Large deflection of annular throttle－slices in shock absorbers

HE Li－ping（贺李平）， GU Liang（顾亮）， XIAO Jie－ping（肖介平）

（1.Beijing Institute of Radio Measurement，Beijing 100854，China；2.School of Mechanical Engineering，Beijing Institute of Technology，Beijing 100081，China；3.Department of CAE and Performance Integration，Beijing Automotive Technology Center，Beijing 100021，China）

The superposed annular throttle－slices of automobile shock absorbers are multilayer annular plates.Their largest deflection has great impact on the characteristics of the shock absorbers［1－3］and need to be accurately calculated.Refs.［4－6］gave the analytical formulas of large deflection for a single throttle－slice by specifying the ratio of inner to outer radius of the throttle－slice.However，these formulas can’t be used in general situations.Another analytical formula of large deflection for a single throttle－slice was deduced in Refs.［7］，but it haven’t been used for superposed throttle－slices yet.Although the annular throttle－slices could be considered as single throttle－slice according to the equivalent thickness method［8］，the method is based on small deflection theory and is not suitable for large deflection situations.

A general analytical formula of large deflection for throttle－slices was deduced in this paper by combining Chien－perturbation method with finite element method （FEM）.The analytical formula was verified and showed accurate results for calculating large deflection of annular throttle－slices in shock absorbers.

1 Mechanical model

Annular throttle－slices in shock absorbers are clamped at inner edge and free at outer edge as shown in Fig.1，whereqis the uniform pressure applied to the throttle－slices，b，athe inner and outer radius respectively，wthe deflection，andnthe number of slices.It is assumed that the forces between the throttle－slices are uniform.

2 Analytical solution

2.1 Von Kármán equations of single annular throttle－slice

For a single annular throttle－slice subjected to uniform pressureq，Von Kármán equations in the form of axial symmetry［7］are

Fig.1 Mechanical models of annular throttle－slices

Theannularthrottle－sliceisclampedatinner edge，andfreeatouteredge，sotheboundaryconditionscanbeexpressedas

2.2 Solution by perturbation method

Wm，the dimensionless deflection of throttleslice at outer edge，is set as the perturbation parameter：

The undetermined functionsφ（y），S（y）andQinEq.（1）canbeexpandedintopowerseriesofWm，asfollows：

Sothestiffnesscurveequationoflargedeflectionofannularthrottle－sliceis

2.3 Analytical solution of superposed annular throttle－slices

Eq.（7）can be rewritten as

SothatEq.（8）canbesimplifiedto

For the first，…kth，…nththrottle slice，Eq.（9）can be written respectively as follows：

Summing up thenformulas above gives

whereHk，Jkcorrespond toHandJrespectively for the thickness of throttle－slicehk.

Eq.（10）is the general formula of large deflection for annular throttle－slices in shock absorbers.For a single throttle－slice，Eq.（10）can be simplified to Eq.（9）.For superposed ones which are composed of the same thicknesshk，Eq.（10）can be simplified as

3 Numerical tests and curve fitting

While it is crucial to get the two undetermined coefficientsβ1andβ2ascanbeseenfromEq.（7），itisdifficulttoobtainananalyticalformulabecauseofthecomplexityofboundaryconditions.Therefore，numericaltestswerecarriedoutbased onFEMinthispaper.Theanalyticalformulaof largedeflectionforannularthrottle－slicescould beobtainedthroughcurvefittingofβ1andβ2.

3.1 Finite element model of annular throttle－slice

Finite element analysis software ANSYS was used for the numerical tests.The 8－node structural shell element（SHELL93）with large deflection characteristic was used for simulation of the annular throttle－slice.SHELL93is the higher order of 4－node quadrangular shell element，which is better for the simulation of curved boundaries，and gets higher accurate results by using fewer elements than the lower order element.The FEM solution will be converged to an exact solution with increasing element density.So the establishment of higher accurate finite element model needs finer element density.

Takea＝20mm，b＝10mm，n＝1，h＝0.3mm，E＝206GPa，μ＝0.3，q＝0.2MPafor example.Takingintoaccountthesymmetry，finiteelementmodelsof1／4throttle－slicewithdifferentelementdensitieswereestablished（Fig.2）.Fig.3showstherelationbetweentheFEMnumeri－calsolutions（we）ofthelargestdeflectionofthe throttle－sliceattheouteredgeandthevariouselementdensities.

Fig.2 Finite element model of1／4 throttle－slice（mesh10×40）

Fig.3 Deflection values at the outer edge for various element densities

Althoughalowelementdensitycanmeetthe requirementsofaccuracy（Fig.3），the1／4throttle－slicewasdividedinto10partsalongradialdirectionand40partsalongcircumferentialdirectionrespectivelyforahigheraccuracy（Fig.2）.

3.2 Finite element numerical tests

Theratioofinnertoouterradiusofthe throttle-slice，α＝b／a，isbetween0.30and0.80usually.Seth＝0.3mm，E＝206GPa，μ＝0.3，q＝0.2MPa，anda＝20or30mmrespectively.Tab.1liststhenumericaltestresults.

3.3 Curve fitting

Whileβ1andβ2arefunctionsofαandthe Poisson’sratioμ，μhaslesseffectonthedeflectionofthrottle－slicerelativity.Sotheerrorsfromμcouldbeignoredbysettingμ＝0.3.Substituting deflectionvaluesinTab.1intoEq.（7）givestwoundeterminedcoefficientsβ1andβ2，aslistedin Tab.2.

Tab.1 Deflection values of throttle－slice in outer edge

Tab.2 Undetermined coefficientsβ1 andβ2

Lorentzianfunctionwasusedtononlinear leastsquarescurvesub－fittingofβ1andβ2for higheraccuracy，whichareshowninFig.4.

Fig.4 Sub－fitting curves ofβ1 andβ2

Meanwhile，thesub－fittingcurveequations canbeobtained：

where the undetermined coefficients are listed in Tab.3.

Substitutingβ1，β2intoEq.（10）giveswm，the largestdeflectionofannularthrottle－slicesattheouteredge.

4 Accuracy verification and discussion

The FEM solution was presented to verify the accuracy of the analytical solution.In this section，the structural parameters and material properties of throttle－slices are set as follows unless otherwise noted：a＝16mm，b＝8mm，α＝0.50，E＝206GPa，μ＝0.3，n＝3，h1＝0.10mm，h2＝0.15mm，h3＝0.20mm，andq＝0.2MPa.

Tab.3 Undetermined coefficients of Lorentzian function

Eachthrottle－sliceshouldbedividedinto10partsalongradialdirectionand40partsalong circumferentialdirection.Frictionbetweenthrottle－sliceswasignoredconsideringtheeffectoflubricatingoilfilm.Takingintoaccountthesymmetry，1／4finiteelementcontactmodelofsuperposedthrottle－sliceswasestablished.

Theanalyticalsolutionswerecomparedwith FEMsolutionswithn，a，α，E，μandqvaryingin acertainrangerespectively，andtheiraccuracy wasverified.

4.1 Different pieces

Seta＝16mm，b＝8mm，E＝206GPa，μ＝0.3，q＝0.2MPa，n＝1－5.Tab.4liststhecomparisonbetweenanalyticalandFEMsolutions.

Tab.4 Deflection values of analytical solutions vs.FEM solutions for various n

InTab.4，Δwistherelativeerrorbetweenthe analyticalandFEMsolutions.

4.2 Same ratio of inner to outer radius

Settingα＝0.5tomakesurebandavaryproportionally.Fig.5showsthecomparisonbetween analyticalandFEMsolutions.

Fig.5 Deflection curves of analytical solutions vs.FEM solutions forα＝0.5

4.3 Same inner radius

Settingb＝8mmtomakesureavariesandα∈［0.3，0.8］.Fig.6showsthecomparisonbetweenanalyticalandFEMsolutions.

Fig.6 Deflection curves of analytical solutions vs.FEM solutions for b＝8 mm

4.4 Other variables

MakesureE，μandqvaryrespectively.Fig.7showsthecomparisonbetweenanalyticalandFEM solutions.

Fig.7bshowsthatthePoisson’sratioμhas lesseffectonthedeflectionofthrottle－slicesthan otherparameters.Sotheerrorscausedbynotconsideringμinβ1andβ2canbeignored.

Thenumericalcomparisonsaboveverifythat theanalyticalformula（Eq.（10））oflargedeflectionforannularthrottle－slicesisaccurate（datashowsthattherelativeerrorislessthan3%）andabletomeettherequirementsofengineeringdesigns.

5 Conclusions

①Ananalyticalformulaoflargedeflection forsinglethrottle－slicewasdeducedbasedon Chien－perturbationmethod，finiteelementnumericaltestsandcurvefitting.Aformulaforsuperposedthrottle－sliceswasfurtherdeducedandgeneralized.Thegeneralanalyticalformulacanbe usedforthedeflectioncalculationofbothasingle throttle－sliceandsuperposedones.

②Theanalyticalformuladeducedinthis papercanalsobeusedforsmalldeflectioncalculationofthrottle－slices，sincethesmalldeflection theoryisthesimplifiedformoflargedeflectiontheory.

Fig.7 Deflection curves of analytical solutions vs.FEM solutions for various E，μ，q values

③Toensuretheaccuracyoftheanalytical formula，finiteelementmodelofthrottle－slices withhigheraccuracywasestablished.The Lorentzianfunctionissuitableforcurvefitting.Thenumericaltestsverifythattheanalyticalformulahassatisfactoryaccuracy.

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（Edited byCai Jianying）