-dimensional finite element analysis for a dry running tilted shaftbushing bearing, Wear 408–409 (2018) 171–179, http://dx.doi.org/10.1016/j. wear.2018.05.008. [17] R. Tandler, N. Bohn, U. Gabbert, E. Woschke, Experimental investigations of the internal friction in automotive bush chain drive systems, Tribol. Int. 140 (2019) http://dx.doi.org/10.1016/j.triboint.2019.105871.
9