Investigation of inerter-based suspension systems for heavy vehicles

Abstract

The inerter is a two-terminal component that can be added to the spring-and-damper configuration of a suspension system. It has the property that the force exerted is proportional to the relative acceleration at its terminals. Studies have demonstrated the inerter's benefit of providing superior vibration isolation when it is used in the vehicle suspension of passenger cars. However, similar benefit on another common vehicle class on the roads, namely heavy vehicles, remain to be shown, as these vehicles have vastly different parameter values than passenger cars. This study is an investigation on the performance improvement brought by an inerter in the suspension of common heavy vehicles. In the study, the parameter values of a truck and a bus were adopted in the quarter vehicle model with two different spring-damper-inerter configurations (parallel and serial inerter), and the improvements in vibration isolation and road holding capability were determined by optimization of inertance. Results show that the inerter is similarly effective in providing the said improvements when implemented on heavy vehicles instead of on passenger cars, judging from reductions in sprung mass acceleration and dynamic tire load. It is also observed that the performance benefit is associated with larger optimum inertance than that for passenger cars. Overall, the inerter has been shown to be beneficial in the parallel and serial configurations, both of which are common and can be practically implemented in the suspension of heavy vehicles.

Full text

RESEARCH ARTICLE
Investigation of inerter-based suspension
systems for heavy vehicles
Ming Foong SoongID
☯
*, Rahizar Ramli
☯
, Ahmad Abdullah Saifizul
☯
, Kah Yin Goh
☯
, Su
Xian LongID
☯
Faculty of Engineering, Department of Mechanical Engineering, Universiti Malaya, Kuala Lumpur, Malaysia
☯These authors contributed equally to this work.
*mfsoong@um.edu.my
Abstract
The inerter is a two-terminal component that can be added to the spring-and-damper con-
figuration of a suspension system. It has the property that the force exerted is proportional
to the relative acceleration at its terminals. Studies have demonstrated the inerter’s benefit
of providing superior vibration isolation when it is used in the vehicle suspension of pas-
senger cars. However, similar benefit on another common vehicle class on the roads,
namely heavy vehicles, remain to be shown, as these vehicles have vastly different
parameter values than passenger cars. This study is an investigation on the performance
improvement brought by an inerter in the suspension of common heavy vehicles. In the
study, the parameter values of a truck and a bus were adopted in the quarter vehicle
model with two different spring-damper-inerter configurations (parallel and serial inerter),
and the improvements in vibration isolation and road holding capability were determined
by optimization of inertance. Results show that the inerter is similarly effective in providing
the said improvements when implemented on heavy vehicles instead of on passenger
cars, judging from reductions in sprung mass acceleration and dynamic tire load. It is also
observed that the performance benefit is associated with larger optimum inertance than
that for passenger cars. Overall, the inerter has been shown to be beneficial in the parallel
and serial configurations, both of which are common and can be practically implemented
in the suspension of heavy vehicles.
Introduction
The suspension system of a vehicle is a system of springs, dampers and linkages which connect
between the wheel and the vehicle body. When designed and tuned accordingly, a vehicle sus-
pension can serve the purposes of isolating the vehicle body from vibrations coming from the
ground due to road irregularities, as well as maintaining consistent contact between the wheel
and the road surface by minimizing the tire’s normal load variations [1]. The former is impor-
tant for ride comfort (passenger-carrying vehicles) or protection of goods (goods-carrying
vehicles), while the latter is important for tire’s road holding ability which indirectly relates to
vehicle handling and safety. Given the extensive usage of road vehicles in the transportation
scene, a well-performing vehicle suspension is especially important.
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OPEN ACCESS
Citation: Soong MF, Ramli R, Saifizul AA, Goh KY,
Long SX (2023) Investigation of inerter-based
suspension systems for heavy vehicles. PLoS ONE
18(1): e0280290. https://doi.org/10.1371/journal.
pone.0280290
Editor: Muhammad Usman, National University of
Sciences and Technology, PAKISTAN
Received: October 25, 2022
Accepted: December 23, 2022
Published: January 20, 2023
Copyright: ©2023 Soong et al. This is an open
access article distributed under the terms of the
Creative Commons Attribution License, which
permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All relevant data are
within the paper and its Supporting information
files.
Funding: This work is supported by Fundamental
Research Grant Scheme (FP086-2020), Ministry of
Higher Education of Malaysia (Recipient: Saifizul, A.
A.), and Faculty Research Grant Scheme
(GPF018A-2018), Universiti Malaya (Recipient:
Ramli, R.). The funders had no role in study
design, data collection and analysis, decision to
publish, or preparation of the manuscript.

Regardless of the complexity of a modern vehicle suspension system, it can currently be
generalized to having two major elements: the spring element and the damper element. The
former supports the static load of a vehicle and temporarily stores the undesirable energy due
to ground vibrations, while the latter dissipates this unwanted energy and literally dampen the
vehicle response. Presently, the design and tuning of a vehicle suspension is largely around
these two components. However, less known is that apart from spring and damper, there is
another two-terminal element known as the inerter [2] that can also join the line-up as a sus-
pension component. Fundamentally, the inerter is a two-terminal, mass-like inertial device
that can provide translational inertia by utilizing the rotational inertia of a flywheel and con-
verting it back to translational effect via motion conversion mechanisms, thereby responding
to the relative motion of the two terminals. With the addition of inerter, this forms the trio of
suspension elements: the spring reacts to relative displacement, the damper reacts to relative
velocity, while the inerter reacts to relative acceleration. More importantly, the addition of
inerter in a suspension stretches the conventional limit of suspension design and tuning,
because of the wider possibility a new element offers, for instance: the various suspension
layouts with inerter and the optimization of inertance which is the defining property of the
inerter.
The inerter has been shown to be applicable in various areas, some of which include train
suspension [3–5], building suspension [6–9], and vehicle suspension [10–16]. In all these
applications, benefits to vibration isolation have been reported. For example, in train suspen-
sion application, early studies showed that performance improvements and lateral stability
could be obtained by employing the inerter [3,4], while the focus was later shifted towards
using inerter-based suspension to improve both passenger comfort and track wear [5]. Mean-
while, in building application, the inerter was initially studied as a suspension to building
which demonstrated effective vibration suppression [6–8]. The implementation eventually
evolved to the tuned mass damper with inerter system as a better alternative to conventional
tuned mass damper [9]. In the context of vehicle suspension, it has been shown that the ride
and road holding performance can be enhanced by the addition of inerter to the original
spring-and-damper setup [10–16]. These studies demonstrate the promising incorporation of
inerter in many vehicle suspensions. However, one thing in common is that these concen-
trated solely on passenger cars, judging from the use of vehicle parameter values that are typi-
cal to this class of vehicles. What this means is that the other major category of vehicles,
commonly termed as heavy vehicles, have not been explored before with inerter. In general,
heavy vehicles consist of several classes of goods-carrying or passenger-carrying vehicles, such
as heavy trucks, single-unit-trucks, buses, etc. Although the suspension’s purposes of achieving
vibration isolation and road holding ability remain the same, heavy vehicles can have different
suspension requirements due to vastly different sprung and unsprung masses and the corre-
sponding different mass ratios. In the context of implementation of inerter which is still new
to heavy vehicles, these include suitable suspension layouts with inerter, the optimum iner-
tance values, and the corresponding ride and road holding performance brought by the addi-
tion of an inerter.
This study investigates the improvements in vibration isolation (ride) and road holding
ability brought by the incorporation of an inerter, specifically concerning heavy vehicles. In
the study, the suspension performance criteria were evaluated theoretically, considering two
heavy vehicle classes (trucks and buses) as well as common suspension layouts. Through the
analysis presented in the sections that follow, it was determined that the inerter, when incorpo-
rated in heavy vehicles, is similarly effective in achieving superior vibration isolation. This is
important as it translates to better passenger comfort (for buses) and prevention of goods dam-
age (for heavy trucks) in the transportation scene.
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Competing interests: The authors have declared
that no competing interests exist.

Inerter in vehicle suspension
In essence, a mechanical-based inerter is a physical device which achieves the mass-like effect
by exploiting the rotational inertia and converting it to two-terminal, translational inertia
effect by means of motion conversion mechanisms, such as ball-screw mechanism [17], rack-
and-pinion mechanism [17], hydraulic mechanism [18], etc. In a more recent realization, the
inertial effect of moving fluid in helical tube has also been adopted [19,20], which allows the
inerter to be controllable [20]. Another recently studied realization uses the crank mechanism
to achieve the intended inertial effect and also to provide a variable negative stiffness [21]. As a
passive element, working principle of an inerter follows the definition as in Eq (1), as first
stated in [2]:
Finerter ¼b a2a1
ð Þ ð1Þ
in which F
inerter
is the force at the terminals, bis the property of the inerter known as the iner-
tance with the unit of kilogram (kg), while a
1
and a
2
are the relative acceleration of the termi-
nals. In other words, an inerter has the property that the force exerted is directly proportional
to the relative acceleration at its terminals. From a microscopic point of view, each of the phys-
ical realizations of inerter device described earlier has its own detailed derivation in which the
inertance is a function of its design variables, and each also has some non-ideal factors apart
from the inertial effect, such as inherent elasticity, friction, and backlash as reported previously
[22], and inherent damping as well for the fluid inerter [19]. However, macroscopically, all
these still follows the ideal force-acceleration relationship, very much like the displacement-
responding behavior of a suspension spring and the velocity-responding behavior of a suspen-
sion damper. This is the fundamental inerter definition that is employed in this study, follow-
ing most of the previous studies concerning inerter in vehicle suspensions [10–14].
In the context of vehicle suspension, there can be many layouts of suspension components
that the addition of an inerter can form, many of which can be seen in [10] and [11]. However,
presently, the most common suspension layouts with inerter remain the simple parallel and
the simple serial inerter layouts. As the name implies, the former has the inerter added in par-
allel to the spring and damper elements to form a suspension, while the latter contains a seri-
ally arranged inerter to the usual vehicle suspension with spring and damper. Within the same
context of study, these layouts are often implemented in a two-degree-of-freedom (DOF)
quarter vehicle model for theoretical performance analysis and optimization [10–16]. Follow-
ing this, these layouts, together with the quarter vehicle model, are illustrated in Fig 1.
In which m
s
,m
u
are the sprung and unsprung masses, kis the suspension stiffness, cis the
suspension damping, bis the inertance of the added inerter, k
t
is the tire stiffness, and z
s
,z
u
,z
g
are vertical sprung mass displacement, unsprung mass displacement and the road input dis-
placement respectively. For serial inerter, m
n
represents the mass of the node that connects
between the damper and the inerter, while z
n
is the corresponding displacement.
The additional inerter component in Fig 1b and 1c expectedly changes the equations of
motion of a quarter vehicle model, albeit slightly. Therefore, when employed in studies involv-
ing vehicle suspension, the equation sets as below are applicable. For parallel inerter layout, the
dynamics of motion are detailed by Eqs (2) and (3):
msz
€
s¼k zuzs
ð Þ þ c_
zu_
zs
ð Þ þ b z
€
uz
€
s
  ð2Þ
muz
€
u¼ktzgzu
 k zuzs
ð Þ  c_
zu_
zs
ð Þ  b z
€
uz
€
s
  ð3Þ
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Meanwhile, for serial inerter layout, Eqs (4) to (6) are applicable:
msz
€
s¼k zuzs
ð Þ þ c_
zn_
zs
ð Þ ð4Þ
mnz
€
n¼b z
€
uz
€
n
 c_
zn_
zs
ð Þ ð5Þ
muz
€
u¼ktzgzu
 k zuzs
ð Þ  b z
€
uz
€
n
  ð6Þ
The aforementioned combinations of vehicle model and suspension layout are the common
setup as seen in various studies concerning inerter in vehicle suspension. In all these studies,
suspension performance improvements have been shown. For instance, in the early work of
[10] and [11], the optimization and analytical solution involving vehicle models with various
suspension layouts involving inerter gave improvements in ride comfort, tire load and suspen-
sion’s ability to carry load, for a wide range of suspension stiffness. Similarly, in another study
[12], the parallel inerter in a quarter vehicle model was subjected to alternative goal program-
ming optimization method, and superior passenger comfort and tire grip were obtained while
maintaining equal suspension deflection compared to conventional passive suspension with-
out inerter. Later, with the implementation of variable or switchable inerter in mathematical
vehicle models, further enhancement in suspension performance has also become achievable
[14]. In a recent study, an inerter-based mechatronic device, consisting of an inerter and an
electric motor with passive load, was used to achieve vehicle vibration suppression, and the
study demonstrated improvements in road holding without diverse effect on ride comfort and
suspension travel [15]. Meanwhile, another study showed the inerter being implemented with
an active suspension to improve ride comfort and reduce actuator force of the active suspen-
sion [16].
Fig 1. A quarter vehicle model with (a) typical spring and damper suspension, (b) parallel inerter suspension
layout, and (c) serial inerter suspension layout.
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While the above, among others, all lead to the consistent point that the inerter is capable of
giving superior performance when adopted in the suspension of passenger cars, presently
there has not been similar performance study that is specifically meant for heavy vehicles. As
mentioned earlier, the implementation of inerter is still new in heavy vehicles. Since heavy
vehicles differ vastly from passenger cars in terms of vehicle parameter values, it is worth to
consider a few heavy vehicle types and investigate the suspension performance due to the
implementation of inerter in these vehicles.
Vehicle modeling and setup of analysis
In line with earlier studies of inerter, the two-DOF quarter vehicle model was taken as the repre-
sentative system of vehicle in this study. This is a lumped-mass, rigid body mathematical model
that is commonly used in studies concerning fundamental suspension analyses, including inerter
and semi-active suspensions [14]. While maintaining simplicity, it can be used to obtain qualita-
tively correct sprung and unsprung mass responses [23] for comparative analyses, and adequate
accuracy are achievable without resorting to higher-DOF models [24]. For comprehensiveness,
the quarter vehicle parameter values of a truck and a bus were adopted, as both these heavy vehi-
cle classes were of interest in the study of inerter’s performance benefit. These are presented in
Table 1 together with those of a typical passenger car as reference [14,25,26]. By quick observa-
tion, it is worth noting that each parameter value of heavy vehicles is generally an order of magni-
tude greater than the corresponding parameter value of a passenger car.
The two vehicle models were constructed in computational software environment (Simu-
link
1
), following the dynamics of parallel inerter layout (Eqs (2) and (3)) and serial inerter lay-
out (Eqs (4) to (6)). Combined with the road profile modeling, the models can be simulated to
obtain the sprung and unsprung mass responses due to the presence of a vertical road displace-
ment input. As shown in Fig 2, two different profiles were chosen as the road input, namely
Table 1. The quarter vehicle parameter values of a truck and a bus with a typical passenger car as comparison.
Vehicle parameter Truck Bus Passenger car
Sprung mass, m
s
(kg) 3400 4000 317.5
Unsprung mass, m
u
(kg) 350 550 45.4
Suspension stiffness, k(Nm
-1
) 300000 200000 22000
Suspension damping, c(Nsm
-1
) 2000 / 20000 (compression / expansion) 10000 1500
Tire stiffness, k
t
(Nm
-1
) 1000000 1700000 192000
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Fig 2. Representation of (a) step road profile of 0.1 m height and (b) random smooth road profile in the study.
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the step road profile and the random road profile. The step road profile is a transient input
with a fixed step height that is also common in control system studies; it was adopted for a sim-
plified emulation of a vehicle hitting an obstacle, such as a curb or a bump. Meanwhile, the
random road profile is a realistically-generated profile based on the road roughness coefficient
from ISO8608:1995 (class A; smooth road classification) [27]; it was used in this study to emu-
late regular smooth-road driving, for example as seen in long-distance expressway transporta-
tion involving heavy vehicles. The use of both road inputs in the study ensures comprehensive
coverage of driving scenarios, since a road is regarded as a combination of isolated transient
road features and continuously distributed profile irregularities [23].
The suspension performance improvements due to inerter were investigated via two sepa-
rate approaches. In the first approach, the inerter was treated solely as an add-on suspension
component to the existing quarter vehicle models to emulate retrofitting a suspension, thus
having the suspension stiffness and damping maintained. This is applicable to both analyses
involving the parallel inerter layout and the serial inerter layout. Meanwhile, in the second
approach, both the inertance and the damping rate were subjected to optimization of suspen-
sion performance criteria to determine the performance benefit. This emulates suspension
tuning from scratch. It is also in accordance to past research which suggests combined consid-
eration of inerter and damping in vehicle suspension implementation [13]. For both
approaches mentioned above, a range of bfrom 0 kg to 1000 kg was tested for the parallel iner-
ter layout, while the range of 0 kg to 3000 kg was applicable to the serial inerter layout. These
are in line with similar past studies which considered inertance to sprung mass ratio of close to
one as the evaluation limit [12–14]. Meanwhile, the parallel inerter layout was assessed with a
smaller range due to small values of optimum inertance reported in past studies involving pas-
senger cars. These ranges remain generally realistic as large inertance values are achievable by
appropriate design values of the design function bwhile keeping reasonable actual device mass
[2]. Meanwhile, the damping took a range from 0 Nsm
-1
to 30000 Nsm
-1
in the optimization
of second approach of study. In general, all these ranges are wider than those adopted in simi-
lar past studies involving passenger cars, since the heavy vehicle parameter values are greater,
and by consistent scaling considering transfer function, the suitable inertance (and damping if
optimized together) is expectedly greater as well.
Suspension performance with inerter
From the computational results corresponding to the two quarter vehicle models with inerter,
the performance of heavy vehicle suspension with inerter was looked into, and any benefit of
incorporating inerter in the suspension of heavy vehicles would be known. In the study, two
suspension performance criteria were considered, namely the root-mean-squared (RMS)
sprung mass acceleration and the RMS dynamic tire load. Basically, the former describes the
effect on the sprung mass (vehicle body) due to road input and is a measure of vehicle ride
comfort, while the latter represents the variation of tire loads on the ground, which affects the
tire-ground contact and therefore is a measure of road holding ability. In both considerations
of implementation (inerter as add-on component and inerter with damping considered),
reductions of RMS sprung mass acceleration and RMS dynamic tire load represent respectively
the improvements in ride and road holding of the tested vehicle models.
Inerter as add-on component
Fig 3 displays the RMS sprung mass acceleration and RMS dynamic tire load with parallel
inertance value bfor the truck and bus models, respectively, under the step road input of 0.1
m. The responses of truck and bus models are of similar trend, for which the RMS sprung
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mass acceleration shows reduction while the RMS dynamic tire load increases with b. It can
be observed from Fig 3a and 3c that there are absolute minimum RMS sprung mass acceler-
ations of 1.296 ms
-2
and 1.883 ms
-2
when the bvalues are 145 kg and 140 kg, respectively,
for the truck and bus models. Meanwhile, there is no minimum RMS dynamic tire load
observed for both models. In fact, at the ride-optimized bvalues, the RMS dynamic tire
loads for the truck and bus models are greater at 8.070 kN and 14.474 kN, respectively, as
displayed in Fig 3b and 3d.
In the context of suspension performance, the add-on parallel inerter with step road input
of 0.1 m has resulted in reductions of 10.67% and 5.54% in RMS sprung mass acceleration,
respectively, for the truck and bus models. These are, in fact, greater improvements compared
to that achieved for some tested passenger car in earlier studies (vehicle parameter values as
stated in Table 1; about 2%) [13–14]. Therefore, it is worth noting that the implementation of
add-on parallel inerter is also effective on improving the ride performance of heavy vehicles in
addition to passenger cars. It is also interesting to note that both the truck and bus models
have optimum bvalues (145 kg and 140 kg) that are two orders of magnitude greater com-
pared to that of a typical passenger car [13]. This can be attributed to the consistent scaling-up
of vehicle parameter values for heavy vehicles relative to passenger cars, which is apparent in
Fig 3. (a) RMS sprung mass acceleration and (b) RMS dynamic tire loadagainst parallel inertance for truck,with
(c) RMS sprung mass acceleration and (d) RMS dynamic tire load variation for bus.
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Table 1. Meanwhile, in term of RMS dynamic tire load, the add-on parallel inerter has caused
increments of 7.53% and 8.92%, respectively, for the truck and bus models. The increments
are slightly greater than that in the relevant earlier studies concerning a passenger car (about
5%), but generally the level of increment in RMS dynamic tire load is almost the same, hence
the road holding ability is relatively maintained.
Table 2 summarizes the suspension performance of truck and bus models with add-on par-
allel and serial inerters under different road inputs. For the add-on parallel inerter with ran-
dom road input, both the truck and bus models perform similarly as in the step road input
situation, where there is an improvement of ride comfort at the expense of road holding ability.
For RMS sprung mass acceleration, the truck and bus models display reductions of 4.19% and
6.20%, respectively. Meanwhile, there are increments of RMS dynamic tire load of 7.11% and
9.88% for the truck and bus models, respectively.
In comparison of the effect of different road inputs, it can be observed that the add-on par-
allel inerter (truck model) achieves greater reduction in RMS sprung mass acceleration with
step road input compared to random road input. This implies that the ride comfort improve-
ment of the truck model is greater in transient scenarios such as road bumps. Conversely, the
ride improvements for the bus model, even though exist for the add-on parallel inerter, are
about the same between the two road inputs. From these comparisons, the performance bene-
fits due to the incorporation of inerter on vehicle suspensions are generally affected by the
combinations of vehicle parameter values that differ for different vehicle categories.
With step road profile as the road input, it is also possible to investigate the ride perfor-
mance from the perspective of transient response characteristics. In particular, reasonably
long rise time and peak time, as well as low percentage overshoot and short settling time, are
some of the characteristics of a vehicle with good ride comfort. Table 3 displays the transient
response characteristics of the truck and bus models with parallel and serial inerter layouts.
The results show that the rise time and peak time can be increased by 8.47% and 2.39%, respec-
tively, while the percentage overshoot and settling time can be reduced by 10.32% and 15.90%,
respectively, for the truck model with add-on parallel inerter. These indicate reasonably good
ride comfort improvement. Meanwhile, the bus model with add-on parallel inerter shows
5.88% increment in rise time, but just a slight increase in peak time. Additionally, the reduc-
tion in percentage overshoot of 2.47% is less than that of the truck model, while the settling
time is increased by 1.19%. Consistent with the earlier deduction that the performance benefits
are dependent on vehicle categories, it is quite obvious here that the truck model has better
ride improvement compared to the bus model.
Table 2. Summary and comparison of heavy vehicle suspension performance due to different road inputs.
RMS sprung mass acceleration (ms
-2
) RMS dynamic tire load (kN)
Truck Reference Optimum Difference (%) Reference Optimum Difference (%)
Step road input Parallel 1.4504 1.2957 -10.67 7.5045 8.0698 7.53
Serial 1.4504 2.9030 100.15 7.5045 12.0476 60.54
Random road input Parallel 0.1767 0.1693 -4.19 0.7695 0.8242 7.11
Serial 0.1767 0.1996 12.96 0.7695 0.9562 24.26
Bus Reference Optimum Difference (%) Reference Optimum Difference (%)
Step road input Parallel 1.9933 1.8828 -5.54 13.2644 14.4474 8.92
Serial 1.9933 2.4342 22.12 13.2644 15.0270 13.29
Random road input Parallel 0.1773 0.1663 -6.20 1.2573 1.3815 9.88
Serial 0.1773 0.1761 -0.68 1.2573 1.3220 5.15
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Conversely, the implementation of serial inerter as add-on component to the suspension of
heavy vehicle is much less effective relative to parallel inerter at the existing damping rate.
There is basically no absolute minimum for RMS sprung mass acceleration and RMS dynamic
tire load across the range of inertance values b. Consequently, in order to compare with the
add-on parallel inerter, the bvalues at which changes in both RMS values become small and
insignificant were chosen, namely 800 kg and 1000 kg for the truck and bus models, respec-
tively. The results show that the add-on serial inerter does not improve the responses due to
road inputs, for both the truck and bus models. From Table 2, increments in RMS sprung
mass acceleration and RMS dynamic tire load are observed, except for a very slight and rather
negligible reduction in RMS sprung mass acceleration corresponding to the random road
input. There is also little improvement in the transient response, as evident from Table 3.
Even though the serial inerter layout does not bring performance benefits when considered
as an add-on component to existing suspensions, it is important not to rule it out for further
investigation. Due to the sequential arrangement between suspension damper and inerter, it is
possible that the add-on approach to existing damping rate is not exactly optimum yet for the
serial inerter layout. In fact, it has been suggested before that the suspension damping cshould
be adjusted together with inertance bto harness more potential from inerter [13]. Thus, it will
be good if both band cvalues are considered in the optimization for ride and road holding
ability. The results are discussed in the next sub-section.
Optimized performance based on inertance and damping
In the last part of the study, both inertance band suspension damping cwere optimized
based on the same suspension performance criteria, namely the RMS sprung mass accelera-
tion, an indication of ride comfort, and the RMS dynamic tire load, an indication of road
holding ability. However, both criteria were analyzed together in the optimization. Admit-
tedly, the nature of ride comfort and road holding ability are known to be conflicting among
each other. Generally, an emphasis towards the former will lead to a compromise of the lat-
ter. Therefore, to optimize, or minimize, both RMS sprung mass acceleration and RMS
dynamic tire load, the Pareto optimization approach considering these two as optimization
objectives were adopted. The idea of Pareto optimization is that the solutions obtained are
non-dominated solutions, as none of these can minimize one objective without worsening
another. It follows that the set of optimum solutions are equally dominant with only different
emphasis on each objective. Using the specified ranges of inertance band damping cas
described in the vehicle modeling section, the optimum solutions considering the two
Table 3. Transient response characteristics of heavy vehicles with inerter due to step road input.
Reference Optimum (parallel) Difference (%) Optimum (serial) Difference (%)
Truck
Rise time (s) 0.177 0.192 8.47 0.149 -15.82
Peak time (s) 0.419 0.429 2.39 0.420 0.24
Maximum overshoot (%) 43.568 39.071 -10.32 92.589 112.52
Settling time (s) 3.888 3.270 -15.90 5.000 28.60
Bus
Rise time (s) 0.119 0.126 5.88 0.129 8.40
Peak time (s) 0.359 0.360 0.28 0.389 8.25
Maximum overshoot (%) 0.748 0.730 -2.47 0.836 11.68
Settling time (s) 4.281 4.332 1.19 5.000 16.80
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objectives are shown as Pareto fronts for the truck and bus models with parallel and serial
layouts of inerter. These are illustrated in Fig 4. They are also compared with the respective
optimum solutions for the reference case without inerter.
In a qualitative comparison, a case is superior to another if its Pareto front is towards the
direction of minimizing the objectives, relative to that of another. Fig 4a and 4b show that the
serial inerter layout for the truck model displays quite promising results, as it reduces the RMS
sprung mass acceleration without worsening the RMS dynamic tire load. Meanwhile, the par-
allel inerter layout shows a shift of Pareto front towards the left but upwards when compared
to the reference. This infers that it is possible to reduce the RMS sprung mass acceleration but
with a corresponding increase in RMS dynamic tire load, which is quite similar to the perfor-
mance of add-on parallel inerter in the previous sub-section. As an overall observation, when
both inertance band damping care allowed to vary, the suspension with serial inerter is supe-
rior to the suspension with parallel inerter which, in turn, is better than the reference suspen-
sion without inerter (the spring-and-damper configuration). This is applicable to step road
input and random road input. Additionally, Pareto fronts for the bus model also display the
same trend as that for the truck model. In general, the implementation of inerter in the suspen-
sion of heavy vehicles does bring better ride comfort while maintaining the road holding
ability.
Finally, to evaluate the suspension performance quantitatively, the middle point of the
Pareto front from each case is selected as the sample solution for comparison, as shown in
Table 4. Each point on the Pareto front corresponds to a set of RMS values which can be
mapped back to some combination of band cvalues. The approach of taking the middle point
from each set of solutions brings a balanced emphasis on the minimization of both objectives.
This allows a fair comparison of performance improvements among different layouts of iner-
ter, as well as different types of heavy vehicle.
The truck model with serial inerter displays the greatest reductions in the RMS sprung
mass acceleration, with 5.72% and 4.05% observed for the step and random road inputs,
respectively, without significant changes in the RMS dynamic tire load. In fact, the RMS
dynamic tire load is actually also reduced by 1.69% with the case involving step road input.
Fig 4. Pareto fronts for truck considering (a) step road input and (b) random road input, with the same
optimization for bus considering (c) step road input and (d) random road input.
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Meanwhile, the truck model with parallel inerter shows 2.45% and 3.22% reductions in the
RMS sprung mass acceleration, with 2.19% and 4.42% increments in RMS dynamic tire load,
respectively, for the step and random road inputs. Thus, the serial inerter layout does indeed
outperform the parallel inerter layout for the truck model, as the former is capable of achieving
better ride comfort and road holding ability. It is now apparent that when the suspension
damping is tuned together with the inertance, the benefit brought by a serial inerter can be
boosted further.
For the bus model, the serial inerter gives slightly more reduction of 4.62% in the RMS
sprung mass acceleration with the step road input, while showing less reduction with the ran-
dom road input (2.01%). The indication that the inerter gives better transient response, as first
pointed out in the previous sub-section, is again observed here. Meanwhile, the reduction of
RMS sprung mass acceleration for the parallel inerter is similar between the two road input
cases (approximately 3.2% reduction is achievable). Comparing by using the bus model, the
serial inerter layout is again superior to the parallel inerter layout as the former has less incre-
ment of the RMS dynamic tire load, hence less compromise to the road holding ability when
the ride is improved.
Conclusion
This study investigates the inerter-based suspension systems for heavy vehicles, and it shows
that the inerter brings superior performance compared to a suspension without inerter. Results
have shown that the inerter is capable of improving two major suspension performance crite-
ria, whether it is implemented in the parallel or serial layout. When the inerter is treated as an
add-on device, the parallel inerter layout improves the sprung mass acceleration at the expense
of dynamic tire load. Meanwhile, when both inertance and suspension damping are consid-
ered, the serial inerter demonstrated superiority with improvements in both sprung mass
acceleration and dynamic tire load, although this comes with narrower Pareto optimal design
points than that of the parallel inerter layout. Regardless of layouts, the level of performance
improvement for heavy vehicles is comparable to that for passenger cars with improvements
of up to 10%, but generally the required optimum inertance for heavy vehicles is two orders of
magnitude greater than that for passenger cars. Overall, this study has demonstrated that the
inerter has similar improvements in vibration isolation and road holding performance when
incorporated in heavy vehicles instead of in passenger cars where the improvements are
already known. The benefits that these translate to, namely better passenger comfort and pre-
vention of goods damage, together with the greater installation space and less critical penalty
Table 4. Summary and comparison of heavy vehicle suspension performance considering optimization of damping and inertance.
RMS sprung mass acceleration (ms
-2
) RMS dynamic tire load (kN)
Truck Reference Optimum Difference (%) Reference Optimum Difference (%)
Step road input Parallel 1.784 1.740 -2.45 7.1843 7.3414 2.19
Serial 1.784 1.682 -5.72 7.1843 7.0627 -1.69
Random road input Parallel 0.163 0.158 -3.22 0.6624 0.6917 4.42
Serial 0.163 0.157 -4.05 0.6624 0.6665 0.62
Bus Reference Optimum Difference (%) Reference Optimum Difference (%)
Step road input Parallel 1.896 1.835 -3.21 11.0058 11.5281 4.75
Serial 1.896 1.808 -4.62 11.0058 11.2114 1.87
Random road input Parallel 0.176 0.170 -3.15 1.0500 1.1030 5.04
Serial 0.176 0.172 -2.01 1.0500 1.0885 3.66
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of weight due to the installation of an additional device, make the mass-adoption of inerter in
heavy vehicles especially promising.
Supporting information
S1 File. Structure of quarter vehicle model with parallel inerter in Simulink
1
.
(PDF)
S2 File. Structure of quarter vehicle model with serial inerter in Simulink
1
.
(PDF)
Acknowledgments
The authors would like to express gratitude to Mr. Ng, B. H. and Dr. Sim, H. Y. for assisting in
this research.
Author Contributions
Conceptualization: Ming Foong Soong.
Formal analysis: Ming Foong Soong, Rahizar Ramli, Kah Yin Goh, Su Xian Long.
Funding acquisition: Rahizar Ramli, Ahmad Abdullah Saifizul.
Methodology: Ming Foong Soong, Rahizar Ramli, Ahmad Abdullah Saifizul, Kah Yin Goh, Su
Xian Long.
Resources: Rahizar Ramli, Ahmad Abdullah Saifizul.
Writing – original draft: Ming Foong Soong.
Writing – review & editing: Ming Foong Soong, Su Xian Long.
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