Modeling and simulation of the equivalent vertical stiffness of leaf spring suspensions

In vehicle multi-body dynamics (MBD) modeling, the stiffness parameterization of leaf spring (LS) is an unavoidable challenge regardless of the selection of modeling methods. On the contrary, the parameterization of Coil Spring (CS) stiffness is easy to achieve by adjusting the scale factor. Therefore, a novel LS stiffness parameterization method by treating the suspension stiffness as an intermediate variable through a CS stiffness is proposed based on the virtual displacement theory. The proposed method is then implemented in the vehicle-level modeling of a commercial Van with front transverse leaf spring suspensions and rear longitudinal parabolic leaf spring suspensions. The MBD model is validated by natural frequency tests and suspension stiffness simulations. Furthermore, the vertical acceleration of the car body is also verified. Results show that the root mean square (RMS) values of body vertical acceleration in the equivalent CS model are just slightly lower than that in the LS suspensions. The applicability and capability of the proposed method are proven to address the limitation of LS stiffness parameterization in MBD modeling. It lays the groundwork for efficiently simulating the LS suspension in vehicle ride and handling design and optimization.


Introduction
Leaf springs (LS) are crucial elastic components in the suspension system of commercial vehicles.The LS has been widely used since it can provide significant high load capacity, good reliability, and cost-effectiveness.Leaf springs used in vehicle suspension systems are primarily categorized into two types: transverse and longitudinal.The transverse leaf spring, a single-leaf spring, boasts a simple structure with excellent damping and anti-body roll performance. 1Longitudinal leaf springs can be categorized based on quantity, with options including fewer-leaf 2,3 and multi-leaf springs. 4In vehicle design, suspension modeling is one of the most important works in vehicle dynamics.However, the LS model is more complicated than other elastic elements due to its structure complexity and interactions between leaf components.To benefit the vehicle dynamics simulation, developing an accurate, reliable, and fast LS model is the key.At present, the LS models mainly include the Finite Element (FE) model, 3,5,6 the discrete beam model, [7][8][9] and the SAE three-link model. 10o solve the complex solving problem of the FE model, Adams/Car developed the Leaf Toolbox based on the discrete beam theory.Although the toolbox can effectively characterize the mechanical properties of LS, the vertical stiffness is still dependent on the crosssectional area S of the discrete beam and the moments of inertia I xx , I yy , and I zz .To parameterize the longitudinal parabolic leaf spring (LPLS) stiffness, an approximate kinematic model is developed using the three-link model connected by either torsion springs or bushings.11,12 The varying width and height of parabolic leaf springs with their length lead to non-constant curvature in the cantilever beam, resulting in highly nonlinear static deflection behavior.To capture this nonlinearity, a five-link LS model for describing the parabolic leaf spring's stiffness characteristics was proposed.13 Despite the ability of the parabolic leaf spring three-link or five-link model to parameterize the LS stiffness, both models ultimately transform the LS stiffness into other bushing stiffness or torsional stiffness of the torsion spring.Thus, directly using LS stiffness as a design parameter remains challenging due to its dependence on geometric structure and material properties, which complicates suspension stiffness design and optimization.
Many scholars have done great work on the optimal matching of LS suspension stiffness.The analytical vehicle dynamics models derived by equations of motion usually use static stiffness as the LS suspension stiffness 14 alternatively.The Bouc-Wen model 15,16 and the generalized Maxwell-slip damper (GMD) model 17 were used to describe the LS hysteresis characteristics caused by friction.However, those methods can describe the elastic characteristics of LS, but they are not capable of stiffness optimization.The analytical models of LS stiffness have also been studied by researchers.In such models, the LS stiffness is usually dependent on the leaf length, thickness, width, elastic modulus, tensile modulus, compressive modulus, shear modulus, etc. [18][19][20] The analytical models are still limited in the direct parameterization of the LS stiffness, which is a unavoidable trouble in vehicle dynamics simulation and suspensions design.
In Adams/Car MBD models, the LS stiffness cannot be directly parameterized due to its complex modeling, while the coil spring (CS) allows stiffness parameterization by adjusting the scale factor.Thus, this paper proposes a novel method to parameterize the LS stiffness using CS in Adams/Car MBD models.The proposed method takes the suspension stiffness as an intermediate variable to develop an equivalent CS kinematic model based on the virtual displacement method.This is a new stiffness equivalent approach for the MBD modeling of LS.The effectiveness of the proposed method is verified through the parallel wheel travel (PWT) simulation, suspension natural frequency tests, and swept sine excitation tests.Finally, vehicle models with whole suspensions are established to compare the RMS values of the vehicle body vertical acceleration on an A-class road to further verify the applicability of the proposed method.This method resolves the issue of LS stiffness parameterization in the MBD modeling.It contributes to the ride and handling analysis of vehicles with LS suspensions.The rest of the paper consists as follows: suspension description, equivalent model of LS vertical stiffness, multibody dynamic model, road excitation, model verification, model discussions, and conclusion.

Suspensions description
The LS suspensions in this study belong to a light commercial van, which uses the transverse leaf spring (TLS) suspension and the LPLS suspension respectively.The TLS acts as the elastic element of the front suspension, as shown in the green part of Figure 1(a).The middle part of the TLS is installed on the subframe with the upper and lower rubber gasket.The end of the TLS is installed on the lower control arm by the LS end mount.The rear LPLS suspension is shown in Figure 1(b).The end of the LPLS is fixed on the body by rubber bushing and shackle.The middle of the LPLS is fixed on the rear axle by the U bolt.
Ignoring the anti-roll bar and the tie rod, the partially simplified diagram of the front TLS suspension is shown in Figure 2(a).A 1 and B 1 are the endpoints of the lower control arm.C 1 is the outer point of the lower control arm.The lower control arm D A 1 B 1 C 1 is simplified to A 1 C 1 .D 1 is the center point of the knuckle.E 1 and F 1 are the installation position of the damper between the knuckle and the body.F 1 C 1 is the virtual kingpin axis.G 1 H 1 stands for the TLS.G 1 is the installation position between the end position of the TLS and the lower control arm.A 1 and F 1 are the fixed position on the body.K l f is the TLS stiffness, which is perpendicular to A 1 C 1 .
Figure 2(b) shows the partially simplified model of the rear LPLS suspension without the anti-roll bar.Since the rear LPLS suspension is non-independent, only half of the rear axle model is shown here.A 2 is the midpoint of the rear differential and the roll center of the rear LPLS suspension.B 2 and C 2 are the installation position of the damper between the rear axle and the body.D 2 is the middle position of the U-bolt.E 2 is the installation position of the front and rear leaf eyes of the LPLS.F 2 is the connection position between the axle and the rear inner wheel hub.G 2 and H 2 are the middle points of the inside and outside wheel centers.K l r is the LPLS stiffness.K s r is the LPLS suspension stiffness.

Equivalent model of LS vertical stiffness
Since the LS stiffness cannot be parameterized during the vehicle ride comfort simulation, a method of equivalently replacing the LS with a CS whose stiffness can be parameterized in Adams is proposed to solve the LS optimal stiffness.For the TLS suspension, the CS is mounted on both ends of the damper to equate the TLS stiffness, as shown in Figure 3(a).For the LPLS suspension, the LPLS is replaced by an equivalent CS, where the installation position is consistent with that in LPLS, as shown in Figure 3(b).In this paper, we introduce the suspension stiffness K s , which enables the parameterization of LS stiffness using equivalent CS stiffness, provided that the LS suspension stiffness K s leaf matches the CS suspension stiffness K s coil .The definitions of each symbol in Figure 3 are shown in Table 1.
In Figure 3(a), force F l f is perpendicular to the lower control arm A 1 C 1 .Based on virtual displacement theory, P23 (Point C 1 ), P35 (Point A 1 ), and P45 (Point F 1 ) are the instantaneous center of velocity between two components.P25 (Point M) is the instantaneous center of parts 2 and 7. Point N is the roll center position of the front suspension, which is located on the symmetrical centerline of the vehicle.Point L is the intersection of the instant center M and the vertical line of the ground.
For the equivalent model in Figure 3, there are relationships as follows: where F f is the ground force on the wheel, F f = K s f Df f .F l f is the force exerted by the lower control arm on the end of the TLS, F l f = K l f Df l f .Df f is the vertical virtual displacement of point J. Df l f is the virtual vertical displacement of the LS along the lower control arm.K s f is the suspension stiffness, which is a virtual spring mounted between the wheel and the body.K l f is the stiffness of TLS.F in is the ground force on the inner wheel, F in = K in Df in .F out is the ground force on the outer wheel,   The virtual angular displacement of the rear suspension l2 The line length of A 2 W 2 b 1 the angle between JM and the ground m1 The line length of The angle between A 2 W 1 and the ground m2 The line length of A 2 W 1 u the angle between the damper axle and the vertical line n1 The line length of A 1 C 1 u the angle between F Coil f and the vertical line f The virtual displacement of MJ along a 1 g The angle between A 2 W 2 and the ground The horizontal length of ME 1 The line length of JL The line length of JM The horizontal length of The front suspension stiffness The TLS stiffness K s r The rear suspension stiffness K l r The LPLS stiffness The front equivalent CS stiffness The rear equivalent CS stiffness K out Rear suspension stiffness at W 2 the virtual displacement of the LPLS along the axis DE.K in and K out are the equivalent CS stiffness between the inner and outer wheels and the body, There is a relationship that K s r = K in + K out .Df , Df in and Df out are obtained by equation ( 2) where f is the virtual displacement of point J along the instantaneous direction of motion.The relationship between the LS virtual compression Df l f and point C virtual displacement Df C is obtained by equation ( 3), Df C = la.
Combining equations ( 1) and ( 3), the relationship between K l and K s is shown in equation (4).
The lengths of l, q, and p depend on the coordinate of the instantaneous center (point M).The coordinates of the M point are obtained through the line equation (5) of straight line MC 1 and the plane equation (6), which is a plane passing through the F 1 point and perpendicular to the straight line F 1 E 1 .
So far, the relationship between K s and K l has been established.Next the relationship between K s and K coil needs to be established.As the installation position of the equivalent CS is the same as that in the LPLS, it follows that K coir r = K l r .For the front equivalent CS suspension, there is: where F = K s f Df .F coil f is the spring force of the equivalent CS along the axis of the damper, The relation of f coil f and Df coil f can be obtained in equation (8).
Therefore, the relationship between K coil f and K s f is shown in equation (9).Combining equations ( 4) and ( 9), the relationship between K coil and K l of the TLS suspension is obtained in equation (10).
Since the instantaneous centers M and A 2 of the TLS and the LPLS suspension change dynamically during the wheel travel process, the equations ( 10) and ( 4) can only represent the instantaneous relationship of K Coil and K l .Therefore, the dynamic proportionality factor r x i ð Þ of K Coil and K leaf must be solved.
Through the suspension PWT simulation of Adams Car, the wheel travel distance X leaf of the LS suspension and the displacement of the equivalent CS Z coil is presented as follows: where x 0 is the initial value of the wheel travel distance, x 0 = 0. m = n + i. i 2 0, n ½ .The suspension stiffness k s x n ð Þ and hub force f x n ð Þ at any position x n are obtained using equation (12).
As for the TLS suspension, when . Since the number of LSs in the LPLS suspension is three, the LPLS suspension stiffness K s r Àx i r ð Þ is the third LS stiffness when f r Àx i r ð Þ)0.When two LSs are in contact, the initial stiffness of the LPLS suspension is recorded as K s r Àz a r ð Þ.When three LSs are in contact, the initial stiffness of the LPLS suspension is recorded as K s r Àz b r ð Þ, a 2 (0, i), b 2 (0, a).The equivalent CS dynamic stiffness K coil Z coil ð Þ and the relationship between k coil z coil ð Þ and K s X ð Þ are shown in equation (13).
The spring force of equivalent CS F coil Z coil ð Þ is obtained as follows: where f coil z 0 ð Þ= 0. The spring force and displacement of equivalent CS can be obtained by equations ( 11)-( 14).Import this information into the Adams spring documentation to aid in replacement.The free length of the CS is determined by the suspension preload.The preload method is utilized to establish the displacement of the hardpoints.The preload of equivalent CS is obtained by equation (15). where DM is the distance between the upper and lower hard points of CS.OFF is the free length of the CS suspension.
The equivalent CS is brought into the suspension model for PWT simulation to obtain the wheel travel X coil and suspension stiffness K s coil .The suspension proportional factor r f x i ð Þ of equivalent CS suspension and LS suspension is calculated by equation (16).
To ensure the stiffness consistency of the CM suspension and the TLS suspension, the CS stiffness K coil f and K coil r needs to be corrected.According to equations (11) and ( 14), the stiffness K Þ of the modified CS can be calculated by equation (17).
Through equations ( 14)-( 17), the displacement Z coil , the spring force F coil Z coil ð Þ and the CS stiffness K coil Z coil ð Þ can be obtained.In summary, the conversion relationship between the LS suspension stiffness K s X ð Þ and the equivalent CS stiffness K coil is illustrated in Figure 4. Firstly, the PWT simulation is carried out on the LS suspension multibody model, and the relationship among wheel travel distance X l , hub force F l X l ð Þ, and suspension stiffness K s l X l ð Þ is obtained.Secondly, the CS stiffness is calculated according to equations (11-13).The spring displacement and free length are obtained through equations ( 14)- (15).Next, the CS is used to replace the LS.The CS suspension MBD model is built, and the PWT simulation is performed again.Then the suspension stiffness K s coil X coil ð Þ of the CS suspension is obtained.The suspension stiffness proportionality factor r x i ð Þ between CS suspension and LS suspension is calculated by equation ( 16).It should be noted that the calculation methods in equation ( 17) of the modified CS stiffness K of the TLS suspension and the modified CS stiffness K (17).Only r x i ð Þø 95% can the CS stiffness calculation end.The method proposed in this paper solves the dynamic proportional coefficient r x i ð Þ of the CS stiffness K coil and the LS suspension stiffness K s l during the wheel travel process.

Suspension model
Based on the proposed kinematic model of equivalent CS suspensions in Figure 3, the front TLS suspension MBD model is replaced by an equivalent CS suspension, as shown in Figure 5(a) and (b).The rear LPLS suspension MBD model is replaced by a TLCS suspension, as shown in Figure 6(a) and (b).Since the LPLS plays a guiding rod in the rear suspension model, it is necessary to increase the guiding link in the rear equivalent CS suspension.The two-link structure plays a guiding role in the rear equivalent CS suspension.The bottom of the rear equivalent CS is fixed to the rear axle and the top is fixed to the body.The K&C PWT tests are carried out on the MBD model of the front and rear suspensions.The wheel travel of the front and rear suspensions in the simulation experiments was set to [250, 50] mm and [250, 100] mm, respectively, based on the travel limits of the suspension bump-stops.Table 2 shows the suspension MBD model parameters based on the supplier-provided parameters of the experimental test sample vehicle.
The TLCS suspension MBD model retains three bushings in the structure model, as shown in Figure 7(a). A, D, and E are the center points of the bushings.B and C are the center points of the U-bolt and the upper LS fixed position.F and G are the connection points of the links and the rear axle.Points I and H are the connection position of the equivalent CS with the body and rear axle.m and n are the length of the front and rear ends of the U-bolt fixing section.The geometrical relationship of the TLCS suspension is given in equation (18).
The topology of the TLCS suspension is shown in Figure 7(b).The front link AF is connected to the body by a bush at point A and connected to the rear axle by a fixed pair at point F. The rear link GD is connected  to the rear axle by a rotating pair at point G and connected to the connecting link ED by a bush at point D.
The connecting link DE is connected to the body by a bush at point E. The upper and lower of the CS are built as the general part.The CS upper part is fixed with the body at point I.The CS lower part is fixed with the rear axle at point H.

Vehicle model
The vehicle MBD model with LS suspensions and equivalent CS suspension is shown in Figure 8.The vehicle model is mainly composed of nine parts: front suspension, front-wheel system, rear suspension, rear-wheel system, steering system, body system, powertrain system, front arb system, and rear arb system.A virtual body is used to represent the vehicle body model.

Road excitation
Road surface excitation has an important impact on vehicle ride comfort research. 21To verify the accuracy of the MBD model, it is crucial to simulate the suspension's natural frequency and the vehicle model.According to National Standard GB/T 4783-1984 22 , the suspension's natural frequency is tested by the rolloff method.A virtual road that corresponds to the test road is built in Adams/Car.The brick-shape obstacle height is 90 mm and the length is 200 mm, as shown in Figure 9(a).Based on Sayers empirical model, 23 the road profile parameters are generated by Adams's random road generator.The white noise PSD amplitude G s is represented as where n 0 is the reference spatial frequency, 24 n 0 = 0:1m À1 .G q n 0 ð Þ is the PSD value of n 0 and called the road roughness coefficient.When Gq n 0 ð Þ is set to 16 3 10 À6 m 3 , Figure 9(a) illustrates the test conducted on a closed circular high-speed road surface at the test site.Accordingly, the simulation test was set to emulate   an A-class road surface, and the displacement power spectrum profile of the A-class road was constructed using Adams Road Builder, as depicted in Figure 9(b).

Model validation
Since the suspension stiffness will increase suddenly after the suspension hits the rubber bump-stop, the change of the suspension stiffness is simulated by the PWT when the bump-stop is not touched, as shown in Figure 10.The limit travel of the front and rear suspensions is set at 50 mm and 100 mm respectively.The suspension stiffness maximum error of the TLS and the LPLS with its equivalent CS suspension is 23.34 and 0.13% respectively.The suspension stiffness at the initial position is listed in Table 4.When the wheel travel is 0 mm, the vertical stiffness error of equivalent front CS suspension and rear TLCS suspension is 0.41% and 0.081% respectively.The vertical stiffness of the equivalent CS suspension is the same as that of the LS suspension, as shown in Figure 10 and Table 4.

Comparison of suspension natural frequency
To verify the suspension MBD model of the light Van vehicle, a natural frequency test is conducted considering the vehicle's large mass.The test involves allowing both wheels of the front and rear suspension simultaneously allowed to freely drop from equal-height bumps while the vehicle is in an engine-off state.The choice of suspension natural frequency test for model verification offers the following advantages: (1) Keeping the engine in a stalled state avoids the influence of engine self-excitation on the body response.This ensures that the obtained vibration responses are free vibrations, which accurately reflect the inherent characteristics of the suspension system.(2) The vehicle free drop test ensures single and approximate pulse excitation as the test excitation, eliminating other input interferences in the vibration response of the suspension and vehicle body.
In this study, the front and rear suspensions of the vehicle were tested for their natural frequencies using the roll-off method.The vehicle's front and rear wheels were positioned on a 90 mm brick platform, as depicted in Figure 9. Triaxial accelerometers were installed on the lower control arm and subframe of the front TLS suspension (Figure 11(a) and (b)), as well as on the axle and body of the rear LPLS suspension (Figure 11(c)  and (d)).
Using Siemens LMS Test.Lab software, the body acceleration time signals of the front and rear suspensions were calculated through FFT (Fast Fourier Transform), as depicted in Figure 12.The front and rear suspension natural frequency data, obtained from

Comparison of swept sine excitation
Using sine waves with amplitudes of 5 and 10 mm and a frequency of 20 Hz as the road excitation, frequency sweep simulation experiments are carried out on the front and rear LS suspensions and equivalent coil spring suspensions, respectively.The power spectral densities of the vertical vibration acceleration of the sprung mass of the front and rear suspensions are shown in Figure 13.The body's vertical frequency remains unchanged after using the equivalent CS.However, the vibration amplitude is reduced.To verify the suitability of the equivalent CS suspension models for vehicle ride comfort analysis, the RMS value of body vertical acceleration on an A-class road is analyzed.

RMS comparison of body vertical acceleration
The PSD of the vehicle body's vertical acceleration is compared between the LS suspension vehicle and the equivalent CS vehicle driving at 40 km/h on an A-class road, as shown in Figure 14(a).The natural frequency of the equivalent CS suspension is slightly lower than that of the LS suspension.Within the low-frequency range (up to 2 Hz), the vertical acceleration PSD amplitude of the equivalent CS suspension body increases, while the amplitude of other frequencies decreases.This difference is attributed to the discrete beam method  used in constructing the LS suspension, which increases the unsprung mass and leads to reduced low-frequency suspension amplitude.Figure 14(b) shows the RMS values of the body center vertical acceleration for LS and CS suspensions on an A-class road at speeds ranging from 10 to 90 km/h.The maximum error for the CS suspension compared to LS suspension is 212.06% at 60 km/h, which falls within an acceptable range.The trend of the RMS values of body vertical vibration accelerations is consistent between the two suspensions.Hence, the equivalent CS suspension effectively reflects the ride comfort of the LS suspension vehicle, providing a basis for parameterizing suspension stiffness in vehicle ride comfort optimization.

Conclusion
This paper proposes an equivalent CS suspension method to parameterize LS stiffness using suspension stiffness as an intermediate variable, addressing the non-parameterization issue of LS stiffness in vehicle MBD models.TLS and LPLS suspensions of a Van vehicle are used to establish suspension and vehicle MBD models.Suspension PWT simulation shows that the vertical stiffness error between the equivalent CS and LS suspension is below 0.5% at the initial wheel travel position.Within the maximum travel of the bump-stop, the vertical stiffness error between the equivalent CS suspension and LS suspension is within 4%.

F
r is the force of the LPLS by Part 3, F l r = K l r Df l r .Df in and Df out are the vertical virtual displacement of the contact point W 1 and W 2 .Df l r is

Figure 3 .
Figure 3.The kinematic model of equivalent CS suspensions: (a) the CS equivalent model for TLS suspension and (b) the CS equivalent model for LPLS suspension.

Figure 4 .
Figure 4. Flow chart of equivalent LS suspension with CS suspension.

Figure 5 .
Figure 5.The MBD model of front suspension: (a) TLS suspension and (b) equivalent CS suspension.

Figure 7 .
Figure 7. Equivalent TLCS model: (a) geometry of TLCS suspension model and (b) topology of TLCS suspension model.

Figure 8 .
Figure 8. Adams MBD models with LS and CS suspensions: (a) the vehicle MBD model with LS suspensions and (b) the vehicle MBD model with equivalent CS suspensions.

igure 9 .
Road excitation: (a) suspension frequency offset test and (b) road A-class profile.

Figure 10 .
Figure 10.Suspension stiffness of LS and equivalent CS suspensions: (a) front suspension stiffness and (b) rear suspension stiffness.

Figure 11 .
Figure 11.The layout positions of the triaxial accelerometers for the suspension natural frequency test: (a) lower control arm for front TLS suspension, (b) body for front TLS suspension, (c) axle for rear LPLS suspension, and (d) body for rear LPLS suspension.

Figure 12 .
Figure 12.Natural frequency of suspensions: (a) front suspension body frequency compared and (b) rear suspension body frequency compared.

Figure 14 .
Figure 14.RMS comparison of body vertical acceleration: (a) body vertical acceleration PSD at 40 km/h and (b) RMS of body vertical acceleration.

Figure 13 .
Figure 13.Body vertical acceleration comparison at swept sine excitation: (a) front suspension and (b) rear suspension.

Table 1 .
The symbol definitions of the equivalent CS suspensions.

Table 2 .
Simulation parameters of the front and rear suspension.

Table 3
displays the setting parameters of the vehicle MBD model, which were determined based on the test sample Catia vehicle assembly drawing and the provided sample test parameters from the supplier.

Table 3 .
Base vehicle specification parameters.

Table 4 .
Suspension stiffness at the initial position of suspensions.
the suspension natural frequency test, are recorded as ''LMS front TLS suspension'' and ''LMS rear TLS suspension,'' respectively.The suspension natural frequency test shows the front and rear suspension natural frequency is 1.56 and 1.95 Hz, respectively.The simulation results validate the accuracy of the LS suspension MBD model and the proposed equivalent CS suspension model by demonstrating their agreement with the experimental data.