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Abstract

The vehicle unsprung mass plays a key role for simulating driving dynamics and ride comfort. Conventional methods for determining the unsprung mass involve dismantling the vehicle axles. This paper presents a disassembly-free method for determining the unsprung masses of a passenger car. For the method, quasi-static and dynamic measurements of one passenger car on a full vehicle test rig and of one tire on a single-axis hydropulser test rig are conducted. Non-linear dynamic mathematical models of the axle and tires are developed, parameterized, and validated. To determine the unsprung masses, the vehicle body is rigidly connected to the full vehicle test rig. The wheels of an axle are dynamically excited with the test rig platforms at the wheel contact patches. The excitation signal corresponds to a vertical in-phase displacement excitation with frequency sweep signals in a frequency range of 5–20 Hz and a constant amplitude of 3 mm in order to excite the natural frequencies of the unsprung masses. The measurements are conducted with a total of four different variations of the unsprung masses. The unsprung masses are obtained by simulative optimization on the dynamic measurements of the full vehicle test rig. The developed tire model can be used for the simulation of similar test rig measurements. The method is performed and validated for a MacPherson axle and is able to determine the unsprung mass of the axle with an accuracy of 99.6%.

Keywords

Numerical simulation vehicle testing vehicle dynamics vehicle comfort tire model

Introduction

The vehicle unsprung masses are important parameters for vehicle dynamics and vehicle comfort models.^1–4 A higher unsprung mass leads to increased wheel load fluctuations and higher suspension loads.^5–7 The correct parameterization of the unsprung masses is of outstanding importance for the simulation of relevant driving maneuvers.

Ries describes a method for determining the unsprung masses in which a vehicle is measured on a full vehicle test rig.⁸ The vehicle body is rigidly connected to the test rig. The vehicle springs and shock absorbers are dismantled from the vehicle and the weight of the remaining unsprung mass is measured.

Bixel and Heydinger present another test rig-based method for determining the sprung and unsprung masses of a passenger car.^9,10 The sprung and unsprung masses are determined by quasi-statically raising and lowering the vehicle body and measuring the distances between the wheel centers and the wheel contacts. The results are compared with the sum of the individual masses of the unsprung components. Deviations between 0.3% and 18% were observed.

Tang presents a method that can determine the sprung mass of a vehicle by means of road tests without knowledge of the road excitation.¹¹ He shows how this method can also be used to determine the unsprung masses of a vehicle, which is currently only possible with an error of 18%. To determine the sprung mass of a vehicle, the vertical accelerations on the vehicle’s wheel carriers and the vertical accelerations on the vehicle body above the wheels are measured. By means of frequency analysis and simulations of a quarter car model, the sprung mass of the vehicle can be determined with an accuracy of 99%. However, the unsprung masses can only be determined with an accuracy of 82%. The main reason for the inaccuracy of the method is said to be the non-linear tire properties.

This literature review shows that it has not yet been possible to develop a method that can determine the unsprung masses of a vehicle with a high degree of accuracy in road tests or by means of test rig measurements without dismantling the axle. This work presents a method that determines the unsprung masses of a vehicle with test rig measurements of the full vehicle and one tire without dismantling the axle. The method was developed and validated on the MacPherson axle of a test vehicle. Dynamic measurements were conducted on the vehicle and on one tire. Non-linear mathematical models of the axle and tire were developed and validated on the basis of the measurements. A simulative optimization determines the unsprung masses and the vertical axle damping. The axle and tire models are presented in the following sections. To validate the method, the unsprung masses were determined by measurements and using a Computer-Aided Design model (CAD model).

Materials and methods

The method for determining the unsprung mass consists of five parts:

Quasi-static and dynamic measurements of the test vehicle on a full vehicle test rig.

Parameterization of the axle model with the quasi-static measurements of the full vehicle test rig.

Measurement of the radial characteristics of one vehicle tire on a single-axis hydropulser.

Parameterization of the dynamic tire model.

Simulative optimization of the unsprung masses and the suspension damping characteristics with the dynamic measurements of the full vehicle test rig.

The quasi-static measurements of the Suspension Motion Simulator (SMS) are used to parameterize the axle model. The axle does not need to be dismantled. The shock absorber characteristics and the tire characteristics are the only dynamic properties of the axle that are necessary for the axle model. In the final step of the method, the shock absorber characteristics are optimized together with the unsprung masses on the dynamic measurements of the full vehicle. Therefore, measuring the shock absorber properties is not necessary for the method. Other dynamic properties of the axle, such as the characteristics of the shock absorber top mounts, do not have to be taken into account for the method. This simplification is presented and validated in the following sections.

The correct representation of the dynamic tire characteristics is the most important part of the method. The Tire Model section describes the development of the necessary tire model. The tire model can thus also be used to simulate similar test rig measurements.

The measurements of the full vehicle were conducted with the SMS.^12,13 The test rig has two actuator platforms that can excite the wheels of a vehicle axle in four degrees of freedom (longitudinal, lateral, vertical and rotational about the z-axis). All forces and torques are measured on the platforms. The platforms can be controlled by displacement as well as by force up to 30 $Hz$ . Table 1 shows the technical specifications of the test rig. It is also possible to excite the vehicle axle on the wheel carriers directly with a wheel replacement system. Furthermore, the movements of the chassis, wheels and platforms can be measured three-dimensionally with an optical measuring system with an accuracy of ±0.04 $mm$ . Figure 1 shows the test rig and the test vehicle.

Table 1.

Suspension motion simulator specifications.

Direction	Controlled by position		Controlled by force
	Quasi-static	At 30 Hz	Quasi-static	At 30 Hz
X	±100 mm	±5 mm	±4 kN	±750 N
Y	±100 mm	±5 mm	±4 kN	±750 N
Z	±100 mm	±5 mm	±15 kN	By request
Rotation axis Z	±5°	±0.7°	±200 Nm	±25 Nm

Figure 1.

Suspension Motion Simulator with test vehicle.

The tire measurements were conducted with a single-axis hydropulser. Figure 2 shows the single-axis hydropulser with the test tire. The test rig is capable of generating signals with a maximum frequency of 100 $Hz$ . The maximum force of the test rig is 50 $kN$ and the maximum deflection is 250 $mm$ .

Figure 2.

Single-axle hydropulser with test tire.

Full-vehicle measurements

The first step to determine the unsprung masses is to conduct quasi-static and dynamic measurements on the SMS. First, the vehicle body is rigidly connected to the test rig at the sill and front end. Figure 3 shows the mounting of the vehicle to the test rig. Due to the rigid connection of the vehicle to the test rig, compliance and thus movement of the vehicle body can be neglected.

Figure 3.

Vehicle mounting to the test rig.

To parameterize the axle model, quasi-static vertical excitations are applied to the wheels of the axle both in-phase and in anti-phase. The displacements of the platforms, the wheels and the fixed vehicle body are measured with the optical measuring system. At the same time, the quasi-static forces and moments in the tire contact patches are measured on the platforms. The vertical stiffness, the friction and the anti-roll bar stiffness of the axle can be determined using the displacements and the vertical wheel forces. To determine the unsprung masses of the test vehicle, excitations of the test rig platforms are defined, which excite the natural frequencies of the unsprung masses. The test rig platforms are excited in-phase and uniaxially in the vertical direction with a frequency sweep with a constant amplitude of 3 $mm$ . The frequency range of the sweep is between 5 and 20 $Hz$ so that the natural frequencies of the unsprung masses are excited. The frequency range can be adjusted depending on the axle characteristics. To validate the results, additional masses in the form of 5 $kg$ steel disks are attached to the wheels. Figure 4 shows a wheel with an additional mass. In addition to the variation without additional mass, three further variations with additional masses of 5, 10, and 15 $kg$ /wheel are measured. Figure 5 shows the amplitude responses of the vertical force at the tire contact patch for the four measured variations over the excitation frequency for one wheel. The figure shows that the natural frequency of the unsprung mass decreases as the additional mass increases.

Figure 4.

Additional wheel mass.

Figure 5.

Amplitude response of the vertical wheel contact forces for different wheel mass variants.

Axle model

To determine the unsprung mass, a mathematical characteristic curve model of the axle is required, which can reproduce the dynamic measurements of the full vehicle with a high degree of accuracy. For model development and model validation, the force-velocity characteristics of the shock absorbers, the characteristics of the shock absorber top mounts and the unsprung masses were identified. Figure 6 shows Axle Model 1. The model consists of a non-linear suspension stiffness ( $c_{S}$ ), a Dahl friction model ( $f_{DF}$ ), a force-velocity characteristic of the shock absorbers ( $b_{D}$ ), an anti-roll bar stiffness ( $c_{ARB}$ ), an anti–roll bar friction ( $f_{DFARB}$ ), the unsprung mass ( $m_{wheel}$ ), a shock absorber top mount model as Kelvin–Voigt element ( $c_{TM}$ , $b_{TM}$ ), piston rod mass ( $m_{PR}$ ), and a Kelvin–Voigt element as tire model ( $c_{T}$ , $b_{T}$ ). The suspension stiffness, the suspension friction and the anti–roll bar stiffness are parameterized with in–phase and anti–phase quasi–static vertical excitation of the suspension on the SMS. The anti–roll bar friction was set equal to the measured axle friction, as it can be neglected for the inphase excitation of both wheels. The force–velocity characteristic of the shock absorbers, the parameters of the top mount model and the tire model are determined by measuring those components on the single–axis hydropulser. The component measurements of the shock absorber and the shock absorber top mount are only used to define and validate the axle model and are not part of the method for determining the unsprung masses. Figure 7 shows the force–velocity characteristics of the shock absorbers.

Figure 6.

Axle Model 1.

Figure 7.

CAD model of the wheel carrier and the MacPherson axle strut.

To validate the unsprung masses, the relevant components were weighed and a CAD model of the axle was set up. Figure 8 shows the CAD model of the axle. Table 2 summarizes the axle components and their masses. Table 3 summarizes all parameters of the axle model from Figure 6.

Figure 8.

Amplitude responses of the vertical wheel contact forces of the measurements without additional wheel mass and different model variants.

Table 2.

Unsprung mass components.

Components	Mass (kg)
Wheel carrier with brake	21.58
Wheel	18.37
Suspension strut	6.69
Coupling rod	0.43
Tie rod	0.40
Total	47.47

Table 3.

Axle model parameters.

Parameter	Value
Unsprung mass $m_{wheel}$ [kg]	47
Vertical suspension stiffness $c_{S}$ [N/mm; nonlinear in model]	29.50
Vertical suspension friction $F_{FrS}$ [N]	109
Dahl friction parameter $δ_{D}$ [/]	1
Dahl friction parameter $σ_{0 Fr}$ [/]	11,043
Shock absorber characteristics $b_{D}$ [nonlinear in model]	Figure 7
Radial tire stiffness $c_{T}$ [N/mm; nonlinear in model]	269
Radial tire damping $b_{T}$ [Ns/m]	251
Anti–roll bar stiffness $c_{ARB}$ [N/mm]	19.44
Anti–roll bar friction $f_{[DF ARB]}$ [N]	109
Shock absorber piston rod mass $m_{PR}$ [kg]	0.8
Top mount stiffness $c_{TM}$ [N/mm]	450
Top mount damping $b_{TM}$ [Ns/m]	1000

Several model variants were compared with each other for model development. Figure 9 shows the amplitude response of the vertical wheel contact forces over the excitation frequency for the measurements of the basic variant without additional wheel mass and the simulation results of Axle Models 1–3. All simulations were conducted with a fixed time step of 1 $ms$ . The figure shows that Axle Model 1 with the simple tire model is not able to reproduce the measured signal. It underestimates the tire force and the natural frequency of the unsprung mass. Since the standard tire stiffness is ∼10 times higher than the vertical axle stiffness, it is obvious that the simple tire model is not able to reproduce the necessary tire characteristics with sufficient accuracy. Therefore, a model was developed which can represent the non–linear properties of the tire. Figure 10 shows Axle Model 2 with the non–linear Tire Model 5. The tire model is explained in the following section. Figure 9 shows that Axle Model 2 is able to reproduce the vertical wheel contact forces of the tests with high accuracy.

Figure 9.

Axle Model 2.

Figure 10.

Axle Model 3.

As the measurement of the shock absorber top mount cannot be conducted as part of the method for determining the unsprung masses, the axle model is simulated without the shock absorber top mount. Figure 11 shows Axle Model 3 without shock absorber top mount. Figure 9 shows that Axle Model 3 slightly underestimates the vertical wheel contact forces, but the natural frequency of the unsprung mass is still very well represented. Axle Model 3 is therefore used for the method for determining the unsprung mass.

Figure 11.

Measured and simulated (Tire Model 5) dynamic stiffness over frequency for preload variation and 5 mm amplitude of the test tire.

Equations (1) and (2) describe the equations of motion of the unsprung masses. Equations (3) to (5) show the calculation of the spring force, the shock absorber force and the anti–roll bar force of the axle. Equation (6) shows the calculation of friction according to Dahl.¹⁴ The Dahl friction model is used for vertical axle friction as well as for anti–roll bar friction and in the tire models as a friction element and as part of the Prantl element:

{\overset{\cdot\cdot}{z}}_{wheel left} = \frac{F_{S} + F_{D} + F_{DF} - F_{ARB}}{m_{wheel}}

(1)

{\overset{\cdot\cdot}{z}}_{wheel right} = \frac{F_{S} + F_{D} + F_{DF} + F_{ARB}}{m_{wheel}}

(2)

F_{S} = c_{S} \cdot z_{wheel}

(3)

F_{D} = b_{D} \cdot {\overset{\cdot}{z}}_{wheel}

(4)

F_{ARB} = (z_{left} - z_{right}) \cdot c_{ARB} + F_{(DF ARB)}

(5)

\begin{array}{l} F_{F r} = \int (\dot{z} \cdot σ_{0 F r} \cdot s g n (1 - \frac{F_{F r}}{F_{F r S}} \cdot s g n (\dot{z})) \cdot \\ {| 1 - s g n (\dot{z}) \cdot \frac{F_{F r}}{F_{F r S}} |}^{δ_{D}}) \end{array}

(6)

Tire model

In the previous section it was shown that the correct representation of the dynamic tire properties is crucial for determining the unsprung mass. In order to analyze the radial tire properties and to develop a model, various measurements were conducted. First, quasi–static measurements were performed to determine the tire stiffness characteristics. Subsequently, harmonic frequency sweeps were performed for different amplitudes and preloads in a frequency range between 1 and 25 $Hz$ . Table 4 shows the specifications of these harmonic measurements.

Table 4.

Specifications of frequency sweeps from 1 to 25 $Hz$ for tire measurements.

Preload (N)	Amplitude (mm)
1500	2
1500	5
1500	10
3000	2
3000	5
3000	10
4500	2
4500	5
4500	10
6000	2
6000	5
6000	10

Research shows that the dynamic stiffness of filled elastomers decreases with increasing excitation amplitude and increases with increasing excitation frequency.¹⁵ Furthermore, the damping of an elastomer decreases with increasing preload. The harmonic measurements, which are listed in Table 4, were therefore conducted with different amplitudes and at different preloads. Castillo Aguilar et. al. performed similar dynamic tire measurements.¹⁶

Figure 12 shows the measured and simulated (Tire Model 5) dynamic tire stiffness over the excitation frequency for harmonic excitations and an amplitude of 5 $mm$ for different preloads.

Figure 12.

Single point vertical tire models from literature: (a) Spring (b) Kelvin-Voigt (c) Gehmann (d) FTire.

For model development and comparison, four tire models were initially selected from the literature and parameterized. Figure 13 shows the rheological models. The models are a non–linear stiffness, a Kelvin–Voigt element, a Gehmann model^17,18 and the FTire model.^19,20 The Kelvin–Voigt model is used as radial force model in the MF-Tire and MF-Swift.^21,22 Equations (7) and (8) show the calculation of the Maxwell element and the Prantl element:

F_{Max} = c_{Max} \cdot (z - \int^{\frac{F_{Max}}{d_{Max}}} dt)

(7)

F_{Prantl} = sgn (\int (\frac{F_{Prantl}}{c_{P}} - z) dt) \cdot f_{P}

(8)

Figure 13.

Extended Universal Bushing model.^24,25

Five new tire models were also developed on the basis of elastomer modeling. The parameterization of all models was done with the same quasi–static measurements and the harmonic sweep measurements. The parameterization was performed in several steps for all models. First, the non–linear stiffness characteristic of the tire was extracted from the quasi–static measurement. The other parameters of the models were then determined using a particle swarm optimizer with 96 particles in 200 iterations on the harmonic sweep signals. The harmonic excitation consisted of frequency sweep signals with up to 25 $Hz$ at 2, 5, and 10 $mm$ amplitude at four different operating points. The signals are listed in Table 4. The root–mean–square error (RMSE) between the measured force ( $F_{m}$ ) and the simulated force ( $F_{s}$ ) was used as the optimization criterion and is shown in equation (9):

RMSE = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(F_{m_{i}} - F_{s_{i}})}^{2}}

(9)

The evaluation of the dynamic tire measurements showed similarities to dynamic elastomer measurements. Elastomeric mount models, like the Universal Bushing (UBS),²³ take these effects into account. In addition to a non–linear spring and a damping element, the UBS consists of three Maxwell elements connected in parallel, which represent the dynamic stiffening of the elastomer. Furthermore, the model has a parallel–connected friction element and a hydro element. The UBS is thus able to represent the characteristics of elastomeric hydromounts. The author of this article was also able to investigate the dynamic characteristics of an engine mount for excitations of frequencies up to 20 $Hz$ and deflections up to 10 $mm$ and developed a model that reproduces the characteristics of elastomeric mounts for excitations of high deflections and frequencies with high accuracy.^24,25 The Extended Universal Bushing (EUB) reproduces the deflection–dependent damping and dynamic hardening of the elastomer by means of deflection–dependent scaling functions of damper and Maxwell elements. Figure 14 shows the developed Extended Universal Bushing (EUB) for uniaxial excitation. Equation (10) shows the displacement–dependent scaling function for damper elements and Maxwell elements:

f (z) = K_{D} \cdot z_{T}^{2} + 1

(10)

Figure 14.

Tire Model 3.

To better represent the dynamic radial properties of the tire, elements of the EUB from Figure 14 are used to build new tire models. Tire Model 1 corresponds to the FTire and has an additional Dahl friction element connected in parallel. Tire Model 2 consists of a parallel connection of a non–linear stiffness characteristic, a scalar damping element, three Maxwell elements connected in parallel, a Prantl element and a Dahl friction element. Figure 15 shows Tire Model 3, which is similar to Tire Model 2, but additionally has a deflection–dependent scaling function in series with a scalar damping parameter. This scaling function represents the decrease in damping at high preloads. Finally, a parallel Dahl friction element is added to represent the hysteresis. The parameterization of the model is conducted in a multi–stage manner. The non–linear stiffness characteristic is taken from the quasi–static measurements and inserted into the model as a lookup table. The other dynamic parameters are optimized on the harmonic sweep signals with a particle swarm optimizer.

Figure 15.

Tire Model 4.

Figure 16 shows Tire Model 4, which is based on Tire Model 3. Unlike Tire Model 3, Tire Model 4 has deflection–dependent scaling functions also in series with all three Maxwell elements. The parameterization of the model is done in three steps. First, Tire Model 3 is parameterized. The parameters of Tire Model 3 are adopted in Tire Model 4 and all four scaling functions are determined with a particle swarm optimizer on the harmonic sweep signals.

Figure 16.

Tire Model 5.

Figure 17 shows Tire Model 5, which is based on Tire Model 3, but has three Prantl elements connected in parallel instead of one.

Figure 17.

RMSE of all tire models for the harmonic sweep signals.

The RMSE of the harmonic signals are used for model validation and comparison, as the excitations of the SMS also consist of harmonic signals. Figure 18 shows the RMSE of all tire models for the harmonic sweep signals. It can be seen that each further model development reduces the simulation error, which is why Tire Model 5 is used in the axle model.

Figure 18.

Simulation results and measurements of the amplitude responses of the vertical wheel contact forces for the four unsprung mass variations.

Optimization of unsprung mass

Figure 11 shows the final version of the axle model including Tire Model 5, which is used to optimize the unsprung mass. The suspension stiffness, the suspension friction, the anti–roll bar stiffness and the anti–roll bar friction are parameterized with the quasi–static full vehicle measurements on the SMS. The tire model is parameterized with the measurements on the single–axis hydropulser. The shock absorber characteristics and the unsprung masses are optimized on the dynamic measurements of the SMS with a particle swarm optimizer. The RMSE between simulations and measurements for the different mass variations is used as optimization criterion. Figure 19 shows the amplitudes of the vertical wheel contact forces of the measurements and the optimized simulation results for the four different mass variations. The figure shows that the natural frequencies of the different variants can be reproduced well by the simulation model. An unsprung mass of 47.65 kg was identified. Table 2 shows that the estimated unsprung mass is 47.47 kg. The result shows that the method can determine the unsprung masses of the axle with an accuracy of 99.6%.

Figure 19.

Measured and optimized force-velocity characteristics of the shock absorbers.

Figure 7 shows the measured and optimized shock absorber characteristics. It can be seen that the shock absorber characteristics cannot be optimized correctly. However, the deviations of the simulated shock absorber forces are negligible for the measurements of the axle and thus for the method for determining the unsprung mass. As a further validation step, the unsprung mass was simulatively optimized using the measured force-velocity characteristic of the shock absorbers. In this optimization, the unsprung mass was deter-mined to be 47.44 kg, which demonstrates the validity of the method.

Discussion

The method is able to determine the unsprung masses of the examined MacPherson axle with high accuracy. It was shown that the Axle Model 3 is sufficiently accurate for the method. The radial tire properties show the highest impact on the method. It can be assumed that the axle model can be applied to other axle concepts.

The measured horizontal forces at the tire contact patches are <1000 $N$ for all mass variants. Other axle concepts could generate larger horizontal forces with the presented excitations, which could influence the tire properties and thus falsify the results. The extension of the tire model to include the influence of horizontal forces on the radial stiffness of the tire could show good results. It is therefore recommended to apply the method to other axle concepts and, if necessary, to extend the tire model to include the influence of horizontal tire forces.

The presented axle models and tire models can also be used for the validation of dynamic test rig measurements on full vehicle test rigs. Examples of applications are the shock absorber inspection methods “Phase-Shift” and “EUSAMA,” which examine the axle damping of passenger cars and are based on a similar test setup.^26–30 The developed tire model can be used to simulative investigate and further develop such tests and inspection methods.

Conclusion

This work presents a method to determine the unsprung masses of passenger cars. The method is based on test rig measurements with a full vehicle test rig and a single–axle hydropulser and can be conducted without dismantling the axle. Quasi–static measurements are carried out to identify the relevant parameters of the e model. Furthermore, the wheels of the vehicle are excited with frequency sweeps in such a way that the natural frequencies of the unsprung masses are measured. One tire of the vehicle is measured quasi–statically and dynamically on a one-axis hydropulser test rig so that a model of the stationary tire can be parameterized which can reproduce the dynamic and non–linear properties of the stationary tire. The unsprung masses are then optimized with the axle model. The method is validated on a MacPherson axle and shows good results with an accuracy of 99.6%. To ensure general validity, the method should be tested on other axle concepts. The presented axle and tire models can be used for the simulation of similar test rig measurements.

Footnotes

Abbreviations

The following abbreviations are used in this manuscript:

CAD Computer-Aided Design

RMSE Root-Mean-Square Error

SMS Suspension Motion Simulator

UBS Universal Bushing

EUB Extended Unoversal Bushing

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Tobias Schramm

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A disassembly-free test rig-based method for determining the unsprung masses of a passenger car based on a non-linear dynamic model of the stationary tire

Abstract

Keywords

Introduction

Materials and methods

Full-vehicle measurements

Axle model

Tire model

Optimization of unsprung mass

Discussion

Conclusion

Footnotes

Abbreviations

Declaration of conflicting interests

Funding

ORCID iD

References