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Materials and Design 149 (2018) 1–14Contents lists available at ScienceDirectMaterials and Designjournal homepage: www.elsevier.com/locate/matdesSqueal analysis of thin-walled lattice brake disc structureAminreza Karamoozian a, Chin An Tan b, Liangmo Wang c,⁎abcDepartment of Automotive Engineering, School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu, ChinaDept. of Mechanical Engineering, Wayne State University, Detroit, MI, USADepartment of Automotive Engineering, School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, Jiangsu, ChinaH I G H L I G H T SG R A P H I C A LA B S T R A C T The dynamic stability of a novel brakedisc design consisting of lattice periodictruss structures is investigated. The lattice brake disc system results inimproved thermal properties and structural stability. The lattice brake disc design has a lowersqueal propensity than the conventionalvanned type in the frequency range ofabout 2 to 8 kHz.a r t i c l ei n f oArticle history:Received 13 November 2017Received in revised form 12 March 2018Accepted 18 March 2018Available online 26 March ake disca b s t r a c tThis paper presents a dynamic stability study of a novel brake disc design consisting of periodic lattice truss substructures. An integrated approach of theoretical modeling, experimental modal analysis, and finite elementsmethods is employed in this investigation to understand the squeal characteristics. The brake system is analytically modeled by a rotating annular disc subjected to in-plane frictional loads. Natural frequencies and forced response of the brake disc are obtained and validated by finite elements results. Experimental modal analysis of thelattice brake rotor/pad system with free-free boundary conditions is performed to obtain the modal properties ofthe brake rotor as inputs to the finite elements model. The FEA also includes models for the heat convection during braking and the non-linear contact forces between the rotor and the pads obtained from simulations of theSAE J2521 drag braking noise test matrix. The likelihood of squeal noise occurrence or squeal propensity forboth the lattice and conventional vanned type brake discs are examined. The propensity is quantified by the standard deviation of the statistical occurrence of brake instability. It is shown that the lattice brake disc design has alower propensity in the low frequency range of about 4 to 8 kHz. 2018 Elsevier Ltd. All rights reserved.1. IntroductionAutomotive disc brake squeal has been investigated as the constantsource of concerns in warranty and insurance issues for many years.⁎ Corresponding author.E-mail addresses: ar.karamoozian@njust.edu.cn, (A. Karamoozian), tan@wayne.edu,(C.A. Tan), liangmo@njust.edu.cn. (L. 0264-1275/ 2018 Elsevier Ltd. All rights reserved.Complaints of brake noise have caused automotive manufacturers considerable profit loss from warranty claims [1]. As a result, over the pastfew decades, numerous studies have been conducted to understand, reduce, and possibly eliminate brake squeal noise by analytical modeling,experimental, numerical and finite element methods. Research hasshown that the main causes for brake noise are due to frictioninduced dynamic instabilities in the brake generated by the roughnessat the contact surfaces and the complex time-varying characteristics of
2A. Karamoozian et al. / Materials and Design 149 (2018) nmmnmPsPdΔγ1reXnmϕsωfω f, aω f, agigN1strut lengthinclination angle between the truss member and thebase of the unit cellvolume with respect to the unit celltotal volume of the solid metal within the unit cellbrazed node width arearelative densitytruss fraction factorhomogenized bending stiffness of the unit cell of I-typelatticemoving distributed contact loadnormalization constant for the correspondingeigenfunctionnatural frequencygeneralized forcesangular velocitymode shape characterized by setting the eigenfunctionwith n nodal diametersgeneralized modal massstatic normal pressuresdynamic normal pressuresstatic deformation of the lining due to Psangle of evaluation of the curvetransverse displacementmodal response amplitudespatial phase anglemoving body frequencyapparent resonant frequencyapparent resonant frequencypopulation sample of instability measurementsaverage value of the whole populationtotal population of instability measuresthe frictional forces [2,3]. Moreover, it has been suggested thatoverheating of the brake system and deterioration of the friction coefficient can also be causes for brake noise [1,4–6]. Therefore, adequatecooling of braking components is critical for the quietness and stabilityof brakes, especially for high-performance passenger vehicles [5–7].Periodic cellular metals (PCMs), which are lightweight materials, areexcellent heat exchange media with open cell topologies and have greatpotentials in applications involving heat dissipation with small spacings[8]. PCMs are often used to design lightweight sandwich panel structures, where unilateral fluid flows are required. Moreover, PCMs are capable of absorbing energies of contact through thermal transport acrossthe faces of sandwich structures due to their high porosity with 20% orless of their interior volumes occupied by metals [9]. PCMs, withthree-dimensional interconnected void spaces, are also regarded as lattice structures which are very effective for thermal transport by providing a high thermal conductivity path and heat transfer for cooling fluidsor coolant flows through the structure by contiguous media [10,11].However, because of the shapes of PCMs, their thermal properties depend on the structural orientations and need to be optimized. Latticestructures can also offer considerable stiffness, strength, and loadcarrying capacity, despite using fewer materials [12,13].Recent advances in manufacturing technologies have expanded theapplications of PCMs [14,15], leading to a new concept of bidirectionalventilated brake disc with more effective thermal capabilities for passenger vehicles [16]. In this novel design, a mechanically strong latticewith high porosity is used as the core of the brake disc to improve thethermal properties of the brake system. The lattice brake disc is now capable of faster heat transfer with increased pumping capacity due to alarger Nusselt number. However, despite the improved thermal convection, there has been no study on the dynamic stability and squeal analysis of these lattice structures for brake disc applications. The objectiveof this paper is to investigate the dynamics of a brake disc systemmodeled by a rotationally periodic lattice structure consisting of a finitenumber of identical substructures. This paper is organized as follows.The governing equations of motion of the brake disc, modeled by a rotating disc, are presented in the principal coordinates. The response ofthe disc to a distributed axial load, which models the frictional load, isthen obtained. Experimental modal analysis (EMA) using an impacthammer test is employed to obtain the brake rotor modal propertieswhich are input parameters to the finite elements analysis. Analyticalresults are combined with finite elements stability analysis using thecomplex eigenvalue method to determine the onset of instability orsqueal for the lattice brake design. The likelihood of squeal noise occurrence or the squeal propensity is quantified by a parameter based on thesystem eigenvalues and the standard deviation of the statistical occurrence of brake instability measurements.2. Literature summary on brake squeal and lattice structuredynamicsResearch to understand brake squeal mechanisms has focused on investigating and predicting vibration mode instabilities. The most common approach is the complex eigenvalue analysis (CEA), a linearmethod which checks for the positive sign of the real part of the eigenvalues as a condition for the onset of mode instability [17–19]. Besidesthe CEA, nonlinear contact pressure analysis has also been applied to investigate brake instabilities [20–25]. However, owing to the complexgeometries of and non-conservative frictional loads in brake systems,the CEA itself is insufficient to accurately predict brake squeal and isthus often integrated with finite elements methods for instability analyses of brake squeal [26–29]. Studies on brake noise also include the effects of brake geometries [30], vehicle speed and brake pressure [31],and properties of the contact materials [32,33]. In summary, the proximity of the squeal frequencies and resemblance of the mode shapesof a rotating brake disc with the natural frequencies and mode shapesof a stationary rotor are generally regarded as conditions for squeal[34,35]. The various frequency ranges of brake squeal noise are summarized in the review paper [36]. In addition to the CEA, dynamic transientanalysis has also been employed to investigate the time response andinstability of brake systems. The time domain responses are transformed to the frequency domain by the fast Fourier transform (FFT)technique to reveal the underlying dynamic characteristics and identifythe conditions for brake vibration instability and squeal [37,38].Vibration analyses of PCMs and lattice structures have focused onmodels of periodic systems with repeated identical substructures.Employing symmetry properties of the substructure dynamic matricesresults in a coupled set of linear differential equations for the responses[39–41]. Eigenfunctions of the resulting periodic unit transfer matrix arethen used to obtain the frequency response of the system. Studies on theforced response a rotationally periodic structure due to a rotating forceare found in [42,43]. In addition to analytical solutions, modeling oflarge repetitive lattice structures as continua for stability analysis hasalso been considered [44]. Other studies include the vibration analysisof arbitrary lattice structures having repetitive geometries in any combination of coordinate directions [45].3. Modeling of lattice brake disc structureA new design of lattice structure (I-type) is introduced, which supports the loads by lining up the truss topology. Modeling parametersthat are critical factors in brake disc instabilities are discussed in thissection. Comparing to traditional lattice designs, the new I-structurehas different types of nodes, which are constrained between the strutmembers and the brazed nodes are positioned between the struts and
A. Karamoozian et al. / Materials and Design 149 (2018) 1–143the face plates. The braze alloy adds an extra weight to the lattice structure and increases the relative density by about 0.2% [46]. Consider theunit cells shown in Fig. 1, the I-type volume, expressed in terms of thelength, geometry and cross-sectional area of the lattice strut member is:shown in Fig. 1 and the methods presented in [49,51], the homogenizedbending stiffness of the unit cell of an I-lattice filled plate with respect toη can be obtained as [53]:V U ¼ 4ðh þ 2ℓ cosθÞðh þ 2ℓ sinθÞℓ sinθ sinω sin2ωK bending;I ¼ð1Þwhere ℓ is the strut length and ω is the inclination angle between thetruss member and the base of the unit cell. The total volume of thesolid metal within the unit cell is the same for all types of structures.For a rectangular cross-section of width w and thickness t (Fig. 1), thetotal volume is:V T ¼ 4tωðh þ 2ℓ cosθ þ bÞð2Þwhere b is the width of the brazed node in which the node area is assumed to be square (it is noted that in an ideal lattice structure, b 0). The relative density of the I-type lattice structure can be found by dividing the volume of occupied metal to the unit cell:ρ¼VTðωt ℓÞ ðh þ 2ℓ cosθ þ bÞ¼V U ðh þ 2ℓ cosθÞðh þ 2ℓ sinθÞ sinθ sinω sin2ωð3ÞNote that there is no contribution from the flat nodes (with thewidth of b) to the stiffness and strength of the lattice truss core structure. However, for non-ideal lattice structures (where b 0), there is astiffness reduction [12,47]. Accordingly, a truss mass fraction factor ηis introduced, which leads to an expression that separates the components of the core topology into those that directly contribute to the stiffness from those that are applied to achieve robust performance[12,47,48]:η¼ðh þ 2ℓ cosθÞðh þ 2ℓ cosθ þ bÞð4ÞThe cellular truss structures are modeled as anisotropic continua. Todefine the lattice stiffness, we model the periodicity cell of lattice platesby employing the periodicity theory of beams and rods. A wellestablished continuum model for repetitive lattice structures can befound in [49]. Other studies have shown that more refined models canbe obtained by combining classical methods of the theory of strengthand the homogenization method [50]. Derivation of equilibrium and kinematic relations and conditions with respect to the joint nodes areshown in [51,52]. Applying variational principles to the periodic cell"#2ððh þ 2ℓ cosθÞ sinωÞ t 20E0 t 0 2 ηððhþ2ℓcosθþbÞsinωÞþ121 ν0 2pffiffiffi 3333 EH tηððh þ 2ℓ cosθ þ bÞ sinωÞ2E0 t 0 þþ w483 1 ν0 2ð5Þwhere E0 and ν0 are the Young's modulus and Poisson's ratio of theplate, respectively, E the Young's modulus of the lattice truss structure,t the thickness of the lattice truss, t0 the plate thickness and H the height,see Fig. 1. Because of the symmetry of the plate system considered, coupling stiffness can be neglected. Hence, we can use the statisticalmethods introduced in [25] and [54,55] to predict the interface contactstiffness by modeling a number of spring elements at the contactinterfaces.4. Problem formulationConsider a rotating disc with lattice periodic structures as shown inFig. 2. The structures are lined within the elastic body which is externally excited by an oscillatory, moving distributed frictional load generated by the relative sliding between the elastic body and the surface ofthe lining. This surface traction is transformed onto the centroid of therotating disc and the lining area, and can be represented by a distributedaxial load μp [54,55]. Here, p is the contact normal pressure and μ is thecoefficient of friction between the moving elastic body and the lining.Various more complex contact and friction models have been proposedbased on the interface tribology and temperature dependence of coefficient of friction [2,56].To examine the conditions of instability of the transverse motion ofthe lattice disc, it is assumed that: (1) there is no loss of contact in theinterface with consideration of conformal contact between the liningand the moving elastic body; (2) the coefficient of friction and materialproperties of the lining area are constant; (3) coupling between theaxial and transverse motions of the lattice disc is neglected. Fig. 2(a,b) show the free-body diagrams of the system. It should be noted thatbecause of the nature of friction, the distributed axial load is a slopedependent, non-conservative follower-type force.Depending on the number of vanes, various kinds of interactions between a rotating disc and the pad have been considered. The dynamicinteractions between a rotating disc and the pad can be modeled byFig. 1. Schematics of a unit cell of the periodic I-type lattice truss structure and the cell periodicity.
4A. Karamoozian et al. / Materials and Design 149 (2018) 1–14two distinct analyses with reference to the source of excitation. In thefirst analysis, it is presumed that the dynamic force excitation is appliedto the disc and rotates with it. A second approach models the excitationas exerted on the stationary pad. In both analyses, it is assumed that vibration between the disc and the pad is transmitted through a liningsurface, resulting in forces that are proportional to the displacement.In the present study, we adopt the approach that the excitation is applied to the rotating disc to investigate the squeal phenomenon. Itshould be mentioned that the effect of the static pressure behind thebackplate of the pad can also cause oscillation in the transverse direction. Hence, this excitation is modeled as a moving distributed contactload fnm(r, θ, t) for the nm-th normal mode. Here, θ is a coordinate thatrotates with the disc; see Fig. 2. Responses of the pair of normalmodes of a rotating disc due to excitation of the dynamic distributedload are then obtained.rotationally periodic lattice structure can be written as: m þ Mnm ω2nm qm ðt Þ ¼ Q nm ðt ÞMnm qð6Þwhere Mnm is the mass normalization constant of the correspondingeigenfunction, ωnm is the natural frequency, and Qnm(t) is the generalized force. The normalization Rωnm (R)/Mnm 1 is used to simplifyand decouple the equations. Consider a rotating disc with angular velocity Ω and a dynamic distributed axial load fnm(r, θ, t) acting at a radius Ron the disc. Assume that the symmetry axis of the disc and the axis ofrotation are the same, the distribution of the forcing function, in termsof the coordinates that rotate with the disc, can be written as: f ðr; θ; t Þ ¼ Fδ½θ nðΩt þ α Þ δðr RÞnmð7Þwhere α is the spatial angle of the applied force and δ[θ n(Ωt α)] isthe Dirac-delta function, defined as:4.1. Basic equationsDenote θ as the spatial variable, t temporal variable, and re the transverse displacement of the beam [54,55]. In terms of the normal coordinates of the eigenfunctions, the governing equations of motion for a8δ½θ nðΩt þ αÞ ¼ 0 Zþ :for θ ðΩt þ αÞδ½θ nðΩt þ αÞ dθ ¼ 1ð8Þ Fig. 2. Schematics of the coordinate system of and forces on the rotating disc and beam model: (a) beam subjected to a distributed axial load due to friction; (b) free body diagram of a beamelement, G 0 and G 0.
A. Karamoozian et al. / Materials and Design 149 (2018) 1–145Fig. 3. Test devices, accelerometer location positions and impact hammer striking positions.Imposing the modeling assumptions discussed earlier, the force distribution can be expressed by: f ðr; θ; t Þ ¼ μP ðr; θ; t Þ δ½θ nðΩt þ α Þ δðr RÞnmcos nθ and φsnm(r, θ) sin nθ. Hence, the generalized forces acting onthe pair of normal modes are:Zr Z2πð9ÞIntegrating the above expression for the distributed force in both directions of r and θ, an analytical expression for the force on the disc isobtained:Q snm ðt Þ ¼Zr Z2π¼μP ðr; θ; t Þ0f nm ðr; θ; t Þ φsnm ðr; θÞ r drdθ0h0isin2 nθ sinnðΩt þ α Þ þ sin n θ cosnθ cosnðΩt þ α Þ r drdθ0ð12Þf nm ðr; θ; t Þ ¼ μP ðr; θ; t Þ sinnθ sinnðΩt þ α Þþ μP ðr; θ; t Þ cosnθ cosnðΩt þ α Þ:ð10ÞSimilarly, for the other mode:The generalized force for the nm-th mode is thus:Zr Z2πQ cnm ðt Þ ¼Zr Z2πQ nm ðt Þ ¼f nm ðr; θ; t Þ φnm ðr; θÞ rdrdθ0ð11Þ0Zr Z2π¼0f nm ðr; θ; t Þ φcnm ðr; θÞ r drdθ00 μP ðr; θ; t Þ sin n θ cosnθ sinnðΩt þ α Þ þ cos2 nθ cosnðΩt þ α Þ r drdθ0ð13Þwhere r and θ are the polar coordinates, φnm(r, θ) is the mode shape ofthe eigenfunction with n nodal diameters and m nodal circles (φnm(r, θ) φcnm(r, θ) φsnm(r, θ)), the φnm(r, θ) is given by φcnm(r, θ) Eqs. (12) and (13) represent the generalized forces for the modes.Employing Eq. (6), the generalized or normal mode responses areImpact hammer FFT plots: Pad350300300250AmplitudeAmplitudeImpact hammer FFT plots: vanned type brake equency (Hz)(a)100000200040006000Frequency (Hz)(b)Fig. 4. Impact hammer results FFT plots for (a) the vanned typed brake disc and (b) the pad.8000
6A. Karamoozian et al. / Materials and Design 149 (2018) 1–14calculated by using the convolution integral for lightly damped systems.Based on our modeling assumptions, the instantaneous distributed normal pressure P can be written as the sum of the static (Ps) and dynamic(Pd) normal pressures [54,55]:P ðr; θ; t Þ ¼ P s þ P d ðr; θ; t Þ:ð14ÞDenote K and G as the transverse and shear moduli of the lining, respectively. It can be shown that the pressure P can be written in terms ofthe relative displacement of the lining as:where ψ1 ωf
Brake disc 1. Introduction Automotive disc brake squeal has been investigated as the constant source of concerns in warranty and insurance issues for many years. Complaints of brake noise have caused automotive manufacturers con-siderable profit loss from warranty claims [1]. As a result, over the past
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