numerical study of the
fluid flow through
automotive
shock-absorber shims
bachelor in mechanical
engineering
Department of Mechanical Engineering
https://mecanica.uniandes.edu.co/
c
2014, May.
website:
http://jmauriciou.wix.com/jurbano
e-mail:
A structural, fluid-flow (CFD) and fluid-structure interaction (FSI) analysis was performed to an automotive hydraulic-damper. Both compression and extension settings were studied in An -sys. The structural analysis consisted on the simulation of the
displacement-pressure response of each shim-stack, consider-ing the contact between the disks. These displacement-curves were used to compute a three-dimensional fluid flow analysis.
The force-velocity predictions for the compression setting im-proved under low velocity regimes, in comparison to an axis-symmetric model with bonded shim-stacks. Furthermore, the
1-Way FSI analysis revealed three-dimensional deformation
pat-terns in the shim stack, which is why a fully coupled simulation is recommended.
R E S U M E N
Se realizó un analisis estructural, de dinámica de fluídos (CFD) y de interacción fluido-estructura (FSI) a un amortiguador hi-dráulico de automóvil. Tanto la carrera en extensión como la de compresión se simularon en Ansys. La simulación
estructu-ral consideró el contacto entre los discos. Las curvas de despla-zamiento presión se utilizaron en el análisis tridimensional de CFD.
Las predicciones de fuerza-velocidad para la configuración en compresión mejoran para velocidades bajas, en comparación con un modelo axis-simétrico con válvulas continuas. Adicio-nalmente, el analisis FSI 1-dirección reveló patrones
tridimen-sionales de deformación, por lo cual se recomienda una simu-lación completamente acoplada.
Introduction viii
1 nonlinear structural analysis 1 1.1 shim structural-modeling 2
1.1.1 Contact modeling 3
1.1.2 Shear deformations 7
1.1.3 Finite-element model 7
1.1.4 Extension-setting results 9
1.1.5 Compression-setting results 13 1.2 shim sack modal-analysis 17
1.2.1 Modal analysis 17
1.2.2 Extension-configuration results 18
1.2.3 Compression-configuration results 18 1.3 conclusions 19
2 three-dimensional fluid flow analysis 21 2.1 fluid flow simulation 22
2.1.1 Geometry generation 23
2.1.2 Mesh generation 25
2.1.3 CFD Setup 27
2.1.4 Simulation strategy, seeOñate[2013] 30 2.2 fluid flow results 32
2.2.1 Extension-setting results 32
2.2.2 Compression-setting results 36 2.3 conclusions 41
3 fluid-structure interaction analysis 44 3.1 fsi modeling 44
3.2 fsi results 45
3.2.1 Extension setting 45
3.2.2 Compression setting 46 3.3 conclusions 48
Conclusion 49
bibliography 51
Figure0.1 MacPhersonφ=48mm hydraulic-damper ix
Figure0.2 Piston assembly. ix
Figure0.3 Piston detail. Compression shim-stack,
piston, extension shim stack. x
Figure1.1 Shim assemblies and flow direction. 2 Figure1.2 Finite-element contact modeling. 5
Figure1.3 Boundary loads for the structural model. 8 Figure1.4 Mesh refinement analysis. Extension
set-ting. 9
Figure1.5 Mesh for the extension shim-stack. 9 Figure1.6 Mesh refinement analysis. Compression
setting. 10
Figure1.7 Mesh for the compression shim-stack. 10 Figure1.8 Displacement-Pressure curve for the
ex-tension setting. 11
Figure1.9 Displacement-Pressure curve for the
ex-tension setting. 11
Figure1.10 Extension setting. von Mises stresses in
the biggest disk. 12
Figure1.11 Extension setting. Contact status. 13 Figure1.12 Extension setting. Contact pressure. 14 Figure1.13 Displacement-Pressure curve for the
com-pression setting. 14
Figure1.14 Compression setting. vonMisesstress. 15 Figure1.15 Compression setting. Contact status. 16 Figure1.16 Compression setting. Contact status. 16 Figure1.17 Mode shapes of the extension shim-stack. 18 Figure1.18 Mode shapes of the compression
shim-stack. 19
Figure2.1 Flow passages in the extension setting. 22 Figure2.2 Flow passages in the compression setting. 23 Figure2.3 Fluid geometry in the extension setting. 23 Figure2.4 Fluid geometry in the compression
set-ting. 24
Figure2.5 Flow passages and initial gap in the
ex-tension setting. 24
Figure2.6 Flow passages and initial gap in the
com-pression setting. 25
Figure2.7 General overview of the extension-setting
mesh. 26
Figure2.8 Mesh refinement at the outlet of the flow
passages in the piston. 26
Figure2.9 Mesh refinement at the upstream and
down-stream of the piston. 27
Figure2.10 Boundary conditions for the extension
set-ting. 28
Figure2.11 Boundary conditions for the compression
setting. 29
Figure2.12 Domain selection for the force
computa-tion. 29
Figure2.13 FSI interface for both damper settings. 30 Figure2.14 Simulation strategy. 31
Figure2.15 Force-velocity curves. Extension setting. 32 Figure2.16 Averaged pressure in shim-stack vs.
ve-locity for the extension setting. 34 Figure2.17 Streamlines in the extension setting. 35 Figure2.18 Pressure in the extension setting. 37 Figure2.19 Velocity profiles in the flow passages.
Ex-tension setting. 38
Figure2.20 Force-velocity curves. Compression
set-ting. 38
Figure2.21 Averaged pressure in shim-stack vs.
ve-locity for the compression setting. 39 Figure2.22 Streamlines in the compression setting. 40 Figure2.23 Static pressure in the compression
set-ting. 42
Figure2.24 Velocity profiles in the flow passages.
Com-pression setting. 43
Figure3.1 Fluid pressure in the extension shim-stack
interface (lower disk). 46
Figure3.2 Displacement in the extension shim-stack
using FSI. 46
Figure3.3 Fluid pressure in the compression
shim-stack interface (lower disk). 47
Figure3.4 Displacement in the compression
Table2.1 Force [N] calculation for the extension
setting. Comparison withOñate[2013]. 33 Table2.2 Error [%] in the force calculation.
Exten-sion setting. 33
Table2.3 Force [N] calculation for the compression
setting. Comparison withOñate[2013]. 37 Table2.4 Error [%] in the force calculation.
Com-pression setting. 38
An automotive hydraulic-damper provides a velocity-dependent force. By controlling flow passages, the pressure drop may be higher or lower. The shim-stacks are an easy and simple mech-anism suited to control those fluid passages. The shim-stacks are based on the cumulative stiffness provided by an assembly of concentric shims. Thus, the design of the force-velocity curve is summarized in choosing the right amount and geometry of those disks.
This pathway has led to a highly empirical approach in damper design. Therefore, the main aim of this project is to contribute in the implementation of a full physically-based model of a damper. This model will give a better understanding of the fluid and structural mechanics inside the damper, and will pro-vide an robust procedure to simulate different shim-stack and piston geometries.
Previously, Oñate [2013] implemented an axis-symmetrical model, considering the structural and fluid mechanics as sep-arate physics. The present investigation refines the structural model of the shim-stacks, performs a three-dimensional anal-ysis of the fluid mechanics, and provides a first insight into model coupling.
geometry of the damper
The damper selected for this investigation is the MacPherson
φ=48mm hydraulic-damper, which is used in the automobile Hyunday Atos in the frontal wheels. The damper geometry
was provided by Gabriel at Colombia. The main view is
pre-sented in Figure0.1.
Figure 0.1:MacPherson φ = 48 mm hydraulic-damper. All mea-sures in mm [A,2005].
The moving part of the damper corresponds to the piston assembly, see Figure 0.2. This assembly consists of the piston-rod, shim-stacks, the piston and the nut.
Piston rod
Extension shim-stack
Compression shim-stack Piston
Nut
Figure 0.2:Piston assembly.
A half-section view of the piston assembly is shown in Fig-ure 0.3(the nut is excluded). This three-part assembly consti-tutes the most important geometry for the investigation. The geometry of the piston generates different flow-passages for the compression and the extension setting, and the shims work as fluid-valves, which open and close according to the flow di-rection.
Compr
ession
Extension
Figure 0.3:Piston detail. Compression shim-stack, piston, extension shim stack. All measures in mm.
scope and objectives of the project
This project will provide a solid understanding of the fluid and structural mechanics in a hydraulic damper. As it was stated earlier, the shim-stacks create a gradual restriction to the fluid-flow in each damper setting. Thus, since the structural behavior of the shim-stacks play such a fundamental role, the displacement-pressure relation of each shim-stack will be inves-tigated first.
As the damper operates in the extension and in the compres-sion stroke, a different fluid analysis is required for both set-tings. The simulation will be considered stationary, and a three dimensional geometry will be generated. As the piston-shim gap is not known a-priori, an iterative process will be made. The force-velocity curve will be calculated for each damper set-ting, and will be compared to the experimental data.
Finally, in a separate analysis, the fluid pressure will be di-rectly imported into the structural simulation. A comparison of the deformation patterns of each shim-stack will be made, first using a uniform pressure, and then by considering the fluid pressure.
Since the simulation is not fully coupled, a manual iteration is required in each operation point. Nonetheless, the actual model will be useful to analyze three-dimensional fluid flow and structural behavior.
Main objective
Perform a stationary numerical study of an automotive hydraulic-damper by performing structural, CFD and FSI numeric simu-lations.
Specific objectives
1. Identify the displacement-pressure curves for each shim
stack by simulating the structural mechanics.
2. Analyze the three dimensional fluid-flow in the damper
by using a CFD analysis.
3. Compare the deformation patterns in the shim-stacks
be-tween an uncoupled formulation and a1-way coupled
for-mulation.
structure of the project
This project will address each specific objective in a different chapter. The outline is as follows:
1. Chapter 1: The structural analysis of the shim-stack is discussed. The details of contact modeling and shear de-formation will be presented. The finite element formu-lation of the model will be shown and the displacement-pressure behavior for each shim-stack will be computed.
2. Chapter 2: The fluid-flow modeling is addressed. For each damper setting, the geometry generation is discussed, and the CFD general setup is presented. The force-velocity curves for each damper setting is calculated, and a de-tailed three-dimensional analysis to the pressure and ve-locity distributions will be provided.
3. Chapter 3: The pressure of the fluid-flow analysis is im-ported to the structural model. A comparison will be made between the independent results vs. the interaction results.
Each chapter begins with a rapid theoretical background of the simulation tools used. The simulation setup is then pre-sented in detail. The results are always divided according to the damper setting. Each chapter closes with a summary of re-sults. The end of the project will be made in Chapter3.3, where concrete conclusions of each objective will be made.
1
N O N L I N E A R S T R U C T U R A L
A N A LY S I S
The shim stacks are responsible for creating a controlled gap for the fluid flow, both in the extension and in the compression setting. This gap generates a velocity gradient, which is in turn responsible for a pressure difference between the fluid cham-bers. Thus, the structural behavior of the shim assemblies have a direct influence on the force-velocity curve of each damper.
Given the complexity of performing experimental investiga-tions on such small structures, most of the shim-structural be-havior has been addressed by using numerical simulations. In the present chapter, the nonlinear structural analysis of the shim assemblies will be treated in detail. This analysis includes a numerical investigation of the structural and modal response of the shim assemblies.
Following the same approach of Oñate [2013] in the struc-tural analysis, the displacement-pressure curves for each damper setting will be computed. These curves will be used in the fluid-flow modeling, which will be shown in the next chapter. In the same manner, the modal shapes and natural frequencies will be the expected result of the modal analysis. These structural properties would be of great usage in case of performing a tran-sient analysis.
1.1
shim structural-modeling
The shim assemblies consist of an arrangement of concentric, thin disks. Each disk may vary within its diameter and width. Its main purpose is to provide a controlled opening for the fluid flow, as a function of a given flow pressure.
There are two shim assemblies in a shim-valve damper. Each one provides resistance to the fluid flow in a different damper setting (extension or compression), see Figure1.1.
(a)Extension setting. (b)Compression setting.
Figure 1.1:Nut - extension shim - piston - compression shim assem-bly. The flow direction is shown with the blue arrows, and the working shim assembly is highlighted in yellow.
As the fluid flows in the extension setting, the compression shim closes certain passages in the piston, and the fluid impacts the extension shim. This impact generates a pressure under the extension shim, and it deforms. In the compression setting, the extension shim closes different passages, and the fluid flows through the piston and impacts with the compression shim, which deforms.
As an initial approach, Oñate [2013] considered the shim as-semblies continuous structures. This assumption led to a linear displacement-pressure relation. It is the objective of this project to further deepen in the shim structural-analysis and to include other physical constraints to the model.
As it has been previously investigated byKulkarniet al.[2013], Kulkarniet al.[2012],Czopet al.[2012], andSatputeet al.[2013], the contact between the shim disks plays a fundamental role in the shim structural behavior. For instance, if the interface
be-tween the disks is treated as a bonded contact, the displacement-pressure curve will behave different as if it would be assumed frictionless.
Furthermore, given the high slenderness ratio of the disks, it is also appropriate to consider shear deformations. For exam-ple, Farjoud et al. [2012] includes the First Order Shear Defor-mation Theory, to particularly consider cross-section rotations and disk sliding.
In this manner, the present structural analysis will include an interface simulation and will consider shear deformations. The detailed structural modeling of these two phenomena will be presented separately. Then, the results for each damper setting will be shown.
1.1.1 Contact modeling
The exact interface condition between the shim disks is un-known. However, certain approaches have been taken. For instance,Kulkarniet al.[2013],Czopet al.[2012] andFarjoudet al. [2012] consider frictionless sliding between the disks. They all agree also in using a linear material.
Due to the constant interaction of the disks with the fluid, it is likely that a thin film of oil is generated in the interface between the disks. Although this may aid in a sliding condition, friction sliding is not discarded.
Thus, the structural simulation will consider the contact be-tween the disks. Three scenarios will be simulated for each damper setting:
1. All interfaces bonded: This scenario simulates the shims as
a bonded shim-assembly. It is expected that the resulting displacement-pressure curve is very similar to that of a continuous structure.
2. All interfaces sliding: This scenario considers frictionless
and frictional sliding in the interfaces. CoulombFriction
This friction coefficient will be varied using the following values: µ = 0.00,0.15,0.30. µ = 0.00 simulates an ideal frictionless contact, µ = 0.15 is an approximate value of the friction coefficient steel-steel under lubricated condi-tions, and µ = 0.30 is an approximate value of the steel-steel friction coefficient under dry conditions.
3. Some interfaces bonded and other sliding: Given that some
interface contact-areas are greater than others, some in-terfaces may experience greater deformations and thus greater lubrication. Thus, it is likely that the contact con-dition is not the same in all interfaces. For this reason, in this scenario some interfaces will be treated as bonded and others as sliding. It is assumed that the disks located near the flow passages may experience a greater sliding than those located at the other end.
As it was mentioned earlier, the simulation will be performed in Ansys. There are many available options to simulate contact
in Ansys. The most relevant for the present study are bonded,
frictionless and frictional contact. There are some considera-tions which must be taken into account regarding the numer-ical solution scheme of the contact. The following guidelines were adapted fromANSYS[2010].
• Contact formulation: The physical contact occurs when two surfaces touch each other. They can not inter-penetrate, nor transmit tensile forces, only compression forces. It is considered that contact is a changing-status nonlinearity, which means that the stiffness of the system continuously depends on the contact status.
The finite-element formulation of contact involves the en-forcement of compatibility. This is a numerical relation-ship to prevent the surfaces from inter-penetrating each other. Using penalty-based methods, a small penetration
xpenetration is allowed and a resulting force is calculated
Fnormal, see Figure1.2. There are four main contact formu-lations used in Ansys. Each one will be discussed briefly.
1. Pure Penalty: The resulting force is obtained using
Equation (1.1).
Figure 1.2:Finite-element contact modeling. Some penetration is
al-lowed in penalty-based methods. Taken from ANSYS
[2010].
This means that for each contact, a corresponding contact stiffnessknormalwill be computed. The higher the stiffness is, the lower the penetration will be. This method usually requires few iterations, allows direct or iterative solvers, and symmetric and asymmetric contact. However, contact penetration is present and uncontrolled.
2. Augmented Lagrange: An additional term is added in
Equation (1.1), which makes the method less sensi-tive to contact stiffness, see Equation (1.2).
Fnormal =knormalxpenetration+λ (1.2)
The penetration is still present but controlled to some extent. One of the main disadvantages is that this method requires more iterations if penetration is too large. However, direct and iterative solvers may be used, as well as symmetric or asymmetric contacts.
3. Normal Lagrange: This formulation replaces the
con-tact stiffness with an additional degree of freedom. This means, that instead of resolving for the contact stiffness knormal, the method, solves for the contact pressureFnormal.
The main advantage consists in that nearly-zero pen-etration is observed, and that it does not require a normal contact stiffness. However, the method re-quires a direct solver and can only be used in asym-metric contacts.
4. MPC: The multi-point constraint formulation uses
additional equations to tie the displacements between the surfaces in contact. Thus, it may only be used in bonded contacts. This is not a penalty-based method, and can only be used in asymmetric contacts.
• Symmetric/Asymmetric behavior: Each contact in Ansys
in-volves a contact and a target surface. Using a symmetric contact, the contact surfaces are restricted from penetrat-ing the target surfaces and vice-versa. By using a asym-metric behavior, only the contact surfaces are allowed to penetrate thetargetsurfaces.
• Interface treatment: This parameter controls which is the initial state of the two surfaces. An offset may be consid-ered (Add offset), or the two surfaces may be configured as initially in contact (Adjust to touch).
• Contact stiffness: The contact stiffness knormal is the most important parameter affecting the convergence. It may be calculated automatically by Ansys. It is important to
update the contact stiffness in each iteration.
As it was addressed earlier, only bonded, frictional and fric-tionless contact will be considered in the shim structural-analysis. It must be reminded that the frictionless contact is simply a fric-tional contact whereµ=0.00. In this manner, two formulations will be chosen, one for the bonded contact and other for the fric-tional contact.
As penetration in any extent is undesirable, the Normal La-grange and MPC methods were considered initially for the bonded contact. Both restrict the interface to only asymmet-ric contacts. However, the MPC formulation converges easily, and it will be used in the bonded-contact modeling.
Given that an uncontrollable penetration is undesirable, the Pure-Penalty Formulation was discarded. The Augmented La-grange Formulation offers a greater numeric accuracy than the Normal Lagrange Formulation because it allows symmetric and asymmetric contacts, and allows contact detection in integra-tion points. This is why it was considered for the fricintegra-tional- frictional-contact modeling.
Whenever possible, the interface treatment will be configured to initial contact (Adjust to touch) and a symmetric contact will be preferred. To assist the accuracy of the contact modeling, the initial gap, penetration and frictional stress will be controlled after the simulation.
1.1.2 Shear deformations
The shim disks are considerably thin in comparison to their di-ameter. Given this high slenderness ratio, it is advisable to con-sider shear deformations. As Farjoud et al. [2012] mentioned, the classical plate theory does not account for the sliding be-tween the disks, thus it is advisable to work with the First-Order Shear Deformation Theory.
This theory is known also as the Mindlin-Reissner Plate
Theory, and is already implemented in Ansys using the
ele-ment Shell-181. Thus, the shim disks will be modeled using
shell elements.
1.1.3 Finite-element model
As it was stated previously, the material will be assumed linear elastic with an elastic modulus ofE=200GPa and a Poisson’s
Ratio of ν = 0.30. The boundary conditions of the model are defined by the loads and supports. Each shim assembly is fix supported in the interior boundary and in the exterior disk (the one which has not a direct contact in the fluid passages). The fluid load will be assumed as an uniform pressure, applied to the disk which is in direct contact to the fluid passages. The boundary conditions for the two damper settings are shown in Figure1.3.
When working with shell elements, it is advisable to create a mesh predominantly with quad elements of uniform size [Wang, 2006]. This was taken into account in the mesh gen-eration in Ansys. Once the geometry, material and boundary
conditions were defined, a mesh sensitivity analysis was per-formed.
(a)Extension setting. (b)Compression setting.
Figure 1.3:Boundary loads for the structural model. Fixed support in the interior and upper boundary, and an uniform pres-sure in the inferior disk.
The selected control-variable was the maximum displacement in the structure after an application of a uniform pressure of 1
MPa. As it is not the objective to analyze the interaction be-tween disks, the contact was assumed to be bonded. The ele-ment size was varied and the changes in maximum displace-ment were studied. A relative error was defined for the control variable in the following manner (see Equation (1.3)):
Relative error = di−di−1
di ×100% (1.3)
Thus, the relative error quantifies how the displacement varies as a percentage of the next iteration. Therefore, a low relative-error in thei−th iteration implies that the refinement from the
i−1−th mesh to the i−mesh is not necessary. The convergence plot for the extension setting is shown in Figure1.4.
The mesh convergence in the extension setting reveals that by using an minimum element-size of0.15 mm, the relative error
reaches a value of 3.5%. With this element size, the mesh has 35 000elements (Half of the structure is considered, given the
symmetry). The mesh for the extension setting is presented in Figure1.5.
The mesh convergence-study for the compression setting is shown in Figure 1.6. In this case, a relative error of 0.1 % is
obtained using a0.2mm element. With this consideration, the
0.0 10000.0 20000.0 30000.0 40000.0 50000.0 60000.0 70000.0 Numer of elements
0.00
20.00
40.00
60.00
80.00
100.00
Relative error [%]
Figure 1.4:Mesh refinement analysis for the extension setting. The relative error of the maximum displacement is presented as a function of the element size.
Figure 1.5:Mesh for the extension shim-stack. Number of elements
35000. Only half of the structure is considered given the symmetry.
1.1.4 Extension-setting results
The extension structural analysis was performed first with shell elements and then with solid elements. The first analysis is presented in Figure1.8.
As can be seen, by using shell elements, and by considering a frictional or frictionless contact between the disks, the observed maximum displacement in the disks is much more greater than the results of Oñate [2013]. It is remarkable that there is prac-tically no difference between the frictional and frictionless con-tacts, if the same condition is applied to all interfaces.
0.0 10000.0 20000.0 30000.0 40000.0 50000.0 Numer of elements
0.00
10.00
20.00
30.00
40.00
50.00
60.00
Relative error [%]
Figure 1.6:Mesh refinement analysis for the compression setting. The relative error of the maximum displacement is pre-sented as a function of the element size.
Figure 1.7:Mesh for the compression shim-stack. Number of ele-ments 10000. Only half of the structure is considered given the symmetry.
However, the observed displacement using shell elements is significantly larger than the bonded, continuous model. As it was verified later in the CFD analysis, this large deformation behavior does not lead to a accurate calculation of the damping force.
It was taken as a fundamental condition for the validity of the structural model, that, weather using solid or shell elements, the expected maximum displacement when considering every contact as bonded should correspond exactly to the continuous model ofOñate[2013]. This was not fulfilled by the shell model,
0.0 1.0 2.0 3.0 Pressure [MPa] 0.000 0.100 0.200 0.300 0.400 Displacement [mm]
mu = 0.30 mu = 0.15 mu = 0.00 Oñate (2013)
Figure 1.8:Displacement-Pressure curve for the extension setting. Bonded, frictionless and frictional contacts compared. All curves are computed using shell elements (Excepting the results ofOñate[2013]).
as it can be seen in Figure1.9 (See curve 6B, which represents
the structural behavior of a six time bonded shell shim stack).
0.0 1.0 2.0 3.0
Pressure [MPa] 0.000 0.100 0.200 0.300 0.400 Displacement [mm] 6B 6F-less 5B+1F-less 4B+2F-less 3B+3F-less Oñate (2013)
Figure 1.9:Displacement-Pressure curve for the extension setting. Bonded and frictionless configurations compared. Ex-cepting the 6 bonded contact-configuration (6B) and the 6-time frictionless configuration (6 F-less), all curves are computed using solid elements.
It is seen in Figure 1.9 that the shell model with six bonded connections represents a very stiff behavior, whose maximum displacement is much more inferior from the results of Oñate [2013]. It was then decided to compute the structural
simula-tion using solid elements, and to configure the shim-stack with different contact conditions in the disks.
In this manner, the smaller disks located at the upper part of the shim-stack were assumed to be bonded. This assump-tion is reasonable since the contact area is inferior, and they are restricted by the nut on one side. On the contrary, the bigger disks were assumed to have a frictionless contact. This condi-tion may work considering that these disks are larger in contact area, and are closer to the fluid, which may aid in lubrication.
Thus, a 5-bonded 1-frictionless configuration was simulated
(5B+1F-less) using solid elements. A very near behavior from
the results of Oñate [2013] is obtained. Next, a 4-bonded and 2-frictionless configuration was simulated, and a slightly less
rigid model was obtained. Finally, a3-bonded and3-frictionless
was computed, resulting in a very flexible shim-stack. Thus, it was decided to work with the4-bonded and2-frictionless solid
configuration.
Since the biggest disk is closest to the fluid, and given that it contains the highest displacement, some additional mechanical parameters where further investigated. Thevon-Misesstresses
are shown in Figure 1.10. The stress distribution in the disk
Figure 1.10:Extension setting. vonMisesstresses in the biggest disk. The maximum value reaches approximately100MPa.
as a circumference, exactly where the next disk is located. In this zone, the average value reaches approximately 100 MPa,
which results in a factor of safety near 4.0. The contact status
was also investigated, see Figure1.11. The entire disk is sliding
Figure 1.11:Extension setting. Contact status. The entire area of the biggest disk slides relatively to the next disk.
relatively to the next disk. This is a expected result, since this disk presents the highest displacement, and the connection is frictionless. Moreover, due to the frictionless interface, there are no frictional stresses. Finally, the contact pressure is shown in Figure2.18. The contact pressure reaches a maximum value of 2 MPa which does not affect the structural integrity of the
disk. Again, the zone of highest contact pressure is located at the circumferential contact between the biggest disk and the next disk.
1.1.5 Compression-setting results
The compression shim-stack was simulated using shell elements. The frictionless and frictional connection-results are shown in Figure1.13.
Figure 1.12: Extension setting. Contact pressure.
0.0 1.0 2.0 3.0
Pressure [MPa] 0.000
0.200 0.400 0.600 0.800 1.000 1.200 1.400
Displacement [mm]
mu = 0.50 mu = 0.25 mu = 0.00 Oñate (2013)
Figure 1.13:Displacement-Pressure curve for the compression set-ting.
A nonlinear behavior is shown in the compression shim-stack. Although the shim-stack is significantly less rigid than the ex-tension shim-stack, the results using shell elements do not dif-fer greatly from the calculations ofOñate[2013].
Furthermore, as the simulation in the extension setting re-vealed, there is no significant difference between the friction-less and frictional connections. Thus, for numeric simplicity, the configuration using frictionless contacts and shell elements was considered for the continuation of the investigation.
The von Mises stress is shown in Figure 1.14. The circum-ferential zone with the highest stresses is again located just un-der the next disk. In the compression setting, the von Mises
stresses reach a value near550MPa. These high values indicate
that there may be some plastic strains affecting the structure. The contact status is presented in Figure 1.15. As the
simula-Figure 1.14:Compression setting. vonMisesstress. The highest val-ues reach approximately550MPa, which may generate plastic strains in the structure.
tion reveals, there is a sliding condition just below the area of the adjacent disk. The contact pressure is shown in Figure2.23. The maximum value reached is approximately 40 MPa, and is
Figure 1.15:Compression setting. Contact status.
1.2
shim sack modal-analysis
As a complement to the structural analysis, the shim modal-analysis will provide an initial insight to a dynamic modal-analysis of the structure. The mode shapes obtained from this study will be valuable when considering further dynamical problems, such as self-induced vibrations (flutter). The present section deals with the simulation of the modal analysis, and presents the mode shapes and frequencies for each shim stack.
1.2.1 Modal analysis
The modal analysis provides a dynamical characterization of a structure. At it simplest form, it provides a solution to the eigenvalue problem given an undamped free vibration. Con-sidering a multi-degree of freedom structure, the equations of motion may be written as shown in Equation (1.4).
[M]U¨ + [C]U˙ + [K]U=F(t) (1.4)
Neglecting structural damping, and assuming a oscillating mo-tion of the formU =φsinωtand ¨U = −ω2φsinωt, the equa-tions of motion reduce to an eigenvalue problem, see Equation (1.5):
[K]φ−ω2[M]φ=0 (1.5)
For a system withn−degrees of freedom, the solution vector φ
containsn− mode shapes, which are associated with n− natu-ral frequencies.
The modal analysis will also be performed in Ansys. The
model requires the definition of the structure supports. These are the same used in the structural analysis,i.e. a fixed support in the interior boundary and in the external disk.
The result of the analysis consists of the mode shapes and natural frequencies. The mode shapes are a representation of the expected deformation under its corresponding natural fre-quency. It must be clearly stated that this type of analysis does not provide information about the amount of displacement erated in the structure, but only a qualitative picture of the gen-eral deformation pattern.
1.2.2 Extension-configuration results
The modal-analysis results for the extension shim-stack are shown in Figure1.17. The three main mode shapes are presented. The first two mode shapes are very similar: both reveal a major part of the assembly unaltered, and only a small sector is deformed downwards and upwards, respectively. The third mode differs in that there are two sectors which simultaneously deform, one upwards and the other downwards. The natural frequencies are very high. Although the assembly is highly deformable, the high natural frequencies may occur given the low mass of the disks. Furthermore, these frequencies enter the human hearing-range. This may suggest that the self-induced vibrations may also generate a disturbing human-audible noise.
(a)Mode1:15329Hz. (b)Mode2: 15548Hz.
(c)Mode3:15599Hz.
Figure 1.17:Mode shapes of the extension shim-stack.
1.2.3 Compression-configuration results
The modal-analysis results for the compression shim-stack are shown in Figure1.18. In this case, the three mode shapes differ from each other. Also, the natural frequencies are lower than the extension shim-stack. This fact is directly associated with a higher mass, in comparison to the extension shim-stack.
(a)Mode1:7814Hz. (b) Mode2:7861Hz.
(c)Mode3:7884Hz.
Figure 1.18: Mode shapes of the compression shim-stack.
1.3
conclusions
Instead of a continuous model of the shim-stack, a disk-assembly was simulated for both damper settings. Special interest was given to the contact interaction between the disks, and the type of element used in the simulation.
The extension setting had a very low stiffness if simulated with shell elements. Even with a bonded contact between the disks, the displacement-pressure curve was much more higher than the continuous model. Thus, solid elements were con-sidered. Several combination of bonded and frictionless disks were proved in the CFD simulations. The model with the four smaller disks bonded and the two bigger disks frictionless showed the best results, and was chosen.
The compression setting is less stiff in comparison to the ex-tension setting. The displacement-pressure curve using shell el-ements depicts a nonlinear behavior, whose displacel-ements are in the same order of magnitude of the continuous model. There is no significant difference in using a frictionless and frictional contact-formulation. Since the frictionless contact is less de-manding to solve, a shell model with frictionless contacts was chosen.
The stresses in the compression assembly are significantly higher than in the extension setting. This higher loading may lead to a rapid failure. The displacement-pressure curves are the most significant results of the structural analysis, and will be fundamental in the fluid-flow simulation.
2
T H R E E - D I M E N S I O N A L
F L U I D F L O W A N A LY S I S
The fluid flow through the shim stacks and piston passages cre-ates a pressure difference in the damper, which makes it pos-sible to generate a velocity-dependent force and to dissipate vibrations in the vehicle. The nature of the generated force depends on two main factors: the resistance to the fluid flow provided by each shim stack; and the pressure gradient gen-erated throughout the fluid-passages. The former factor was addressed in the previous chapter, and the latter phenomena will be treated in detail in the present chapter.
In this manner, the fluid flow analysis ultimately summarizes in computing the force-velocity curve for each damper setting. Previously, Oñate [2013] performed an axis-symmetric fluid flow analysis, where the force-velocity curves agreed within an acceptable range. Oñate [2013] also computed pressure-gradient and velocity-pressure-gradient curves, to further understand how the force developed in the damper. One of the main objec-tives of the present investigation is to take advantage of a full three-dimensional modeling, and to improve the damper-force prediction.
This chapter is organized as follows. The first section presents the fluid-flow simulation in detail. The geometry generation will be discussed, as well as the CFD simulation setup. Next, the results for the two damper settings will be shown in the second section. Finally, the results will be summarized in the third section.
2.1
fluid flow simulation
The fluid flow occurs in opposite manners in each damper set-ting. In the extension setting, the fluid flows parallel to the pis-ton rod. It flows through the compression shim-stack, enters the piston passages, and then the fluid impacts the extension shim-stack, which deforms. The flow passages in the extension setting are shown in Figure 2.1. In the extension setting, the
Figure 2.1:Flow passages in the extension setting.
compression shim-stack is pressed against the piston, which closes some fluid-passages at the interior of the piston. Given that the compression shim-stack is supported, the pressure ex-erted by the fluid causes no major deformation in it. Thus, it is reasonable to treat it as a rigid body.
The compression-setting flow is analogous. The flow passes first around the nut, and then it presses the extension shim-stack against the piston. At this moment, the fluid-passages used in the extension-setting are closed by the extension shim-stack. Thus, the flow enters into another passages, from where the fluid will impact the compression shim-stack. Thereafter, the flow returns parallel to the piston rod, and flows into the main chamber. This process is illustrated in Figure 2.2. In this manner, the fluid performs two different trajectories. This leads to the simulation of the two damper settings indepen-dently. It must be stated that this flow description follows from a moving inertial-frame located at the piston. This assump-tion greatly simplifies the simulaassump-tion, since no re-meshing nor moving-meshes are required. Although the flow geometry dif-fers in the two damper settings, the boundary conditions and CFD-setup remains the same for the two settings.
Figure 2.2: Flow passages in the compression setting.
2.1.1 Geometry generation
As a special consideration in the geometry generation, it must be taken into account that the fluid domain must be long enough to capture the recirculation zones downstream in both damper settings.
As the simulation is defined as a steady state analysis, the fluid completely fills the cylinder. Furthermore, the working shim-stack is assumed to be initially deformed. This allows a continuous fluid geometry across the working shim-stack. The fluid geometry for the extension setting is shown in Figure2.3 and for the compression setting in Figure2.4.
(a)Lateral view.
(b)Isometric view. (c)Detail.
(a)Lateral view.
(b)Isometric view. (c)Detail.
Figure 2.4: Fluid geometry in the compression setting.
To generate a continuous fluid geometry for both damper settings,the definition of an initial shim-displacement is first required. It must be taken into account that this gap will have a direct incidence in the damping force. This force will increase quadratically if the shim gap is reduced. This initial gap is illustrated in Figure2.5for the extension setting and in Figure 2.6for the compression setting.
(a)Flow passages in the extension shim-stack.
(b) Detail of the initial gap in the extension setting.
Figure 2.5:Flow passages and initial gap in the extension setting.
Given that it is not possible to compute the shim displace-ment exclusively as an explicit output of the CFD analysis, it will be calculated iteratively. Oñate [2013] proposed the
fol-(a)Flow passages in the compres-sion shim-stack.
(b) Detail of the initial gap in the compression setting.
Figure 2.6:Flow passages and initial gap in the compression setting.
lowing procedure. First, an initial shim displacement will be assumed, then the simulation will be performed and the sure over the working shim-stack will be computed. This pres-sure will be used as an input parameter in the displacement-pressure curve (computed for each damper setting in the struc-tural analysis).
In this manner, using the method of Oñate [2013], the as-sumed initial displacement will be compared against the struc-tural displacement. For each operation point, an iteration must be performed until the initial displacement and the structural displacement coincide within a good range. This simulation strategy will be further illustrated in the coming sections.
2.1.2 Mesh generation
The mesh generation follows similarly in both damper settings, see Figure2.7. Two key factors are relevant: a clear representa-tion of the pressure drop at the exit if the fluid passage, and a clear identification of the recirculation zones downstream.
For the first condition, the element size of the fluid-structure interface (shim which is in contact to the piston passages) was
Figure 2.7:General overview of the extension-setting mesh. 1.5·106 quadriedral elements.
controlled. The element in this boundary was restricted to an element size of1·10−5 m and a maximum growth rate of 1.20. As a result, the refinement of this region is depicted in Figure 2.8.
Figure 2.8:Mesh refinement at the outlet of the flow passages in the piston. Element size at the fluid-structure interface: 1· 10−5 m; Maximum element growth-rate:1.20.
Finally, the mesh was refined at the upstream and down-stream of the geometry. The mesh was generated using the in-flation tool of Ansys. In these regions, the mesh is also mapped
around the circumference of the cylinder. A layer growth-rate of 1.25 was selected, and the maximum number of layers was set to10, see Figure2.9.
The layer-refinement was configured at the outer cylinder and at the piston-rod cylinder. The final mesh consisted of
Figure 2.9:Mesh refinement at the upstream and downstream of the piston. Inflation tool used. Layer growth rate: 1.25; Maxi-mum number of layers: 10.
2.1.3 CFD Setup
The fluid-flow analysis is performed in Ansys- CFX. The
work-ing fluid is a Polar Oil, and was rheologicaly tested by Oñate [2013]. The flow is considered laminar throughout the entire damper. However, in the flow passages located in the piston, the flow reaches a slightly transition regime.
Considering the oil density ρ=870kg/m3 and the dynamic viscosity to be ν = 0.020Pa·s [Oñate, 2013]. The minimum passage diameter is0.15cm, and the approximate velocity is50
m/s. The ReynoldsNumber may be calculated as follows:
Re = ρvD
ν
= 870kg/m 3×
50 m/s×0.15cm
0.020Pa·s
= 3260 (2.1)
The Reynolds Number is slightly larger than 3000. This
in-dicates that the flow in the piston passage may be considered as transitional-turbulent. However, this condition is at most
present in the piston passage and in the shim-stack initial gap. The rest of the fluid geometry experiences velocities lower than
10 m/s, which is a laminar flow. Moreover, it will be proved
that the force component coming from the shear-stress is less than 5%. For the aforementioned reasons, the flow is
consid-ered fully laminar.
Each operation point of the damper is defined by an inlet velocity. Thus, the inlet boundary-condition was defined as a normal speed vin. The outlet condition was defined as having
an average static pressure of 0.00 Pa (averaged over the whole
outlet). As it was mentioned in the geometry section, halve of the fluid is considered, since there exists a symmetry. This symmetry is also defined as a boundary condition. All other boundaries are configured as no-slip walls. The boundary con-ditions for the extension setting are shown in Figure2.10, and for the compression setting in the Figure2.11. Two additional
Figure 2.10: Boundary contidions for the extension setting.
boundaries are defined. The first is the boundary required for the force calculation. The force is computed as the area sum-mation of the pressure and shear stress, see Equation (2.2).
F=
Z
Area
PdA+
Z
Area
τdA (2.2)
The area of integration corresponds to every wall the fluid touches in the moving assembly. That is, the walls of the piston rod, both shim stacks, the piston and the nut. To investigate the developing of the force separately in the piston, three com-ponents are defined:
Figure 2.11: Boundary contidions for the compression setting.
1. Force1: Force developed in the piston rod and in the
com-pression shim-stack (see Figure 2.12a).
2. Force2: Force developed in the piston and in the extension
shim-stack (see Figure 2.12b).
3. Force3: Force developed in the nut (see Figure 2.12c).
(a)Domain selection forForce1. (b)Domain selection forForce2.
(c)Domain selection forForce3.
Figure 2.12:Domain selection for the force computation.
The second boundary corresponds to the fluid-structure in-terface to the working shim-stack, that is, the low boundary
of the inferior disk of the working shim-stack. This boundary will define the area from which the pressure of the fluid will be imported to the structure. This boundary is shown in Fig-ure 2.13a for the extension setting and in Figure 2.13b for the compression setting.
(a)Fluid-structure interface for the extension setting.
(b)Fluid-structure interface for the compression setting.
Figure 2.13: FSI interface for both damper settings.
The fluid was assumed as incompressible, isothermal and laminar. Hence, the solver is required only to solve the continu-ity and the momentum equations. For both equations, a RMS residual was requested to be lower than1e-5. The force and the
pressure at the FSI interface where monitored during the entire simulation.
2.1.4 Simulation strategy, see Oñate [2013]
As the initial gap in each damper configuration depends on both structural and fluid-dynamics physics, it can not be explic-itly solved. This is why for each operation point, an iteration must be performed. The simulation strategy is shown in Figure 2.14.
Each operation point is defined as an input velocity Vin.
Fur-thermore, an initial gap din is assumed. With the initial gap, the fluid geometry is constructed, and the CFD simulation is computed. As an output, a fluid pressure over the FSI interface and the fluid force will be calculated. The fluid pressure will be area-averaged and used in the corresponding d−P curve to determine the associated shim displacement dout. This
dis-placement will be compared against the initial gap. If both coincide within an error < 5%, the simulation will have
con-CFD
d - P Curve
FSI
V
ind
ind
outd
fsiP
F
Error d
in< 5%
No
Yes
F=F
out
Figure 2.14: Simulation strategy.
verged. Otherwise, a new input gap is set in the geometry, and the CFD analysis is again performed.
Once the simulation has converged, the fluid force will corre-spond to the operation-point force. Likewise, the fluid pressure over the FSI-interface will be used as an input parameter to a new structural simulation. This procedure will be explained in the next chapter.
Its worth mentioning that the fluid force is very sensitive to the initial gap. The dependence of a force increment in relation to a gap decrease is quadratical. In the extension setting, the gap was computed with an accuracy of 0.001 mm, and in the
compression setting with an accuracy of0.005mm.
For each operation point, approximately four iterations were performed. Each one took ca. 10 minutes to converge. The
simulation was computed in a computer cluster, however, the simulation times were almost the same in a Windows 7 Intel
Core i5-2400 CPU computer, with 3.10 GHz and 8.00 GB of
2.2
fluid flow results
The fluid-flow results will be analyzed separately for each damper setting. As it was explained in the previous section, each op-eration point requires an itop-eration over the initial gap. Thus, only the simulations where the initial gap and the correspond-ing maximum shim displacement coincide within an error less than5% will be considered for a detailed fluid flow analysis.
First, the the force-velocity curves will be analyzed. Then, the operation point which is closest to 1.00 m/s will be taken
into account for each damper setting. A corresponding fluid analysis will be performed exclusively in the selected operation-point. This analysis discusses the velocity and pressure distri-bution in the damper.
2.2.1 Extension-setting results
Force calculation
The force-velocity curves are shown in Figure2.15. The numer-ical data is shown in Table 2.1. The error in comparison with the experimental data is shown both in the operation curve and in Table2.2.
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s] 0 400 800 1200 1600 2000 Force [N] Gabriel (2013) Oñate (2013) Urbano (2014)
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s] 0.00 10.00 20.00 30.00 40.00 50.00
Error [%] Oñate (2013)Urbano (2014)
Table 2.1: Force [N] calculation for the extension setting. Compari-son withOñate[2013].
Velocity [m/s] Gabriel (2013) Oñate[2013] Urbano (2014)
0.300 849 1200 527
0.501 1089 1464 815
0.644 1275 1550 1070 0.889 1652 1830 1550 0.983 1788 1923 1770 1.134 2017 1998 2122
Table 2.2: Error [%] in the force calculation. Extension setting.
Velocity [m/s] Oñate [2013] Urbano (2014)
0.300 41.34 37.93 0.501 34.44 25.16 0.644 21.57 16.08 0.889 10.77 6.17 0.983 7.55 1.01 1.134 0.94 5.21
The present simulation differs from the results ofOñate[2013] in that a new structural model is considered, and in that fluid geometry is three dimensional. In the extension setting the re-sults ofOñate [2013] overestimate the fluid force for low veloc-ities. In the present model the force is underestimated for low velocities. Nonetheless, the relative error does not differ greatly between both models. The error is, excepting for the last opera-tion point, always greater than10%, and also decreases almost
linearly.
The main assumption for the error consists in the structural description of the shim-stack. The pressure generated at the inferior part of the extension shim-stack is shown in Figure2.16
The fluid pressure reaches2MPa in the extension shim-stack.
Using the extension displacement-pressure curve (see Figure 1.9), the expected displacement is in the order of0.60mm.
If the model were more rigid, the expected displacement would be inferior, and the fluid force would be greater. How-ever, the force near 1.00 m/s is relatively good described by
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s] 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 Pressure [Mpa]
Figure 2.16:Averaged pressure in shim-stack vs. velocity for the ex-tension setting.
structural model, a new highly-nonlinear formulation would be required.
The stiffness of the structure should be greater under low pressures, and should converge to the present-model stiffness at a pressure near2.00MPa.
Flow analysis
For the flow analysis in the extension setting, the last opera-tion point was chosen. An inlet velocity of Vin = 1.134 m/s was defined, and the converged shim displacement/shim gap is din = 0.068 mm. The streamlines are presented in Figure 2.17.A very remarkable velocity increase occurs in the extension flow passages. At the inlet, the velocity reaches approximately
1 m/s, but in the contraction, the velocity rises to ca 50 m/s
(see for example Figure 2.17b). Downstream the valve, a large recirculation zone is observed. The recirculation patterns are three-dimensional, see Figure2.17c.
In the simulation, a large downstream region was generated. This supported the assumption of a constant-pressure bound-ary condition. However, in the working condition of the damper, the downstream region gets shorter as the piston rod moves. Thus, at full extension, the downstream region is much more shorter than at the beginning of the extension stroke. Given the
(a)Lateral view.
(b) Flow passages in the piston. (c)Recirculation zones down-stream.
large recirculation zone, the zero-pressure boundary condition is thus only acceptable at the beginning of the extension stroke.
The pressure distribution is shown in Figure 2.18. A clear pressure difference is observed between the flow before and af-ter the piston. This gradient enlightens the working principle of the hydraulic damper. The difference reaches approximately
6 MPa. This pressure gradient occurs mainly in two
contrac-tions: the piston passages and the shim-piston gap (see Figure 2.18b).
As the fluid flows through the piston passages, a first con-traction is encountered. This concon-traction causes the pressure to drop from nearly 6 MPa to almost 3 MPa. The final pressure
drop is seen at the shim initial-gap. Given that this contraction is in the order of 0.01 mm, the highest velocity is developed,
and the pressure drops to 0 MPa, which corresponds to the
outlet condition.
The depicted situation obviously corresponds to a stationary analysis. The piston-shim gap is time dependent, and so is the second pressure-drop. The detailed geometry of the piston-shim gap is also assumed to be axis-symmetric, where the gap corresponds to the maximum shim deflection. The validity of this assumption will be further discussed in the next chapter.
A velocity contour-plot is shown in Figure 2.19. Again, the highest velocities are developed through the piston passages and in the piston-shim gap.
2.2.2 Compression-setting results
Force calculation
The force-velocity curves are shown in Figure 2.20. The nu-meric data is shown in Table2.3. The error in comparison with the experimental data is shown both in the operation curve and in Table2.4.
The force increases linearly with the velocity. The simulation ofOñate [2013] underestimates the fluid force. The present
in-(a)Lateral view.
(b) Pressure in the piston faces. (c)Pressure in the flow passages.
Figure 2.18: Static pressure in the extension setting.
Table 2.3: Force [N] calculation for the compression setting. Compar-ison withOñate[2013].
Velocity [m/s] Gabriel (2013) Oñate[2013] Urbano (2014)
0.500 645 496 610
0.647 710 570 680
0.890 848 706 820
0.988 923 795 850
Figure 2.19:Velocity profiles in the flow passages. Extension setting.
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s]
0 200 400 600 800 1000
Force [N]
Gabriel (2013) Oñate (2013) Urbano (2014)
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s]
0.00 5.00 10.00 15.00 20.00 25.00
Error [%] Oñate (2013)Urbano (2014)
Figure 2.20:Force-velocity curves. Compression setting.
Table 2.4:Error [%] in the force calculation. Compression setting.
Velocity [m/s] Oñate [2013] Urbano (2014)
0.500 23.10 5.43 0.647 19.72 4.23 0.890 16.75 3.30 0.988 13.87 7.91 1.175 10.75 12.39
vestigation shows also a underestimated fluid force, but in the low velocities it corresponds relatively well to the experimental data.
By using the new structural model and by implementing the three-dimensional CFD analysis, the error is diminished under
10% for every operation point, excepting the operation point at
Vin =1.20m/s.
In the present model, the two operation points at Vin =1.00
m/s andVin =1.00 m/s show an increasing error. This occurs because the predicted force increases at a lower rate in velocities near1.00m/s. The pressure - velocity curve is shown in Figure
2.21.
0.000 0.200 0.400 0.600 0.800 1.000 1.200 Velocity [m/s]
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
Pressure [Mpa]
Figure 2.21:Averaged pressure in shim-stack vs. velocity for the compression setting.
The pressure is one order of magnitude inferior in the com-pression setting than in the extension setting. By examining again the displacement-pressure curve for the compression shim-stack (see Figure 1.13), the pressure regime developed by the fluid does imply a nonlinear behavior in the displacement, since the pressure is extremely low (P < 0.20 MPa).
A further analysis could be conducted to describe more pre-cisely the structural behavior of the compression shim-stack un-der low pressures. Particularly, it is of interest to improve the calculation around 2.00 MPa, which corresponds to the
Flow analysis
An operation point with an inlet velocity of Vin = 0.890 m/s was chosen. The piston-shim gap reached din =0.15mm after the simulation converged. The general flow behavior is shown in Figure2.22.
As in the extension setting, the highest fluid-velocity is devel-oped through the piston passages and in the piston-shim gap. However, the velocity increase is not as high as in the extension setting. For instance, at an inlet velocity of Vin = 0.890 m/s, a maximum velocity of approximately20 m/s is developed in
the piston passages.
(a)Lateral view.
(b) Flow passages in the piston. (c)Recirculation zones down-stream.
Figure 2.22:Streamlines in the compression setting.
The highest velocity is developed in the piston-shim gap (see for example Figure 2.22c). There, the velocity reaches 50 m/s.
These are however much more smaller than those developed in the extension setting.
The outlet boundary condition is also zero pressure. Given that the downstream region does not change with the piston-rod displacement, no corrections would be needed for the out-let boundary-condition.
The pressure gradient is shown in Figure 2.23. The overall pressure-drop is approximately 1.6 MPa, which is much
infe-rior than in the extension-setting. This fact explains why the fluid force in the compression setting is lower than in the exten-sion setting.
The pressure drop occurs in two ways. The two piston-passages account for a small pressure drop (from approximately1.3MPa
to1.0MPa). The biggest pressure-gradient is observed in some
small connections, that carry the fluid through the extension fluid-passages (see Figure2.23c). In those connections, the pres-sure falls to approximately0.2MPa. Finally, a velocity
contour-plot is shown in Figure 2.24. The velocity inside the piston passages is in the order of15m/s.
2.3
conclusions
The most important assumption in the CFD analysis is the ini-tial piston-shim gap. An iniini-tial gap was assumed, the CFD analysis was solved and the pressure below the working shim-stack was averaged. This pressure was then used in the cor-responding displacement-pressure curve to find an output dis-placement. When both displacements coincided below < 5%, the simulation had converged.
The force-velocity curve in the extension setting does not im-prove significantly in comparison to the model ofOñate [2013]. Under low velocities a high error is observed. This may imply that the structural model is much more stiffer under low veloci-ties. A great recirculation zone is observed downstream. As the piston rod displaces in the extension setting, the downstream domain is reduced. This is not considered in the simulation,
(a)Lateral view.
(b) Pressure in the piston faces. (c)Pressure in the flow passages.
Figure 2.24:Velocity profiles in the flow passages. Compression set-ting.
and may have an important influence in the boundary condi-tion.
Using a three dimensional flow-analysis and a new structural model, the force-velocity prediction in the compression setting improved for low velocities. The recirculation zone is smaller than in the extension setting, and it is not affected by the piston-rod displacement.
The pressure drop develops in a two-step process. The first contraction occurs in the piston passages, where the pressure decreases slightly. The final step is the piston-shim gap, where the velocity increase and pressure drop are highest. The com-pression setting exhibits a lower pressure drop than the exten-sion setting. This occurs because in the compresexten-sion setting the fluid flows also through the extension passages.
3
F L U I D - S T R U C T U R E
I N T E R A C T I O N A N A LY S I S
The structural behavior of the shim-stacks was deeply investi-gated in Chapter 1 and the three-dimensional fluid flow was analyzed in Chapter 2. In the present chapter, the interaction between both physics will be discussed.
The structural analysis revealed that by considering the contact-modeling between the disks, a nonlinear behavior could be ob-tained under high pressures. As a result, the displacement-pressures for each damper setting were computed. The CFD analysis used these curves to force the convergence of the piston-shim gap. The average pressure in the corresponding working-shim was computed, and taken as an input parameter for the displacement-pressure curve.
Thus, the maximum displacement of the shim (structural analysis) was set as the initial piston-shim gap (fluid-flow anal-ysis). The initial piston-shim gap was generated uniformly and axis-symmetrically. Thus, it is assumed that the deformation pattern of the shim stacks is at least axis-symmetric. This as-sumption will be proved in the present chapter.
3.1
fsi modeling
Two formulations for the FSI modeling are available in Ansys.
The first one is the 1-Way approach. Here, the fluid pressure
is imported as an external load to the structure, and a regular structural-simulation is preformed. This procedure has the ad-vantage that the pressure is not considered uniformly, but as a realistic fluid load. This allows the computation of a more accurate description of the stresses and strains in the structure.
The 1-Way approach has the disadvantage that it does not
couples the two physics, i.e. the influence of the structure is not considered in the fluid analysis. This coupling is taken into account in the2-Way approach.
This second formulation allows a feedback between the struc-tural and fluid-flow physics: the deformation of the structure will deform the fluid mesh, and an iterative process is followed until the results of both simulations fully coincide. This proce-dure is much more demanding.
For the present investigation, the1-Way formulation was
cho-sen. Although it does not represent a coupling between the two physics, it will provide a greater insight to the interaction of the fluid and the shim-stack loading. Thus, the FSI modeling was considered as an external analysis, and was not taken as a decisive factor in the simulation convergence of the operation points.
3.2
fsi results
3.2.1 Extension setting
An operation point with an inlet velocity of Vin = 1.134 m/s was simulated. The fluid pressure in the inferior disk of the extension shim-stack is shown in Figure3.1.
The pressure distribution is highly uniform, and has an av-erage value of ¯P = 2.20 MPa. The three circular zones of the piston-passages depict higher pressures, near 5 MPa. Given
that these high-pressure zones have a relative small area, it is ex-pected that the displacement patterns will not differ much from a uniform pressure. The comparison between the displacement patterns is shown in Figure3.2.
As expected, the displacement patterns are practically iden-tical, and the pattern is axis-symmetrical. Therefore, in the ex-tension setting the assumption of an uniform, axis-symmetrical piston-shim gap is reasonable.
Figure 3.1:Fluid pressure in the extension shim-stack interface (lower disk). Vin =1.134m/s; ¯P =2.20MPa
Figure 3.2:Displacement [m] in the extension shim-stack using FSI. Fluid pressure (left) and uniform average-pressure (right). Vin =1.134m/s; ¯P=2.20 MPa.
3.2.2 Compression setting
The operation point has an inlet velocity of Vin = 0.890 m/s. The fluid pressure over the first disk in the compression
shim-stack is shown in Figure 3.3. In this case, the pressure is not
Figure 3.3:Fluid pressure in the compression shim-stack interface (lower disk). Vin =0.890m/s; ¯P =0.132MPa.
completely uniform. The pressure difference between the piston-passages and the rest of the disk is very significant. The fluid ex-erts a pressure near1.2MPa in the piston passages. In the rest
of the disk, the pressure reaches approximately0.12MPa. Thus,
it is likely that the area-averaged pressure provides a wrong de-scription of the deformation pattern.
The comparison between the FSI and the averaged pressure models is shown in Figure3.4.
Figure 3.4:Displacement [m] in the compression shim-stack using FSI. Fluid pressure (left) and uniform average-pressure (right). Vin =0.890m/s; ¯P =0.132MPa.
As expected, the displacement patterns differ greatly by us-ing a FSI approach. By importus-ing the fluid pressure, the areas near the piston-passages deform much more. The three dimen-sional representation reveals that the displacement pattern is not axis-symmetrical.
This conclusion implies that the uniform gap assumption must be reconsidered. The displacement of the first disk varies considerably in the disk circumference. A nonuniform-width contraction would lead to different velocities and pressures in the tangential direction.
3.3
conclusions
The fluid-structure simulation revealed a different displacement pattern in the compression shim-stack. In this case, the fluid pressure is not entirely uniform, which causes some regions to deform much more than by assuming an uniform-average pressure.
The displacement profile is not axis-symmetric, which inval-idates the assumption of an uniform, axis-symmetric piston-shim gap in the compression setting. Although the force-velocity curves agree to the experimental data at a good extent, a2-Way
conclusions regarding the structural
analysis
The shim model is highly sensitive to element type, contact formulation and contact type. In the extension setting, a model with shell elements predicted a very large displacement in com-parison to a solid-element model. The difference in stiffness from a totally bonded model and a totally frictionless model is too large.
Additionally, the modal analysis revealed that the extension shim-stack model is highly nonlinear. The linear structural be-havior of the extension setting leads to a linearly-falling error in the in the force-velocity curve. Therefore, only a highly nonlin-ear structural behavior would reduce the error at low velocities.
By making a relation between the CFD and the structural study, the compression shim-stack model behaves well under low velocities. The compression model does not require a highly nonlinear structural behavior under low velocities. Near1m/s
however, both the present and the results from Oñate [2013] underestimate the experimental pressure.
However, further analysis is required. Although the contact between the interfaces was found to profoundly influence the structural behavior, no phenomenological approach was strictly followed to define shim-stack structural model. Thus a further analysis is required to fully understand the contact simulation, and to define a physically-based model.