Consideration of Restoration in Yaw Resonance

Abstract

Yaw resonances cause the rear-end of the vehicle to swing, which is related to the feeling of handling. As a basis for improving this motion, this paper considers the restoration of yaw resonance. The equilibrium position of the yaw resonance is the extension of the velocity vector at the “heading point,” where the vehicle median line is perpendicular to the turning radius in a steady state turn. Toward this position, the center of percussion at the rear-end of the vehicle travels. This travel is the restoration of yaw. Observing the vehicle behavior from the earth-fixed coordinate system at the moment when the heading point changes direction of travel, the heading point and the center of percussion travel in their respective directions. Each motion continues for a distance from the rear wheel to the heading point to reach the equilibrium position. This continues time equals “yaw lead time constant.” Therefore, when the yaw lead time constant is small, the vehicle is restored in a short time.

1 Introduction

The swinging motion of the rear-end of the vehicle, as in the hand gesture that begins at the 223-second point in a video [1], interferes with the pleasantness of the handling. This paper proves that this swing is the yaw resonance, discusses its mechanism, and proposes a method to improve it.

2 Vehicle Model and Its Yaw Resonance

The vehicle planar motion, represented by the yaw velocity “r” and the sideslip angle at C.G. “$\beta$,” are represented by a linear model shown in Fig. 1. Its vehicle mass m is represented by portion masses divided into front and rear masses [2],

$$\begin{aligned} m_\textrm{f} = \frac{ l_\textrm{r} }{l} m, \ \ \ \ m_\textrm{r} = \frac{ l_\textrm{f} }{l} m \ \ \ \mathrm {(kg)} \end{aligned}$$

(1)

according to the vertical loads. The yaw moment of inertia $I_z$ is expressed as

$$\begin{aligned} I_\textrm{z} = m _\textrm{f} (k_\textrm{N} l_\textrm{f} )^2+ m _\textrm{r}(k_\textrm{N} l_\textrm{r} )^2 ={k_\textrm{N}}^2m l_\textrm{f} l_\textrm{r} \ \ \ \mathrm {(kgm}^2\mathrm {)} \end{aligned}$$

(2)

where ${k_\textrm{N}}^2$ is a coefficient called dynamic index [3].

Fig. 1.

A planar 2-DOF model for center of gravity

Newton’s second law equations of the model are described by

$$\begin{aligned} {k_\textrm{N}}^2m l_\textrm{f} l_\textrm{r}\dot{r} = & 2 F_\textrm{f} l_\textrm{f}-2 F_\textrm{r} l_\textrm{r} \ \ \ \mathrm {(Nm)} \end{aligned}$$

(3)

$$\begin{aligned} mV(r+ \dot{\beta })= & 2 F_\textrm{f}+2 F_\textrm{r} \ \ \ \mathrm {(N)} . \end{aligned}$$

(4)

The front and rear cornering forces $2 F_\textrm{f}$ and $2 F_\textrm{r}$ are respectively

$$\begin{aligned} 2 F_\textrm{f} = -C_\textrm{f} m_\textrm{f} \alpha _\textrm{f} , \ \ \ \ \ 2 F_\textrm{r} = -C_\textrm{r} m_\textrm{r} \alpha _\textrm{r} \ \ \ \ \ \ \ \mathrm {(N)} \end{aligned}$$

(5)

where $C_\textrm{f}$ and $C_\textrm{r}$ are the cornering coefficients in the acceleration dimension, respectively. Their typical values are $C_\textrm{f}=100$ and $C_\textrm{r}=200$ [(m/s$^2$)/rad], respectively. The front slip angle $\alpha _\textrm{f}$ and the rear slip angle $\alpha _\textrm{r}$ are written as

$$\begin{aligned} \alpha _\textrm{f} = & \beta _\textrm{f} -\delta ,\ \ \ \ \ \alpha _\textrm{r} =\beta _\textrm{r}\end{aligned}$$

(6)

$$\begin{aligned} \beta _\textrm{f} = & \beta +\frac{ l_\textrm{f} }{V}r \ \ \ \beta _\textrm{r} = \beta -\frac{ l_\textrm{r} }{V}r \ \ \ \mathrm {(rad)} . \end{aligned}$$

(7)

Algebraic manipulation of the above equations yields

$$\begin{aligned} \dot{r} = & - \frac{ C_\textrm{f} l_\textrm{f}+ C_\textrm{r} l_\textrm{r}}{{k_\textrm{N}}^2 l V} r -\frac{ C_\textrm{f}- C_\textrm{r}}{{k_\textrm{N}}^2 l}\beta +\frac{ C_\textrm{f}}{{k_\textrm{N}}^2 l} \delta \ \ \ \ \ \ \ \ \ \ \ \mathrm {(rad/s}^2) \end{aligned}$$

(8)

(9)

The yaw natural frequency “$\omega _\textrm{n}$” of the model is described by

$$\begin{aligned} \omega _\textrm{n} = \sqrt{ \frac{ C_\textrm{r}+\left( \dfrac{ C_\textrm{r} l }{V^2}-1 \right) C_\textrm{f} }{ {k_\textrm{N}}^2 l}} \ \ \mathrm {(rad/s)} . \end{aligned}$$

(10)

3 Motion Equations Representing the Resonance Explicitly

To understand resonances, that of a single pendulum is suitable. Its Newton’s second law shown in Fig. 2a is represented in Fig. 2b. This figure expresses the phenomenon that $\ddot{\theta }$ occurs in the opposite direction of $\theta$ in proportion to $\theta$ as shown in Fig. 2c. Therefore, the equilibrium position is at $\theta =0$. Thus transforming the equations of motion of the vehicle into a form similar to that in Fig. 2b can reveal its equilibrium position and the variable that accelerates toward it.

Fig. 2.

The pendulum

The final term of Eq. (8) is eliminated by replacing $\beta$ with $\beta _\textrm{rpc}$ shown in Fig. 1. Subtracting ${k_\textrm{N}}^2 l_\textrm{r}/V$ times Eq. (8) from Eq. (9) yields

$$\begin{aligned} \dfrac{\textrm{d}}{\textrm{d}t} \left( \beta -\dfrac{{k_\textrm{N}}^2 l_\textrm{r}}{V} r \right) = -\left( 1-\dfrac{ C_\textrm{r} l_\textrm{r} }{V^2} \right) r-\frac{ C_\textrm{r}}{V}\beta . \end{aligned}$$

(11)

The term in round brackets on the left-hand side means the sideslip angle “$\beta _\textrm{rpc}$” at rpc behind ${k_\textrm{N}}^2 l_\textrm{r}$ from the center of gravity. Hence

$$\begin{aligned} \beta _\textrm{rpc}=\beta -\frac{{k_\textrm{N}}^2 l_\textrm{r}}{V}r. \end{aligned}$$

(12)

Position rpc is the center of percussion where no acceleration occurs when a percussion force is applied to the front wheel position in the lateral direction of the vehicle [3]. By using Eq. (12), eliminating $\beta$ from Eqs. (8) and (9) obtains

$$\begin{aligned} \dot{r} = & - \left[ 1+ \frac{({k_\textrm{N}}^2-1)l_\textrm{r}}{l}\frac{ C_\textrm{f}- C_\textrm{r}}{ C_\textrm{f}} \right] \frac{C_\textrm{f}}{{k_\textrm{N}}^2\,V} r -\frac{ C_\textrm{r}- C_\textrm{f}}{{k_\textrm{N}}^2l}\beta _\textrm{rpc} +\frac{ C_\textrm{f}}{{k_\textrm{N}}^2l} \delta \end{aligned}$$

(13)

(14)

where $T_r$ is yaw lead time constant [4] expressed as

(15)

The time constant is the coefficient of Laplace variable s in the numerator of the transfer function relating the steer angle to the yaw rate.

The first term on the right-hand side of Eq. (14) is eliminated by replacing r with sideslip angle “$\beta _\textrm{H}$” at Heading Point “H.P.” where the radius is perpendicular to the vehicle median line in a steady state turn. H.P. is located near the rear wheels at very low speeds and moves forward with increasing vehicle speed. As typical examples, H.P. is at the midpoint of the wheelbase at 60 (km/h) and on the front axle at 80 (km/h). H.P. is represented as $l _\textrm{H}$ ahead of the rear wheels, as shown in Fig. 1. The length $l _\textrm{H}$ is expressed as

(16)

Using $\beta _\textrm{H}$ and $\beta _\textrm{rpc}$, yow velocity is written as

$$\begin{aligned} r=\frac{\beta _\textrm{H}-\beta _\textrm{rpc}}{ l_\textrm{H}+ ({k_\textrm{N}}^2-1)l_\textrm{r}}V \ \ \ \mathrm {(rad/s)}. \end{aligned}$$

(17)

Substituting this expression into the expressions (8) and (9) and performing the algebraic operations yields

(18)

(19)

where $\zeta$ is yaw damping ratio.

The first and last terms on the right-hand side of Eq. (18) are eliminated by assuming a steering where they cancel each other out. That is

(20)

Numerical simulation results are omitted, but under non-zero initial conditions, the steering proportional to $\beta _\textrm{H}$ causes the yaw resonance to continue. Under this steering, Eq. (19) is reduced as

$$\begin{aligned} \dot{\beta }_{\textrm{H}} = \omega _{\textrm{n}}^{2}T_r \beta _{\textrm{rpc}} .\end{aligned}$$

(21)

Comparing Eqs. (19) and (21) with Fig. 2b with attention to the negative sign, we obtain Fig. 3a.

Fig. 3.

Resonance and Restoration of Yaw: $\omega _\textrm{n}$ and $T_r$ denote the yaw natural frequency and yaw lead time constant, respectively.

4 Yaw Resonance and Its Restoration

Figure 3a means that the yaw resonance equilibrium position is at $\beta _\textrm{H}=0$ and $\dot{\beta }_\textrm{rpc}$ is restored toward this position, as shown in Fig. 3b.

Yaw resonance is a phenomenon in which the rear wheels accelerates toward the extension of H.P.’s vehicle speed vector. The meaning of Fig. 3 is represented by the behavior of the vehicle in Fig. 4. The equilibrium state of the yaw resonance is position B, where $\beta _\textrm{H}=0$ as in steady-state turning. At this time, the rear wheels position r is on the extension of H.P.’s vehicle speed vector $\overrightarrow{V_\textrm{H}}$. Hence, $\overrightarrow{V_\textrm{H}}$ is located at position r in the extension of $\overrightarrow{V_\textrm{H}}$ is a balanced state. Therefore, yaw resonance is a phenomenon in which rcp accelerates toward the extension of $\overrightarrow{V_\textrm{H}}$ like positions A and C.

Fig. 4.

Yaw resonance mode

This motion rotates the vehicle around H.P. It is shown in the H.P. path, which is closer to a straight line than the front or rear wheel paths in Fig. 5, which shows a restoring behavior after an unbalanced condition are given. For concise analysis, rpc is assumed to be on the rear wheel.

Intuitively, the restoration is the behavior in which rpc maintains its former motion to the vicinity of the ground position where H.P. was at the moment of imbalance, and then follows the trajectory of H.P. This phenomenon is seen in Fig. 5, where rpc moves almost straight ahead to the vicinity of the initial H.P. ground position, and then follows the trajectory of H.P.

The restoration mechanism is represented in schematic Fig. 6. The schematic shows the result of H.P. and rpc continuing to travel $l_\textrm{H}$(m) in each direction at the moment of disproportionation. The vehicle median line is on the vehicle speed vector at H.P., thus $\beta _\textrm{H}=0$, which means the equilibrium position, as shown at position B in Fig. 4.

The time required for the vehicle to travel $l_\textrm{H}$(m) is $T_r$(s). $T_r$ is located in the block $-1/T_r$ that represents the strength of restoration in Fig. 3a. This block corresponds to the $-g/L$ block, which represents the restoration of the pendulum, shown in Fig. 2b. Thus, a small $T_r$ or $l_\textrm{H}$ immediately restore the vehicle.

Fig. 5.

Vehicle trajectories (initial conditions: $\beta _\textrm{H}\ne 0$, $\beta _\textrm{rpc}=0$ and $\delta =0$)

Fig. 6.

Schematic of the restoration in yaw resonance implied in Fig. 5

5 Handling Improvement

To emphasize the restoration, yaw lead time constant and the length can be shortened by increasing the rear cornering coefficient from Eq. (16).

To suppress the manifestation of yaw resonance pointed out in the movie [1], it is necessary to increase the yaw damping ratio, which can be achieved by increasing the front cornering coefficient [5].

6 Conclusion

This research has revealed that the swaying phenomenon at the rear-end of the vehicle is a manifestation of yaw resonance. Restoration of the yaw resonance is the running of the rpc to the H.P. position at the moment of imbalance. This suggests that the vehicle planar motion can be improved by controls that focus on the heading point and rpc.