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Frequency-domain method for

evaluating the ride comfort of a
motorcycle
a a a
Vittore Cossalter , Alberto Doria , Stefano Garbin & Roberto
a
Lot
a
Department of Mechanical Engineering , University of Padova ,
Via Venezia 1, 35131, Padova, Italy
Published online: 16 Feb 2007.

To cite this article: Vittore Cossalter , Alberto Doria , Stefano Garbin & Roberto Lot (2006)
Frequency-domain method for evaluating the ride comfort of a motorcycle, Vehicle System
Dynamics: International Journal of Vehicle Mechanics and Mobility, 44:4, 339-355, DOI:
10.1080/00423110500420712

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 Vehicle System Dynamics
Vol. 44, No. 4, April 2006, 339–355

Frequency-domain method for evaluating the

ride comfort of a motorcycle
VITTORE COSSALTER, ALBERTO DORIA*, STEFANO GARBIN and ROBERTO LOT
Downloaded by [University of Connecticut] at 18:07 10 October 2014

Department of Mechanical Engineering, University of Padova, Via Venezia 1, 35131 Padova, Italy

In many European towns, the demand for fast and efficient mobility is frequently satisfied by means of
two-wheeled vehicles. The improvement of comfort of two-wheeled vehicles used by tired and busy
workers can increase safety in ground transport. Nowadays, multibody codes make it possible to predict
the ride comfort of two-wheeled vehicles by means of time-domain or frequency-domain simulations.
Comfort indices can be developed by post-processing the results of numerical simulations. This task
is difficult, because the indices should depend on vehicle characteristics and should be independent of
road quality and vehicle speed. Poor quality roads may generate nonlinear effects. Speed influences the
trim of the vehicle and the wheelbase filtering, which takes place because the same road unevenness
excites the front and rear wheel with a time delay which depends on the vehicle’s speed.
In this paper, the comfort of two-wheeled vehicles is studied by means of a frequency-domain
approach. The wheelbase filtering is averaged considering typical missions of the vehicle. The missions
are journeys with a forward speed that assumes different values according to a probability density
function. Indices of comfort are calculated taking into account the human sensitivity. The examples
show that the proposed comfort indices depend on suspensions’ characteristics and, hence, are useful
design tools. Finally, some time-domain calculations are carried out to give emphasis to nonlinear
effects and to show the limits of the frequency-domain analysis.

Keywords: Comfort; Motorcycle; Multi-body; Ride

1. Introduction

Actual roads are characterized by a random fluctuation of surface elevation about the nominal
geometry, which is called road unevenness. When a two-wheeled vehicle runs, road unevenness
forces the vertical displacements of both wheels generating vibrations, which are transmit-
ted through the suspensions to the frame, rider and passenger. Vibrations cause discomfort,
noise and, in the worst cases, the failure of mechanical components or electronic equipment.
Moreover, road unevenness causes variations in tyre load, and hence tyre adhesion may be
impaired.
The spectrum of the vehicle vibrations induced by road unevenness may be divided into three
ranges of frequencies: the quasi-static range (frequency ν < 0.5 Hz), the ride range (0.5 <
ν < 20 Hz) and the acoustic range (20 < ν < 20 000 Hz). The quasi-static range corresponds
to the passage through hills and slopes, and the frequency of excitation is low enough in

*Corresponding author. Tel.: +39 049 827 6803; Fax: +39 049 827 6785; Email: alberto.doria@unipd.it

Vehicle System Dynamics

ISSN 0042-3114 print/ISSN 1744-5159 online © 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/00423110500420712
 340 V. Cossalter et al.

relation to the natural frequencies of the vehicle to assume the motorcycle as a static system.
The ride range roughly corresponds to the passage on undulations of the road. In this band of
frequency, the modes of vibration of the vehicle are characterized by the rigid motion of the
front and rear frames, by the travel of the suspensions and by the deformation of the tyres. A
full dynamic model is required to study the vehicle behaviour in this frequency band. The ride
range is the most important from the point of view of comfort for the following reasons. The
human sensitivities to whole-body vibrations and to arm-hand vibrations peaks in the ranges
1–8 Hz [1] and 12–16 Hz, respectively [2]. In the ride range of frequencies, the spectrum
of vibrations generated by road unevenness, which is a monotonically decreasing function,
still shows relevant amplitudes. Finally, the acoustic range corresponds to the effect of road
roughness. The response of the motorcycle in this frequency range has to be calculated taking
into account the structural deformation of the front and rear frames, because the structural
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modes of vibration belong to this frequency band [3, 4]. The human body is less sensitive to
high frequency vibrations, but these vibrations generate noise that is perceived by the rider.
In two-wheeled vehicles, there are other sources of excitation: engine shaking forces, rotat-
ing imbalance and aerodynamic forces. The frequencies of engine shaking forces are in the
range 40–400 Hz, which belong to the acoustic frequency range. The forces generated by
rotating imbalance of the wheels belong to the ride range only if the speed is small and the
wheel radius is large. Finally, aerodynamic forces are important only if the speed of the vehicle
is high.
The comfort of vehicles can be studied numerically both in the time domain and in the
frequency domain. The time-domain approach [5] makes it possible to take into account
many nonlinear phenomena that are present in vehicle dynamics. They are related to the
vehicle’s kinematics, stiffness and damping characteristics of suspensions, presence of end-
stroke pads, tyre and aerodynamic forces. Actually, by means of a detailed vehicle model and
a time-domain analysis, the comfort can be evaluated in extreme conditions too; for example,
off-road journeys. If a nonlinear model is used, the superposition principle does not hold true
and the comfort of the vehicle in different roads and in different conditions can be evaluated
only by means of different simulations. This is the main limitation of the time-domain analysis,
as many cumbersome simulations have to be carried out to study vehicle comfort.
This inconvenience is not present in the frequency-domain approach, which is based on
linear models of the vehicle. The superposition principle makes it easy to evaluate the vehicle’s
behaviour in different conditions such as different kinds of roads or different speeds. The
frequency-domain approach gives the possibility of formulating comfort indices, as proposed
in this paper, but nonlinear phenomena are neglected.
This paper deals with the numerical study of motorcycle comfort in the ride range using the
frequency-domain method. The paper focusses on the excitation caused by road unevenness
and on the importance of suspensions, whereas other excitation sources (i.e. engine shak-
ing forces) are not considered. The analysis is carried out by means of a multibody model
of the vehicle which includes rigid bodies, suspensions and deformable tyres. At a given
speed, the steady-state configuration of the vehicle is evaluated using the full nonlinear model,
then the equations of motion are linearized and the analysis is carried out in the frequency
domain. Some additional simulations in the time domain are presented to assess the validity
and show the limitations of the proposed frequency-domain analysis. The frame compliance
is not included in the model because the ‘structural’ modes of vibration do not belong to the
ride range of frequencies [3] and are less excited than the ‘rigid’ modes.
The study of comfort becomes more complex owing to the wheelbase filtering [6, 7]. This
phenomenon takes place because, when a two-wheeled vehicle follows a path, both the wheels
pass over the same road unevenness but at different times. The excitation of the rear wheel is
the same as the front wheel, but has a time delay which is proportional to the wheelbase and
 Method for evaluating the ride comfort of a motorcycle 341

inversely proportional to the speed. The significance of the wheelbase filtering is discussed and
an original method for estimating the average behaviour of the vehicle for different speeds is
developed. Finally, indices for comfort evaluation are proposed and are used for investigating
the influence of the suspensions stiffness and damping on the ride comfort.

2. Motorcycle response to road unevenness

The human body is sensitive to accelerations. Studies dealing with vehicle’s comfort are based
on the analysis of the accelerations of the vehicle sprung mass when the vehicle runs at constant
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speed on an uneven road surface. In this paper, the motorcycle model described in [8] is used.
The vehicle was modelled with four rigid bodies (the frame with the rigidly attached rider,
the swinging arm and the wheels, see figure 1) and the nonlinear equations of motion were
derived. In the framework of this research, the out-of-plane degrees of freedom are not used.
For any given vehicle speed V, the steady-state conditions are calculated according to the
nonlinear equations of motion. From this configuration, the equations of motion are numer-
ically linearized. Because of the presence of the road excitation, the linearized equations of
motion become

Mq̈ + Cq̇ + Kq = C∗ ḟ + K∗ f, (1)

in which f = {ff , fr }T is the vector of road unevenness that excites the front and rear wheels,
q is a vector of independent coordinates, M, C and K are the mass, stiffness and damping
matrices and C∗ and K∗ are the superimposed motion damping and stiffness matrix.At constant
speed, all matrices of equation (1) are time-constant. If the drag force is neglected, the trim,
and therefore these matrices, do not depend on the vehicle speed.
The longitudinal and vertical displacement of the saddle (x, z) can be calculated as linear
combinations of coordinates q through matrix D:


x
y= = Dq. (2)
z

Figure 1. Two-wheeled vehicle model.

 342 V. Cossalter et al.

2.1 Frequency response functions

The horizontal and vertical accelerations can be obtained by transforming equations (1) and
(2) into the frequency domain and rearranging them as follows:

[−ω2 M + iωC + K]Q(ω) = [iωC∗ + K∗ ]F(ω), (3)

−1 ∗ ∗
Q(ω) = [−ω M + iωC + K] [iωC + K ]F(ω),
2
(4)
Y(ω) = DQ(ω), (5)

where F(ω), Q(ω) and Y(ω) correspond to f, q and y in the time domain. The frequency
response functions (FRFs) in terms of accelerations may be calculated as follows:
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Ÿ(ω) = −ω2 Y(ω) = H(ω)F(ω), (6)

in which H(ω) is the FRFs matrix of the system:

 
Hx,f (ω) Hx,r (ω)
H(ω) = = −ω2 D(−ω2 M + iωC + K)−1 (iωC∗ + K∗ ). (7)
Hz,f (ω) Hz,r (ω)

H(ω) has dimension 2 × 2 and the rows correspond to the outputs ẍ, z̈ (first index), whereas
the columns correspond to the inputs ff , fr (second index).
Throughout this paper, the numerical calculations are performed by means of the FastBike
code [8] considering a commercial scooter with 150 cc displacement. The trim of the scooter,
which is the geometric configuration that the motorcycle assumes in steady state condition
[9], was calculated assuming a constant forward speed of 20 m/s, then the equations of motion
were linearized and the FRFs matrix was calculated.
Plots (a) and (b) of figure 2 show the magnitudes and phases of the FRFs between the
horizontal acceleration and the front wheel displacement Hx,f (ω) and between the horizontal
acceleration and rear wheel displacement Hx,r (ω). With the front input, the magnitude of
the horizontal acceleration response regularly increases with the frequency and reaches a
maximum near the front hop mode natural frequency (the modal characteristics of the vehicle
are summarized in table 1), whereas with the rear input, the magnitude of the response suddenly
increases near the pitch mode natural frequency. In the whole range of frequencies, the phase
difference between the two FRFs is about π, this result agrees with the physical behaviour
that horizontal vibrations due to front and rear excitation have opposite directions.
Plot (c) of figure 2 shows the magnitude and phase of the FRF between the vertical accel-
eration of the saddle and the front wheel displacement Hz,f (ω). It highlights two peaks at 3.2
and 15.7 Hz, the first one is close to the natural frequencies of the bounce and pitch modes,
the second one is close to the frequency of the front hop mode.
Plot (d) shows the FRF between the vertical acceleration of the saddle and the rear wheel
displacement Hz,r (ω). This FRF has a larger magnitude at low frequency than the previous
one; in particular, the magnitude is large in the range of frequencies that includes the bounce,
pitch and rear hop modes. The phase diagrams of Hz,f (ω) and Hz,r (ω) are similar and show that
at low frequency the vehicle acceleration is in opposition with the contact point displacement,
whereas at higher frequency the vehicle acceleration is almost in phase with the contact point
displacement.
It is worth highlighting that, at frequency higher than wheel hop resonances, the magnitude
of acceleration response decreases as the frequency increases in the FRFs shown in figure 2.
In order to model the vibration of the vehicle when it runs at different speeds, the nonlinear
equations of motion should be linearized in every condition and the FRFs matrix should be
 Method for evaluating the ride comfort of a motorcycle 343
Downloaded by [University of Connecticut] at 18:07 10 October 2014

Figure 2. FRFs of the scooter.

recalculated because the variations in speed may cause variations in the trim. In this paper,
the FRFs matrix calculated at the mean speed is assumed to hold true for a range of speeds,
as the variations in the trim are negligible. Some calculations were carried out to determine
the range of speeds in which the frequency response matrix is approximately constant; these
results will be shown in section 3.5.

Table 1. Modal characteristics of the reference

vehicle.

Natural frequency
of the damped
Mode type system (Hz) Modal damping %

Bounce 2.74 30
Pitch 3.53 33
Rear hop 7.10 70
Front hop 13.78 39
 344 V. Cossalter et al.

2.2 Wheelbase filtering

Even if the motorcycle has two separate excitations, when it follows a path, both wheels pass
on the same road unevenness w(t) and the rear wheel input is the same as the front one with
a time delay, which is proportional to the wheelbase p and inversely proportional to the speed
V as follows:  

  w(t) 
ff (t) 
= , (8)
fr (t) w t − p 
V
or in the frequency domain:
 
1
F (ω) = −iω(p/V ) W (ω) = T(ω)W (ω).
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(9)
e

By substituting equation (9) into equation (6), one obtains

Ÿ(ω) = H(ω)T(ω)W (ω) = H∗ (ω, V )W (ω) (10)

Because the time delay correlates the rear input to the front one, the new functions H∗ (ω, V )
are called correlated FRFs.

∗

Hx Hx,f + Hx,r e−iω(p/V )
= . (11)
Hz∗ Hz,f + Hz,r e−iω(p/V )

This correlation between front and rear excitation is also called wheelbase filtering. In order
to understand the physical meaning of wheelbase filtering, it is useful to consider the genesis
of the correlated FRFs H∗ (ω, V ) by means of a Nyquist plot, i.e. the locus described in the
complex plane by a vector having the magnitude and phase angle of the FRF. The term e−iωp/V
in equation (11) has the meaning of a rotational operator in the complex plane; therefore, a
correlated FRF is the resultant of the vector due to the front excitation and the vector due to the
rear excitation rotated by angle −ωp/V , which is caused by the wheelbase effect. Figure 3
refers to the saddle vertical acceleration. Plot (a) is the Nyquist plots of Hz,f (ω), Hz,r (ω);
the complex vectors at a frequency ν = 13.2 Hz are also represented, the phase difference is
about π/2. Plot (b) shows the correlated FRF Hz∗ (ω, V ) which is given by the composition
of the previous two FRFs when the forward speed is 20 m/s. The phase difference between
the vectors corresponding to front and rear excitations is strongly modified by the wheelbase
effect: the two components that give Hz∗ (ω, V ) are almost in phase.

Figure 3. Saddle acceleration. Nyquist plot of the front and rear FRFs and of the correlated FRF.
 Method for evaluating the ride comfort of a motorcycle 345

The influence of the vehicle speed and excitation frequency on the wheelbase filtering may
be highlighted by calculating the magnitudes of the correlated FRFs as follows:
 pω
|Hx∗ (ω, V )| = |Hx,f (ω)|2 + |Hx,r (ω)|2 + 2|Hx,f (ω)||Hx,r (ω)| cos + ϕx (ω) ,
V
(12)
 pω
|Hz∗ (ω, V )| = |Hz,f (ω)|2 + |Hz,r (ω)|2 + 2|Hz,f (ω)||Hz,r (ω)| cos + ϕz (ω) , (13)
V
in which ϕx (ω) is the phase difference between Hx,f (ω) and Hx,r (ω) and ϕz (ω) is the
phase difference between Hz,f (ω) and Hz,r (ω). Equations (12) and (13) show that the response
reaches the maximum value when the argument of cosine is 2 kπ (k is an integer number) and
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reaches the minimum value when the argument is (2 k + 1)π .

It is interesting to analyse the behaviour of an ideal symmetric vehicle, in which the front
and rear excitations cause the opposite effect on the horizontal motion and the same effect on
the vertical motion, i.e.

Hx,f (ω) = −Hx,r (ω) = Hx (ω) =⇒ ϕx = π, (14)

Hz,f (ω) = Hz,r (ω) = Hz (ω) =⇒ ϕz = 0. (15)

By substituting the equations above in the expressions (12) and (13), one obtains
  pω 
|Hx∗ (ω, V )| = |Hx (ω)| 2 1 − cos , (16)
V
  pω 
|Hz∗ (ω, V )| = |Hz (ω)| 2 1 + cos . (17)
V
These equations highlight that the response of the vehicle is modulated with a constant
frequency ν = V /p and the maximums of horizontal acceleration correspond to the
minimums of vertical acceleration and vice versa.
These considerations are still valid for real vehicles. Indeed, figure 4 shows the magnitude
of the correlated FRFs of the longitudinal and vertical saddle acceleration for three different
values of the forward speed (5, 20 and 35 m/s). The wheelbase modulation is clear and in
agreement with the above considerations. In addition, it may be noticed that, as the number
of maximums and minimums decreases when the forward speed increases, the FRFs are more
regular at a high speed than at a low speed.

Figure 4. Magnitude of the correlated FRFs: (a) saddle horizontal acceleration, (b) saddle vertical acceleration.
 346 V. Cossalter et al.

2.3 Motorcycle response to road unevenness

The elevation of a road surface measured along its length is different for every road and
has random characteristics [10]. However, experimental tests show that random profiles of
different roads have common properties if they are represented by means of their power spectral
density (PSD) function, which represents the magnitude of the irregularities as a function of
their wavelength. These tests also showed that actual roads have a PSD that rapidly decreases
when the wavelength decreases [11, 12]. The ISO/TC 108 WG-9 standard [13] suggests to
represent the road PSD as a function of the wavenumber k = 2π/λ, which has the physical
meaning of the number of wavelengths λ contained in a distance of 2π . According to this
standard, the PSD of asphalted roads can be described by means of the following two gradients
function:
 n 
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k0 n = n1 k ≤ k0
Srr (k) = S0 (18)
k n = n2 k > k0

in which k0 = 1 is the cut-off wavenumber, n1 = 2, n2 = 1.5 are exponents and S0 is a

constant that depends on the quality of the road, see table 2.
If the speed of the vehicle is constant, the PSD in the wavenumber domain is transformed
into a new PSD in time domain by means of the following equation [14]:
 ω
 n n = n1 ≤ k0
S0 V k0
Srr (ω, V ) = ωV (19)
V ω n = n2 > k0 .
V

If forward speed V increases, road unevenness having a given range of wavelengths are
transformed into wider ranges of frequencies and for this reason the factor 1/V is present
in equation (19). At the same time, the magnitude of the excitation at a given frequency
increases with the speed because it corresponds to a shorter wavenumber and larger amplitude
unevenness. The second effect is dominant because n > 1 and Srr (ω, V ) increases as V
increases, see figure 5.

Table 2. Road quality constants according to ISO/TC 108 WG9.

Road class Very good Good Average Poor Very poor

S0 [m2 /(rad m−1 )] 4 × 10−6 16 × 10−6 64 × 10−6 256 × 10−6 1024 × 10−6

Figure 5. PSD of road profile. Average quality road.

 Method for evaluating the ride comfort of a motorcycle 347

Figure 6. PSD of saddle acceleration, average quality road: (a) saddle horizontal acceleration, (b) saddle vertical
Downloaded by [University of Connecticut] at 18:07 10 October 2014

acceleration.

Ride quality may be evaluated by looking at the PSD of vehicle acceleration. As the cor-
related FRFs have been calculated in the previous section, the PSDs of the horizontal and
vertical saddle acceleration can be calculated as follows [14]:
Sxx (ω, V ) = |Hx∗ (ω)|2 Srr (ω, V ), (20)
Szz (ω, V ) = |Hz∗ (ω)|2 Srr (ω, V ). (21)
Figure 6 shows the PSDs of the horizontal and vertical saddle acceleration of the scooter that
runs at different speeds on an average quality road. Some phenomena are evident. The PSD
of horizontal acceleration at low frequency (<5 Hz) is much smaller than the PSD of vertical
acceleration, because at low frequency the FRFs of the horizontal acceleration are smaller
than the ones of the vertical acceleration. The wheelbase filtering found in the FRFs is still
present in the PSDs of accelerations. As the frequency increases, the PSDs of accelerations
decreases because the road excitation strongly decreases. When the forward speed increases,
the PSDs of accelerations increases because road excitation is larger.

3. Evaluation of comfort

3.1 Averaging of the wheelbase filtering

The main purpose of this paper is the development of indices of comfort. Figure 6 shows that
when the vehicle moves along similar roads the PSDs of accelerations are functions of forward
speed. In order to define a comfort index, the statistic behaviour of a vehicle that performs a
typical mission with different speeds has to be considered.
The procedure that is presented herein differs from the one developed by Hunt [6] for
modelling traffic-induced ground vibration, because in the present analysis the wheelbase is
constant and the forward speed is variable.
The mission of the vehicle is described in terms of a probability density function p(V ),
which gives the probability of speed V taking a particular value during the journey. Then, the
mean values of PSDs of accelerations are calculated by integrating the product of the accel-
eration PSD times the probability density function. The following equations hold for saddle
vertical acceleration (the most important component). Similar equations hold for horizontal
acceleration
 Vmax  Vmax
S̃zz (ω) = Szz (ω, V )p(V )dV = |Hz∗ (ω, V )|2 Srr (ω, V )p(V )dV . (22)
Vmin Vmin
 348 V. Cossalter et al.

The particular structure of |Hz∗ (ω, V )|2 makes it possible to separate the variables and simplify
the calculations. Squaring the correlated FRF equation (13) yields the following:

|Hz∗ (ω, V )|2 = |Hz,f (ω)|2 + |Hz,r (ω)|2 + 2|Hz,f (ω)||Hz,r (ω)|
 pω pω 
× cos ϕz (ω) cos − sin ϕz (ω) sin . (23)
V V
If equation (23) is introduced into equation (22), the magnitudes and phases of the FRFs can
be moved outside the integral because they do not depend on the speed:

S̃zz (ω) = (|Hz,f (ω)|2 + |Hz,r (ω)|2 )wa (ω)

+ 2|Hz,f (ω)||Hz,r (ω)|[cos ϕz (ω)wr (ω) − sin ϕz (ω)wi (ω)], (24)
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with
 Vmax  pω
wr (ω) = Srr (ω, V ) cos p(V ) dV , (25)
Vmin V
 Vmax  pω
wi (ω) = Srr (ω, V ) sin p(V ) dV , (26)
Vmin V
 Vmax
wa (ω) = Srr (ω, V )p(V ) dV . (27)
Vmin

It is important to observe that wr (ω), wi (ω), wa (ω) are not related to any particular transfer
function of the vehicle, but they depend only on the mission. A similar expression holds true
for the horizontal acceleration

S̃xx (ω) = (|Hx,f (ω)|2 + |Hx,r (ω)|2 )wa (ω)

+ 2|Hx,f (ω)||Hx,r (ω)|[cos ϕx (ω)wr (ω) − sin ϕx (ω)wi (ω)] (28)

and for the acceleration of any other point which is relevant to comfort (e.g. a point on
the handle-bars). Therefore, the weight functions wr (ω), wi (ω), wa (ω) make it possible to
evaluate the statistic behaviour of the vehicle moving at different speeds.
It is worth to mention that the integration does not eliminate the wheelbase filtering, but
calculates a ‘mean’ wheelbase filtering. If the vehicle operates in the range of high speeds,
pω/V is about zero in the whole frequency range and the integration of equation (22) leads to

S̃zz (ω) ≈ (|Hz,f (ω)|2 + |Hz,r (ω)|2 + 2|Hz,f (ω)||Hz,r (ω)| cos ϕz (ω))wa (ω), (29)

because wr (ω) ≈ wa (ω) and wi (ω) ≈ 0. Equation (29) has the meaning of PSD of vertical
saddle acceleration neglecting the wheelbase filtering.

3.2 Mission profile

The probability density function p(V ) of the forward speed for a two-wheeled vehicle that
performs a typical mission has to be defined. The lower limit of the speed range is not zero,
because at very low speed, the vehicle tends to be unstable [9]. The upper limit depends on
the power of the engine.
The simplest, but not trivial, p(V ) is the uniform distribution: the probability of forward
speed taking every value of the range of speeds is equal. Figure 7 shows the mean PSD of the
horizontal and vertical acceleration considering an uniform distribution in the range 5–35 m/s.
 Method for evaluating the ride comfort of a motorcycle 349

Figure 7. Mean PSD of saddle acceleration, average quality road and uniform distribution: (a) horizontal
Downloaded by [University of Connecticut] at 18:07 10 October 2014

acceleration, (b) vertical acceleration.

The mean PSD of horizontal acceleration (solid line) shows a large maximum at ∼5 Hz. The
vertical acceleration shows a large maximum at low frequency (close to the frequencies of the
pitch and bounce modes) and a secondary maximum at ∼12 Hz. The shape of the curve and
the values are similar to the ones of vertical acceleration PSD at the average speed of 20 m/s
(figure 6), but the sequence of maximums and minimums typical of the wheelbase filtering is
less evident.
In figure 7, two limit curves are represented as well. They are the mean PSD of accelerations
that are calculated neglecting the wheelbase filtering (equation (29) for the vertical acceler-
ation). The differences between the mean curves in the presence of the wheelbase filtering
and the limit curves highlight that equation (24) does not eliminate the wheelbase filtering. In
particular, figure 7 shows that the wheelbase filtering decreases the vertical acceleration and
contributes to generate the horizontal acceleration.
The β probability density function [15] is well suited to define a mission profile of a vehicle,
because it can be defined on the interval [Vmin Vmax ] and the mean value can be moved within
the interval. The mean value and the variance of the β probability density function are defined
by setting the values of two parameters (a, b). Figure 8 shows the β probability density
function for different choices of a, b. It is worth highlighting that, if a = b, the probability
density function is symmetric, the mean coincides with the centre of the interval and the
variance decreases if a increases. If a = b = 1, the β probability density function coincides
with the uniform probability density function.
From a practical point of view, the missions with symmetric β probability density function
are well suited to study the comfort of a vehicle which runs principally at speeds that are close

Figure 8. β probability density function of forward speed with different exponents.

 350 V. Cossalter et al.

Figure 9. Mean PSD of saddle vertical acceleration, average quality road and some β distributions: (a) horizontal
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acceleration, (b) vertical acceleration.

to the mean of the range of possible speeds of the vehicle (mean speed mission). However,
some users of the same vehicle may run, on average, at lower speeds (urban mission) or at
higher speeds (main-road mission). In these cases, a non-symmetric β probability density
function is more suited to study the comfort.
Figure 9 deals with three missions: a mission with mean speed 20 m/s and symmetric
probability density function (a = b = 4); a main-road mission with mean speed 26 m/s
(a = 4.59, b = 1.97); an urban mission with mean speed 14 m/s (a = 1.97, b = 4.59). The
variances of the distributions for each mission are equal, the matrix of FRFs is calculated at
the mean speed. The symmetric β probability density function gives results similar to the ones
obtained with the uniform probability density function, but the undulations due to the wheel-
base filtering are more evident. The mean curves are fairly similar to the curves of acceleration
PSD calculated at the mean speed (20 m/s, figure 6).
The main-road mission, with large probability densities of high speeds, leads to an enlarge-
ment of the peak of the horizontal acceleration PSD and to an increase of the peak of the
vertical acceleration PSD. These effects take place because the road PSD at high speed is
large (figure 5). The urban mission, with large probability densities at low speeds, leads to a
reduction of main peak of both the vertical and the horizontal acceleration PSDs.

3.3 Human sensitivity to vibrations

Ride comfort is related to the acceleration perceived by the rider. Actually, the body behaves
like a filter. Its sensitivity is large in a narrow band of frequencies (1–2 Hz for back-to-chest and
side-to-side direction, 4–8 Hz for buttocks-to-head direction) and decreases sharply outside
this band [1].
International standards give curves to weigh the accelerations according to human sensi-
tivity. These curves refer to people in upright or seated position and may be considered a
first estimation of the sensitivity of the rider or passenger of a two-wheeled vehicle. In the
framework of this research, the weight curves proposed by ISO 2631 are used [1].
Figure 10 deals with a mission with speed range 5–35 m/s and symmetric β probability
density function (a = b = 4). Plot (a) makes a comparison between the curve of the mean
PSD of the horizontal acceleration of the saddle and the curve of the mean PSD of the weighted
horizontal acceleration. The latter is calculated multiplying the PSD of acceleration by the
back-to-chest weighting curve wx (ω) squared. There is a very large difference between the
two curves, because the weighting curve for the horizontal acceleration wx (ω) is unity at very
low frequency (<2 Hz) and then decreases sharply. Therefore, the horizontal accelerations
 Method for evaluating the ride comfort of a motorcycle 351

Figure 10. Effect of human sensitivity: (a) horizontal acceleration, (b) vertical acceleration.
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have a small influence on comfort. Plot (b) of figure 10 makes the same comparison for the
saddle vertical acceleration. The weighting curve wz (ω) slightly modifies the main peak of
the PSD of the vertical acceleration, which is close to the maximum sensitivity band (4–8 Hz)
and reduces high-frequency accelerations.

3.4 Definition of comfort indices

The magnitude of the acceleration transmitted to the suspended frame strongly depends on
the frequency. The comparison between different vehicles in a narrow band of frequencies is
not correct, because a vehicle may transmit a smaller acceleration than another vehicle in the
considered frequency band but may transmit larger accelerations in different bands.
A possible solution of this problem is the evaluation of the overall level of acceleration in
the whole frequency range, i.e. the definition of the following comfort index:

ωR
I= kx wx2 (ω) S̃xx (ω) + kz wz2 (ω) S̃zz (ω) dω. (30)
0

In which ωR is the upper limit of the ride range. kx = 1.4 and kz = 1 are the coefficient proposed
in ISO 2631 standard [1] for combining weighted accelerations in the two directions.
The proposed comfort index takes into account the vehicle dynamic behaviour, road quality
and mission profile and also has the physical meaning of the root-mean-square value of the
acceleration of the time history which corresponds to the mission.
Because a linear behaviour of the vehicle is assumed, the quality of the road influences the
accelerations only through the constant S0 (see equation (19)), which can be moved outside the
integrals. Therefore, a slightly different definition of the comfort index that does not depend
on the road quality is the following:
  ωR
1
I0 = kx wx2 (ω) S̃xx (ω) + kz wz2 (ω) S̃zz (ω) dω. (31)
S0 0

This index is useful to compare vehicle comfort characteristics regardless of the quality of the
road, but it loses its meaning as the overall level of accelerations.
The peaks of the PSD have a limited influence on the index based on the overall level, if
their bandwidth is narrow. Hence, a vehicle with a small overall level and relevant resonance
peaks may be perceived as very comfortable. In order to overcome this problem, quadratic
 352 V. Cossalter et al.

indices that are more sensitive to the PSD peaks may be defined using a second power of the
PSD

ωR  2  2
Q=4 kx wx2 (ω)S̃xx (ω) + kz wz2 (ω)S̃zz (ω) dω (32)
0
 
1 ωR  2  2
Q0 = 4 kx wx2 (ω)S̃xx (ω) + kz wz2 (ω)S̃zz (ω) dω. (33)
S02 0

A second power of acceleration PSD has the dimensions of a fourth power of acceleration and
it is worth to mention that this power of acceleration is currently used in the definition of the
root-mean-quad value of acceleration in the evaluation of vehicle discomfort [16].
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Table 3 shows the influence of the kind of mission on the comfort indices. The speed range
is always 5–35 m/s. The missions with uniform distribution and β symmetric distribution give
very similar indices. The non-symmetric β distribution (a = 4.59, b = 1.97) that corresponds
to a main-road mission shows the largest indices, whereas the non-symmetric β distribution
(a = 1.97, b = 4.59) that corresponds to a urban mission shows the smallest indices. The
quadratic index Q0 is more sensitive to the kind of mission, because figure 9 shows that the
most important difference between the uniform, urban and main-road mission is the height of
the main peak.

3.5 Nonlinear effects

The proposed method for the evaluation of the riding comfort is straightforward, but it is based
on the assumption of linear dynamic behaviour of the vehicle. In this section, the influence of
the most important nonlinear phenomena is examined.
The dependence of the trim of the vehicle on the aerodynamic drag force and, hence, on the
vehicle speed is a first nonlinear effect. This dependence implies that the linearized equations
of motion depend on the speed and therefore the FRFs matrix is not constant. Actually, the
variations in the FRF matrix have a negligible effect on the overall process. The numerical
results show very small differences in the comfort index (<1%) when the trim is calculated
considering the maximum (or minimum) speed instead of the mean speed.
A second, more important, nonlinear effect is due to the nonlinear dynamic behavior of
the motorcycle. The main source of nonlinearity is due to suspension friction and the shock
absorber behaviour, which may be relevant even for irregularities having small amplitude. For
higher amplitude road irregularities, there are additional non-linear effects due to the gross
variations in the geometry of the vehicle. Moreover, abrupt variations in the vehicle speed
influence the dynamics in the vertical plane. In order to estimate these effects, a time-domain
analysis was performed by means of the FastBike code [8]. The simulations were carried out
considering the scooter in straight motion on rough roads having the statistical properties of
table 2. The duration of the simulations was 60 s and the forward speed varied from 5 to 35 m/s
in order to obtain a β distribution of speed with a = b = 4. The time-domain histories of the

Table 3. Typical missions and comfort indices.

Mission I0 Q0

Uniform 315.0 16.6

Mean speed (β: a = 4, b = 4) 316.8 16.3
Main road (β: a = 4.59, b = 1.97) 345.4 19.0
Urban (β: a = 1.97, b = 4.59) 280.1 13.8
 Method for evaluating the ride comfort of a motorcycle 353

Figure 11. Comparison between linear and nonlinear analysis, mean speed mission: (a) horizontal acceleration,
(b) vertical acceleration.
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saddle vertical acceleration and pitch angular acceleration were elaborated by means of the
FFT algorithm to obtain the PSDs. No weighting curve was used.
Figure 11 shows the comparisons between the results obtained by means of nonlinear
analysis and those obtained with the method proposed in this paper. A logarithmic axis is used
to plot the results that refer to very good quality roads and average quality roads on the same
graph.
There is a good agreement between the two sets of results both in terms of numerical values
of PSD at a certain frequency and in terms of the shape of the curves. The linear analysis
slightly underestimates the main peak of the horizontal acceleration PSD and overestimates
the main peak of the vertical acceleration PSD. The nonlinear analysis, like the linear analysis,
predicts PSD values that are roughly proportional to the road quality; the curves that derive
from the nonlinear analysis show irregularities which may be related with the excitation of
nonlinear phenomena at certain frequencies.

3.6 Significance of suspension stiffness and damping on the ride comfort

It is well known that the suspensions’ characteristics have a large influence on the com-
fort. Therefore, if the comfort indices are properly defined, they must show a relevant
dependence on the stiffness and damping characteristics of the suspensions. To study this
important subject, some frequency-domain simulations were carried out considering paramet-
ric variations in the suspensions’ characteristics and calculating the comfort index (I0 ) that
corresponds to a main-road mission. This index takes into account both vertical and horizontal
vibrations.
Figure 12 deals with the numerical results. Plot (a) is the contour plot of I0 against the
front and rear values of damping. Because a linear model is used, the damping values are the
averages of the values in compression and extension. The front and rear suspension stiffnesses
are held at the reference values of 17650 and 16000 N/m, respectively. The comfort index
depends more on the front suspension damping than on the rear suspension damping and
reaches the minimum value when the front suspension damping is small. The black circle
shows the reference setting of damping values and the corresponding value of the comfort
index, which is close to the minimum value. Plot (b) of figure 12 is the contour plot of I0
against the front and rear suspension stiffnesses. The front and rear values of damping are
held at the reference values of 750 and 1350 Ns/m, respectively. The black circle shows the
reference setting of stiffnesses. The rear suspension stiffness is close to the optimum value,
whereas the front suspension stiffness has to be reduced to improve the comfort. Actually,
 354 V. Cossalter et al.
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Figure 12. Effect of suspensions’ characteristics on the comfort index: (a) effect of damping, (b) effect of stiffness.

the choice of suspensions’ characteristics has to take into account other aspects of motorcycle
behaviour (trim variations, handling properties) and the front suspension stiffness cannot be
decreased too much.

4. Conclusions

The frequency-domain method proposed in this paper makes use of four FRFs, which describe
the longitudinal and vertical response of the saddle to road excitation at the front and rear wheel.
The wheelbase filtering is taken into account by combining the complex FRFs due to front and
rear excitation. The PSD of saddle acceleration is then calculated assuming random properties
of the road profile.
The comfort is evaluated by introducing the concept of vehicle mission, which is a typical
journey with a forward speed that assumes different values according to a probability density
function.
The proposed method is very fast and makes it possible to define a numeric comfort index
that depends on the characteristics of the suspensions and can be used for optimization pur-
poses. With simple modifications, the method may take into account hand-arm vibrations, foot
vibrations and the vibrations transmitted to the passenger. The proposed method can be used
for elaborating the FRFs measured in laboratory by means of shakers and accelerometers, so it
is able to predict the comfort on the road starting from stationary tests. As the numerical code
also calculates the FRF between tyre loads and wheels’ displacements, the proposed method
makes possible the definition of adhesion indices based on the variations in tyre loads during
typical missions.
The main limit of the method coincides with the limit of linear analysis: the method is well
suited to study the comfort on asphalt roads, but cannot be used for studying the comfort in
off-road missions, which are characterized by large suspension travels.

Acknowledgement

This research was partially supported by funds from the University of Padova (2003 Research
Projects) and from the Ministry of Education, University and Research (PRIN 2004).
 Method for evaluating the ride comfort of a motorcycle 355

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