Analysis of Tire Acoustic Cavity Resonance Energy Transmission Characteristics in Wheels Based on Power Flow Method

applied sciences

Article Analysis of Tire Acoustic Cavity Resonance Energy Transmission Characteristics in Wheels Based on Power Flow Method

Wei Zhao , Yuting Liu, Xiandong Liu *, Yingchun Shan and Xiaojun Hu

School of Transportation Science and Engineering, Beihang University, 37 Xueyuan Road, Haidian District, Beijing 100191, China; [email protected] (W.Z.); [email protected] (Y.L.); [email protected] (Y.S.); [email protected] (X.H.) * Correspondence: [email protected]

Abstract: As a kind of low-frequency vehicle interior noise, tire acoustic cavity resonance noise plays an important role, since the other noise (e.g., engine noise, wind noise and friction noise) has been largely suppressed. For the suspension system, wheels stand first in the propagation path of this energy. Therefore, it is of great significance to study the influence of wheel design on the transmission characteristics of this vibration energy. However, currently the related research has not received enough attention. In this paper, two sizes of aluminum alloy wheel finite element models are constructed, and their modal characteristics are analyzed and verified by experimental tests simultaneously. A mathematically fitting sound pressure load model arising from the tire acoustic cavity resonance acting on the rim is first put forward. Then, the power flow method is applied to investigate the resonance energy distribution and transmission characteristics in the wheels. The structure intensity distribution and energy transmission efficiency can be described and analyzed clearly. Furthermore, the effects of material structure damping and the wheel spoke number on the Citation: Zhao, W.; Liu, Y.; Liu, X.; energy transmission are also discussed. Shan, Y.; Hu, X. Analysis of Tire Acoustic Cavity Resonance Energy Keywords: tire acoustic cavity resonance; structural power flow; energy transmission; finite ele- Transmission Characteristics in ment method Wheels Based on Power Flow Method. Appl. Sci. 2021, 11, 3979. https://doi.org/10.3390/app11093979

Academic Editor: Nicola Bosso 1. Introduction Tire acoustic cavity resonance (TACR) noise is well known for its large effect on vehicle Received: 1 April 2021 ride comfort. Researchers have found a distinct peak in the frequency spectrum of interior Accepted: 25 April 2021 noise that coincides with the natural frequency of the tire acoustic cavity [1–3]. Since the Published: 27 April 2021 importance of this resonance phenomenon was discovered, efforts have been made to reveal the vibration and acoustic properties of the tire acoustic cavity’s coupled structure [4–7]. Publisher’s Note: MDPI stays neutral In respect to noise reduction, previous studies mostly used sound absorbing materials, with regard to jurisdictional claims in resonators or damping structures to suppress the resonance [8–10]. Haverkamp [11] found published maps and institutional affil- that filling the tire cavity with mineral fibers could reduce the noise sound pressure level by iations. 20 dB, while Fernandez [12] studied the noise reduction effects of various sound-absorbing materials including fabric fibers and aluminum foam. Kamiyama [13] used a Helmholtz resonator to dissipate the resonance energy and applied the device to industrial models. The wheel stands first in the propagation path of TACR energy into the car cabin and is Copyright: © 2021 by the authors. highly designable. Therefore, the wheel has high potential for resonance noise suppression. Licensee MDPI, Basel, Switzerland. Ni et al. [14] considered the wheel’s optimal design as the most economical and practical This article is an open access article method to suppress the propagation of TACR noise. Since then, research on modifying distributed under the terms and wheel structures has appeared. Yang et al. [15] studied the influence of fiber-reinforced conditions of the Creative Commons composite wheels on tire cavity noise, Mohamed et al. [16] used the inner trim to reduce Attribution (CC BY) license (https:// the resonance. However, up to now, there has not yet been a complete wheel design theory creativecommons.org/licenses/by/ and method to suppress the energy transmission. 4.0/).

Appl. Sci. 2021, 11, 3979. https://doi.org/10.3390/app11093979 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 3979 2 of 19

Appl. Sci. 2021, 11, 3979 2 of 18 to reduce the resonance. However, up to now, there has not yet been a complete wheel design theory and method to suppress the energy transmission. TACR energy propagates from the tire cavity to the wheel, ultimately reaching the car cabin.TACR As energyfor the propagatesinvestigation from of the the energy tire cavity propagation to the wheel, in a structure, ultimately the reaching structural the powercar cabin. flow As method for the is investigation very helpful of [17–19] the energy and propagation can be used in to a describe structure, the the vibration structural energypower distribution. flow method Bolognani is very helpful et al. [20] [17 studied–19] and the can power be used flow toof describea coupled the cylindrical vibration shell–plateenergy distribution. structure with Bolognani four types et al. of [ 20coupling] studied springs. the power Goyder flow et of al. a coupled[21] calculated cylindrical the powershell–plate flow of structure infinitely with long four beams types and of slab coupling structures springs. under Goyder force et and al. torque [21] calculated excitation. the Inpower recent flow years, of infinitelythe research long on beams the structural and slab structurespower flow under method force in and engineering torque excitation. design In recent years, the research on the structural power flow method in engineering design has has increased and expanded to various areas [22]. Al et al. [23] used this method to increased and expanded to various areas [22]. Al et al. [23] used this method to evaluate evaluate the energy in locally resonant metamaterials. However, there is no previous the energy in locally resonant metamaterials. However, there is no previous research on research on the application of the power flow method for investigating the energy the application of the power flow method for investigating the energy dissipation of TACR dissipation of TACR in the propagation process. in the propagation process. In this paper, a mathematically fitting load model arising from TACR energy acting In this paper, a mathematically fitting load model arising from TACR energy acting on the rim is put forward, which is first based on experimental and simulation results. on the rim is put forward, which is first based on experimental and simulation results. The The power flow method is applied to calculate the resonance energy transmission power flow method is applied to calculate the resonance energy transmission characteristics characteristics in different aluminum alloy wheels. Then, the effects of structure damping in different aluminum alloy wheels. Then, the effects of structure damping and the wheel and the wheel spoke number are also studied. The path of this work’s technique is shown spoke number are also studied. The path of this work’s technique is shown in Figure1. inThese Figure works 1. These may helpworks further may help design further the wheel design structure the wheel to reduce structure the propagationto reduce the of propagationTACR energy of intoTACR the energy cabin. into the cabin.

Figure 1. Flowchart of the technique’s path. Figure 1. Flowchart of the technique’s path. Appl. Sci. 2021, 11, 3979 3 of 19

Appl. Sci. 2021, 11, 3979 3 of 18

2. Simulation Method 2.1.2. SimulationSound Pressure Method Load Modeling and Verification 2.1.As Sound shown Pressure in Figure Load Modeling 2, the two and‐dimensional Verification and three‐dimensional finite element modelsAs (2 shown‐D and in3‐D Figure FEMs)2, theof a two-dimensional 185/60 R15 tire were and established three-dimensional to analyze finite the element sound fieldmodels generated (2-D and by 3-D TACR. FEMs) The of awhole 185/60 assembly R15 tire weremodel established was built to in analyze ABAQUS the soundsoftware, field whichgenerated consisted by TACR. of the Thetire with whole hyperelastic assembly model rubber was and built reinforcements, in ABAQUS the software, air medium, which rimconsisted and road of thesurface. tire with The hyperelastic material properties rubber and were reinforcements, provided by the the tire air medium, manufacturers. rim and Theroad rubber surface. part The was material simulated properties by continuum were provided elements by theCGAX3H tire manufacturers. and CGAX4H, The the rubber air mediumpart was was simulated meshed by by continuum ACAX3 elements, elements and CGAX3H the reinforcements and CGAX4H, were the defined air medium by rebar was elementsmeshed with by ACAX3 the type elements, of SFMGAX1. and the Since reinforcements the tire cavity were was defined mainly by rebarfocused elements on in this with work,the type the rim of SFMGAX1. and road face Since were the regarded tire cavity as wasrigid mainly bodies focuseddue to the on higher in this stiffness work, the than rim theand tire road and face air medium. were regarded The rim as and rigid tire bodies were due fixed, to theand higher vertical stiffness displacement than the constraints tire and air andmedium. loads Thewere rim applied and tire through were fixed, the androad vertical surface. displacement In the whole constraints assembly and model, loads werethe numberapplied of through elements the roadwas surface.36,001, and In the the whole number assembly of nodes model, was the 40,622. number The of elements modal characteristicswas 36,001, and of the the tire number cavity of coupling nodes was FEM 40,622. agreed The well modal with characteristics the experimental of the test tire results.cavity couplingThe specific FEM modeling agreed well and with verification the experimental process could test results. refer to The authors’ specific previous modeling workand verification[24]. process could refer to authors’ previous work [24].

(a) (b)

FigureFigure 2. 2. FEMFEM of of a a185/60 185/60 R15 R15 tire. tire. (a) ( a2)‐D 2-D FEM. FEM. (b) ( b3)‐D 3-D FEM. FEM.

InIn the the research research aboutabout thethe TACR TACR problem, problem, the the most most concerning concerning aspect aspect is the is first the natural first naturalfrequency frequency of the tireof the cavity, tire cavity, which canwhich be can calculated be calculated in Equation in Equation (1) as (1) as 𝑐c f𝑓 = (1)(1) 1 l𝑙 where c represents the speed of sound in the acoustic medium and l is the median where c represents the speed of sound in the acoustic medium and l is the median circum- circumference of the tire cavity. ference of the tire cavity. In order to simulate TACR conditions, modal analysis of the tire acoustic cavity was In order to simulate TACR conditions, modal analysis of the tire acoustic cavity was carried out first. The free acoustic modal shapes were obtained in the case of a 2.5 bar carried out first. The free acoustic modal shapes were obtained in the case of a 2.5 bar inflationinflation pressure, pressure, as as shown shown in in Figure Figure 3.3 .The The two two modes modes differed differed from from each each other other in in the the circumferentialcircumferential direction direction (90 (90 degrees) degrees) but belongedbelonged to to the the same same natural natural frequency frequency of of 237 237 Hz. Hz. It is well known that under the road load, the first natural frequency of the tire cavity splits into two values, low and high, corresponding to the fore-and-aft and vertical modal shapes, respectively, as shown in Figure4. Since the resonance in the vertical direction is the major cause of the spindle vibration, only the TACR sound pressure distribution at the higher natural frequency was analyzed in the following study. Appl. Sci. 2021, 11, 3979 4 of 19

Appl. Sci. 2021, 11, 3979 4 of 18 Appl. Sci. 2021, 11, 3979 4 of 19

Figure 3. Free acoustic modal shapes of the tire cavity.

It is well known that under the road load, the first natural frequency of the tire cavity splits into two values, low and high, corresponding to the fore‐and‐aft and vertical modal shapes, respectively, as shown in Figure 4. Since the resonance in the vertical direction is the major cause of the spindle vibration, only the TACR sound pressure distribution at the higher natural frequency was analyzed in the following study. FigureFigure 3.3. Free acoustic modalmodal shapesshapes ofof thethe tiretire cavity.cavity.

FigureFigure 4.4. Acoustic modal shapesshapes ofof thethe tiretire cavitiescavities underunder roadroad load.load.

TheThe frequencyfrequency responseresponse characteristicscharacteristics of the tiretire acousticacoustic cavitycavity werewere acquiredacquired byby sweepsweep frequencyfrequency analysis.analysis. The excitation was set byby applyingapplying aa uniformuniform panelpanel velocityvelocity acousticacoustic boundaryboundary conditioncondition (1(1 mm/smm/s in Z direction) atat thethe contactcontact patch.patch. TheThe simulationsimulation conditions were set with different road loads and inflation pressures (road load range = conditions were set with different road loads and inflation pressures (road load range = 3000–45003000–4500 N;N; inflationinflation pressurepressure range range = = 1.9–2.51.9–2.5 bar).bar). TheThe dynamicdynamic equationequation ofof thethe acousticacoustic Figure 4. Acoustic modal shapes of the tire cavities under road load. structurestructure couplingcoupling systemsystem isis describeddescribed asas followsfollows [[24]:24]: " # .. " # The frequency[M ][ response0]𝑀 characteristics 0 𝑢 [K of𝐾] the− [tireS]𝑆 acoustic𝑢 cavity𝑓 were acquired by s u s {u} { fs} (2) sweep frequencyT analysis.h 𝜌i𝑆 The 𝑀 excitation.. +𝑝 was0 set h by𝐾i applying 𝑝 a uniform=0 panel velocity(2) ρ[S] M f p [0] K f {p} {0} acoustic boundary condition (1 mm/s in Z direction) at the contact patch. The simulation where 𝑀 and 𝑀 are the acoustic and structure mass matrices, respectively, p is the conditions were seth withi different road loads and inflation pressures (road load range = where3000–4500sound [pressure,Ms ]N;and inflation 𝐾Mf pressureandare the𝐾 acousticrange indicate = 1.9–2.5 and the structure acousticbar). The massand dynamic structure matrices, equation stiffness respectively, of the matrices, acousticp is respectively, and 𝑆h is ithe acoustic structure coupling matrix showing the interaction thestructure sound coupling pressure, systemK f and is described[Ks] indicate as follows the acoustic [24]: and structure stiffness matrices, between the fluid and the structure. respectively, and [S] is the𝑀 acoustic0 structure𝐾 coupling 𝑆 matrix showing the interaction Taking the 2.5 bar inflation pressure𝑢 and 3000 N road𝑢 load𝑓 as an example, the between the fluid and the structure. (2) simulation results of the𝜌 𝑆TACR 𝑀 sound 𝑝 pressure0 amplitude𝐾 𝑝 distribution0 under the road Taking the 2.5 bar inflation pressure and 3000 N road load as an example, the simula- load are shown in Figure 5a, and the sound pressure amplitude distribution in the tionwhere results 𝑀 ofand the 𝑀 TACR are sound the acoustic pressure and amplitude structure distribution mass matrices, under respectively, the road load p is arethe circumferential direction of the tire cavity can be more intuitively represented in the polar shownsound inpressure, Figure5 a,𝐾 and and the sound𝐾 indicate pressure the amplitude acoustic distribution and structure in the stiffness circumferential matrices, coordinate system, shown in Figure 5b. The typical TACR generates two antinodes which directionrespectively, of the and tire 𝑆 cavity is the can acoustic be more structure intuitively coupling represented matrix in theshowing polar coordinatethe interaction system,between shown the in fluid Figure and5b. the The structure. typical TACR generates two antinodes which are 180 degrees out ofTaking phase tothe each 2.5 otherbar inflation and located pressure at the topand and 3000 bottom N road in theload vertical as an direction. example, the simulationConsidering results the of soundthe TACR pressure sound amplitude pressure changeamplitude trend distribution along the circumferentialunder the road direction,load are shown a cosine in function Figure was5a, selectedand the tosound approximate pressure the amplitude sound field distribution in the tire cavity.in the Incircumferential this paper, we direction only considered of the tire the cavity standing can wavebe more generated intuitively in the represented vertical direction in the polar and coordinate system, shown in Figure 5b. The typical TACR generates two antinodes which Appl. Sci. 2021, 11, 3979 5 of 18

ignored the mode deﬂection effect caused by rotation. The sound pressure distribution function resulting from the cavity resonance is approximately expressed as

Appl. Sci. 2021, 11, 3979 = ( ) 5 of 19 Ps Ap f cos θ sin 2π f1t (3)

where Ps is the sound pressure at each node, Ap f is the maximum sound pressure amplitude of theare sound 180 degrees ﬁeld in out the of simulation phase to each process, otherθ andis the located angular at positionthe top and of the bottom node in in the a polar vertical coordinatedirection. system and f1 is the ﬁrst natural frequency of the tire’s acoustic cavity.

(a) (b) Figure 5. Sound pressure amplitude distribution in tire cavity under the road load. (a) Sound pressure amplitude Figure 5. Sound pressure amplitude distribution in tire cavity under the road load. (a) Sound pressure amplitude distribution in the FEM. (b) Sound pressure amplitude distribution in the polar coordinate system. distribution in the FEM. (b) Sound pressure amplitude distribution in the polar coordinate system.

In fact,Considering the standing the sound wave soundpressure field amplitude caused by change the cavity trend resonance along the wascircumferential also the loaddirection, distribution a cosine acting function on the was rim. selected To obtain to the approximate TACR sound the pressuresound field amplitude in the tire in thecavity. actualIn this working paper, conditions, we only considered the tire bench the test standing was performed. wave generated The tire in and the wheel vertical assembly direction wasand fixed ignored on the testthe basemode and deflection driven to effect rotate caused by the drum.by rotation. Excitation The was sound applied pressure to thedistribution tire tread through function a rotating resulting drum. from Athe pressure-tolerable cavity resonance is microphone approximately installed expressed inside as the cavity rotated with the tire and transmitted the measured sound pressure value to the 𝑃 𝐴 cos 𝜃 sin2𝜋𝑓𝑡 (3) signal acquisition and processing system through a wireless telemetry instrument. The specificwhere experimental 𝑃 is the sound equipment pressure device at parameters each node, are 𝐴 shown is the in Tablemaximum1, and sound the detailed pressure testamplitude process can of be the referred sound tofield in [ in25 ].the simulation process, 𝜃 is the angular position of the node in a polar coordinate system and 𝑓 is the first natural frequency of the tire’s Tableacoustic 1. Experiment cavity. devices. In fact, the standing wave sound field caused by the cavity resonance was also the load distributionName acting on the rim. To obtain the TACR sound Instructions pressure amplitude in the actual working conditions,Tire the tire bench test was performed. 185/60 The R15 tire 88H and wheel assembly was fixedWireless on the telemetry test base system and driven to rotate by the drum. JM3870Excitation was applied to the tireThe tread customized through sound a rotating pressure drum. sensor A pressure‐tolerable microphone CHZ-212 installed inside the Tire-testing machine C-YLSN-1112 cavity rotated withWheel the tire and transmitted the measured 5.5 sound J × 15 pressure value to the signal acquisitionBattery and processing system through a wireless 12 V 1.telemetry 3 AH instrument. The specific experimental equipment device parameters are shown in Table 1, and the detailed test process can be referred to in [25]. The angular position of the sound pressure sensor was determined based on its time historyTable data, 1. Experiment obtained devices. by the wheel angle sensor. Then, with both the pressure data and angular position data, the sound cavity pressure distribution could be processed. The test equipment is shown in Figure6.Name In the case of the 2.2 bar inflation pressureInstructions and the vehicle speed being 50 km/h, the time historyTire of the sound pressure in the tire185/60 cavity R15 resulting 88H from TACR is shownWireless in Figure telemetry7. It was system the bandpass filtering for the originalJM3870 signal in a range of 210–260The customized Hz which sound covered pressure the first sensor natural frequency of theCHZ tire’s‐212 acoustic cavity. From the plot, it canTire be‐testing seen that machine the period of the curve was 0.14 s,C and‐YLSN the‐1112 highest sound pressure was about 100 Pa.Wheel The Ap f in the load model could be determined5.5 J × 15 by the experimental test results. Battery 12 V 1. 3 AH

The angular position of the sound pressure sensor was determined based on its time history data, obtained by the wheel angle sensor. Then, with both the pressure data and angular position data, the sound cavity pressure distribution could be processed. The test equipment is shown in Figure 6. In the case of the 2.2 bar inflation pressure and the vehicle speed being 50 km/h, the time history of the sound pressure in the tire cavity resulting Appl.Appl. Sci.Sci. 20212021,, 1111,, 39793979 66 ofof 1919

fromfrom TACRTACR isis shownshown inin FigureFigure 7.7. ItIt waswas thethe bandpassbandpass filteringfiltering forfor thethe originaloriginal signalsignal inin aa rangerange ofof 210–260210–260 HzHz whichwhich coveredcovered thethe firstfirst naturalnatural frequencyfrequency ofof thethe tire’stire’s acousticacoustic cavity.cavity. FromFrom thethe plot,plot, itit cancan bebe seenseen thatthat thethe periodperiod ofof thethe curvecurve waswas 0.140.14 s,s, andand thethe highesthighest soundsound Appl. Sci. 2021, 11, 3979 pressurepressure waswas aboutabout 100100 Pa.Pa. TheThe 𝐴𝐴 inin thethe loadload modelmodel couldcould bebe determineddetermined byby thethe6 of 18 experimentalexperimental testtest results.results.

((aa)) ((bb))

FigureFigureFigure 6.6. 6. TireTireTire acousticacoustic acoustic cavitycavity cavity soundsound sound pressurepressure pressure testtest test [25].[25]. [25 ((aa].)) Experimental (Experimentala) Experimental equipmentequipment equipment onon site.site. on ( site.(bb)) (b) Main Main experimental devices. Mainexperimental experimental devices. devices.

FigureFigureFigure 7.7. 7. TimeTimeTime historyhistory history ofof of thethe the soundsound sound pressurepressure pressure inin thethe in the tiretire tire cavity.cavity. cavity.

ForForFor thethe the 185/60185/60 185/60 R15R15 R15 tire,tire, tire, thethe the finitefinite finite elementelement element simulationsimulation simulation resultsresults results andand andfittingfitting fitting loadload loadmodelmodel model ofof of thethethe soundsound sound pressurepressure pressure werewere were drawndrawn drawn inin in thethe the samesame same polarpolar polar coordinatecoordinate coordinate system,system, system, asas shown asshown shown inin FigureFigure in Figure 8. 8.8.It ItIt can cancan be bebe seen seenseen thatthat they they werewere were inin in goodgood good agreementagreement agreement withwith with eacheach each other;other; other; thatthat that isis toto is say,say, to the say,the soundsound the sound pressurepressurepressure distributiondistribution distribution underunder under thethe the firstfirst first TACRTACR TACR frequencyfrequency frequency excitationexcitation excitation waswas was consistentconsistent consistent withwith with thethe the Appl. Sci. 2021, 11, 3979 cosinecosine function. function. The The load load model model was was verified verified and and could could be used be used to calculate to calculate the7 of power19 the power cosine function. The load model was verified and could be used to calculate the power flowflowflow inin in thethe the wheels.wheels. wheels.

Figure 8. 8. ComparisonComparison of the of finite the ﬁniteelement element simulation simulation results and results the fitting and load the model ﬁtting of load the model of the soundsound pressure. pressure.

2.2. Wheel Modeling and Verification When analyzing the vibration response of the wheel under the TACR sound pressure load, the wheel should be regarded as an elastomer rather than a rigid body. In order to establish the accurate wheel FEMs, the wheel free modal experiment involved a hammer‐ hitting excitation method, as shown in Figure 9. The types of impact hammer, accelerometer and acquisition system are an LC series with an ICP sensor, INVYJ9A4017 and INV306U.

Figure 9. Wheel modal experiment test.

Based on the test results, two FEMs of the 14‐ and 15‐inch aluminum alloy wheels were constructed. Taking the 14‐inch wheel as an example, the type of the wheel was 5.5 J × 15. The material of the wheel was cast A356‐T6 aluminum alloy, and the density, Young’s modulus and Poisson’s ratio were 2.7 × 103 kg/m3, 72 GPa and 0.3, respectively. The element type was C3D10, and the numbers of elements and nodes in the wheel’s FEM were 117,905 and 202,448, respectively. The comparison between the finite element simulation and the experiment test results is shown in Table 2. Both the first three natural frequencies and the modal shapes were highly consistent, which verifies the validity of the wheel FEMs. Appl. Sci. 2021, 11, 3979 7 of 19

Appl. Sci. 2021, 11, 3979 Figure 8. Comparison of the finite element simulation results and the fitting load model of the7 of 18 sound pressure.

2.2. Wheel Modeling and Verification 2.2. Wheel Modeling and Veriﬁcation When analyzing the vibration response of the wheel under the TACR sound pressure load,When the wheel analyzing should the be vibration regarded response as an elastomer of the wheel rather under than the a rigid TACR body. sound In pressure order to load,establish the wheelthe accurate should wheel be regarded FEMs, the as anwheel elastomer free modal rather experiment than a rigid involved body. In a hammer order to‐ establishhitting excitation the accurate method, wheel FEMs,as shown the wheel in Figure free modal 9. The experiment types of involved impact a hammer-hammer, hittingaccelerometer excitation and method, acquisition as shown system in Figure are an9 .LC The series types with of impact an ICP hammer, sensor, accelerometer INVYJ9A4017 andand acquisitionINV306U. system are an LC series with an ICP sensor, INVYJ9A4017 and INV306U.

FigureFigure 9.9. WheelWheel modalmodal experimentexperiment test.test.

BasedBased onon thethe testtest results,results, twotwo FEMsFEMs ofof thethe 14- 14‐ andand 15-inch 15‐inch aluminum aluminum alloy alloy wheels wheels werewere constructed.constructed. Taking Taking the the 14 14-inch‐inch wheel wheel as as an an example, example, the the type type of ofthe the wheel wheel was was 5.5 5.5J × J15.× 15.The The material material of ofthe the wheel wheel was was cast cast A356 A356-T6‐T6 aluminum aluminum alloy, alloy, and and the density,density, Young’sYoung’s modulus modulus and and Poisson’s Poisson’s ratio ratio were were 2.7 2.7× ×10 1033 kg/mkg/m3,, 72 72 GPa GPa and 0.3, respectively. TheThe element type was was C3D10, C3D10, and and the the numbers numbers of ofelements elements and and nodes nodes in the in wheel’s the wheel’s FEM FEMwere were 117,905 117,905 and and 202,448, 202,448, respectively. respectively. The The comparison comparison between between the the finitefinite elementelement Appl. Sci. 2021, 11, 3979 simulationsimulation andand thethe experimentexperiment testtest resultsresults is is shown shown in in TableTable2 .2. Both Both the the first first three three natural natural8 of 19 Appl. Sci. 2021, 11, 3979 8 of 19 frequenciesfrequencies and and the the modal modal shapes shapes were were highly highly consistent, consistent, which which verifies verifies the validitythe validity of the of wheelthe wheel FEMs. FEMs. Table 2. Comparison between experiment tests and finite element simulation. TableTable 2. Comparison2. Comparison between between experiment experiment tests tests and and finite finite element element simulation. simulation. Modal Shapes and Modal Shapes Shapes and and Experiment Test Finite Element Simulation ExperimentExperiment TestTest Finite Element Element Simulation Simulation NaturalNatural Frequencies Frequencies Frequencies

1st1st1st mode mode mode

474 Hz 476476 Hz (+0.4%)(+0.4%) 474474 Hz Hz 476 Hz (+0.4%)

2nd2nd mode mode

1062 Hz (−0.7%) 10701070 Hz Hz 1062 Hz (−0.7%)

3rd3rd mode mode

1704 Hz (−1.6%) 17311731 Hz Hz 1704 Hz (−1.6%)

2.3.2.3. Vibration Vibration Energy Energy Transmission Transmission Characteristics Characteristics DirectDirect‐solution‐solution steady steady state state dynamic dynamic analysis analysis was was applied applied to to analyze analyze the the vibration vibration responseresponse and and structure structure intensity intensity distribution distribution of of the the wheels wheels under under TACR TACR excitation. excitation. The The vibrationvibration energy energy generated generated by by TACR TACR first first impacted impacted the the wheel wheel rim, rim, then then the the wheel wheel spokes, spokes, andand then then went went through through the the bolts bolts at at the the hub hub into into the the suspension suspension system, system, and and how how this this energyenergy propagated propagated in in the the wheels wheels was was focused focused on on in in this this work. work. The The cylindrical cylindrical coordinate coordinate withwith the the wheel wheel center center as as the the origin origin was was used used for for analysis. analysis. As As Figure Figure 10 10 shows, shows, along along the the spokespoke is is the the R R direction, direction, and and along along the the rim rim circumference circumference is is the the θ θ direction. direction. Appl.Appl. Sci. Sci. 2021 2021, 11, 11, 3979, 3979 8 8of of 19 19

Appl.Appl. Sci. Sci. 2021 2021, 11, 11, 3979, 3979 8 8of of 19 19

TableTable 2. 2. Comparison Comparison between between experiment experiment tests tests and and finite finite element element simulation. simulation. TableTable 2. 2. Comparison Comparison between between experiment experiment tests tests and and finite finite element element simulation. simulation. ModalModal Shapes Shapes and and ExperimentExperiment Test Test FiniteFinite Element Element Simulation Simulation NaturalModalNaturalModal ShapesFrequencies FrequenciesShapes and and ExperimentExperiment Test Test FiniteFinite Element Element Simulation Simulation NaturalNatural Frequencies Frequencies

1st1st mode mode Appl. Sci. 2021, 11, 3979 1st1st mode mode 8 of 18

Table 2. Cont. 474474 Hz Hz 476476 Hz Hz (+0.4%) (+0.4%) Modal Shapes and Experiment474 Hz Test Finite476 Element476 Hz Hz (+0.4%) Simulation(+0.4%) Natural Frequencies 474 Hz

2nd2nd mode mode 2nd2nd2nd mode mode mode

107010701070 Hz Hz 10621062 Hz Hz ( −(0.7%)−0.7%) 10701070 Hz Hz 10621062 HzHz Hz ( −(− (0.7%)0.7%)−0.7%)

3rd3rd mode mode mode 3rd3rd mode mode

1731 Hz − 17311731 Hz Hz 170417041704 HzHz Hz ( ( −(1.6%)−1.6%) 17311731 Hz Hz 17041704 Hz Hz (− (1.6%)−1.6%) 2.3.2.3.2.3. Vibration Vibration Vibration Energy Energy Energy Transmission Transmission Transmission Characteristics Characteristics Characteristics 2.3.2.3. Direct-solutionVibrationDirect VibrationDirect‐solution‐ solutionEnergy Energy steady steadyTransmission steadyTransmission state state state dynamic dynamicCharacteristics dynamicCharacteristics analysis analysis analysis was was was applied applied applied to to to analyze analyze analyze the the the vibration vibration vibration responseresponseresponseDirectDirect and and and‐solution‐ structuresolution structure structure steady steady intensity intensity intensity state state distribution distributiondynamic distributiondynamic analysis analysis of of of the the the wheelswas wheels waswheels applied applied under under under to TACRto TACRanalyze TACRanalyze excitation. excitation. excitation.the the vibration vibration The The The vibrationvibrationresponsevibrationresponse energy energyand energyand structure structure generated generated generated intensity intensity by by by TACR TACR TACR distribution distribution ﬁrst first first impacted impacted impacted of of the the the the wheelsthe wheels wheel wheel wheel under rim, rim,under rim, then then TACRthen TACR the the the wheelexcitation. wheel excitation. wheel spokes, spokes, spokes, The The andandvibrationandvibration then then then went energywent wentenergy through through throughgenerated generated the the the by bolts boltsby bolts TACR TACR at at at the thefirst thefirst hub hub impactedhub impacted into into into the the thethe the suspension suspension wheelsuspension wheel rim, rim, system, thensystem, thensystem, the the and wheeland andwheel how how how spokes, spokes, this this this energyenergyandenergyand then then propagated propagated propagated went went through through in in in the the the wheelsthe wheelsthe wheels bolts bolts was was wasat at focused thefocused thefocused hub hub on intoon intoon in in in thisthe thisthe this suspension work. work.suspension work. The The The cylindrical cylindricalsystem, cylindricalsystem, and and coordinate coordinate coordinatehow how this this Appl. Sci. 2021, 11, 3979 9 of 19 withwithenergywithenergy the the the propagated wheel wheelpropagated wheel center center center in in as theas asthe the the wheelsthe wheels origin origin origin was was waswas was focused usedfocused used used for for onfor on analysis. analysis.in analysis. in this this work. As work.As As Figure Figure TheFigure The cylindrical 10 cylindrical10 10shows, shows, shows, alongcoordinate alongcoordinate along the the the spokespokewithspokewith the is isthe theis thewheel the wheel R R direction,R direction, direction,center center as and asand theand the along alongorigin alongorigin the the wasthe was rim rim rimused circumferenceused circumference circumference for for analysis. analysis. is is theAsis theAs the Figureθ Figureθdirection. θ direction. direction. 10 10 shows, shows, along along the the spokespoke is is the the R R direction, direction, and and along along the the rim rim circumference circumference is is the the θ θdirection. direction.

FigureFigure 10. 10. TACRTACR sound sound pressure pressure load load model model acting acting on on the the wheel. wheel. Considering the TACR energy transmission path, simulation of the connection be- Considering the TACR energy transmission path, simulation of the connection tween the wheel and suspension system at the hub was required. The radial spring and between the wheel and suspension system at the hub was required. The radial spring and damping were added in the center of the hub to simulate the stiffness and damping of the damping were added in the center of the hub to simulate the stiffness and damping of the suspension system. All freedoms of the bolts were constrained except the R direction in suspension system. All freedoms of the bolts were constrained except the R direction in the cylindrical coordinate system, and the radial springs were set to simulate the action the cylindrical coordinate system, and the radial springs were set to simulate the action of the suspension system on the wheel, since the force from sound pressure in the cavity resulting from the acoustic cavity resonance was normal to the surface of the rim; that is, it was normal along the R direction. The four parallel spring dampers with uniform circumferential distribution were set as shown in the Figure 11.

Figure 11. Springs and dampers simulating the suspension’s stiffness and damping.

Based on vibration theory, the suspension damping can be expressed as 𝐶 𝜉 (4) 2√𝐾𝑀 where ξ is the damping ratio of the suspension system (generally about 0.2–0.45), C represents the equivalent damping coefficient of the suspension shock absorber, K is the suspension stiffness and M indicates the unsprung mass loaded on the suspension system. An ordinary working condition was chosen where K was 40 kN/m, C was 1386 Ns/m and M was 300 kg, referring to [26]. In the future, we will establish different kinds of suspension systems to cover more details. Based on above load models and constraint conditions, simulations of the 14‐inch and 15‐inch aluminum wheels under sound pressure excitation were performed. Figure 12 shows high stress values near the spokes and bolt holes, as the green‐colored areas illustrate. For the 14‐inch and 15‐inch wheels, the maximum von Mises stresses were 89.39 Pa and 152.8 Pa, which were both located at the bolt holes, and the high structure intensity and power flow at these areas could be inferred as well. Appl. Sci. 2021, 11, 3979 9 of 19

Figure 10. TACR sound pressure load model acting on the wheel.

Considering the TACR energy transmission path, simulation of the connection between the wheel and suspension system at the hub was required. The radial spring and Appl. Sci. 2021, 11, 3979 9 of 18 damping were added in the center of the hub to simulate the stiffness and damping of the suspension system. All freedoms of the bolts were constrained except the R direction in the cylindrical coordinate system, and the radial springs were set to simulate the action of theof the suspension suspension system system on onthe the wheel, wheel, since since the the force force from from sound sound pressure pressure in in the the cavity cavity resultingresulting from from the the acoustic acoustic cavity cavity resonance resonance was was normal normal to tothe the surface surface of ofthe the rim; rim; that that is, itis, was it was normal normal along along the the R Rdirection. direction. The The four four parallel parallel spring spring dampers dampers with with uniform circumferentialcircumferential distribution distribution were were set set as shown shown in the Figure 11.11.

FigureFigure 11. 11. SpringsSprings and and dampers dampers simulating simulating the the suspension’s suspension’s stiffness stiffness and and damping. damping.

BasedBased on on vibration vibration theory, theory, the suspension damping can be expressed as C𝐶 ξ =𝜉√ (4)(4) 2 2√KM𝐾𝑀 wherewhere ξξ isis the damping ratio of the suspension system (generally about 0.2–0.45), CC representsrepresents the the equivalent equivalent damping damping coefficient coefﬁcient of of the the suspension suspension shock shock absorber, absorber, KK isis the the suspensionsuspension stiffness stiffness and and MM indicatesindicates the the unsprung unsprung mass mass loaded loaded on on the the suspension suspension system. system. AnAn ordinary ordinary working working condition was chosen wherewhere KK waswas 40 40 kN/m, kN/m, C was 1386 Ns/m andand MM waswas 300 300 kg, kg, referring referring to to [26]. [26]. In In the the future, future, we we will will establish establish different different kinds kinds of of suspensionsuspension systems systems to to cover cover more more details. details. BasedBased on on above above load load models models and and constraint conditions, simulations simulations of of the the 14 14-inch‐inch andand 15 15-inch‐inch aluminum wheelswheels underunder sound sound pressure pressure excitation excitation were were performed. performed.Figure Figure 12 12shows shows high high stress stress values values near near the spokes the spokes and bolt and holes, bolt asholes, the green-colored as the green‐ areascolored illustrate. areas illustrate.For the 14-inch For the and 14‐inch 15-inch and wheels, 15‐inch thewheels, maximum the maximum von Mises von stresses Mises stresses were 89.39 were Pa 89.39 and Appl. Sci. 2021, 11, 3979 10 of 19 Pa152.8 and Pa, 152.8 which Pa, which were both were located both located at the at bolt the holes, bolt holes, and the and high the high structure structure intensity intensity and andpower power ﬂow flow at these at these areas areas could could be inferred be inferred as well. as well.

(a) (b)

FigureFigure 12. 12. StressStress distributions distributions of ofthe the two two wheels: wheels: (a) ( a14) 14-inch‐inch wheel wheel and and (b ()b 15) 15-inch‐inch wheel. wheel.

TheThe power power flow flow represents represents the the ability ability of of external external forces forces to to work work or or structures structures to to dissipatedissipate energy energy per per unit unit of of time. time. The The instantaneous instantaneous power power flow flow is isdefined defined as as the the product product ofof the the force force and and the the velocity velocity in in the the same same direction direction and and phase: phase:

𝑃𝐹P = F(𝑡t)∙𝑉·V(t𝑡) (5)(5) where P represents the power flow, F(t) is the external force and V(t) indicates the response velocity. For vibration analysis, the average power flow in a certain period of time can better reflect the energy intensity of the excited structure and can be expressed as follows: 1 𝑃 lim 𝐹𝑡 ∙𝑉𝑡𝑑𝑡 (6) 𝑇 → For complex excitation 𝐹𝑡 𝐹𝑒, the corresponding velocity response is 𝑉𝑡 𝑉𝑒. Thus, the time‐averaged power flow expression is 1 1 𝑃 lim Re𝐹𝑒 ∙Re𝑉𝑒𝑑𝑡 Re𝐹𝑉∗ (7) 𝑇 → 2 where F and V are the complex amplitudes of force and velocity, respectively, * means the conjugation of a complex number and Re indicates the real part of a complex number. Structure damping is one method of energy dissipation. It was necessary to consider that in this work because there is no vibration system without structure damping in practice. In calculating the power flow, the phases of the force and velocity vector were vertical if there was no wheel structure damping, and according to Equation (7), there would be no power flow. When it comes to finite element simulation, the stress state at any point in the elastic body could be represented by six stress components, namely normal stress 𝜎, 𝜎, 𝜎 and shear stress 𝜏, 𝜏, 𝜏. For an arbitrary direction n, the structure intensity field of a microelement in the elastic body was as shown in Figure 13.

Figure 13. Structure intensity field of a microelement in the elastic body. Appl. Sci. 2021, 11, 3979 10 of 19

(a) (b) Figure 12. Stress distributions of the two wheels: (a) 14‐inch wheel and (b) 15‐inch wheel.

The power flow represents the ability of external forces to work or structures to Appl. Sci. 2021, 11, 3979 dissipate energy per unit of time. The instantaneous power flow is defined as the product10 of 18 of the force and the velocity in the same direction and phase: 𝑃𝐹𝑡 ∙𝑉𝑡 (5) where P represents the power flow, F(t) is the external force and V(t) indicates the response where P represents the power flow, F(t) is the external force and V(t) indicates the velocity. response velocity. For vibration analysis, the average power flow in a certain period of time can better For vibration analysis, the average power flow in a certain period of time can better reflect the energy intensity of the excited structure and can be expressed as follows: reflect the energy intensity of the excited structure and can be expressed as follows: 11 Z T P𝑃= limlim 𝐹F(𝑡t)·∙𝑉V(t𝑡)dt𝑑𝑡 (6) T T→∞ 0 (6) 𝑇 → ForFor complex complex excitation excitation 𝐹F(𝑡t)𝐹𝑒= Feiωt,, the the corresponding velocity velocity response response is is 𝑉V(𝑡t) = 𝑉𝑒Veiωt.. Thus, Thus, the the time time-averaged‐averaged power power flow flow expression expression is is T 1 1 Z n o n o 11 ∗ =𝑃 lim Re𝐹𝑒iωt ·∙Re𝑉𝑒iωt 𝑑𝑡 = Re{𝐹𝑉 ∗ } (7) P lim→ Re Fe Re Ve dt Re FV (7) T T𝑇→∞ 0 22 wherewhere FF andand VV areare the the complex complex amplitudes amplitudes of of force force and and velocity, velocity, respectively, respectively, * * means means the the conjugationconjugation of of a a complex complex number number and and Re Re indicates indicates the the real real part part of of a a complex complex number. number. StructureStructure damping damping is is one one method method of of energy energy dissipation. dissipation. It It was was necessary necessary to to consider consider thatthat in thisthis work work because because there there is no is vibrationno vibration system system without without structure structure damping damping in practice. in practice.In calculating In calculating the power the flow, power the flow, phases the of phases the force of the and force velocity and vector velocity were vector vertical were if verticalthere was if there no wheel was structureno wheel damping, structure and damping, according and to according Equation to (7), Equation there would (7), there be no wouldpower be flow. no power flow. WhenWhen it it comes comes to to finite finite element element simulation, simulation, the the stress stress state state at at any any point point in in the the elastic elastic bodybody could could be be represented represented by by six six stress stress components, components, namely namely normal normal stress stress 𝜎σ, x𝜎,σ,y ,𝜎σ zandand shearshear stress stress 𝜏τxy,, 𝜏τyz,, τ𝜏zx. For For an an arbitrary arbitrary direction direction nn,, the the structure structure intensity intensity field field of of a a microelementmicroelement in in the the elastic elastic body body was was as as shown shown in in Figure Figure 13.13.

FigureFigure 13. 13. StructureStructure intensity intensity field ﬁeld of of a a microelement microelement in in the the elastic elastic body. body.

The structure intensity, deﬁned as the power ﬂow per unit area, can be expressed as follows [27]: 1 p = − Re(σ v∗ + τ v∗ + τ v∗) (8) n 2 n n n1 1 n2 2

where σn represents the normal stress in the normal n direction, τn1 and τn2 are the shear ∗ ∗ ∗ stress in the direction of 1 and 2, respectively, and vn, v1 and v2 indicate the complex conjugate of velocities in the normal n, 1 and 2 directions, respectively. As parameters like the element stress and displacement were gained from harmonic response analysis, the structure intensity could then be calculated. Solid elements had degrees of freedom in three directions, namely x, y and z. The structure intensity could then be expressed by the stress and displacement parameters as ω p = − Imσ u∗ + τ v∗ + τ w∗ (9) x 2 x xy xz ω p = − Imτ u∗ + σ v∗ + τ w∗ (10) y 2 yx y yz ω p = − Imτ u∗ + τ v∗ + σ w∗ (11) z 2 zx zy z Appl. Sci. 2021, 11, 3979 11 of 19

The structure intensity, defined as the power flow per unit area, can be expressed as follows [27]: 1 𝑝 Re𝜎 𝑣∗ 𝜏 𝑣∗ 𝜏 𝑣∗ (8) 2

where 𝜎 represents the normal stress in the normal n direction, 𝜏 and 𝜏 are the ∗ ∗ ∗ shear stress in the direction of 1 and 2, respectively, and 𝑣 , 𝑣 and 𝑣 indicate the complex conjugate of velocities in the normal n, 1 and 2 directions, respectively. As parameters like the element stress and displacement were gained from harmonic response analysis, the structure intensity could then be calculated. Solid elements had degrees of freedom in three directions, namely x, y and z. The structure intensity could then be expressed by the stress and displacement parameters as 𝜔 𝑝 Im𝜎 𝑢∗ 𝜏 𝑣∗ 𝜏 𝑤∗ (9) 2 𝜔 𝑝 Im𝜏 𝑢∗ 𝜎 𝑣∗ 𝜏 𝑤∗ (10) 2 Appl. Sci. 2021, 11, 3979 11 of 18 𝜔 𝑝 Im𝜏 𝑢∗ 𝜏 𝑣∗ 𝜎 𝑤∗ (11) 2

where 𝑝 , 𝑝 and 𝑝 indicate the structure intensity in the x, y and z directions, where px, py and pz indicate the structure intensity in the x, y and z directions, respectively; 𝑢∗ 𝑣∗ 𝑤∗ respectively;ω is the circular ω is frequency; the circular and ufrequency;∗, v∗ and w and∗ indicate , the and complex indicate conjugate the of velocitiescomplex conjugatein the x, y ofand velocitiesz directions, in the x respectively., y and z directions, According respectively. to the results According of the to ﬁnite the results element of thesimulation finite element in Figure simulation 12, the complex in Figure stress and12, displacementthe complex distributionsstress and displacement under TACR distributionsfrequency excitation under TACR were obtained. frequency To excitation calculate were the structure obtained. intensity To calculate of each the node structure of the intensityFEMs, an of in-house each node code of the including FEMs, an Matlab in‐house and code Python including scripts Matlab was developed. and Python The scripts ﬂow waschart developed. of the calculation The flow process chart of was the as calculation given in Figure process 14 .was as given in Figure 14.

Appl. Sci. 2021, 11, 3979 12 of 19

Figure 14. Power flow calculation process. Figure 14. Power flow calculation process. AfterAfter the the calculations calculations above, above, we we could could get get the visualizedvisualized structurestructure intensityintensity distri- distributionsbutions of of the the 14-inch 14‐inch and and 15-inch 15‐inch wheels,wheels, as shown shown in in Figure Figure 15. 15 It. can It can be beseen seen that that in inwheels wheels ofof differentdifferent sizes,sizes, there were significant significant differences differences in in terms terms of of energy energy levels levels and andlocations locations of of the the maximum maximum structure structure intensity. intensity. Therefore, Therefore, it it is is possible possible to to control control the the energy energydistribution distribution and and transmission transmission of powerof power flow flow by by changing changing the the wheel wheel structure,structure, so so as to asachieve to achieve an an optimal optimal design. design.

(a) (b)

FigureFigure 15. 15. StructureStructure intensity intensity distributions distributions of the of thetwo two wheels: wheels: (a) 14 (a‐)inch 14-inch wheel wheel and and (b) 15 (b‐)inch 15-inch wheel. wheel.

As the structure intensity calculated in Section 2.3 was the power flow per unit area, to compare the energy transmission ability of different structures, the power flow calculation in the whole area was required. Based on the wheel simulation results at different nodes, the average power flow at the given area was defined as

∑ 𝑝 𝑝 𝑆 (12) 𝑁

where 𝑝 represents the structure intensity at the ith node, S represents the input/output cross‐sectional area, N indicates the input/output number of nodes and 𝑝 means the power flow through a certain area. The power flow input area was defined as the rim surface where the TACR energy first impacted the wheel, and the output area was defined as the bolt holes where the energy transferred into the suspension system. Then, as the power flow of the input and output areas in the wheel was calculated, the transmission efficiency of the TACR energy was defined as the ratio of the input and output power flow in this paper, which was calculated as

𝑃 𝜑 100% (13) 𝑃

where 𝑃 represents the power flow at the rim where excitation force is applied and 𝑃 indicates the power flow at the bolt holes where the wheel and suspension connect. The transmission efficiency could help analyze and compare the energy transmission characteristics between wheels of different structures, as it could determine the ability of the structure to dissipate energy through the wheels and other structures intuitively and conveniently. Here, the power flow method was used in the TACR energy transmission characteristic investigation. The concept of using transmission efficiency to determine the structure energy dissipation ability could guide wheel structure optimization in further research. Additionally, later in this paper, this method is used to investigate the influence of several wheel design parameters on the TACR energy transmission characteristics. Table 3 shows the calculation results of the input and output power flow for the 14‐inch and 15‐inch aluminum alloy wheels.

Appl. Sci. 2021, 11, 3979 12 of 18

As the structure intensity calculated in Section 2.3 was the power flow per unit area, to compare the energy transmission ability of different structures, the power flow calculation in the whole area was required. Based on the wheel simulation results at different nodes, the average power flow at the given area was defined as

∑ p p = si S (12) p f N

where psi represents the structure intensity at the ith node, S represents the input/output cross-sectional area, N indicates the input/output number of nodes and pp f means the power flow through a certain area. The power flow input area was defined as the rim surface where the TACR energy first impacted the wheel, and the output area was defined as the bolt holes where the energy transferred into the suspension system. Then, as the power flow of the input and output areas in the wheel was calculated, the transmission efficiency of the TACR energy was defined as the ratio of the input and output power flow in this paper, which was calculated as

P ϕ = out × 100% (13) Pin

where Pin represents the power flow at the rim where excitation force is applied and Pout indicates the power flow at the bolt holes where the wheel and suspension connect. The transmission efficiency could help analyze and compare the energy transmission characteristics between wheels of different structures, as it could determine the ability of the structure to dissipate energy through the wheels and other structures intuitively and conveniently. Here, the power flow method was used in the TACR energy transmission characteristic investigation. The concept of using transmission efficiency to determine the structure energy dissipation ability could guide wheel structure optimization in further research. Additionally, later in this paper, this method is used to investigate the influence of several wheel design parameters on the TACR energy transmission characteristics. Table3 shows the calculation results of the input and output power flow for the 14-inch and 15-inch aluminum alloy wheels.

Table 3. Input and output power ﬂow of the 14-inch and 15-inch wheels.

Title 2 Title 3 Type Input Output Input Output −3 2 ∑ psi (10 W/mm ) 48.26 3.94 145.4 0.58 N 14,580 480 18,868 600 S (mm2) 58,273 1214 64,004 1649 −3 Pp f (10 W) 192.9 9.967 493.2 1.586 ϕ 5.2% 0.3%

The transmission efﬁciencies of the 14-inch and 15-inch wheels were both low. The TACR energy entered the wheel from the rim to the hub, and most of the energy was dissipated by structure damping and wave interference. Since the elastic waves had the same frequency but different phases, they would interfere with each other during propagation in different positions in the wheels. Additionally, the much smaller output cross-sectional area compared with the input one also led to a low output efﬁciency. The huge difference shown in Table3 can be explained from the perspective of the wheel structure difference. The locations of the structure intensity maximum values in the two sizes of the wheels are shown in Figure 16, and it can be seen that the maximum value in the 14-inch wheel was located at the bolt holes, while the maximum value in the 15-inch wheel was located at the rim outside, and the former was higher than the latter. For a 14-inch wheel, the TACR energy could propagate to the hub through the spokes, while Appl. Sci. 2021, 11, 3979 13 of 19

Table 3. Input and output power flow of the 14‐inch and 15‐inch wheels.

Title 2 Title 3 Type Input Output Input Output −3 2 ∑ 𝑝 (10 W/mm ) 48.26 3.94 145.4 0.58 N 14,580 480 18,868 600 S (mm2) 58,273 1214 64,004 1649 −3 𝑃 (10 W) 192.9 9.967 493.2 1.586 𝜑 5.2% 0.3%

The transmission efficiencies of the 14‐inch and 15‐inch wheels were both low. The TACR energy entered the wheel from the rim to the hub, and most of the energy was dissipated by structure damping and wave interference. Since the elastic waves had the same frequency but different phases, they would interfere with each other during propagation in different positions in the wheels. Additionally, the much smaller output cross‐sectional area compared with the input one also led to a low output efficiency. The huge difference shown in Table 3 can be explained from the perspective of the wheel structure difference. The locations of the structure intensity maximum values in the two sizes of the wheels are shown in Figure 16, and it can be seen that the maximum value Appl. Sci. 2021, 11, 3979 13 of 18 in the 14‐inch wheel was located at the bolt holes, while the maximum value in the 15‐ inch wheel was located at the rim outside, and the former was higher than the latter. For a 14‐inch wheel, the TACR energy could propagate to the hub through the spokes, while forfor a a15 15-inch‐inch wheel, wheel, most most of of the the energy energy could could not not reach reach the the hub hub (for (for the the wheels wheels involved involved inin this this paper). paper). This This further further shows shows that that the the wheel wheel structure structure has has a ahuge huge design design space space for for suppressingsuppressing the the propagation propagation of of TACR TACR energy, energy, which which is is worth worth discussing. discussing.

(a) (b)

FigureFigure 16. 16. LocationsLocations of of the the structure structure intensity intensity max max values values in in the the two two wheels: wheels: (a) ( a14) 14-inch‐inch wheel wheel and and (b) 15‐inch wheel. (b) 15-inch wheel.

3.3. Influence Influence Factors Factors Discussion Discussion 3.1.3.1. Structure Structure Damping Damping StructureStructure damping damping is is a a parameter reflectingreflecting the the energy energy dissipation dissipation in in the the vibration vibration pro- process,cess, the the effect effect of which of which on TACR on TACR energy energy transmission transmission is studied is instudied this section. in this Aluminum section. Aluminumalloy wheel alloy structure wheel damping structure generally damping varies generally from 1% varies to 2%; from thus, 1% the structureto 2%; thus, damping the structurewas set to damping be 0.01, 0.015 was or set 0.02 to in be the 0.01, simulation, 0.015 or while 0.02 other in the parameters simulation, remained while constant. other Appl. Sci. 2021, 11, 3979 14 of 19 parametersThe input andremained output constant. power flow The calculationinput and output results power in the flow two sizescalculation of wheels results are in shown the twoin Figuresizes of 17 wheels. are shown in Figure 17.

(a) (b)

FigureFigure 17.17. InputInput and output power flowflow inin thethe wheelswheels versusversus structurestructure damping:damping: (a) inputinput power flowflow andand ((bb)) outputoutput powerpower flow.flow.

These results indicate that with the increase in structurestructure damping,damping, the inputinput andand output power ﬂowsflows both decreased,decreased, asas expected.expected. The power flow ﬂow change and rate ofof decrease areare shown shown in in Table Table4. Damping 4. Damping represents represents the energy the energy dissipation dissipation ability, ability, so TACR so energyTACR energy loss in loss the in transmission the transmission process process was greater was greater under under the same the same conditions conditions with with the the increase in structure damping. There are several effective ways to increase the wheel structure damping, such as additional components or using fiber‐reinforced composite materials [28,29].

Table 4. Input and output power flow in the wheels under different structure damping.

Power Flow (10−3 W) Structure 14‐Inch Wheel 15‐Inch Wheel Damping Input Output Input Output 0.01 192.87 (100%) 9.9672 (100%) 493.16 (100%) 1.5855 (100%) 0.015 130.01 (−32.6%) 6.8411 (−31.7%) 309.23 (−37.3%) 1.1138 (−29.8%) 0.02 99.444 (−48.4%) 5.2128 (−47.7%) 229.95 (−53.4%) 0.8527 (46.2%)

3.2. Spoke Number Wheels consist of spokes and rims. Since the rim follows specific standards, the spokes, as the connection between the rim and hub, are highly designable. In this section, the effect of the number of spokes on power flow transmission is investigated. When adjusting the design scheme, the shape of the rim remained the same while the hub was simplified as a support, and the spokes number was adjusted to four, five, six, seven, eight or nine. To keep the total wheel mass unchanged, the center angle each spoke covered varied with the spoke number. For the wheel with four spokes, the center angle of each spoke was 45°. As the spoke number increased, the center angle of a single spoke changed to 36°, 30°, 25.7°, 22.5° or 20°. Simplified wheel models with different numbers of spokes are shown in Figure 18. Appl. Sci. 2021, 11, 3979 14 of 18

increase in structure damping. There are several effective ways to increase the wheel structure damping, such as additional components or using ﬁber-reinforced composite materials [28,29].

Table 4. Input and output power ﬂow in the wheels under different structure damping.

Power Flow (10−3 W) Structure 14-Inch Wheel 15-Inch Wheel Damping Input Output Input Output 0.01 192.87 (100%) 9.9672 (100%) 493.16 (100%) 1.5855 (100%) 0.015 130.01 (−32.6%) 6.8411 (−31.7%) 309.23 (−37.3%) 1.1138 (−29.8%) 0.02 99.444 (−48.4%) 5.2128 (−47.7%) 229.95 (−53.4%) 0.8527 (46.2%)

3.2. Spoke Number Wheels consist of spokes and rims. Since the rim follows specific standards, the spokes, as the connection between the rim and hub, are highly designable. In this section, the effect of the number of spokes on power flow transmission is investigated. When adjusting the design scheme, the shape of the rim remained the same while the hub was simplified as a support, and the spokes number was adjusted to four, five, six, seven, eight or nine. To keep the total wheel mass unchanged, the center angle each spoke covered varied with the spoke number. For the wheel with four spokes, the center angle of each spoke was 45◦. As the spoke number increased, the center angle of a single spoke changed to 36◦, 30◦, Appl. Sci. 2021, 11, 3979 ◦ ◦ ◦ 15 of 19 25.7 , 22.5 or 20 . Simplified wheel models with different numbers of spokes are shown in Figure 18.

FigureFigure 18. 18.Simpliﬁed Simplified wheel wheel models models with with spokes spokes of of different different numbers. numbers.

Then,Then, the the input input and and output output power power flows flows of of those those simplified simplified wheels wheels were were calculated. calculated. TakingTaking the the 15-inch 15‐inch aluminum aluminum alloy alloy wheels wheels as an exampleas an example in Figure in 19 Figure(a similar 19 conclusion (a similar couldconclusion be drawn could from be thedrawn 14-inch from wheels), the 14 the‐inch wheels wheels), with the odd-numbered wheels with spokes odd‐numbered showed greaterspokes inputshowed and greater output input power and flowsoutput than power the flows ones than with the even-numbered ones with even spokes.‐numbered In addition,spokes. In the addition, input and the output input powerand output flows power increased flows in theincreased case of in the the higher case oddof the number higher ofodd spokes. number of spokes. As the wheels with even-numbered spokes showed a lower transmission efficiency of vibration energy, therefore, in order to suppress TACR energy propagation, the wheels with even-numbered spokes should be adopted for the wheel structure in this work. According to the above results, the wheel with four or eight spokes should be chosen in the case of the same wheel size and structure. Then, the structure intensity distributions of the 15-inch wheel with 5, 7 or 9 spokes were given as in Figure 20. The maximum values of the structure intensity in those wheels increased as the number of spokes varied from 5 to 9, and the maximum values were located near the inboard flanges, which were located at the node of the cosine load model. The greater deformation was due to the lower stiffness of the inboard flanges than the outboard ones. Additionally, it can be seen that the structure intensity decreased as the energy went through the spokes to the bolt holes, where the wheel and suspension connect together.

(a) (b) Figure 19. Input and output power flow in the simplified 15‐inch wheels: (a) input power flow and (b) output power flow.

Figure 18. Simplified wheel models with spokes of different numbers.

Then, the input and output power flows of those simplified wheels were calculated. Taking the 15‐inch aluminum alloy wheels as an example in Figure 19 (a similar conclusion could be drawn from the 14‐inch wheels), the wheels with odd‐numbered spokes showed greater input and output power flows than the ones with even‐numbered Appl. Sci. 11 2021, , 3979 spokes. In addition, the input and output power flows increased in the case of the higher15 of 18 odd number of spokes.

Appl. Sci. 20212021,, 1111,, 39793979 1616 ofof 1919 (a) (b) FigureFigure 19. Input and output power flowflow in the simplified simplified 15-inch15‐inch wheels: (a) input power flow flow and ( b) output power flow. flow.

As the wheels with even‐numbered spokes showed a lower transmission efficiency of vibration energy, therefore, in order to suppress TACR energy propagation, the wheels with even‐numbered spokes should be adopted for the wheel structure in this work. According to the above results, the wheel with four or eight spokes should be chosen in the case of the same wheel size and structure. Then, the structure intensity distributions of the 15‐inch wheel with 5, 7 or 9 spokes were given as in Figure 20. The maximum values of the structure intensity in those wheels increased as the number of spokes varied from 5 to 9, and the maximum values were located near the inboard flanges, which were located at the node of the cosine load model. ((a)) Five‐‐spokespoke wheelThe greater deformation((b)) Seven was‐‐spokespoke due wheelto the lower stiffness of((c ))the Nine inboard‐‐spokespoke flangeswheel than the outboard ones. Additionally, it can be seen that the structure intensity decreased as the FigureFigure 20.20. StructureStructure intensity intensityintensity distributions distributions in inin simplified simplifiedsimplified wheels wheels with with spokes spokesspokes of of different different numbers: numbers: (a) (( five-spokea)) fivefive‐‐spokespoke wheel; wheel; (b) ((b)) sevenseven‐‐spokespoke wheel; and energy ((cc)) nine ‐‐spokewentspoke wheel.through the spokes to the bolt holes, where the wheel and suspension seven-spoke wheel; and (c) nine-spokeconnect together. wheel. To knowknow howhow thethethe powerpower flowflowflow propagatedpropagated ininin thethethe wheelswheels andand clarifyclarify thethethe influenceinfluenceinfluence mechanismmechanism of the spokespoke number,number, thethe simplifiedsimplifiedsimplified roundroundround diskdisk modelsmodels (RDMs)(RDMs)(RDMs) withoutwithout andand withwith spokesspokes of of different different numbers numbers were were constructed constructed as asshownshown shown inin Figure in Figure 21. The21. RDMs The RDMs with witha 10 mm a 10 thicknessthickness mm thickness had thethe had samesame the cross same‐‐sectionsection cross-section shapeshape as shape thethe simplifiedsimplified as the simplified wheel inin Figure wheel 20, in Figurewhich 20was, which used was toto usedanalyze to analyze thethe influenceinfluence the influence of thethe of spoke thespoke spoke number number on on thethe the power power flowflowflow transmission.transmission. Vector graphsgraphs ofof theirtheir structurestructure intensitiesintensitiesintensities underunder thethethe samesamesame soundsoundsound pressurepressure excitationexcitation areare shownshown inin FigureFigure 2222..

FigureFigure 21.21. SimplifiedSimpliﬁed RDMsRDMs withoutwithout andand withwith spokesspokes ofof differentdifferent numbers.numbers.

In the vector graphs, the length and the arrow direction of the vector represent the amount and direction of the power flow at a specific point, respectively, so the power flow distribution in the wheel can be clearly observed. From the comparison of the

((a)) ((b)) ((c)) Figure 22. Vector graphs of thethe structurestructure intensitiesintensities inin RDMs: ((a)) zero‐‐spokespoke RDM; ((b)) fourfour‐‐spokespoke RDM; and ((cc)) fivefive‐‐spokespoke RDM.

InIn thethe vector graphs, thethe lengthlength and thethe arrow direction of thethe vector representrepresent thethe amount and direction of thethe power flowflow at a specificspecific point, respectively,respectively, soso thethe power flowflow distribution inin thethe wheel can be clearly observed. From thethe comparison of thethe severalseveral simulationsimulation resultsresults inin Figure 22, itit can be seenseen thatthat thethe structurestructure intensityintensity gradually attenuated fromfrom thethe spokespoke edge toto thethe wheel center area, and thethe structurestructure intensityintensity along Appl. Sci. 2021, 11, 3979 16 of 19

(a) Five‐spoke wheel (b) Seven‐spoke wheel (c) Nine‐spoke wheel Figure 20. Structure intensity distributions in simplified wheels with spokes of different numbers: (a) five‐spoke wheel; (b) seven‐spoke wheel; and (c) nine‐spoke wheel.

To know how the power flow propagated in the wheels and clarify the influence mechanism of the spoke number, the simplified round disk models (RDMs) without and with spokes of different numbers were constructed as shown in Figure 21. The RDMs with a 10 mm thickness had the same cross‐section shape as the simplified wheel in Figure 20, Appl. Sci. 2021, 11, 3979 which was used to analyze the influence of the spoke number on the power16 flow of 18 transmission. Vector graphs of their structure intensities under the same sound pressure excitation are shown in Figure 22. several simulation results in Figure 22, it can be seen that the structure intensity gradually attenuated from the spoke edge to the wheel center area, and the structure intensity along the load direction was significantly higher than the other areas. In the RDM with four and five spokes, several significantly high stress areas appeared at the spoke edge, which may have been due to the abrupt change of the round disk section. In the local zoom of the top spoke in the four-spoke RDM shown in Figure 23, the propagation direction of power flow can be observed more clearly. It can be seen that in one single spoke, the power flowed toward and away from the wheel center at the same time. Additionally, the length of the vector became smaller from the spoke edge to the wheel center. Therefore, this indicates that the interference of waves with the same frequency but Figuredifferent 21. phasesSimplified was RDMs one reasonwithout that and thewith energy spokes couldof different not reach numbers. the hub.

Appl. Sci. 2021, 11, 3979 17 of 19

the load direction was significantly higher than the other areas. In the RDM with four and five spokes, several significantly high stress areas appeared at the spoke edge, which may have been due to the abrupt change of the round disk section. In the local zoom of the top spoke in the four‐spoke RDM shown in Figure 23, the propagation direction of power flow can be observed more clearly. It can be seen that in one single spoke, the power flowed toward and away from the wheel center at the same (a) time. Additionally, the length(b) of the vector became smaller from(c) the spoke edge to the wheel center. Therefore, this indicates that the interference of waves with the same Figure 22. 22. VectorVector graphs graphs of of the the structure structure intensities intensities in RDMs: in RDMs: (a) zero (a) zero-spoke‐spoke RDM; RDM; (b) four (b)‐ four-spokespoke RDM; RDM; and (c and) five (‐cspoke) ﬁve- frequency but different phases was one reason that the energy could not reach the hub. RDM.spoke RDM.

In the vector graphs, the length and the arrow direction of the vector represent the amount and direction of the power flow at a specific point, respectively, so the power flow distribution in the wheel can be clearly observed. From the comparison of the several simulation results in Figure 22, it can be seen that the structure intensity gradually attenuated from the spoke edge to the wheel center area, and the structure intensity along

FigureFigure 23.23. LocalLocal zoomzoom ofof thethe toptop spokespoke inin thethe four-spokefour‐spoke RDM. RDM.

4.4. ConclusionsConclusions InIn thisthis paper,paper, 14-inch 14‐inch and and 15-inch 15‐inch aluminum aluminum alloy alloy wheel wheel FEMs FEMs were were constructed constructed to to analyzeanalyze the the transmissiontransmission characteristics characteristics of of the the TACR TACR energy energy in in wheels wheels based based on on the the power power flowflow method.method. Then,Then, thethe effectseffects ofof structurestructure dampingdamping andand thethe number number of of spokes spokes were were also also studied.studied. OurOur conclusionsconclusions areare asas follows:follows: 1.1. The sound pressure distributiondistribution inin thethe acousticacoustic loadload modelmodel generatedgenerated byby TACRTACR was was put forward firstfirst andand provenproven toto bebe feasible.feasible. TheThe powerpower flowflow method,method, whenwhen first first used used in the investigation of energy transmissiontransmission characteristicscharacteristics inin wheels,wheels, is is helpful helpful to to the the quantitative description of energy propagation;propagation; 2. The distribution of TACR energy during propagation was different for different structures of wheels, and the values of the input and output power flows varied greatly. Therefore, it is feasible to reduce the resonance energy to propagate into the suspension system and further influence passengers through improving the design of the wheel; 3. When the number of spokes was odd, both the input and output of power flows under the same excitation were larger than for wheels with even‐numbered spokes. Among the wheels with 4–9 spokes, wheels with four or eight spokes should be chosen to reduce the TACR energy propagation. The current work is an exploration of suppressing the TACR noise transmission into the car cabin. For the wheel design, this work only considered the influence of structure damping and the number of spokes. The other factors, such as the rim profile shape, spoke thickness and hub connection geometry, can be studied by the power flow method in subsequent works.

2. The distribution of TACR energy during propagation was different for different structures of wheels, and the values of the input and output power flows varied greatly. Therefore, it is feasible to reduce the resonance energy to propagate into the suspension system and further influence passengers through improving the design of the wheel; 3. When the number of spokes was odd, both the input and output of power flows under the same excitation were larger than for wheels with even-numbered spokes. Among the wheels with 4–9 spokes, wheels with four or eight spokes should be chosen to reduce the TACR energy propagation. The current work is an exploration of suppressing the TACR noise transmission into the car cabin. For the wheel design, this work only considered the influence of structure damping and the number of spokes. The other factors, such as the rim profile shape, spoke thickness and hub connection geometry, can be studied by the power flow method in subsequent works.

Author Contributions: Conceptualization, X.L. and W.Z.; methodology, Y.L. and W.Z.; software, Y.L., W.Z. and Y.S.; validation, W.Z. and Y.S.; writing—original draft preparation, Y.L. and W.Z.; writing—review and editing, X.L., Y.S. and X.H. All authors have read and agreed to the published version of the manuscript. Funding: This study was supported by the National Natural Science Foundation of China, grant number 51675021. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: We thank all the participants in this study. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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