T H E U N I V E R S I T Y 0 F M I C H I G A N COLLEGE OF ENGINEERING Department of Engineering Mechanics Tire and Suspension Systems Research Group Translation No. 1 ON THE CALCULATION OF THE CRITICAL ROLLING SPEED OF A PNEUMATIC TIRE V. L. Biderman translated. by James Robbins Richard N. Dodge Project Directors: S. K. Clark and R. A. Dodge UMRI Project 02957 administered by: THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR January 1960

kI'\Q \R' ' I I I -1,, ~ ~ t~ -, t? I ", I, I'll

The original was published as a treatise in Volume 3 of Trudy Nauchno-Issledovatel'skogo Instituta Shinnoi Promyshlennosti of The Scientific Research Institute of the Tire Industry (Moscow) in 1957. The Tire and Suspension Systems Research Group at The University of Michigan is sponsored by: FIRESTONE TIRE AND RUBBER COMPANY GENERAL TIRE AND RUBBER COMPANY B. F. GOODRICH TIRE COMPANY GOODYEAR TIRE AND RUBBER COMPANY UNITED STATES RUBBER COMPANY

ON THE CALCULATION OF THEE CRITICAL ROLLING-SPEED OF A PNEUMATIC TIRE V. L. Biderman As the angular velocity of a pneumatic tire increases, the state of rolling of the tire changes abruptly at a certain limit. Waves appear on the surface of the tire (Fig.l), stationary in the area, but moving with the speed of rolling relative to the tire. The speed at which waves appear on the surface of the tire may be called the critical speed of rolling of the tire. As the rolling speed of the tire approaches the critical point, rolling loss (It is not possible in the tire abruptly increases, and correto secure a satis- spondingly the heating of the tire infactory reproduction creases. The life of the tire at a speed of this photograph.) near the critical point, therefore, is quite short. In this way, phenomena appearing at the critical speed restrict the use of tires at high speeds. Consequently, it Fig. 1. Wave-like deformations of is necessary to plan special tires with a 7.50-20 tire during the time of a 7.50-20 tireduring the time of possibly a higher critical speed for rolling on a stand at a speed of speeding automobiles. 180 kmn/hr. General trends of the design of tires used at high speeds are important. An increase of the critical rolling speed of a tire is attained with an increase of internal pressure in the tire and with a reduction of its mass. However, the quantitative influence of these factors on critical speed has not been sufficiently investigated. There is also a lack of information on the dependence of critical speed on the stiffness of rubber and cord, and on the angle of cord fibers. and. other constructional features of the tire. Consequently it has been of practical interest to develop a method for the calculation of the critical rolling speed of a pneumatic tire. In a published work it was demonstrated that the appearance of the critical speed of a tire is completely analagous to the appearance of lateral oscil*S. D. Ponomarev, V. L. Biderman, K. K. Lixarez, V. M. Makyshin, N. N. Malinin, V. I. Feodosev, Fundamental contemporary methods of calculation on reliability in mechanical engineering, Mashgiz, 1952. 1

lations of turbine-driven discs. Oscillations of a tire appear when its rolling speed becomes equal to the speed of travel of a running wave of deformation along the circumference of the tire. Moreover, forces of resilience of the tire are equalized with forces of inertia, and therefore the exterior load may be thought of as due to the presence of internal friction in the material. With this is associated the increasing loss of power as the rolling speed of the tire approaches the critical point. Thus the critical rolling speed of the tire is equal to the minimum speed of travel of the wave along its circumference. The tire represents a rubber-cord shell with extremely strong anisotropic elastic properties. Deformation in the circumferential and meridional directions of this rubber-cord shell are relatively small due to a change in rhombus angles (formed by fiber cords of adjacent layers), and the deformation is restricted in the direction of the fiber cords due to the greater rigidity of the fiber cords. Using this as a basis, we can now take as a principle of calculation a hypothesis in which fiber cords do not elongate during the time of membranous deformation of the shell. Investigation of tire deformation during the time at which waves appear (see Fig. 1) shows that, in a given cross section of a tire, normal movements have an identical sign, that is, the increase of diameter of a tire in a given cross section matches the simultaneous increase of profile width. The same sort of wave deformations may extend not only along the toroidal-shaped tire casing, but also along a straight rubber-cord sleeve. It is evident that the speed of propagation of waves along the tire may in the first approximation be assumed to be equal to the speed of the waves in a sleeve of the same cross section and in the presence of the same internal pressure. However, this does not take into account the curvature near the rim of the tire. The influence of this factor may be evaluated by approximate corrections. Let us examine the progress of the deformation waves along the straight rubber-cord cylindrical shell (Fig. 2). Fig. 2. Two-layered rubber-cord hose: 1. Airtight inward layer; 2,3. Layer of cord; 4. Protective device. 2

The critical speed is calculated with the aid of Rayleigh's method. This method, as is well known, is based upon equating values of potehtial and kinetic energy of the system. Component parts of the potential energy system are: (1) Energy of compressed air in the inner cavity of the shell at the time of its oscillation; (2) Energy of extension of the walls of the shell in connection with its deformation; (3) Bending energy of the walls of the shell. To calculate the value of potential and kinetic energy, we must get expressions for the displaced points of the shell. Interest is directed to the case where there is no tangential deformation of the shell, inasmuch as, as was shown above, longitudinal nodal lines are absent at the time of the extension of waves in the tire. In this case of deformation in which the tangential component of deformation is zero, any point of the shell may be defined by two dimensions: radial displacement c and axial displacement u (Fig. 3, a). k' dsj ~~d ~ W+ Kax (b) u+ ax ax Fig. 3. On the derivation of conditions of nonextensibility in tire cords. Inasmuch as the hypothesis of unextensible tire cord has dimensions u and c cannot be independent. been accepted, Let us examine the element dU of the length of tire cord, the projection of which on circumferential and axial directions equals ds and dx, respectively, until the shell begins to deform. After deformation, ds changes to ds' (Fig. 3b): 1

ds' ds(l + but dx becomes (Fig. 3b) drx' = x 1+ a +( ( 72 The new length of element di is determined from the equation~ (dd)2 = d or '2 =ds2(1+ *) )+ dx [(l+ ax) ~ 6) TX Dividing both parts of the equation by dt2 and taking into account that ds/d alignment of profile), we find: dn ds 1 +, + dx x 2] (+Pjcs + ax) (ax) 2 or after simplification, Equation; c2(os2:+ [28 +(k)2+( J)] sin2.(1> EquatDividing both espresses the relationships between u and taki when the length of t;he tire cord is constant, Jn the presence of small movements, quadratie terms cosin Eq(1) appear small in omparison with linear termsd by fibers oefore in the first approximation, Eq) (1) gives the functiond = 6u 2 1 = tan1 (la) P ax HoEquatiwever, this expression does not appear exact enough for the problem with which we are faced Substituting the approximate expression for ts, (la), we fin the expression for in corrected for temns of second order aso aux 2- 1 U2 P ax ( 2 12x 2 6X2

We let u express the sinusoidal running waves u = uo sin 2 (x - ct) (3) L where L = the length of the wave; c = the speed of propagation; uo = amplitude dimension of displacement, Then we obtain a suitable expression for: - = -uo -t- tan Cosn)cs 2 (x - p S2 ct) ot 1- tau0 2+ tan2B)cos2 2- (x ct) O0LL 2 L L 1 p2u2 164 tan6s sin2 2(x - t) (4) 2 O L4 L( In this expression the first term is dominant, while the remaining terms are small. They must be taken into account only for calculation of compressed air effects, inasmuch as in this case the first term drops out. Let us now proceed to the calculation of kinetic energy of the oscillating shell. The element of a shell with the dimension dx'ds has a mass dm qdxods, where q is mass per unit surface area. Components of the velocity are, omitting a few terms of higher order: a -c -u cos (x - ct) t -c-u L pc 4 tan2p sin 2 (x - ct).The incremental kinetic energy is given by: d 1.Tu= 16 2 / 2 LKt) ( t) j = 1 qc2u2 42 7os2 2t (x - ct) + 42 p2tan4 sin2 22 (x - ct) dxds 2 0 L 2 s L2 L Integrating dT over s from zero to 2ip and over x from zero to L, we find the kinetic energy in the limits of one wave of oscillation: 2Tc3 2 2 ( 4T2 p2tani TP =-. — c pquo + ptan) (5) L L2 5

The potential energy of the mation UB = -pAV where AV is the air pressure changes during the time of deforchange of the volume of the shell. The volume of the cylindrical length dx (Fig. 4) before deformation is: dV = -p2dx, 0 b~,j i dv0~~ EdVO dxt L L x(i+ - ) and after deformation, dV~' = A(p +W)2dx(l + MA r 6x/ The change of volume is: AdV = op2 Fig. 4. On the determination of the change of air volume during deformation of the hose. j2 L + +. p ax 2 J p dx 6x1~ )u (''X2 p dx p Substituting the value of o and u and integrating x between the limits of zero to L, we find a change of volume within the confines of one wave of oscillation: 2 2 22 (2 AV = -UoIp L 4 112p2 4 LJ tan2 According to this value of AV, the potential energy of pressure equals: UB = pu2i 2 2t2(3 + 4 tan4 tan L L2 /B~tan28 (6) Let us now examine the deformation energy of the shell; this energy is made up of two parts: (1) Energy UM of deformation of the membrane type, and (2) Energy Ue of bending deformation. This leads to the supposition that, in the presence of membrane deformation, tire cords remain unextensible, and all energy in this case will be consumed by the deformation of the rubber. Average deformation of the shell is determined by using the terms circumferential deformation ct = W/p and lengthwise deformation cx = ~u/6x These expressions represent rubber deformation in the tread and interlayer areas. In layers where the rubber occupies only space between the fibers, ac6

tual deformation of the rubber is greater and approkimately' Etp = kit; Exp = kex, where k = 1/1 - X id where: i = end count per centimeter d = diameter of tire cord The volume of rubber in the layers on one surface of the shell is: 1 d - n k where n = the number of cord layers and the volume of the rubber in the tread and interlayers is: (h - d. n) where h = the overall thickness of the shell walls. Deformation energy of a single volume of rubber in the presence of plane stress conditions and deformation, st and EX, amounts to: a = 2(1 = - I( 2) x + Et + 2peX ~ et ' where Ep = modulus elasticity of the rubber; p.= Poisson's Ratio. The energy of deformation of the tread and interlayers over an element of the shell's surface, dx'ds, is: E 2 dU1 = (h - d - n)2) (e +t +E 2Ce.et) dxds, 2(1 - and the energy of the rubber in the plies is: **Derivation of a reduced formula based. on a substitution of an actual cross section of fiber with a like size taken at right angles, d.(dl/4). ***Translator's Note: The reader is referred.to an article by Bleich, H. St and DiMaggio, F., '"A Strain-Energy Expression for Thin Cylindrical Shells," Jouro Appl. Mech., 20, n. 3 (Sept., 1953), 448, for a discussion of the implications of this expression. 7

Ep 2 2 dU2 - kd ~ n 2(1 2 (x + Et + 2xt ) dxds Summing up these values, we find: EPh (E2 + et + 2kExEt) dxds.7) 2(1 - 2) x where h = the reduced thickness of the rubber; * ad. in (7a) h* = h + (k - l)d n = h +(7a) 4 - iid Substituting the values Ex and Et in expression (7) and integrating s from zero to 2scp, and x from zero to L, we find the energy of membrane deformation of a shell within the limbits of one wave: Ephh 4)'3p uo(l + tan 4 - 2k tan2 ) (8) 2(l - L2) L Now let us calculate the energy of bending of the shell walls. As a result of deformation of the shell, its generatrix, not having curvature originally, acquires a curvature approximately equalling: x,1 = 2x2 but the circumference, which originally had a curvature l/p, acquires a new curvature l/pS-o; in this way the change of curvature of the circumference amounts to: X21 1 1 p + p p2 At a distance y from the neutral surface of deflection in relation to the change of curvature, deformations occur: a2W x = Xly = -- Y ax2 CO ct = X2Y = _ y p2 In view of the rigidity of the cords.s, the middle surface of the tire carcass will be used as the neutral axis and y measured from thereO During the time of deflection it is necessary to take into account not only the deformation of the rubber, but also the deformation of the cordso ****Translator 's Note: Within the assumption of small displacements. 8

In the layer found at a distance y from the neutral surface, tire cord is elongated: k = Ex sin+t + Ct s2 = - sin2B + cos y o The deformation energy of tire cord of this layer on the section dxds of the surface amounts to: E e2 E 2 dU 1 k dxds =ik y22W t2 + x cos4 dxds k 2 where Ek = the ratio of stress in the cord to its deformation. Summarizing energy for all layers, we find: dUk = 1 i Ek Zy2 2 tan2 + 2 COS4 dxds. (9) k 2 k a 2 Whereupon Zy2 —this is the sum of the squares of the distance of all cord layers from the neutral layer. Substituting in expression (9) the values c and a2o/ax2 and integrating s from zero to 2tp and x froip zero to L, we find the energy of deformation of tire cord during the time of deflection: Uk = Ak 2s3 6 _ 4~4p2 tan u) 2 (10) pL 2L where Ak = i Ek Zy2 sin4 Deformation energy of rubber during the time of deflection of the shell may be determined in the following way: The energy of deformation of the element ds dx dy amounts to: E 2 2 2(1 -Ep2 (cx + et + 2~eXet) ds dx dy 2(1_ 2) [( + +2po, a y2 ds dx dy Disregarding the influence of cords on the deformation of rubber in the layers of the carcass and performing the integration over y, we find: dUe Ap 1 k /21 dsx, (11) ep P2L()+ P4 P2,x2] 9

where A Ep(hl + h3) (ha) 3(1 _ l 2) h, and h2 are the distances from the neutral layer to the exterior and interior surface of the shell. Substituting in expression (11) the values C and a2w/ax2 and performing the integration over s and x, we find the deformation energy of rubber inside the limits of one wave due to deflection as: Uep = Ap 2 tan t (1 + 16 - 84 2 (12) We find the velocity of wave travel by equating the kinetic energy change to total potential energy: T = UB + UM + UK + Uep ' where T is determined according to formula (5), but the remaining values according to formulas (6), (8), (10), and (12), respectively0 Thus we find: c2 pp a + b_ 2 + c4 q 1 + d ' ) where = = 2 2p/L 9 a = 5 tan + + (1 + tan - 2 tan) + Ak- + tan4, pp( 1 - 2) p3 p b = tan 2 tan tan 2 tan pp p0 Ak + Ap tan4, and pp3 d = tan4. From expression (13) it is clear that the speed of wave propagation depends on its wavelength, since X is the ratio of the perimeter of a cross section of the shell to the length of the waver(Fig. 5). From expression (13) we find that the minimum value of c occurs when = = -bd + ad2)/cd l/d. (14) 10

C From expression (14) the length of the wave L* in the presence of oscillation is determined as: L* = 2rp The speed of the wave is determined by substituting x2 from expression (14) into formula (13). Fig. 5. The relation of wave speed to the ratio of the perimeter of a cross section of the shell to the length of the wave. For the calculation of the critical speed of a rolling tire, according to the previous expressions, an effective value of specific tire mass q and of the angle of cord fibers P must be derived. By means of an example we are going to show how to calculate the critical speed of a rolling tire of size 7.50 x 16 in the presence of an internal pressure p = 2.5 kg/cm2. In this example, the radius of the tire profile is p = 8 cm; average angle of tire cord can be taken as P = 45~. The tire has 6 layers of cord 9T; the average size of cord layer in the tire tread amounts to 1.4 mm, end-count of the fibers is i = 8 fiber/cm, and the size of cord fibers d = 0.8 mm. The average the thickness of material y = 1.5 mass per unit surface a wall on the average * 10-3 kg/cm3. Then area of the tire is found, assuming that equals 2 cm, and the specific weight of q = 92(1.15 x )0-3) 981 2.35 x 10o6 kg-sec2 cm Inasmuch as the fiber in the casing under the effect of inner pressure has initial deformation of the order of 2%, the modulus of the fiber must be determined at about this value of elongation, whereupon the following formula may be used: Ek " (N3 - N2)100, where N3 = stress in the fiber in the presence of 3% elongation; N2 = stress in the fiber in the presence of 2% elongation. For cord 9T, (N3 - N2) = i kg and consequently Ek = 100 kg/fiber. of rubber resilience Ep = 10kg/cm2; Poisson's Ratio = 0.5. p kg/cm2;n' R t o i 0 5 Modulus Let us calculate dimensions necessary for the computation of critical speed. According to formula (7a) we find the equivalent thickness of the rubber: 11

td2i ~ n h* = h + 4 - -d 4 - -aid = 2 + (8)2 8 * 6 - 2.48 cm 4 - c. 8. o.8 According to formula (10a), rigidity at deflection of cord layers is determined as: Ak = iEk Zy2 sin4, where y is the distance of each of the layers from the neutral surface. Figuring that the neutral surface runs through the center between the third and fourth layers, we get Y1 = 0.35 cm; y4 = 0.07 cm; Y2 = 0.21 cm; Y5 =-0.21 cm; y3 = 0.07 cm; Y6 = 0.36 cm..Then Ak = 8(100)(2)(0.352 + 0.212 + 0.072) = 68 kg-cm The rigidity of the rubber is determined according to Eq. (Ila), realizing that hl = 0.5 cm, h2 = 1.5 cm; = Ep(h3 + h3) 3 (1 - t2) 10[(0.5)3 + (1.5)3] 5( - 0.6 kg - cm 3(0-75) The coefficients a, b, c, and d that appear in Eq. (13) are determined: Eph* ( +t-2A + + ta4 A a=3tan2p+ P( - (1 + tan ' - 2k tan2P)+ p+tan4 = 3(1)2 +_10(2)(48)( [1 + (1)4 - 2(0.5)(1)2] + 68 5 (2 5)(8)(.?75) (2.5)(8)3 15.6 (1)4 (2.5)(8)3 = 4.7; Ak Ap b = tan6- 2 tan2 - p 22 tan tan4 pp3 p3 = (1)6 68(2)(1)2 15.6 (2)(0 15)(1)4 = 88; (2.5)(8) ~-(2.5)(8) Ak + AP tan4 68 + 15.6 (1) = 0.065; ='PP3 an =(2.5)(8)3 d - tan 4 1. k2 is found according to (14): 2 /c - bd + ad2 1 - C cd2 d o/0.065 - 0.88(1) + 4.7(1). o.065(1) - 1 = 6.74. 12

.* = 2.6 Length of wave deformation: L = 2p 2(28) = 19.3 cm In 2.6 The critical speed is found according to (13): 2 pp 2. a 4.7 + c0 4 2.5(8 4.88(6.74) + 0.065(6.74)2 (. x) 1 + 1(6.74) q 1 + d%2 (2.35 x 108) 1 + 1(6-74)2 135.1 x 106 cm2/sec2; c = 3.62 x 103 cm/sec = 36.2 m/sec = 130 km/hr The actual critical rolling speed of this tire, found experimentally by V. I. Novopolsk, amounts to 160 km/hr, and the wave length was L = 17 cm. It is possible to assume some increase of critical speed and decrease of wavelength compared to calculated values primarily due to the influence of fixity at the rim. The expressions derived allow us to analyze the dependence of the critical rolling speed of a tire on inner pressure, angle of tire cord, and rigidity of rubber. Appropriately calculated graphs are presented in Figs. 6, 7, and 8. From the graphs it is obvious that to increase the critical speed one can increase 200 180 E 160 O 140 120 100 I I I I 1 2 3 4 5 6 p, kg/cm2 40 45 50 55 B3* 60 Fig. 6. Dependence of critical speed on inner pressure. Fig. 7. Dependence of critical speed on tire cord angle. the inner pressure and lower the tires' mass, as well as to increase the angle of cord fibers and to-raise the rigidity of the rubber. 13

I,% E 140 o120 I I I The method just used for the determination of critical speed may be developed without using a simplified hypothesis about the equivalence of the rubber-cord shell and tire. However, inasmuch as deformation will of necessity include tangential movement, the mathematical expressions are considerably more complicated. In conclusion it should be pointed out that the influence of dynamic processes on the performance of a tire begin to tell in the presence of speed essentially less than critical. In view of this, the permissible operation speed of a rolling tire must be designated as 0 5 10 15 Ep, kg/cm2 20 Fig. 8. Dependence of critical speed on modulus of rubber resilience. Vmax = Tr Ckp, where ir is the coefficient of allowance, a number smaller than one. The value of the coefficient rT must be chosen with due regard to tire load and the required mileage in the presence of speed V max. 14

UNIVERSITY OF MICHIGAN 31111111111111111111111111 0 11111114 1 7 3 9015 02514 7755