National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001

NASA Technical Paper 3586

Quasi-Static and Dynamic ResponseCharacteristics of F-4 Bias-Ply andRadial-Belted Main Gear TiresPamela A. DavisLangley Research Center • Hampton, Virginia

February 1997

Printed copies available from the following:

NASA Center for AeroSpace Information National Technical Information Service (NTIS)800 Elkridge Landing Road 5285 Port Royal RoadLinthicum Heights, MD 21090-2934 Springfield, VA 22161-2171(301) 621-0390 (703) 487-4650

Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html

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Contents

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Review of Previous Pertinent Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Construction of Bias-Ply and Radial-Belted Aircraft Tires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Bias-Ply Aircraft Tires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Radial-Belted Aircraft Tires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Test Apparatus and Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Test Apparatus No. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Test Apparatus No. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Test Apparatus No. 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Damping Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Viscous Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Structural Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Coulomb Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Data Reduction and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Spring Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Quasi-static tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Dynamic tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Damping Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Quasi-static tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Dynamic tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Energy Loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Footprint Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Quasi-Static Pure Vertical Load Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Load deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Quasi-Static Combined Vertical-Lateral Load Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Load deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Quasi-Static Combined Vertical Fore-and-Aft Load Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Load deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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Free-Vibration Combined Vertical-Lateral Load Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Time-history plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Free-Vibration Combined Vertical Fore-and-Aft Load Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Time-history plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Energy loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Quasi-Static and Free-Vibration Lateral Load Data Comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Quasi-Static and Free-Vibration Fore-and-Aft Load Data Comparison . . . . . . . . . . . . . . . . . . . . . . . 15Spring rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Damping factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Energy loss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Footprint Geometrical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Moment-of-Inertia Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Nomenclature

A constant

Agrooves tread groove area, in2

Agross area of entire footprint, in2

Anet net footprint area, in2

B constant

c damper constant

C constant

Cc critical damping constant

Ceq equivalent viscous damping factor

Ecyc energy dissipated per cycle, in-lb

f frequency of oscillation, Hz

fd equivalent displacement, in.

F external load applied to a system, lb

Fd damping/friction force, lb

Fmax load at maximum deflection, lb

F0 initial applied-force magnitude, lb

Fx=0 load at zero deflection, lb

J mass moment of inertia, in-lb-sec2

k spring rate, lb/in.

kc spring rate from cable interaction, lb/in.

kp platen spring rate, lb/in.

kt tire spring rate, lb/in.

k/m frequency parameter

L length of support cables, ft

m mass of system, lbmmp mass of platen, lbmr radius of circle, in.

R radial distance from center of plate to support cables, in.

t time, sec

wp platen weight, lb

W weight of object, lb

x displacement, in.

X maximum displacement amplitude, in.

α acceleration, ft/sec2

β structural damping factor

∆ increment of change

δ log decrement

µ friction coefficient

µk kinetic friction coefficient

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φ phase angle of damped oscillation, deg

τ period of oscillation, sec

θ angle of circle, rad

ω circular forcing frequency, Hz

ωd frequency of damped free vibration, Hz

ωn natural frequency of vibration, Hz

ζ viscous damping factor

Introduction

When the Wright brothers made their first flight in1903, tires were not part of the design. The first wheeledlanding-gear flight was made in Europe in October 1906by Santos-Dumont’s “No. 14 bis” aircraft (ref.1). Sincethen the bias-ply aircraft tire has gone through manychanges which have enhanced its performance. Thus,there has been little desire in the aircraft landing-gearindustry to change from the bias-ply tire to the newerradial-belted tire. For more than 30 years (Europe in the1960’s and the United States in the 1970’s), the automo-tive industry has found that radial-belted tires heightenvehicle performance and offer many advantages over thetraditional bias-ply tire (ref.2). Despite the benefits ofradial-belted automotive tires (longer tread life, cooleroperating temperatures, and improved friction character-istics), the general belief in the aircraft landing-gearindustry, until 1980, was that the mechanical propertiesof these radial-belted tires were unacceptable for aircraftuse. For example, landing-gear designers had severalconcerns about radial tires that included the following:(1) lateral forces during crosswind landings might exceedtire-wheel coupling capability, (2) tire fore-and-aft

stiffness properties might degrade the performancecharacteristics of aircraft antiskid braking systems, and(3) greatly reduced tire vertical stiffness characteristicsmight adversely affect aircraft landing dynamicresponse.

Understandably, the aircraft tire industry cautiouslyapproached the production of radial-belted tires. How-ever, the idea that aircraft operating costs could be low-ered by increasing tread life, reducing tire weight, andimproving safety margins over the current bias-ply air-craft tire were sufficient reasons to continue efforts toachieve a radial-belted aircraft tire. Several tire manufac-turers, both in Europe and in the United States, havedeveloped radial-belted aircraft tire designs which appearto overcome many of the previously mentioned concernsand which have been successfully tested on several dif-ferent aircraft. Today, radial-belted tires have been certi-fied and are being used on commercial aircraft such asthe Dassault Aviation Falcon 900, Airbus Industries air-craft, the French ATR 42 transport, and on military air-craft such as the U.S. Air Force F-15E, the FrenchMirage, and the British Tornado (ref. 3).

Abstract

An investigation was conducted at Langley Research Center to determine thequasi-static and dynamic response characteristics of the U.S. Air Force F-4 fighter30×11.5-14.5/26PR bias-ply and radial-belted main gear tires. Tire properties weremeasured by the application of vertical, lateral, and fore-and-aft loads. Massmoment-of-inertia data were also obtained. The results of the study include quasi-static load-deflection curves, free-vibration time-history plots, energy loss associatedwith hysteresis, stiffness and damping characteristics, footprint geometry, and inertiaproperties of each type of tire. The difference between bias-ply and radial-belted tireconstruction is given, as well as the advantages and disadvantages of each tiredesign. Three simple damping models representing viscous, structural, and Coulombfriction are presented and compared with the experimental data. The conclusions thatare discussed contain a summary of test observations. Results of this study show thatradial-belted tire vertical stiffness values are comparable to the bias-ply tire stiffnessand that use of this radial-belted tire on aircraft should not affect aircraft landingdynamics. Lateral and fore-and-aft stiffness properties were diminished in the radial-belted tire, thus leading to the possibility of increased tire shimmy and raising thequestion of compatibility with the existing antiskid braking system for this tire. Lat-eral and fore-and-aft damping were increased in the free-vibration tests when theywere compared with damping from the quasi-static tests, suggesting not only the pres-ence of the assumed structural damping but also the presence of viscous damping.Coulomb friction characteristics were not applicable to the tests that were conducted.Footprint geometrical data suggest that footprint aspect ratio effects may interferewith improved hydroplaning potential that is associated with this radial-belted tirethat is operated at a higher inflation pressure than the bias-ply tire. Moment-of-inertia values were lower for the radial-belted tire than for its bias-ply counterpart;the lower values indicate that less energy is needed during spin-up operations andshould result in less tread wear for this radial-belted tire.

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In order to analyze tire mechanical properties associ-ated with aircraft takeoff, landing, and taxi operations,tire and landing-gear designers must have accurate,up-to-date tire data available. Still used extensively,NASA Technical Report R-64 by Smiley and Horne(ref. 4) summarizes the state of knowledge of themechanical characteristics of aircraft tires as they existedover 30years ago. Because this report deals exclusivelywith bias-ply tires, a need exists for a similar database ofradial-belted aircraft tire mechanical properties for use inthe prediction of landing-gear response characteristicsfor modern aircraft.

This research effort was initiated to study themechanical properties of radial-belted and bias-ply30×11.5-14.5/26PR aircraft tires (the main gear tire onthe U.S. Air Force F-4 fighter) under a variety of testconditions and to develop a radial-belted tire propertiesdatabase. In this investigation, the bias-ply tire was testedat the rated inflation pressure of 245 psi, and the radial-belted tire was tested at 310 psi. These inflation pressurescorrespond to a 35-percent tire vertical deflection at therated vertical load of 25000 lb. Both tires have a 26-plyrating. The two tire types used in this investigation areshown, uninflated and unmounted, in figure 1, and theirprincipal characteristics are given in table 1. All tireswere furnished by the U.S. Air Force and were procuredfrom different tire manufacturers. Prior to testing bythe National Aeronautics and Space Administration(NASA), all tires were preconditioned at the WrightResearch Development Center, Dayton, Ohio with2 miles of taxi tests at 26 knots at rated load and inflationpressure.

The objectives of the work presented here are (a) todetermine, to evaluate, and to compare certain quasi-static and dynamic response characteristics of the30×11.5-14.5/26PR aircraft main gear tire of a bias-plyand a radial-belted design; (b) to use these properties tohelp define tire performance during taxi, takeoff, andlanding operations; and (c) to define the suitability of theradial-belted tire as a replacement for the bias-ply aircrafttire. To accomplish these objectives, vertical, lateral,fore-and-aft, and inertial tire properties were determined,as well as the type of damping that was present duringstatic and dynamic testing. The study involved the fol-lowing tasks:

1. Measurement of vertical, lateral, and fore-and-aft loaddeflection, stiffness and hysteretic properties of eachtire type

2. Measurement of tire footprint geometrical and inertialproperties

3. Definition of viscous, structural, and Coulomb frictiondamping models

4. Application of these simple damping models to thetwo tire designs tested

5. Comparison and analysis of experimental test results

Review of Previous Pertinent Work

From the simple horsedrawn cart to the high-speedtransport aircraft of today, wheels and tires have becomean essential part of our transportation system. As a result,landing-gear designers and engineers have been optimiz-ing methods of defining tire mechanical properties andsolving such problems as landing-gear vibration, tirewear, and tire heating. A major contribution to thelanding-gear industry, which set the standard for analyti-cally defining tire properties, was a publication bySmiley and Horne (ref.4) on mechanical properties ofpneumatic tires. The database for NASA Report R-64,however, was various sizes of bias-ply pneumatic tiresonly.

Tanner (ref. 5) did early testing with radial-beltedaircraft tires in 1974 in which he defined tire fore-and-aftelastic characteristics and noted lower tread wear andheat dissipation for the radial-belted tire compared withbias-ply and bias-belted tires. He also concluded that thelower fore-and-aft stiffness properties of the radial-beltedtire, compared with the bias-ply tire, could have animpact on aircraft antiskid system performance. Tiremanufacturers published some of their findings on radial-belted tire characteristics in the 1980’s. The data pre-sented were limited but invaluable to tire designers andengineers (refs. 6–8). Both NASA and the U.S. Air Forcehave taken the lead in quasi-static and dynamic testing ofradial-belted aircraft tires. References 5–20 compare theproperties of radial-belted and bias-ply aircraft tires.These publications have helped aircraft tire manufactur-ers optimize the radial-belted design.

In addition to defining tire mechanical properties,there has been interest in the damping characteristics ofaircraft tires, especially radial-belted tires. Shimmy is aproblem for aircraft landing gear just as it is for the frontend of automobiles. Shimmy, defined as the self-excitedoscillation of wheels about their axis, was evaluatedearly in aviation history for aircraft nose gear byVon Schlippe and Dietrich (refs. 21 and 22) and later byMoreland (ref. 23), de Carbon (ref. 24), and Collins andBlack (refs. 25 and 26). Although these theories dorequire certain tire properties such as static lateralstiffness and the viscous damping factor to be initiallydetermined, none completely addresses the issue of tiredamping and its contributions to landing-gear shimmybehavior. Many works have been published that addressstructural (hysteretic) and viscous damping(refs. 4,11–13, 19, and 27–34), but few tire analysts have usedthese developed models to define precisely the damping

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mechanism in the tire-wheel assembly under both quasi-static and dynamic test conditions.

Construction of Bias-Ply and Radial-BeltedAircraft Tires

The aircraft tire is a mechanical system that has threeprimary functions: (1) to provide lateral stability to theaircraft by the generation of lateral forces while underdeformation, (2) to act as a mechanical buffer and shockabsorber while under load and in contact with theground, and (3) to provide a mechanism for braking toreduce ground speed (ref. 35).

The generation of lateral forces is an important func-tion of the aircraft tire. As an airplane approaches a run-way, it is supported laterally in the air and can approachat an angle in a crosswind. Airplane aerodynamics con-tinue to provide lateral support to the aircraft at touch-down where a minimum lateral tire force is required. Themaximum lateral force from the tires is required as theaircraft slows down during rollout and during taxi opera-tions. Thus, aircraft tire manufacturers had to design aradial-belted tire that would not generate lateral forcesthat were too high during touchdown, which, in turn,could cause lateral instability that would transmit highloads to the landing gear (ref. 7).

While acting as a buffer and shock absorber for theaircraft during the landing impact, the tire must also beable to carry extremely heavy loads at high speeds. Arace car tire may require this speed capability, but com-pared with an aircraft tire, a race car tire is lightly loaded.In contrast, military and commercial aircraft operate athigh speeds but can be loaded up to 50 000 lb per tire.

During braking the tire must be able to reduceground speed while surviving within and resisting envi-ronmental factors to decelerate the aircraft (ref. 8). Envi-ronmental factors influenced by weather are snow, ice,and rain; factors that result from runway conditions arebumps, potholes, and foreign objects.

The following sections describe the design of thebias-ply and radial-belted aircraft tires, examine the dif-ferences between the two tire types, and highlight theunique radial-belted design.

Bias-Ply Aircraft Tires

The bias-ply tire is constructed of numerous lami-nates of rubber-textile plies that alternate at variousangles from 60° near the wire beads to 30° near thecrown area of the tire (fig. 2). Additional plies are laid atsome specified angle between the tire carcass and thetread to provide tread reinforcement. Multiple bead wireson each side of the tire hold together the large number of

plies and form a torus-like shell. The highly stiffenedshell and the weight of the tire are a result of the plyassembly and the multiple-ply casing. At the bead heel,chaffer strips are added for additional protection.Because of its cross-ply construction, the bias-ply tirehas greater interply friction than its radial-belted tirecounterpart, which leads to significant structural stressesduring tire flexing. These stresses result in severe heatbuildup in the tire casing and tread rubber during groundoperations. This heat buildup can be limited by addingreinforcing plies, but at the expense of additional weight.The size of the bias-ply tire required for an aircraft appli-cation is determined by both the load-carrying require-ments and the acceptable tire pressure (ref. 7).

Radial-Belted Aircraft Tires

The radial-belted aircraft tire has carcasspliesapproximately oriented in the plane of the tire cross sec-tion (fig. 3). Its size and load-carrying requirementsdetermine the number of plies in the tire. Fewer plies ofhigher denier textile cords result in a weight and volumesavings over the comparable bias-ply tire. The weightand volume savings result in an increase in payload forcommercial aircraft and an increase in armament for mil-itary aircraft. For the radial-belted tire tested in thisresearch effort, there was a 20-percent weight savingsover the bias-ply tire. Generally, only one steel-bead wireis needed on each side of the tire, as opposed to multiplebead wires in the bias-ply tire. For the radial-belted tiresused in this investigation, a textile cord belt surroundsthe casing of the tire circumferentially, and a woven steelbelt surrounds the cord belt and acts as a protector ply.The separation of the casing and belt package contributesto the uniqueness of the radial-belted tire. When the cas-ing flexes under load, the tread is stabilized by the beltpackage, and there is less tire scrubbing (lateral sliding ofthe tire on the pavement). Shear stresses in the rubbermatrix are minimized, and loads are efficiently distrib-uted throughout the tire structure. There is less interplyfriction and thus less heat buildup during ground opera-tions. Therefore, the radial-belted tire can have a layer ofunreinforced wear rubber (no reinforcing plies) beneaththe tread. This design reduces the tendency of the tire tochunk and increases its wear resistance. (See refs. 2, 3, 5,7, 8, 14, and 35.)

Test Apparatus and Procedures

Quasi-static pure vertical-loading tests, combinedvertical-lateral loading tests, and combined vertical fore-and-aft loading tests were conductedto measure tireload-deflection properties, spring-rate values, dampingfactors, and energy loss associated with hysteresis. Foot-print geometrical properties for the two tire designswere also measured. Free-vibration tests of combined

4

vertical-lateral and vertical fore-and-aft tests were com-pletedto measure tire stiffness values, damping charac-teristics, and energy loss associated with dampedharmonic motion. Finally, moment-of-inertia tests wereobtained for each tire type.

The bias-ply tire was tested at an inflation pressureof 245 psi, and the radial-belted tire was tested at aninflation pressure of 310 psi. The inflation pressures cor-respond to a 35-percent tire vertical deflection at therated vertical load of 25000 lb. An inflation pressure of245 psi for the radial-belted tire yielded a 52-percent tiredeflection. Although it is preferable to compare tires atthe same inflation pressure, it was more important thatthey be tested at the same percent tire deflection becausethis percentage is a major requirement for tire mixabilityon landing gear. An increase in tire deflection can causetire overheating. Tests were conducted on the radial-belted tire at the lower inflation pressure of 245 psi, andthe results indicate greatly reduced quasi-static anddynamic mechanical properties. References 11 and 20contain further information and data on these tests at thelower inflation pressure of 245 psi for this radial-beltedtire design. The following sections describe the test appa-ratus and procedures used to obtain the desired tireproperties.

Test Apparatus No. 1

The test apparatus used to measure the tire verticalmechanical properties under quasi-static loading condi-tions is shown in figure 4. The tire wheel-axle assemblyis mounted on a dynamometer that is instrumented withfive strain-gauge beams. Two beams are used for mea-suring vertical load and two are used for fore-and-aftload; one beam is used for measuring lateral load. Thetest fixture has one hydraulic cylinder which applies ver-tical load to the dynamometer and loads the tire onto thesurface of a 40-in-square frictionless table or bearingplate. The table is instrumented with three load cellsmounted underneath to measure the applied vertical load.Each load cell has a span of 0 to 50000 lb with a load-ing accuracy of lb. A displacement transducer ismounted parallel to the hydraulic cylinder to measure tirevertical displacements. The displacement transducer hasa span of 0 to 5.5 in. with a measuring accuracy of±0.02 in.

For the quasi-static pure vertical-load tests, load wasapplied hydraulically to the tire until the maximum ratedvertical load of 25000 lb was reached; the applied loadwas then gradually reduced to zero. Output data from thevarious instruments were recorded by a computer in1000-lb increments for these tests. Measurements weretaken at four different peripheral positions around eachtire. Vertical load and deflection were continuously mon-

itored during the loading and unloading cycle to producea load-deflection curve or hysteresis loop. Such data pro-vided an indication of tire vertical-loading behavior anddefined the vertical spring rate and energy loss character-istics. Subsequent to the tests, the data were convertedinto engineering units and saved for further analysis andevaluation.

Test Apparatus No. 2

A second test apparatus used for quasi-static andfree-vibration lateral and fore-and-aft load tests is shownschematically in figure 5. It consists of a main structurewith two three-bay portal frames joined overhead by fourbeams and along the floor by a thick plate. The tire-wheel assembly is mounted between two aluminumadapter plates which are each fixed to a vertical beamsuspended from the upper part of the structure. Theadapter plates prevent axial rotation of the wheel andsupport the tire-wheel assembly. A steel platen 26-in.square and 6-in. thick, suspended from four 0.5-in-diameter vertical cables 6.5-ft long is used to apply thevertical load to the tire. Each cable is suspended from aload cell (with a span of 0 to 50000 lb and a loadingaccuracy of lb) connected to a screw jack that ismechanically driven by an electric motor. The fourcables move simultaneously and displace the platen inthe vertical direction to load or unload the tire. A grittyfilm is applied to the platen surface to reduce tire foot-print slippage. A two-degree-of-freedom analysis of theplaten motion that confirms that the cable-suspended sys-tem has no significant coupling between the pitching andtranslating motions is given in reference 30.

Quasi-static lateral loading and quasi-static fore-and-aft loading of the tire were attained by displacing theplaten in the lateral or fore-and-aft direction by means ofa hydraulic cylinder after a vertical load was applied.Lateral (side) forces were measured by a load cell (with aspan of 0 to 50000 lb and a loading accuracy of±30 lb)connected in series with the hydraulic cylinder. A similarinstrumentation configuration was used to measure fore-and-aft (braking) forces and is shown in figure 6. Verti-cal, lateral, and fore-and-aft displacements of the platenwere measured by displacement transducers (with a spanof 0 to 5.5 in. and a measurement accuracy of±0.02 in.).Platen displacements were considered to be equal to tirefootprint displacements. Axial rotational movementsbetween the wheel and the adapter plates were measuredwith a direct current displacement transducer (DCDT).An extensiometer fabricated from a 3-in. copper beryl-lium strain-gauge arch capable of measuring small dis-placements was used to measure tire-wheel slippage,which sometimes occurred during testing of the radial-belted tire. This tire-wheel slippage may be caused by thedifference between the radial-belted and bias-ply tire

30±

30±

5

bead zone, which along with lower structural stiffnesscharacteristics of the radial-belted tire, could inducedifferent contact pressures and stress fields in the wheel(ref. 36).

For the quasi-static combined vertical-lateral andcombined vertical fore-and-aft tests, vertical load wasapplied to the tire and maintained at the tire-rated load of25000 lb. A side load was gradually applied up to themaximum load of 3000 lb and then reduced to zero. Thisside force was restricted to a level at which no tire foot-print slippage was discernible and which corresponded to12 percent of the vertical load. This loading process wasrepeated in the opposite direction and resulted in fourcycles that formed a hysteresis loop. The data acquisitionsystem was manually triggered at side-load increments of250 lb. Data from the various sensors were fed into acomputer and afterwards saved for further reduction andanalysis. Similar loading and data reduction procedureswere used for the combined vertical fore-and-aft tests.

For the free-vibration tests, a quick-release mecha-nism in series with a hydraulic cylinder is connected tothe platen by a 0.24-in-diameter steel cable. This hydrau-lic cylinder is used to apply a lateral or fore-and-aft loadto the tire and then the load is released, causing the tirewheel-platen assembly to vibrate. A displacement trans-ducer is used to determine lateral and fore-and-aft foot-print deflections, and a 25g accelerometer is used tomeasure tire-platen accelerations. Figure 7 shows thistest fixture in the fore-and-aft free-vibration testconfiguration.

Dynamic tire characteristics were obtained fromsimple, single degree-of-freedom free-vibration tests.These tests were conducted through a vertical load rangeof 5000 to 25000 lb in 5000-lb increments. The specifiedvertical load was applied and then a side or braking forcewas applied up to 3000 lb. A quick-release mechanismwas activated, causing the tire wheel-platen assembly tofreely vibrate. The instrumentation output data wererecorded by a computer that was manually triggered justprior to the activation of the quick-release mechanism.The data were saved for further reduction and analysis.

Test apparatus no. 2 was also used to obtain tire foot-print geometrical properties for each tire design. Thefootprints were obtained from static vertical loads rang-ing from 5000 to 25000 lb in 5000-lb increments. Theprocess to obtain the footprints involved coating the tiretread with ink or chalk and applying the desired verticalload to the tire on a cardboard sheet located between thetire and the platen. Geometrical properties were obtainedwith a computerized planimeter from the resulting foot-print silhouettes.

Test Apparatus No. 3

Mass moment-of-inertia properties of the tires testedwere measured by using the torsional pendulum shown infigure 8. The torsional pendulum used for these tests con-sists of two 24-in-diameter circular plates bolted togetherand suspended in parallel by three equally tensioned,17-ft wire cables that are attached to an overhead sup-port. An index in 5° increments up to±20° is scribed onthe plate. A pointer is aligned with the index at 0° as areference mark.

The tare moment of inertia of the two plates was ini-tially determined by rotating the plate assembly througha predetermined angle and releasing it. The plate assem-bly was rotated 3 times at approximately 10°, 7.5°, 5°,and 2.5° for 20 oscillations each, and the time wasrecorded with a standard stopwatch. Similar tests wereconducted with the inflated tire, with the wheel assem-bly, and with the rotors suspended between the plates.The weights of the plates and the tire-wheel assemblywere recorded.

Damping Models

To optimize the performance of landing gear,designers and engineers must analyze all landing-gearparts that play key roles. The tires are one key element.Understanding the dynamic behavior of aircraft tires isimportant for the optimum design and operation of theaircraft and its landing gear. Antiskid braking systemsare affected by the tire’s elastic responses; cross-windlandings are affected by the tire’s lateral dynamic behav-ior; shimmy is influenced by the tire's torsional behavior.Analysis of the damping properties and how differenttypes of damping can influence tire performance isimportant in correcting these potential problems.

This section reviews three simple damping models:viscous, structural, and Coulomb friction. These modelswere chosen because of their simplicity and because aglobal view of damping was desired. The models onlyprovide some insight into the damping mechanisms thatmight be present during quasi-static and dynamic tiretesting and are not presented as accurate models for char-acterizing aircraft tire damping. (See refs. 30 and 37–40.)

Viscous Damping

In the study of vibrations, the viscous damper (dash-pot) is the most common type of energy-dissipating ele-ment and is defined as a resistive force exerted on a bodymoving in a viscous fluid. An example of a viscousdamper in aircraft is the landing-gear shock absorber.

In developing the viscous damping equations, it wasassumed that the tire is represented by a linear elasticspring in parallel with a damping element to obtain an

6

approximation of its damping behavior. To more realisti-cally represent the tire, a viscoelastic model should beused with an additional spring placed in series with thedamper. However, this model is less attractive analyti-cally since the in-series spring and damper elements havefrequency-dependent parameters (ref. 34).

The viscous damping factorζ, which is the ratio ofthe actual damping present in the system to the criticaldamping constant for the system, is given by

(1)

The tire wheel-platen system was of the dampedfree-vibration type in which energy was dissipated fromthe system. In this study, the response of the free-vibration system was assumed to be underdamped so that0 < ζ < 1 andc < 2mωn:

(2)

Equation (2) can be interpreted as oscillatory motionwith constant frequencyωd andphase angleφ but withexponentially decaying amplitude as seen infigure 9.

The rate at which the maximum amplitude decays isexpressed in terms of the natural log of an amplituderatio that is known as the log decrement denoted byδ:

(3)

(4)

(5)

Table 2 gives a summary of the viscous dampingparameters.

Structural Damping

Solid materials exhibit structural or hysteretic damp-ing. This type of damping is caused by internal friction inthe material as internal planes slip or slide during defor-mation and is commonly seen as a hysteresis loop inwhich a phase lag between force and deflection exists.The area enclosed in the hysteresis loop represents theenergy loss per loading cycle and can be written as

(6)

Although structural damping is the most commontype of damping, in free-vibration studies structuraldamping becomes indistinguishable from viscous damp-ing and is difficult to treat analytically since it is definedin terms of energy loss and a nonlinear function of dis-placement. It is more convenient to express structuraldamping in terms of an equivalent viscous dampingfactor.

For the free-vibration case, the damping force isdefined by the damping constant and the first derivativeof displacement:

(7)

SubstitutingF from equation (7) into equation (6) yields

(8)

(9)

(10)

Systems possessing structural damping that are sub-jected to harmonic excitation may be treated as if theywere subjected to equivalent viscous damping:

(11)

The energy dissipated per cycle for structural damp-ing is independent of the frequency and proportionalto the stiffness of the material and the square of thedisplacement amplitude. SubstitutingCeq from equa-tion (11) forc in equation (10) yields

(12)

Then β can be determined by using the log decre-ment method for a free-vibration system as was done inthe viscous damping case. The energy equation for thehalf cycle between t1 andt2 is

(13)

(14)

(15)

ζ cCc------ c

2mωn--------------= =

x t( ) Ae ζωnt– ωdt φ–( )cos=

Ae ζωnt–

δx1

x2-----

ln ζωnτ 2πζ

1 ζ2–( )-----------------------= = =

ζ δ

2π( )2 δ2+[ ]-----------------------------------=

ζ δ2π------≅

∆Ecyc F xdcyc∫ Fx td

02π ω⁄∫= =

F cx cωX ωt φ+( )cos= =

∆Ecyc cω2X2 12--- 1 2 ωt φ+( )cos+[ ]

td02π ω⁄∫=

∆Ecyc cω2X2 t2---

1ω----

ωt φcossin+0

2π ω⁄=

∆Ecyc cωπX2=

Ceqβkω------=

∆Ecyc πβkX2=

kX12

2----------

πβkX12

4-----------------

πβkX1.52

4---------------------––

kX1.52

2-------------=

kX12

2----------

πβkX12

4-----------------–

kX1.52

2-------------

πβkX1.52

4---------------------+=

kX12

2---------- 1 πβ

2------–

kX1.52

2------------- 1 πβ

2------+

=

7

The energy loss in a half-cycle is assumed to be pro-portional to that of a full cycle. Simplifying equation (15)yields

(16)

Looking at the next half-cycle,

(17)

Combining equations (16) and (17) gives a ratio of twosuccessive amplitudes:

(18)

where β is very small for most materials and thus can bewritten as

(19)

The log decrement is thus

(20)

(21)

In the case of a quasi-static test, structural dampingcan be mathematically written in terms of the viscousdamping factorζ by the complex stiffness equation

(22)

where

F external force on systemζ viscous damping factork total spring ratex complex displacement

The complex sinusoidal force is given by thefollowing:

(23)

where

F0 initial applied-force magnitudeω circular forcing frequency

SubstitutingF from equation (22) into equation (23)yields

(24)

Simplifying equation (24) and solving forx,

(25)

(26)

Converting (1 –i2ζ) to polar notation by using figure 10and the following relationship ofθ to ζ

(27)

yields, for small angles,

(28)

(29)

(30)

Substitutingθ from equation (28) andr from equation(30) into equation (29),

(31)

Substituting (1− i2ζ) from equation (31) intoequa-tion (26) yields

(32)

A plot of displacement with respect to applied forceyields a hysteresis loop whose width increases withζ.The relationship of the width of the hysteresis loop to thedamping factor can be derived by using the real part ofthe complex applied forceF and displacement x:

(33)

(34)

Substitutingeiωt from equation (33) into equation(23)gives

(35)

X12

X1.52

----------1 πβ

2------+

1 πβ2

------–----------------=

X1.52

X22

----------1 πβ

2------+

1 πβ2

------–----------------=

X1

X2------

1 πβ2

------+

1 πβ2

------–----------------=

X1

X2------ 1 πβ+≅

δX1

X2------

ln 1 πβ+( )ln πβ≅= =

β δπ---=

F 1 i2ζ+( )kx=

F F0ei ωt=

F0ei ωt 1 i2ζ+( )kx=

xF0 k⁄

1 i2ζ+( )----------------------ei ωt=

xF0 k⁄

1 4ζ2+( )----------------------- 1 i2ζ–( )ei ωt=

tan 1– 2ζ–1

--------- θ=

θ 2ζ–=

1 i2ζ–( ) rei θ=

r 1 4ζ2+=

1 i2ζ–( ) 1 4ζ2+( ) ei 2ζ–( )=

xF0 k⁄

1 4ζ2+( )---------------------------ei ωt 2ζ–( )=

ei ωt ωtcos=

ei ωt 2ζ–( ) ωt 2ζ–( )cos i ωt 2ζ–( )sin–=

F F0 ωtcos=

8

Substituting ei(ωt − 2ζ) from equation (34) into equa-tion (32) yields

(36)

For x = 0, cos(ωt − 2ζ) = 0. Therefore,

(37)

From equation (35) the applied force at zero dis-placement can be defined for small damping by thefollowing:

(38)

For small angles,

(39)

and

(40)

Equation (40) represents structural damping in termsof viscous damping, and the static damping factor isa function of the load values at zero displacementand at maximum displacement (depicted graphicallyin figure 11). Table 3 presents the structural dampingparameters.

Coulomb Friction

Coulomb (dry friction) damping occurs when bodiesslide on dry surfaces. Motion begins when there is aforce acting upon the body that overcomes the frictionthat resists this motion. This dry friction force is parallelto the surface and proportional to the normal force to thesurface:

(41)

The decay is linear with time (fig. 12), and themotion stops when the displacement is not sufficient forthe spring's restoring force to overcome static friction.Decay occurs at the end of the half cycle when the ampli-tude is smaller than 2fd:

(42)

Repeating for the next half-cycle,

(43)

For the static test case, the determination of theCoulomb friction parameters is shown in figure 13,where the damping force is defined as

(44)

From figure 13,

(45)

Stiffness properties can be determined from the springrate

(46)

Table 4 presents the Coulomb friction parameters.

Data Reduction and Analysis

Basic data reduction techniques used in this investi-gation that enabled the comparison and analysis of staticand dynamic (free-vibration) data are presented in thefollowing sections.

Spring Rate

Spring-rate (stiffness) values give an indication ofthe types of vertical perturbations, elastic responses, andlateral stability characteristics of an aircraft tire. Spring-rate values for each tire design were computed fromexperimental data for both the quasi-static and free-vibration lateral and the fore-and-aft load tests, as well asfor the quasi-static vertical-load tests.

Quasi-static tests.The quasi-static vertical, lateral,and fore-and-aft spring rates were determined by measur-ing the slope at specific points of load applicationandinitial load relief that are defined by the vertical,lateral, and fore-and-aft load-deflection curves. Theupper limit of the initial load-relief curve was not consid-ered because of the uncertain value of the turnaroundpoint as a result of the mechanical system limitation. Thefollowing steps were used for calculating the slopes ofthe load application and initial load-relief curves. Theslopes are represented by the solid lines that are tangentto the curves, as shown in figure 14.

1. A second-degree polynomial curve fit was chosen foreach loading cycle, and a third-degree polynomialcurve fit was chosen for each unloading cycle of thehysteresis loop.

xF0 k⁄

1 4ζ2+---------------------- ωt 2ζ–( )cos=

ωtπ2--- 2ζ+ 3π

2------ 2ζ+ …, ,=

Fx=0 F0π2--- 2ζ+

cos=

Fx=0 2ζF0=

ζ 12---

Fx 0=

Fmax--------------

=

Fd µkW=

x1 x 1––( )2Fd

k---------- 2 f d= =

x1 x2–( )4Fd

k---------- 4 f d= =

Fd µW=

A 2Fd 2µW= =

kBC----

2Fd

2X----------= =

9

2. The start and end points for calculating the slope onthe loading and unloading curves were selected byusing a computerized digitizer.

3. The slope was calculated at discrete points along theloading or unloading curve within the bounds of theselected start and stop points.

The method described previously is normally usedto obtain the tire spring rate from a quasi-static load-deflection curve. However, in order to compare thequasi-static and dynamic spring-rate values for the lateraland fore-and-aft tests, a technique that considers theeffect of cable interaction was used. The slope of thequasi-static force-displacement hysteresis-loop axis (thedashed line connecting the loop extremes, as shown infig. 14) defines the total stiffness applied to the platen kp.The tire spring ratekt is then determined by subtractingthe cable interaction stiffnesskc from the platen springrate. (See ref. 30.)

(47)

(48)

where

F load applied to the system, lblc free swing cable length = 6.5 ftkt tire spring rate, lb/in.kp platen spring rate, lb/in.kc spring rate from cable interaction, lb/in.

Dynamic tests.For the lateral and fore-and-aft free-vibration tests, the tire spring rate is also a function of theplaten spring rate minus the spring rate from cable inter-action. Since the frequency is a function of the platenspring rate and the mass of the platenmp, the spring ratekp may be calculated as follows (ref. 30):

(49)

(50)

where

f frequency, Hz

τ period of oscillation, sec

The period is measured from the displacement time-history plots from which the frequency parameter is

determined. The total spring rate acting on the platen isthen

(51)

(52)

where

wp platen weight = 1182 lbα acceleration = 32.2 ft/sec2

As in the quasi-static case, the spring rate, caused bycable interactionkc must be subtracted fromkp, and thetire spring rate then becomes

(53)

Damping Factor

Damping factors were determined for both the quasi-static and dynamic lateral tests and also for the fore-and-aft tests that were conducted. These damping factorsgave a first approximation of the damping type in thesystem and whether one tire type exhibited more damp-ing than the other.

In this investigation, viscous and structural dampingwere determined based on system response characteris-tics. Coulomb friction, however, was not applicable tothe system because the limited loads achieved in the lat-eral and fore-and-aft directions did not allow for large-scale (measurable) tire footprint slippage. If these testswere repeated at higher lateral and fore-and-aft loads,Coulomb friction might occur.

Quasi-static tests.For the quasi-static tests, thestructural damping factor written in terms of the viscousdamping factorζ is determined from the lateral and fore-and-aft load values at zero and maximum tire deflections(fig. 11).

(54)

where

Fx=0 load at zero deflectionFmax load at maximum deflection

Dynamic tests.For the dynamic tests, the viscousdamping factorζ is determined from the logarithmic

kcFlc----=

kt kp kc–=

f1

2π------ k

m----=

km---- 2πf( )2 2π

τ------

2= =

km---- frequency parameter

2πτ

------ 2

=

kp 2πf( )2mp=

kpkm----

wp

α-------

=

kt kp kc–=

ζ 12---

Fx=0

Fmax------------

=

10

decrement of the decaying displacement amplitude of adisplacement time-history plot

(55)

However, in these tests it was difficult to determinethe logarithmic decrement directly from the time-historyplots because of the drifting equilibrium level (X-axis). Amore accurate viscous damping factor can be determinedby computing a double amplitude from the differencebetween the spline-fitted displacement envelope thatpasses through the displacement extremes, as shown infigure 9. The logarithmic decrementδ and viscous damp-ing factorζ can then be expressed as follows:

(56)

(57)

In the free-vibration case, the structural dampingfactor β can be written in terms of the viscous dampingfactorζ.

(58)

Energy Loss

The energy loss associated with hysteresis is anotherindicator of structural damping. For the quasi-static load-deflection curves, the energy loss is typically determinedby the area enclosed in the hysteresis loop. To have amore precise comparison between the quasi-static anddynamic tests, the energy loss per cycle is computed as afunction of the structural damping factor, the tire springrate, and the maximum displacement amplitude for bothtest conditions

(59)

where

β structural damping factor = 2ζkt tire spring rate, lb/in.X maximum displacement amplitude, in.

Footprint Geometry

Footprint geometrical properties were determinedfrom the net footprint area that was calculated from thefollowing equation:

(60)

where

Agross area of entire footprint, in2

Agrooves area of tread grooves, in2

Moment of Inertia

The mass moment of inertia for each tire design andthe tare inertia for the two pendulum plates were calcu-lated using the following equation:

(61)

where

J mass moment of inertia, in-lb-sec2

τ average period of system oscillation, secW weight of object being measured, lbR radial distance from center of plate to support

cables, in.L length of support cables, 17 ft

The mass moment of inertia of the tire and rotating partswas calculated using the following equation:

(62)

Results and Discussions

Several characteristics of the radial-belted tire havedelayed its acceptance into the aircraft industry. The keyto its gaining acceptance lies in understanding the differ-ences between bias-ply and radial-belted tires. The fol-lowing sections review the results of the vertical, thecombined vertical-lateral, and the combined verticalfore-and-aft tests and make observations of the similari-ties and differences between the F-4 radial-belted aircrafttire and its bias-ply counterpart.

Quasi-Static Pure Vertical Load Tests

Quasi-static (nonrolling) vertical load tests werecompleted with a bias-ply and a radial-belted F-4 aircrafttire. Results of these tests are presented in the form ofvertical load-deflection curves, vertical spring rate, andenergy loss associated with hysteresis.

Load deflection.As each tire came into contact withthe frictionless table, vertical load was applied until thedesired load of 25000 lb was reached. The applied loadwas then reduced to zero. The resulting load-deflectioncurve, or hysteresis loop, is indicative of tire vertical-loading behavior and provides information which definesthe tire vertical spring rate and energy loss. Four load-deflection curves were generated for each tire at four dif-ferent tire peripheral positions.

ζ 12π------

X1

X2------

ln=

δ2X1

2X2----------

ln=

ζ δ2π------=

β 2ζ=

∆Ecyc πβktX2=

Anet Agross Agrooves–=

Jτ2W R2

4π2L-----------------=

Jtireτ2W R2

4π2L----------------- Jplates–=

11

Figure 15 shows typical vertical load-deflectioncurves for the bias-ply and radial-belted tires: the lowerbound of tire vertical stiffness is represented by the load-application curve, and the upper bound of tire verticalstiffness is represented by the initial load-relief curve.For the bias-ply tire, the data show that the verticaldeflection is nonlinear at the initial load application butthat it is linear over the remaining load-application rangeto the maximum load of 25000 lb. Analogous load-deflection characteristics apply to the load relief range.The loading-unloading process shows some hysteresis asa result of energy loss during this process. The maximumtire vertical displacement is 2.48 in.

For the radial-belted tire, the load deflection is non-linear during the entire load application and the loadrelief cycles. The maximum tire vertical displacement is2.53 in., which is 2 percent higher than that of the bias-ply tire.

The vertical load-deflection characteristics of radial-belted and bias-ply tires are of interest to strut designers.In general, the load-deflection characteristics are similarfor the two tire designs tested and suggest that thereshould be no impact on strut valving and on the loadstransmitted to the airframe for aircraft equipped with thisradial-belted tire (ref. 6).

Spring rate.In order to cover the range of stiffnessvalues that the tire would experience as a result of verti-cal perturbations during aircraft taxi, takeoff, and landingmaneuvers, vertical stiffness data are presented in termsof tire spring rate (fig. 16). The lower bound of tire verti-cal stiffness for the bias-ply tire is represented by theload-application curve denoted by the square symbols,and the upper bound of vertical stiffness is representedby the load-relief curve denoted by the triangular sym-bols. Vertical spring-rate values were obtained by mea-suring the instantaneous slope along the vertical load-deflection curve. A regression analysis technique wasused to fit a curve through the spring-rate data. At initialload application, the spring rate of the bias-ply tireincreases linearly from 7121 lb/in. to 11515 lb/in. andremains constant at this higher value for the remainder ofthe load application process. A maximum spring rate of14432 lb/in. is observed at load relief, which becomesnonlinear as the load decreases to 11 515 lb/in.

Vertical spring-rate values for the radial-belted tireare also shown in figure 16; the lower bound of tire verti-cal stiffness is represented by the load-application curvedenoted by the diamond symbols, and the upper bound ofvertical stiffness is represented by the load-relief curvedenoted by the circular symbols. During load application,the tire spring rate is nonlinear and increases from7170lb/in. to 12340 lb/in. The maximum spring rate for

the radial-belted tire is 14113 lb/in. at initial load reliefand continues to decrease nonlinearly.

In figure 16, it can be observed that the vertical stiff-ness characteristics of the bias-ply and radial-belted tiresare similar. One implication of this similarity is that thelanding dynamics characteristics of an aircraft equippedwith this radial-belted tire would be similar to an aircraftequipped with standard bias-ply tires.

Energy loss.Energy loss associated with hysteresisduring the loading and unloading period is representedby the enclosed area of the load-deflection curve (fig. 15)and was measured using a computerized planimeter. Thishysteresis loop arises largely as a consequence of thestructural or hysteretic damping forces that oppose thetire deformation. The average energy loss is 2819 in-lbfor the bias-ply tire and 2547 in-lb for the radial-beltedtire under vertical-loading conditions.

The lower energy loss for the radial-belted tire sug-gests that there is less heat generated in the radial-beltedtire during tire cyclic deformation; therefore, it should bea cooler operating tire. This suggested heat reductionmay be a result of the tire construction, which produceslower internal shear stress and thus energy loss. The heatreduction could lead to improved tire durability andallow for shorter aircraft turnaround times (ref. 8).

Quasi-Static Combined Vertical-Lateral LoadTests

Quasi-static combined vertical-lateral load tests wereconducted to determine the stiffness and damping char-acteristics of the two tire designs tested. These results arerepresented by load-deflection curves and spring-ratecurves that are analyzed in the following sections.

Load deflection.An initial vertical load of 25000 lbwas applied to the tire, which was then subjected to aload of 3000 lb perpendicular to the wheel plane. Fiveindividual curves (load application and load relief) weregenerated to obtain a complete hysteresis loop.Lateral(side) force and tire footprint displacement data wererecorded at 250-lb side-load increments. Four load-deflection curves (hysteresis loops), each one corre-sponding to a different location 90° around the peripheryof the tire, were developed. Typical load-deflectioncurves for the bias-ply and the radial-belted tires areshown in figure 17. Based on observed nonlinear trends,a second degree polynomial was chosen to curve fit thedata points obtained during load application, and a thirddegree polynomial was chosen for the load relief data.

Results shown in figure 17 represent load-deflectioncharacteristics of the bias-ply and radial-belted tires anddemonstrate the hysteretic nature of the loading and

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unloading cycles. A maximum deflection of 0.38 in. wasobtained for the bias-ply tire under a 3000-lb side force.Under combined vertical-lateral loading conditions, thebias-ply tire exhibits linear characteristics during theentire load application cycle and nonlinear characteristicsduring initial load relief.

Similar lateral load-deflection characteristics wereobtained for the radial-belted tire for both the loading andunloading cycles. The maximum lateral displacementwas 0.52, which is a 37-percent increase in tire footprintdisplacement compared with its bias-ply tire counterpart.This increase in footprint displacement is a consequenceof the radial-belted tire construction.

Spring rate. Lateral spring-rate values wereobtained by evaluating the instantaneous slope along theloading and unloading curves. Lateral spring-rate valuesare plotted as a function of lateral displacements for allfour peripheral tire positions for both the bias-ply and theradial-belted designs (fig. 18). Bias-ply tire spring-ratevalues decrease slightly from 7311 lb/in. to 7200 lb/in.during the load application process represented by thesquare symbols. During the initial load-relief process,spring-rate values, denoted by the triangular symbols,decrease from a maximum value of 10622 lb/in. to7600 lb/in.

The radial-belted tire has slightly decreasing spring-rate values (symbolized by diamonds) from 5624 lb/in.to 5602 lb/in. during the load application cycle, as shownin figure 18. The maximum spring rate (represented bycircles) of 8098 lb/in. is noted at initial load reliefdecreasing to 6098 lb/in.

Lateral stiffness characteristics of the tires aredirectly related to cornering and yaw response and arethus important to shimmy analysis as well as to the land-ing dynamics in a crosswind landing. The radial-beltedtire’s stiffness values are on average 24 percent lowerthan the comparable bias-ply tire. This reduction in lat-eral stiffness may indicate an increase in the incidence oflanding-gear shimmy when this radial-belted tirereplaces its bias-ply counterpart. Landing dynamics in acrosswind with this radial-belted tire may be affected aswell.

Damping factor.The quasi-static structural dampingfactors, in terms of viscous damping, are determinedfrom the maximum and minimum lateral load at zero dis-placement on the load-deflection curve (fig.17). Thecomputed static damping factors are 0.042 and 0.038 forthe bias-ply and the radial-belted tires, respectively. Thestructural damping for the radial-belted tire is within10 percent of that for the bias-ply tire, indicating that

structural damping is similar for the two tires testedunder quasi-static lateral loading conditions.

Energy loss.The area enclosed in the lateral load-deflection curves for the bias-ply and radial-belted tiresis a measurement of the energy loss as a result of hystere-sis (fig. 17). The average energy loss is 289 in-lb for thebias-ply tire and 300 in-lb for the radial-belted tire. Theenergy loss because of hysteresis during the lateral load-ing and unloading cycles is similar for the two tires thatwere tested, and thus the structural damping properties ofthe tires are similar under the given loading conditions.

Quasi-Static Combined Vertical Fore-and-AftLoad Tests

Quasi-static combined vertical fore-and-aft load testswere conducted to determine the fore-and-aft stiffnessand damping characteristics from the resulting load-deflection curves. Results from these tests are given inthe following sections.

Load deflection.Load deflection tests involvedapplying a maximum vertical load of 25000 lb and afore-and-aft load of 3000 lb to the tire. Braking force andtire footprint displacement data were recorded at 250-lbload increments. Four load-deflection curves were gener-ated, each one corresponding to a different location 90°around the periphery of the tire. A second degree polyno-mial was chosen to curve fit the data points obtained dur-ing load application in the fore or aft directions, and athird degree polynomial was chosen for load relief data.Typical plots showing data with curve fits for each tiretested are shown in figure 19.

Results shown in figure 19 for the bias-ply tire repre-sent fore-and-aft load-deflection characteristics that arecorrected for wheel adapter plate slippage and demon-strate the hysteretic nature of the loading and unloadingprocess. A maximum deflection of 0.19 in. was obtainedfor the bias-ply tire under a 3000-lb fore-and-aft (brak-ing) force. The tire exhibits linear characteristics duringthe entire load application cycle and nonlinear character-istics during initial load relief.

Similar load-deflection characteristics were obtainedfor the radial-belted tire that were corrected for tire-wheel slippage as well as for wheel adapter plate slip-page (ref. 12). The radial-belted tire exhibits characteris-tics similar to the bias-ply tire during both loadapplication and load relief. The maximum fore-and-aftdisplacement is 0.27 in. This 42-percent difference in tiredeflection between the bias-ply and the radial-belted tirescan be attributed to the difference in the elastic propertiesof the tire designs (ref. 5).

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Spring rate.Fore-and-aft spring-rate values wereobtained by evaluating the instantaneous slope of eachloading and unloading interval along the load-deflectioncurves. Fore-and-aft spring-rate values are plotted as afunction of fore-and-aft footprint displacements for bothtire designs in figure 20. Bias-ply tire spring-rate valueslinearly decrease from 15140 lb/in. to 12944 lb/in. dur-ing the load-application process and are represented bythe square symbols. During initial load relief, spring-ratevalues (triangular symbols) decrease from a maximumvalue of 26495 lb/in. to a minimum value of 18131 lb/in.

The radial-belted tire has linear spring-rate trendsduring load application (diamond symbols) where thespring rate decreases by 1.6 percent from 11343 lb/in. to11157 lb/in. The maximum spring rate of 15833 lb/in. isat initial load relief and decreases to 12870 lb/in. Thesevalues are represented by the circular symbols.

The stiffness values for the radial-belted tire were14 to 40 percent lower than the stiffness values for thebias-ply tire. The lower fore-and-aft stiffness values ofthe radial-belted tire may introduce a lag between thebraking effort and the ground reaction that could affectthe dynamics of antiskid braking systems used with thisradial-belted tire but that are “tuned” for bias-ply tires(ref. 5).

Damping factor.Structural damping factors forquasi-static tests were determined from the fore-and-aftmaximum and minimum load at zero displacement on theload-deflection curves (fig. 19). The computed quasi-static structural damping factors are 0.068 and 0.044 forthe bias-ply and the radial-belted tires, respectively.These data indicate that there is 35 percent more damp-ing occurring in the bias-ply tire than in the radial-beltedtire during quasi-static fore-and-aft tests.

Energy loss.The static fore-and-aft energy lossassociated with hysteresis for the bias-ply and the radial-belted tires is represented by the area enclosed in thefore-and-aft load-deflection curves (fig. 19). The averageenergy loss is 229 in-lb for the bias-ply tire and 169 in-lbfor the radial-belted tire, which is 26 percent lower thanthe bias-ply tire. These results are in agreement with thevalues determined for the structural damping factor andsuggest that the radial-belted tire should operate at lowertemperatures than the comparable bias-ply tire. Theresults also suggest the possibility of less wear for theradial-belted tire during cyclic braking operations.

Free-Vibration Combined Vertical-Lateral LoadTests

Results are presented in this section from the free-vibration, combined vertical-lateral load tests for each

tire design. Lateral spring rate, damping factor, andenergy loss values were obtained from the displacementtime-history plots.

Time-history plots.Ten displacement time-historyplots and 10 acceleration time-history plots were gener-ated for each tire type tested at two different tire periph-eral positions. A specified vertical load was applied tothe tire, and then a side load of 3000 lb was applied andreleased, resulting inthe displacement and accelerationresponse of the platen to a free-vibration test, as shownin figure 21. Final reference displacement levels areshown along with the displacement and accelerationenvelopes. This shift in equilibrium level was attributedto tire creep (ref. 30). Tests were conducted at verticalloads of 5000 lb up to 25000 lb in 5000-lb increments.

From the bias-ply tire time-history plots for verticalloads of 5000 lb to 25000 lb, the system was under-damped, exponentially decaying, and had a decreasingfrequency from 8 to 6 Hz as the vertical load decreased,thus indicating that the period of oscillation issensitiveto changes in vertical load. As shown in figure21 at25000-lb vertical load, the maximum bias-ply tire dis-placement amplitude is 0.32 in. and the maximum accel-eration is 2.14g.

The time-history plots for the radial-belted tire at25000-lb vertical load are also shown in figure 21. Themaximum tire displacement is 0.37 in. and maximumacceleration is 2.12g. The frequency of oscillationdecreases from 8 Hz to 6 Hz as the vertical loaddecreases from 25000 lb to 5000 lb.

During the lateral free-vibration tests, the bias-plyand radial-belted tire had similar acceleration and fre-quency characteristics. However, the radial-belted tirefootprint deflection, which was attributed to tire con-struction, was 16 percent greater than that of the bias-plytire.

Spring rate.The lateral spring-rate values weredetermined from the frequency of vibration, the weightof the platen, and the frequency parameter for each tiredesign and are plotted as a function of the vertical load infigure 22. In general, the spring-rate values increase asvertical load increases for each tire design. Both tiresexhibit nonlinear spring-rate characteristics. The bias-ply tire has higher spring-rate values that range from5225lb/in. to 7585 lb/in. for the same vertical loads asthe radial-belted tire, whose tire spring-rate values rangefrom 4757 lb/in. to 6676 lb/in. The spring-rate values ofthe bias-ply tire are 9 to 12 percent higher than those ofthe radial-belted tire and were expected to be because ofthe bias-ply tire’s stiffer sidewall construction.

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Tire lateral stiffness measurements are importantproperties in the dynamic analysis of aircraft wheelshimmy. The lower stiffness values of the radial-beltedtire imply that tire wheel shimmy conditions may existwhen this tire is used.

Damping factor.Lateral viscous damping factorswere determined from the tire displacement amplitudesof the time-history plots by using the log decrementmethod. The viscous damping factor, as a function ofvertical load for each tire, is shown in figure 23. As thevertical load increases, the damping factor decreases forboth tire designs and thus is sensitive to the vertical loadrange that is applied during these free-vibration tests.The viscous damping factor values for both tire designsare lowest at 20000 lb of vertical load. The bias-ply tireshows higher viscous damping factor values that rangefrom 0.088 to 0.047; the values of the radial-belted tirerange from 0.07 to 0.031. Thus, viscous damping isgreater in the bias-ply tire than in the radial-belted tireunder these loading conditions.

In order to verify the calculation of the damping fac-tor when the log decrement method is used, a semilogplot of displacement amplitude, as a function of the num-ber of cycles, is shown in figure 24. The linear character-istics of both tires indicate that the log decrement methodyields consistent results for the first four cycles of thesetests.

The lateral structural damping factor in terms of vis-cous damping was determined from the tire displacementtime-history plots. Bias-ply structural damping factorvalues range from 0.092 to 0.176, and the radial-beltedtire values range from 0.062 to 0.14. The lower dampingvalues of the radial-belted tire indicate that there is lessstructural damping under these loading conditions.

Energy loss.The dynamic lateral energy loss percycle for each tire was calculated. The bias-ply tire has acalculated energy loss of 217 in-lb and the radial-beltedtire has an energy loss per cycle of 175 in-lb, both at25000 lb of vertical load. Again, this result suggests thatthe radial-belted tire should be a cooler operating tirethan the comparable bias-ply tire.

Free-Vibration Combined Vertical Fore-and-AftLoad Tests

Results from the free-vibration combined verticalfore-and-aft load tests for each tire design are presentedin this section. Fore-and-aft spring rate, damping factor,and energy loss values were obtained from the displace-ment time-history plots generated from free-vibrationtests.

Time-history plots.Ten displacement time-historyplots and 10 acceleration time-history plots were gener-ated for each tire at two different tire positions. A speci-fied vertical load was applied to the tire and then abraking load of 3000 lb was applied and released. Thesetests were conducted at vertical loads from 5000 lb to25000 lb in 5000-lb increments.

Typical displacement time-history and accelerationtime-history plots for the bias-ply and the radial-beltedtires at 25000-lb vertical load are shown in figure 25.The time-history plots show that the system is under-damped and is decaying exponentially. As in thedynamic lateral load tests, the displacement time-historyplots exhibit a shift in the equilibrium level that is attrib-uted to tire creep.

The maximum bias-ply tire displacement amplitudeis 0.14 in. and the maximum acceleration is 2.0g. Thefrequency of vibration decreases from 13 Hz to 9 Hz asthe vertical load decreases and indicates that the fre-quency is sensitive to variations in vertical load. Themaximum radial-belted tire displacement is 0.16 in. andthe maximum acceleration is 2.0g. The frequency ofvibration decreases from 10 Hz to 7 Hz as the verticalload decreases.

Both tires have similar acceleration and frequencycharacteristics. The increase in radial-belted tire footprintdisplacement suggests a more elastic tire than the bias-ply tire under braking conditions. This increase in radial-belted tire elasticity could adversely affect the operationof an antiskid braking system designed for the less elasticbias-ply tire.

Spring rate.The fore-and-aft spring-rate values foreach tire design are plotted as a function of the verticalload in figure 26. In general, the spring rate increases asvertical load increases for each tire design, and both tiresshow nonlinear spring-rate characteristics. The bias-ply tire has higher spring-rate values that range from9201lb/in. to 20610 lb/in. for the same vertical loads asthe radial-belted tire. The radial-belted tire spring-ratevalues range from 5921 lb/in. to 11 025 lb/in. The spring-rate values of the radial-belted tire are 36 to 47 percentlower than those of the bias-ply tire, and these lowerspring rate values may have an adverse affect on the anti-skid braking system performance when this radial-beltedtire is used.

Damping factor.Viscous damping factors weredetermined from the tire displacement amplitudes of thetime-history plots. The damping factors, as a function ofvertical load for each tire, are shown in figure 27. As thevertical load increases, the viscous damping factor

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decreases for both tire designs. The bias-ply tire hashigher viscous damping factors, ranging from 0.086to 0.127, than those of the radial-belted tire, which rangefrom 0.064 to 0.106. Structural damping in terms ofviscous damping yields bias-ply tire values of 0.172to 0.254 and radial-belted tire values of 0.128 and 0.212.Up to 26 percent more structural and viscous dampingoccurred in the bias-ply tire during these free-vibrationtests.

Energy loss.The dynamic energy loss associatedwith hysteresis for each tire is calculated from the tirestructural damping factor, the spring rate, and the maxi-mum displacement amplitude. The bias-ply tire has a cal-culated energy loss of 204 in-lb; the radial-belted tire hasan energy loss per cycle of 136 in-lb. These data suggestthat the radial-belted tire generates less heat during thefree-vibration tests, which should result in a lower oper-ating temperature for this tire. Tires with lower operatingtemperatures tend to have less wear during normal brak-ing operations.

Quasi-Static and Free-Vibration Lateral LoadData Comparison

A comparison of the quasi-static and free-vibrationlateral load data for each tire design is presented in thissection. Since the quasi-static tests were only conductedat the rated vertical load of 25000 lb, the comparisonwith the dynamic data was only at this vertical load.Table 5presents the quasi-static and dynamic lateral loadtire properties.

Spring rate.The bias-ply and the radial-belted tiresexhibit very similar stiffness values in the quasi-staticand dynamic data. The bias-ply quasi-static spring rate is7214 lb/in. as compared with 5674 lb/in. for the radial-belted tire. The dynamic spring rates are 7585 lb/in. forthe bias-ply tire and 6676 lb/in. for the radial-belted tire.The bias-ply tire has higher stiffness values than theradial-belted tire under both loading conditions becauseof its stiffer carcass. The increased stiffness values underfree-vibration tests for the tires that were tested may beattributed to the viscoelastic nature of the tire material.

In the quasi-static tests, the stiffness of the radial-belted tire is 21 percent lower than that of the comparablebias-ply tire. However, the free-vibration tests give amore realistic view of what to expect under actual operat-ing conditions, and in this case the radial-belted tireexhibits stiffness values that are 12 percent lower thanthose of the bias-ply tire. Although the radial-belted tirestill exhibits lower lateral stiffness characteristics underfree-vibration testing conditions, this 12-percent differ-ence is encouraging, given the concerns in the aircraft

tire industry about the lateral force capabilities of theradial-belted tire.

Damping factor.The calculated quasi-static struc-tural damping factors in terms of viscous damping are0.044 and 0.038 for the bias-ply and radial-belted tires,respectively. The viscous damping factor for the free-vibration test is 0.047 for the bias-ply tire and 0.031 forthe radial-belted tire. The damping factor values are sim-ilar under both test conditions for both tire types.

Energy loss.The calculated quasi-static energy lossfor the bias-ply and radial-belted tires are similar:319 in-lb and 339 in-lb, respectively. For the free-vibration tests, the computed energy loss is 217 in-lb forthe bias-ply tire and 175 in-lb for the radial-belted tire.The energy loss values in the free-vibration case werelower than those calculated under static testing condi-tions (32 percent for the bias-ply tire and 48 percent forthe radial-belted tire). The quasi-static test conditionsproduced more tire hysteresis than the free-vibration testconditions produced.

Quasi-Static and Free-Vibration Fore-and-AftLoad Data Comparison

A comparison of the quasi-static and free-vibrationfore-and-aft load data for each tire design is presented inthis section. Since the quasi-static data were gatheredonly at the rated vertical load of 25000 lb, the compari-son with the dynamic data was conducted only at thisvertical load. Table 6 presents both quasi-static anddynamic fore-and-aft load data.

Spring rate.The quasi-static spring-rate values are16145 lb/in. and 9394 lb/in. for the bias-ply and theradial-belted tires, respectively. The spring rates for thefree-vibration tests are 20601 lb/in. and 11025 lb/in. forthe bias-ply and the radial-belted tires, respectively. Theresults show that both tires are stiffer during fore-and-aftfree-vibration tests than during quasi-static tests; theincreased stiffness may be partly attributed to the vis-coelastic nature of the tires.

Damping factor.The calculated quasi-static struc-tural damping factors in terms of viscous damping are0.055 and 0.036 for the bias-ply and the radial-beltedtires, respectively. The viscous damping factor for thefree-vibration tests is 0.086 for the bias-ply tire and 0.064for the radial-belted tire. The quasi-static tests resulted inlower structural damping factors, in terms of viscousdamping, than the free-vibration tests viscous dampingfactors. These data suggest that possibly some viscousdamping, as well as the assumed structural damping,occurs under free-vibration testing conditions.

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Energy loss.The calculated quasi-static energy lossfor the bias-ply and the radial-belted tires is 222 in-lb and128 in-lb, respectively. For the free-vibration tests, thecomputed energy loss is 204 in-lb for the bias-ply tireand 135 in-lb for the radial-belted tire. The bias-ply tire’scalculated quasi-static energy loss is 9 percent higherthan its dynamic energy loss. The calculated quasi-staticenergy loss for the radial-belted tire is 5 percent lowerthan its dynamic energy loss. These differences betweenthe quasi-static and dynamic energy loss measurementsfor each tire are small and indicate that the tires havesimilar hysteretic characteristics under both testingconditions.

Footprint Geometrical Properties

Footprint geometrical properties help predict tirehydroplaning characteristics. Results from tire footprintmeasurements for each tire design are presented in thissection.

Tire footprints were obtained for each tire under var-ious vertical loads. Geometric parameters such as foot-print area, length, and width were measured from theresulting footprint silhouettes. A plot of net footprint areaas a function of vertical load is shown in figure 28. Thebias-ply tire has a greater footprint area than the radial-belted tire for the same vertical loads. This larger foot-print area is also evident in the tire footprint silhouettesthat are shown in figure 29 and that were taken at a max-imum rated load of 25000 lb. The bias-ply tire has amore rectangular footprint, and the radial-belted tire hasa more oval footprint. At 245 psi, the predicted hydro-planing speed of the bias-ply tire is 141 knots; the pre-dicted hydroplaning speed of the radial-belted tire at310 psi is 159 knots (ref. 41). However, the increasedhydroplaning speed for the radial-belted tire at the higherinflation pressure may not be realized because of adverseeffects associated with the aspect ratio (height-to-widthratio of the tire footprint)of the nearly circular footprintof the radial-belted tire (ref. 42). In general, these testssuggest that the bias-ply tire may perform more favor-ably under wet runway conditions.

Moment-of-Inertia Properties

Inertia properties of tires help to define the tire’sspin-up characteristics during touchdown. Moment-of-inertia tests were conducted for both tire designs and theresults are presented.

The mass moment of inertia for each tire designand the tare inertia for the two pendulum plates werecalculated. The moment of inertia is 4.35 in-lb-sec2 forthe plates,31.5 in-lb-sec2 for the bias-ply tire, and27.7 in-lb-sec2 for the radial-belted tire. The moment ofinertia of the radial-belted tire is 12 percent lower than

that of the bias-ply tire. This difference implies that theradial-belted tire would require slightly less energy tospin up during the landing touchdown than the bias-plytire would. Less energy needed in the spin-up portion ofa touchdown could mean reduced tread wear duringhigh-speed landings.

Concluding Remarks

An investigation at the Langley Research Center wasconducted to determine, evaluate, and compare certainquasi-static and dynamic mechanical properties of theU.S. Air Force F-4 military fighter 30×11.5-14.5/26PRbias-ply and radial-belted main gear tires that define theirperformance during taxi, takeoff, and landing operationsand to define the suitability of the radial-belted tire as areplacement for the bias-ply aircraft tire. These proper-ties were obtained from quasi-static vertical, lateral, andfore-and-aft load-deflection data; lateral and fore-and-aftfree-vibration time-history plots; tire footprint measure-ments; and moment-of-inertia tests. Three dampingmodels: viscous, structural, and Coulomb friction werepresented that gave an insight into the type of dampingthat occurs under both quasi-static and free-vibration testconditions.

The results of this investigation indicate the follow-ing observations:

1. In general, the radial-belted tire has vertical load char-acteristics that are similar to those of the conventionalbias-ply tire. However, significant differences areobserved between the bias-ply and the radial-beltedtires’ lateral and fore-and-aft load characteristics.

2. Vertical load-deflection characteristics obtained forthe radial-belted and the bias-ply tires for the givenload range are similar. Under lateral and fore-and-aftload conditions, the radial-belted tire has greater foot-print displacements. This radial-belted tire has a morecircular footprint and less tread in contact with thesurface than its bias-ply tire counterpart.

3. Vertical stiffness characteristics obtained for theradial-belted and bias-ply tires are similar for thegiven load range considered in this study. Radial-belted stiffness values are lower under quasi-static anddynamic lateral and fore-and-aft testing conditions.Test results indicate that both tire designs are stifferduring free-vibration tests than during quasi-statictests.

4. Energy loss associated with hysteresis of the radial-belted tire is less than that of the bias-ply tire undervertical and fore-and-aft quasi-static test conditions,as well as under lateral and fore-and-aft free-vibrationtest conditions. Damping characteristics are similarfor both tires under quasi-static lateral loads.

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Quasi-static tests resulted in lower damping at therated vertical load of 25000 lb than the free-vibrationtests had at that load. The radial-belted tire has lowermoment of inertia values than the bias-ply tire.

The following conclusions are made from the aboveobservations:

1. Similar vertical load stiffness characteristics betweenthe two tire designs imply that there should be noimpact on strut valving, on the loads transmitted to theairframe, and on the landing dynamics for aircraftequipped with this radial-belted tire under normaloperating conditions.

2. Lateral and fore-and-aft stiffness properties of thisradial-belted tire may result in an increase in tireshimmy and may affect the performance of an anti-skid braking system “tuned” for bias-ply tires. Theincreased overall stiffness properties of the two tiredesigns during free-vibration tests, compared with thequasi-static tests, may be attributed in part to theviscoelastic nature of the tires. Footprint geometricalproperties of the radial-belted tire suggest that it mightbe more sensitive to hydroplaning conditions than itsbias-ply tire counterpart.

3. The energy loss measured for the radial-belted tiresuggests that lower operating temperatures during nor-mal ground and braking operations may lead toimproved tire durability. The lower temperatures ofthe radial-belted tire could allow for shorter aircraftturnaround time and less tread wear. Similar tempera-ture profiles may occur during cornering maneuvers.A comparison of the higher damping factors underfore-and-aft free-vibration tests with quasi-static testsfor both tire designs suggests that some viscous damp-ing is present, as well as the assumed structural damp-ing. Moment-of-inertia tests indicate that the radial-belted tire requires less energy to spin up duringtouchdown, and that less energy may result in reducedtread wear and reduced heating during high-speedlandings.

NASA Langley Research CenterHampton, VA 23681-0001May 22, 1996

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17. Davis, Pamela A.: Evaluation of 26×6.6 Radial-Belted andBias-Ply Aircraft Tires.Presented at the Tire Industry Confer-ence, (Greenville, South Carolina) Oct. 1991.

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18

21. Von Schlippe, B.; and Dietrich, R.:Shimmying of a PneumaticWheel. WPAFB F-TS-3727-RE, Oct. 1941.

22. Von Schlippe, B.; and Dietrich, R.:Shimmying of a PneumaticWheel. Papers on Shimmy and Rolling Behavior of LandingGears Presented at Stuttgart Conference Oct. 16 and 17, 1941,NACA TM-1365, 1954, pp. 125–160.

23. Moreland, William J.: The Story of Shimmy.J. Aeronaut. Sci.,vol. 21, no. 12, Dec. 1954, pp. 793–808.

24. de Carbon, Christian Bourcier:Analytical Study of Shimmy ofAirplane Wheels. NACA TM-1337, 1952.

25. Collins, R. L.: Theories on the Mechanics of Tires and TheirApplications to Shimmy Analysis.J. Aircr., vol. 8, no. 4, April1971, pp. 271–277.

26. Collins, R. L.; and Black, R. J.: Experimental Determinationof Tire Parameters for Aircraft Landing Gear Shimmy Stabil-ity Studies. AIAA-68-311, April 1968.

27. Smiley, Robert F.: Some Considerations of Hysteresis Effectson Tire Motion and Wheel Shimmy. NACA TN 4001, 1957.

28. Horne, Walter B.: Static Force-Deflection Characteristics ofSix Aircraft Tires Under Combined Loading. NACA TN 2926,1953.

29. Tanner, John A.; Stubbs, Sandy M.; and McCarthy, John L.:Static and Yawed-Rolling Mechanical Properties of Two,Type VII, Aircraft Tires. NASA TP-1863, 1981.

30. Sleeper, Robert K.; and Dreher, Robert C.: Tire Stiffness andDamping Determined From Static and Free-Vibration Tests.NASA TP-1671, 1980.

31. Dodge, R. N.; and Clark, S. K.:Comparison of Some Staticand Dynamic Mechanical Properties of 18×5.5 and 49×17

Type VII Aircraft Tires as Measured by Three Test Facilities.NASA CR-165720, 1981.

32. Harvay, Gabriel; and Crowe, R. E.: Spring and DampingConstants of Helicopter Tires. McDonnell Aircraft Corp.,H-122-1, July 17, 1945.

33. Clark, S. K.; Dodge, R. N.; and Nybakken, G. H.: DynamicProperties of Aircraft Tires. J. Aircr., vol. 11, no. 3, Mar. 1974,pp. 166–172.

34. Thompson, A. G.: The Effect of Tyre Damping on the Perfor-mance of Vibration Absorbers in an Active Suspension.J. Sound & Vib., vol. 133, no. 3, 1989, pp. 457–465.

35. Brahney, James H., ed.: Radial Tires for Aircraft? Aerosp.Eng., May 1988, pp. 21–23.

36. Panikkar, T. P. K.: Dunlop Aircraft Radial Tyres. DunlopAerosp. Group Newsl., vol. 2, no. 3, Aug. 1986.

37. Hutton, David V.: Applied Mechanical Vibrations. McGraw-Hill, 1981, pp. 59–69.

38. Meirovitch, Leonard: Elements of Vibration Analysis.McGraw-Hill, 1975, pp. 1–87.

39. Meirovitch, Leonard: Analytical Methods in Vibrations.MacMillian Co., 1967, pp. 1–123, 388–390.

40. Thomson, William Tyrrell: Theory of Vibration with Applica-tions.Prentice-Hall, Inc., 1972, pp. 1–78.

41. Horne, Walter B.; and Dreher, Robert C.:Phenomena of Pneu-matic Tire Hydroplaning. NASA TN D-2056, 1963.

42. Horne, Walter B.; Yager, Thomas J.; and Ivey, Don L.: RecentStudies to Investigate Effects of Tire Footprint Aspect Ratio onDynamic Hydroplaning Speed.The Tire Pavement Interface,STEM STEP 929, M. G. Potting and T. J. Yager, eds., ASTM,1986.

19

Table 1. Characteristics of 30×11.5-14.5/26PR Aircraft Tires

Parameter Bias-ply Radial-beltedSize . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30×11.5-14.5 30×11.5-14.5Ply rating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 26Weight, lb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68.75 55.50Rated vertical load, lb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25000 25000Rated inflation pressure at 35-percent load deflection, psi . . . . . . . . . . . . 245 310Outside diameter of unloaded tire, in. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 30Maximum carcass width of unloaded tire, in. . . . . . . . . . . . . . . . . . . . . . . 8 8Tread grooves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4

Table 2. Viscous Damping Parameters

Symbol Definition

Viscous damping factor

Viscous damping factor for smallδ

Log decrement

Damped frequency of oscillation

Spring rate

Table 3. Structural Damping Parameters

Symbol Definition

Structural damping factor

Equivalent viscous damping factor

Log decrement

Energy loss per cycle

Spring rate

Damped frequency of oscillation

ζ δ

2π2 δ2–-------------------------=

ζ δ2π------=

δx1

x2-----ln=

ωd ωn 1 ζ2–=

k mωn2 m

2πτ

------ 2

= =

β 2ζ=

Ceqβkω------=

δx1

x2-----ln πβ≅=

∆Ecyc πβkX2=

k mωn2 m

2πτ

------ 2

= =

ωd ωn 1 ζ2–=

20

Table 4. Coulomb Friction Parameters

Symbol Definition

Damping or friction force

Equivalent displacement

Spring rate

Table 5. Quasi-static and Dynamic Lateral Load Test Data

TireMaximum

deflection, in.Spring rate, lb/in. Structural

dampingViscousdamping

Energy loss, in-lbSlope Axis Measurement Calculation

Quasi-static data:Bias 0.38 10622 7214 0.042 — 289 319Radial 0.52 8098 5674 0.038 — 300 339

Dynamic data:Bias 0.32 — 7585 0.092 0.047 — 217Radial 0.37 — 6676 0.062 0.031 — 175

Table 6. Static and Dynamic Fore-and-Aft Load Test Data

TireMaximum

deflection, in.Spring rate, lb/in. Structural

dampingViscousdamping

Energy loss, in-lbSlope Axis Measurement Calculation

Quasi-static data:Bias 0.19 26495 16145 0.068 — 229 222Radial 0.27 15833 9394 0.044 — 169 128

Dynamic data:Bias 0.14 — 20610 0.172 0.086 — 204Radial 0.16 — 11025 0.128 0.064 — 136

Fd µkW=

f d

Fd

k------=

k2Fd

2X----------=

21

L-86-10,087Figure 1. The 30×11.5-14.5/26PR aircraft tires.

Bias-ply tire Radial-belted tire

22

L-89-13674Figure 2. Bias-ply aircraft tire construction.

Figure 3. Radial-belted aircraft tire construction.

Breakers

Tread-reinforcing ply

Inner liner

Cord body

Bead heel

Bead toe

Apex strip

Wire beads

Flippers

Ply turnups

Chaffers

Plies

Sidewall

Tread

Steel protector ply

Tread

Belts

Carcass plies

Liner

Turnup

Flipper

Wire bead

23

L-86-174Figure 4. Test apparatus no. 1 (used for quasi-static vertical load tests).

Frictionless tableLoad cells

Hydraulic cylinder

Strain-gauge beam

Dynamometer

Test tire

24

Figure 5. Test apparatus no. 2 (used for quasi-static and dynamic lateral and fore-and-aft load tests).

Figure 6. Quasi-static fore-and-aft test setup for test apparatus no. 2.

Screw jacks

Load cells

Cables

Test tire

Adapter plates

Platen

Cable

Extensiometer

Wheel

Displacement transducer

Platen

Hydraulic cylinder

Load cell

DCDT

Tire

25

Figure 7. Free-vibration fore-and-aft test setup for test apparatus no. 2.

L-86-1586Figure 8. Torsional pendulum apparatus used for moment-of-inertia tests.

To ratchet mechanism

Load cell

Quick- release device

Cable

Pulley

TireCable

Locked wheel

Displacement transducer

Platen

Lower plate

Support cables

Index

Pointer

Upper plate

26

Figure 9. Response of underdamped system, 0 <ζ < 1.

Figure 10. Polar notation of damping parameter.

Displacement

Time

2X

2X1

2

τDisplacement envelope

t t1 2

θ

2ζ

1

r

Imaginary

Real

27

Figure 11. Quasi-static damping factor components.

Figure 12. Response of system with Coulomb friction.

Deflection

Force

2F max2F

x=0

Displacement

Time

4f d

X -2-1X

2X1X

28

Figure 13. Response of system under quasi-static load with Coulomb friction.

Figure 14. Spring rate defined by hysteresis loop.

Force

Deflection

C

A B

Deflection

Force

Slope of load application

Slope of initial load relief

Slope of hysteresis loop

29

Figure 15. Vertical load-deflection curve of bias-ply and radial-belted tires.

Figure 16. Vertical stiffness characteristics of bias-ply and radial-belted tires.

30 000

20 000

Vertical load, lb

10 000

0 1 2 3

Vertical deflection, in.

Bias-ply tire

Radial-belted tire

20 000

15 000

Vertical spring rate,

lb/in.10 000

5 000

0 1 2 3

Vertical deflection, in.

Load application

Initial load reliefBias-ply tire Radial-belted tire

30

Figure 17. Lateral load-deflection curve of bias-ply and radial-belted tires.

Figure 18. Lateral stiffness characteristics of bias-ply and radial-belted tires.

4000

2000

-2000

-4000

Lateral load, lb

-.6 -.4 -.2 .2 .4 .6

Lateral deflection, in.

Bias-ply tire

Radial-belted tire

12 000

11 000

10 000

9 000

8 000

7 000

6 000

5 000

Lateral spring rate, lb/in.

0 .1 .2 .3 .4 .6

Lateral deflection, in.

Load application

Initial load relief

.5

Radial-belted tire

Bias-ply tire

,

,

31

Figure 19. Fore-and-aft load-deflection curve of bias-ply and radial-belted tires.

Figure 20. Fore-and-aft stiffness characteristics of bias-ply and radial-belted tires.

4000

2000

-2000

-4000

Fore-and-aft load, lb

-.3 -.2 -.1 .1 .2 .3

Fore-and-aft deflection, in.

Bias-ply tire

Radial-belted tire

30 000

25 000

20 000

15 000

10 000

5 0000 .1 .2 .3

Fore-and-aft deflection, in.

Load application

Initial load relief

Fore-and-aft spring rate,

lb/in.

Bias-ply tire

Radial-belted tire

,

,

32

Fig

ure

21.

Late

ral f

ree-

vibr

atio

n tim

e-hi

stor

y pl

ots

of b

ias-

ply

and

radi

al-b

elte

d tir

es.

-3-2-10123

Acc

eler

atio

n,

g

.5 -.50

Dis

plac

emen

t, in

.

.51.

01.

52.

02.

5

Tim

e, s

ec

Dis

plac

emen

t env

elop

e of

bi

as-p

ly ti

re

Fina

l eq

uilib

rium

le

vels

Acc

eler

atio

n en

velo

pe o

f bi

as-p

ly ti

re

Rad

ial-

belte

d tir

e

Rad

ial-

belte

d tir

e

Bia

s-pl

y tir

e

Bia

s-pl

y tir

e

33

Figure 22. Lateral free-vibration stiffness characteristics of bias-ply and radial-belted tires.

Figure 23. Lateral free-vibration damping characteristics of bias-ply and radial-belted tires.

8000

7000

6000

5000

4000

Lateral spring rate,

lb/in.

0 10 000 20 000 30 000Vertical load, lb

Bias-ply tire

Radial-belted tire

.10

.08

.06

.04

.02

10 0000 20 000 30 000

Bias-ply tire

Vertical load, lb

Damping factor

Radial-belted tire

34

Figure 24. Frequency effects of displacement amplitude.

1.0

Displacement amplitude, in.

Bias-ply tire

0 1 2 3 4 5Number of cycles

.1

Radial-belted tire

35

Fig

ure

25.

For

e-an

d-af

t fre

e-vi

brat

ion

time-

hist

ory

plot

s of

bia

s-pl

y an

d ra

dial

-bel

ted

tires

.

2.5

0 -2.5

Acc

eler

atio

n,

g

Acc

eler

atio

n en

velo

pe o

f ra

dial

-bel

ted

tire

.32 .1

6

0 - .1

6

- .3

2 1.0

1.5

2.0

Tim

e, s

ec

Dis

plac

emen

t env

elop

e of

bi

as-p

ly ti

re

Fina

l equ

ilibr

ium

le

vels

Dis

plac

emen

t, in

.

Rad

ial-

belte

d tir

e

Bia

s-pl

y tir

e

Bia

s-pl

y tir

e

Rad

ial-

belte

d tir

e

36

Figure 26. Fore-and-aft free-vibration stiffness characteristics of bias-ply and radial-belted tires.

Figure 27. Fore-and-aft free-vibration damping characteristics of bias-ply and radial-belted tires.

30 000

20 000

10 000

Fore-and-aft spring rate,

lb/in.

10 0000 20 000 30 000

Vertical load, lb

Bias-ply tire

Radial-belted tire

.14

.12

.10

.08

.06

.040 10 000 20 000 30 000

Damping factor

Vertical load, lb

Bias-ply tire

Radial-belted tire

37

Figure 28. Net tire footprint area as function of vertical load of bias-ply and radial-belted tires.

Figure 29. The 30×11.5-14.5/26PR tire footprint silhouettes at 25000-lb vertical load.

100

80

60

40

20

10 0000 20 000 30 000

Vertical load, lb

Bias-ply tire

Net footprint area, in2

Radial-belted tire

Bias-ply tire Radial-belted tire

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February 1997 Technical Paper

Quasi-Static and Dynamic Response Characteristics of F-4 Bias-Ply andRadial-Belted Main Gear Tires WU 505-63-10-02

Pamela A. Davis

L-17433

NASA TP-3586

An investigation was conducted at Langley Research Center to determine the quasi-static and dynamic responsecharacteristics of F-4 military fighter 30×11.5-14.5/26PR bias-ply and radial-belted main gear tires. Tire propertieswere measured by the application of vertical, lateral, and fore-and-aft loads. Mass moment-of-inertia data were alsoobtained. The results of the study include quasi-static load-deflection curves, free-vibration time-history plots,energy loss associated with hysteresis, stiffness and damping characteristics, footprint geometry, and inertia proper-ties of each type of tire. The difference between bias-ply and radial-belted tire construction is given, as well as theadvantages and disadvantages of each tire design. Three simple damping models representing viscous, structural,and Coulomb friction are presented and compared with the experimental data. The conclusions discussed contain asummary of test observations.

Radial belted; Bias-ply; Aircraft tires; Hysteresis; Free vibration; Load deflection;Energy loss; Footprint geometry; Damping; Stiffness

42

A03

NASA Langley Research CenterHampton, VA 23681-0001

National Aeronautics and Space AdministrationWashington, DC 20546-0001

Unclassified–UnlimitedSubject Category 05Availability: NASA CASI (301) 621-0390

Unclassified Unclassified Unclassified