Full Terms & Conditions of access and use can be found at
http://www.tandfonline.com/action/journalInformation?journalCode=nvsd20
Download by: [University of Nebraska, Lincoln] Date: 06 February 2017, At: 09:39
Vehicle System Dynamics
International Journal of Vehicle Mechanics and Mobility
ISSN: 0042-3114 (Print) 1744-5159 (Online) Journal homepage: http://www.tandfonline.com/loi/nvsd20
An efficient sensitivity analysis method for
modified geometry of Macpherson suspension based on Pearson correlation coefficient
Mohammad Hasan Shojaeefard, Abolfazl Khalkhali & Sadegh Yarmohammadisatri
To cite this article: Mohammad Hasan Shojaeefard, Abolfazl Khalkhali & Sadegh
Yarmohammadisatri (2017): An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson correlation coefficient, Vehicle System Dynamics, DOI: 10.1080/00423114.2017.1283046
To link to this article: http://dx.doi.org/10.1080/00423114.2017.1283046
Published online: 06 Feb 2017.
Submit your article to this journal
View related articles
View Crossmark data
http://dx.doi.org/10.1080/00423114.2017.1283046
An efficient sensitivity analysis method for modified geometry of Macpherson suspension based on Pearson correlation coefficient
Mohammad Hasan Shojaeefard, Abolfazl Khalkhali and Sadegh Yarmohammadisatri
Automotive Simulation and Optimal Design Research Laboratory, School of Automotive Engineering, Iran University of Science and Technology, Tehran, Iran
ABSTRACT
The main purpose of this paper is to propose a new method for designing Macpherson suspension, based on the Sobol indices in terms of Pearson correlation which determines the importance of each member on the behaviour of vehicle suspension. The formula- tion of dynamic analysis of Macpherson suspension system is devel- oped using the suspension members as the modified links in order to achieve the desired kinematic behaviour. The mechanical system is replaced with an equivalent constrained links and then kinematic laws are utilised to obtain a new modified geometry of Macpherson suspension. The equivalent mechanism of Macpherson suspension increased the speed of analysis and reduced its complexity. The ADAMS/CAR software is utilised to simulate a full vehicle, Renault Logan car, in order to analyse the accuracy of modified geometry model. An experimental 4-poster test rig is considered for validating both ADAMS/CAR simulation and analytical geometry model. Pear- son correlation coefficient is applied to analyse the sensitivity of each suspension member according to vehicle objective functions such as sprung mass acceleration, etc. Besides this matter, the estimation of Pearson correlation coefficient between variables is analysed in this method. It is understood that the Pearson correlation coefficient is an efficient method for analysing the vehicle suspension which leads to a better design of Macpherson suspension system.
ARTICLE HISTORY Received 22 August 2016 Revised 6 January 2017 Accepted 11 January 2017 KEYWORDS
Sensitivity analysis; Pearson correlation coefficient;
geometry of suspension system; Macpherson
Nomenclature
ms sprung mass (kg) mu unsprung mass (kg)
cp damping coefficient (N s/m) kt tyre stiffness (N/m)
ks spring stiffness (N/m) zr road profile (m)
ϕ control arm rotation angle (deg) ϕ0 initial control arm rotation angle (deg)
CONTACT Abolfazl Khalkhali [email protected]
© 2017 Informa UK Limited, trading as Taylor & Francis Group
zs sprung mass displacement (m) zu unsprung mass displacement (m) ξa length of suspension arm (m) (Figure6) ξb length of suspension arm (m) (Figure6) ξc length of control arm (m) (Figure6) acc.zs sprung mass acceleration (m/s2)
acc.ϕ control arm angular acceleration (deg/s2) ϕ roll angle of control arm (deg)
d relative displacement of sprung mass and unsprung mass (m)
˜V relative velocity of sprung mass and unsprung mass (m/s) ξ0 dimension in reference frameξ0τ0ψ0
ξ1 dimension in reference frameξ1τ1ψ1
τ0 dimension in reference frameξ0τ0ψ0
τ1 dimension in reference frameξ1τ1ψ1
ψ0 dimension in reference frameξ0τ0ψ0
ψ1 dimension in reference frameξ1τ1ψ1
u value of point P in ξ direction v value of point P in τ direction w value of point P in ψ direction
zw value of point w in z direction (m) (Figure6) zp value of point p in z direction (m) (Figure6) zq value of point q in z direction (m) (Figure6) yw value of point w in y direction (m) (Figure6) yp value of point p in y direction (m) (Figure6) yq value of point q in y direction (m) (Figure6) σ angle between ow and y axis (deg) (see Figure6) Fxk the kth external force in x direction (N)
Fyk the kth external force in y direction (N) Fzk the kth external force in z direction (N) qj the jth general coordinate in Lagrange equation V potential energy in Lagrange equation
T kinematic energy in Lagrange equation D energy of damping in Lagrange equation
Qj(n) the general force corresponding to the general coordinate qj t time domain in Lagrange equation (s)
λ(t) white noise function
ψ spectral density of white noise σ2 variance of road roughness
ζ0 constant value in linear regression equation ζ1 constant value in linear regression equation
νi the regression coefficients in linear regression equation
1. Introduction
Most of the research papers related to the suspension systems have been concentrated on the modelling of vehicle suspension and controlling strategies which lead to improved ride
and handling. However, the sensitivity of vehicle suspension parameters according to vehi- cle objective functions such as sprung mass acceleration, suspension roll angle, suspension roll rate, relative displacement of tyre and sprung mass and relative velocity of tyre and sprung mass has not been studied extensively.
The Macpherson suspension configuration mainly consists of a spring, strut and wheel coordination [1,2]. Basically, there are four fundamental members for a commonly used Macpherson suspension system including a lower control arm, telescopic strut, which is connected to the chassis of the vehicle and to the wheel carrier, wheel and spindle [3]. The vibrations related to road profile are decreased by suspension system [4]. For analysing the suspension system behaviour, different models have been utilised. The quarter model is a linear model which has been widely used to analyse the performance of suspension system. However, the geometry of suspension system such as camber, caster, kingpin incli- nation and etc. is not considered in this case [5]. Kim and Ro [6] and Mantaras et al.
[7] considered geometry of vehicle suspension and analysed the kinematic and dynamic of vehicle suspension. Chen and Beale [8] used a three-dimensional (3D) model of the Macpherson suspension in order to estimate its dynamic parameters. These models did not consider the control design in suspension behaviour. Hu et al. [9] investigated the passive suspensions by considering multiple performance requirements including ride comfort, suspension deflection and tyre grip. Nonlinear model of the Macpherson strut suspension system studied by Nemeth and Gaspar [10].
For improving vehicle suspension, the fundamentals of design method should be con- sidered. Layout and kinematic analysis are the main elements which should be studied in the design process [11]. Seeking to meet the performance requirements often leads to a conflicting situation and requires a compromise considering the kinematic and dynamic properties [12]. Different papers studied the conflict between the ride and handling in the behaviour of the vehicle but the effects of vehicle parameters on each other and suspension objective functions have not been considered yet. One of the suitable methods for this pur- pose is Monte Carlo simulation. The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis [13]. Monte Carlo simulations are typically characterised by a large number of unknown parameters, many of which are difficult to obtain experimentally. The parameter sensitivity analysis (SA) for Monte Carlo simulations is complicated by the stochastic nature of the simulations, making it difficult to isolate responses from background noise [14].
As the first step, the Monte Carlo method of sensitivity analyses, selects a random set of input data values drawn from their individual probability distributions. These values are then used in the simulation model to obtain values of output variable. This process is repeated many times and the model calibration must be valid for the input data values selected. The last results are a probability distribution of the model output variables and system performance indices which are obtained from the variations and possible values of all of the input values [15]. In Monte Carlo analysis, amount of each parameter set is selected randomly from the distributions which are describing each individual. Then, the suspension system is simulated to obtain estimations for the selected performance indices.
This must be done many times (often well over 100) to obtain a statistical description of system’s performance variability. Vehicle SA method is used to quantify the effects of the unknown suspension parameters on the vehicle performance. This information can be used to decide which parameters should be optimised or determined more accurately
through further experimental or simulation studies [16]. It is used in both point esti- mate and probabilistic approaches to identify and rank important sources of variability as well as the important sources of uncertainty [17]. Lai and Ehsani [18] utilised state of charge for analysing the sensitivity of vehicle performance to transmission gears for a parallel hybrid electric vehicle-based dynamic programming. Othaganont et al. [19] has presented an extended technique for analysing sensitivity parameters in modern road vehi- cles. The new techniques consider first-order and second-order effects, showing both the effects on individual parameters and also the cross-coupling between different parameter sensitivities.
Large numbers of elements affect the vehicle suspension performance and cause a large design space. This paper concentrated on the kinematic response of a Macpher- son suspension under step input and random vibration road profile which has been utilised for sensitive analysing. The outputs or responses of the mechanism, to this input, present suspension behaviour such as sprung mass acceleration, and kingpin location and orientation.
Most of the research papers reduce the effects of caster angle and kingpin angle on the suspension design. This paper attempts to compensate this weakness and presents a proper method based on considering the modified Macpherson mechanism and its sensitivity.
In this paper, an enhanced model for geometry of suspension is presented which decreases the complexity in geometry model and enhances the speed of analysis. This mod- ified model is more suitable for the SA approach. Then, a Renault Logan car is simulated in Adams/Car for verifying the correctness of the proposed model. Experimental tests have been done in one of the Iranian automaker companies (Saipa) in order to analyse the cor- rectness of the Adams/car Model and the modified model. In this case a 4-poster test rig is utilised for analysing the results of suspension which is subjected to the road profile.
Then, the sensitivity analyses of vehicle suspension are considered using Pearson corre- lation coefficient technique based on the Monte Carlo simulation. The main goal of this study is to determine the most sensitive design parameter for the vehicle suspension. The most and least sensitive parameters of suspension are obtained using Pearson correlation coefficient. Moreover, Pearson correlation coefficient is utilised between input parameters which determined a high correlation rate about 0.06 between ktandξaand also between ξcand ks.
2. Experimentally verified ADAMS/CAR Renault Logan simulation
In this section, simulation of Renault Logan car is carried out using Adams software. In order to verify the simulation, the results will be compared with experimental results. Main objective of such experiment and computer simulation tests is evaluation of the modi- fied geometry model which will be completely explained in the next section. The results of modified geometry model will be compared with both experimental test and ADAMS simulation results represented in this section.
The ADAMS model consists of 87 moving parts, 1 flexible body, 6 cylindrical joints, 9 revolute joints, 6 spherical joints and 2 Hooke joints. In Figure1, the connections between strut and vehicle chassis and also between control arm and vehicle chassis, which are through bushings, are shown. A tie rod, the link which connects steering gear to the front
Figure 1.Full vehicle simulation of Renault Logan car in ADAMS.
Figure 2.Experimental test of Renault Logan car for determining centre of gravity.
wheel, is considered as well. For determining the exact value of the vehicle centre of gravity, an experimental test has been done as shown in Figure2.
The coordinates of vehicle gravity centre, which are obtained by the experimental test, are reported in Table1. These values are inserted in ADAMS model of the vehicle. The
Table 1.Basic dimensions of Renault Rogan.
Vehicle parameter Value Unit
Centre of gravity x = 972.4 mm
y = 11.1 z = 370.2
Mass 1060.0 (kg)
Wheelbase 2630 (mm)
Unsprung mass for suspension 22.5 kg
Sprung mass for suspension 328 kg
Strut sprung coefficient 30,000 N/m
Strut damping coefficient 1350 Ns/m
Tyre stiffness 210,000 N/m
ϕ0 3 degree
α 74 degree
ζA 0.658 m
ζB 0.262 m
ζC 0.31 m
values for other characteristics of vehicle suspension, which are obtained by test and mea- surement, are also reported in Table1. The vehicle is subjected to step input so-called as heave test [20,21] with amplitude of 12.5 mm. Result of chassis vertical acceleration obtained from ADAMS simulation is depicted in Figure3. According to Figure3, firstly, sudden change in sprung mass acceleration is occurred then it oscillates for a period of time and finally oscillations are damped.
As mentioned earlier, in order to verify the results of ADAMS/CAR simulation, the experimental test was performed. In this case, a 4-poster road simulator is utilised in one of main Iranian automakers (SAIPA) (see Figure4). The four post-test is a test rig basically consisting of 4 hydraulic actuators, one for each wheel. Vibrating of the actuators, based on the step input (12.5 mm), simulates the road surface and forces exerted by the road on the wheels. The movements of the system are tightly controlled by a digital test con- troller [22,23]. The data, obtained from experimental tests (see Table1) have been utilised in analysis of the Logan car suspension.
Figure 3.Comparing results of sprung mass acceleration in experimental test, ADAMS and modified geometry model.
Figure 4.4-post experimental test in one of main Iranian automaker (SAIPA).
3. Modified geometry model of Macpherson suspension
The Macpherson is a popular suspension mechanism for most of the commercial cars which is commonly used in the front suspension system. The kinematic modelling of the suspension mechanism is combination of elements which simulate the behaviour of sus- pension members such as control bar, strut, etc. The 3D model for front suspension and Full vehicle model of Renault Logan which is developed in ADAMS software is shown in Figures1and5.
In comparison to other suspension models, geometry models can compute suspen- sion geometric outputs such as camber, caster and kingpin inclination angles due to deep focus on suspension mechanism. The assumptions for the proposed modified model are presented as follow:
• The vehicle chassis is assumed to have vertical displacement and its motion has been neglected in the other directions.
• The unsprung mass is linked to the chassis through strut and control arm.
• The spring and damping coefficients are considered to have a linear behaviour.
• The masses of strut and control arm is smaller than sprung and unsprung masses so that it is neglected.
The mechanism of Macpherson suspension has two degree-of–freedom (DOF): One of them expresses angle of control arm rotation (ϕ) and the other one represents the vertical displacement of the sprung mass (zs). These two parameters are generalised coor- dinates of the Macpherson suspension. Parameters zuand zrrepresent the unsprung mass displacement and road profile, respectively.
The modified geometry model of this work is based on the model represented by Fallah et al. [1] which is depicted in Figure6. The generalised key points of the modified Macpher- son suspension indicated as o, p, q, t and w. The global reference frame {o} is considered for determining the exact position of the key points. In the presented modified model firstly, the intersection of strut line (wt) with the control arm (oq) is found and presented as point
Figure 5.Front suspension model of Renault Logan car in Catia.
Figure 6.Schematic of modified Macpherson mechanism.
p. Utilising point p instead of point t, which has been used by Fallah et al. [1], reduces the complexity and also solving run time of dynamic equations.
This modified model not only facilitates analysing parameters of the suspension kine- matic such as king-pin inclination, caster and camber angles on the ride vibrations, but also, presents a more precise model of the Macpherson suspension system for enhancing the ride quality.
In common quarter-car model, various semi-active and active control algorithms have been developed to enhance the ride comfort and handling performance based on a two DOF model. However, it should be declared that without considering the impact of the suspension kinematics and related linkage, the quarter model could not be significantly useful. Thus, the demand for a precise model of the Macpherson suspension system to explore the effect of suspension kinematics on the dynamic performance of the suspension becomes increasingly important for ride control design.
Another weakness of the conventional suspension model is that it cannot facilitate the evaluation of the effect of variation of the kinematic parameters, which is useful in improv- ing the stability and handling performances of the vehicle. Some of those parameters that play an important role in chassis design are caster angle, camber angle, track and king-pin inclination angle. Variations of camber angle enhance tyre wear and generate lateral forces acting on the wheel, causing the vehicle to steer to one side. Caster angles and kingpin inclination alterations affect the self-aligning moment and as a result, affect the stability and handling of the vehicle when wheels bounce or rebound. When the wheels moves on a road with rough profile, the track alteration makes the rolling tyre to slip, generating lateral forces and even influencing the steering.
The vehicle suspension key points will be changed when they are subjected to road pro- file. For transforming the key point of the suspension to the new position, the rotational transformations matrixes have been utilised. The new key points could be parameterised by multiplying the old points and rotational transformations matrix of
as presented in
Equation (1) [24].
P0=
⎡
⎣(uξ1+ vτ1+ wψ1).ξ0
(uξ1+ vτ1+ wψ1).τ0
(uξ1+ vτ1+ wψ1).ψ0
⎤
⎦ =
⎡
⎣ uξ1.ξ0+ vτ1.ξ0+ wψ1.ξ0
uξ1.τ0+ vτ1.τ0+ wψ1.τ0
uξ1.ψ0+ vτ1.ψ0+ wψ1.ψ0
⎤
⎦ , (1)
P0=
⎡
⎣ ξ1.ξ0+ τ1.ξ0+ ψ1.ξ0
ξ1.τ0+ τ1.τ0+ ψ1.τ0
ξ1.ψ0+ τ1.ψ0+ ψ1.ψ0
⎤
⎦
⎡
⎣u v w
⎤
⎦ ,
P0=
0 1
P1. (2)
In the Equation (2),1
0presents the rotational matrix which has been utilised for trans- ferring the key point P0 from reference frame ofξ0τ0ψ0to P1from reference frame of ξ1τ1ψ1. P1expressed as (u, v, w)tand is presented in the following equation:
P = u.ξ + v.τ + w.ψ. (3)
With consideration of transformation matrix (Equation (1)) and simplifying it, the vehicle suspension key points are obtained through Equation (4).
zw= zs
yp= ξB(cos(ϕ − ϕ0) − cos(−ϕ0)) zp= zs+ ξB(sin(ϕ − ϕ0) − sin(−ϕ0)) yq= ξC(cos(ϕ − ϕ0) − cos(−ϕ0)) zq= ξC(sin(ϕ − ϕ0) − sin(−ϕ0)) + zs,
(4)
whereϕ0presents the initial angular displacement of the control arm at the equilibrium point.ξA,ξB andξCare the lengths of the suspension arms as indicated in Figure6. zs
presents the vertical displacement of sprung mass. The lateral displacement of sprung mass is considered to be zero. zw, zp, zq, yw, ypand yqare vertical and lateral displacements of points w, p and q, respectively.
3.1. Dynamic equations of vehicle suspension
The equations of the modified vehicle suspension model can be obtained by utilising Lagrange mechanics.
d dt
∂T
∂ ˙qj
− ∂T
∂qj +∂V
∂qj = Qj(n) Qj(n) =
Fxk∂xk
qj + Fyk∂yk
qj + Fzk∂zk qj
,
(5)
T = 1
2ms˙z2s + 1
2mu(˙y2q+ ˙z2q) V = 1
2ks( ξ)2+ 1
2kt(zq− zr)2,
(6)
where T, V are kinematic energy and potential energy, respectively (Equation (6)). In Equation (6), ξ and zq− zrobtained by considering the geometry of vehicle suspension as follows:
By considering the triangle OWP, Equation (7) presented as:
ξ = (ξA2+ ξB2− 2ξAξBcosσ)1/2
ξ= (ξA2+ ξB2− 2ξAξBcos(σ− ϕ))1/2, (7) where
σ= σ − ϕ0 (8)
ξ and ξare the initial length of control link wp (see Figure6) by considering the rotation ofϕ. σ is the angle between ow and y axis which is presented in Figure6.
( ξ)2= (ξ − ξ)2= ξA2+ ξB2− 2ξAξBcosα+ ξA2+ ξB2− 2ξAξB cos(σ− ϕ)
− 2(ξA2+ ξB2− 2ξAξBcosσ)1/2(ξA2+ ξB2− 2ξAξBcos(σ− ϕ))1/2, (9) aξ = ξA2+ ξB2,
bξ = 2ξAξB,
( ξ)2= (ξ − ξ)2= 2aξ − bξ(cos σ+ cos(σ− ϕ))
− 2{a2ξ − aξbξ(cos σ+ cos(σ− ϕ) + b2ξcosαcos(σ− ϕ))}1/2, (10)
˙ ξ = ˙ξ − ˙ξ= bξ sin(σ− ϕ) ˙ϕ
2(aξ − bξcos(σ− ϕ))1/2, (11)
zq− zr= ξc(sin(ϕ − ϕ0) − sin(−ϕ0)) − zr. (12) The Equations (9)–(12) are substituted in Equation (6) which is presented in Appendix.
The Equations (4)–(12) indicate the transformed key points and basic parameters of vehicle suspension which are utilised to obtain state equations of vehicle suspension. The state equations of vehicle suspension are used for analysing the vehicle suspension behaviour under different conditions.
Lagrange theory (Equation (5)) is utilised for obtaining equations of motion in Macpherson suspension which is indicated in Equation (13).
η1¨zs+ η2¨ϕ + A1+ A2= A3→ η1¨zs+ η2¨ϕ = A3− A1− A2= P1,
η3¨zs+ η4¨ϕ + A4+ A5= A6→ η3¨zs+ η4¨ϕ = A6− A4− A5= P2, (13) where
A1= −∂T
∂zs, A2= ∂V
∂zs, A3= −∂D
∂zs
A4= −∂T
∂ϕ, A5= ∂V
∂ϕ, A6= −∂D
∂ϕ,
(14)
where D is energy of damping and can be calculated from Equation (15)
D = 12cp( ˙ ξ)2. (15)
The equations of motion will be presented as:
η1¨zs+ η2¨ϕ = P1
η4¨zs+ η3¨ϕ = P
×(−η4)
−−−−→
×(η1)
−η4η1¨zs+ −η4η2¨ϕ = −η4P1,
η1η4¨zs+ η1η3¨ϕ = η1P2, (16)
¨ϕ = η1P2− η4P1
η1η3− η2η4,
¨zs= η2P2− η3P1
η2η4− η1η3.
(17)
By considering the Equation (16), Lagrange equation will turn into the state function as presented in Equation (17). The parameters for equations of state expresses in Equation (18).
[x1 x2 x3 x4]T = [zs ˙zs ϕ ˙ϕ]T, (18)
˙x1= x2,
˙x2= f1(x1, x2, x3, x4, zr),
˙x3= x4,
˙x4= f2(x1, x2, x3, x4, zr).
(19)
Equation (19) will be considered in the equilibrium state xe = (x1e, x2e, x3e, x4e) = (0,0,0,0). The state equations will be obtained by utilising Equation (19). By substituting Equation (17) in Equation (19) the equation of state function is presented as:
f1= 1 D1
muξc2 sin(x3− ϕ0)x24−1
2kssin(σ− x3) cos( x3+ ϕ0)(x3) + cp (x3) ˙ϕ − ktξcsin2(x3− ϕ0)z(0)
, (20)
f2= − 1 D2
m2uξc2sin(x3− ϕ0) cos(x3− ϕ0)x24+ (ms+ mu)cp (x3)x4
− 1
2(ms+ mu)ks sin(σ− x3)(x3) + msktξccos( x3− ϕ0) z(0)
, (21)
where D1and D2are computed by Equation (22).
D1= msξC+ muξCsin2(x3− ϕ0),
D2= msmuξc2+ m2uξc2sin2(x3− ϕ0). (22) (x3), (x3), z(0), cξ and dξ, which have been used in Equations (20)–(21), are computed through Equations (23)–(26).
(x3) = bξ + dξ
(cξ − dξcos(σ− x3))1/2, (23)
(x3) = b2ξsin2(σ− x3)
4(aξ − bξcos(σ− x3)), (24)
z(0) = z(x1, x2, zr) = x1+ ξc(sin(x3− ϕ0) − sin(−ϕ0)) − zr, (25) cξ = a2ξ − aξbξcos(σ + ϕ0),
dξ = aξbξ − b2ξcos(σ + ϕ0). (26)
Finally, the linear model will be presented as:
˙x = Ax(t) + Bzr(t), x(0) = x0,
A =
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣
0 1 0 0
∂f1
∂x1
∂f1
∂x2
∂f1
∂x3
∂f1
∂x4
0 0 0 1
∂f2
∂x1
∂f2
∂x2
∂f2
∂x3
∂f2
∂x4
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
. (27)
The parameters of the matrix are calculated as presented in the appendix.
B =
0 ∂f∂z1r 0 ∂z∂f2r
T
zr=0 =
⎡
⎢⎢
⎢⎢
⎢⎢
⎢⎣
0 ktξcsin2(−ϕ0) msξc+ muξcsin2(−ϕ0)
0 msktξccos(−ϕ0) msmuξc2+ m2uξc2sin2(−ϕ0)
⎤
⎥⎥
⎥⎥
⎥⎥
⎥⎦
. (28)
3.2. Response to the step input
To evaluate the modified geometry model represented in this paper; the results of such model are compared with those obtained from ADAMS and experimental tests of Renault Logan car. Such comparison is demonstrated in Figure3, which shows a good accuracy of the modified geometry model.
It is desired to find other outputs using the modified geometry model which works fast and low cost comparing to the ADAMS model and also experimental test. Different accel- eration and displacement sensors were installed on the vehicle to record the results of road vibrations on the vehicle parts. In this experimental test, seven acceleration sensor and two displacement sensors have been utilised. Two of these accelerometers have been installed on the shock absorber of the rear wheel which determines the acceleration which is inserted on the rear sprung mass of vehicle. Two of them are installed on the front shock absorber of the vehicle. One of the accelerometer is under the driver seat for detecting the acceleration which is inserted to the driver and one in the approximate place of the vehicle centre of gravity.
By exciting all wheels of the vehicle under bounce pulse as road profile in 4-poster test rig sprung mass displacement, acceleration of sprung mass, acceleration of unsprung mass, control arm angle, acceleration of control arm angle, caster angle and kingpin angle are
Figure 7.Results of modified geometry suspension under step input road profile.
obtained from modified geometry model, which are depicted in Figure7. It is noted that the actuators of 4-poster test rig generate road input with amplitude, maximum displacement, of 12.5 mm and frequency of 0.2 Hz.
As presented in Figure7, the caster of suspension geometry differed from 3.9° to 4.1°.
The kingpin angle had a range of 10.9°–11.4°. The heave, roll and pitch of vehicle body motion must be investigated to analyse the vehicle ride and handling.
3.3. Response to the random road profile
The road surface can be supposed as a stationary stochastic process provided that the velocity of the vehicle is constant [25]. There are different methods for obtaining the corre- sponding random signal in time and frequency domains [25–27]. In this research, shaping filter method has been used for generation of the random road profiles [26]. For a vehicle, moving with the constant velocity of V, the road profile signal zr(t) is obtained using the linear filter described by the following differential equation:
d
dtzr(t) = −α.v.zr(t) + λ(t), (29) whereλ(t) is the white noise function with the spectral density of ψ= 2αVσ∧ 2,σ2indicates the variance of the road roughness, v is the vehicle velocity and α is a constant depending on the type of road surface. Based on many works, theσ2= 4(0) has been considered in this research. The reference values of power spectral density (PSD) for0 = 1 (rad/m),
(0), are provided based on standard of ISO 8608 as given in Table2.
The differential Equation (29) can schematically be presented in MATLAB-SIMULINK as presented in Figure8(a). By solving this equation, the random road profile signal in time domain (zr(t)) is obtained as shown in Figure8(b).
3.4. Vibration model of the vehicle suspension system
Modified geometry model of vehicle suspension is shown in Figure6. By considering zr
as inputs and using Laplace transformation, transformation functions can also be derived.
When the input excitation of the system is the PSD function, the output PSD function will be as [28]:
−→ G(s)F(s) −→X(s) X(jw) = G(jw)F(jw)
Sxx(jw) = |G(jw)|2SFF(jw). (30) For assessment of the PSD function, one can utilise the root mean square (rms) values.
For a random signal described by the PSD function, it can be acquired by Confalonieri et al. [29].
Table 2.Road roughness values classified by ISO 2631.
Road class σ (10−3) (0) (10−3),0= 1 α (rad/m)
A (very good) 2 1 0.127
B (good) 4 4 0.127
C (Average) 8 16 0.127
D (poor) 16 64 0.127
E (very poor) 32 256 0.127
(a)
(b)
Figure 8.(a) MATLAB-SIMULINK diagram for obtaining random vibration road profile. (b) Random vibration road profile.
In this paper, excitation function in the frequency domain for motorway road profile introduced in [24] which has been obtained using shape filtering and afterwards, dynamic response of the chassis [24].
xrms=
Sg(f ) df . (31)
For the frequency domain solution, Laplace transform is utilised in the case that initial conditions are zero. These equations are presented as follows:
[M]¨z + [c]˙z + [k]z = zr, (32)
[M]s2Z(s) + [c]sZ(s) + [k]Z(s) = [F]zr(s). (33) In the case that the tyre is under excitation of zr(t), ZR is the road profile, the dynamic equation is presented as Equation (34):
L(zr(t)) = [0 kt]TZr(s). (34)
Zr(s) is the random road profile in the frequency domain as system input. So that system transform function is presented as follow:
Z(s) = ([M]s2+ [c]s + [k])−1[0 kp]TZp(s). (35) Frequency domain is a special mode of Laplace which jw is substituted with s. Vertical displacement of passenger seat in frequency domain is presented in Equation (36).
[D]
Z ϕ
= B2.Zr, (36)
D11 = j2w2+ ktξcsin(ϕ − ϕ0)2
(ξcmusin(ϕ − ϕ0)2+ ξcms), (37) D21 = ktξcmsCos(ϕ − ϕ0)
ξc2msmu+ ξc2m2uSin(ϕ − ϕ0)2, (38) B212 = ktξcmsCos[−ϕ0+ x3]
ξc2msmu+ ξc2m2uSin[−ϕ0+ x3]2. (39) In this paper quarter model is used for verification of modified geometry model too.
The quarter model is a model which commonly used for simulating suspension perfor- mance. However, this model does not consider suspension geometry. The sprung mass, acceleration of sprung mass, versus time, transmissibility ratios and ratio of sprung mass
Figure 9.Comparing results of quarter model and modified geometry model.
acceleration and road profile displacement of modified geometry model and quarter model compared with each other based on the vehicle dimension presented in Table1. As pre- sented in Figure9, there is a good agreement between the sprung mass, acceleration of sprung mass, versus time, transmissibility ratios and ratio of sprung mass acceleration and road profile displacement of these two models, which verifies the correction of the modified suspension model. In the quarter model zsand zuare the parameters which are measured and analysed. However, in geometry model zsandϕ are the effecting parameters.
4. Sensitivity analysis
The sensitivity of vehicle suspension model, which is subjected to the random vibration road profile, is a key point in the vehicle operational analysis such as ride and handling [30].
In General, SA is a crucial tool which concerns the mathematical model performance. It is applied to the suspension model in order to identify the influence of input parameters on the output variations, and rank input parameter based on the vehicle performance sensi- tivity [30–33]. In this paper, the main goals of SA are assessing the complicated suspension models, decreasing complexity in analysis of model by omitting useless parameters, and finally, verifying the model by considering the simulation of underlying real problem [31].
The Monte Carlo simulation is utilised to produce random sampling which is applied in the SA. The risk assessment based on the Monte Carlo method is applied to identify the uncertainty in the outputs which are related to the vehicle suspension inputs [30]. Semenov analysed the Pearson correlation and other outlier correlation estimators based on Monte Carlo simulation [32]. This comportment is related to the uncertainty analysis which is supplement with SA and determines the most important parameters. The considered num- ber of sample size and random variants are derived, according to the norms of correlation which are obtained based on probabilistic distributions.
Different methods could be utilised in SA of suspension model. One of the methods, which is based on variance-based global technique, is Sobol’s approach. Relative impor- tance of input variants is considered in this technique [33]. The main goal in this approach is determining uncertainty in the model and its critical variants [33–35]. This approach is appropriate for multiple criteria analyses [36] such as suspension model. Glen considered the Pearson correlation in Sobol indices and developed this approach to modify spuri- ous correlations which are existed in this model [33]. The bounds of correlation errors must be considered in correlation control approach. This subject determines the stop- ping criteria in correlation control technique. A correlation estimator is applied in the model in order to determine errors of the presented correlation sample. In this case an approach which is called Latin hypercube sampling (LHS) using Monte Carlo simulation for variance-reduction sampling.
4.1. Pearson correlation coefficient
SA approach is represented by means of a coefficient which is called SA index. The SA index ranks input parameters based on their influence on the outputs. It is utilised in statistical model. The variation and predicted amount of objective function in the model and ranking the contributions to the variance is an application for statistical SA [37].
In this paper vehicle suspension model is considered which consists of different inde- pendent variables, X = (X1,. . . , Xn) which expressed as (kt, mu, ms, cp,ξa,ξb,ξc, ks), and several dependent variable, Y = (acczs, accϕ,ϕ, d, ˜V), in the case that Y = f (X).
The uncertainty in vehicle suspension is determined by the variance of objective func- tion Y = (acc.zs, acc.ϕ, ϕ, d, ˜V). In continue sensitivity of Macpherson suspension is computed by the variance in Xi, weighted by the first-order partial of Y with respect to Xi.
Determining a formulation for the variance that each input parameter of vehicle suspen- sion Xi related to the variance of vehicle suspension objective function V(Y) is the main goal of utilising variance-based approach in this case.
The vehicle suspension model has a function which is described as Y = f (X), where Y denotes the output and x1,. . . , xkare independent parameters of suspension geometry.
As vehicle suspension is a complex model, different objective functions are considered for analysing most important input parameters. Acceleration of sprung mass (acczs), angular acceleration of control arm (acc.ϕ), roll angle (ϕ), relative displacement of sprung mass and unsprung mass (d) and relative velocity of sprung mass and unsprung mass ( ˜V) are considered simultaneously for determining the most important input parameters which have large effects on the suspension behaviour. In this method, an upper and lower limits are considered for each input variables which varying over its probability density function.
In the case that an exact value ofx is considered for input parameters, the problem is ranking the vehicle suspension input based on their variance. By taking the variance of all suspension parameters except
Xi, the conditional variance of Y given Xi =xiis presented as Var(Y/Xi =xi). In geom- etry suspension model, it is not possible to determine an exact value for the geometry parameters. In this case, average of conditional variance is considered for all the value which could be determined asx of Xi, that is, E[Var(Y | Xi)] is the estimated value through the overall variation of suspension parameter Xi.
In the case that, there are unconditional variance of outputs, Var(Y), in the model and by considering Var(Y) = Var(E[Y | Xi])+ E[Var(Y | Xi)], the conditional variance would be determined as Vi = V(E[Y | Xi]). The presented equation is utilised as importance index of Xion the variance of Y. Normalising this equation presents the SA by Sobol (ηi) [38]:
ηi2= var[E(Y/Xi)]
var(Y) (40)
E(Y/Xi = x) and Var[E(Y/Xi = x)] present the regression function and its variance, respectively [31]. Equation (40) represents variance of output as a result of input parame- ters (Xi).
The regression theory is used to match the input data to an estimation equation that gives the output values with a minimum mistake. The least squares linear regression is a commonly used approach in this case. Thus the input N.k data is fitted to an estimated linear equation( ˆYi= ζ0+k
j=1ζ1Xij+ νi). In this equation ζ0,ζ1andνiare the constant value and the regression coefficients which should be specified and an error because of the assumptions, respectively. The constraint of minimising the sum of the squared difference between the line and the data points in Y is applied to the model.
In continue a coefficient is presented to measure how the regression model fit to the original data.
2=
N
i=1( ˆYi− ¯Y)2
N
i=1(Yi− ¯Y)2, (41)
where ˆY, Yiand Y are the assumed output based on regression model, observed values and their mean, respectively. The Equation (41) presents the accuracy of the regression model and in the case that it is value is so close to 1, the estimated model is a suitable one.ζ j, j = 1, . . . , k, indicated the contribution of input parameters and the output. In the case that the coefficient is related to the X and Y, the regression model is normalised in order to omit the unit from the SA coefficient. In Equation (42),ζjˆηj/ˆη presents the SA coefficient.
ˆYi− ¯Yi ˆη =k
j=1
ζjˆηj ˆη
Xij− ¯Xj
ˆηj , (42)
ˆη =
N
i=1
(Yi− ¯Y)2 N − 1
1/2
,
ˆηj =
N
i=1
(Xij− ¯Xj)2 N − 1
1/2
. (43)
Equation (43) presents Pearson correlation coefficient [30,31,39] which is a suitable method for SA of vehicle suspension.
The stochastic approaches of{xξ}ξ=1;...;Nsample and{yξ}ξ=1;...;Nsample are selected in the case that xξ is uniform and independent and yξ is output of suspension. The Pearson correlation coefficient is combination of xtand ytwhich is presented as:
PCC= E[(xξ − δx)(yξ − δy)]
ψx2
ψy2 , (44)
which δx = E(xξ), δy = E(yξ), ψx2= E[(xξ− δx)2] and ψy2= E[(yξ − δy)2]. The esti- mated PCC is dependent; Thus, the sample number could be used instead of E and the expression would be written as follows:
P ˆCC= Nsample1
Nsample
ξ=1 [(xξ − ˆδx)(yξ − ˆδy)]
ˆψx2
ˆψy2 , (45)
which ˆδx= (1/Nsample)Nsample
ξ=1 xξ, ˆδy= (1/Nsample)Nsample
ξ=1 yξ, ˆψx2= (1/Nsample)
Nsample
ξ=1 (xξ − ˆδx)2 and ˆψy2= (1/Nsample)Nsample
ξ=1 (yξ − ˆδy)2. The mentioned Pearson correlation coefficient is obtained by median of the samples [39–41].
In this paper an approach for estimating Pearson correlation (Pij) which utilised two random variables xi,ξand xj,ξ[40]. In this case, Pearson correlation coefficient provided an efficient method for declaring the strength of the linear relationship between two variables
whereξ = 1, . . . , Nsample, Nsampleis the sample number, i = 1, . . . , Nvarand Nvaris the variable number.
Pi,j=
Nsample
ξ=1 (xi,ξ − M(xi,ξ))(xj,ξ − M(xj,ξ))
Nsample
ξ=1 (xi,ξ− M(xi,ξ))2Nsample
ξ=1 (xj,ξ− M(xj,ξ))2
, (46)
M(xi,ξ) = 1 Nsample
Nsample ξ=1
xi,ξ, M(xj,ξ) = 1 Nsample
Nsample ξ=1
xj,ξ.
Tables1and3represent the suspension parameters and its variations, respectively. In the sensitivity analysing method, which has been presented in this paper, some of vehicle parameters values considered as constant parameters such asϕ0,α and some are deter- mined through a distribution technique. The method of minimum and maximum ranges was determined as the distribution technique which is shown in Table3.
In the optimisation model, some parameters have large influences on the vehicle behaviour whereas some have a little influence on the vehicle performance. A typical opti- misation technique does not consider rate of the input parameters on the output of the model and use all design parameters as equal. This matter puts some weakness in the opti- misation procedure which causes the application of sensitivity algorithm in order to present an accurate optimised design [42]. The parameter sensitivities can also be useful for deter- mining where future computational and experimental efforts should be focused; it is not necessary to spend a lot of effort for calculating or measuring physical parameters that do not have a significant effect on the product quality [16].
The importance level of each parameter according to each objective function indi- cated in Table4. Increasing the value of Pearson correlation coefficient declares the higher importance of the parameter.
Table4indicates a measure for sensitivity of suspension geometry according to the parameters. The final output of SA is obtained by summing the absolute values of SA in each parameter for all the objective functions. As presented in this table the greatest SA with respect to sprung mass acceleration, control arm rotating acceleration, control arm rotating angle, relative wheel displacement and relative wheel velocity is related to the damping coefficient. This effect is related to the ability of dampers in absorbing the vibra- tions. The pair-wise scatter plot samples and Pearson correlation matrix for the parameters are presented in Figure10and Table5, respectively.
Figure10 and Table5indicate a measure of the strength for the linear relationship between two variables. For some parameters, strong correlations can be observed. For example, there is a near 0.06 correlation between the stiffness wheel coefficient, Kt and the distance of origin and upper strut mountξa. The same correlation is between theξc
Table 3.Variation of suspension parameters.
Front spring stiffness (ks)
Front damper coefficient (cs)
Sprung mass (ms)
Unsprung
mass (mu) ζA ζB ζC kt
Uniform Min 25,000 1350 300 20 0.6 0.27 0.3 1.9e5
Max 35,000 5500 350 25 0.7 0.37 0.4 2.3e5