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This part of the content is the third section of Chapter 5 of Automobile Theory. I have done some sorting and summary.
1. Two degrees of freedom
The two degrees of freedom initially refer to the two degrees of freedom: lateral and yaw.
The figure below shows a vehicle coordinate system with six degrees of freedom:
- Movement along the x-axis, forward movement
- Movement along the y-axis, sideways movement
- Movement along the z-axis, vertical movement
- Rotation around the x-axis, tilting motion
- Rotation around the y-axis, pitching motion
- Rotation around the z-axis, yaw movement
So, how do you limit the car's degrees of freedom to two?
Automobile theory makes the following assumptions:
- Ignore the effect of the suspension.
- The car body cannot rely on shock absorbers and elastic elements to achieve movement along the z-axis and cannot vibrate up and down.
- There is no so-called independent suspension and non-independent suspension, and it cannot rock left and right, that is, roll movement around the x-axis .
- The pitching motion around the y-axis cannot be completed without the elastic element .
- The car's forward speed remains unchanged.
- There is also no need to consider motion along the x-axis . Because in the future, automobile theory will use kinematics and dynamics to establish simultaneous equations (the content of theoretical mechanics), and the constant velocity along the x-axis means that the acceleration in the x-axis direction is 0, and there is no need to participate in the simultaneous equations. .
The above two assumptions define four degrees of freedom, and the remaining ones are the lateral motion along the y-axis and the yaw motion around the z-axis , which are the two degrees of freedom of the car.
2. Two-wheeled car model
The picture below is a classic simplified two-wheeled car model. The center of mass is O, the one on the left is the rear wheel, and the "wheelbase" from the center of mass is b; the right is the front wheel, and the "wheelbase" from the center of mass is a. The car has to turn left.
So, why can it be simplified to the following model? The main assumptions are three
- Ignoring the role of the suspension, the car body can be regarded as only moving in a plane parallel to the ground.
- The lateral acceleration of the car is ay≤0.4gay≤0.4ga_y≤0.4g, and the tire side deflection characteristics are within the linear range. This article explains that the yaw stiffness of the left and right wheels of the front (or rear) wheel is equal. The left and right wheels can be flattened and regarded as one wheel , and the yaw stiffness is twice that of the original wheel. (The role of the suspension is ignored here, so the vertical load on the left and right wheels is equal, and the vertical load has a certain impact on the cornering stiffness)
- The effects of changes in ground tangential force FXFXF_X, camber lateral force FYγFYγF_{Yγ}, righting moment TZTZT_Z, and vertical load on tire cornering stiffness are not taken into account.
3. Kinematic analysis
The three blue lines in the figure are the vehicle coordinate system, and the entire plane is the geodetic coordinate system. The two vehicle coordinate systems on the lower right are at time t and t+Δt. The car turns left and the center of mass moves to the left. The
vehicle coordinate system in the upper left corner is special and is used for analysis. The x and y coordinate axes of the dotted line are at time t, and the speed of the blue line is at time t+Δt. From time t to time t+Δt, the velocity component along the y-axis of the coordinate system changes as
(v+Δv)cosΔθ−v+(u+Δu)sinΔθ(v+Δv)cosΔθ−v+(u+Δu)sinΔθ(v+Δv)cosΔ\theta-v+(u+Δu)sinΔ\theta due to ΔθΔθΔ\
theta很小，so there is
cosΔθ≈1, sinΔθ≈Δθ≈0cosΔθ≈1, sinΔθ≈Δθ≈0cosΔ\theta\approx1,
sinΔ\theta\approxΔ\theta\approx0If
the second-order trace is ignored, the change in the velocity component along the x-axis of the coordinate system can be reduced Jianwei
Δv+uΔθΔv+uΔθΔv+uΔ\theta
Divide the above formula by Δt and take the limit, which is the component of the absolute acceleration of the car's center of mass on the Oy axis of the vehicle coordinate system
ay=dvdt+udθdt=v⋅+uwray=dvdt+udθdt=v·+uwra_y=\frac{dv}{dt}+u\frac{d\theta}{dt}=\overset{·}{v}+ uw_rwhere
wrwrw_r is the yaw angular velocity.
4. Kinetic analysis
The picture below is a top view of the two-degree-of-freedom car model.
Here are some explanations of the model:
- δδ\delta is the front wheel angle (caused by steering wheel input)
- α1α1\alpha_1 is the side slip angle of the front wheel, α2α2\alpha_2 is the side slip angle of the rear wheel
- ξξ\xi是航向角，ξ=δ−ax=δ−α\xi=\delta-\alpha
- u1u1u_1 is the front wheel speed, u2u2u_2 is the rear wheel speed
- FY1FY1F_{Y1} and FY2FY2F_{Y2} are the side deflection forces of the front and rear wheels, which are perpendicular to their respective wheel planes.
- Point O' is the instantaneous center of the two wheels at this time, and is the intersection point of the vertical lines u1u1u_1 and u2u2u_2.
- v1v1v_1 is the absolute speed of the car, and the direction is the vertical direction determined by the oo' connection.
- Side slip angle of the center of mass β=v/uβ=v/u\beta=v/u, vvv is the velocity component of the center of mass along the y-axis, uuu is the velocity component of the center of mass along the x-axis
The sum of the external force on the car along the y-axis direction and the moment about the center of mass is:
{∑FY=FY1cosδ+FY2∑MZ=αFY1cosδ−bFY2{∑FY=FY1cosδ+FY2∑MZ=αFY1cosδ−bFY2\begin{cases} \
sum F_Y = F_{Y1}cos\delta + F_{Y2}\ \
sum M_Z = \alpha F_{Y1}cos\delta - bF_{Y2}
\end{cases}
Considering that δδ\delta is small, and there are FY1=k1α1FY1=k1α1F_{Y1}=k_1\alpha_1 and FY2=k2α2FY2=k2α2F_{ Y2}=k_2\alpha_2, so the above formula can be written as:
{∑FY=k1α1+k2α2∑MZ=αk1α1−bk2α2{∑FY=k1α1+k2α2∑MZ=αk1α1−bk2α2\begin{cases}
\sum F_Y = k_1\alpha_1+k_2\alpha_2\
\sum M_Z = \alpha k_1 \alpha_1- bk_2\alpha_2
\end{cases}
The heading angle can be approximated as the tangent of the front wheel speed. The v direction can be regarded as the velocity vector relative to the center of mass plus a rotating tangential velocity (the content of theoretical mechanics~). Expressed as follows:
ξ≈tanξ=v+awru=β+awruξ≈tanξ=v+awru=β+awru\xi \approx tan\xi = \frac{v+a w_r}{u}=\beta+\frac{a w_r}{ u}
Then the side slip angle of the front and rear wheels can be expressed:
⎧⎩⎨⎪⎪α1=−(δ−ξ)=β+avru−δα2=v−bwru=β−bwru{α1=−(δ−ξ)=β+avru−δα2=v−bwru=β−bwru \begin{cases}
\alpha_1=-(\delta-\xi)=\beta + \dfrac{a w_r}{u}-\delta\ \
alpha_2=\dfrac{v-bw_r}{u}=\beta- \dfrac{bw_r}{u}
\end{cases}
From this, you can get the relationship between automobile external force, external force moment, and automobile motion parameters:
⎧⎩⎨⎪⎪⎪⎪∑FY=k1(β+monkey−δ)+k2(β−dog)∑MZ=αk1(β+dog−δ)−bk2(β−dog){∑FY=k1(β +month−δ)+k2(β−year)∑MZ=αk1(β+year−δ)−bk2(β−year)\begin{cases} \sum F_Y = k_1(\beta
+ \dfrac{a w_r} {u}-\delta)+k_2(\beta-\dfraction{bw_r}{u})\\
sum M_Z = \alpha k_1(\beta + \dfraction w_r}{u}-\delta)- bk_2( \beta-\dfrac{bw_r}{u})
\end{cases}
5. Differential equations of motion
The simultaneous kinematics and dynamics equations are:
⎧⎩⎨⎪⎪⎪⎪k1(β+cow−δ)+k2(β−cow)=m(v⋅+cow)αk1(β+cow−δ)−bk2(β−cow)=IZwr⋅{k1 (β+bw−δ)+k2(β−bwr)=m(v·+bwr)αk1(β+bw−δ)−bk2(β−bwr)=IZwr·\begin{cases} k_1(\beta
+ \dfraction w_r}{u}-\delta)+k_2(\beta-\dfraction{bw_r}{u})=m(\translated{·}{v}+uw_r)\ \alpha k_1(\beta
+ \dfrac{a w_r}{u}-\delta)- bk_2(\beta-\dfrac{bw_r}{u})=I_Z\translated{·}{w_r} \end{cases}in the IZIZI_Z range
of
values Default settings, wr⋅wr·\overset{·}{w_r}have a slightly different background.
Organize and obtain the two-degree-of-freedom automobile motion differential equation:
⎧⎩⎨⎪⎪⎪⎪(k1+k2)β+1u(ak1−bk2)wr−k1δm(v⋅+uwr)(ak1−bk2)β+1u(a2k1+b2k2)wr−ak1δ=IZwr⋅{(k1+k2)β+1u(ak1−bk2)wr−k1δm(v·+uwr)(ak1−bk2)β+1u(a2k1+b2k2)wr−ak1δ=IZwr·\begin{cases} (k_1+k_2)
\beta+\dfrac{1}{u}(ak_1-bk_2 )w_r-k_1\delta==m(\overset{·}{v}+uw_r)\ (
ak_1-bk_2)\beta+\dfrac{1}{u}(a 2k_1+b 2k_2)w_r-ak_1\delta= I_Z\overset{·}{w_r}
\end{cases}
This simultaneous equation includes parameters of vehicle mass and tire cornering stiffness, which can reflect the basic characteristics of the car's motion curve.