Hindawi Applied Bionics and Biomechanics Volume 2021, Article ID 9976488, 16 pages https://doi.org/10.1155/2021/9976488 Research Article Research on Vibration Reduction Method of Nonpneumatic Tire Spoke Based on the Mechanical Properties of Domestic cat’s Paw Pads 1 1 1 1 2 Haichao Zhou , Huiyun Li , Ye Mei , Guolin Wang, Congzhen Liu, and Lingxin Zhang School of Automotive and Traﬃc Engineering, Jiangsu University, Zhenjiang 212013, China School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255000, China AEOLUS Tyre Co. Ltd., Jiaozuo 454003, China Correspondence should be addressed to Ye Mei; email@example.com Received 24 March 2021; Revised 8 April 2021; Accepted 6 May 2021; Published 17 May 2021 Academic Editor: Donato Romano Copyright © 2021 Haichao Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Although there is no risk of puncture, the vibration problem caused by discontinuous structures limits nonpneumatic tire development (NPT). The vibration reduction of nonpneumatic tires is a solvable urgent problem. This current study analyzed the dynamic grounding characteristics and the vibration reduction mechanism of the cat’s paw pads and then applied the mechanical properties to the bionic design of nonpneumatic tire spokes to solve the vibration problem. Domestic cats’ paw pads’ dynamic grounding characteristics were determined using the pressure-sensitive walkway, high-speed camera, and VIC-2D. The results indicated that the mechanical characteristics of swing deformation of paw pads during the grounding process attenuated the grounding stress and buﬀered the energy storage to achieve the vibration reduction eﬀect. According to the similarity transformation, a ﬁnite element model of NPT that could accurately reconstruct the structure and realistically reﬂect the load deformation was employed. The structure design of asymmetric arcs on the spokes’ side edges was proposed, and it can eﬀectively reduce the radial excitation force of NPT. The three parameters, the asymmetric arc, the thickness, and the curvature of spokes, were used as design variables to maximize the vibration reduction. The orthogonal experimental, the Kriging approximate model, and the genetic algorithm were carefully selected for optimal solutions. Compared with the original tire, the results showed that peak amplitude 1, peak amplitude 2, and the root square of the optimized tire’s amplitudes were reduced by 76.07%, 52.88%, and 51.65%, respectively. These research results oﬀer great potential guidance in the design of low-vibration NPT. 1. Introduction launched by Bridgestone have emerged . These include the honeycomb structure tires which are jointly developed As the only direct contact medium between the vehicle by Resilient Technologies and the University of Wisconsin- and the road, the tire directly aﬀects the steering stability, Madison Polymer Engineering Center  and i-Flex non- driving the vehicle’s safety and riding comfort. However, pneumatic safety tire designed by Hankook . Similarly, traditional pneumatic tires have safety hazards such as “N-wheel” nonpneumatic tire with negative Poisson’s ratio punctures and air leaks, which seriously aﬀect safe vehicle spoke structure was developed by Tianqu Non-Pneumatic driving . Therefore, nonpneumatic tires (NPT) have Wheel Technology Co., Ltd. and BAIC Group . Mechani- demonstrated development advantages regarding safety, cal elastic wheel was also proposed by Zhao Youqun of economy, environmental protection, and wear resistance Nanjing University of Aeronautics and Astronautics [8, 9]. . Since the advent of Michelin’s Tweel tire , a variety And nonpneumatic tires with diﬀerent support structures such as truss, octagonal and gradient elasticity [10–12] are of nonpneumatic tires such as the Air Free Concept 2 Applied Bionics and Biomechanics Glass plate White light e walkway High-speed camera e high-speed camera (a) (b) Figure 1: The contact tests of paw pads: (a) contact pressure test and (b) contact strain test. but few of the evolution in nonpneumatic tire research, th design, and development. 5 Peculiar to all these innovations is a prominent vibration problem associated with nonpneumatic tires, which limits th the speed of vehicles running on nonpneumatic tires. This Palm pad remains a challenge in the development of nonpneumatic rd tires. Consequently, vibration reduction has become one of the key areas of improvement in the development of non- nd pneumatic tires. Compared with the better uniformity of mass distribution of pneumatic tires, the discontinuous sup- port structures of nonpneumatic tires introduce nonuniform Figure 2: The paw pad of the cat. mass distribution, causing nonuniform stiﬀness that results in a local vibration eﬀect [13–15]. As a replacement for pneu- ceived stimuli, making them greatly adaptable and compliant matic tires’ air pressure, the nonpneumatic tires’ spokes play the role of supporting, cushioning, damping, and providing to environmental changes, to be eﬀective and reliable. The force [16, 17]. Hence, this has made the design of spokes a locust’s strong jumping ability allows it to avoid predators key focus of various studies on nonpneumatic tires. Manga and start ﬂying, and the combined action of the rigid claws  found that the spoke vibration was not a forced vibration and the adhesive pads ensures that the static contact between tarsus and ground, which can achieve a smooth jump on a related to the rolling speed but a resonance excited by the buckling and rebound phenomena when the spoke entered smooth surface; in addition, the large take-oﬀ angles also and left the contact zone. Bezgam  obtained the contribu- allow locusts to achieve better performance on smooth sur- tion of spoke shape parameters to spoke vibration through faces, which provides inspiration for the jumping robot orthogonal experiment, adjusted the thickness and curvature design . When catching prey, cats need to have a strong ability to reduce vibration, so as to weaken the impact from of adjacent spoke pairs based on this, and proposed the design concept of alternate spoke pairs to reduce the ampli- the ground and achieve characteristic silence. As the only tudes of spokes and ground vibration. Proddaturi  body part in contact with the ground, the paw pads play an proved in his research that the spokes’ length and curvature important role in vibration reduction realization. Mei et al. have the greatest inﬂuence on vibration, followed by the  obtained the mechanical and grounding shape represen- tations by conducting ground reaction force and contact thickness of the shear beam, the thickness of spokes, and the number of spokes. The thickness of the inner and outer strain experiments on the paw pads of domestic cats in vari- coverages and the inner and outer DeRad were reported to ous gaits, so as to explore the adaptive adjustment of the have less inﬂuence. When adjusting the shear modulus of mechanical characteristics and shape of the paws in various the spokes, Narasimhan  concluded that the change of gaits. Biewener  concluded that, during exercise, cats’ paw pads can eﬀectively buﬀer the ground’s vertical reaction the material led to the change of stiﬀness, and the increase of the stiﬀness caused the spoke vibration to decrease. Mean- force that is 2-3 times the value of their body weight. Zhang while, the spoke material’s change had a greater impact on et al.  carried out a theoretical analysis of the paw pads’ the spoke vibration than the shear band material’s change. vibration reduction characteristics according to the changing In the long-term evolution of animals, a variety of biolog- law of the vertical ground reaction force as the cat fell on the ground and constructed a mass-spring viscoelastic mechani- ical structures and functional characteristics highly adaptable to nature have been formed. Researchers use these principles cal model. For the bionic research of nonpneumatic tires, a to invent and innovate technologies. For example, Romano nonpneumatic tire developed by the Madison Polymer et al.  found in the study of the escape direction of Research Center, Wisconsin, USA, uses a bionic honeycomb Locusta migratoria a high plasticity of those escape motor structure , and the hexagonal honeycomb-like structure is recognized in the ﬁeld of coupled bionics as a structure outputs that are occurring almost in real time with the per- Applied Bionics and Biomechanics 3 Table 1: Peak vertical ground reaction force in each area of the fore and hind paw pads of cats. Peak vertical ground reaction force (%BW) Pads nd rd th th 2 toe pad 3 toe pad 4 toe pad 5 toe pad Palm pad 10:9±1:314:6±2:311:7±1:88:7±1:137:2±4:4 Fore paw pad Hind paw pad 8:3±0:912:1±1:510:2±1:73:9±0:625:1±3:2 e1 [%] – lagrange 15.85 th 14.63 th 13.4 12.17 rd 10.94 9.717 nd 8.489 7.262 t = 0.105 s Reference diagram t = 0.055 s 6.035 4.808 3.581 2.353 1.126 –0.1009 –1.328 –2.555 –3.782 t = 0.235 s t = 0.290 s t = 0.180 s Figure 3: The contact strain results of fore paw pad (t =0:300 s). total with good buﬀering and energy absorption characteristics natural vibration reduction under normal walking gait and [28, 29]. Huang et al.  took advantage of the silent applied this bionic concept to modify spoke-type nonpneu- characteristics of the stripe structure in the feathers of owls matic tire spoke structures. The asymmetric arc design was to add nonsmooth structural units on the surface of the carried out on both sides of the spokes to achieve a vibration spokes to reduce the aerodynamic noise of nonpneumatic reduction eﬀect similar to that of the domestic cat’s paw pads tires. According to the good shock absorption and buﬀering to optimize the tire’s radial vibration characteristics. Then, characteristics of a kangaroo’s lower limb structure, Zhang the vibration reduction eﬀect of the asymmetric arc tire and et al.  made a bionic modiﬁcation design to the spoke the original tire is compared. With vibration reduction being the optimization goal, the bionic modiﬁed optimized param- structure. They conﬁrmed that the bionic nonpneumatic tire’s performance is better than that of a pneumatic tire rel- eters of the spokes were achieved by performing optimization ative to radial stiﬀness, lateral stiﬀness, longitudinal stiﬀness, analysis relative to other structural parameters to obtain the torsional stiﬀness, and tire ground pressure under diﬀerent optimal vibration reduction spoke structure. loads. An in-depth exploration of bionics’ functional charac- teristics and mechanism could improve its accuracy and 2. Research on the Vibration Reduction eﬀectiveness and its possible application on nonpneumatic Mechanism of the Paw Pads of Domestic Cats tires. Therefore, with the aim of making the extremely strong vibration reduction characteristics of cat’s paw pads when in 2.1. The Mechanical Test of the Contact between the Paw Pads contact with the ground be applied to the spokes of the non- and the Ground. The purpose of the mechanical test of the pneumatic tire, it is necessary to conduct research on the contact between the paw pads and the ground is to obtain vibration reduction mechanism of the cat’s paw pad and get the paw pads’ vertical reaction force and the strain charac- an improved method suitable for the spokes. teristics during the contact and normal walking gait Existing researches on vibration reduction of spokes of (v =0:4~0.8 m/s) of domestic cats. The test subjects were four nonpneumatic tires are limited to exploring the impact vibra- healthy, nondefective domestic cats aged between 4 and 7 tion by changing the spokes’ structural parameters and the years, whose weights ranged from 3.8 to 5.6 kg, and having use of materials to ﬁnd a relatively optimized damping solu- a shoulder height between 20 and 28 cm. The mechanical test tion. Structural parameters play a limited guiding role for real site was provided by the Graduate Laboratory of Tire and vehicle applications of nonpneumatic tire development in the Vehicle Rubber of Jiangsu University, China. future. Hence, this paper conducts grounding mechanical During the test, a pressure-sensitive walkway (Walkway tests on domestic cats’ paw pads to analyze how they achieve A101; Tekscan, the USA) was used to measure the paw pads’ 4 Applied Bionics and Biomechanics 0.00 0.03 0.07 0.10 0.13 0.17 0.20 0.23 0.27 0.30 0.00 0.03 0.07 0.10 0.13 0.17 0.20 0.23 0.27 0.30 Time (s) Time (s) E E x x E E y y (a) (b) rd Figure 4: Distribution of strain values in X and Y directions in the (a) 3 toe pad area and (b) palm pad area. vertical reaction force when the domestic cats walked across axes show the paw pad forward and inner directions, respec- the pressure plate in a straight line at diﬀerent speeds. In tively. In terms of the principal strain directions, the four toe order to obtain the strain characteristics of the domestic cats’ pads did not change during the whole grounding process with the ground, black spots and speckles were applied on (Y-oriented tensile deformation). The palm pad area was their paw pads as they walk straight on the glass plate, as mainly under tensile strain in the Y direction before shown in Figure 1(a). A high-speed camera (Olympus i- 0.18 s, and it was primarily in the X direction after 0.18 s, SPEED 3, Japan) installed under the glass plate was used to which indicated that the palm pad had a swing deformation record the motion of the paw pads, and then, the images were phenomenon in the contact surface. And the main strain digitally processed using the VIC-2D of the CSI company in values of the four toe pads’ continued to increase during the entire contact process. The palm pad’s strain value the United States to obtain the strain and related information of the contact between the paw pads and the ground, as increased ﬁrst and then decreased alternately in the inner shown in Figure 1(b). and outer regions. The palm pad’s maximum main strain value was signiﬁcantly lower than that of the toe pads, which was caused by the change of the strain directions. 2.2. The Mechanical Analysis of the Contact between the Paw To further clarify the vibration reduction eﬀect of the Pads and the Ground. For each domestic cat, six valid data palm pad’s swing deformation, the variation trends of the were taken for processing. Each part’s peak vertical ground rd strain values in the X and Y directions of the 3 toe pad reaction force is expressed as a percentage of the domestic and palm pad during the grounding process are extracted, cat’s body weight (%BW); recorded as mean value ± and the results are depicted in Figure 4. Considering that standard deviation. As shown in Figure 2, the domestic cat the incompressibility of the paw pad would cause its local paw pad is divided into a palm pad and four toe pads; the compression in the X or Y directions to be transformed into second, third, fourth, and ﬁfth toe pads, respectively. Peak a tensile deformation in the Y or X directions, the strain vertical ground reaction force in each area of the fore and rd values greater than zero in the X and Y directions of the 3 hind paw pads of cats is depicted in Table 1. As shown in toe pad and palm pad were averaged to characterize the Table 1, the peaks of the toe pads and palm pads of the fore strain values in the X and Y directions, and the strain values paw pads are higher than those of the hind paw pads, and were recorded as E and E , respectively. It can be seen from x y the palm pad area bears the maximum peak value of the rd Figure 4 that both E and E of the 3 toe pad generally show x y entire paw pad area. Accordingly, the palm pad area of the increasing trends as a whole, while the ﬂuctuating changes of fore paw pads of the cat is the key area for realizing the vibra- the opposite trends of E and E in the palm pad can achieve tion reduction function. x y The main strain ﬁeld’s distribution and the strain direc- the strain attenuation value. tions of the domestic cat’s fore paw pad during the whole A careful analysis of the toe and palm pads’ main strain grounding process are displayed in Figure 3. The X and Y changing characteristics shows that a vibration-damping Strain value (%) Strain value (%) Applied Bionics and Biomechanics 5 Hub Spoke Inside coverage Shear layer Ring Reinforcement Outside coverage Tread Aluminum-alloy Steel PU Rubber Figure 6: Structural and material compositions of the nonpneumatic tire. whole wheel consists of 25 pairs of spokes, and the material of the spokes is polyurethane. The ﬂexible ring is divided into three parts by two reinforcements. From the inside to the outsides, there are the inner coverage, the shear layer, and the outer coverage using a polyurethane material. The shear band between the two reinforcements mainly bears the shear Figure 5: Early prototype of Tweel. force when the tire is rolling under load. The reinforcements are made of high-strength steel, providing high rigidity and eﬀect is realized when the palm pad deformation swings in strength in the circumferential direction. The tread is made the X and Y directions; that is front-rear and left-right of rubber to ensure that the tire has excellent friction and swinging deformation. better road gripping ability, 3.1.2. Establishment and Veriﬁcation of Finite Element Model. 3. Bionic Vibration Reduction Design for The ﬁnite element model is displayed in Figure 7. The poly- Spokes of the Nonpneumatic Tire urethane material is modeled using the Marlow model, while the rubber is modeled with the Neo-Hookean model. The The vibration characteristics of tires are an important factor speciﬁc material properties settings is adopted from [19, 20, aﬀecting the NVH of vehicles . The vibration source in 33]. Using the Abaqus/Standard solution method, the road the nonpneumatic tire during the rolling process is primarily surface is deﬁned as a rigid analytical body and is ﬁxed. A from the buckling and rebound of the spokes under tension radial force of 3665 N (a quarter of the rated load of the when entering and leaving the contact area, the interaction nonpneumatic tire) is applied to the rim’s center to simulate between the discrete spokes and the ring, the interaction the tire grounding process. The Coulomb friction model when the ring is in contact with the ground, and the force describes the contact characteristics between the tire and of the ground and the vibration between the ring and the the road surface. spoke transmitted to the hub . Therefore, the spokes have Figure 8 displays the load deﬂection curve (vertical stiﬀ- a great inﬂuence on the vibration of nonpneumatic tires. ness curve) between the simulation value of the ﬁnite element Based on the vibration reduction mechanism of the swing model and the analysis result of the Akshay Narasimhan deformation of the domestic cat’s paw pads, the spokes can curve . The stiﬀness curves of the two are relatively close, be designed with bionics to improve the vibration charac- and the error for a radially loaded of 3665 N tire is only teristics of the nonpneumatic tires and enhance the NVH 0.97%. The comparison results illustrate that the ﬁnite ele- performance of the vehicle. ment model established in this paper can accurately reﬂect the mechanical characteristics of Tweel for further research. 3.1. Finite Element Simulation Analysis of the Nonpneumatic Tire 3.2. Bionic Design of Spokes. Ramachandran et al.  carried out a study on the spoke vibration and concluded that under 3.1.1. The Geometry of the Nonpneumatic Tire. In this paper, Michelin’s Tweel nonpneumatic tire (Figure 5) is selected the same conditions, the vertical middle node vibrates more because of its relatively established and wide application. violently than the upper and lower quarter nodes in the radial The geometric parameters of Tweel are selected from the direction of the spoke. The vibration gradually increases research of Bezgam . The structural composition and from the middle position towards the edges on both sides in the axial direction. Therefore, a scallop-shaped treatment material usage of the three-dimensional geometric model of the nonpneumatic tire is shown in Figure 6. Tweel is mainly method on the side edge of the spokes is proposed: it involves composed of four parts: a rigid hub, deformable spokes, a a precise cutting out of the areas with severe vibrations to ﬂexible ring with reinforcements, and a tread. The wheel reduce the vibration amplitude. Figures 9 and 10 show the hub is made of aluminum alloy, which supports the tire spoke vibration marker nodes and the scallop-shaped edges, respectively. and is assembled with the shaft. The spokes are in pairs, the 6 Applied Bionics and Biomechanics z x Figure 7: Finite element model. 4000 Horizontal middle node Top node Upper quarter node Vertical middle node Lower quarter node Bottom node Le quarter node Right quarter node Figure 9: Vibration marker nodes for spoke. necting line (edge line) of the top and bottom points, which is 02468 10 12 15 mm here. The shape of the arc is constructed based on the Displacement (mm) method of cubic interpolation spline curve. The left-right and Narasimhan up-down swing of the tire during rolling is realized by dis- Simulation value tributing the arcs asymmetrically at both sides of the spoke to the axial and radial bisecting lines accordingly. And the Figure 8: Comparison of radial stiﬀness curve for simulation and front-rear swing is also realized by the asymmetric arrange- Narasimhan analysis value. ment of the adjacent spokes. In ensuring the stability of tire bearing capacity, the spokes are staggered along the circum- ferential direction of the tire in accordance with the asym- Inspired by the scallop-shaped spoke edges treatment and the domestic cat’s paw pad vibration reduction mecha- metric arc of spokes 1 and 2, as displayed in Figure 12. nism, an asymmetric structural design is carried out on the Consequently, the asymmetrical arc not only cuts oﬀ a part spokes to enable the tire to achieve similar swing deformation of the spokes where the vibration is relatively large but also characteristics in the rolling process. The bionic modiﬁcation dissipates the impact from the ground in the swing deforma- tion of the spokes to attain the overall vibration reduction design is presented in Figure 11. The vertices (P) of the asym- metric arc are on the horizontal straight line where the upper eﬀect. and lower quarter nodes are located. The size of the arc is To eliminate the inﬂuence of the bionic structural design determined by the vertical distance h from the P to the con- and to assess the vibration reduction performance of the Reaction force (N) Applied Bionics and Biomechanics 7 Figure 10: The scallop-shaped treatment on the side edge of the spokes. Axial bisecting line Spoke 1 Figure 13: Symmetrical arc structure of spoke. Radial bisecting line Edge line linear hexahedral three-dimensional stress element with Spoke 2 reduced integration (C3D8R). The reinforcement layer has elements of 4-node and quadrilateral with reduced integra- tion (SFM3D4R). For simulation analysis, a radial load of 3665 N and a speed of 60 km/h are applied to the rim. And the road condition is considered ﬁxed. Then, the radial exci- tation force of the road in the time domain of 0.12 s (steady rolling 1.06 cycles) when the tire is rolling in a steady state Figure 11: Bionic modiﬁed design of spoke. is extracted. The comparison of the radial excitation force of three kinds of tires in the time domain is shown in Figure 14. The radial excitation force of the three types of tires ﬂuctuates up and down at the applied load of 3665 N. Spoke 1 The ﬂuctuation of the symmetrical arc tire is the most obvi- ous, followed by the original tire. In contrast, the radial exci- tation force of the asymmetric arc tire has been reduced, especially at the peak, and the excitation force ﬂuctuates more uniformly in the entire time domain. Therefore, the application of asymmetric arcs on the spokes can signiﬁ- cantly reduce the radial vibration of the tire. To further demonstrate that the asymmetric arc tire is Spoke 2 superior to both the symmetric arc tire and original tire in vibration reduction and further clarify the reason why asym- Figure 12: Asymmetrical arc structure of spoke pair. metric arc tire can reduce the radial excitation force of the road surface, the FFT function in MATLAB is used to convert the excitation force in the time domain to the amplitude asymmetric arc swing deformation, a comparison of asym- change in the frequency domain. Since the sound pressure metric and symmetric arc tires of the same weight is made. level (SPL) with a frequency lower than 100 Hz has no signif- As depicted in Figure 13, the P of the symmetric arc is on icant eﬀect on human perception of noise , and when the the horizontal line where the vertical middle node is located, frequency is greater than 1500 Hz, the amplitudes are small, and the vertical distance h from the P to the edge line repre- and there are no signiﬁcant peak amplitudes. So 100 Hz- sents the size of the symmetric arc, which is 15 mm. 1500 Hz is taken as the range of analysis in this paper. 3.3. Finite Element Analysis and Discussion of Bionic Design. Figure 15 shows the spectrum comparison between the orig- The vibration and noise of tires are closely related to the inal tire and the asymmetric arc tire. Figures 15 and 16 , radial excitation force of the road surface during rolling, respectively, show the comparison of the spectrum between the original tire and the asymmetric arc tire and the compar- and the larger the excitation force value is, the higher the vibration and noise value will be [32, 35]. With the aid of ison of the asymmetric arc tire and the symmetric arc tire. 1 2 the Abaqus/Explicit method, the hub, spokes, inner and And the PA (peak amplitude 1, lower frequency) and PA outer coverages, shear layer, and tread adopt an 8-point (peak amplitude 2, higher frequency) have been marked. 8 Applied Bionics and Biomechanics 4.38 4.40 4.42 4.44 4.46 4.48 4.50 Time (s) Original tire Asymmetric arc tire Symmetric arc tire Figure 14: Comparison of time-domain distribution of radial excitation force of three kinds of tires. 250 1 1 PA 350 PA PA PA 100 PA PA 2 PA PA 200 400 600 800 1000 1200 1400 200 400 600 800 1000 1200 1400 Frequency (Hz) Frequency (Hz) Asymmetric arc tire Original tire Symmetric arc tire Asymmetric arc tire Figure 16: The spectrum comparison between the asymmetrical arc Figure 15: The spectrum comparison between the asymmetrical arc tire and the symmetric tire. tire and the original tire. bionic modiﬁcation of spokes. The comparison of the ampli- 1 2 tudes of the three tires is shown in Table 2, and the formula Through comparison, it is found that the PA ,PA , and for calculating the root mean square value is deﬁned as amplitudes corresponding to most frequencies in the entire frequency domain of the asymmetric arc tire are smaller than vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ those of the original tire and the symmetric arc tire. Since the root mean square (RMS) value can reﬂect the t 2 RMS = 〠ðÞ x , ð1Þ amplitudes of the overall vibration in the entire frequency i=1 range, and the peak amplitude reﬂects the vibration intensity at the local frequency; this paper uses both the root mean square value and the peak amplitude to quantify the ampli- where N is the total number of intervals in the step and x is tudes of the ground response to further clarify the eﬀect of the data on the ith interval. Magnitude Force (N) Magnitude Applied Bionics and Biomechanics 9 (3) Finite element analysis numerical test: a model data- Table 2: The comparison of vibration amplitudes of three kinds of tires. base was established based on the DOE method, and ABAQUS simulations were performed on each Amplitude type Tire type model. The selection of the model simulation method 1 2 PA PA RMS and the calculation settings remained the same as Original tire 226.65 77.93 32.33 before Asymmetric arc tire 215.37 44.24 27.01 (4) Approximate model (AM) method: the AM method Symmetric arc tire 349.45 63.34 34.97 is a mathematical model that approximates a set of input variables (independent variables) and output variables (response variables) through a mathemat- As can be seen from Table 2 that the peak amplitude and ical model. It is established according to the rela- root mean square value of the asymmetric arc tire are signif- tionship between the design variables and the 1 2 icantly reduced compared to the original tire. The PA ,PA , simulation response. Here, the Kriging model was and RMS are reduced by 4.98%, 43.23%, and 16.46%, respec- selected to build an approximate model tively, indicating that the asymmetric arc can eﬀectively reduce the radial vibration of the tire. Although the PA of (5) Optimization calculation: after constructing the the symmetrical arc tire is lower than that of the original tire, approximate model, a reasonable algorithm is used the PA at the low-frequency band that has a greater impact to solve the objective function to obtain the optimal on tire vibration is 35.14% higher than that of the original design parameters. In this paper, a genetic algorithm tire, signiﬁcantly increasing the local vibration intensity and (GA) was employed to obtain the optimal solution its RMS is higher than that of the original tire. 4.1. Design of Experiments Method. The design of the exper- In the comparison between the asymmetric arc tire, the iment method can identify key experimental factors, deter- symmetric arc tire, and the original tire, it is found that not mine the best combination of parameters, analyze the only does the peak amplitudes of the asymmetric arc tire relationship between independent and dependent variables, decrease but also the overall vibration amplitudes decrease. and provide sample data for constructing an approximate This indicates that the asymmetric arc tire can weaken the model . In the process of designing the experiment, the impact through the characteristics of swing deformation to design points of the experiment should cover the design achieve a better vibration reduction eﬀect and authenticates space evenly and avoid the repetition of sampling points as the feasibility of bionic vibration reduction. far as possible; the number of test analysis should be mini- mized to ensure that the calculation cost is not too high 4. Optimization for Vibration under the premise of ensuring accuracy. All the above stated Reduction of Spokes practical requirements were satisﬁed by applying the orthog- onal array (OA) method since it considers both the interac- The feasibility of applying the vibration reduction mecha- tion and the test accuracy and delivers an advantage of high nism of cat’s paw pad to the spokes of nonpneumatic tires eﬃciency, speed, and economy. was veriﬁed. However, considering that other parameters of According to the requirement of orthogonality, an OA the spokes will also have a certain impact on the vibration table in the form of L ðE Þ is generated to design the exper- of the tires, and therefore without changing the inner and iment, where L is the table, n is the total number of design outer diameters of the tire (the length of the spokes remains solutions required (the number of rows in the table), E is unchanged), the asymmetric arc is combined with the thick- the level of the factors, and P is the number of factors. In this ness and curvature of the spokes to optimize the design of the experiment, 3 factors and 3 levels were considered, so an spokes to achieve a better damping eﬀect. The speciﬁc orthogonal table of L ð3 Þ was used. process of the design optimization of the spokes structure is The three factors of the orthogonal test are the size of the illustrated in Figure 17. asymmetric arc (A), the thickness of the spoke (B), and the curvature of the spoke (C). The deﬁnition of the thickness (1) Parameterization: the optimization design include and curvature of the spoke are shown in Figure 18; h depicts choosing the right design variables and reducing the the vertical distance from the vertical middle node to the line number of design variables to reduce complications connecting the top and bottom nodes. The size of the dis- and cost. . The spoke could be parameterized by tance expresses the magnitude of curvature, the thickness of three variables: the size of the asymmetric arc, the the spoke is represented byh , and the size of the asymmetric thickness, and the spoke’s curvature arc was described in Section 3.2. (2) Design of experiments (DOE) method: the DOE The original values of the asymmetric arc is 15 mm, while method provides a reasonable and eﬀective method the thickness and curvature of the spoke are 4.2 mm and to obtain information and data, which directly aﬀects 8 mm, respectively. The factors and level design of the the quality of the approximation model and is one of orthogonal experiment are indicated in Table 3. The orthog- the most important statistical methods in the optimi- onal table generated according to the factors and levels and zation process. Here, the orthogonal array (OA) was the simulation results (RMS) of the nine groups of design chosen to generate the sample points schemes are shown in Table 4. 10 Applied Bionics and Biomechanics Design cycle FEA numerical tests with ABAQUS Parameterization Design of experiments (DOE) Optimization cycle Approximation models Optimization algorithm Output Figure 17: Flowchart of the design optimization Top node Table 4: Schemes for the OA method. No. ABC RMS 1 11.25 3.15 6.00 27.38 2 15.00 4.20 6.00 18.38 3 18.75 5.25 6.00 17.00 Vertical 4 15.00 3.15 8.00 23.06 middle node 5 18.75 4.20 8.00 25.27 6 11.25 5.25 8.00 23.55 7 18.75 3.15 10.00 29.33 8 11.25 4.20 10.00 32.07 9 15.00 5.25 10.00 30.79 Bottom node Figure 18: The deﬁnition of the thickness and curvature of the spoke. spokes can eﬀectively deliver a better vibration reduction eﬀect. Table 3: Factors and levels of OAs. With the view of ascertaining the inﬂuence of design var- iables on the RMS value and the degree of contribution, a Level (mm) Pareto chart as shown in Figure 19 is drawn. It can be seen Factor from the ﬁgure that for a single design parameter, the curva- A 11.25 15.00 18.75 ture of the spoke has the greatest inﬂuence on the RMS value, B 3.15 4.20 5.25 with a contribution rate of 31.71%, and the increase of the C 6.00 8.00 10.00 curvature will increase the RMS value, followed by the size of the asymmetric arc and the thickness of the spoke, whose contribution rates are 8.69% and 3.75%, respectively, and as the size of the asymmetric arc and the thickness of the spoke increase, the RMS values decrease. The nonlinear inﬂuence of As can be seen from Table 4, compared with the original a single variable on the RMS value is dominant. For example, tire, the RMS values of all the nine schemes are reduced, with the contribution of C to the RMS value is about 17.81%, that the smallest and the largest decreasing values being 4.21% is, the inﬂuence on the RMS value is quadratic. Figure 20 fur- and 47.42%; compared with asymmetric arc tires, the RSM ther illustrates the eﬀect of design variables on the RMS values of the four groups of design schemes are reduced. The minimum decrease is 6.44%, and the maximum decrease value. As can be seen from Figure 20 that the curve of C is 37.06%. The result proves that a proper combination of the has a larger curvature, the trends of the curves of A and B size of the asymmetrical arc, thickness, and curvature of the are more synchronized, which explains that the size of the Applied Bionics and Biomechanics 11 A-B A-C –10 –5 0 5 10 15 20 25 30 35 Total eﬀect on RMS (%) Figure 19: Pareto chart for the RMS. 1 1.2 1.4 1.6 1.8 2 Leveis Figure 20: Main eﬀect plot for the RMS. asymmetrical arc and the thickness of the spoke have a rela- network model, orthogonal polynomial, and Kriging model. tively similar eﬀect on the RMS value. In addition, the Pareto However, since RSM model is not capable of describing non- chart also provides the correlation between the design vari- linear problems, the RBF model takes a long time to build a ables and the target variables, in which, the correlation model, and considering that the problem studied in this between A and B has the greatest impact on the RMS value; paper does not only have a high degree of nonlinearity but a value of about 18.57%, indicating that both changes have also has random errors, the Kriging model was selected for the greatest inﬂuence. A and C have the least inﬂuence on the construction of an approximate model . The Kriging the RMS value (about -1.33%). model can be expressed as yx ðÞ =fx ðÞ +Zx ðÞ, ð2Þ 4.2. Approximate Model Method 4.2.1. Kriging Approximation Model. Approximate models where yðxÞ is the unknown deterministic function, f ðxÞ is a include the response surface model (RSM), RBF/EBF neural known approximation function, ZðxÞ is the realization of a RMS 12 Applied Bionics and Biomechanics Table 5: Kriging model validation. Group A (mm) B (mm) C (mm) Kriging model ABAQUS Error (%) 1 13.13 3.68 7 22.37 21.84 2.43% 2 16.88 4.73 9 27.47 26.47 3.78% stochastic process with mean zero, variance σ , and nonzero Therefore, the constant term of the Kriging model is used for the global portion, while the Gaussian correlation func- covariance f ðxÞ provides a global approximation model of the design space, and ZðxÞ creates localized deviations so that tion (4) is used for the local deviations. the Kriging model can interpolate the sample points . In 4.2.2. Error Analysis. When constructing the Kriging model, many cases, f ðxÞ is taken as a constant, and β is also employed there will be errors caused by the polynomial model itself in the design of spokes structure of nonpneumatic tires. or ﬁtting. Therefore, the squared multiple correlation coeﬃ- The covariance matrix of ZðxÞ is formulated as cient R is introduced to verify the reliability of the Kriging i j 2 i j model. The closer R is to 1, the more accurate the ﬁtting will cov Zx ,Zx = σ MRx , x , ð3Þ be. The ﬁnal result shows that R =0:99, for which, a conclu- sion can be drawn that, the Kriging model has suﬃcient accu- i j where Rðx , x Þ is the correlation function between any two racy to interpolate these 9 sample points for optimization i j input points x and x of n observed points, and M is the calculations. n × n correlation matrix with values along the diagonal In this case, two groups of variables were randomly . Gaussian correlation function was used to calculate selected, and the ABAQUS and Kriging models were, respec- i j Rðx , x Þ and is given by: tively, used to obtain the RSM values. As illustrated in Table 5, the error between the calculation results of the "# Kriging model and those of ABAQUS is small, which further i j i Rx , x = exp −〠 θ x − x , ð4Þ k k veriﬁes the accuracy of the Kriging model. k=1 4.3. Optimization Calculation. The genetic algorithm (GA) is where x and x are the kth components of sample points, a global optimization method that mainly uses the laws of k k biological evolution to solve optimization problems. GA and θ are the unknown correlation parameters, which can encodes the individuals and then performs the genetic oper- be obtained by the maximum-likelihood estimation (MLE) ations of selection, crossover, and mutation on the encoded  method according to individual to seek the optimal solution . In this study, 1 the multi-island genetic algorithm (MIGA), which can be max Φθ = − n ln σ∧ +ln M : ð5Þ ½ ðÞ jj regarded as an improved genetic algorithm, was used to solve θ >0 2 the optimal solution. MIGA divides a large population into several subpopulations, each of which carries out genetic While any value for θ creates an interpolative Kriging operations independently, and the individuals on each island model, the ‘best’ Kriging model is found by solving the k- transfer to other islands in a certain proportion periodically dimensional unconstrained nonlinear optimization problem to complete the periodic exchange of information . given by equation (5) . For a given θ, the closed-form Objective constrained optimization problem can be solution for the optimal values of β and σ can be obtained deﬁned as follows: and formulated as Objective function: minimize RMS Design variables with limits: −1 T −1 T −1 β = I M I I M I Y, (i)7:50 ≤ A ≤ 22:50 ð6Þ (ii)2:10 ≤ B ≤ 6:30 2 T −1 σ∧ = Y − Iβ∧ M Y − Iβ , ðÞ (iii)4:00 ≤ C ≤ 12:00 The RMS value was optimized by MIGA. The size of the subpopulation is 100, the number of islands is 100, and the where I is a d-dimensional unit vector and Y = ½yðx Þ, ⋯, number of evolutionary generations is 10. The optimization yðx Þ is the vector of true limit state function values . result is A =17:31, B =4:85, and C =6:04. The RMS value Predicted estimates, y at untried values of x, are given by for the optimization result is 16.30, which is 4.12% better than the optimal value in the OA table. And the optimization T −1 b b ̂yx ðÞ = β + r ðÞ x M Y − Iβ , ð7Þ values of A, B, and C are reduced by 7.68% and 7.62% and increased by 0.67%, respectively, compared with the design variables of the optimal vibration reduction in the orthogonal where r is the correlation vector given by experiment. The values of design variables and RMS of the two groups are not much diﬀerent, which veriﬁes the reliabil- T 1 i r ðÞ x = Rx, x , ⋯,Rx, x ðÞ i =1, ⋯, n : ð8Þ ity of the Kriging model from another perspective. Applied Bionics and Biomechanics 13 4.38 4.40 4.42 4.44 4.46 4.48 4.50 Time (s) Original tire Asymmetric arc tire Optimized tire Figure 21: Comparison of the distribution of radial excitation force of three tires. PA PA PA PA PA 2 PA 200 400 600 800 1000 1200 1400 Frequency (Hz) Original tire Asymmetric arc tire Optimized tire Figure 22: The spectrum comparison between the optimized tire, original tire, and asymmetric arc tire. The values of optimized design variables are used for sim- Figure 22, which indicates that the PA value of the opti- ulation analysis and compared with the original tire and the mized tire is much smaller than that of the original tire and asymmetric arc tire, and the distribution diagram of the asymmetric arc tire, and the PA value and RMS value are radial excitation force of the three is obtained, as depicted also signiﬁcantly reduced. The result of the comparison in Figure 21. It manifests that although several peaks of the proves that the optimized tire has a more prominent advan- optimized tire are increased, the local ﬂuctuations around tage of vibration reduction. the peaks are reduced; and the green markers illustrate that According to the simulation results, the RMS value of the the ﬂuctuation curve is relatively straight, therefore, the optimized tire is 15.63, while the value obtained by MIGA is ﬂuctuations during the whole cycle are greatly reduced. The 16.30, with an error of only 4.11%. In addition, since the orig- spectrum comparison of the three tires is shown in inal tire and the asymmetric arc tire both use the results of Force (N) Magnitude 14 Applied Bionics and Biomechanics the four toe pads showed a cumulative increasing Table 6: The comparison of amplitudes. trend over time, while in the palm rest area, E and Amplitude type E did not show a trend of increase over time, and Tire type 1 2 y PA PA RMS each exhibited ﬂuctuating changes and the trends Original tire 226.65 77.93 32.33 were opposite to each other; in other words, when Asymmetric arc tire 215.37 44.24 27.01 E increases or decreases, E correspondingly x y Optimized tire 54.24 36.72 15.63 decreases or increases, which means that the palm pad through front-rear, left-right swing deformation to weaken the ground impact to achieve the purpose of vibration reduction Table 7: The comparison of spoke parameters. (2) First of all, the ﬁnite element model of the spoke-type Parameter Tire type nonpneumatic tire was established, and its stiﬀness AB C curve was compared with those in the reference. Asymmetric arc tire 15 mm 4.2 mm 8 mm The small error between the two veriﬁed the feasibil- Optimized tire 17.31 mm 4.85 mm 6.04 mm ity of the model. After that, based on the principle of Increase or decrease ↑15.40% ↑15.48% ↓24.50% bionics, the spoke structure of the asymmetric arc was proposed. Through the comparative analysis of radial vibration, it was found that the peak amplitude values and root mean square value of the asymmetric simulation calculations, the simulation values of the opti- arc tire were distinctly lower than those of the sym- mized tire are used for comparative analysis. Table 6 shows metric arc tire and original tire, which proved that 1 2 the comparison of the values of PA ,PA , and RMS value asymmetric arc tire had signiﬁcant vibration reduc- among the original tire, asymmetric arc tire and optimized tion characteristics. Finally, it can be concluded that 1 2 tire. Compared with the original tire, the PA ,PA , and the swing deformation vibration reduction mecha- RMS values of the optimized tire are reduced by 76.07%, nism of the cat’s paw pads had a positive vibration 1 2 52.88%, and 51.65%, respectively. The values of PA ,PA , reduction eﬀect when applied to the spokes of non- and RMS of the optimized tire are 74.82%, 17.00%, and pneumatic tires 42.13% lower than those of the asymmetric arc tire, respec- (3) To maximize the vibration reduction performance of tively. Results show that using the size of the asymmetric the structural design of the spokes, based on the arc, the thickness, and curvature of the tire as design variables bionic design of the asymmetric arc, an optimization to optimize the design has an excellent vibration reduction for vibration reduction using the OA method, the eﬀect. And combined with the comparison of the structural Kriging approximate model, and the MIGA was parameters of the asymmetric arc tire and the optimized tire employed to obtain the optimal design parameters in Table 7, it is found that increasing the size of the asym- of the spokes. The DOE analysis revealed the curva- metric arc and the thickness of the spoke appropriately ture of the spokes as the most key parameter for and reducing the curvature of the spoke will have a better vibration reduction, followed by the size of the asym- vibration reduction eﬀect. metric arc and the thickness of the spokes, while increasing the size of the asymmetric arc and the 5. Conclusion thickness of the spokes and decreasing the curvature of the spokes appropriately will obtain a better vibra- In this paper, the spokes of the nonpneumatic tire were tion reduction eﬀect. Results showed that the optimal treated with asymmetric arc using the vibration reduction 1 combination of the design variables can reduce PA , mechanism of domestic cat’s paw pads, and the vibration PA , and RMS values by 74.82%, 17.00%, and characteristics of the asymmetric arc tire, symmetric arc tire, 42.13%, respectively, compared with the asymmetric and original tire under rolling conditions were compared and arc tire, and 76.07%, 52.88%, and 51.65% lower than analyzed. Furthermore, the size of the asymmetric arc, the those of the original tire, respectively thickness and curvature of spokes were used as design vari- ables for vibration reduction optimization, and the following conclusions were drawn. Data Availability (1) Using the pressure-sensitive walkway, high-speed The data used to support the ﬁndings of this study are camera, and VIC-2D to carry out the grounding available from the corresponding author upon request. mechanical tests of the paw pads of domestic cats, it was found that the peak vertical ground reaction force of the fore paw pad was greater than that of Conflicts of Interest the hind paw pad, and there were signiﬁcant diﬀer- ences in strain between the four toe pads and the The author states that there are no conﬂicts of interest related palm pad. The strains in the X and Y directions of to the publication of this article. Applied Bionics and Biomechanics 15  Y. Q. Zhao, H. X. Fu, F. Lin, and Y. Q. Li, “Advancement of Acknowledgments non-pneumatic wheels and mechanical characteristics,” Jour- This research is funded by the National Natural Science nal of Jiangsu University, vol. 37, no. 6, pp. 621–627, 2016. 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Applied Bionics and Biomechanics – Hindawi Publishing Corporation
Published: May 17, 2021