International journal of automotive technology, tập 12, số 5, 2011

International Journal of Automotive Technology, Vol 12, No 5, pp 631−638 (2011) DOI 10.1007/s12239−011−0073−1 Copyright © 2011 KSAE 1229−9138/2011/060−01 ESTIMATION OF VALVE SPRING SURGE AMPLITUDE USING THE VARIABLE NATURAL FREQUENCY AND THE DAMPING RATIO H LIU1) and D KIM2)* 1) Graduate School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan 680-749, Korea 2) Department of Mechanical and Automotive Engineering, University of Ulsan, Ulsan 680-749, Korea (Received March 2010; Revised December 2010) ABSTRACT−Internal vibration of the valve spring is a critical factor in determining the dynamic characteristics of highspeed valve train systems Because precise prediction of the spring surge amplitude is a difficult problem, especially for nonlinear variable-pitch springs, the development stage requires a process of trial and error In the present study, a new method that considers the variable natural frequency and variable damping ratio is proposed to predict the spring surge amplitude First, the change in the natural frequency and damping ratio caused by compression is predicted from the initially given pitch curve at the free height Second, the spring surge amplitude is estimated by solving the wave equation with nonlinear variable coefficients The surge amplitudes of typical valve springs are also measured using a motoring test rig and are compared with theoretical results predicted by the spring drawing and cam profile data KEY WORDS : Valve train dynamics, Valve spring, Spring surge, Variable natural frequency, Variable damping ratio NOMENCLATURE train dynamics The valve spring surge is the internal vibration of the spring itself caused by periodic cam profile excitation Because the valve spring surge amplitude is increased when the spring's natural frequency coincides with one of the cam profile's harmonic components, the internal vibration of the valve spring often causes premature failure and malfunctions in the valve motion, especially at high engine speeds Therefore, the valve spring surge is regarded as a critical factor in determining the dynamic characteristics of the entire valve train system In general, three methods are used in the analysis of valve spring surge: the discrete model, the distributed parameter model, and the finite element model Seidlitz (1989) and Schamel et al (1993) proposed a discrete model to predict spring surge characteristics and coil clash phenomena The valve spring was described as a series of mass-stiffness-damper elements, and the coil clash was determined by examining the free space between the adjacent coils Because the discrete model is based on the discretization of the spring coils, the accuracy of this method is highly dependent on the number of discretization segments Usually, four to eight segments per turn are recommended in the ADAMS\Engine software (Ortmann and Skovbjerg, 2000) Some errors in the natural frequency are inevitable due to this relatively rough discretization Pisano and Freudenstein (1983) developed a distributed parameter model for the analysis of a high-speed camfollower system The internal vibration of the valve spring was expressed by the wave equation Phlips et al (1989) also presented a distributed parameter model in the surge d R c k m t x L : spring wire diameter [mm] : spring mean coil radius [mm] : speed of wave propagation [m/s] : spring stiffness [N/mm] : mass of active coils [kg] : time [s] : distance along spring axis [mm] : spring length [mm] φ : camshaft rotation angle [deg] n : order of spring natural frequency Gn(φ) : dynamic component of spring internal vibration, function of camshaft angle [mm] ωn(φ) : natural frequency, function of camshaft angle [Hz] ζ (φ) : damping ratio, function of camshaft angle : number of active coils Nα E : Young’s modulus of spring material [Pa] G : shear modulus of spring material [Pa] ρ : density of spring material [kg/m3] υ : Poisson’s ratio γ : shear coefficient of wire cross section INTRODUCTION The valve spring is one of the most important valve train components in determining the characteristics of valve *Corresponding author e-mail: djkim@ulsan.ac.kr 631 632 H LIU and D KIM analysis of a progressive valve spring Because the natural frequency of the valve spring is directly input, the distributed parameter model usually gives a good estimation of the valve spring surge However, accurate estimation of the spring's natural frequency is a troublesome problem In the finite element approach, Kim and David (1990) modeled a valve spring using Galerkin’s method and estimated the valve spring surge by calculating the frequency response function between the valve motion and the valve spring reaction force Even though many analytical models have been proposed for estimating spring surge amplitudes, researchers continue to try to improve the accuracy of the model (Blair et al., 2010 and Huber et al., 2010) In this paper, an algorithm is developed to estimate valve spring surge characteristics by considering the variable natural frequency and the variable damping ratio The change in the spring's natural frequency and damping ratio during compression is estimated from the initially given pitch curve In the simulation, the end coil effect and the clearance between close turns are taken into account Finally, the spring surge amplitude is estimated by solving the nonlinear wave equation with variable coefficients, and the estimated spring surge amplitudes are compared with the measured data MEASUREMENT OF THE VALVE SPRING NATURAL FREQUENCY AND THE DAMPING RATIO slightly, with only a small clearance between them This clearance causes periodic separation and contact with valve motion The vibration energy is dissipated in this process, and the magnitude of the absorbed energy increases the damping ratio of the spring Therefore, the damping is not constant but rather varies with the valve motion To understand the nonlinear properties of the variable natural frequency and damping ratio, we performed multiple experiments An experimental setup was prepared as shown in Figure A valve spring was compressed to a known height and excited by a vibration exciter A strain gauge was fixed at 0.5 turn points from the top boundary point First, the resonance frequency was attained by the frequency response function (FRF) between the acceleration (input) and strain on the spring wire (output) Then, the damping ratio could be determined by the half-power Figure Half-power bandwidth method In many spring surge analysis methods, a constant pitch valve spring is assumed to have a constant natural frequency, and the natural frequency of a two-step variable pitch spring is assumed to change suddenly when the narrow pitch coils collapse together The damping ratio of the valve spring is also treated as a constant value and is assumed to be in the range of 0.02-0.03 (Lee and Patterson, 1997) To suppress the surge amplitude, most valve springs are purposefully designed to have variable pitches At the spring install height, narrow pitch coils contact each other Figure Schematic diagram of the experimental setup Figure Measured natural frequency and damping ratio ESTIMATION OF VALVE SPRING SURGE AMPLITUDE USING THE VARIABLE NATURAL FREQUENCY bandwidth method In this method, the FRF amplitude of the system is obtained first For each natural frequency, there are two points corresponding to the half-power points dB down from the peak, as shown in Figure Finally, the damping ratio is calculated using Equation (1) f -2∆f2 – f1 - = = 2ζ fn fn (1) where ∆f is half-power bandwidth Figure shows the measured natural frequency and the damping ratio results with respect to different spring heights For a constant pitch valve spring, the natural frequency increases gradually, but the damping ratio shows a sudden increase at the transition region of the natural frequency In a two-step variable pitch spring, there are two peaks in the measured results for the damping ratio Furthermore, the two peaks are located just before the jump in natural frequency This phenomenon can help to clarify the relationship between the natural frequency and the damping ratio The natural frequency increases as a result of the reduced number of active coils Before the coils collapse together, some of the coils maintain a small clearance and undergo contact and separation with the spring's internal vibration Because of the energy dissipation that occurs during the contact and separation of the coils, peaks exist in the damping ratio As the spring compression continues, the slight clearance collapses to reduce the number of active coils Subsequently, the natural frequency is increased, and the damping ratio returns to the normal value SPRING SURGE ESTIMATION Estimation of the natural frequency is the first step in predicting the valve spring surge phenomena Liu and Kim (2009) presented a paper presenting an efficient way to estimate the natural frequencies of valve springs by considering the end coil effects The paper revealed the reason for the mismatch between the calculated and measured natural frequencies To consider the end coil 633 effects, the valve spring is described by active coils and two torsional springs at both extremities, as shown in Figure As the valve spring height is different for the open and closed states of the valve, its natural frequency changes significantly during operation This periodic change in the natural frequency makes it difficult to predict spring surge phenomena The natural frequency changes arise mostly from the change in the number of active coils due to compression In general, the number of active coils decreases during compression because some of the active coils collapse into dead coils at both ends The change in the number of active coils can be determined using the stepwise compression procedure In addition, the nonlinear load-deflection relation of the valve spring can also be attained Effective end coils are determined for each step The stiffness of the equivalent torsional spring is calculated using the updated end coils With the updated number of active coils and the equivalent torsional stiffness corresponding to each spring height, the change in natural frequency caused by compression could be predicted successfully Because the valve spring is the most flexible component in the valve train system, the natural frequencies of the internal vibration of the valve train are usually much higher than the frequency of the valve spring The effect of the vibrations in the valve train on valve spring surge is negligible because the vibration amplitude is very small and because its frequency is much higher than that of the valve spring The excitation amplitudes in the lower frequency range, which can excite valve spring surge, mostly come from kinematic valve motion Therefore, the valve spring can be removed from the valve train system and studied independently (Kim and David, 1990) In this paper, a distributed model approach is employed for estimating the spring surge properties The internal vibration is governed by the wave equation (Schamel et al., 1993), ∂2y ( x, t ) ∂2y ( x, t ) -= c2 -2 ∂t ∂x2 (2) where y(x, t) is the particle displacement, and c2 = L2k ⁄ m is the speed of wave propagation along the direction of the spring axis After some manipulation, the internal vibration of the valve spring can be obtained, as in Equation (3), in the domain of the camshaft rotation angle Gn is the dynamic component of the spring's internal vibration, and φ is the camshaft rotation angle The variable natural frequency and damping ratio are represented as coefficients of the equation d -G-n ωn( φ )- dG ω2n2 d2 y -n + -+ 2ζ( φ ) -Gn = – 2 n ω d φ π dφ cs dφ ωcs Figure Valve spring model considering end coil effects (3) To find the solution of this equation, the harmonic balance method is employed for the approximation (Tolstov, 1962) The solution for the internal vibration of 634 H LIU and D KIM the valve spring is assumed to have the form of a Fourier series ∞ Gn ( φ ) = ∑ ( pk cos k φ + qk sin k φ ) (4) k=0 The first and second derivatives of Equation (4) can also be represented in the form of a Fourier series dG-n ∞ -= k( –pk sin kφ + qk cos kφ ) dφ k∑ =0 (5) d -Gn ( φ ) ∞ = ∑ –k ( pk cos kφ + qk sin kφ ) d φ2 k=0 (6) Because the valve motion is periodic, the variable natural frequency, variable damping ratio, and valve acceleration are also periodic and can be expressed in the form of a Fourier series ∞ ζ ( φ ) = ∑ ( ck cos kφ + bk sin kφ ) Figure Direct acting type valve train system (7) k=0 ∞ ωn ( φ ) = ∑ ( ck cos k φ + dk sin k φ ) (8) k=0 ∞ d2y = ( ek cos kφ + fk sin kφ ) dφ2 k∑ =0 (9) Finally, by substituting Equations (4)~(9) into Equation (3), the unknown coefficients in Equation (4) can be calculated The spring reaction force is represented as both static and dynamic terms, as follows: F = Fs + F d (10) In general, the static term of the reaction force can be found from the load-deflection curve If the spring is linear, the static term is simply the spring install load plus the spring stiffness times the valve lift Fs = F0 + ky( φ ) (11) where F0 is the spring install load The dynamic term of the reaction force is related to the internal vibration of the valve spring Fd = π k∑ nGn ( φ ) (12) n TESTING SPRING SURGE AMPLITUDE To check the validity of the present method, a direct acting type valve train system was investigated, as shown in Figure Because the system stiffness of the direct acting type valve train is relatively high, the error between the cam profile and valve motion is small Therefore, the effect Figure Schematic of valve spring surge test rig of the system's internal vibrations on the valve spring surge amplitude is reduced To obtain the valve spring surge signals, a motoring test rig was set up as shown in Figure The cylinder head of a direct acting type OHC engine was mounted on the workbench, and the camshaft was directly driven by a speed-controlled electric motor through a flexible disc coupling A rotation encoder was mounted on the cam shaft All components of the valve train system were assembled inside the cylinder head Lubricating oil was pumped into the cylinder head galleries by an external pump Because the engine block was removed from the test rig, no gas force or crankshaft vibrations were involved in the experiment The internal vibrations of the valve spring were examined by a strain gauge fixed at 0.5 turns from the top boundary point of the spring Figure shows a spring with the strain gauge employed in the test Because the longitudinal motion of the valve spring caused torsional deformation of the spring wire, the internal vibration of the valve spring could be measured by a shear-type strain gauge Figure shows a typical measured spring force result at 2,200 camshaft rpms The surge signal in the spring force represents only one dominant harmonic component Therefore, the first mode of the spring's ESTIMATION OF VALVE SPRING SURGE AMPLITUDE USING THE VARIABLE NATURAL FREQUENCY 635 Figure 10 Valve lift and valve acceleration Figure Valve spring with strain gauge Table Harmonic amplitudes of the valve acceleration n en -0.0035783 Figure Typical measured results for the dynamic spring force fn 0.0 n en fn 11 0.024633 -0.0053618 -0.0090806 4.0852e-05 12 -0.020348 0.0061671 0.031039 -0.0028949 -0.00036838 13 0.008547 -0.045734 0.0010844 14 0.0018228 -0.0009056 0.042278 -0.0019226 15 -0.0065835 0.0024861 -0.018885 0.0019701 16 0.0057925 -0.0016873 -0.012754 -0.00021443 17 -0.0022975 0.0002425 0.035858 -0.0032508 18 -0.0007358 0.00031063 -0.038616 0.0064722 19 0.0016971 9.0003e-005 0.020235 -0.0065023 20 -0.00096377 -0.00048241 10 0.010203 0.0016672 ILLUSTRATIVE EXAMPLES Figure Definition of spring surge amplitude longitudinal vibration is enough to predict the spring surge properties The ratio of the surge amplitude is usually used as the criterion to judge the spring surge property, which is defined as the ratio of the amplitude of the dynamic spring force to that of static spring force, as shown in Figure and Equation (13) It indicates the surge level of the spring and also indirectly indicates the load fluctuation of the valve spring This relative surge amplitude was predicted using the simulation method and then compared with the test results B Samp = A (13) To verify our method, two types of valve springs were simulated and tested in the same valve train One spring was the constant pitch spring, and the other spring was the two-step variable pitch spring The spring surge amplitudes were estimated using our method and then compared with the test results As in Equation (3), the valve spring surge was excited by periodic valve acceleration Figure 10 shows the kinematic valve displacement and acceleration curves used in the experiment The harmonic amplitudes of the acceleration curve are listed in Table 5.1 Estimation of the Surge Amplitude of a Constant Pitch Valve Spring A constant pitch valve spring was investigated in this example The properties of the valve spring were as follows Figure 11 shows the initial pitch curve of the constant pitch valve spring Nα = 5.9, d = 3.4 mm, R = 10.35 mm, ρ = 7850 kg/m3, υ = 0.3, γ = 0.8863 636 H LIU and D KIM Figure 11 Initial pitch curve of constant pitch spring Figure 14 Comparison of measured and calculated spring surge amplitudes (assumed constant natural frequency) Figure 12 Measured natural frequency and damping ratio of a constant pitch valve spring Figure 15 Comparison of calculated surge amplitudes with different damping ratios Figure13 Comparison of measured and calculated spring surge amplitudes (assumed variable natural frequency) Figure 12 shows the measured natural frequency and damping ratio of the spring In this example, the spring amplitudes were estimated using the measured natural frequency and the damping ratio, and the surge amplitude result was compared with the test data in Figure 13 The simulated result agrees well with the test data, which indicates that the proposed method can predict the surge amplitude in the cam shaft speed operating range In most valve train analyses, the natural frequency and damping ratio of a constant pitch spring are assumed to be constant The same simulation was performed, but with a constant natural frequency and damping ratio instead of the varying data in Figure 12 The constant natural frequency was assumed to be 490 Hz, which was calculated by assuming fixed boundary conditions at the install height The damping ratio was assumed to be 0.016 Figure 14 shows the predicted and tested spring surge amplitudes The calculated results of the spring surge amplitude changed with large fluctuations, especially at high cam shaft speeds These results indicate that variation in the natural frequency is important in estimating the spring surge amplitude, even in the case of a constant pitch valve spring Because the two-step valve spring is purposely designed to have varying stiffness, variations in the natural frequency will be highly important in the spring surge estimation The damping ratio is a critical factor in determining the amplitude of the vibration response To check the effect of a varying damping ratio, the same simulation was also performed with a constant damping ratio of 0.016 The natural frequency was varied, as shown in Figure 12 The simulated results for a constant damping ratio were nearly identical to the results for a varying damping ratio, as shown in Figure 15 These findings indicate that variation in the damping ratio does not significantly change the tendency of the spring surge amplitude but does limit the amplitudes at the resonance frequency Therefore, it is ESTIMATION OF VALVE SPRING SURGE AMPLITUDE USING THE VARIABLE NATURAL FREQUENCY 637 possible to estimate the spring surge properties using a constant damping ratio in the initial stages of valve spring design Test data shows that the recommended value of the damping ratio for a constant pitch valve spring is 0.0160.02, and for a two-step variable valve spring is 0.02-0.025 5.2 Estimation of Surge Amplitude of a Two-step Variable Pitch Spring Two-step variable pitch springs are most commonly used in automotive engine valve trains In this example, the constant pitch spring from Example was simply replaced by a two-step variable pitch spring with the same install height The spring was excited by the same valve motion In estimating the spring surge amplitudes, only the spring design parameters, which can be found in the spring drawing, were used The initial pitch curve of the two-step variable pitch spring is given in Figure 16, and the spring design parameters used were as follows: Nα = 6.9, d = 2.9 mm, R = 10.1 mm, ρ = 7850 kg/m3, υ = 0.3, γ = 0.8863 First, the change in the natural frequency was estimated by considering the end coil effects The spring stiffness was attained simultaneously with step-by-step compression The predicted natural frequencies are given in Figure 17 The result shows a jump in the natural frequency at the install height, where the small pitch in Figure 16 collapsed Figure 18 Comparison of measured and calculated spring surge amplitudes of the two-step variable pitch spring As the compression continued, the natural frequency increased gradually, like the natural frequency change in a constant pitch spring In this example, the damping ratio was assumed to be a constant value, 0.02 Finally, the spring surge amplitudes were calculated using the predicted natural frequencies and were compared with the test results in Figure 18 The results show that the spring surge amplitudes can be predicted without any experimental data, which makes valve spring development more efficient Furthermore, the simulated spring surge amplitude also gives feedback to the spring designer, which can be used in modifying the pitch curve and improving the spring surge properties in the design stage CONCLUSIONS Figure 16 Initial pitch of the two-step variable pitch valve spring Figure 17 Predicted natural frequency of two-step variable pitch valve spring In this paper, a method was presented to predict the valve spring surge amplitude by considering the variable natural frequency and damping ratio Prior to estimating the surge amplitude, the theoretical change in natural frequency during the operation was also calculated by considering the end coil effects The following results were obtained: (1) The effect of the natural frequency change on the surge amplitude was significant When the natural frequency was assumed to be constant, the resonance peak of the spring surge drifted to another engine speed (2) The damping ratio of the spring dramatically increased just before the natural frequency increased because of the collapse of close coils However, the increased damping ratio quickly returned to the original value when the close coils completely collapsed Therefore, the effect of damping change on the spring surge amplitude was relatively small The recommended value of the damping ratio for constant pitch valve springs was found to be 0.016-0.02 and was 0.02-0.025 for two-step variable pitch valve springs (3) The valve spring surge amplitude could be predicted from only the valve train design variables without any experimental data The variable natural frequency was calculated from the spring dimensions and pitch curve 638 H LIU and D KIM The equivalent damping ratio of a multi-step pitch spring could also be estimated from the number of close coils The predicted surge amplitude agreed well with the test results, and the ability to make this prediction will make valve spring development more efficient REFERENCES Blair, B G., McCartan, C and Cahoon, W (2010) Measurement and computation of the characteristics of progressive valve springs SAE Paper No 2010-01-1056 Huber, R., Clauberg, J and Ulbrich, H (2010) An efficient spring model based on a curved beam with non-smooth contact mechanics for valve train simulations SAE Paper No 2010-01-1057 Kim, D and David, J W (1990) A combined model for high speed valve train dynamics (partly linear and partly nonlinear) SAE Paper No 901726 Liu, H and Kim, D (2009) Effects of end coils on the natural frequency of automotive engine valve springs Int J Automotive Technology 10, 4, 413−420 Lee, J and Patterson, D J (1997) Nonlinear valve train dynamics simulation with a distributed parameter model of valve springs J Engineering for Gas Turbines and Power, 119, 692−698 Ortmann, C and Skovbjerg, H (2000) Powertrain analysis applications using Adams/Engine powered by FEV Part I: Valve Spring Int ADAMS Users Conf., Rome Pisano, A P and Freudenstein, F (1983) An experimental and analytical investigation of the dynamic response of a high-speed cam-follower system Part 2: A combined, lumped/distributed parameter dynamic model J Mechanisms, Transmissions, and Automation in Design, 105, 699−704 Phlips, P J., Schamel, A R and Meyer, J (1989) An efficient model for valvetrain and spring dynamics SAE Paper No 890619 Schamel, A R., Hammacher, J and Utsch, D (1993) Modeling and measurement techniques for valve spring dynamics in high revving internal combustion engines SAE Paper No 930615 Seidlitz, S (1989) Valve train dynamics – A computer study SAE Paper No 890620 Tolstov, G P (1962) Fourier Series Dover Publications, Inc New York International Journal of Automotive Technology, Vol 12, No 5, pp 639−644 (2011) DOI 10.1007/s12239−011−0074−0 Copyright © 2011 KSAE 1229−9138/2011/060−02 EFFECT OF A 2-STAGE INJECTION STRATEGY ON THE COMBUSTION AND FLAME CHARACTERISTICS IN A PCCI ENGINE Y J KIM1), K B KIM2) and K H LEE1)* 1) Department of Mechanical Engineering, Hanyang University, Gyeonggi 426-791, Korea School of Mechanical Engineering, Chungbuk National University, Chungbuk 361-763, Korea 2) (Received 12 March 2010; Revised April 2011) ABSTRACT−Recently, to reduce environmental pollution and the waste of limited energy resources, there is an increasing requirement for higher engine efficiency and lower levels of harmful emissions A premixed charge compression ignition (PCCI) engine, which uses a 2-stage type injection, has drawn attention because this combustion system can simultaneously reduce the amount of NOx and PM exhausted from diesel engines It is well known that the fuel injection timing and the spray angle in a PCCI engine affect the mixture formation and the combustion To acquire two optimal injection timings, the combustion and emission characteristics of the PCCI engine were analyzed with various injection conditions The flame visualization was performed to validate the result obtained from the engine test This study reveals that the optimum injection timings are BTDC 60° for the first injection and ATDC 5° for the second injection In addition, the injection ratio of to showed the best NOx and PM emission results KEY WORDS : Flame visualization, Emissions, Premixed charge compression ignition (PCCI), NOx, PM INTRODUCTION a well-mixed lean mixture and no diffusion flame Its combustion is initiated at the same time throughout the entire area of the combustion chamber The homogeneous mixture restricts the emissions of PM, which is generated due to locally excessive fuel richness The thermal efficiency is also expected to increase because the time loss decreases when the diffusion combustion is absent Hino Motors conducted many studies on a split injection strategy in a single cylinder direct injection engine to improve its emission performance (Yokota et al., 1997) Toyota Motor Co studied a smokeless mechanism by reducing the temperature (Akihama et al., 2001) UNIBUS (UNIform BUlky combustion System) of Toyota is a technology using an early (BTDC 50o) and a late injection (ATDC 13o) (Hasegawa et al., 2003) In this study, a 4-cylinder diesel engine was used to analyze the combustion and emission characteristics of the PCCI strategy with a 2-stage injection In addition, the flame visualization technique was employed to observe the PM formation and to support the engine test analysis Based on these data, we obtained the optimum injection timings and injection ratio for the two injections, which allowed a simultaneous decrease in NOx and PM in the PCCI engine It is essential to reduce the amount of NOx and particulate matter (PM) emitted from diesel engines because they are harmful to the environment Many ongoing studies focus on high pressure fuel injection systems (e.g., the common-rail system) and after-treatment techniques of diesel engines to reduce the emission of NOx and PM simultaneously; however, such a goal is undoubtedly difficult to attain Compared to conventional diesel engines, the premixed charge compression ignition (PCCI) diesel engine presents a new combustion concept with lower emissions and higher thermal efficiency (Kathi et al., 2002; Neely et al., 2004) This combustion mechanism is designed to achieve compression ignition by a fully premixed homogeneous charge instead of the diffusion combustion A 2-stage injection strategy, including an early injection and a late injection, gives PCCI a leaner and more homogeneous mixture and a lower combustion temperature than the conventional diesel engines, which leads to a decrease in the NOx and PM levels However, the PCCI engine still has some drawbacks such as the limited driving range, the difficult ignition timing control and the increase in CO and HC due to decreasing the in-cylinder pressure and temperature at the early injection timing Such issues must be overcome for practical use Technically, the combustion principle of PCCI relies on METHOD OF MEASUREMENT 2.1 Engine Performance Test Figure shows the schematic of a common rail direct *Corresponding author e-mail: hylee@hanyang.ac.kr 639 770 J K LEE and J K SHIM is parallel to the driving direction of the vehicle In their research, they report that their method does not give an exact solution for the roll axis of the vehicle body in its initial symmetric position In this paper, a three-dimensional method to determine the kinematic roll axis of a full-vehicle model in an initial symmetric position is proposed In this research, the kinematic roll axis, which will be referred to as the roll twist axis, is defined as the instantaneous screw axis of the vehicle body with respect to the ground in roll motion Also, contrary to the conventional definitions of the roll axis and roll center, the kinematic roll centers of the front and rear suspension systems are defined as the intersections of the kinematic roll axis of the vehicle with the respective transverse vertical planes through the front and rear wheel contact points The proposed method is based on the theory of screws, which is a mathematical tool widely used in spatial kinematics and robotics The results of the proposed analysis method are compared with the conventional planar methods, and the validity and limitations of the conventional roll center determination methods are reviewed from the standpoint of three-dimensional kinematics SCREW THEORY The theory of screws provides a useful tool for representing finite and infinitesimal displacements, instantaneous motion properties of a rigid body in kinematics, and equivalent systems of forces acting on a rigid body in statics (Ball, 1900; Roth, 1984) A screw is a line or axis with an associated pitch In kinematics, any displacement of a rigid body can be achieved by a rotation about a unique axis combined with a translation parallel to that axis The unique axis of this screw displacement or twist is referred to as the screw axis of the displacement, and the ratio of the translation to the rotation is called the pitch of the screw The amplitude or magnitude of the twist is the angular displacement about the axis Similarly, any instantaneous motion of a rigid body can be described by a screw motion about an instantaneous screw axis, and the pitch of this motion is the ratio of the translational velocity along the screw axis to the angular velocity about the axis The screw motion or twist is referred to as a twisting motion in this case, and the amplitude of the twist is the angular velocity In statics, any system of forces and couplings acting on a rigid body can be uniquely reduced to a single force along a unique axis and a coupling about the same axis This unique axis is called the screw axis, and the combination of the coaxial force and coupling is called a wrench acting on a screw The pitch of the wrench is the ratio of the moment to the force, and the amplitude or intensity of the wrench is the magnitude of the force A screw, $, can be used to describe either a twist or a wrench and can be represented conveniently by nonnormalized or general screw coordinates (Yuan and Freudenstein, 1971; Hunt, 1978; Davidson and Hunt, 2004) defined by a pair of vectors as follows: $ = [ S;So ] = [ L, M, N ;P*, Q*, R* ] (1) where S is the direction vector along the screw axis The vector So is defined by So = r × S + hS (2) where r is the position vector of any point on the screw axis and h is the pitch of the screw, which is given by S ⋅ S LP* + MQ* + NR-* h = -0 = -S⋅S L2 + M2 + N2 (3) By setting the amplitude or intensity of the screw to ρ = S ⋅ S = L2 + M2 + N2 (4) the screw, $, can be written in terms of its amplitude ρ and its unit screw, $ˆ , as $ = ρ$ˆ = ρ [ s;s0 ] = [ ρ l, ρ m, ρn ;ρ p*, ρ q*, ρ r* ] = ρ[ l, m, n;p*, q*, r* ] (5) where l2 + m2 + n2 = (6) If a twist represented by the screw coordinates describes the velocity of a rigid body, then the first three components of the twist represent the angular velocity, and the last three components represent the linear velocity of a point that is instantaneously coincident with the origin of the reference frame However, if a screw represents a wrench acting on a rigid body, then the first three components of the wrench represent the resulting force, and the last three components represent the resulting moment due to the combined effects of the force and the couple about the origin of the reference frame (Tsai, 1999) Therefore, if a wrench $1= [S1; S01]= ρ1[s1; s01]=ρ1 [l1, m1, n1;p*1, q*1, r*1 ] acts on a rigid body that undergoes a twisting motion represented by a twist $2= [S2; S02]=ρ2[s2; s02]=ρ2 [ l2, m2, n2;p*2, q*2, r*2 ] , then the rate of working, dW/dt, of the wrench on the rigid body (Davidson and Hunt, 2004) is given by dW = S1 ⋅ S02 + S2 ⋅ S01 dt = ρ1 ρ2 ( l1p*2 + m1 q*2 + n1 r*2 + l2 p*1 + m2 q*1 + n2r*1 ) (7) If the rate of working is zero, then the two screws $1 and $2 are said to be reciprocal (Ball, 1900), and Equation (7) becomes dW = l1 p*2 + m1 q*2 + n1r*2 + l2p*1 + m2q*1 + n2 r*1 = dt (8) Note that the amplitudes of screws not affect the VALIDITY AND LIMITATIONS OF THE KINEMATIC ROLL CENTER CONCEPT 771 reciprocity of screws A unit screw depends on five parameters: one for the pitch and four for the axis Hence, there exists a quadruple infinity (∞4) of screws reciprocal to a given screw If five independent screws, $1 to $5, are given, then writing Equation (8) five times gives (9) Equation (9) with the unity condition l2 + m2 + n2 = (10) will yield a unique unit screw $ˆ that is reciprocal to the given five screws The rate of working of a force F on a body that has a velocity v can also be obtained by dW = F ⋅ v dt (11) Equation (11) states that if the force and the velocity are perpendicular to each other, then the rate of working of the force is zero Hence, if the velocity of a point of interest is given, the reciprocal wrenches acting on the point can be obtained: these wrenches pass through the point and lie in the plane normal to the velocity vector FULL-VEHICLE MODEL In this section, the equivalent full-vehicle kinematic model in its initial symmetric position is constructed, and a threedimensional kinematic method to determine the kinematic roll axis of the vehicle model using screw theory is proposed In general, a suspension mechanism has one degree of freedom when the steering motion is constrained The instantaneous motion of the wheel with respect to the vehicle body can be represented by a twist; hence, a suspension mechanism can be modeled as a screw or an instantaneous helical (H) joint that connects the wheel to the vehicle body For example, Figure shows a double wishbone suspension mechanism and its instantaneously equivalent kinematic model in which the suspension is replaced by an instantaneous helical joint placed at the position of the twist Using this approach, a full vehicle that has four wheels can be modeled as a vehicle body with four wheels, each of which is connected to the body by a helical joint To perform a roll axis analysis of the vehicle model, the contact between the wheel and the ground needs to be modeled with a proper equivalent joint Assuming that the wheel always remains in contact with the ground, the contact between the wheel and the ground can be modeled as a joint that has five degrees of freedom: three rotational Figure Double wishbone suspension and its instantaneously equivalent mechanism with a helical joint and two translational In the initial symmetric position of the vehicle, however, the wheel contact points should have zero translational velocity to ensure the symmetry and continuity of the twist axis of the wheel with respect to the ground (Lee, 2007) Hence, the wheel contact point can be modeled as a spherical (S) joint Now, the full-vehicle model in an initial symmetric position is intermediately modeled as a vehicle body connected to the ground by four H-S links When the vehicle model is under a lateral force that causes the vehicle body to roll, the reaction force at each wheel acts through the suspension system, and hence the vehicle body is acted upon by four wrenches, which will be referred to as the suspension wrenches in this paper If the four suspension wrenches are determined and an additional wrench corresponds to the roll condition of the vehicle is specified, the roll twist axis that is reciprocal to the five wrenches acting on the vehicle body can be determined using Equations (9) and (10) In the kinematic inversion of the model with the vehicle body to be fixed, let the velocity of the spherical joint with respect to the helical joint be vS From Equation (11), to make the rate of working zero, the suspension wrench wS acting through the spherical joint must be perpendicular to the velocity vS, and hence wS must lie in the plane perpendicular to vs This plane is referred to as the wrench plane The helical-spherical (H-S) link, its wrench plane, and the planar pencils of possible wrenches are shown in Figure When the vehicle in its initial symmetric position is under a lateral force, only the Y and Z components of the wheel reaction force exist because there is no pitch motion; hence, the suspension wrench wS acts on the YZ plane through the center of the spherical joint Therefore, wS lies on the line of intersection between the wrench plane and the YZ plane, as shown in Figure Now, the relationships among the wrench wS, the velocity vS, and the projection of vS on the YZ plane, vS/YZ, can be observed Because the velocity of the spherical joint vS is normal to the wrench plane, and the wrench wS lies on the line that is common to the wench plane and the YZ plane and passes through the spherical joint, wS and vS/YZ are perpendicular to each other by the theorem of three perpendiculars, as shown in Figure 4, which is drawn in a rotated position to aid visualization This observation proves the validity of the conventional planar roll center determination method in which the kinematic roll center is 772 J K LEE and J K SHIM Figure Velocity method Figure H-S link, wrench plane and possible wrenches Figure Projection method (Reimpell et al., 2001) Figure Suspension wrench wS acting through the H-S link in the YZ plane Figure Graphical construction of the theorem of three perpendiculars determined as the intersection of the vehicle body’s center line and the trace normal or velocity normal line of the wheel contact point in symmetric situations Figure shows the graphical construction of the method that uses the line normal to vC/YZ, which is the projected velocity of the wheel contact point velocity vC onto the YZ plane, to determine the roll center position In this research, this method is referred to as the velocity method The velocity method is readily used not only for planar suspension mechanisms but also for spatial suspension mechanisms if the direction normal to the projected velocity of the wheel contact point with respect to the vehicle body onto the YZ plane is obtained Figure shows a roll center determination method that can be used if the suspension control arm rotation axes are inclined at an angle to one another (Reimpell et al., 2001) In this method, when the suspension geometry is projected onto the XZ plane as shown in the left of Figure 6, the points of intersection a2 and b2 of the vertical lines drawn from the joints a1 and b1 on the wheel and the respective inclined control arm rotation axes are used to determine the equivalent planar suspension mechanism when viewed from the rear, as shown in the right of Figure This method is referred to as the projection method in this research Because there exist five wrenches between the vehicle body and the wheel in the given suspension geometry, the twist axis of the wheel with respect to the vehicle body can be determined (Lee, 2007) Using the twist axis of the wheel with respect to the vehicle body and the wheel contact point, the suspension wrench acting on the vehicle body through each wheel can be obtained in the initial symmetric position of the vehicle model Up to this point, four suspension wrenches acting on a vehicle body with four wheels have been determined To determine the roll twist axis of the vehicle body in roll motion using Equations (9) and (10), one more wrench must be specified When the vehicle in its initial symmetric position is acted upon by a lateral force that causes the vehicle body to undergo roll motion, the center of gravity of the vehicle body does not have velocity in the longitudinal direction Hence, it is proper to assume that the fifth wrench is acting on the center of gravity in the longitudinal direction Now, the roll twist axis of the vehicle body that is reciprocal to the five wrenches can be determined for the vehicle body in its initial symmetric position NUMERICAL EXAMPLE As a numerical example of the proposed method of determining the roll twist axis, a full-vehicle model with double wishbone suspensions (Figure 7(a)) in the front and multi-link suspensions (Figure 7(b)) in the rear is analyzed, and the results are compared with the two conventional methods explained in Section 3: the projection method and the velocity method In Figure (a), the two revolute joints of the double wishbone suspension, whose directions of rotation are ua and ub respectively, are replaced by two spherical joints on the corresponding axes of rotation This VALIDITY AND LIMITATIONS OF THE KINEMATIC ROLL CENTER CONCEPT Figure (a) Double wishbone suspension (b) Multi-link suspension Table Joint positions of a double wishbone suspension Joint a0 Body joint positions X 20.000 Y Z 350.000 210.000 ' 340.000 370.000 220.000 b0 -160.000 420.000 680.000 a b ' 60.000 c0 Joint 420.000 660.000 160.000 370.000 330.000 Wheel joint positions X Y Z a1 -10.000 740.000 220.000 b1 20.000 660.000 690.000 c1 130.000 700.000 330.000 Table Joint positions of a 5-SS multi-link suspension Joint Body joint positions X Y Z Joint Wheel joint positions X Y Z 773 Figure Modified projection method for multi-link suspension equivalent to the original suspension mechanism The projection method shown in Figure 6, however, cannot be directly applied to multi-link suspensions Figure shows a modified projection method used for the multi-link suspension mechanism in this paper In this projection method, when viewed from the side, the intersection of the extension lines of the upper links connecting the wheel to the body is considered to be a virtual joint Then, the method shown in Figure is applied to determine the rear roll center of the full-vehicle model The second method compared in this paper is the velocity method The velocity of the wheel contact point with respect to the vehicle body can be calculated after the twist axis of the wheel with respect to the vehicle body is obtained The third method is the method proposed in this paper The front and rear roll centers of the full-vehicle model are determined as the points of intersection between the roll twist axis of the vehicle body and the transverse vertical planes through the front and rear wheel contact points, respectively a0 2400.000 550.000 290.000 a1 2600.000 700.000 220.000 b0 2700.000 270.000 230.000 b1 2700.000 720.000 190.000 c0 2800.000 250.000 270.000 c1 2840.000 730.000 240.000 RESULTS d0 2600.000 500.000 680.000 e0 2840.000 470.000 680.000 d1 2710.000 680.000 700.000 In this section, the results of roll center analyses using each of the three methods are presented Table shows the calculated screw coordinates of the suspension wrenches and the roll twist axis of the full-vehicle model in its initial symmetric position using the proposed three-dimensional method Table shows the kinematic roll center positions obtained by all three different methods Table shows that the roll center positions obtained by the velocity method and the proposed method are the same, as explained in Section The result obtained by the projection method has slight differences in the positions of the front and rear roll centers, possibly because the planar mechanism obtained from the projection of the three-dimensional suspension mechanism does not accurately represent the motion of the original mechanisms Nevertheless, the result might be acceptable because the differences are between 10-1mm to 100mm while the scale of the model is 103mm Figure 10 shows the vehicle model and the roll twist axis obtained by the proposed method Table Wheel contact points Types of suspension Double wishbone Multi-link Wheel contact point X Y Z -10.000 770.000 0.000 2690.000 760.000 0.000 replacement does not change the degrees of freedom of the suspension mechanism and its motion Tables to list the positions of the joints and wheel contact points of the righthand side suspension mechanisms of the full-vehicle model viewed from the rear The XYZ coordinate system used in this paper is shown in Figure 10 The joint positions of the left-hand side suspension systems can be obtained by changing the signs of the Y values in Tables to The analysis results of the three different methods are compared in the following section The first method is the projection method shown in Figure 6, which uses the projections of the suspension geometry onto the XZ plane and the YZ plane to determine the planar mechanism CONCLUSIONS In this research, the conventional definition of the kinematic roll axis is reviewed using the theory of screws, 774 J K LEE and J K SHIM Table Screw coordinates of the instantaneous screw axes Suspension wrenches & Roll twist axis Screw coordinates l m n p* q* r* Front Left 0.000 0.997 0.076 58.777 -0.763 9.971 Front Right 0.000 -0.997 0.076 -58.777 -0.763 -9.971 Rear Left 0.000 0.987 0.158 119.836 424.155 -2656.349 Rear Right 0.000 -0.987 0.158 -119.836 424.155 2656.349 Roll twist axis 0.999 0.000 0.023 0.000 -59.164 0.000 Table Roll center positions Method Front roll center Rear roll center X Y Z X Y Z Projection method -10.000 0.000 58.396 2690.000 0.000 119.724 Velocity method -10.000 0.000 58.949 2690.000 0.000 121.354 Proposed method -10.000 0.000 58.949 2690.000 0.000 121.354 Figure 10 Full-vehicle model and its roll twist axis which is a useful three-dimensional analysis tool in kinematics The kinematic roll axis, referred to as the roll twist axis in this paper, is the instantaneous screw or twist axis of the vehicle body with respect to the ground in roll motion With the aid of screw theory and the concept of the rate of working, the proposed method first determines the five wrenches acting on the vehicle body in roll motion in its initial symmetric position, and the roll twist axis of the vehicle is then determined from the five wrenches using the reciprocity of screws After the roll twist axis is obtained, the front and rear roll centers are determined as the intersections of the roll twist axis of the vehicle with the respective transverse vertical planes through the front and rear wheel contact points This definition is contrary to the conventional definition of the roll axis in which the roll axis is defined as the line joining the front and rear roll centers In this research, the validity of the velocity method is verified using screw theory It is also found the projection method yields quite accurate roll center positions However, the conventional planar methods may have limitations in practice as the results may be valid only when the vehicle is in symmetric situations When applying the proposed screw method to the vehicle model in symmetric situations, the wrenches of the left and right suspensions intersect each other because they exist in the vertical planes passing through the wheel contact points, and the intersection points in the front and rear planes are the front and rear roll centers, respectively These results are the same as those of the conventional methods When the vehicle is not in symmetric situations, the conventional planar methods applied to a three-dimensional suspension mechanism may yield errors because the lines normal to the projected velocities of the wheel contact points onto the XZ plane not generally intersect The proposed method is based upon a three-dimensional kinematic analysis technique and directly determines the roll twist axis rather than determining it from the front and rear roll centers; hence, it can be extended to vehicle models in asymmetric situations ACKNOWLEDGEMENT−This research was supported by a Korea University Grant and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No 2009-0063169) REFERENCES Ball, R S (1900) A Treatise on the Theory of Screws VALIDITY AND LIMITATIONS OF THE KINEMATIC ROLL CENTER CONCEPT Cambridge University Press New York Davidson, J K and Hunt, K H (2004) Robots and Screw Theory: Applications of Kinematics and Statics to Robotics Oxford University Press New York Dixon, J C (1987) The roll-centre concept in vehicle handling dynamics Proc Institution of Mechanical Engineers 201, D1, Part D, Transport Engineering, 69− 78 Dixon, J C (1996) Tire, Suspension and Handling 2nd Edn SAE Int Warrendale Gerrard, M B (1999) Roll centres and jacking forces in independent suspensions – A first principles explanation and a designer’s toolkit SAE Paper No 1999-01-0046 Gillespie, T D (1992) Fundamentals of Vehicle Dynamics SAE Inc Warrendale Goodsell, D (1995) Dictionary of Automotive Engineering, 2nd edn SAE Inc Warrendale Hunt, K H (1978) Kinematic Geometry of Mechanisms Clarendon Press Oxford Jones, R A (1999) Understanding vehicle roll using mechanism simulation software SAE Paper No 199901-0030 Jung, J., Shim, T and Gertsch, J (2009) A vehicle rollstability indicator incorporating roll-center movements IEEE Trans Vehic Tech., 58, 4078−4087 Lee, J K (2007) Screw Axis Theory and its Applications to the Behavior Analysis of Full-Vehicle Models and the Synthesis of Suspension System Ph D Dissertation Korea University Seoul Korea Lee, J K and Shim, J K (2006) Roll center analysis of a half-car model using pole for small displacement Int J Automotive Technology 7, 7, 833−839 775 Lee, J K and Shim, J K (2006) Roll motion analysis of a full vehicle model using screw theory 31th FISITA World Automotive Cong., Yokohama, Japan, F2006V114 Milliken, W F and Milliken, D L (1995) Race Car Vehicle Dynamics SAE Int Warrendale Mitchell, W C (1998) Asymmetric roll centers SAE Paper No 983085 Mitchell, W C (2006) Force-based roll centers and an improved kinematic roll center SAE Paper No 2006-013617 Moline, D., Vaduri, S and Law, E H (2000) Fidelity of vehicle models using roll center principles SAE Paper No 2000-01-0693 Morse, P and Starkey, J M (1996) A force-based roll center model for vehicle suspensions SAE Paper No 962536 Raghavan, M (2005) Suspension synthesis for N:1 Roll center motion ASME J Mech Des., 127, 673−678 Reimpell, J., Stoll, H and Betzler, J W (2001) The Automotive Chassis 2nd Edn SAE Inc Warrendale Roth, B (1984) Screws, motors, and wrenches that cannot be bought in a hardware store Proc 1st Int Symp Robotics Research, 679−693 Suh, C H (1991) Suspension analysis with instant screw axis theory SAE Paper No 910017 Tsai, L W (1999) Robot Analysis: The Mechanics of Serial and Parallel Manipulators John Wiley & Sons New York Yuan, M S C and Freudenstein, F (1971) Kinematic analysis of spatial mechanisms by means of screw coordinates Part 1-Screw coordinates Trans ASME, J Eng For Industry, 93, 27−33 International Journal of Automotive Technology, Vol 12, No 5, pp 777−785 (2011) DOI 10.1007/s12239−011−0090−0 Copyright © 2011 KSAE 1229−9138/2011/060−18 VEHICLE INTERIOR NOISE MODEL BASED ON A POWER LAW H.-S KOOK1)*, D LEE2) and K.-D IH2) 1) Department of Automotive Engineering, Kookmin University, Seoul 136-702, Korea Test Team 1, Test Center, Hyundai Motor Co., 772-1 Jangdeok-dong, Hwaseong-si, Gyeonggi 445-706, Korea 2) (Received 17 November 2010; Revised 22 March 2011) ABSTRACT−Identifying the components of a vehicle’s interior noise is important in many phases of the noise, vibration, and harshness (NVH) development process Many test methods that have been widely used in the automobile industry to separate noise sources are based on system identification methods in the frequency domain However, none of the frequency response function-based methods can directly estimate the wind noise component In this article, an analytical model for the interior noise level based on a simple power law was developed It was assumed that the mean squared acoustic pressure for the interior noise could be obtained by summing up those of the wind noise, road noise, and background noise The wind noise and road noise were further assumed to depend only on wind speed and the vehicle’s driving speed, respectively, and to follow a simple power law The resulting analytical model includes five parameters that can be optimized for the vehicle and the road The validity of the model was verified by using data obtained from cruise tests performed on a proving ground for cruise speeds ranging from 40 km/h to 130 km/h The model is applied to the overall and 1/3-octave bands of interior noise and is shown to describe the data trends fairly well For the test vehicle used in the present work, the overall mean squared pressures for the wind and road noise components are shown to be proportional to the wind speed to the 5.8 power and to the driving speed to the 3.4 power, respectively KEY WORDS : Wind noise, Road noise, Separation of noise INTRODUCTION underbody Finally, engine noise includes intake/exhaust noise as well as engine noise itself transmitted through the mounts, the driveshaft, and airborne paths The dominant noise source in a frequency band varies and shifts as the cruise and wind speeds change Typically, road noise is known to dominate the interior noise up to a mid-frequency range of approximately 500 Hz for relatively low cruise speeds At a higher cruise speed of 140 km/h, wind noise has been observed to dominate the interior noise at almost all audible frequency ranges for a passenger vehicle (Lindener et al., 2007) The proportion of the engine noise component is relatively small compared to that of other noise sources because power train noise is acoustically managed relatively well by insulation However, during rapid acceleration, engine noise may dominate the interior noise in the 1/3-octave bands associated with the engine’s rotational speed and its harmonics The source of any abnormal noises should be identified so that the research and development department responsible for the source can take appropriate action Separation of the interior noise into different noise sources is also important in terms of balancing noises in the design and development stage: Appropriate subjective or objective NVH targets can be made at the module and component levels However, for road and wind noises, it is not easy for even a trained ear to distinguish one noise from another because they are generally broadband Moreover, because these noises are A vehicle’s interior noise and/or sound is an important cause of consumer complaints and a determinant of consumer satisfaction with a chosen vehicle As vehicles are required to be lighter to improve fuel economy, more engineering efforts must be devoted to reducing interior noise However, quietness is not always the main objective Drivers sometimes expect interior noise to be dynamic and sporty, depending on the type of vehicle The characteristics of a vehicle’s interior noise are currently believed to be one of the most important factors determining the vehicle’s brand image Motor companies are therefore making more efforts than ever to improve the noise, vibration, and harshness (NVH) characteristics to gain a competitive edge in the market A vehicle’s interior noise is the sum of all of the noises caused by many sources transmitted through various paths Noise sources can be grouped into three categories: road noise, wind noise, and engine noise Road noise is caused by tire/road interaction mechanisms such as tread impact, stick/slip on the tire/road interface, and air-pumping of the tread Wind noise is associated with turbulent pressure fluctuations exerted by the air flow on surface areas of the vehicle such as the windshield, A-pillar, side mirrors, and *Corresponding author e-mail: kook@kookmin.ac.kr 777 778 H.-S KOOK, D LEE and K.-D IH transmitted to the cabin by various paths, most sound localization techniques (i.e., acoustic holography techniques and acoustic beamforming techniques) are not useful for separating road noise from wind noise in low-frequency bands Many test methods that have been widely used in the automobile industry to separate noise sources are based on system identification methods in the frequency domain These methods identify frequency response functions in a multiple-input/single-output model Such methods include the multiple coherence function method (Bendat, 1976; Bendat and Piersol, 2000), transfer path analysis (Van der Auweraer et al., 2007), and operational transfer path analysis (de Klerk and Ossipov, 2010) One critical drawback of the frequency response function-based methods is that they cannot yield reliable results for the contribution of wind noise Because wind noise is not properly described by combinations of linear processes of a limited number of input signals, its contribution cannot be directly estimated by frequency response function-based methods The contribution of wind noise is therefore sometimes indirectly estimated using the residual spectral component, which cannot be described by the input signals However, this could lead to serious overestimation of the wind noise when wind noise is not the dominant noise source For instance, a 3-dB error in the estimation of the spectral level of road and engine noise is not unusual in the aforementioned methods The spectral components based on the measured inputs are often underestimated due to nonlinearity in the measurement system (especially in the higher-frequency bands, where wind noise is often dominant) If the spectral density of the sum of road and engine noise is dB (i.e., 50%) lower than that of the interior noise due to system nonlinearity, the indirect method would erroneously result in a wind noise contribution as large as the sum of those of the road and engine noise, although in reality the wind noise component is negligible This paper proposes a new method that can separate the wind noise contribution from those of other noises The new method is not based on a frequency response function model but on a simple power law model The present work is motivated by Romberg and Lajoie (1977), who, more than three decades ago, first introduced a test and analysis method that can estimate the wind noise component from the total interior noise measured in a vehicle in operation on the road In section 2, the theoretical background of the power law model presented in Romberg and Lajoie’s work and the new method proposed in the present work are described Section presents and discusses the results of an empirical test of the method POWER LAW MODEL FOR A VEHICLE’S INTERIOR NOISE Most of the acoustic power of each of the three major noise sources is determined primarily by one or at most two relevant parameters These parameters are the vehicle driving speed for road noise, the wind speed and wind yaw angle for wind noise, and the engine rpm and load for engine noise In some cases, a simple relationship between the sound pressure level of a noise source and the associated parameters can be established using experimental data or a mathematical analogy, as is the case for wind noise The acoustic power of the wind noise is generally considered to be proportional to the wind speed to some power (for example, 5.5 in Harris, 1957, 6.11 in Oettle et al., 2010, and in Mousley et al., 2001) The power-law-based model explained here is based on such a relationship 2.1 Previous Power Law Model In a cruise test performed on proving grounds, the interior noise level is known to not be precisely repeatable but to vary to some extent from one test to another, even though road tests are repeated with care on the same road and at the same driving speed The interior noise level varies from one test to another because continuously varying wind conditions change the wind noise component, whereas the other noise components remain almost unchanged Romberg and Lajoie (1977) developed a simple method for estimating the wind noise component in a vehicle’s interior cabin by using the sensitivity of the interior noise level to changes in wind conditions during cruise tests Their model is based on the assumption that the mean squared acoustic pressure of the total interior noise can be obtained by summing up those of the wind noise and the residual noise (that is, road and engine noise) This is a reasonable assumption because the wind noise is uncorrelated with the residual noise They also assumed, following Harris (1957), that the mean squared acoustic pressure of the wind noise component is proportional to the dynamic pressure to the 2.75 power The mean squared acoustic pressure of the total interior noise can thus be modeled as: 2 p = Kq2.75 + pres , (1) where K is a proportional constant for the wind noise, q is the dynamic pressure measured in the in-operation vehicle on the road, and the second term on the right-hand side is the mean squared acoustic pressure of the residual noise, which is assumed to be approximately constant for a constant driving speed The dependence of the mean squared acoustic pressure on the dynamic pressure is equivalent to a 5.5 power dependence on wind speed because the dynamic pressure is the wind speed squared Note that the sensitivity of the wind noise to the wind yaw angle was not modeled in Equation (1) However, the increase in the wind noise level has been observed to be at most 1.5 dB for a yaw angle of 20o for a vehicle tested in a wind tunnel Other investigators (Oettle et al., 2010; Peric et al., 1997) also observed experimentally that the wind noise level increases up to approximately to 1.5 dB for VEHICLE INTERIOR NOISE MODEL BASED ON A POWER LAW wind yaw angles within ±10o as compared to zero yaw angle conditions on the road Note also that the residual noise resulting from road and engine noise is assumed to be approximately constant for a particular cruise speed This assumption is valid only when the engine rpm varies approximately proportionally with the cruise speed and when varying wind conditions not impose serious changes in engine loading This condition may not hold for cruise tests at low driving speeds or for lightweight vehicles with low engine power To obtain the wind speed relative to the vehicle during cruise tests on the proving ground, Romberg and Lajoie attached a pitot tube to the test vehicle and continuously monitored the dynamic pressure it measured To obtain test data over a wider range of wind speeds, tests were performed in both upwind and downwind conditions on an oval track on the proving ground Scattered data for the mean squared acoustic pressure of the interior noise were obtained for a cruise test with a nominally constant driving speed and were plotted as a function of the dynamic pressure to the 2.75 power The proportional parameter K could then be identified as the slope of the straight line that best fits the data in the plot in the least squares sense The residual component in Equation (1) is the intercept of the straight line with the ordinate of the plot and indicates the mean squared acoustic pressure of the road and engine noise at the specific cruise speed tested In theory, by repeating the procedure mentioned above for each set of cruise tests performed over a range of cruise speeds, not only the wind noise component but also the residual component can be obtained as functions of the wind and vehicle speeds, respectively The analytical method was shown to be applicable to octave band levels as well as to overall levels In practice, the analytical method proposed by Romberg and Lajoie has some limitations A straight line can be relatively easily determined when the wind noise dominates the interior noise However, when this is not the case, the straight line fitting is subject to considerable variation, and the accuracy of the estimation of the wind noise contribution as a function of wind speed could be questionable As a result, the proportional constant K estimated by the analytical method proposed by Romberg and Lajoie could vary greatly from one set of cruise tests to another at a different cruise speed, which was in fact observed by the present authors in data obtained from a cruise test similar to theirs 2.2 Improved Power Law Model A new power law model is proposed in this article to overcome some of the limitations of Romberg and Lajoie’s model In the new model, the assumption that wind noise is more sensitive to changes in wind speed than to changes in the wind yaw angle for typical wind conditions on the road is used, as in Romberg and Lajoie’s model In the new model, the mean squared acoustic pressure of the vehicle’s interior noise is represented as: p = Kw ( Vw )α + Kr ( Vr )β + pbgn , 2 779 (2) where VW and Vr indicate the wind and vehicle speeds, respectively; Kw and Kr are proportional constants for the wind and road noise components, respectively, α and β are speed powers, and the last term on the right-hand side is a mean squared acoustic pressure for the background noise In Equation (2), the measured variables are p2 , VW, and Vr, and the parameters to be estimated are Kw, Kr, α, β, and pbgn One difference between this model and that of Romberg and Lajoie as expressed in Equation (1) is that the mean squared acoustic pressure of the road noise component is now modeled to be proportional to the driving speed to some power, similar to the concept on which the wind noise model is based The surface roughness signal experienced by a vehicle running on a typical road may be considered to be a realization of a homogeneous and isotropic random process (Dodds and Robson, 1973) The characteristics of road surface roughness can then be represented by the autopower spectral density as a function of wavenumber The auto-power spectral density generally decays as the wavenumber increases and can be approximated for a limited range of wavenumbers by a single profile of a straight line with a negative slope in a log-log-scaled spectral density plot (Robson, 1979) That is, G( k ) = ck–γ , (3) where k is the wavenumber, γ is the slope in the log-log plot, and c is a proportional constant For a broader range of wavenumbers, the spectral density may be better represented by double or multiple profiles with different slopes that are appropriate for each range If the spectral density is well represented by a single profile, the power spectral density of the excitation force arising from the tire/road interaction at a given frequency would increase proportionally to the driving speed to the γ power because the wavenumber k and frequency w are related by k = ω/v, where v is the driving speed Because the dependence of the road noise component on the vehicle driving speed is modeled explicitly in Equation (2), the data obtained from cruise tests, which were obtained at different cruise speeds, can be applied as a whole to globally estimate the parameters in Equation (2), instead of being applied one by one for each cruise speed, as was the case for the model in the previous study Another difference is that the exponents α and β are not fixed constants but are now parameters to be found This is understandable for road noise because the exponent β would obviously depend on the choice of road and the type of tire For wind noise, parameterizing the exponent could also be useful because the exponent may be not universal but dependent on a vehicle’s surface shape Another reason is worth mentioning Let us consider, in the case of road noise, why parameterization of the exponents is necessary 780 H.-S KOOK, D LEE and K.-D IH Similar arguments may be applicable to wind noise If the auto-power spectral density for road excitation is well represented by a single profile (i.e., one slope) for the entire frequency range of interest, the exponent β would be determined as the single constant γ expressed in Equation (3) for the entire frequency range However, if the autopower spectral density is better represented by multiple profiles, the exponent β could vary with frequency and be determined as the slope of the profile defined in the corresponding wavenumber range associated with a specific frequency In this case, the road noise would be better represented by an exponent that varies with frequency than by a fixed exponent This could be useful, especially when the power law model is applied to each octave or 1/3octave bandpass-filtered signal Another difference is that the interior noise model now includes a background noise component The background noise term is a constant term and is independent of both the wind speed and the driving speed Note that the seemingly constant term for the residual noise component in Romberg and Lajoie’s model expressed in Equation (1) is in fact a function of the driving speed The background noise level in typical proving grounds is very low, especially in the highfrequency bands When the level of either the wind noise or the road noise component is far greater than the background noise level, even at the lowest cruise speed tested, it is not possible to accurately estimate the power of the background noise by any means, and the estimation of the background noise level would therefore be meaningless However, if either the wind noise or road noise component, which is masked by the background noise at lower cruise speeds, gradually increases and finally dominates the interior noise as the vehicle’s cruise speed increases, inclusion of the background noise term is useful for more accurately estimating the parameters for the noise component Note that the engine noise is not explicitly modeled in Equation (2) As noted above, most of the excitation power of the engine noise is in narrow frequency bands associated with the engine rotational speed and its harmonic components Therefore, a 1/3-octave bandpass-filtered engine noise level plotted as a function of engine rpm does not monotonically increase with rpm but shows small peaks as the engine rotational speed passes through a range of rpm associated with the central frequency of the applied 1/ 3-octave band A simple power-law-based model therefore does not fit the engine noise characteristics However, in the interior noise data obtained in this work, the effect of the engine noise deviating from a simple power law was observed to be minimal for 1/3-octave bands with central frequencies greater than 125 Hz, and the interior noise was observed to increase monotonically with the vehicle’s cruise speed Equation (2) was found to fit the overall sound pressure level and 1/3-octave band level at higher central frequencies fairly well The road noise component estimated in Equation (2) therefore also includes the engine noise component Figure Pinwheel mounted on the roof of an SUV to measure wind speed and wind yaw angle during cruise tests on the road RESULTS AND DISCUSSION 3.1 Empirical Setup and Test The test vehicle used in the experiment was a compactsized hatchback car with an automatic transmission For the test and analysis method proposed here, only three parameters need to be measured during the test: interior noise, vehicle speed, and wind speed A B&K 1/2” microphone (type 4189) was used to measure the interior noise at the driver’s left ear The vehicle’s speed was measured by a GPS (MicroSAT, Corrsys-Datron); the wind yaw angle and the wind speed over the roof of the test vehicle were measured by using a pinwheel (R.M.Young, Model 05305) during the test The pinwheel used in the test is shown in Figure attached to an SUV (not the vehicle used in the present work) The pinwheel measures the total wind speed in the horizontal plane To obtain data that may be necessary in a future analysis using a frequency response function-based method, additional sensors were used during the test, including four PCB three-axis accelerometers glued to the knuckles of each wheel, four B&K surface microphones (type 4949) attached inside the wheel housings (one for each), one PCB three-axis accelerometer mounted on the engine block, and one pulse tachometer (Ono Sokki, SE152) to measure the voltage input to a spark plug The number of required input channels was therefore 24, and a portable 64-channel data acquisition device (Muller-BBM, PAK-Mobile MK II system) was used to capture the time signals during the test The cruise test was performed on a windy day on the proving ground at the Hyundai Kia Motors R&D Center Nineteen nominally constant cruise speeds ranging from 40 km/h to 130 km/h in increments of km/h were tested The cruise test was repeated for upwind and downwind conditions for each cruise speed on the proving ground’s oval track The variation in the wind speed measured by the pinwheel on the roof of the vehicle is shown in Figure for a cruise test at a nominal cruise speed of 130 km/h Because the proving ground has no roadside obstacles that can cause intermittent cross winds during cruise tests, the variation in the wind yaw angle at driving speeds higher VEHICLE INTERIOR NOISE MODEL BASED ON A POWER LAW Figure Typical variations in vehicle and wind speeds during cruise tests performed at a nominally constant speed of 130 km/h on a proving ground Cruise tests were performed under both upwind (solid lines) and downwind (dashed lines) conditions The lines in the middle denote the vehicle’s speed; the upper and lower lines denote the wind speed measured on the roof of the vehicle during the tests than 90 km/h was observed to be minimal (less than ±10o) for most of the test cases As shown in Figure 2, some variation in the vehicle’s speed about the nominal cruise speed appeared, even though the test driver tried to keep the speed as constant as possible during the test In the analysis method proposed here, a constant cruise speed need not be strictly maintained because the sensitivity of the interior noise to the driving speed is already explicitly modeled, as shown in Equation (2) The vehicle was driven in the “D” (drive) mode, and the engine rpm was observed to vary in proportion to the vehicle’s speed at high cruise speeds However, for cruise speeds of 50, 55, and 60 km/h in upwind conditions and 55, 60, and 65 km/h in downwind conditions, the automatic transmission frequently shifted gears up and down during the cruise test depending on the instantaneous change in load The sampling frequency was 32,768 Hz, and the cruise test lasted at least 25 seconds to capture the time signals for each case In addition to cruise tests on a proving ground, a wind tunnel test was performed to obtain interior noise data to be compared with the estimated wind noise level from the road driving test The test was performed in the Hyundai Aeroacoustic Wind Tunnel (HAWT) at ten different wind speeds (50, 70, 90, 110, 130, 140, 150, 160, 170 and 180 km/h) 3.2 Data Processing Most of the acoustic power of a vehicle’s interior noise is generally in low-frequency bands, which the power-lawbased model expressed in Equation (2) may not fit well due to the dominance of engine noise To obtain a better fit in the overall sound pressure level, an A-weighted interior noise signal was used in the present study A-weighted sound pressure levels are widely used in the automobile industry because they indicate the subjective noise level more directly A-weighted signals were obtained by convoluting the interior noise signal with an A-weighting 781 filter in the time domain The A-weighted interior noise signal was then convoluted with each of the 1/3-octave bandpass filters with central frequencies ranging from 25 Hz to 10 kHz (a total of 27 bandpass filters) For all of the 1/3-octave bandpass-filtered signals, the mean squared acoustic pressure was calculated by averaging the squared pressure for non-overlapping one-second periods Mean values for the vehicle’s speed and the wind speed were also calculated by averaging the vehicle speed and wind speed signals, respectively, for the same time period Once the mean values for each of the 1/3-octave bands were obtained, the overall mean squared acoustic pressure could be obtained by summing up the mean squared pressures calculated from each band The frequency range for the overall noise therefore covers the acoustic power from 22 Hz to 11.3 kHz These overall and 1/3-octave band mean values represent the variations in the interior noise level as a function of both the vehicle’s mean speed and the mean wind speed as scattered data points in a plot Because 25-second-long signals were captured under upwind and downwind conditions, the number of data points obtained for each frequency band was 950 (i.e., × 25 × 19), and the number of frequency bands was 28 (27 1/ 3-octave bands plus overall frequency band) To optimize the five parameters in Equation (2) for each frequency band so that the mean squared pressure calculated by Equation (2) could yield the best fit with the 950 data points for each frequency band, a Matlab® function that solves nonlinear least squares problems was used 3.3 Results The scattered data points obtained for the overall interior Figure Sound pressure levels for the vehicle’s interior noise measured near the driver’s left ear as a function of wind speed (log-scaled in the figure) during the cruise test: data points for cruise speeds at 40 km/h (downward-pointing triangles), 55 km/h (upward-pointing triangles), 70 km/h (circles), 85 km/h (squares), 100 km/h (pentagrams), 115 km/h (left-pointing triangles), and 130 km/h (hexagrams) and curve fits by Equation (2) Sound pressure levels were calculated by averaging the sound power for non-overlapping one-second periods from the 25-second-long interior signals measured for upwind and downwind conditions, respectively 782 H.-S KOOK, D LEE and K.-D IH noise are plotted in Figure as a function of the mean wind speed Only the data points from every third cruise speed tested are shown in the figure for clarity of presentation, although all of the data points were used in the parameter optimization procedure Each group of data points associated with a cruise speed is denoted by an identical marker and can be seen as divided into two sets: one for upwind conditions and the other for downwind conditions The interior noise is insensitive to changes in wind speed at a cruise speed of 40 km/h and becomes more sensitive as the wind speed increases The sensitivity of the interior noise level to changes in the vehicle’s speed can be observed from the level offset between the two adjacent groups of data points that differ in cruise speed but share similar wind speeds The solid lines in the figure represent the curve fitting based on Equation (2) The trend in the data points is fairly well described by the model The individual noise components in Equation (2) are plotted as a function of the relevant speed as straight lines in Figure The wind noise component is negligible at lower cruise speeds but dominates the interior noise at speeds above 120 km/h due to its steeper slope The slope of the wind noise was found to be 58.4 dB/decade (i.e., α = 5.84), and that of the road noise was 34 dB/decade (i.e., β = 3.4) The curved solid line in the figure is the total interior noise as a function of the cruise speed if the wind is still (i.e., if the wind speed is the same as the driving speed) The total interior noise predicted for the still wind condition is shown to be between the two sets of experimental data obtained under upwind and downwind conditions, respectively Also shown in Figure is the interior noise level obtained from the Figure Overall A-weighted levels of the wind noise (straight solid line), road noise (dashed line), and background noise (dashed-dotted line) components as functions of wind speed (vehicle speed in the case of road noise) calculated from the optimized parameters in the curve fitted by Equation (2) The curved solid line represents the total overall A-weighted noise level calculated by Equation (2); the data points represent the overall A-weighted noise level measured in each of the nominally constant-speed cruise tests under upwind (downward-pointing triangles) and downwind (upward-pointing triangles) conditions The interior noise level obtained from an aeroacoustic wind tunnel test is also shown for comparison (circles) Figure Comparison of data points for selected 1/3-octave bands with curve fits calculated by Equation (2) for central frequencies of (a) 63 Hz, (b) 125 Hz, (c) 250 Hz, (d) 500 Hz, (e) kHz, and (f) kHz VEHICLE INTERIOR NOISE MODEL BASED ON A POWER LAW aeroacoustic wind tunnel test, which was obtained at various wind speeds at zero yaw angle Figures and show that the interior noise model proposed here describes the data obtained for a wide range of cruise speeds fairly well The interior noise model was applied to 1/3-octave band data; the results are shown in Figure 5(a–f) and Figure 6(a– f) for selected bands Noticeable discrepancies appear between the curve fits and the data obtained at low cruise speeds, especially for the 63-Hz band, as shown in Figure 5(a) This discrepancy can also be identified in the corresponding plot in Figure 6(a) This suggests that optimization processes should be performed for a narrower speed range if a better accuracy is desired at lowerfrequency bands In Figure 6(a), a small peak in the sound pressure level of the data appears around 75 km/h The peak is attributed to the second-order component of the engine excitation because the engine rpm varied between 1800 and 1850 rpm during the cruise test The interior noise model based on a simple power law does not describe the data very well in such cases However, the model generally describes the interior noise with reasonable 783 accuracy for the 1/3-octave bands, considering the simplicity of the model The slopes of the wind noise and road noise that were found for the 1/3-octave bands shown in Figures and are summarized in Table The root mean squared error between the data and the value estimated by Equation (2) is also shown in the table The optimized slopes for both the wind noise and the road noise vary depending on the frequency band Note that estimated parameters for individual noise components could be inaccurate in some cases, even though the total interior noise level agrees well with the model This occurs when the individual noise components are negligibly small for all of the cruise speed ranges tested The background noise component in the 63Hz, 500-Hz, and 1-kHz bands, shown in Figures 6(a), 6(d), and 6(e), respectively; the wind noise component for the 1kHz band, shown in Figure 6(e); and the road noise component for the 4-kHz band, shown in Figure 6(f), not dominate the interior noise for the speed range tested, and thus, the estimated parameters associated with them could be inaccurate Figure 1/3-Octave band levels of the wind noise (straight solid line), road noise (dashed line), and background noise (dashed-dotted line) components as functions of wind speed (vehicle speed in the case of road noise) calculated from the optimized parameters in the curve fitted by Equation (2) The curved solid line represents the total 1/3-octave band noise level calculated by Equation (2); the data points represent the 1/3-octave band noise level measured in each of the nominally constant-speed cruise tests under upwind (downward-pointing triangles) and downwind (upward-pointing triangles) conditions The interior noise level obtained from an aeroacoustic wind tunnel test is also shown for comparison (circles) The selected central frequencies shown in the figure are (a) 63 Hz, (b) 125 Hz, (c) 250 Hz, (d) 500 Hz, (e) kHz, and (f) kHz 784 H.-S KOOK, D LEE and K.-D IH Table Estimated slopes and root mean squared error for the overall level and the 1/3-octave bands shown in Figures and 1/3-Octave band center frequency Wind noise Road noise Root mean squared error slope slope (dB) (dB/decade) (dB/decade) 63 Hz 41.9 18.5 2.29 125 Hz 56.8 40.9 1.21 250 Hz 54.7 17.7 0.94 500 Hz 58.6 20.7 0.63 kHz 55 33.5 0.58 kHz 58 37.7 0.19 Overall 58.4 34 0.48 Figure Color map for the levels and proportions of the contribution of the noise components to the total interior noise The abscissa represents the 1/3-octave band central frequencies (“Ovl” denotes the A-weighted overall level); the ordinate represents the cruise speed Noise components are differentiated by color: wind noise (blue), road noise (orange), and background noise (gray) Each color’s degree of saturation denotes the sound pressure level on an Aweighted decibel scale (i.e., the lower the sound pressure level is, the whiter the color is) The proportion of the contribution of each noise component can be identified by its width The proportions and levels of the three noise components at each cruise speed (if the wind is still) were calculated by using the interior noise model for the overall and 1/3octave bands and are represented as a color map in Figure For each frequency band in the figure, the leftmost stripe is the wind noise component, the stripe in the middle is the road noise, and the rightmost stripe is the background noise The relative width of each stripe represents the proportion of the noise component in the total interior noise; the sound pressure level is denoted by the saturation of the color The A-weighted sound pressure levels in the 125-Hz to 250-Hz 1/3-octave bands dominated the interior noise The color map shows that at cruise speeds higher than 120 km/h, the wind noise component dominates the lower-frequency bands (those with central frequencies from 50 Hz to 630 Hz) as well as the frequency bands with central frequencies higher than kHz and that the road noise component dominates the higher-frequency bands (those with central frequencies from kHz to kHz) This observation is somewhat contrary to what is generally believed in the automotive industry CONCLUSION The simple model developed in this work successfully separated a vehicle’s interior noise into three components: one that depends on the wind speed, another that depends on the vehicle’s speed, and a third component that is independent of both of these speeds Power-law-based models have an advantage over traditional methods based on frequency response functions because they can estimate the wind noise component of the interior noise more directly Of course, tests in an aeroacoustic wind tunnel could yield informative data on the wind noise component because other noise components are suppressed during such tests The model and analysis method proposed here, however, have merits even relative to wind tunnel tests Wind noise can be estimated for real road-driving conditions, such as airflows under the under-floors and around the wheels in the wheel wells and the characteristics of acoustic reflections from the surroundings The testing and analysis method proposed here does not replace but rather complements frequency response function-based methods An analysis using the power law model requires one windy test day and the addition of a few sensors for measuring signals such as the wind speed and vehicle speed to the traditional test set-up for frequency response function-based methods ACKNOWLEDGEMENT−This work was supported by research program 2009 of Kookmin University in Korea and a 2008-2009 Research Grant from Hyundai Motor Company REFERENCES Bendat, J S (1976) System identification from multiple input/output data J Sound and Vibration, 49, 293−308 Bendat, J S and Piersol, A (2000) 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