A Novel Approach To The Calculation of Pothole-Induced Contact Forces in MDOF Vehicle Models
A Novel Approach To The Calculation of Pothole-Induced Contact Forces in MDOF Vehicle Models
A Novel Approach To The Calculation of Pothole-Induced Contact Forces in MDOF Vehicle Models
Abstract
A technique is developed to predict the dynamic contact forces arising after passing
road surface irregularities by a vehicle modeled as an undamped multiple-degrees-
of-freedom (MDOF) system. An MDOF system moving along an uneven profile is
decomposed into an aggregate of independent oscillators in the modal space, such
that the response of each oscillator can be calculated independently. An equation re-
lating the contact forces in the physical space to the modal forces is established. The
technique developed is applied to the calculation of the coefficients of the harmonic
components of the contact forces arising after the passage of a cosine pothole.
The application of the technique to various problems, such as evaluation of the ef-
fect of parameter modifications on the vehicle dynamics and reduction of vehicle
models in bridge related problems, as well as its extension to the damped case, are
also discussed. One interesting phenomenon reported in the DIVINE project [1],
regarding the replacement of a steel suspension by an air suspension resulted in
increase of the maximum response of short-span bridges is explained by applying
the technique suggested. The discussion is amply illustrated by examples of the ap-
plication of the technique to the calculation of the tire forces due to a pothole for
two simplequarter-car and half-carvehicle models.
The paper is concerned with the assessment of dynamic tire forces that arise
after the passage of a road surface irregularity by a vehicle. This problem is
of great importance as it is well known that dynamic loads produced by ve-
hicles considerably affect damage of the infrastructure (pavement or bridges),
a significant portion of which in many countries is either aging or reaching
the end of its life. The recently concluded multinational DIVINE (Dynamic
Interaction Between Vehicles and Infrastructure Experiment) project [1] indi-
cates that trucks wear pavements at a rate which is dependent not only on
the static load carried by the vehicle, but also on the dynamic performance of
the vehicle, on the longitudinal profile of the road and on the structural vari-
ability of the pavement. The outcomes of this project suggest that current
understanding of the dynamic interactions between moving vehicles and the
infrastructure carrying them is inadequate.
(dyn static )
DI = 100%,
static
where dyn and static are peak dynamic and static deflections, respectively.
Such great values of the DI cannot be explained in the framework of simple
moving force or moving massvehicle models. Indeed, the dependence of the
maximum deflection of a simply supported beam on the speed of the traveling
force presented in [4] shows that the maximum DI (about 70%) occurs at
very high speed and, for the vehicle speeds of interest, it does not exceed 10
15%. The inertia effect of the moving mass in this speed range is also small
and can be neglected. Moreover, as can be concluded from many publica-
tions, as well as from our numerical experiments, such great values of the DI
cannot be obtained in the framework of more complicatedmultiple-degrees-
of-freedommodels if the bridge surface is assumed to have flat longitudinal
profile (as noted in [1], for a smooth profile, the influence of the truck sus-
pension is insignificant). The above arguments lead us to speculate that the
high values of the DI measured in some field experiments can be explained only
by the presence of road irregularities on the bridge and its approaches. Then
it follows that the examination of the effect of uneven road profile is crucial
in the analysis of high-magnitude bridge vibration.
The effect of road surface irregularities on the bridge vibration has been ex-
amined by many researchers (see, e.g., [13,58] and references therein), and
2
many methods for numerical solving the problem of a vehicle moving along a
bridge with an uneven surface have been developed (e.g., [611]). The main
difficulty associated with this problem is in the large number of parameters
involved. As a result, the majority of studies reported in the literature are
confined to extensive numerical modeling or field experiments. An obvious
disadvantage of these approaches is that results of numerical or field exper-
iments are often valid only for a particular bridge and vehicle and cannot
easily be generalized to other configurations. It is then not surprising that
results reported in the literature are sometimes contradictory. This point is
well illustrated by the following examples. In both [2] and [5], short-span
bridges are considered. However, [2] shows large values of the dynamic incre-
ment measured in field experiments (up to 137%), whereas [5] reports that
the analytical simulations and field tests showed that DLF is considerably
lower than code-specified values (the latter are around 30% depending on
the code and bridge length) and recommends to reduce the design values of
DLF. Another example of this kind can be found in [3]. The authors explore
how different factors influence bridge behavior. Based on results of numerical
experiments with elaborate finite-dimensional models of a bridge and vehicle,
they conclude that, in terms of the maximum bridge deflection, the initial
vehicle suspension oscillation had the greatest effect and the bridge-surface
roughness was found to have negligible effect. On the other hand, they justly
note that road surface irregularities on the bridge approaches is the princi-
pal cause of the initial suspension oscillation, which means, in fact, that it is
road roughness (road surface irregularities) that had the greatest effect on the
maximum bridge deflection.
3
of the vehicle oscillations due to the irregularity contains frequencies that
match fundamental frequency of the bridge and whether the magnitudes of
the corresponding harmonic forces are sizeable.
The problem of calculation of the dynamic forces arising after passing an irreg-
ularity is very important also in studies related to pavement damage [1,1216].
Based on experimental results reported in the literature, Potter et al. [12] con-
clude that the peak damage due to dynamic loads can be between 1.5 and 12
times the level of damage caused by a static load and note that, at highway
speeds, the parameter which causes the greatest variation in dynamic tyre
forces, and the largest changes in ranking of suspensions, is the road roughness
level. Moreover, as indicated in [14], there is an evidence that fatigue failure
of pavements is likely to be governed by peak dynamic forces, and not by the
average dynamic forces. Then it follows that, both in bridge and pavement-
damage related applications, it is critically important to establish dependence
of the peak tire forces arising after passing an irregularity on the irregularity
parameters, suspension characteristics, and vehicle speed.
The general idea of the approach discussed in this paper is to decouple equa-
tions governing vibration of an MDOF vehicle model moving along uneven
road, to solve the uncoupled equations in the modal space, and to transform
back the results obtained into the physical space. The fact that the model is
decomposed into independent SDOF oscillators makes it possible to find so-
lution for each oscillator analytically (or semi-analytically). The crucial step,
when transforming back to the physical space, is to calculate not the contact
forces themselves but rather the Fourier coefficients of their harmonic compo-
nents. This allows us to represent the results obtained in a form suitable for
subsequent analysis. Moreover, in many cases (e.g., when applied to problems
of bridge vibration), the Fourier coefficients of the harmonic components of
the contact forces give us more valuable information than just magnitudes of
the total dynamic forces.
In this paper, we consider the case of undamped vehicle models, although the
approach discussed is applicable to damped vehicles as well (see Section 5.2).
The format of the presentation is as follows. In Section 2, we reduce the prob-
lem of vibration of an MDOF vehicle moving along uneven road to that of
independent SDOF oscillators in the modal space. A technique for the calcu-
lation of contact forces arising after the passage of an isolated cosine pothole
is presented in Section 3. In Section 4, an interesting phenomenon reported in
[1,2] is discussed and explained by applying the technique suggested. Section 5
discusses extensions of the approach suggested to damped vehicle models, to
local irregularities of different types, and to bridge-related problems.
To conclude the introduction, let us cite the DIVINE report [1]: There is now
a need for a higher level of scientific knowledge about the interaction between
4
trucks and pavements, and between trucks and bridges, in order to introduce
regulations based on vehicle performance in terms of road friendliness. It is
achieving this goal that the work presented is aimed at.
For the half-car model (Fig. 2b), the parameters are as follows:
k2 0 0 0 1 0
n = 4, m = 2, l1 = 0, l2 = l, Kint = , Sv = .
0 k2 0001
The vehicle mass and stiffness matrices are written in a standard way and
not presented here. The order of numbering the coordinates can be easily
understood from the form of the matrices Sv .
The free vertical vibration of the vehicle resting on the rigid foundation is
governed by the equation
5
for the stiffness matrix of the supported vehicle, equation (1) reduces to
Let now the vehicle move with a speed v along a road with a longitudinal
profile r(x). In this case, the vehicle is subject to external forces acting on it
at the contact points due to variation in the road profile r(x). Introduce the
notation Sr for the operator defined by the relation
r(x l1 )
r(x l2 )
Sr r(x) = .. . (4)
.
r(x l
m )
Then, it can be checked directly that the equation governing vertical vibration
of the moving vehicle is given by
and
re(x) = ASr r(x), (8)
where A is a dimensionless nm matrix, we rewrite equation (6) in the form
f + K
M f =K
fre(vt), (9)
m e ( re (vt)), i = 1, . . . , n,
fi i = k (10)
i i i
6
Remark 1 Note that the case where the vehicle traverses a beam with an
uneven profile r(x) is treated in exactly the same way. In this case, the function
r(x) in all equations is to be replaced by w(x, t) + r(x), where w(x, t) is the
displacement of the beam point x at time t.
As can be easily seen, the ith equation in (10) governs vibration of the 1-DOF
oscillator with the modal mass m fi and the spring coefficient k e moving along
i
the profile rei (x). The matrix A transforms the input profile for the original
MDOF vehicle model into the profiles rei (x) for the independent oscillators and
is further referred to as model scaling matrix. Thus, we reduced the problem
of an MDOF system moving along a profile r(x) to n independent problems
for 1-DOF oscillators, with the profiles rei (x) being different for each oscillator.
Solving n independent equations (10), we find the vector of modal coordinates
.
To derive a relationship between the vector of the dynamic contact (tire) forces
Lemma 1 For any linear vehicle model, the following matrix identity holds:
f K .
AT KA (12)
int
7
(3) Transform the input Sr r(x) of the original problem into that for the
uncoupled problem by means of the matrix A: re(x) = ASr r(x).
(4) Solve uncoupled system of equations (9) to get the forces Fe .
(5) Transform these forces to the contact ones by means of the matrix AT :
Fc = AT Fe .
To solve the problem, we will apply the technique discussed in the previous
section. Denote by Ai the ith row of the scaling matrix A and by aij its entries.
Consider the ith equation (10). The longitudinal profile for the ith oscillator
can be written as
8
After the passage of a pothole, the modal forces Fej are harmonic ones. De-
note by |Fej | their amplitudes. Expanding the vector of contact forces into the
Fourier series,
n
X
Fc = Cj cos( j t + j ) (16)
j=1
It will be shown later in this section that m potholes of the same length can
be replaced by one equivalent pothole. Thus, the key point in finding the
dynamic effect of a pothole on an MDOF vehicle is to be able to efficiently
calculate the magnitude of the contact force arising after passing one pothole
by an SDOF oscillator.
b0 bf0
= , (19)
2v v
0 (f0 = 0 /2) and k0 are the oscillator eigenfrequency and spring coefficient,
respectively. The function
| sin |
() = , (20)
| 2 1|
called the dynamic amplification factor for a pothole (see [17] for detail), is
shown in Fig. 3. By means of the dynamic amplification factor, one can imme-
diately estimate magnitude of the contact force acting on the road from the
oscillator after passing the pothole. As can be seen, it linearly depends on the
spring stiffness and the pothole depth, and the remaining three parameters
(v, b, and 0 or f0 ) are combined through the function of one variable ().
9
3.3 An MDOF vehicle with one contact point passing a pothole
We begin the examination of an MDOF vehicle model with the simplest case
where a vehicle has one contact point, m = 1. In this case, the contact force
Fc is a scalar, and A is a vector of length n: A = [a11 , . . . , an1 ]T . The jth
independent oscillator passes one pothole of width b and depth ae j = aj1 a, and
we can immediately apply the results of [17] to find amplitudes of the forces
Fej by equation (18),
|Fej | = ae j ke j (j ), (21)
where () is given by (20) and
fj b
j = . (22)
v
The Fourier coefficients Cj given by (17) are calculated analytically as
Thus, given the parameters of the uncoupled system, the function (), shown
in Fig. 3, bears all required information about the behavior of the MDOF
system after passing a pothole.
Note that in the case of an MDOF vehicle, it is more convenient to plot all
functions Cj in one figure in order to get better idea of the contribution of
each oscillator into the dynamics of vehicle vibration. These curves can be
plotted versus the parameter b/v for a fixed value of the pothole depth a (the
dependence on which is trivial).
As an illustration, consider the quarter-car model (Fig. 2a) with the fol-
lowing parameters: m1 = 3.6104 kg, m2 = 2.0103 kg [7,18], k1 = 4106 ,
k2 = 1.2107 N/m, and c1 = c2 = 0. Applying the technique described in the
previous section, we obtain masses and frequencies of the modal oscillators,
f1 = 3.404104 kg, m
m f2 = 0.403104 kg, f1 = 2.05 Hz, f2 = 14.3 Hz, and the
entries of the scaling matrix A, a11 = 1.05 and a21 = 0.74.
10
3.4 General case of an MDOF vehicle
Let us show that several potholes of identical widths can be replaced by one
equivalent pothole, which reduces the problem to that considered in the
previous section. We consider an oscillator moving along the horizontal rigid
surface with m potholes (bumps) as shown in Fig. 5 and set the problem of
finding the amplitude of the oscillator free vibration after passing all potholes.
The potholes are assumed to be of form (14), have the same width b but
different depths aj , and the jth pothole is located at the distance lj from the
first one. If lj+1 lj < b, the (j + 1)th and jth potholes overlap.
Denote by Ts the moment when the oscillator leaves the sth pothole, Ts =
T + ls /v, where T = b/v. For t Ts , rs (vt) = 0, and the solution to equation
(26) represents the oscillator free vibration,
11
It has been proven in [17] that
Zs = as (), (28)
where and () are defined by (19) and (20), respectively. The phase angle 0
is determined by the oscillator parameters and the pothole width but does not
depend on as ; hence, 0 is the same for all zs (t). When t Tm , representation
(27) is valid for all s and we have
m
" m
#
X X
z(t) = Zs Re[ei(0 (tT )+0 0 ls /v) ] = Re () ei(0 (tT )+0 ) as ei0 ls /v .
s=1 s=1
(29)
Then, it follows that the magnitude of the oscillator free vibration after passing
m potholes is
m
X
Z= as ei0 ls /v (). (30)
s=1
Equation (24) suggests the following way of calculation of the depth of the
equivalent pothole. Each of the m potholes is assigned a complex depth by
multiplying the real value as by exp(i0 ls /v), which accounts for the time
lag between the passage of the pothole by the sth and first contact points.
The depth of the equivalent pothole is then the magnitude of the sum of the
complex depths obtained.
Then, the modal forces |Fej | are calculated by (21) with the substitution of |aej |
for aej and the vector Cj of the Fourier coefficients, by equation (17).
Remark 2 The scalar equation (23) obtained for the case of one contact point
can be used to calculate components of the vector Cj . The pth component of Cj
(the jth Fourier coefficient of the pth contact force) is obtained by substitution
of ajp for aj1 into (23).
12
3.4.2 Multiple eigenfrequencies
Consider free vibration of the j1 th and j2 th oscillators in the modal space after
passing all m potholes. By equation (29), we have
h i
jk (t) = Re ae jk ei(j (tT )+j ) (j ), k = 1, 2,
Noting that the right-hand side of the last equation represents the harmonic
function with the amplitude |ATj1 kej1 aej1 + ATj2 kej2 ae j2 |(j ) and extending this
to the case of arbitrary multiplicity of a repeated eigenfrequency, we arrive at
the following theorem.
The last result implies that the case of repeated eigenfrequencies presents, in
fact, almost no additional difficulties. Indeed, in the general case, we simply
need to perform all calculations in the complex plane: to find complex depths
aej (rather then only their magnitudes |aej |) for all oscillators in the modal
space and to calculate complex Fourier coefficients as ATj kej aej (j ) for all n
harmonics (without regard to whether they are single or multiple). Then, if
some eigenfrequencies are identical (or close to each other), we add the corre-
sponding complex Fourier coefficients. And only after this, we take absolute
13
values of the Fourier coefficients obtained to get real coefficients of the expan-
sion (16).
Step 2. Calculate complex depths of the equivalent potholes for all modal
oscillators by (31).
The basic difference of the general case from the case of one contact point
is that the Fourier coefficients in the former case depend on two parameters
rather than on one parameter as it was in the latter case. Indeed, (j ) is a
function of the ratio b/v, and the depth of the equivalent pothole given by (31)
is a function of vehicle speed v. The shape of each curve Cjr plotted versus
b/v and the abscissa of its peak depend only on the eigenfrequency of the
corresponding oscillator and are determined by the function (j ). However,
the height of the curve is determined by the ae j , which now depends on the
speed. Thus, if we want to examine the dependence of the Fourier coefficients
on both vehicle speed and pothole width, we have to consider a family of plots
parametrized by the values of the vehicle speed. Note that one can choose a
14
different pair of independent parameters, e.g., b and v; however, the pair b/v
and v seems to be more convenient.
3.5 Example
Figures 6 and 7 show the amplitudes of the pitch (dashed line), body-bounce
(bold solid line), and axle-hop (thin solid line) components of the first contact
force after passing a pothole of depth a = 1 cm versus b/v for two values of
the vehicle speed: 10 and 30 m/s, respectively. In view of the model symmetry,
the results related to the second contact force are the same and not presented.
As can be seen, the Fourier coefficient corresponding to the repeated axle-hop
eigenfrequency is not affected by the vehicle speed and depends only on the
parameter b/v, whereas the pitch and axle hop do depend on both parameters.
15
hop force in the whole interval of speed values of interest. On the contrary, in
the case of the long pothole, the body-bounce and pitch forces are considerably
greater than the axle-hop force. Figure 9 also demonstrates that, although the
peak values of the body-bounce and pitch forces are almost the same for the
given pothole, the contributions of these forces in the total contact force in
different intervals of speed are considerably different.
There is another way to represent the results, which seems to be more appro-
priate when we want to examine a wide range of vehicle speeds and to avoid
drawing many figures. Indeed, the jth Fourier coefficient can be written in the
form
Cj (b, v) = ATj kej aej (j ) j (v)j (b/v). (33)
(Here, Cj (b, v) and j (v) are m-vectors; however, in the following discussion,
we consider (33) as a scalar equation associated with a certain contact point,
in which, for simplicity of notation, the subscript denoting a particular con-
tact point is dropped). Thus, each Fourier coefficient is obtained from the
unique function () by scaling the variable b/v, j (b/v) (j ), where
j = fj b/v, and by multiplying it by the corresponding speed-dependent co-
efficient j (v) = ATj kej aej . Then, instead of drawing figures of the Fourier coef-
ficients for different speeds (or different potholes), we can confine ourselves to
two figures: one figure with the plots of the functions j (b/v), j = 1, . . . , nd
and the other with the plots of the multipliers j (v). Under such a represen-
tation, the first figure shows the shape and relative locations of the curves
representing the Fourier coefficients. In particular, it shows the regions of the
parameter b/v where the harmonic forces take their maximum values or, vice
versa, can be neglected. The second figure shows dependence of the multipliers
j on the vehicle speed. The use of both figure allows us to accurately evaluate
the Fourier coefficients for any values of the pothole width b and vehicle speed
v.
16
4 On one phenomenon reported in the DIVINE
The DIVINE report [1] defines air suspension as more road-friendly than steel
suspension and recommends using it instead of the latter, but notes that For
short-span bridges (10 meters) with poor profiles, large dynamic responses
occur for both air-suspended and steel-suspended vehicles. By taking into
account that the basic difference between two suspensions is in the body-
bounce natural frequencies whereas the vibration of short-span bridges are
affected by axle hop (fundamental frequencies of such bridges are in the range
of axle-hop frequencies), this observation sounds quite natural.
It is further noted, however, that [1, p. 11] The highest measured responses
were for short-span bridges . . . traversed by air-suspended vehicles where axle
hop was excited by short-wavelength roughness. This observation seems to
rely on results of field experiments reported in the work [2], which also states:
Generally the peak bridge deflections were smaller when the air suspensions
were fitted except when axle hop was induced by roughness. At first glance,
the phenomenon observed in [2] sounds strange and raises the question: How
could softening of the suspension (reduction of the body-bounce frequency)
increase the bridge response affected by axle hop? The conclusion of the paper
[2] that vehicles fitted with air suspension can couple with short span bridges
does not answer the question and explains nothing.
As noted in [1, p. 53], in the case of short-span bridges, the dynamics of the
bridgevehicle system is completely different from that in the case of medium-
to long-span bridges, and true interaction no longer occurs. Under these
conditions, the model of a bridge being forced to vibrate by external forces
i.e., dynamic wheel loadswithout taking the vehicle masses into account
should be adopted. Then, in view of matching of the fundamental frequency
of the bridge and the axle-hop frequency, the increase in the bridge vibration
can be explained by an increase in the axle-hop force. We applied the technique
developed in this paper to check whether the replacement of a steel suspension
by an air suspension results in an increase in the axle-hop force.
In terms of the 2-DOF model shown in Fig 2a, replacement of a steel suspen-
sion by an air suspension is modeled by softening the spring k1 supporting
the vehicle body. The other spring k2 is not changed since we assume that
the tires remain the same. Considering the 2-DOF model with the body-
bounce frequency 2.05 Hz used in the experiment described in Section 3.3
(Fig. 4) as steel-suspended, we reduced the spring coefficient k1 by two
times, k1 = 2 106 N/m, which, in turn, reduced the body-bounce and axle-
hop frequencies to 1.55 Hz and 13.3 Hz, respectively. The modified model
was assumed to represent the air-suspended vehicle. The masses of the
f1 = 3.53104 kg and
modal oscillators for the air-suspended model are m
17
f2 = 0.401104 kg, and the entries of the scaling matrix A are a11 = 1.02 and
m
a21 = 0.85. The amplitudes of the body-bounce and axle-hop forces for the
air-suspended model are depicted in Fig. 12 by the dashed and solid lines,
respectively. As could be predicted, the reduction of the suspension frequency
considerably reduced the force associated with the body bounce. However, at
the same time, this increased the magnitude of the high-frequency force asso-
ciated with the axle-hop by about 15% (in spite of the fact that the axle-hop
frequency diminished!). Since the axle-hop and bridge eigenfrequencies are as-
sumed to match well, the increase of the axle-hop force immediately results in
the increase of the bridge response.
Note that the result obtained is not specific to the example considered but
is rather general. Softening of vehicle suspension decreases magnitude of the
low-frequency force associated with the body bounce but increases the ampli-
tude of the high-frequency axle-hop force. From the physical standpoint this
phenomenon can be explained as follows. By softening the suspension spring
coefficient, we permit the axle (which vibrates between the road and vehicle
body) vibrate with greater amplitude. Since the force transmitted to the road
is determined by the amplitude of the axle vibration and by the spring coeffi-
cient k2 , which has not been changed (the tires are the same), its magnitude
increases. This implies that an air-suspended vehicle is potentially dangerous
for short-span bridges with fundamental frequencies in the range of vehicle
axle-hop frequencies. Moreover, although air-suspended vehicles are consid-
ered as road-friendly ones, they can produce a greater pavement damage
compared to steel-suspended vehicles in the case of uneven road surface with
short-wavelength irregularities, which excite the axle-hope vibration.
The results discussed in Section 3 are not specific to the pothole described
by function (14). We used this particular pothole simply because its dynamic
amplification factor, the function (), is available in the analytical form [17].
18
As can be seen, the technique is easily adopted to any other local irregularity
if its dynamic amplification factor, which shows the dependence of an SDOF
oscillator response on the oscillator and irregularity parameters, can be cal-
culated. The only thing that is required to do when considering a pothole of
a different form is to replace one function () by another. In particular, one
can take advantage of the dynamic amplification factor
4
s () = | cos |
|1 4 2 |
given in [17] to obtain the Fourier coefficients of the contact forces due to
passage of a half-sine pothole
a sin x
b
, 0 x b,
r(x) =
0, x < 0, x > b
(which differs from the cosine pothole (14) in that the derivative of r(x) at
x = 0 and x = b have jumps) by an MDOF vehicle.
The key point in the approach suggested is the decomposition of the moving
MDOF system into an aggregate of independent moving oscillators. In the
undamped case, the governing equations can always be uncoupled by trans-
forming to modal coordinates. In the damped case, this transformation does
not uncouple the equations, except for a special case of proportional damping,
and we have two following possibilities.
19
practical standpoint (the system dynamics is determined, in the first turn,
by the diagonal elements of the modal damping matrix). The quality of the
approximation depends not on the level of damping in the system but rather
on the degree of damping nonproportionality (this notion can be defined
in strict terms), such that even a highly damped system can perfectly be
approximated in this way.
Both above approaches have been already implemented and, at the moment,
are under numerical verification. The advantage of the first approach is in
its physical clearness. Still, the second approach seems to be more promising
since it is exact (no approximations are involved) and because of convenience
of analytical calculations in complex arithmetic.
Note also that, in the damped case, the contact forces are expanded in the
series of the functions ej t cos( j t + j ), where j is the damping coefficient
of the jth modal oscillator. Thus, the Fourier coefficients to be calculated are
magnitudes of the exponentially decaying harmonic functions.
In the problems related to bridge vibration, the use of an SDOF vehicle model
is often justified in view of the fact that only vehicle vibration at a frequency
close to the bridge fundamental frequency considerably affects vibration of the
bridge. The use of an SDOF vehicle model simplifies the analysis and, thus,
is more convenient for the designer. Then, one faces the following problem.
Given an MDOF vehicle model and a bridge, what oscillator is to be chosen
to adequately represent the vehicle model? The technique developed in Sec-
tion 2 perfectly suits this goal. In certain circumstances, especially when stress
calculations are required, it may be advisable to use a reduced system with
more than one degrees of freedom, i.e., to retain some modal oscillators with
eigenfrequencies not matching the fundamental frequency of the bridge that
20
produce sizable contact forces for a given road surface profile. The additional
information provided by the plots of the Fourier coefficients of the contact
forces due to road surface irregularities can be used to create an elaborate
reduced vehicle model. A technique for the reduction of an MDOF vehicle
model based on the method suggested in this paper is discussed in [19].
6 Conclusions
Acknowledgements
The authors wish to acknowledge the support of the Civil and Mechanical
Systems Division of the National Science Foundation through grant number
CMS-9800136.
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21
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23
List of Figures
Figure 6. Amplitudes of the pitch (dashed line), body-bounce (bold line), and
axle-hop (thin solid line) components of the first contact force after passing
the pothole of depth a = 1 cm for the 4-DOF model moving at v = 10 m/s.
Figure 7. Amplitudes of the pitch (dashed line), body-bounce (bold line), and
axle-hop (thin solid line) components of the first contact force after passing
the pothole of depth a = 1 cm for the 4-DOF model moving at v = 30 m/s.
Figure 10. Functions 1 (b/v) (dashed line), 2 (b/v) (bold line), and 3 (b/v)
(thin solid line) for the 4-DOF model.
Figure 11. Magnitudes of the harmonic forces 1 (v) (dashed line), 2 (v) (bold
line), and 3 (v) (thin solid line) for the 4-DOF model.
24
Mnn Knn
m 2 1
Kint
l2
lm
Fig. 1.
25
Fig. 2.
26
1.8
1.6
1.4
1.2
1
()
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4
Fig. 3.
27
5
x 10
3
2.5
Amplitudes of harmonics (N)
1.5
0.5
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
b/v
Fig. 4.
28
v
m0
k0
b a
m
a
2
a1
l2
lm
Fig. 5.
29
4
x 10
15
Amplitudes of harmonics (N)
10
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
b/v
Fig. 6.
30
4
x 10
15
Amplitudes of harmonics (N)
10
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
b/v
Fig. 7.
31
4
x 10
15
Amplitudes of harmonics (N)
10
0
10 15 20 25 30 35 40 45 50
Vehicle velocity, v (m/s)
Fig. 8.
32
4
x 10
4
3.5
3
Amplitudes of harmonics (N)
2.5
1.5
0.5
0
10 15 20 25 30 35 40 45 50
Vehicle velocity, v (m/s)
Fig. 9.
33
1.8
1.6
1.4
1.2
( ), j=1, 2, 3
0.8
j
0.6
0.4
0.2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b/v
Fig. 10.
34
4
x 10
9
6
j(v), j=1, 2, 3, (N)
0
5 10 15 20 25 30 35 40 45 50
v (m/s)
Fig. 11.
35
5
x 10
3.5
2.5
Amplitudes of harmonics (N)
1.5
0.5
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
b/v
Fig. 12.
36