Abstract
A kinetodynamic model of a crosslinkage air seatsuspension system is formulated to obtain relations for effective vertical suspension stiffness and damping characteristics. A twostage optimization methodology is proposed to derive vehiclespecific 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 driverselected seated height. Vehiclespecific 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 kinetodynamic model and optimization method could thus serve as an important tool for designing vehiclespecific suspension seats.
1. Introduction
Drivers of offroad vehicles are occupationally exposed to comprehensive magnitudes of low frequency wholebody 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 seatsuspension to limit transmission of WBV to the seated driver. Vibration attenuation performance of seatsuspensions is strongly dependent upon magnitude and frequency contents of vehicle vibration, which may be characterized in three categories [2]: (i) suspension lockup 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 endstops. 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 seatsuspension yields best performance when adjusted to midride 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 driverselected 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 semiactive and active seatsuspensions have also been proposed for achieving improved performance under varying excitations and body mass [5, 6].
The aforementioned studies on passive, semiactive and active seatsuspension 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 crosslinkage 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 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 kinetodynamic model of a seatsuspension system comprising an air spring and a hydraulic damper within a crosslinkage mechanism is formulated. The model is subsequently used to identify optimal coordinates of the air spring attachments and optimal damping requirements for vehiclespecific suspension seat designs.
2. Kinetodynamic model formulations
Fig. 1(a) illustrates planar representation of a seatsuspensionoccupant system comprising an air spring and a hydraulic damper with two pairs of cross links of length $L$, 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 $={l}_{1}$; OC $={l}_{2}$). Suspension travel in extension and compression is limited by two elastic endstops, I and J, respectively. The airspring is attached to the cross links via arm OS (${l}_{5}$), while the hydraulic damper is mounted between F and E via arms CF (${l}_{3}$) and OE (${l}_{4}$), respectively, attached to the crosslinks. Table 1 summarizes the geometric and suspension parameters shown in Fig. 1, and ${H}_{0}$ is midride height. Damper is modeled using twostage properties, where ${C}_{c1}$ and ${C}_{e1}$ are lowspeed damping constants in compression and rebound, and ${C}_{c2}$ and ${C}_{e2}$ are respective highspeed coefficients, while transitions between low and highspeed damping occur at ${V}_{c}$ and ${V}_{e}$. Nominal suspension considered in the study employed singlestage damping in compression [2]. The cushion is modeled using equivalent linear stiffness ${k}_{c}$ and damping ${c}_{c}$. The force developed by air spring along its axis ${F}_{s}$ is derived from the air pressure assuming polytropic gas process, such that: ${F}_{s}={P}_{0}{V}_{0}^{n}{A}_{e}/{\left({V}_{0}+{A}_{e}{\delta}_{s}\right)}^{n}$, where ${P}_{0}$ and ${V}_{0}$ are pressure volume of air at equilibrium position, ${A}_{e}$ is effective area, ${\delta}_{s}$ is spring deflection along its axis and $n$ is polytropic constant.
Fig. 1a) Kineto dynamic and b) equivalent vertical dynamic models of seatsuspensionoccupant system
a)
b)
Suspension seats are invariably modeled using equivalent vertical properties, as shown in Fig. 1(b) [2]. Kinematics of the crosslinks together with inclinations of air spring and damper, however, yield highly nonlinear variations in vertical spring ${F}_{se}$ and damping ${F}_{de}$ forces during a deflection cycle. In the model, the seated body is represented by a rigid mass ${m}_{b}$, 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 ${m}_{s}$, while endstops are modeled as clearance springs with effective vertical endstops forces as ${F}_{ee}$. The equations of motion of the twoDOF seatoccupant model, incorporating suspension kinematics, can be expressed as:
where ${z}_{1}$ and ${z}_{2}$ are displacements of suspension and the seated body masses, respectively, and ${F}_{cu}={k}_{c}\left({z}_{2}{z}_{1}\right)+{c}_{c}\left({\dot{z}}_{2}{\dot{z}}_{1}\right)$ is force developed by the cushion. The effective vertical forces due to air spring force ${F}_{se}$, hydraulic damper force ${F}_{de}$ and endstops force ${F}_{ee}$ are derived considering the suspension kinematics, as:
where ${F}_{d}$ is force developed by damper along its axis and ${F}_{end}$ is horizontal force due to endstop 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 frequencyweighted rms acceleration at the seatoccupant 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:
where ${\ddot{z}}_{iw}\left(t\right)$ is frequencyweighted acceleration of the seat mass/base, which is obtained upon using ${W}_{k}$ frequencyweighting defined in ISO2631/1 and $T$ is the simulation period.
Table 1Parameters of the seatsuspension used for model validation
Parameter  Value  Parameter  Value  Parameter  Value 
${l}_{1}$, m  0.1792  ${l}_{2}$, m  0.1792  $\gamma $, deg  31.02 
${l}_{3}$, m  0.0324  ${l}_{4}$, m  0.0554  $\beta $, deg  41.59 
${l}_{5}$, m  0.0472  ${l}_{6}$, m  0.135  Travel, mm  140 
${l}_{7}$, m  0.2089  ${H}_{0}$ (midride), m  0.160  $\eta $, deg  79.88 
${k}_{c}$, N/mm  29.977  ${c}_{c}$, Ns/mm  0.938  ${m}_{s}$, kg  5 
${m}_{b}$, kg  77  ${P}_{0}$, kPa  689.47  ${V}_{0}$, cm^{3}  276.87 
${A}_{e}$, cm^{2}  28  $n$  1.38  ${V}_{e}$, m/s  0.11563 
${C}_{e1}$, Ns/mm  4.83  ${C}_{e2}$, Ns/mm  6.13  ${C}_{c1}={C}_{c2}$, Ns/mm  2.45 
3. Vehiclespecific seatsuspension design optimization
A twostage optimization problem is formulated to seek vehiclespecific optimal designs considering suspension kinematics. In the firststage, 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 ${K}_{e}$ over the entire suspension travel, such that:
where $\chi =\left\{{l}_{5},{l}_{6},\eta \right\}$ is the design vector related to air spring coordinates (Fig. 1). The above is solved subject to limit constraints: ${\text{0}<l}_{5}$, ${l}_{6}<\text{0.1792}$ and $0<\eta <\pi $. Moreover, natural frequency (excluding the cushion) is permitted to vary in the 1.25 to 1.35 Hz range.
In the second stage, optimal suspension damping is identified to realize an improved compromise between SEAT and VDV ratios. It is evident that effective vertical mode damping is strongly dependent upon coordinates of the damper attachments, including damper orientations ($\gamma $, $\beta $), and lengths of arms OE (${l}_{4}$) and CF (${l}_{3}$), apart from the low and highspeed damping coefficients. A methodology is thus formulated to identify vehiclespecific 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:
where $\chi =\left\{{l}_{3},{l}_{4},\beta ,\gamma ,{C}_{e1},{C}_{c1},{C}_{e2},{C}_{c2},{V}_{e},{V}_{c}\right\}$ is the design vector, and ${\gamma}_{1}$_{}and ${\gamma}_{2}$_{}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: $\chi >0$ with the exception of rebound transition velocity, limited to ${V}_{e}>$–0.06 m/s; and ${V}_{c}<$0.06 m/s, while the static height of the suspension as taken at the midride 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 endstop impacts, such that: $\mathrm{max}\left(\left{z}_{s}\right\right)<$ 70 mm.
4. Results and discussions
The optimal coordinates of air spring attachments were obtained as: ${l}_{5}=\mathrm{}$0.01 m, ${l}_{6}=\mathrm{}$0.05 m and $\eta =$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 midposition), 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.
Fig. 2a) 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)
a)
b)
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). While the nominal suspension travel exceeds permissible free travel, the optimal suspension limits the deflection within the free travel and thereby eliminates endstop impacts.
Table 2Optimal vehiclespecific optimal suspension damper parameters
Parameter  Optimal parameters for specific vehicle class excitation superimposed with filtered shock pulse  
EM1  EM4  EM6  EM9  
${l}_{3}$, mm  0.024  0.012  0.064  0.023 
${l}_{4}$, mm  0.025  0.024  0.016  0.046 
${\beta}_{2}$, rad  0.66  0.674  0.815  0.258 
${\gamma}_{2}$, rad  0.789  0.71  0.142  0.255 
${C}_{e1}$, Ns/m  9.01  10.01  2.32  5.78 
${C}_{c1}$, Ns/m  4.23  2.3  1.54  4.21 
${C}_{e2}$, Ns/m  2.19  6.5  3.39  3.63 
${C}_{c2}$, Ns/m  1.23  3.4  3.23  2.15 
${V}_{e}$, m/s  0.05  0.033  0.02  0.01 
${V}_{c}$, 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 crosslinkage contributes to considerable variations in effective vertical spring rate and damping characteristics of the seatsuspension system during a deflection cycle. Variations in driverselected 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 vibrationspecific damping characteristics and its coordinates/orientation. Such a vehiclespecific optimal design could yield substantial reductions in SEAT and VDV ratio responses.
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