A kineto-dynamic model of a cross-linkage air seat-suspension system is formulated to obtain relations for effective vertical suspension stiffness and damping characteristics. A two-stage optimization methodology is proposed to derive vehicle-specific optimal designs considering different classes of earthmoving vehicles. The results show that optimal air spring coordinates can yield nearly constant natural frequency during the deflection cycle, irrespective of the seated body mass and driver-selected seated height. Vehicle-specific optimal damping characteristics, identified in the second stage, provided substantial reductions in seat effective amplitude transmissibility (SEAT) and vibration dose values (VDV) for all classes of earthmoving vehicles considered in the study. The proposed kineto-dynamic model and optimization method could thus serve as an important tool for designing vehicle-specific suspension seats

Subhash Rakheja

Wenbin Shangguan

Yijie Shui

English

Vibroengineering PROCEDIA

 304 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
Kineto-dynamic optimization of seat-suspension 
Yijie Shui1, Subhash Rakheja2, Wenbin Shangguan3 
1, 2, 3School of Mechanical and Automotive Engineering, South China University of Technology, 
Guangzhou, China 
2CONCAVE Research Center, Mechanical and Industrial Engineering, Concordia University, Montreal, 
Canada 
2Corresponding author 
E-mail: 1shuiyijie0530@163.com, 2subhash.rakheja@concordia.ca, 3sgwb@scut.edu.cn 
(Received 7 May 2016; accepted 16 May 2016) 
Abstract. A kineto-dynamic model of a cross-linkage air seat-suspension system is formulated to 
obtain relations for effective vertical suspension stiffness and damping characteristics. A 
two-stage optimization methodology is proposed to derive vehicle-specific optimal designs 
considering different classes of earthmoving vehicles. The results show that optimal air spring 
coordinates can yield nearly constant natural frequency during the deflection cycle, irrespective 
of the seated body mass and driver-selected seated height. Vehicle-specific optimal damping 
characteristics, identified in the second stage, provided substantial reductions in seat effective 
amplitude transmissibility (SEAT) and vibration dose values (VDV) for all classes of earthmoving 
vehicles considered in the study. The proposed kineto-dynamic model and optimization method 
could thus serve as an important tool for designing vehicle-specific suspension seats. 
Keywords: kineto-dynamic seat-suspension, optimization method, vehicle-specific design. 
1. Introduction 
Drivers of off-road vehicles are occupationally exposed to comprehensive magnitudes of low 
frequency whole-body vibration (WBV) and intermittent shocks. Apart from discomfort, fatigue 
and poor performance rate, WBV is associated with greater risks of low back pain and 
degenerative changes in spine among exposed drivers [1]. Such vehicles generally employ a 
seat-suspension to limit transmission of WBV to the seated driver. Vibration attenuation 
performance of seat-suspensions is strongly dependent upon magnitude and frequency contents of 
vehicle vibration, which may be characterized in three categories [2]: (i) suspension lock-up under 
low levels of vehicle vibration due to friction; (ii) attenuation or amplification under medium to 
high levels of continuous vibration leading to suspension travel within the permissible free travel 
depending upon the frequency contents of vehicle vibration and suspension design; and 
(iii) amplification of vibration and shock motions when suspension travel exceeds its free travel 
leading to impacts against elastic end-stops. Suspension designs within the last two categories 
pose conflicting requirements, particularly for the suspension damping. 
The design of suspension seats also involves additional challenges associated with varying 
body mass and seated height. Variations in body mass may affect suspension natural frequency 
and thereby the vibration isolation performance [3]. A seat-suspension yields best performance 
when adjusted to mid-ride position so as permit maximum suspension travel in compression and 
rebound. Effective suspension stiffness, especially for air suspensions, and the permissible 
suspension travel, however, are affected by driver-selected seat height, which may cause impacts 
with motion limiting stops. A number of studies have identified optimal designs to address 
conflicting requirements for shock and vibration attenuation [4]. A number of semi-active and 
active seat-suspensions have also been proposed for achieving improved performance under 
varying excitations and body mass [5, 6].  
The aforementioned studies on passive, semi-active and active seat-suspension systems have 
invariably employed equivalent vertical suspension stiffness and damping, while neglecting 
contributions due to suspension kinematics. The vast majority of the suspensions employ a 
cross-linkage mechanism with rollers to ensure pure vertical motion of the seat. The orientations 
of the air/mechanical spring and damper, generally attached to the cross links, thus vary 
KINETO-DYNAMIC OPTIMIZATION OF SEAT-SUSPENSION.  
YIJIE SHUI, SUBHASH RAKHEJA, WENBIN SHANGGUAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 305 
considerably during a vibration cycle, which lead to nonlinear variations in effective stiffness and 
damping with the nature of base vibration. Furthermore, the effects of motion limiting stops within 
the suspension, variations in the body mass as well suspension height, and friction due to rollers 
are generally, which are known to strongly alter the suspension performance.  
In this paper, a kineto-dynamic model of a seat-suspension system comprising an air spring 
and a hydraulic damper within a cross-linkage mechanism is formulated. The model is 
subsequently used to identify optimal coordinates of the air spring attachments and optimal 
damping requirements for vehicle-specific suspension seat designs. 
2. Kineto-dynamic model formulations  
Fig. 1(a) illustrates planar representation of a seat-suspension-occupant system comprising an 
air spring and a hydraulic damper with two pairs of cross links of length ܮ, AC and BD. Links AC 
and BD are pinned to seat pan and base at A and D, respectively, and supported on guiding rollers 
at B and C, constrained along the horizontal direction. The two cross links are also attached at O 
via a pin joint (AO = ݈ଵ; OC = ݈ଶ). Suspension travel in extension and compression is limited by 
two elastic end-stops, I and J, respectively. The air-spring is attached to the cross links via arm 
OS (݈ହ), while the hydraulic damper is mounted between F and E via arms CF (݈ଷ) and OE (݈ସ), 
respectively, attached to the cross-links. Table 1 summarizes the geometric and suspension 
parameters shown in Fig. 1, and ܪ଴  is mid-ride height. Damper is modeled using two-stage 
properties, where ܥ௖ଵ and ܥ௘ଵ are low-speed damping constants in compression and rebound, and 
ܥ௖ଶ and ܥ௘ଶ are respective high-speed coefficients, while transitions between low- and high-speed 
damping occur at ௖ܸ and ௘ܸ. Nominal suspension considered in the study employed single-stage 
damping in compression [2]. The cushion is modeled using equivalent linear stiffness ݇௖  and 
damping ܿ௖. The force developed by air spring along its axis ܨ௦ is derived from the air pressure 
assuming polytropic gas process, such that: ܨ௦ = ଴ܲ ଴ܸ௡ܣ௘ ( ଴ܸ + ܣ௘ߜ௦)௡⁄ , where ଴ܲ  and ଴ܸ  are 
pressure volume of air at equilibrium position, ܣ௘ is effective area, ߜ௦ is spring deflection along 
its axis and ݊ is polytropic constant. 




6l
h
7l
 
a) 
ck cc
bm
0z
seF deF
2z
1z
eeF
sm
 
b) 
Fig. 1. a) Kineto dynamic and b) equivalent vertical dynamic models of seat-suspension-occupant system 
Suspension seats are invariably modeled using equivalent vertical properties, as shown in 
Fig. 1(b) [2]. Kinematics of the cross-links together with inclinations of air spring and damper, 
however, yield highly nonlinear variations in vertical spring ܨ௦௘ and damping ܨௗ௘ forces during a 
deflection cycle. In the model, the seated body is represented by a rigid mass ݉௕ , since the 
contributions of seated body biodynamics are known to be small for low natural frequency 
suspensions. The seat pan is denoted by a rigid mass ݉௦, while end-stops are modeled as clearance 
springs with effective vertical end-stops forces as ܨ௘௘. The equations of motion of the two-DOF 
seat-occupant model, incorporating suspension kinematics, can be expressed as: 
݉௦ݖሷଵ + ܨ௦ + ܨௗ௘ − ܨ௖௨ + ܨ௘௘ = 0, ݉௕ݖሷଶ + ܨ௖௨ = 0, (1)
KINETO-DYNAMIC OPTIMIZATION OF SEAT-SUSPENSION.  
YIJIE SHUI, SUBHASH RAKHEJA, WENBIN SHANGGUAN 
306 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
where ݖଵ and ݖଶ are displacements of suspension and the seated body masses, respectively, and 
ܨ௖௨ = ݇௖(ݖଶ − ݖଵ) + ܿ௖(ݖሶଶ − ݖሶଵ) is force developed by the cushion. The effective vertical forces 
due to air spring force ܨ௦௘ , hydraulic damper force ܨௗ௘  and end-stops force ܨ௘௘  are derived 
considering the suspension kinematics, as: 
ܨ௦௘ =
ሾ݈ଶ sin(߮ − ߙ) + ݈ହ sin(߮ + ߟ − ߙ)ሿ(݂ sin ߙ − cos ߙ)
(݈ଵ − ݈ଶ)݂ sin ߙ cos ߙ − (݈ଵ + ݈ଶ)cosଶߙ ܨ௦, (2)
ܨௗ௘ =
൬ ሾ݈ଶ sin(ߠ − ߙ) − ݈ଷ sin(ߠ + ߚ − ߙ)ሿ ×(݂sinߙ − cosߙ) − ݈ସsin(ߠ + ߛ − ߙ)(݂sinߙ + cosߙ)൰
(݈ଵ − ݈ଶ)݂sinߙcosߙ − (݈ଵ + ݈ଶ)cosଶߙ ܨௗ
(3)
ܨ௘௘ =
݈ଶ sin ߙ
(݈ଵ − ݈ଶ)݂ sin ߙ cos ߙ − (݈ଵ + ݈ଶ)cosଶߙ ܨ௘௡ௗ, (4)
where ܨௗ is force developed by damper along its axis and ܨ௘௡ௗ is horizontal force due to end-stop 
impacts. Above equations are solved to obtain vibration and shock isolation of the suspension 
under given excitation and body mass. The vibration isolation performance is evaluated in terms 
of seat effective amplitude transmissibility (SEAT), defined as ratio of overall frequency-weighted 
rms acceleration at the seat-occupant interface to that at the seat base [3]. The shock isolation 
performance is evaluated in terms of vibration dose value (VDV) ratio, ratio of VDV of seat mass 
vibration to that of the base vibration, while VDV is computed from: 
ܸܦ ௜ܸ = ቈන ݖ௜௪ସ
்
଴
 ݀ݐ቉
଴.ଶହ
, ݅ = 0, 2, (5)
where ݖሷ௜௪(ݐ) is frequency-weighted acceleration of the seat mass/base, which is obtained upon 
using ௞ܹ frequency-weighting defined in ISO-2631/1 and ܶ is the simulation period. 
Table 1. Parameters of the seat-suspension used for model validation 
Parameter Value Parameter Value Parameter Value 
݈ଵ, m 0.1792 ݈ଶ, m 0.1792 ߛ, deg 31.02 
݈ଷ, m 0.0324 ݈ସ, m  0.0554 ߚ, deg 41.59 
݈ହ, m 0.0472 ݈଺, m 0.135 Travel, mm 140 
݈଻, m 0.2089 ܪ଴ (mid-ride), m 0.160 ߟ, deg  79.88 
݇௖, N/mm 29.977 ܿ௖, Ns/mm 0.938 ݉௦, kg 5 
݉௕, kg 77 ଴ܲ, kPa 689.47 ଴ܸ, cm3 276.87 
ܣ௘, cm2 28 ݊ 1.38 ௘ܸ, m/s 0.11563 
ܥ௘ଵ, Ns/mm 4.83 ܥ௘ଶ, Ns/mm 6.13 ܥ௖ଵ = ܥ௖ଶ, Ns/mm 2.45 
3. Vehicle-specific seat-suspension design optimization  
A two-stage optimization problem is formulated to seek vehicle-specific optimal designs 
considering suspension kinematics. In the first-stage, optimal spring coordinates are identified to 
obtain constant stiffness over the suspension stroke, and constant natural frequency for the range 
of body masses and suspension heights. The minimization problem is formulated to minimize 
maximum difference in effective stiffness ܭ௘ over the entire suspension travel, such that: 
ܷ(߯) = minimizeሼmaxሾܭ௘ሿ − minሾܭ௘ሿ ሽ, (6)
where ߯ = ሼ݈ହ, ݈଺, ߟሽ is the design vector related to air spring coordinates (Fig. 1). The above is 
solved subject to limit constraints: 0 < ݈ହ , ݈଺ < 0.1792  and 0 < ߟ < ߨ . Moreover, natural 
frequency (excluding the cushion) is permitted to vary in the 1.25 to 1.35 Hz range. 
In the second stage, optimal suspension damping is identified to realize an improved 
KINETO-DYNAMIC OPTIMIZATION OF SEAT-SUSPENSION.  
YIJIE SHUI, SUBHASH RAKHEJA, WENBIN SHANGGUAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533 307 
compromise between SEAT and VDV ratios. It is evident that effective vertical mode damping is 
strongly dependent upon coordinates of the damper attachments, including damper orientations 
(ߛ, ߚ), and lengths of arms OE (݈ସ) and CF (݈ଷ), apart from the low- and high-speed damping 
coefficients. A methodology is thus formulated to identify vehicle-specific optimal suspension 
damping under vibration spectra of different earthmoving vehicles (EM1, EM4, EM6, EM9) 
superimposed with a filtered shock pulse [7]. The minimization problem is formulated to reduce 
sum of SEAT and VDV ratio under each vehicular excitation, such that: 
ܷ(߯) = minimize(ߛଵܵܧܣܶ + ߛଶܸܦܸݎܽݐ݅݋), (7)
where ߯ = ሼ݈ଷ, ݈ସ, ߚ, ߛ, ܥ௘ଵ, ܥ௖ଵ, ܥ௘ଶ, ܥ௖ଶ, ௘ܸ, ௖ܸሽ is the design vector, and ߛଵ  and ߛଶ  are weighting 
constants, which are considered identical so as to equally emphasize the vibration and shock 
isolation. The above minimization problem is solved subject to constraints: ߯ > 0  with the 
exception of rebound transition velocity, limited to ௘ܸ > –0.06 m/s; and ௖ܸ < 0.06 m/s, while the 
static height of the suspension as taken at the mid-ride level, and the seated body mass is limited 
to that 50th percentile subject. Peak suspension deflection is further limited to free travel so as 
minimize the occurrence of end-stop impacts, such that: max(|ݖ௦|) < 70 mm. 
4. Results and discussions  
The optimal coordinates of air spring attachments were obtained as: ݈ହ = 0.01 m, ݈଺ = 0.05 m 
and ߟ = 1.46 rad. Fig. 2(a) compares effective stiffness variations of optimal suspension with that 
of the nominal design. It is evident that the identified optimal spring coordinates can yield nearly 
constant effective stiffness over the entire suspension travel. The stiffness variations were further 
obtained for three different seat heights (±10 mm and ±20 mm from mid-position), and natural 
frequencies were estimated for three different seat loads (41.25, 56.25 and 67.5 kg), representing 
75 % of 5th, 50th and 95th percentile body masses, while neglecting the effect of cushion. The 
results showed nearly constant natural frequency near 1.33 Hz, irrespective of the seat mass and 
the suspension height. 
 
a) 
 
b) 
Fig. 2. a) Variations in effective vertical stiffness of optimal and nominal suspensions;  
b) relative displacement response of the optimal and nominal suspension  
under EM1 excitation. (····Permissible suspension travel) 
Table 2 summarizes optimal suspension damping parameters and coordinates of damper 
mounts for selected classes of vehicle vibrations superimposed with filtered shock pulses. Table 
also presents vibration and shock isolation performance of the optimal and nominal suspensions 
under selected excitations in terms of SEAT and VDV ratios. Results show substantially lower 
SEAT and VDV ratios of the optimal suspension compared to the nominal suspension. The VDV 
ratios are greater than SEAT values in all cases, which is due to presence of shock motion. As an 
example, Fig. 2(b) illustrates relative suspension deflection response under excitation (EM1). 
KINETO-DYNAMIC OPTIMIZATION OF SEAT-SUSPENSION.  
YIJIE SHUI, SUBHASH RAKHEJA, WENBIN SHANGGUAN 
308 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. DEC 2016, VOL. 10. ISSN 2345-0533  
While the nominal suspension travel exceeds permissible free travel, the optimal suspension limits 
the deflection within the free travel and thereby eliminates end-stop impacts. 
Table 2. Optimal vehicle-specific optimal suspension damper parameters 
Parameter 
Optimal parameters for specific vehicle class excitation 
superimposed with filtered shock pulse 
EM1 EM4 EM6 EM9 
݈ଷ, mm 0.024 0.012 0.064 0.023 
݈ସ, mm 0.025 0.024 0.016 0.046 
ߚଶ, rad 0.66 0.674 0.815 0.258 
ߛଶ, rad 0.789 0.71 0.142 0.255 
ܥ௘ଵ, Ns/m 9.01 10.01 2.32 5.78 
ܥ௖ଵ, Ns/m 4.23 2.3 1.54 4.21 
ܥ௘ଶ, Ns/m 2.19 6.5 3.39 3.63 
ܥ௖ଶ, Ns/m 1.23 3.4 3.23 2.15 
௘ܸ, m/s 0.05 0.033 0.02 0.01 
௖ܸ, m/s 0.055 0.035 0.015 0.015 
SEAT 0.46 0.56 0.30 0.18 
VDV ratio 0.62 0.72 0.51 0.32 
 Nominal suspension 
SEAT 1.20 1.23 0.33 0.43 
VDV ratio 1.31 1.50 0.56 0.55 
5. Conclusions 
Kinematics of the cross-linkage contributes to considerable variations in effective vertical 
spring rate and damping characteristics of the seat-suspension system during a deflection cycle. 
Variations in driver-selected seat height also yield considerable changes in suspension stiffness 
and thus the natural frequency. Optimal coordinates of air spring mounting could yield nearly 
constant stiffness and natural frequency over the entire suspension travel, irrespective of 
suspension height and seated body mass. A good compromise between the shock and vibration 
isolation performance can be achieved via optimal vehicle vibration-specific damping 
characteristics and its coordinates/orientation. Such a vehicle-specific optimal design could yield 
substantial reductions in SEAT and VDV ratio responses.  
References 
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[3] Ma X. Q., Rakheja S., Su C.-Y. Damping requirement of a suspension seat subject to low frequency vehicle 
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[4] Duke M., Goss G. Investigation of tractor driver seat performance with non-linear stiffness and on-off 
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Kineto-dynamic optimization of seat-suspension

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