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

Abstract

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. ¹ Longitudinal leaf springs can be categorized based on quantity, with options including fewer-leaf^2,3 and multi-leaf springs. ⁴ In 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,^7–9 and the SAE three-link model. ¹⁰

To 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 cross-sectional area $S$ of the discrete beam and the moments of inertia $I_{\mathrm{xx}}$ , $I_{\mathrm{yy}}$ , and $I_{\mathrm{zz}}$ . 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.^18–20 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.

Figure 1.

Typical Van commercial vehicle suspensions: (a) front TLS suspension and (b) rear LPLS suspension.

Ignoring the anti-roll bar and the tie rod, the partially simplified diagram of the front TLS suspension is shown in Figure 2(a). A₁ and B₁ are the endpoints of the lower control arm. C₁ is the outer point of the lower control arm. The lower control arm $\mathrm{\Delta }$ A₁B₁C₁ is simplified to A₁C₁. D₁ is the center point of the knuckle. E₁ and F₁ are the installation position of the damper between the knuckle and the body. F₁C₁ is the virtual kingpin axis. G₁H₁ stands for the TLS. G₁ is the installation position between the end position of the TLS and the lower control arm. A₁ and F₁ are the fixed position on the body. $K_{l\_f}$ is the TLS stiffness, which is perpendicular to A₁C₁.

Figure 2.

Simplified Schematic of LS suspension: (a) TLS suspension and (b) LPLS suspension.

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. A₂ is the midpoint of the rear differential and the roll center of the rear LPLS suspension. B₂ and C₂ are the installation position of the damper between the rear axle and the body. D₂ is the middle position of the U-bolt. E₂ is the installation position of the front and rear leaf eyes of the LPLS. F₂ is the connection position between the axle and the rear inner wheel hub. G₂ and H₂ 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 $K_{s\_\mathrm{leaf}}$ matches the CS suspension stiffness $K_{s\_\mathrm{coil}}$ . The definitions of each symbol in Figure 3 are shown in Table 1.

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.

Table 1.

The symbol definitions of the equivalent CS suspensions.

Symbol	Definition	Symbol	Definition
${\alpha }_1$	The virtual angular displacement of the front suspension	l1	The line length of C1M1
${\alpha }_2$	The virtual angular displacement of the rear suspension	l2	The line length of A2W2
${\beta }_1$	the angle between JM and the ground	m1	The line length of A1G1
${\beta }_2$	The angle between A2W1 and the ground	m2	The line length of A2W1
$\theta$	the angle between the damper axle and the vertical line	n1	The line length of A1C1
$\phi$	the angle between $F_{\mathrm{Coil}\_f}$ and the vertical line	$f$	The virtual displacement of MJ along α1
$\gamma$	The angle between A2W2 and the ground	$\mathrm{\Delta }f$	Virtual compression of $K_{s\_f}$
a1	The line length of ME1	$\mathrm{\Delta }{f}_{l\_f}$	Virtual compression of $K_{l\_f}$
a2	The horizontal length of A2W2	$f_{\mathrm{coil}\_f}$	The virtual displacement of ME1 along α1
b1	The horizontal length of ME1	$\mathrm{\Delta }{f}_{\mathrm{coil}\_f}$	Virtual compression of $K_{\mathrm{coil}\_f}$
b2	The horizontal length of A2W1	$\mathrm{\Delta }{f}_{\mathrm{coil}\_r}$	Virtual compression of $K_{\mathrm{coil}\_r}$
B	the wheelbase	$f_{\mathrm{in}}$	The virtual displacement of A2W1 along α2
p	The line length of JL	$\mathrm{\Delta }{f}_{\mathrm{in}}$	Virtual compression of $K_{\mathrm{in}}$
q1	The line length of JM	$f_{\mathrm{out}}$	The virtual displacement of A2W2 along α2
q2	The horizontal length of A2D2	$\mathrm{\Delta }{f}_{\mathrm{out}}$	Virtual compression of $K_{\mathrm{out}}$
$K_{s\_f}$	The front suspension stiffness	$K_{l\_f}$	The TLS stiffness
$K_{s\_r}$	The rear suspension stiffness	$K_{l\_r}$	The LPLS stiffness
$K_{\mathrm{coil}\_f}$	The front equivalent CS stiffness	$K_{\mathrm{in}}$	Rear suspension stiffness at W1
$K_{\mathrm{coil}\_r}$	The rear equivalent CS stiffness	$K_{\mathrm{out}}$	Rear suspension stiffness at W2

In Figure 3(a), force $F_{l\_f}$ is perpendicular to the lower control arm A₁C₁. Based on virtual displacement theory, P23 (Point C₁), P35 (Point A₁), and P45 (Point F₁) 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:

$$\begin{array}{c}{F}_f\mathrm{\Delta }{f}_f=F_{l\mathrm{\_}f}\mathrm{\Delta }{f}_{l\mathrm{\_}f}\hfill \\ F_{\mathrm{in}}\mathrm{\Delta }{f}_{\mathrm{in}}+F_{\mathrm{out}}\mathrm{\Delta }{f}_{\mathrm{out}}=F_{l\mathrm{\_}r}\mathrm{\Delta }{f}_{l\mathrm{\_}r}\hfill \end{array}$$

(1)

where $F_f$ is the ground force on the wheel, $F_f=K_{s\_f}\mathrm{\Delta }{f}_f$ . $F_{l\_f}$ is the force exerted by the lower control arm on the end of the TLS, $F_{l\_f}=K_{l\_f}\mathrm{\Delta }{f}_{l\_f}$ . $\mathrm{\Delta }{f}_f$ is the vertical virtual displacement of point J. $\mathrm{\Delta }{f}_{l\_f}$ 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. $F_{\mathrm{in}}$ is the ground force on the inner wheel, $F_{\mathrm{in}}=K_{\mathrm{in}}\mathrm{\Delta }{f}_{\mathrm{in}}$ . $F_{\mathrm{out}}$ is the ground force on the outer wheel, $F_{\mathrm{out}}=K_{\mathrm{out}}\mathrm{\Delta }{f}_{\mathrm{out}}$ . The relation is $F_{\mathrm{in}}=F_{\mathrm{out}}$ . $F_{l\_r}$ is the force of the LPLS by Part 3, $F_{l\_r}=K_{l\_r}\mathrm{\Delta }{f}_{l\_r}$ . $\mathrm{\Delta }{f}_{\mathrm{in}}$ and $\mathrm{\Delta }{f}_{\mathrm{out}}$ are the vertical virtual displacement of the contact point W₁ and W₂. $\mathrm{\Delta }{f}_{l\_r}$ is the virtual displacement of the LPLS along the axis DE. $K_{\mathrm{in}}$ and $K_{\mathrm{out}}$ are the equivalent CS stiffness between the inner and outer wheels and the body, $K_{\mathrm{in}}=K_{\mathrm{out}}$ . There is a relationship that $K_{s\_r}=K_{\mathrm{in}}+K_{\mathrm{out}}$ . $\mathrm{\Delta }f$ , $\mathrm{\Delta }{f}_{\mathrm{in}}$ and $\mathrm{\Delta }{f}_{\mathrm{out}}$ are obtained by equation (2)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Delta }f=f\mathit{cos}\beta =q\alpha \mathit{cos}\beta =\left(q\mathit{cos}\beta \right)\alpha =p\alpha }\\ \multicolumn{1}{c}{\mathrm{\Delta }{f}_{\mathrm{in}}=f_{\mathrm{in}}\mathit{cos}\beta =\alpha l\mathit{cos}\beta =\left(l\mathit{cos}\beta \right)\alpha =a·\alpha }\\ \multicolumn{1}{c}{\mathrm{\Delta }{f}_{\mathrm{out}}=f_{\mathrm{out}}\mathit{cos}\gamma =\alpha m\mathit{cos}\gamma =\left(m\mathit{cos}\gamma \right)\alpha =b·\alpha }\end{array}$$

(2)

where $f$ is the virtual displacement of point J along the instantaneous direction of motion. The relationship between the LS virtual compression $\mathrm{\Delta }{f}_{l\_f}$ and point C virtual displacement $\mathrm{\Delta }{f}_C$ is obtained by equation (3), $\mathrm{\Delta }{f}_C=l\alpha$ .

$$\mathrm{\Delta }{f}_{l}m=\mathrm{\Delta }{f}_{c}n$$

(3)

Combining equations (1) and (3), the relationship between $K_l$ and $K_s$ is shown in equation (4).

$$\begin{array}{c}{K}_{s\mathrm{\_}f}=K_{l\mathrm{\_}f}{\left(\frac{\ln }{\mathrm{mp}}\right)}^2\hfill \\ K_{s\mathrm{\_}r}=K_{l\mathrm{\_}r}\left(\frac{2q^2}{a^2+b^2}\right)\hfill \end{array}$$

(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 MC₁ and the plane equation (6), which is a plane passing through the F₁ point and perpendicular to the straight line F₁E₁.

$$\frac{x_M-x_C}{x_A-x_C}=\frac{y_M-y_C}{y_A-y_C}=\frac{z_M-z_C}{z_A-z_C}$$

(5)

$$\begin{array}{c}(x_E-x_F)(x_M-x_F)+(y_E-y_F)(y_M-y_F)\hfill \\ +(z_E-z_F)(z_M-z_F)=0\hfill \end{array}$$

(6)

So far, the relationship between $K_s$ and $K_l$ has been established. Next the relationship between $K_s$ and $K_{\mathrm{coil}}$ needs to be established. As the installation position of the equivalent CS is the same as that in the LPLS, it follows that $K_{\mathrm{coir}\_r}=K_{l\_r}$ . For the front equivalent CS suspension, there is:

$$F\mathrm{\Delta }f=F_{\mathrm{coil}\mathrm{\_}f}\mathrm{\Delta }{f}_{\mathrm{coil}\mathrm{\_}f}$$

(7)

where $F=K_{s\_f}\mathrm{\Delta }f$ . $F_{\mathrm{coil}\_f}$ is the spring force of the equivalent CS along the axis of the damper, $F_{\mathrm{coil}\_f}=K_{\mathrm{coil}\_f}\mathrm{\Delta }{f}_{\mathrm{coil}\_f}$ . $\mathrm{\Delta }{f}_{\mathrm{coil}\_f}$ is the virtual displacement of the CS along with the axis FE. $f_{\mathrm{coil}\_f}$ is the virtual displacement of point E₁, $f_{\mathrm{coil}\_f}={\alpha }_1{a}_1$ . The relation of $f_{\mathrm{coil}\_f}$ and $\mathrm{\Delta }{f}_{\mathrm{coil}\_f}$ can be obtained in equation (8).

$$\mathrm{\Delta }{f}_{\mathrm{coil}\mathrm{\_}f}=\frac{f_{\mathrm{coil}}\mathit{cos}\phi }{\mathit{cos}\theta }=\frac{{\alpha }_1{a}_1\mathit{cos}\phi }{\mathit{cos}\theta }=\frac{{\alpha }_1\left(a_1\mathit{cos}\phi \right)}{\mathit{cos}\theta }=\frac{{\alpha }_1{b}_1}{\mathit{cos}\theta }$$

(8)

Therefore, the relationship between $K_{\mathrm{coil}\_f}$ and $K_{s\_f}$ is shown in equation (9). Combining equations (4) and (9), the relationship between $K_{\mathrm{coil}}$ and $K_l$ of the TLS suspension is obtained in equation (10).

$$K_{\mathrm{coil}\mathrm{\_}f}=K_{s\mathrm{\_}f}{\left(\frac{p\mathit{cos}\theta }{b_1}\right)}^2$$

(9)

$$K_{\mathrm{coil}\mathrm{\_}f}=K_{l\mathrm{\_}f}{\left(\frac{l_1·n_1·\mathit{cos}\theta }{m_1·b_1}\right)}^2$$

(10)

Since the instantaneous centers M and A₂ 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 $K_{\mathrm{Coil}}$ and $K_l$ . Therefore, the dynamic proportionality factor $r\left(x_i\right)$ of $K_{\mathrm{Coil}}$ and $K_{\mathrm{leaf}}$ must be solved.

Through the suspension PWT simulation of Adams Car, the wheel travel distance $X_{\mathrm{leaf}}$ of the LS suspension and the displacement of the equivalent CS $Z_{\mathrm{coil}}$ is presented as follows:

$$\begin{array}{c}{X}_{\mathrm{leaf}}=[-x_n,\cdots -x_i\cdots ,x_0,\cdots x_i\cdots ,x_n]\hfill \\ Z_{\mathrm{coil}}=[-z_n,\cdots -z_1,z_0,z_1\cdots z_n,\cdots z_m]\hfill \end{array}$$

(11)

where $x_0$ is the initial value of the wheel travel distance, $x_0=0$ . $m=n+i$ . $i\in [0,n]$ . The suspension stiffness $k_s\left(x_n\right)$ and hub force $f\left(x_n\right)$ at any position $x_n$ are obtained using equation (12).

$$\begin{array}{c}{K}_s\left(X\right)=\hfill \\ [k_s(-x_n)\cdots k_s(-x_i)\cdots k_s(-x_1),k_s\left(x_0\right),k_s\left(x_1\right)\cdots k_s\left(x_i\right)\cdots k_s\left(x_n\right)]\hfill \\ F\left(X\right)=\hfill \\ [f(-x_n)\cdots f(-x_i)\cdots f(-x_1),f\left(x_0\right),f\left(x_1\right)\cdots f\left(x_i\right)\cdots f\left(x_n\right)]\hfill \end{array}$$

(12)

As for the TLS suspension, when $f_f(-x_{i\_f})\Rightarrow 0$ , $K_{s\_f}(-x_{i\_f})$ is recorded as $k_{\mathrm{coil}\_f}\left(z_{0\_f}\right)$ at the initial position of the equivalent CS, $K_{s\_f}(-x_{i\_f})=k_{\mathrm{coil}\_f}\left(z_{0\_f}\right)$ . Since the number of LSs in the LPLS suspension is three, the LPLS suspension stiffness $K_{s\_r}(-x_{i\_r})$ is the third LS stiffness when $f_r(-x_{i\_r})\Rightarrow 0$ . When two LSs are in contact, the initial stiffness of the LPLS suspension is recorded as $K_{s\_r}(-z_{a\_r})$ . When three LSs are in contact, the initial stiffness of the LPLS suspension is recorded as $K_{s\_r}(-z_{b\_r})$ , $a\in (0,i)$ , $b\in (0,a)$ .

The equivalent CS dynamic stiffness $K_{\mathrm{coil}}\left(Z_{\mathrm{coil}}\right)$ and the relationship between $k_{\mathrm{coil}}\left(z_{\mathrm{coil}}\right)$ and $K_s\left(X\right)$ are shown in equation (13).

$$\begin{array}{c}{K}_{\mathrm{coil}}\left(Z_{\mathrm{coil}}\right)=\hfill \\ [k_{\mathrm{coil}}(-z_n),\cdots k_{\mathrm{coil}}(-z_1),k_{\mathrm{coil}}\left(z_0\right),k_{\mathrm{coil}}\left(z_1\right)\cdots k_{\mathrm{coil}}\left(z_n\right)\cdots k_{\mathrm{coil}}\left(z_m\right)]\hfill \\ k_{\mathrm{coil}\_f}\left(z_j\right)=\{\begin{array}{cc}{k}_{s_{-}f}(-x_{n_{-}f}),\hfill & j\in [-n,-(n-i))\hfill \\ \multicolumn{1}{c}{k_{s_{-}f}(-x_{f_{-}i}),}& i\prime \in [-n,-i),j\in [-(n-i),0)\hfill \\ \multicolumn{1}{c}{k_{s_{-}f}\left(x_{f_{-}i}\right),}& i\prime \in [-i,n],j\in [0,m]\hfill \end{array}\hfill \\ k_{\mathrm{coil}\_r}\left(z_j\right)=\{\begin{array}{ccc}{k}_{s_{-}r}(-x_{i_{-}r}),\hfill & i\in [0,n],\hfill & j\in [-n,0]\hfill \\ \multicolumn{1}{c}{k_{s_{-}r}(-x_{i_{-}r}),}& i\prime \in (-i,-a],\hfill & j\in [0,i-a)\hfill \\ \multicolumn{1}{c}{k_{s_{-}r}(-x_{i_{-}r}),}& i\prime \in (-a,-b],\hfill & j\in [i-a,a-b]\hfill \\ \multicolumn{1}{c}{k_{s_{-}r}(-x_{i_{-}r}),}& i\prime \in (-b,0],\hfill & j\in [a-b,b]\hfill \\ \multicolumn{1}{c}{k_{s_{-}r}\left(x_{i-r}\right),}& i\prime \in (0,n],\hfill & j\in [b,m]\hfill \end{array}\hfill \end{array}$$

(13)

The spring force of equivalent CS $F_{\mathrm{coil}}\left(Z_{\mathrm{coil}}\right)$ is obtained as follows:

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{f}_{\mathrm{coil}}(-z_j)=(z_{j-1}-z_j)·k_{\mathrm{coil}}(-z_j)+f_{\mathrm{coil}}(-z_{j-1}),j\in [-n,0)\hfill \\ \multicolumn{1}{c}{f_{\mathrm{coil}}\left(z_j\right)=(z_j-z_{j-1})·k_{\mathrm{coil}}\left(z_j\right)+f_{\mathrm{coil}}\left(z_{j-1}\right),j\in [0,m]}\end{array}}\end{array}$$

(14)

where $f_{\mathrm{coil}}\left(z_0\right)=0$ . 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).

$$F=-K(\mathrm{DM}-\mathrm{OFF})$$

(15)

where $F=f\left(x_0\right)$ , $K=k_{\mathrm{coil}}\left(x_0\right)$ . $\mathrm{DM}$ is the distance between the upper and lower hard points of CS. $\mathrm{OFF}$ 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 $X_{\mathrm{coil}}$ and suspension stiffness $K_{s\_\mathrm{coil}}$ . The suspension proportional factor $r_f\left(x_i\right)$ of equivalent CS suspension and LS suspension is calculated by equation (16).

$$r\left(x_i\right)=k_{\mathrm{coil}}\left(x_i\right)/k_{s\mathrm{\_}l}\left(x_i\right)\mathrm{i}\in [-n,n]$$

(16)

To ensure the stiffness consistency of the CM suspension and the TLS suspension, the CS stiffness $K_{\mathrm{coil}\_f}$ and $K_{\mathrm{coil}\_r}$ needs to be corrected. According to equations (11) and (14), the stiffness $K_{\mathrm{coil}\_f}^{\prime }\left(Z_{\mathrm{coil}\_f}\right)$ and $K_{\mathrm{coil}\_r}^{\prime }\left(Z_{\mathrm{coil}\_r}\right)$ of the modified CS can be calculated by equation (17).

$$\begin{array}{c}{k\prime }_{{\mathrm{coil}}_{-}f}\left(z_j\right)=\{\begin{array}{cc}{k}_{{\mathrm{coil}}_{-}f}\left(z_j\right)/r(-x_n),\hfill & j\in [-n,-(n-i))\hfill \\ \multicolumn{1}{c}{k_{{\mathrm{coil}}_{-}f}\left(z_j\right)/r\left(x_i\right),}& i\in [-n,n],j\in [-(n-i),m]\hfill \end{array}\hfill \\ {k\prime }_{{\mathrm{coil}}_{-}r}\left(z_j\right)=\{\begin{array}{cc}{k}_{{\mathrm{coil}}_{-}r}\left(z_j\right)/r(-x_a),\hfill & \mathrm{a}\in (0,i],j\in [-n,i-a)\hfill \\ \multicolumn{1}{c}{k_{{\mathrm{coil}}_{-}r}\left(z_j\right)/r\left(x_i\right),}& i\in (-a,-b),j\in [i-a,a-b)\hfill \\ \multicolumn{1}{c}{k_{{\mathrm{coil}}_{-}r}\left(z_j\right)/r\left(x_b\right),}& i\in [-b,n],j\in [a-b,m)\hfill \end{array}\hfill \end{array}$$

(17)

Through equations (14)–(17), the displacement $Z_{\mathrm{coil}}$ , the spring force $F_{\mathrm{coil}}\left(Z_{\mathrm{coil}}\right)$ and the CS stiffness $K_{\mathrm{coil}}\left(Z_{\mathrm{coil}}\right)$ can be obtained.

In summary, the conversion relationship between the LS suspension stiffness $K_s\left(X\right)$ and the equivalent CS stiffness $K_{\mathrm{coil}}$ 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 $F_l\left(X_l\right)$ , and suspension stiffness $K_{s\_l}\left(X_l\right)$ 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 $K_{s\_\mathrm{coil}}\left(X_{\mathrm{coil}}\right)$ of the CS suspension is obtained. The suspension stiffness proportionality factor $r\left(x_i\right)$ 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 $K_{\mathrm{coil}\_f}^{\prime }\left(Z_{\mathrm{coil}\_f}\right)$ of the TLS suspension and the modified CS stiffness $K_{\mathrm{coil}\_r}^{\prime }\left(Z_{\mathrm{coil}\_r}\right)$ of the LPLS suspension is different. When $r\left(x_i\right)<95\%$ , the equivalent CS stiffness $K_{\mathrm{coil}}^{\prime }\left(Z_{\mathrm{coil}}\right)$ is recalculated according to equation (17). Only $r\left(x_i\right)\ge 95\%$ can the CS stiffness calculation end. The method proposed in this paper solves the dynamic proportional coefficient $r\left(x_i\right)$ of the CS stiffness $K_{\mathrm{coil}}$ and the LS suspension stiffness $K_{s\_l}$ during the wheel travel process.

Figure 4.

Flow chart of equivalent LS suspension with CS suspension.

Multibody dynamic model

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 [−50, 50] mm and [−50, 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.

Figure 5.

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

Figure 6.

The MBD model of rear suspension: (a) LPLS suspension and (b) equivalent TLCS suspension.

Table 2.

Simulation parameters of the front and rear suspension.

Parameter	Unit	Value
Sprung mass	kg	3452
CG height	mm	518.872
Wheelbase	mm	4325
Tire unloaded radius	mm	349.45
Tire Stiffness	N/mm	220
Drive ratio (Rear)	%	100
Brake ratio (Front)	%	70

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).

$$\begin{array}{c}\mathrm{AF}=\sqrt{A{B}^2-B{F}^2}\hfill \\ \mathrm{FG}=\mathrm{BC}=m+n\hfill \\ \mathrm{GD}=\sqrt{C{D}^2-C{G}^2}\hfill \end{array}$$

(18)

Figure 7.

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

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. 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.

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.

Table 3.

Base vehicle specification parameters.

Base vehicle data	Values	Base vehicle data	Values
Wheelbase (mm)	4235	Total mass	3712 kg
Tread (mm)
Front	1690	Front sprung mass	1436
Rear	1516
Vehicle length (mm)	7940	Rear axle sprung mass	2015
Vehicle width (mm)	2000	Centroid coordinates	2582.8,38.5,518.9
Vehicle height (mm)	2675	Camber (°)	0.35° ± 0.333°
Moment of inertia (kg.mm2)
Ixx	1.89 × 109	Toe angle (°)	0.283° ± 0.167°
Iyy	1.35 × 109	Caster angle	4° ± 0.5°
Izz	1.24 × 109	Kingpin inclination angle	13.417° ± 0.5°

Road excitation

Road surface excitation has an important impact on vehicle ride comfort research. ²¹ To 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 roll-off 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

$$G_s={\left(2\pi n_0\right)}^2{G}_q\left(n_0\right)$$

(19)

where $n_0$ is the reference spatial frequency, ²⁴ $n_0=0.1m^{-1}$ . $G_q\left(n_0\right)$ is the PSD value of $n_0$ and called the road roughness coefficient. When $\mathrm{Gq}\left(n_0\right)$ is set to $16\times {10}^{-6}{m}^3$ , 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).

Figure 9.

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

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 −3.34 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.

Figure 10.

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

Table 4.

Suspension stiffness at the initial position of suspensions.

suspension	Multibody suspension models	Suspension stiffness (N/mm)	Error (%)
Front	TLS	64.189	/
	CS	64.4533	0.41
Rear	LPLS	171.4629	/
	Two-Link CS	171.5776	0.081

Model discussions

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)).

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.

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 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.

Figure 12.

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

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.

Figure 13.

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

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.

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

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 −12.06% 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%.

Suspension natural frequency testing showed that the LS suspension and equivalent CS suspension had the same natural frequencies as the experimental values. The equivalent method preserves the suspension’s natural frequency. However, near the suspension’s natural frequency, the body’s vertical acceleration amplitude of the equivalent CS suspension is slightly higher than that of the LS suspension. In other frequency bands, the body’s vertical acceleration amplitude of the equivalent CS suspension is lower than that of the LS suspension. The suspension sweep frequency sine excitation test confirmed the CS equivalent method maintains the body’s vertical frequency while reducing the PSD amplitude of the body’s vertical acceleration, consistent with the suspension’s natural frequency test.

The simulation on the A-class road shows that the RMS of the body vertical acceleration with the equivalent CS suspension matches the trend of the LS suspension. In the low-frequency band (<2 Hz), the PSD amplitude of the equivalent CS suspension’s vertical acceleration is higher than the LS suspension, but lower in other frequency bands. This aligns with the frequency sweep test and is due to the discrete beam modeling of the LS, which includes its mass. These experiments confirm the suitability of the proposed CS parameterization method for studying vehicle ride comfort. It provides a foundation for efficiently matching suspension stiffness in vehicles with LS suspension.