Robust design of multi-body model of steering mechanism based on uncertainties of suspension parameters

Abstract

This paper presents a modified multi-body dynamics model which can evaluate the performance of the steering and suspension systems simultaneously. The modified model investigates the effects of suspension geometry on the steering mechanism. Moreover, it evaluates the influence of the kingpin axis through steering process. This approach consists of three steps; in the first step, the effects of suspension geometry on steering returnability and ride quality are considered utilizing overturning moment and sprung mass acceleration. In other words, this model considers the suspension parameters as inputs of geometry suspension model and evaluates their influences on vehicle behavior. In the second step, the output data of the geometry model is considered as input data for steering system in order to evaluate the effects of suspension parameters on steering mechanism. The accuracy of the proposed model is verified by experimental tests. In this case, Renault Logan is subjected to a constant cornering test (ISO 7401) and double-lane change test (ISO 4138). In the third step, a probabilistic approach is used by developing NSGA II in order to design a robust Pareto multi-objective optimum algorithm. This modified algorithm is used for robust design of a modified suspension-steering mechanism, as an application, considering probabilistic and uncertain variables. The results achieved by the new model are compared with Adams Car simulation in order to evaluate the accuracy of the model.

Appendices

Appendix 1

By using the triangle OWF, the following equation can be written as:

$$ \begin{aligned} \lambda & = \left( {\lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \Phi^{\prime}} \right)^{1/2} \\ \lambda^{\prime} & = \left( {\lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \left( {\Phi^{\prime} - \Omega } \right)} \right)^{1/2} \\ \end{aligned} $$

(24)

where

$$ \Phi^{\prime} = \Phi - \Omega_{0} $$

(25)

In Eq. (24), λ and $\lambda^{\prime}$ represent the initial length of control member wf (illustrated in Fig. 9). Ω indicates the angle between the ow-axis and y-axis which is presented in Fig. 5.

$$ \begin{aligned} \left( {\Delta \lambda } \right)^{2} & = \left( {\lambda - \lambda^{\prime}} \right)^{2} \\ & = \lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \Phi^{\prime} \\ & \quad + \lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \left( {\Phi^{\prime} - \Omega } \right) \\ & \quad - 2\left( {\lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \Phi^{\prime}} \right)^{1/2} \\ & \quad \left( {\lambda_{A}^{2} + \lambda_{B}^{2} - 2\lambda_{A} \lambda_{B} \cos \left( {\Phi^{\prime} - \Omega } \right)} \right)^{1/2} \\ \end{aligned} $$

(26)

$$ \begin{aligned} a_{\lambda } & = \lambda_{A}^{2} + \lambda_{B}^{2} \\ b_{\lambda } & = 2\lambda_{A} \lambda_{B} \\ \end{aligned} $$

Δλ and z_g–z_r (Eqs. 27 and 29) are calculated by considering the geometry of vehicle suspension as shown below:

$$ \begin{aligned} \left( {\Delta \lambda } \right)^{2} & = \left( {\lambda - \lambda^{\prime}} \right)^{2} \\ & = 2a_{\lambda } - b_{\lambda } \left( {\cos \Phi^{\prime} + \cos \left( {\Phi^{\prime} - \Omega } \right)} \right) \\ & \quad - 2\left\{ {a_{\lambda }^{2} } \right. - a_{\lambda } b_{\lambda } (\cos \sigma^{\prime} + \cos (\Phi^{\prime} - \Omega ) \\ & \quad + \left. {b_{\xi }^{2} \cos \alpha^{\prime}\cos (\Phi^{\prime} - \Omega ))} \right\}^{1/2} \\ \end{aligned} $$

(27)

$$ \dot{\Delta }\xi = \dot{\lambda } - \dot{\lambda }^{\prime} = \frac{{b_{\lambda } \sin \left( {\Phi^{\prime} - \Omega } \right)\dot{\Omega }}}{{2\left( {a_{\lambda } - b_{\lambda } \cos \left( {\Phi^{\prime} - \Omega } \right)} \right)^{1/2} }} $$

(28)

$$ z_{g} - z_{r} = \lambda_{c} \left( {\sin \left( {\Omega - \Omega_{0} } \right) - \sin \left( { - \Omega_{0} } \right)} \right) - z_{r} $$

(29)

Equations (1), (24)–(29) indicate the transformed of suspension key points, based on original variables of the vehicle suspension. These points are utilized to derive the state equations of the suspension geometry. The state equations of geometry suspension are then used to evaluate the vehicle suspension behavior under different conditions. Lagrangian mechanics are used to achieve the equations of motion of the Macpherson suspension, as indicated in Eq. (30).

$$ \begin{aligned} \eta_{1} \ddot{z}_{s1} + \eta_{2} \ddot{\Omega } + A_{1} + A_{2} & = A_{3} \to \eta_{1} \ddot{z}_{s1} + \eta_{2} \ddot{\Omega } \\ & = A_{3} - A_{1} - A_{2} = P_{1} \\ \eta_{3} \ddot{z}_{s1} + \eta_{4} \ddot{\Omega } + A_{4} + A_{5} & = A_{6} \to \eta_{3} \ddot{z}_{s1} + \eta_{4} \ddot{\Omega } \\ & = A_{6} - A_{4} - A_{5} = P_{2} \\ \end{aligned} $$

(30)

where

$$ \begin{aligned} A_{1} & = - \frac{\partial T}{{\partial z_{{{\text{sm}}1}} }},\;A_{2} = \frac{\partial V}{{\partial z_{sm1} }}, \\ A_{3} & = - \frac{\partial D}{{\partial z_{sm1} }} \\ A_{4} & = - \frac{\partial T}{{\partial \Omega }},\;A_{5} = \frac{\partial V}{{\partial \Omega }}, \\ A_{6} & = - \frac{\partial D}{{\partial \Omega }} \\ \end{aligned} $$

(31)

where T, V and D, respectively, represent the kinetic energy, potential energy and the damping energy. The equations of motion and its associated parameters are as follows:

$$ \begin{aligned} \ddot{\Omega } & = \frac{{\eta_{1} P_{2} - \eta_{4} P_{1} }}{{\eta_{1} \eta_{3} - \eta_{2} \eta_{4} }} \\ \ddot{z}_{{{\text{sm}}1}} & = \frac{{\eta_{2} P_{2} - \eta_{3} P_{1} }}{{\eta_{2} \eta_{4} - \eta_{1} \eta_{3} }} \\ \end{aligned} $$

(32)

Appendix 2

By substitution of Eq. (32) into Eq. (3), f₁ and f₂ are calculated as:

$$ \begin{aligned} f_{1} & = \frac{1}{{D_{1} }}\left[ {m_{{{\text{um}}1}} \lambda_{c}^{2} \sin (\varepsilon_{3} - \Omega_{0} )\varepsilon_{4}^{2} } \right. \\ & \quad - \frac{1}{2}k_{{{\text{st}}1}} \sin \left( {\Phi^{\prime} - \varepsilon_{3} } \right)\cos \left( {\varepsilon_{3} + \Omega_{0} } \right)\Lambda (\varepsilon_{3} ) \\ & \quad \left. { + c_{{{\text{st}}1}} {\rm H}\left( {\varepsilon_{3} } \right)\dot{\Omega } - k_{t1} \lambda_{c} \sin^{2} \left( {\varepsilon_{3} - \Omega_{0} } \right)z\left( 0 \right)} \right] \\ \end{aligned} $$

(33)

$$ \begin{aligned} f_{2} & = - \frac{1}{{D_{2} }}\left[ {m_{{{\text{um}}1}}^{2} \lambda_{c}^{2} \sin (\varepsilon_{3} - \Omega_{0} )\cos (\varepsilon_{3} - \Omega_{0} )\varepsilon_{4}^{2} } \right. \\ & \quad + \left( {m_{{{\text{sm}}1}} + m_{{{\text{um}}1}} } \right)c_{s1} {\rm H}\left( {\varepsilon_{3} } \right)\varepsilon_{4} \\ & \quad - \frac{1}{2}\left( {m_{{{\text{sm}}1}} + m_{{{\text{um}}1}} } \right)k_{{{\text{sm}}1}} \sin \left( {\Phi^{\prime} - \varepsilon_{3} } \right)\Lambda \left( {\varepsilon_{3} } \right) \\ & \quad + m_{{{\text{sm}}}} k_{t1} \lambda_{c} \cos \left( {\varepsilon_{3} - \Omega_{0} } \right)z\left( 0 \right)] \\ \end{aligned} $$

(34)

where D₁ and D₂ are given by Eq. (35).

$$ \begin{aligned} D_{1} & = m_{{{\text{sm}}1}} \lambda_{C} + m_{{{\text{um}}1}} \lambda_{C} \sin^{2} \left( {\varepsilon_{3} - \Omega_{0} } \right) \\ D_{2} & = m_{{{\text{sm}}1}} m_{{{\text{um}}1}} \lambda_{c}^{2} + m_{{{\text{um}}1}}^{2} \lambda_{c}^{2} \sin^{2} \left( {\varepsilon_{3} - \Omega_{0} } \right) \\ \end{aligned} $$

(35)

Λ(x₃), H(x₃), c_λ and d_λ are achieved by Eqs. (36)–(39).

$$ \Lambda (x_{3} ) = b_{\lambda } + \frac{{d_{\lambda } }}{{\left( {c_{\lambda } - d_{\lambda } \cos \left( {\Phi^{\prime} - \varepsilon_{3} } \right)} \right)^{1/2} }} $$

(36)

$$ {\rm H}(x_{3} ) = \frac{{b_{v}^{2} \sin^{2} (\Phi^{\prime} - x_{3} )}}{{4\left( {a_{\lambda } - b_{\lambda } \cos \left( {\Phi^{\prime} - \varepsilon_{3} } \right)} \right)}} $$

(37)

$$ \begin{aligned} z\left( 0 \right) & = z\left( {x_{1} ,x_{2} ,z_{r} } \right) = x_{1} \\ & \quad + \lambda_{c} \left( {\sin \left( {x_{3} - \Omega_{0} } \right) - \sin \left( { - \Omega_{0} } \right)} \right) - z_{r1} \\ \end{aligned} $$

(38)

$$ \begin{aligned} c_{\lambda } & = a_{\lambda }^{2} - a_{\lambda } b_{\lambda } \cos \left( {\Phi + \Omega_{0} } \right) \\ d_{\lambda } & = a_{\lambda } b_{\lambda } - b_{\lambda }^{2} \cos \left( {\Phi + \Omega_{0} } \right) \\ \end{aligned} $$

(39)

Appendix 3

$$ \begin{aligned} x_{wp} & = s_{b} \left( {\frac{\cos \vartheta \cos \beta \sin \delta }{{\sqrt {\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \varphi } }}} \right. \\ & \quad + \left. {\frac{1}{4}\frac{\sin 2\vartheta \sin 2\beta (1 - \cos \delta )}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }}} \right) \\ & \quad + s_{a} \left( {1 - \frac{{\cos^{2} \vartheta \sin^{2} \beta }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }}} \right)\left( {1 - \cos \delta } \right) \\ \end{aligned} $$

(40)

$$ \begin{aligned} y_{wp} & = - s_{a} \left( {\frac{\cos \vartheta \cos \beta \sin \delta }{{\sqrt {\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta } }}} \right. \\ & \quad - \left. {\frac{1}{4}\frac{\sin 2\vartheta \sin 2\beta (1 - \cos \delta )}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }}} \right) \\ & \quad + s_{b} \left( {1 - \frac{{\cos^{2} \beta \sin^{2} \vartheta }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }}} \right)\left( {1 - \cos \delta } \right) \\ \end{aligned} $$

(41)

$$ \begin{aligned} z_{wp} & = - R_{w} \\ & \quad - \frac{{s_{b} \cos \vartheta \sin \beta + s_{a} \cos \beta \sin \vartheta }}{{\sqrt {\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta } }}\sin \delta \\ & \quad + \frac{1}{2}\frac{{s_{b} \cos^{2} \beta \sin 2\vartheta - s_{a} \cos^{2} \vartheta \sin 2\beta }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }}\left( {1 - \cos \delta } \right) \\ \end{aligned} $$

(42)

$$ \begin{aligned} x_{tc} & = s_{a} \left( {1 - \cos \delta } \right) \\ & \quad + \frac{{\left( {(1/2)R_{w} \sin 2\beta - s_{a} \sin^{2} \beta } \right)\cos^{2} \vartheta + (1/4)s_{b} \sin 2\vartheta \sin 2\beta }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }} \\ & \quad \left( {1 - \cos \delta } \right) \\ & \quad + \frac{{s_{b} \cos \vartheta - R_{w} \sin \vartheta }}{{\sqrt {\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta } }}\cos \beta \sin \delta \\ \end{aligned} $$

(43)

$$ \begin{aligned} y_{tc} & = s_{b} \left( {1 - \cos \delta } \right) \\ & \quad - \frac{{\left( {(1/2)R_{w} \sin 2\vartheta - s_{b} \sin^{2} \vartheta } \right)\cos^{2} \beta - (1/4)s_{a} \sin 2\vartheta \sin 2\beta }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }} \\ & \quad \left( {1 - \cos \delta } \right) \\ & \quad - \frac{{R_{w} \sin \beta + s_{a} \cos \beta }}{{\sqrt {\cos^{2} \beta + \cos^{2} \vartheta \sin^{2} \beta } }}\cos \vartheta \sin \delta \\ \end{aligned} $$

(44)

$$ \begin{aligned} z_{tc} & = - R_{w} \left( {1 - \cos \delta } \right) \\ & \quad - \frac{{\left( {R_{w} \cos^{2} \vartheta + (1/2)s_{b} \sin 2\vartheta } \right)\cos^{2} \beta - (1/2)s_{a} \cos^{2} \vartheta \sin 2\varphi }}{{\cos^{2} \vartheta \sin^{2} \beta + \cos^{2} \beta }} \\ & \quad \beta \left( {1 - \cos \delta } \right) \\ & \quad \left( {1 - \cos \delta } \right) - \frac{{s_{a} \cos \beta \sin \vartheta + s_{b} \cos \vartheta \sin \beta }}{{\sqrt {\cos^{2} \beta + \cos^{2} \vartheta \sin^{2} \beta } }}\sin \delta \\ \end{aligned} $$

(45)

Appendix 4

$$ \begin{aligned} F_{y} & = \left( {\frac{{a_{2} }}{{v_{x} }}C_{\alpha r} - \frac{{a_{1} }}{{v_{x} }}C_{\alpha f} } \right)r \\ & \quad + \left( {\frac{{C_{\alpha f} C_{\beta f} }}{{v_{x} }} + \frac{{C_{\alpha r} C_{\beta r} }}{{v_{x} }}} \right)p \\ & \quad + \left( { - C_{\alpha f} - C_{\alpha r} } \right)\beta \\ & \quad + \left( {C_{\alpha f} C_{\delta \varphi f} - C_{\varphi r} - C_{\varphi f} + C_{\alpha r} C_{\delta \varphi r} } \right)\varphi \\ & \quad + C_{\alpha f} \delta \\ \end{aligned} $$

(46)

$$ \begin{aligned} M_{z} & = \left( { - \frac{{a^{2}_{2} }}{{v_{x} }}C_{\alpha r} - \frac{{a^{2}_{1} }}{{v_{x} }}C_{\alpha f} } \right)r \\ & \quad + \left( {\frac{{a_{1} C_{\alpha f} C_{\beta f} }}{{v_{x} }} - \frac{{a_{2} C_{\alpha r} C_{\beta r} }}{{v_{x} }}} \right)p \\ & \quad + \left( {a_{2} C_{\alpha r} - a_{1} C_{\alpha f} } \right)\beta \\ & \quad + \left( {a_{2} \left( { - C_{\alpha r} C_{\delta \varphi r} + C_{\varphi r} } \right) - a_{1} \left( { + C_{\varphi f} - C_{\alpha f} C_{\delta \varphi f} } \right)} \right)\varphi \\ & \quad + a_{1} C_{\alpha f\delta } \\ \end{aligned} $$

(47)

$$ \begin{aligned} M_{x} & = \left( {\frac{{a_{2} }}{{v_{x} }}C_{Tr} C_{\alpha r} - \frac{{a_{1} }}{{v_{x} }}C_{Tf} C_{\alpha f} } \right)r \\ & \quad + \left( {\frac{{C_{\alpha f} C_{\beta f} C_{Tf} }}{{v_{x} }} + \frac{{C_{Tr} C_{\alpha r} C_{\beta r} }}{{v_{x} }} - c_{\varphi } } \right)p \\ & \quad + \left( { - C_{Tr} \left( { - C_{\alpha r} C_{\delta \varphi r} + C_{\varphi r} } \right) - C_{Tf} \left( { + C_{\varphi f} - C_{\alpha f} C_{\delta \varphi f} } \right)} \right)\varphi \\ & \quad + C_{Tf} C_{\alpha f} \delta \\ \end{aligned} $$

(48)

The sum of the camber and slip moments generates the roll moment (M_x). This moment is dependent on the lateral force, as indicated in Fig. 6.

$$ \begin{aligned} M_{xf} & = C_{Tf} F_{yf} \\ M_{xr} & = C_{Tr} F_{yr} \\ \end{aligned} $$

(49)

The rear and front coefficients of tire torque are indicated as follows:

$$ \begin{aligned} C_{Tf} & = \frac{{dM_{xf} }}{{dF_{yf} }} \\ C_{Tr} & = \frac{{dM_{xr} }}{{dF_{yr} }} \\ \end{aligned} $$

(50)

The roll moments, which are generated by difference between the vertical forces exerted on the right and left wheels, are related to the roll angle of the car.

$$ \begin{aligned} M_{xk} & = - k_{\phi } \varphi \\ M_{xc} & = - c_{\phi } \dot{\varphi } \\ \end{aligned} $$

(51)

In these equations, c_αf and c_αr are sum of left and right wheels sideslip coefficients for front and rear wheels. C_Tf and C_Tr are front and rear overall torque coefficient. k_ϕ and c_ϕ are the roll stiffness and roll damping of the vehicle which are calculated as:

$$ \begin{aligned} k_{\phi } & = w_{k} = w(k_{f} + \, k_{r} ) \\ c_{\phi } & = w_{c} = w(c_{f} + \, c_{r} ) \\ \end{aligned} $$

(52)

w is the vehicle track; k_f, k_r and c_f, c_r are front and rear spring stiffness and damping coefficient, respectively.

$$ \begin{aligned} C_{r} & = \frac{{\partial F_{y} }}{\partial r} = \frac{{a_{2} }}{{v_{x} }}C_{\alpha r} - \frac{{a_{1} }}{{v_{x} }}C_{\alpha f} \\ C_{p} & = \frac{{C_{\alpha f} C_{\beta f} }}{{v_{x} }} + \frac{{C_{\alpha r} C_{\beta r} }}{{v_{x} }} \\ C_{\beta } & = \left( { - C_{\alpha f} - C_{\alpha r} } \right) \\ C_{\varphi } & = \frac{{\partial F_{y} }}{\partial \varphi } = C_{\alpha f} C_{\delta \varphi f} - C_{\varphi r} - C_{\varphi f} + C_{\alpha r} C_{\delta \varphi r} \\ C_{\delta } & = \frac{{\partial F_{y} }}{\partial \delta } = C_{\alpha f} \\ \end{aligned} $$

(53)

$$ \begin{aligned} E_{r} & = \frac{{\partial M_{x} }}{\partial r} = \frac{{a_{2} }}{{v_{x} }}C_{Tr} C_{\alpha r} - \frac{{a_{1} }}{{v_{x} }}C_{Tf} C_{\alpha f} \\ E_{p} & = \frac{{\partial M_{x} }}{\partial p} = \frac{{C_{\alpha f} C_{\beta f} C_{Tf} }}{{v_{x} }} + \frac{{C_{Tr} C_{\alpha r} C_{\beta r} }}{{v_{x} }} - c_{\varphi } \\ E_{\beta } & = \frac{{\partial M_{x} }}{\partial \beta } = - C_{Tr} C_{\alpha r} - C_{Tf} C_{\alpha f} \\ E_{\varphi } & = \frac{{\partial M_{x} }}{\partial \varphi } = - C_{Tr} \left( { - C_{\alpha r} C_{\delta \varphi r} + C_{\varphi r} } \right) \\ & \quad - C_{Tf} \left( { + C_{\varphi f} - C_{\alpha f} C_{\delta \varphi f} } \right) - k_{\varphi } \\ E_{\delta } & = \frac{{\partial M_{x} }}{\partial \delta } = C_{Tf} C_{\alpha f} \\ \end{aligned} $$

(54)

$$ \begin{aligned} D_{r} & = \frac{{\partial M_{z} }}{\partial r} = - \frac{{a^{2}_{2} }}{{v_{x} }}C_{\alpha r} - \frac{{a^{2}_{1} }}{{v_{x} }}C_{\alpha f} \\ D_{p} & = \frac{{\partial M_{z} }}{\partial p} = \frac{{a_{1} C_{\alpha f} C_{\beta f} }}{{v_{x} }} - \frac{{a_{2} C_{\alpha r} C_{\beta r} }}{{v_{x} }} \\ D_{\beta } & = \frac{{\partial M_{z} }}{\partial \beta } = a_{2} C_{\alpha r} - a_{1} C_{\alpha f} \\ D_{\varphi } & = \frac{{\partial M_{z} }}{\partial \varphi } = a_{2} \left( { - C_{\alpha r} C_{\delta \varphi r} + C_{\varphi r} } \right) \\ & \quad - a_{1} \left( { + C_{\varphi f} - C_{\alpha f} C_{\delta \varphi f} } \right) \\ D_{\delta } & = \frac{{\partial M_{z} }}{\partial \delta } = a_{1} C_{\alpha f} \\ \end{aligned} $$

(55)