Vehicle vibrating on a soft compacting soil half-space: Ground vibrations, terrain damage, and vehicle vibrations

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Journal of Terramechanics Journal of Terramechanics 45 (2008) 121–136 www.elsevier.com/locate/jterra

Vehicle vibrating on a soft compacting soil half-space: Ground vibrations, terrain damage, and vehicle vibrations Robert Hildebrand a,*, Erno Keskinen a, Jose´ Antonio Romero Navarrete b b

a Tampere University of Technology, Box 589, 33101 Tampere, Finland University of Quere´taro, Rio Moctezuma 249, San Juan del Rio, Qro., Mexico

Received 12 February 2007; received in revised form 29 July 2008; accepted 22 September 2008

Abstract The point of departure of the present work may be either an interest in vehicle vibrations themselves, or in ground vibrations and terrain damage due to vehicles traveling off-road. The vibrations of a vehicle traversing dry, soft terrain, which is either rough or undulating, may be significantly modified by the dynamic interaction of the vehicle with the soil, particularly due to losses of energy by soil compaction and as elastic waves. The present work provides a prediction methodology for both vehicle and soil vibrations, accounting for the effects mentioned above. An expedient linear method is compared to a rheologically-based non-linear method. In the linear method, the soil compaction is incorporated as a loss factor in the dynamic stiffness of the otherwise elastic half-space; the imaginary part of that dynamic stiffness already includes the effects of wave damping. The non-linear model treats the compaction using a general rheological model for soils exhibiting both viscous and thixotropic effects, and requires iterative solution. A key feature of the latter model is the hypothesis that the stress distribution may be approximately regarded as quasi-static when calculating compaction losses; that approximation is expected to hold at low frequencies, since the P-wavelength in the soil is then much greater than the dimensions of the zone in which most compaction occurs. The methods predict that the soil compaction and excited ground vibrations have maxima at the vehicle bounce and hop resonances, and at high frequencies at which the Rayleigh wavelength approaches the order of the contact patch diameter. Moreover, sufficiently soft, compactable soils, but fully realizable in nature, control the vehicle response at the hop resonance, and possibly also at the bounce resonance. Ó 2008 ISTVS. Published by Elsevier Ltd. All rights reserved.

1. Introduction A vehicle traveling off-road, whether or not it be an off-road vehicle by design, is excited into vibration by surface roughness of the terrain it crosses, and in turn excites the soil into vibration as well. Such a process is ‘‘shortcircuited” if the soil stiffness is very small compared to the running gear, in which case the surface roughness is instead ‘‘flattened out”. However, the soil tends to be stiff, relative to the vehicle, in the case of longer-wavelength profile variations, as well as at specific characteristic * Corresponding author. Present address: Lake Superior State University, 650 W. Easterday Ave., Sault Ste Marie, MI 49783, USA. Tel.: +1 906 635 2139; fax: +1 906 635 6663. E-mail address: [email protected] (R. Hildebrand).

frequencies at which the vehicle (as a resonant system) has a low dynamic stiffness; in those situations the running gear is forced to follow the surface profile, and vibrations are excited. The large dynamic stresses then induced in the soil in the immediate vicinity of the tires also bring about permanent deformations of the soil, in the form of a wheel rut; see Fig. 1 for an overview of all of the phenomena involved. The entire process just described may be of interest for several different reasons. From the environmental perspective, one may be interested in the effect of vehicle traffic on natural terrain. From the perspective of vehicle performance, the same interest in vibrations exists for off-road travel as it does for on-road travel, the latter having long been given due attention in the automotive, heavy truck, and other wheeled-vehicle industries.

0022-4898/$34.00 Ó 2008 ISTVS. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jterra.2008.09.003

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Compacted Rut

Vehicle Vibrations

Ground Vibrations (surface waves) Ground Vibrations (p and s waves)

Fig. 1. The dynamically interacting phenomena involved in the problem considered include the vibrations of a vehicle as it crosses rough terrain, vibrations of the ground in reaction (which propagate away as waves into the medium and along its surface), and the compaction of a rut into the soil behind the vehicle.

Consider, firstly, the environmental aspect. Both soil compaction and ground vibration are forms of environmental degradation that may be understood in the context of the vehicle–soil interaction process considered. Issues that come to mind, in which these phenomena have a bearing, include the use of recreational off-road vehicles in national parks, petroleum-industry exploration vehicles on arctic tundra as discussed by Abele et al. [2], or forestry vehicles in boreal forests. Soil compaction is a form of permanent damage to a natural terrain. It serves as an impediment to new plant growth by reducing the void ratio, and in turn the permeability of water, the capillary water storage capacity, aeration, and root penetrability (see Kohnke and Franzmeier [12]; Canillas and Salokhe [6], Barnes et al. [4]). Moreover, the reduced water permeability and localized surface depression which result from compaction can create runoff paths for water, leading to soil erosion. Ground vibrations are a more subtle environmental factor; one concern is that they may induce very localized slope erosion because of the reduced stability of loose, upper-layer soils as their apparent angle of internal friction is reduced (Rantama¨ki et al.[21]; Keedwell [11]). This implies an unnatural erosion process. Additionally, there may be direct damage to plants themselves from ground vibrations. Consider, next, the vehicle performance aspect. For one thing, vehicle vibrations are a durability issue. For another, they are a comfort issue (‘‘ride”), i.e., the vibrations passing through the suspension may be sufficient to constitute a disturbance to the occupants. Potential negative effects include driver fatigue, comfort degradation, cabin noise, wayside noise, and cargo damage. Finally, cornering properties of vehicles are known to be degraded by vibrations, and, for trucks, there is an increased risk of cargo shifting (adding to jackknifing or rollover risks). Some existing treatments of vehicles vibrating on terrain focus on the vehicle itself, with the terrain only included as an irregular profile, but lacking compliance. On the other hand, these incorporate very refined tire models. Ahmed and Goupillon [3] discuss an unsuspended agricultural tractor model (although there are non-linear cab and seat

suspensions) in which the tire stiffness and damping are dependent on the tire rotational speed, and the soil roughness profile is low-pass filtered to account for the distributed tire–soil contact. Lehtonen [15] also discusses a tractor with an advanced tire model and similar contact patch filtering, and considering seat, cab, and front-axle suspensions. Each of these models make good predictions of vibrations in tractors on test tracks, but soil properties are not reported, and probably represented well-compacted paths that differ from the undisturbed terrains which are of interest in the present work. A soft undisturbed soil surface, on the other hand, is compliant and susceptible to compaction, and the vehicle response is not independent of the response of the surface it traverses. Relatively compliant soils can be expected to strongly influence the vehicle’s vibration amplitude, especially near vehicle resonances. That being so, other treatments of the vibrations of off-road vehicles have sometimes accounted for a compliant soil. Park et al. [20] model a truck traversing a locally-reacting soil with a non-linear stiffness based on Bekker’s pressure–sinkage relationship, but with discretization of the soil surface in the contact zone; the local deformation of the soil directly under the tire filters the roughness profile exciting the vehicle. However, the present work hypothesizes that other significant phenomena may be present, which can only be accounted for by allowing wave propagation in the soil and also allowing for permanent deformation (‘‘compaction”) in the soil behavior. A portion of the soil compaction in a vehicle passage over terrain is due to the quasi-static loading of the soil by the weight of the vehicle traversing it; this compaction is that implied when predicting the depth of a rut using the Bekker pressure–sinkage relation. Examples are provided for agricultural soils by Canillas and Salokhe [6], and for subgrades of highways by Garnica et al.[8] and railways by Li and Selig [16]. More importantly, the dynamic tire–soil interaction results in vibrations, which, in turn, induce an additional component of soil compaction considerably exceeding the quasi-static component – the effectiveness of vibrations to compact soil, albeit intentionally – rather than incidentally-induced vibrations, are discussed in Muro et al.[18] and Rico and Del Castillo [22] for example. Moreover, the vibrations are transmitted both through the suspension into the vehicle and through the ground to the wayside, where, in each case, they may constitute a disturbance. Several damping mechanisms are hypothesized for vehicle–soil interaction in general. Radiation damping is the process by which elastic waves, e.g., p-, s-, and Rayleigh waves (and, in a layered soil, other wave forms), carry energy away from the tire–soil contact zone. Compaction damping refers to the expenditure of energy to compact the soil, i.e., reduce its porosity, due to relative motions of the soil particles. Other mechanisms could be active in a more nearly saturated soil, such as energy lost due to liquefaction or (perhaps in very coarse grained soils) consolidation. The relative importance of the damping mecha-

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

nisms would likely depend on the degree of saturation of the soil, as well as the loading amplitudes. Compaction is maximized at a certain optimal degree of saturation which is characteristic for the soil, typically in the 5–30% range, with clays and silts tending towards the high end of that range, and sands towards the low end (at higher saturation levels, the dynamically induced pore pressure tends to impede compaction). The working hypothesis in this paper is that wave losses and compaction are adequate to describe the energy losses in the vibration problem if the soils are relatively dry (liquefaction and consolidation are ignored), and that compaction (besides its damping role) also represents the mechanism for terrain damage. Given this working hypothesis, the objective of the present work is to provide a predictive model of both vehicle vibrations and soil vibration, as well as the dynamically-induced compaction of the soil, taking as inputs the surface profile variation (and its wavelength-dependence), vehicle tire and suspension characteristics, and soil properties (compatibility and wave speeds). In what follows, an expedient linear model, permitting frequency-domain analysis, is first proposed. Thereafter, a rheologically-motivated non-linear model is presented; this model is believed to better represent the physical process, but is restricted to time-domain analysis. Neither model attempts to account for complete losses of tire– ground contact, which could be important if the profile is sufficiently rough. A brief discussion is then made of how either model can be used to predict ground vibrations at a distance from the vehicle, and, finally, the most significant conclusions of the study are summarized. 2. Linear model This section presents, and makes use of, a linear predictive model, the primary advantage of which is that it provides frequency-domain results. The basic idea is to handle the soil compaction in the vibration problem by means of a loss factor – a kind of equivalent vibration damping. As described below, the loss factor is approximated using the energy loss of a cycle in a standardized test called the modified Proctor compaction test – probably an overestimation, but more realistic for high strain conditions near the tire than so-called seismic loss factors. This section will first present a method to model the dynamic vehicle–soil interaction problem. That is followed by the presentation of sample results that illustrate credible circumstances in which the soil has a large effect on the vehicle vibrations, the frequency bands in which such effects are most pronounced, and the resulting compaction damage to the terrain itself. Symbols used in the development to follow are summarized in Table 1, below. 2.1. Method In an overall sense, the method involves finding the frequency-dependent dynamic stiffness, meaning the ratio of

123

the excitation force to the displacement amplitude at a given frequency (compare to the static concept of stiffness k in Hooke’s law), of both the soil surface and the vehicle, each at the point of contact with the other – the tire–soil contact zone. Insofar as the surface profile irregularity (‘‘roughness”) must be accommodated by some combination of soil displacement and vehicle (tire) displacement, and the contact forces on the soil and tire are equal and opposite, then the ratio of their respective displacement responses is therefore in inverse proportion to the ratio of their dynamic stiffnesses. This process of surface irregularity inducing vibrations in both the vehicle and the ground is illustrated conceptually in Fig. 2. Hence, given the surface profile, and each dynamic stiffness, at a given frequency, the vibration of each is thereby determined. The problem therefore reduces to that of finding the two dynamic stiffnesses. While the vehicle dynamic stiffness is straightforward and conventional, regarding it as a system of masses coupled by linear springs and dashpots, the dynamic stiffness of the soil is complicated both by its distributed mass and elasticity, and by the phenomenon of compaction. To treat that, the model makes use of a classical integral solution to the dynamic stiffness of a half-space, which already has an imaginary part (i.e., a loss factor) that accounts for wave losses, but adds to that an additional imaginary part to account for compaction. Because the damping provided by soil compaction is expected to strongly influence the tire vibration amplitude, especially near the tire-on-soil resonance frequency, then its description is an essential aspect of the predictive model. Typically, for dynamic purposes, seismic methods are used to obtain a loss factor for the soil (using, for instance, the decay rate of surface waves with distance, above and beyond that decay which is merely due to geometric spreading). The problem, however, with using a loss factor obtained in that way, for the present application, is that it represents conditions of very low strain. Seismic measurements at any considerable distance from an artificial source are likely to involve strains of less than 3  104 (Kramer [13]). The compaction that occurs under a vehicle tire is likely to involve considerably larger strains. A better approximation of the loss factor can therefore be obtained from the Proctor compaction test, which represents high strains. In fact, that approach may represent the opposite extreme, but is probably much closer to the truth than the seismic approach. Thus, the hypothesis underlying the present (linear) model is that volumetric compaction strain per cycle is proportional to elastic strain per cycle, with the same constant of proportionality as in the Proctor test (hence, the designation ‘‘linear model”). An interpolation of the loss factor for the true amplitudes of the tire–soil interaction problem, where that interpolation is based on soil rheological models, would of course be an even better approach; such an approach is outlined in Section 3 ‘‘Non-linear Model”, below.

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Table 1 Symbol key. a Acyl b

Soil rheological parameter (see Eq. (36)) Cylinder cross-sectional area in Proctor test Exponential decay rate of surface roughness spectral density with increasing frequency (see Eq. (34)) Soil rheological parameter (see Eq. (36)) P-wave speed of soil

w W  W

Water content of soil Power dissipated Time-averaged power dissipation

x x1, x2 y ye Y

Csusp

Rayleigh surface wave speed of soil s-wave speed of soil Equivalent linear damping coefficient due to soil compaction (lumping that into a dashpot) Suspension damping coefficient

Distance on ground surface from tire–soil contact zone Distances of geophone from source in seismic falling weight test (e.g., SASW type) Total displacement of a Maxwell element Elastic component of the displacement of a Maxwell element Fourier transform of surface profile roughness

D

Bulk modulus of elasticity of soil

Yfar

E Ec,Proc Ee,Proc f fcrit

YS Ysoil YU Yveh z

fsusp F

Young’s modulus of soil Specific energy per blow (Proctor test) Specific elastic strain energy per blow (Proctor test) Frequency, in Hz Frequency above which the surface roughness amplitude is filtered (as wavelength falls below contact patch dimensions) (see Eq. (36)) Suspension natural frequency, in Hz Tire–soil contact force

g

Gravitational constant, 9.8 m/s2

ahs

G hcyl i J1() kcyl

Shear modulus of soil Length dimension of cylindrical soil sample in Proctor test Unit imaginary number first-order Bessel function of the first kind Equivalent linear spring rate of soil sample volume in cylinder, in Proctor test P-wave number (= x/cP = 2p/kp), in rad/m Tire stiffness (static, i.e., equivalent spring rate) Soil dynamic stiffness Dynamic stiffness of soil and vehicle in series Vehicle dynamic stiffness

d Df e ec,proc ee,proc

Rate of decay of Rayleigh surface waves due to spreading loss (a function of distance x and freq x) Soil rheological parameter (see Eq. (36)) Also: Hammer mass in Proctor test (4.54 kg) Sprung mass (vehicle body mass) Unsprung mass (tire & axle mass) Mass of an unsuspended vehicle Radius of tire–soil contact zone Radial polar coordinate of a point in the earth with respect to the center of the tire–soil contact zone Residue Spectral density of tire–soil contact force Spatial spectral density of surface profile rougness (function of X) Temporal spectral density of surface profile roughness (function of f) Spectral density of non-recoverable part of soil displacment (soil compaction displacement) Spectral density of soil displacement Spectral density of vehicle displacement

C()

Rate of force application in force-controlled Oedometer test The function 1/3 – m/12 Specific weight of soil Dry specific weight of soil Dry specific weight of soil attainable in a modified Proctor compaction test Function specified in Eq. (11)

g

Loss factor of soil medium

gtire j kp m h q r r_ rproc

Tire loss factor Ratio of p- to s-wave speed (an elastic property) P-wavelength of soil (=cP/f) Poisson’s ratio of soil Angular polar coordinate of a point in the earth with respect to the center of the tire–soil contact zone Density of soil Normal stress in soil Rate of change of normal stress in soil Peak normal stress per cycle (Proctor test)

s

A time interval (in Eq. (39))

x xveh X W f f0

Circular frequency, in rad/s Natural frequency of vehicle mass on tire compliance (unsuspended vehicles), in rad/s Spatial frequency, rad/m along surface Rate of test head closure in constant-velocity Oedometer test Variable of integration Root of f = 0 (see Eq. (11))

fsusp

Suspension damping ratio

B cP cR cS Ceq

kP ktire Ksoil Ksys Kveh L(x,x) m mS MU Mveh r R Res SFF Sxx SYY SYcYc SYsoilYsoil SYvehYveh Szz t T m1 ; m2 Vveh

Spectral density of z time (as an independent variable) Period of a harmonic oscillation Particle velocity amplitudes on ground surface in seismic test at distances x1 and x2 from source (resp.) Vehicle speed

Yc soil

zrms a

a U(m) c cd cd,max

Non-recoverable part of soil displacement (soil compaction displacement) Soil surface displacement amplitude far from tire (due to Rayleigh surface waves) Sprung mass displacement response Soil displacement response Unsprung mass displacement response Vehicle displacement response Sinkage of wheel due to dynamically-induced compaction

RMS-value of the sinkage z Point receptance (frequency-dependent) of soil for excitation under tire by a force distributed uniformly over contact zone Point receptance of a half-space to a force distributed uniformly over circular area Vanishing parameter (d ? 0) Bandwidth, in Hz Total strain: Maxwell element (simulating Proctor test) Non-recoverable strain (compaction) per cycle (Proctor) Elastic strain per cycle (Proctor test)

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Vehicle

125

2.1.1. Measurement of soil properties Soil properties and/or tests needed for this model include:

vehicle velocity Vveh

Ground

Vehicle Vehicle Vveh

Ground Fig. 2. To illustrate the mechanism by which vibrations arise in both the soil and vehicle, consider the crude model of the vehicle and soil represented by the schematic diagram (top). In this model, the soil is a bed of independent springs, and the vehicle is a mass-and-roller. A roughness profile is present in the ground, sinusoidal for the sake of simplicity. Evidently, as the vehicle traverses the ground, then either (1) the vehicle moves up-and-down to follow the roughness profile if the ground springs are very stiff (vehicle strongly vibrates); (2) the vehicle moves in an almost horizontal path and the springs deform if the vehicle is massive and the springs soft (ground vibrates); or (3) in intermediate cases, there is a combination of both vertical vehicle motion and spring deformation, the relative amounts depending on the spring stiffness and vehicle massiveness. In the latter two cases, as a soil spring is deformed, strain in the ground is thereby implied; in an actual ground in which the ‘‘springs” are not independent (a continuous medium), strains would then propagate away as waves, resulting in ground vibrations far away from the vehicle path. This is all more clearly evident in the equivalent model (bottom), in which a rigid strip is passed between the horizontallyfixed vehicle and ground, at the same speed. The ground surface and the vehicle must each, to an extent depending on their relative properties, move vertically to accommodate the entire roughness profile – thus, both vehicle and ground vibrations are induced. Note that the actual computational model of the paper is considerably more refined than the model sketched: (1) the ground ‘‘springs” are not independent, but rather part of a continuous medium; (2) the ground actually has distributed mass and damping properties, as well as elasticity; (3) the vehicle has additional degrees-of-freedom, including possibly an unsprung mass, and tire and suspension compliances (with damping); (4) the roughness profile may be random (in the linear model version). Nevertheless, these additional effects are more difficult to represent schematically, and the added complexity is unnecessary for explaining the underlying mechanism by which vibrations are induced in both the ground and the vehicle.

Breaking down the model into steps, corresponding to the subsections that follow, the method begins by enumerating the various input parameters required to obtain the soil dynamic stiffness – those required for the wave speeds, and those required for the additional loss factor due to compaction. The Proctor test is then used to estimate that loss factor. Given the compaction loss factor, it is then added to the elastic half-space dynamic stiffness. Thereafter a simple linear model is selected for the vehicle, after which the responses of both the soil and the vehicle follow from the principle that they are in inverse proportion to the dynamic stiffnesses.

1. Dry specific weight cd. 2. Water content w (or wet specific weight c = cd[1 + w/ 100]). 3. Modified Proctor test: Gives the maximum dry specific weight cd,max, i.e., the specific weight of the soil in a ‘‘fully compacted” state (to the extent that that is attained by this particular standardized test). The test involves delivering 125 blows of a falling mass to a sample of soil constrained in a cylinder. The specific energy per blow (of the 125 called for) is Ecproc = .022 J/cm3/ blow. The attribute ‘‘modified” will not be used further in this paper, but always understood. See, for example, Rantama¨ki et al.[21], Rico and Del Castillo [22], or other soil mechanics texts for a description of this standardized test. 4. SASW and/or refraction seismics: Gives the elastic wave speeds cP and cS, using geophones. See, for example, Kramer [13] or Parasnis [19]. 5. Profile test: To obtain the spectral density of roughness Sxx(X) of the surface, where X is the spatial frequency.

2.1.2. High-strain loss factor Consider the idealization of the stress–strain cycle experienced by the soil in a single blow of the Proctor test, presented in Fig. 3. This idealization assumes perfectly elastic and perfectly plastic stress–strain regimes. Assuming, moreover, that the strain energy endured by the soil is uniformly distributed among the 125 blows called for in that test (.022 J/cm3 per blow, as noted above, by consideration of the kinetic energy developed in the mass of the hammer, given the distance it drops), then theffi compffiffiffiffiffiffiffi paction strain per cycle is the total divided by 125 (since strain energy is proportional to the square of the strain)   1 c : ð1Þ ec;proc ¼ pffiffiffiffiffiffiffiffi 1  d cd;max 125 The elastic properties of the soil, which control the first and third parts of the cycle as shown in Fig. 3, are determined by the given seismic wave velocities. Using these velocities, and the in-situ specific weight, the following derived properties are obtained: c Soil density q ¼ ; g

ð2Þ

c2p  2c2s ; Poisson’s ratio c ¼  2 c2p  cs2

ð3Þ

Shear modulus G ¼ qc2S :

ð4Þ

(Note that both G and m are defined here as derived variables in terms of the seismic wave speeds cP and cS, on the assumption that the wave speeds rather than the elastic

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Ec, proc

stress

σp roc E e, proc

strain

ε c, p roc

ε e, p roc

Fig. 3. Idealized (elastic–plastic) stress–strain cycle in a single Proctor test blow.

constants are our ‘‘given” input information for the model, since a seismic survey using geophones is presumed to be available. In fact, it is more common in works on elasticity to define the wave speeds cP, cS in terms of the elastic constants G, m rather than the other way around; see, for example, Graff [9] or Cremer et al. [7]. One can, however, easily reverse the definitions and arrive at (3) and (4).) Young’s modulus E ¼ 2ð1 þ mÞG; Bulk modulus D ¼

and;

ð5Þ

Eð1  mÞ ; see Cremer etal: ½7: ð1 þ mÞð1  2mÞ ð6Þ

By geometric reasoning using the idealized cycle in Fig. 3, and noting that the constrained lateral expansion in the Proctor test implies that re,proc = Deproc, then the elastic strain and elastic strain energy per blow are, respectively, rproc Ec;proc ¼ D Dec;proc

ð7Þ

E2c;proc 1 Ee;proc ¼ ee;proc rproc ¼ : 2De2c;proc 2

ð8Þ

ee;proc ¼ and

Finally, the loss factor g represented by the cycle depicted in Fig. 3 is g

Ec;proc D e2c;proc ¼ 2pEe;proc p Ec;proc

ð9Þ

using a standard definition of the loss factor (see Cremer et al. [7]), and simply applying it to this particular mechanism for energy expenditure. The loss factor can be regarded as the fraction of the total kinetic and potential energies lost per cycle of vibration (in other words, one way to provide a measure of the amount of damping present in a vibrating system). As noted above, this particular loss factor is appropriate for high strain levels, and therefore more relevant than the seismically obtained loss factor to describe the tire–soil interaction problem. However, it does not contain the ‘‘radiation” damping; that will be incorporated in Section 2.1.3.

2.1.3. Dynamic stiffness of the soil Miller et al. [17] provide a solution to the dynamic stiffness of an elastic half-space excited by a harmonically oscillating pressure load, uniformly distributed over a circular area of radius r, small in comparison to the Pwavelength; this is an extension of the classical theory of Lamb [14]. For the frequencies of relevance in the present problem, especially for soft soils, the wavelength is always much larger than the dimensions of the tire– soil contact zone, so that the Miller–Pursey theory is applicable. Evidently, r describes the radius of the tire– soil contact zone, which is approximated as circular (area estimates can be obtained from Wong [23], depending on the type of tire); the previous condition is thus r << kp. The point receptance (inverse of dynamic stiffness) is given by the integral pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2c2P 1 f2  1 ½J1 ðfk P rÞ2 df; ð10Þ a¼ 4 fk P r CðfÞ qcS 0 where kP is the P-wave number, J1 is a Bessel function of the first-order, and 2

1

1

CðfÞ  ð2f2  j2 Þ  4f2 ðf2  1Þ2 ðf2  j2 Þ2 ¼ 0;

ð11Þ

cP cS

in which j  : This integral requires numerical solution. Moreover, because of the singularity in the integrand at the root f0 of Cðf0 Þ ¼ 0, care must be taken to include the residue occurring there. Thus, the result may be obtained as (Z pffiffiffiffiffiffiffiffiffiffiffiffiffi f0 d 2c2P f2  1 ½J1 ðfk P rÞ2 df a¼ 4 fk P r CðfÞ qcS 0 ) ffi Z 1 pffiffiffiffiffiffiffiffiffiffiffiffi 2 f2  1 ½J1 ðfk P rÞ df þ Res ; ð12Þ  fk P r CðfÞ f0 þd where d is very small, the residue ‘‘Res” is qffiffiffiffiffiffiffiffiffiffiffiffiffi pi f20  1 J1 ðf0 k P rÞ2 Res ¼  ; C0 ðf0 Þ f0 k P r

ð13Þ

and the first two terms are integrated numerically. Finally the soil’s dynamic stiffness Ksoil, incorporating the compaction loss factor, is 1 þ ig : ð14Þ a This quantity may be thought of as an effective spring rate at a given radian frequency x (since a is frequencydependent); i.e., the ratio of force at frequency x to displacement response at the same frequency. The small imaginary component, moreover, represents damping. (Thinking of force and displacement as phasor-type quantities, i.e., in each case an amplitude multiplied by eixt, then velocity – as the derivative of displacement – is the displacement phasor multiplied by ix. Thus the small imaginary term in (14) really represents a forceto-velocity rather than force-to-displacement ratio, as is

K soil ¼

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

also the case for other familiar elements we recognize as providing damping in vibrating systems, such as a purely viscous damper, for instance, which has a law of the type F = Cv.) 2.1.4. Dynamic stiffness of the vehicle The dynamic stiffness of a quarter vehicle is needed for excitation (and response) at the point where it makes contact with the soil, i.e., the point dynamic stiffness at the bottom of a tire. Some possible vehicle models are 1. Rigid wheel and half-axle: K veh ¼ M u x2 ;

ð15Þ

where Mu is the ‘‘unsprung mass”, i.e., the mass of the tire and half-axle. 2. Unsuspended vehicle, rigid wheels: K veh ¼ M veh x2 ;

ð16Þ

where Mveh is the vehicle mass. 3. Unsuspended vehicle, with compliant, damped tires: K veh

k tire ð1 þ igtire Þx2 ¼ ; x2  x2veh

ð17Þ

where xveh

sffiffiffiffiffiffiffiffiffiffi k tire : ¼ M veh

ð18Þ

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2.1.5. Response of wheel–soil system The temporal spectral density SYY(f), as a function of temporal frequency f, is more conveniently used than the spatial spectral density, directly measured as called for in Section 2.1.1. This is S YY ðf Þ ¼

S xx ðXÞ S xx ðf =V veh Þ ¼ ; V veh V veh

ð22Þ

for vehicle velocity Vveh. It implies a roughness amplitude, at frequency f = x/2p, of Y ðxÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Df S YY ðf Þ;

ð23Þ

for a frequency bandwidth Df. The road surface roughness can be regarded as an ideal displacement source, with the soil and tire stiffnesses, in series, forced to accommodate it. Thus, the system’s dynamic stiffness is K soil K veh ; K soil þ K veh

K sys ðxÞ ¼

ð24Þ

and the power spectra and complex amplitudes of the tire– soil contact force, the soil response and the vehicle response, respectively, are 2

S FF ðxÞ ¼ jK sys ðxÞj S YY ðxÞ;

and

F ðxÞ ¼ K sys ðxÞY ðxÞ; ð25a; bÞ

4. Suspended vehicle, with compliant, damped tires:

K veh

!   k tire ð1 þ igtire Þ k susp þ ixC susp  M S x2 ¼ k tire ð1 þ igtire Þ 1      2 ; k susp þ ixC susp þ k tire ð1 þ igtire Þ  M U x2 k susp þ ixC susp  M S x2  k susp þ ixC susp

where ktire is tire stiffness, gtire is tire loss factor, ksusp is the suspension stiffness, Csusp is suspension damping, Ms is the sprung mass, and MU is the unsprung mass. Alternatively, as input suspension parameters, one may specify the suspension damping ratio fsusp and suspension frequency fsusp, from which 2

k susp ¼ 4p

2 M S fsusp

S Y soil Y soil ðxÞ ¼

S FF ðxÞ jK soil ðxÞj

;

and

Y soil ðxÞ ¼

F ðxÞ ; K soil ðxÞ ð26a; bÞ

and

ð20Þ

S Y veh Y veh ðxÞ ¼

S FF ðxÞ 2

jK veh ðxÞj

;

and Y veh ðxÞ ¼

F ðxÞ : K veh ðxÞ ð27a; bÞ

and C susp ¼ 4pfsusp M S fsusp :

2

ð19Þ

ð21Þ

The model to select depends on the situation. The rigid wheel model may be adequate for a very soft suspension at high frequencies, and with a tire much more rigid than the soil. The unsuspended models are applicable to some vehicles, such as agricultural tractors, which lack a suspension, but may also give a good approximation to a suspended vehicle below the bounce resonance. The suspended vehicle model is the most general, of course.

2.1.6. Response of the vehicle masses For vehicle models 1 and 2, the response of the vehicle (body) mass is Y S ¼ Y veh :

ð28Þ

For vehicle model 3, the response of the vehicle mass is YS ¼

1

h

Y veh

x2 x2veh ½1þgtire 

i:

ð29Þ

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For vehicle model 4, the responses of the unsprung and sprung masses, respectively, are Y veh

YU ¼ h

k veh k tire ð1þigtire Þ

i

and 1

YS ¼

Y U x2 ðx2S  x2 Þ þ

C i MsuspS

: ð30Þ

2.1.7. (Dynamic) component of sinkage Volumetric compaction strain per cycle is assumed proportional to elastic strain per cycle, with the same constant of proportionality as in the Proctor test (hence, the designation ‘‘linear model”); this is, again, the same as the assumption of the ideal cycle illustrated in Fig. 3. Thus the net compaction per cycle directly under the tire is proportional to the vibration amplitude at the same frequency x, i.e., the compaction displacement Yc has the power spectrum and complex amplitude ec;proc 2 S Y Y ðxÞ; and S Y c Y c ðxÞ ¼ ee;proc soil soil ec;proc Y soil ðxÞ: ð31a; bÞ Y c ðxÞ ¼ ee;proc When inside the tire–soil contact zone, a point on the roadway is loaded with the stress reixt, where r ¼ F =ðpr2 Þ, assuming a uniform stress distribution over the contact area. When outside of that zone, but in the tire’s path, i.e., when the tire has not yet passed or has already passed the point in question, the vertical stress is zero. As for points underneath the surface not directly under the tire, they may indeed have non-zero vertical stresses, and therefore undergo compaction, due to the propagation through these points of elastic waves excited by the tire. However, the compaction is expected to be dominated by the higher stresses in the time interval during which these points are directly under the tire. Thus, the sinkage z induced by the dynamic soil compaction is approximated zðxÞ  ðcompaction per cycleÞ  ðcycles per secÞ  ðtime duration under contact patchÞ ¼ Y c ðxÞ

x 2r ; 2p V veh

ð32Þ

where Vveh is the vehicle speed, which has a power spectrum of the form x 2r 2 : S zz ðxÞ ¼ S Y C Y C ðxÞ 2p V veh Finally, the root mean square value of the dynamically-induced sinkage along the vehicle path is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 S zz ðxÞdx; ð33Þ zrms ¼ 0

where sinkage induced by quasi-static compaction is ignored.

2.2. Example cases 2.2.1. Input parameters Consider, as an example case, a soil with the properties: w 25%, 10 ton/m3, cd cd,max 17 ton/m3, 300 m/s, cP 60 m/s; cS the wave speeds and density are representative of a welldocumented site at Ledsga˚rd in western Sweden (see Andreasson [1], Bahrekazemi et al. [5], or Hildebrand [10]), but the compacted density is a typical handbook value. Assume, moreover, a tire–soil contact area of 1000 cm2 = .1 m2 (see fig. 2.7 of Wong [23], for a ‘‘fairly dense, moist soil”); thus, r ¼ :178m; assuming a circular contact zone. The temporal spatial density of roughness, on 1–100 Hz, is S xx ¼ 3  104 Xbðf Þ m2 =cycle; where X = f/Vveh is spatial frequency, and where ( 1:6; f < fcrit ; bðf Þ ¼ 1:6 þ 10ff ð5  1:6Þ; f P fcrit

ð34Þ

ð35Þ

crit

in which fcrit = Vveh/2r. Selecting b(f) = 1.6 for all f would correspond, for example, to a typical pasture, as indicated in Wong [23], p. 381. The modification to b(f) above fcrit, however, is an attempt to represent a ‘‘contact patch filter”, i.e., a reduction in the excitation from roughness components of wavelengths of the order of (or smaller than) the contact zone diameter. This contact patch filter is rather arbitrarily selected, but that is accepted since it occurs at frequencies above the main range of interest in this application; more refined contact filtering methods can be found in other works, such as Ahmed and Goupillon [3] which uses a rigid tread band model and Park et al. [20] which had a deformable tread model. The suspended vehicle model is used, with the parameters 800 kN/m ktire .1 gtire 1.5 Hz fsusp .15 fsusp 400 kg, Mu 3600 kg, and Ms Vveh 5.56 m/s (20 km/h); these are approximately the same as used in Park et al. [20] to describe a truck, but with the addition of tire damping and with slight deviation in the suspension parameters in order to use ‘‘round” figures for the suspension frequency and damping ratio.

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

129

2.2.2. Calculated results The loss factor due to compaction is found to be g = 2.25. Because the model is linear, this is frequency independent. The dynamic compaction is z = 2.3 cm, accounting for the frequency interval .1–150 Hz. The frequency dependencies of the compaction, tire and soil vibrations are presented in Fig. 4, and the dynamic stiffnesses of the vehicle, soil, and composite system in Fig. 5. Several observations in connection with these results are that:

 Throughout most of the frequency range illustrated, the soil stiffness is greater than the vehicle stiffness. The general trend is that, for the size of the contact zone over which the load is applied, the soil stiffness slowly falls off as the frequency rises, but remains above the vehicle stiffness (except as noted below). That being so, the vehicle largely responds as if it were on an equally rough pavement.  The soil stiffness is less than the vehicle stiffness at: o the hop resonance around 9 Hz; and, o frequencies above about 94 Hz.

 The loss factor is very large (g = 2.25), calling into question the applicability of the linear approach for this case. However, this case is nevertheless retained, both because the comparison to the non-linear model in Section 3 will be interesting, and because several other interesting qualitative phenomena that require a very soft soil (giving the high loss factor) can be demonstrated.

Thus, at those frequencies, the amplitude of vehicle vibration is controlled by the soil stiffness. Here, where the vehicle is stiffer, the soil will be put into large motions and consume considerable energy, very effectively damping out the vehicle resonances (and undergoing damaging compaction in the process).

compaction (passage) compaction (per cycle) soil vibration vehicle vibration roughness

1.E+00

Displacement, [m]

1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 0.1

1

10

100

1000

Frequency, [Hz] Fig. 4. Soil and vehicle vibrations, and dynamically-induced soil compaction, from the passage of a quarter vehicle over a very soft clay soil; input parameters as in Section 2.2.1. ‘‘Vehicle vibration” refers, specifically, to the response of the sprung mass.

Dynamic Stiffness, [N/m]

1.E+09 Vehicle dynamic stiffness

1.E+08

System dynamic stiffness Soil dynamic stiffness

1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 0.1

1

10

100

1000

Frequency, [Hz] Fig. 5. Vehicle, soil, and composite-system stiffnesses; input parameters as in Section 2.2.1.

 The second such frequency region is evidently due to the steeper slope in the soil stiffness that begins a little below 100 Hz. A possible explanation is the coincidence at about that frequency of the Rayleigh wavelength with the diameter of the tire–soil contact patch.  The soil vibration is evidently controlled by soil stiffness at all frequencies, since Y soil ðxÞ ¼ F ðxÞ=K soil ðxÞ. Evidently, ground vibrations, which are transmitted to the wayside, have maxima at the vehicle bounce and hop resonances, around 1–1.5 Hz and around 9 Hz, respectively, and again above 94 Hz. The latter frequency is again postulated to be due to the Rayleigh wavelength – contact patch diameter coincidence.  The curves for soil vibration and compaction per cycle have the same forms, separated only by a constant spacing in the logarithmic amplitude scale, once again recalling the assumption that the vertical compaction is always in proportion to the vertical vibration.  At low frequencies, the compaction per cycle, directly under the tire, is greater than that due to the entire passage; this is because full cycles are not completed under the tire at such frequencies. When the tire is away from the point of interest, the compaction per cycle is reduced too much to compensate that deficiency, even as cycles are completed.  The vehicle dynamic stiffness has maxima, rather than minima, at the bounce and hop resonances, simply because it is measured from beneath the tire rather than the body of the vehicle. The point dynamic stiffness at the sprung mass does indeed have minima there, consistent with the definition of a resonance for a vehicle. Evidently, however, the displacements of the unsprung mass are large when a resonance occurs, and therefore large forces develop in the lumped stiffness element beneath the unsprung mass, representing the tire stiffness; hence, a dynamic stiffness maximum results.

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R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136 compaction (passage) compaction (per cycle) soil vibration vehicle vibration roughness

1.E+00

As noted, it is considerably more dense than the other soil prior to the vehicle passage, and therefore has less susceptibility to being compacted. This results in a considerably smaller loss factor g = .45, and a dynamicallyinduced sinkage of only 6.2 mm. The dynamic stiffness of the vehicle is less than that of the soil, even at the vehicle resonances, and except for the high frequencies above the Rayleigh wave – contact patch diameter coincidence. Thus,

Dynamic Stiffness, [N/m]

1.E+09 Vehicle dynamic stiffness System dynamic stiffness

1.E+08

Soil dynamic stiffness

1.E+07 1.E+06 1.E+05 1.E+04

1.E+03 0.1

1

10

100

1000

Frequency, [Hz] Fig. 6. Vehicle, soil, and composite-system stiffnesses, for input parameters as in Section 2.2.1, except that the sprung mass is doubled.

Displacement, [m]

1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E-09 1.E-10 0.1

1

10

100

1000

Frequency, [Hz] Fig. 7. Compaction, vehicle vibration (sprung mass), and soil vibration under the tire, for a hypothetical loam soil, and the same vehicle and profile as in the reference case (Section 2.2.1).

1.E+11

Dynamic stiffness, [N/m]

2.2.3. Two other cases for comparison Two other cases are presented for comparison before moving on to the non-linear model. With respect to the ‘‘reference case” studied above, the first comparison case involves a modification to the vehicle (but the same soil), the second case a different soil (but the same vehicle). Thus, in Fig. 6, dynamic stiffness results are presented for the case in which the sprung mass of the vehicle is doubled from that of the reference case, perhaps as if the same truck were loaded with a cargo, but all other parameters are identical. In this case, at both the bounce resonance, a little above 1 Hz, and the hop resonance near 9 Hz, the soil dynamic stiffness is less than the stiffness of the vehicle. The implication is that the soil will control the amplitude of the vehicle (sprung mass) vibration at both resonances, which are incidentally the most important parts of the spectrum from the vehicle vibration perspective. Figs. 7 and 8 present the vibration and compaction amplitudes, and dynamic stiffnesses, for the case of a considerably denser hypothetical loam soil, but using the same vehicle and roughness spectrum as in the reference case. The loam soil has the properties: w 15%, 15 ton/m3, cd cd,max 17 ton/m3, cp 400 m/s, 163 m/s and, cs r .178 m.

Vehicle dynamic stiffness

1.E+10

System dynamic stiffness Soil dynamic stiffness

1.E+09 1.E+08 1.E+07 1.E+06 1.E+05 1.E+04 1.E+03 0.1

1

10

100

1000

Frequency, [Hz] Fig. 8. Vehicle, soil, and composite-system stiffnesses, for a hypothetical loam soil, and the same vehicle and profile as in the reference case (Section 2.2.1).

for this harder soil with less susceptibility to compaction, the vehicle vibrations are independent of the soil, except at high frequencies. 3. Non-linear model This section describes, and then applies, a refined version of the model presented above, in Section 2. Its purpose is the same as that of Section 2, i.e., to permit a modeling of soil vibrations, vehicle vibrations, and soil compaction resulting from a vehicle traversing terrain with surface profile roughness. However, it seeks to address some of the deficiencies of the linear model. Specifically, it retains its validity, even for very high damping levels (which occur on the softest soils, as became apparent in the examples given above in Section 2.2) at which the concept of the loss factor, used above, loses its validity. It provides, moreover, a more reliable treatment of the true damping levels. The compensating drawback, of course, is the added complexity of the model, as well as the necessity to apply it (as a non-linear model) in the time-domain rather than the frequency-domain.

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

Section 3.1, below, will give a description of the physical assumptions made in this non-linear model, and also present arguments as to why these assumptions lead to a model that is representative of real soils. Then, Section 3.2 provides a detailed formulation (referring back to Section 2, to avoid repetition whenever possible). Finally, Section 3.3 addresses how the rather uncommon soil parameters required for this more refined model would be extracted from the results of relatively common soil tests, and then provides sample results using the model, which are compared to those from Section 2, above. 3.1. Rheological model Because an actual vehicle–tire interaction may, in fact, represent some regime between the low strains of seismic tests and the high strains of Proctor tests, then a model that permits the loss factor to fall between those extremes would be desirable. The approach to be used here is to model the soil as a non-linear viscous–thixotropic dashpot in series with a linear dynamic stiffness element both reactive and resistive (see Fig. 9). The non-linear dashpot, based on the theory of Keedwell [11], represents the deformations due to compaction, i.e., relative motions of grains in the soil in a small zone near the tire; its velocity depends (non-linearly) on both the stress and the rate of stress application as m ¼ Bjr_ m j sinh ar:

ð36Þ

The linear dynamic stiffness element represents the elastic response of the half-space of soil surrounding (and supporting) the highly-deformed zone near the tire, and is based on the theory of Miller et al. [17] for the point impedance of an elastic half-space. This is the same representation of the half-space as in Section 2 of this paper, but without incorporating a compaction loss factor into it. The series of the two elements exhibits the following behaviors known to be consistent with the behaviors exhibited by real soils: 1. The soil can undergo a permanent compaction, because the dashpot does not restore after a force is released.

131

2. The behavior approaches that of a linear elastic solid (distributed as a half-space) at small strains, because the dashpot impedance r/v becomes infinite as r_ ! 0, so that the stiffness of the series approaches that of the linear element alone. 3. For high strains, slowly applied at a small constant _ as in a load-controlled oedometer test, the value of r, spring- non-linear dashpot series has a strain softening behavior consistent with that of typical oedometer tests on soils; see Fig. 10. The dashpot alone has the behavior of an arccosh function, approaching a finite strain at zero stress. However, the in-series elasticity permits a linear elastic behavior at small strains. The behavior, in total, has the appearance of typical oedometer test result. 4. The loss factor depends on frequency, as can be seen in many seismic tests, because the stress rate increases with frequency of harmonic excitation, if the stress itself is held constant. 5. Soil compaction is more pronounced if there is a dynamic component of stress than that due to a static loading alone. In the model, this behavior is incorporated by the stress rate dependence of the dashpot element. Moreover, this approach has the advantages that 1. It is motivated by microstructural behavior of soils (see Keedwell [11]). 2. It accounts for coarse-grained soils being more susceptible to dynamic compaction than fine-grained soils, since the rheological parameter m is small for the former, large for the latter.

3.2. Formulation Suppose Keedwell’s rheological law for soils m ¼ Bjr_ m j sinh ar

ð36; repeatedÞ

holds. Harmonic excitation with weak non-linearity are also assumed to imply that r_ ¼ ixr:

ð37Þ

σ

Fe i ω t Y soil Y hs

C eq (F, ω )

local (high deformation) soil nonlinear thixo-viscous response

K hs (1+i η rad ) = α hs -1 ( ω )

elastic halfspace

Fig. 9. Dynamic stiffness of the ground, representing a compacting (nonlinear thixo–viscous) element immediately under the tire, in series with an elastic half-space.

ε series of elements

dasphot alone (arccosh function)

Fig. 10. Stress–strain behavior of the linear spring and non-linear dashpot _ in series, for a constant stress application rate of r.

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The power dissipated by the thixo–viscous damping component, locally, using a series expansion of the hyperbolic sine function in the rheological law, is W ðtÞ ¼ Reðreixt ÞReðveixt Þ 

 1 3 3 3ixt m m m ixt ixt ¼ Reðre ÞRe B  i x r  are þ a r e þ  3!   3 mp a B 4þm m r x ¼ aBr2þm xm cos xt cos xt þ þ 2 3!   mp  cosxt cos 3xt þ þ : 2 ð38Þ For m – 0, the time-average power dissipation is then found as Z s  ¼ lim 1 W ðtÞdt W s!1 s 0 Z h 1 T mpi ¼ aBr2þm xm cos xt cos xt þ dt T 0 2 Z 1 T mp aBr2þm xm cos2 xt cos dt ¼ T 0 2 1 mp : ð39Þ ¼ aBr2þm xm cos 2 2 The present concern is with the compaction that occurs in the local zone of large soil deformation near the source. At the low frequencies of interest in this type of problem, that region is small compared to the excited wavelength in the soil half-space. Thus, the region in which most compaction occurs is approximately in phase with the excitation itself, so that its response may reasonably be approximated as quasi-static. Thus, considering a Boussinesq (static) distribution of the soil stress in the vicinity of the source 3F cos h; 2pR2 and integrating over a region R > r, h 2 ð0; p2Þ, then Z p=2 Z 1  R>r ¼ 1 aBxm cos mp r2þm ðh; RÞ  2pR2 dR W 2 2 0 r r¼

ð40Þ

 sin hdh  2þm Z 1 Z p=2 1 mp 3F 2pR2 m dR ¼ aBx cos 2 2 2p R2þm r 0  cos2þm h sin hdh 2þm

1 mp ð3F Þ ¼ aBxm cos 2 2 ð2pÞ1þm



 r12m  /ðmÞ 1 þ 2m

ð41Þ

is the energy dissipated outside the hemisphere R = r, where Z p=2 1 m ð42Þ cos2þm h sin hdh   : /ðmÞ  3 12 0

The approximation merely represents a linear interpolation between the known solutions at m = 0 and m = 1: /ð0Þ ¼ 13, and /ð1Þ ¼ 14, respectively. A numerical study revealed that the approximation has a maximum error of just over 2%, which occurs at m ¼ 12 . The dissipation on R 6 r can be approximated by assuming a hydrostatic stress distribution, i.e., the pressure throughout is the same as the average pressure p over the tire–soil contact zone. Then,    R6r ¼ 1 aBxm cos mp p2þm 2 pr3 : ð43Þ W 2 2 3 Total dissipation is  ¼W  R>r þ W  R6r ; W

ð44Þ

from which the equivalent linear damping coefficient 2

C eq ¼

1 jF j  ; 2 W

ð45Þ

is found as a function of the amplitude (of which contact force F serves as a measure).Incorporating the equivalent damper, representing compaction in the local, large deformation zone, in-series with the elastic half-space, the dynamic stiffness of the soil is thus

1 1 ; ð46Þ K soil ¼ ahs ðxÞ þ ixC eq ðF ; xÞ which is now a function of both F and x (unlike the linear model, which was only a function of x). Thus, an iterative procedure, illustrated in Fig. 11, is suggested. 3.3. Determination of the rheological parameters, from standard tests To choose the rheological parameters a, B, and m to represent a particular soil, one can select such parameters that best fit the results of standard soil tests taken on that soil. Appendices A, B and C, discuss the fitting of such data from, respectively, Proctor tests, oedometer/triaxial tests, and seismic tests giving the loss factor frequency-dependence. Comparing the vibrations from the non-linear model to those obtained earlier from the linear model, in Fig. 12, the striking difference is the extent to which the resonances and antiresonances are damped out in the former. Probably, the very high loss factor found in applying the linear model to the same input data invalidated assumptions underlying the linear loss factor model (in using a hysteresis loss factor g, i.e., an imaginary part of an elastic modulus, the value of g is supposed to be small – that was not the case for the linear model data presented in Fig. 12, although it was true for the other linear example case, in which g was only 0.45). 4. Ground vibrations away from the vehicle Ground vibrations, excited at the tire and transmitted over the surface of the earth, may also be of interest. The level of such vibrations may be of interest if, for example,

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

133

PSD surface roughness Sxx(ω), vehicle speed v

elastic (loss-less) soil half-space receptance αhs (ω)

dynamic stiffness of soil Ksoil(ω), including compaction losses

soil vibration Ysoil(ω), vehicle vibration Yveh(ω)

equivalent linear viscous damping coefficient, representing high deformation zone Ceq(ω)

tire-soil contact force F (ω)

desired output

soil rheological parameters a, B, m

Fig. 11. Iterative scheme for the non-linear model.

1.E+01

Displacement, [m]

1.E-01 1.E-03 1.E-05 1.E-07 1.E-09 1.E-11

Linear model: soil vibration Linear model: vehicle vibration Non-linear model: soil vibration

1.E-13

Non-linear model: vehicle vibration

1.E-15 0.1

1

10

100

1000

Frequency, [Hz] Fig. 12. Non-linear results for soil and tire vibrations from the passage of a quarter vehicle over the very soft clay soil in black, compared to the corresponding linear results (repeated from Fig. 3) in gray.

they cause disturbances to residents of dwellings near vehicle paths (at least if the vehicle passages are a common occurrence), disturbances to precision manufacturing processes or medical procedures in buildings near vehicle paths, promote erosion in natural environments, or disturb wildlife. The vibration amplitude at a distance from the tire can be calculated using the Lamb solution (see Lamb [14]), as in Section 2.1.6 of the linear model. Usually, only the Rayleigh wave component will be significant at any considerable distance away from the tire. Thus, the vibration on the surface, at a distance x from the source, is Y farsoil ðx; xÞ ¼ Lðx; xÞY soil ðxÞeikR x ;

ð47Þ

where kR is the Rayleigh wave number, and L(x, x) is Lðx; xÞ ¼ ðk R xÞ1=2 : 5. Conclusions For a sufficiently soft and compactable soil, the vehicle and soil responses are not independent, especially at the

hop resonance, but possibly also at the bounce resonance and at very high frequencies (above about 100 Hz). This has implications for soil compaction induced, ground vibrations excited, and vehicle vibrations excited. The damping mechanisms considered in this paper are energy losses due to elastic wave propagation, and energy losses due to the soil compaction itself. Other mechanisms are proposed as well, but not included in the model in its present state of development. Most of the compaction is predicted to occur at the bounce and hop resonances. Ground vibrations also have maxima at the vehicle resonances, and again around 100 Hz, possibly due to a softening of the soil half-space as the Rayleigh wavelength approaches the order of the contact patch diameter. As for vehicle vibrations, these can be considerably modified at the bounce and, especially, the hop resonance frequencies if the soil is very soft and compactable. However, an already compacted soil has little influence on the vehicle response. Appendix A. Simulation of Proctor test to estimate rheological parameters The Proctor test is simulated by replacing the cylindrical test sample by a Maxwell element consisting of an equivalent spring stiffness, obtained from the soil’s elastic properties, in series with an equivalent damper, the net deflection of which represents the compaction, or permanent strain. The hammer mass m = 4.54 kg is dropped on the Maxwell element; in the modified Proctor test, it attains a velocity of 2.274 m/s just as it reaches the upper surface of the soil cylinder. That is equivalent to considering the hammer mass attached to the Maxwell element, but given an initial velocity of 2.274 m/s as illustrated in Fig. A1, in which y is total deformation and ye is the elastic component of that deformation.

134

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136 .

C eq , y , y , y e

m

(1) .

y ( t + Δt ) = y ( t ) + y ( t ) Δt y . .. . y ( t + Δt ) = y ( t ) + y ( t ) Δt . y e ( t + Δt ) = y e ( t ) + y e ( t ) Δt . eq.(6) : Ceq (t + Δt ) = f ( y e (t ), ye ( t ) )

y Ceq

kcyl

..

(2) .

ye

identity .

y

m , s .. y ( 0) = y e ( 0) = y ( 0) = 0 .

.

y (0) = ye (0) = 2.276

kcyl

(9)

Fig. A2. Numerical difference scheme to simulate the Proctor test.

ye

Fig. A1. Model of the soil sample in the modified Proctor test.

Simulated Proctor test: Ledsgård gyttja clay 0.04 0.035

Equilibrium of the mass demands that ðA-1Þ

and equilibrium at the connection point between the damper and the spring that Acyl r ¼ F ¼ k cyl y e ¼ C eq ð_y  y_ e Þ ¼ C eq e_ c hcyl ;

Strain

M€y þ F ¼ 0;

0.025 0.02 0.015

ðA-2Þ

0.01

where Acyl is the cross-sectional area of the cylindrical soil sample in the Proctor test, hcyl is the cylinder’s height dimension, kcyl is its soil sample’s equivalent spring rate, and Ceq is its equivalent damping coefficient. The equivalent damper is found from the same rheological theory as is used in modeling the tire–soil interaction, i.e.,

0.005

C eq ¼

est. final compaction strain = .0366

0.03

r Acyl r Acyl  ¼ ¼ f ðy e ; y_ e ; k cyl Þ:  e_ c hcyl Br_ sinh ar hcyl

ðA-3Þ

The normal stress and stress rate are

elastic strain component

compaction strain total strain

0 0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Time (s) Fig. A3. Simulated Proctor test, with parameters a, B, and m selected so as to give the final compaction strain corresponding to the soft clay material described pin ffiffiffiffiffiffiffiffi Section 2.2.1. It compacts to a strain ec;proc ¼ ð1  10=17Þ= 125 ¼ :0368. By trial and error, and using the algorithm described in this appendix, the parameters a = B = 1.716  10–5 and m = 1/2 produce nearly that compaction strain.

in terms of the displacements y and ye. Since the total strain is directly proportional to y, i.e., y ; ðA-7Þ e¼ hcyl

trated in Fig. A2 is used to numerically simulate the modified Proctor test. For the soft clay material used in the vehicle–soil simulation, the Proctor test simulation is displayed in Fig. A3. An evident deficiency of the simulation is that it allows the compaction strain to oscillate, while it should be monotonically increasing in a real Proctor test. The oscillation occurs because, for simplicity, the loss of contact between the mass and the soil is not allowed in the simulation, i.e., a tensile stress can be transmitted across the interface between the mass and the soil. However, since the oscillations in the compaction strain are small compared to the asymptotic value, the approximation can be tolerated.

then the force acting on the soil column (or top of the Maxwell element) can also be expressed in terms of y as

Appendix B. Simulation of oedometer test to estimate rheological parameters

r¼

F k cyl y e ¼ Acyl Acyl

and

k cyl y_ e F_ r_ ¼ ¼ ; Acyl Acyl

ðA-4; 5Þ

respectively. Thus, the equivalent damper reduces to C eq ¼

Acyl y e Bhcyl y_ e sinh

F ¼ Acyl r ¼

ak cyl y e Acyl

Acyl Dy  k cyl y: h

ðA-6Þ

ðA-8Þ

Thus, the equivalent stiffness of the soil cylinder is k cyl ¼

Acyl D : hcyl

ðA-9Þ

Using equations (1), (2), (6), and (9), as well as difference equations in time to update y, ye, and y_ , the scheme illus-

B.1. Force-controlled oedometer test In this case, the rate of force application is constant F_ ¼ /:

ðB-1Þ

Then, the dashpot element compresses at a velocity v ¼ B/m sinh a/t:

ðB-2Þ

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

Integrating, with the initial displacement taken to be zero, and including the elastic displacement, gives the net compression of the sample as a function of time x¼

B /t cosh a/t þ : k sample a/1m

ðB-3Þ

The curve of F(=/t) vs. x has the same form as that of Fig. 10, and the same appearance as an experimentallyobtained Oedometer curve. Thus, a means of obtaining the rheological parameters a, B, and m, or a subset of those if other test data or physical reasoning is used to establish some of them in advance, is to select them so as to provide a curve best fitting the experimentally-obtained curve. B.2. Displacement-controlled oedometer test In this case, the rate of closing of the test heads is constant v ¼ w:

ðB-4Þ

Assuming that the elastic part of the response is small enough to be ignored, then w ¼ BF_ m sinh aF : Algebraic manipulation gives  m1 1 1 w wm1 dF ¼ 1 w¼ sinhm aF ; B dt Bm

ðB-5Þ

ðB-6Þ

1

1

w ¼ Bm w1m

1 dF sinhm aF : dt

The displacement is then Z F 1 1 1 sinhm aUdU: x ¼ Bm w1m

At each geophone, the entire time response (i.e., not just the first-arrival of the P-wave as in the refraction test) must be recorded, and transformed by an FFT into the frequencydomain, as in SASW. However, it is now the response magnitude rather than (as in SASW) the phase, that is needed from the FFT. That response magnitude should decrease with distance from the source, the rate of decrease depending on the frequency. Assuming that surface waves are the dominant part of the soil response away from the source, then it is possible to infer a frequency-dependent material damping coefficient a(f), by considering the following law for the decay of surface waves excited by a point force:

12 v2 ðf Þ x2 ¼ eaðf Þ½x2 x1  ðC-1Þ x1 v1 ðf Þ where v1(f) and v2(f) are the frequency-dependent particle velocities at distances x1 and x1 2, respectively, from the point source. The term ðx2 =x1 Þ2 is a geometric spreading loss (surface waves from a point source decay as cylindrical waves), while the exponential term is due to material damping. Finally, it is preferable to express the attenuation due to material damping as a loss factor. The frequencydependent loss factor can be interpreted as the imaginary part of the elastic moduli, i.e., to first-order, 2 times the imaginary part of the wave number (see Cremer et al. [7]). Thus, gðf Þ ¼

and thus, ðB-7Þ

ðB-8Þ

0

Thus, curves of F vs. x can be constructed, given trial values a, B, and m, and compared to experimentallyobtained curves. This may continue until the trial rheological parameters yield a constructed curve sufficiently wellmatched to the experimental curve. Appendix C. Loss factor in seismic tests, and relation to rheological parameter m The loss factor’s frequency-dependence, as observed in seismic tests, may be used to determine m, which would then reduce the number of degrees-of-freedom in the curve-fitting methods of Appendices A and B. Seismic tests involving series of geophones lined up in on the soil surface, excited by an impulsive source (such as a fall weight), can give the type of information that is needed. The measurement set-ups for a conventional subsoil refraction survey, or for SASW, fall into this category (the reader is referred to Kramer [13], for example, for a detailed description of these types of tests). The signal processing proceeds in a different manner than in those tests, however.

135

2aðf Þ : kðf Þ

ðC-2Þ

The dispersion curve k(f) of wave number vs. frequency can be obtained by the SASW test, or, if the soil is to be approximated as homogeneous, then it simply takes the form k(f) = 2pf/cR where cR is the Rayleigh wave speed of the homogenous soil. Having found the frequency-dependence of the loss factor by seismic measurements, consider a low-strain excitation of a soil modeled by the rheological law v ¼ BF_ m sinh aF :

ðC-3Þ

At low enough strains, the linear relations jF_ j ¼ xF and aF  sinh aF approximately hold; that is assumed to be the case at the strains typical of the seismic test. Thus, jvj  aBxm jF jmþ1 ;

ðC-4Þ

from which jF j 

1 m m x jF j jvj ¼ C eq jvj; aB

ðC-5Þ

where Ceq is an equivalent damping coefficient. Noting that Ceq 1 x-m, and that a loss factor and damping coefficient are related as g 1 xCeq, then g / x1m / f 1m :

ðC-6Þ

Thus, the value of m can be selected to give the best fit to the experimentally-obtained loss factor frequencydependence.

136

R. Hildebrand et al. / Journal of Terramechanics 45 (2008) 121–136

References [1] Andreasson B. Ledsga˚rd geotekniska fo¨rsta¨rkningsa˚tga¨rder: Go¨teborg-Fja¨ra˚s, Anneberg-Kungsbacka; fo¨rfra˚gningsunderlag. AB Jacobson & Widemark. Report to the Swedish National Rail Administration Banverket; 2000. [2] Abele G, Brown J, Brewer MC. Long-term effects of off-road vehicle traffic on tundra terrain. J Terramech 1984;21(3):283–94. [3] Ahmed OB, Goupillon JF. Predicting the ride vibration of an agricultural tractor. J Terramech 1997;34(1):1–11. [4] Barnes KK, Carleton WW, Taylor HM, Throckmorton RI, Vanden Berg GE, editors. Compaction of agricultural soils. American Society of Agricultural Engineers; 1971. [5] Bahrekazemi M, Hildebrand R, Bodare A, Carlsson U. Freightvib: Measures to reduce railway ground vibrations for increased axle loads in freight traffic, Joint technical report, Div. Soil & Rock Mechanics and the Department of Vehicle Engineering, KTH Royal Inst Technology, Stockholm; 2001. [6] Canillas E, Salokhe M. Modeling compaction in agricultural soils. J Terramech 2002;39:71–84. [7] Cremer L, Heckl M, Ungar EE. Structure-borne sound: structural vibrations and sound radiation at audio frequencies. 2nd ed. Springer-Verlag; 1988. [8] Garnica Anguas P, Pe´rez Garcı´a N, Ruiz Amaya G. Influencia de las condiciones de compactacio´n en la magnitud de la deformacio´n permanente inducida por carga cı´clica en suelos cohesivos compactados. XI Congreso Panamericano de Meca´nica de Suelos e Ingenierı´a Geotecnica. Foz do Iguacßu, Brasil; 1999. [9] Graff K. Wave motion in elastic solids. Dover Publications; 1975, ISBN 0-486-66745-6. [10] Hildebrand R. Effect of soil stabilization on audible band railway ground vibration. Soil Dyn. Earthquake Eng 2004;24:411–24.

[11] Keedwell MJ. Rheology and soil mechanics. E & F N Spon; 1984, ISBN 0853342857. [12] Kohnke H, Franzmeier DP. Soil science simplified, 4th ed. Waveland Press Inc.; 1995, ISBN 0-88133-813-3. [13] Kramer SL. Geotechnical earthquake engineering. Prentice-Hall International Series in Civil Engineering and Engineering Mechanics 1996. [14] Lamb H. On the propagation of tremors over the surface of an elastic solid. Philos Trans R Soc (Lond) 1904;A203:1–42. [15] Lehtonen TJ. Validation of an agricultural tractor MBS model. Int J Heavy Vehicle Syst 2005;12(1):16–27. [16] Li D, Selig ET. Cumulative plastic deformation for fine-grained subgrade soils. J Geotech Eng 1996;122(12):1006–13. [17] Miller GF, Pursey H. The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid. Proc R Soc Lond A Math Phys Sci 1954;223(1155):521–41. [18] Muro T, Kawahara S, Mitsubayashi T. Comparison between the centrifugal and vertical vibro-compaction of high-lifted decomposed weathered granite sandy soil using a tracked vehicle. J Terramech 2001;38:15–45. [19] Parasnis DS. Principles of applied geophysics, 5th ed. Chapman & Hall; 1997, ISBN 0-412-80250-3. [20] Park DS, Popov AA, Cole DJ. Influence of soil deformation on offroad heavy vehicle suspension vibration. J Terramech 2004;41:41–68. [21] Rantama¨ki M, Ja¨a¨kela¨inen R, Tammirinne M. Geotekniikka. Oy Yliopistokustannus/Otatieto University Press Finland; 1979, ISBN 951-672-257-1. [22] Rico A, Del Castillo H. La Ingenierı´a de Suelos en las Vı´as Terrestres: Carreteras, Ferrocarriles y Aeropistas, Editorial Limusa, Me´xico, D.F., Me´xico; 1989, ISBN 968-18-0079-6. [23] Wong JY. Theory of ground vehicles. John Wiley & Sons; 1978, ISBN 0-471-03470-3.