Details of steering characteristics: Calculate the slip angles of the front and rear tires.

In §11 and 12, we set the simplest possible motion situation and gave a lecture on the outline so that you can understand the principle of turning motion.However, in actual driving while accelerating and turning, weight transfer occurs between the inner and outer wheels and the front and rear wheels, resulting in (slightly) changes in tire characteristics.On the other hand, the mass of the car and the position of the center of gravity do not change as characteristics unique to the object, so the centrifugal force acting on the front and rear wheels and the ratio of CF (cornering force) that balances it do not change.In this way, the steering characteristics, which are one of the main themes of dynamic sensitivity related to turning motion, are related to what changes depending on the situation and what does not change.Therefore, a little more complicated work is required to verify the SA (slip angle) to obtain the required CF and to determine the resulting steering characteristics.
Therefore, in this chapter, we will also review the turning mechanism that we have lectured so far, calculate the acceleration acting on the car and the corresponding SA, and confirm the effect on the steering characteristics.Of course, in the strict sense, it's an about number that ignores detailed design elements, and esoteric formulas are simplified.
It may be knowledge that ordinary people do not need to drive an ordinary car normally, but to concretely imagine the effect of steering and accelerator operation on the behavior of the car is aimed at tuning. It should be useful for you.If the sensibility to enjoy driving is combined with the intelligence to understand the movement of the car, that is what makes a demon a gold rod.I think it will create a closer sense of human-horse unity.Troublesome reasoning and calculations will continue this time as well, but please do your best and stay with us until the end.

■ Basic knowledge of circular motion
In order to confirm the effect of acceleration acting on a turning vehicle on steering characteristics (change in SA of front and rear wheels), it is necessary to calculate the centrifugal force applied to the vehicle body, the amount of load transfer, and the CF required for the tires of the front and rear wheels. I have.First of all, I will explain the basic knowledge for that.

First, the "angular velocity" and "radian method" required when calculating the centrifugal force.The speed of linear motion is "distance / time", while the angular velocity is the angle of rotation per hour, that is, "angle / time".Generally, the angle of rotation uses the "degree method" to express one rotation in 1 °, but in the technical field, the "radian method", which is convenient for various calculations, is used.Since it is a ratio, there is no unit, but the value is to be defined as rad (radian).Therefore, the angular velocity is "rad / sec".
To elaborate a little more, rad is the ratio of distance to radius on the circumference.It is a numerical value that the arc sandwiched by that angle is several times the radius.The length of the arc (circumference) of the entire circle (360 °) is 360πr, so dividing it by r gives 2 ° = 180πrad.Therefore, 57.3rad = XNUMX ° ÷ π is about XNUMX °.Using this relationship, the angular velocity can be obtained by dividing the distance (= peripheral velocity: m / sec) that travels on the arc in one second by the radius of gyration (m).

・ Angular velocity (rad / sec) = peripheral velocity (m / sec) ÷ radius of gyration (m)　

The angular velocity is proportional to the moving speed on the circumference and inversely proportional to the radius.Conversely, velocity is the product of angular velocity and radius, which is proportional to each other.You can see that the technical calculation becomes easier.

Next is the "centrifugal acceleration" required to determine the centrifugal force.Originally, "centrifugal force" is a fictitious force, which is the reaction force of "centripetal force" generated toward the center of turning.Since the vectors are of the same size but in opposite directions, we will start by calculating the centripetal force.

I will explain the concept of "centric acceleration".
It is easy to think that acceleration is not applied to a car that makes a circular motion at a constant speed as shown in the figure on the right.However, in reality, the inertial force (blue arrow) that tries to move in a straight line is affected by the acceleration (yellow arrow) that constantly tries to change direction toward the center of turning.
As a result, the car will travel along the circumference in the direction of the green arrow.Think of the thread as pulling the weight when you connect the weight to the thread and turn it.

The magnitude of the acceleration (I will omit it because the theory in the middle is difficult) can be obtained by the following formula.

・ Centripetal acceleration (m / sec)²) = Turning radius (m) x angular velocity (rad / sec)²

Next is the calculation of "centripetal force".
The magnitude of the force is the same as the equation of motion on a straight line, which is mass x acceleration (in this case, the above centripetal acceleration).

・ Centripetal force (N: Newton) = vehicle mass (kg) x centripetal acceleration (m / sec)²)

It is the CF of the tire that creates this centripetal force.Generally, it is expressed as "CF required to counter centrifugal force", but based on this calculation formula, the centrifugal force we feel is "reaction force of centripetal force by CF". You can see that.However, since it is not necessary to be particular about it, we will continue to treat the force in the opposite direction of this centripetal force as the centrifugal force, and proceed with the calculation with the CF required for the rotational movement as the force in the opposite direction equal to the centrifugal force. increase.

■ Setting driving conditions
As an example of concretely verifying the SA of front and rear tires that affect steering characteristics, let's set a daily driving scene.The figure below is a sample of a highway interchange that accelerates from the ETC gate and joins the main line.The actual interchange is composed of more complicated curves, but here, in order to explain the changes in centrifugal force and acceleration state as easily as possible, we made it a configuration in which four arcs with different turning radii are connected.The conditions such as radius and vehicle speed of each section are shown in the table below, and the vehicle specifications required for calculation are also listed.This car has the same weight distribution on the front and rear wheels, so it is basically NS (neutral steer) characteristics.
As I mentioned at the beginning, in reality, calculations incorporating more elements and corresponding engineering ingenuity are involved, but if all of them are covered, the story becomes too complicated and indigestion occurs. Since there is a danger, take the subsequent calculations as a simplified simulation.

Item	unit	location
		a	→	b	→	c	→	d	→	e	→	Main line
Radius of gyration	m	50		50		120		400		1000		Straight line
speed	km / h	20	Isokinetic	20	accelerate	40	accelerate	60	accelerate	100	Isokinetic	100
	m / sec	5.56		5.56		11.11		16.6		27.78		27.78
elapsed time	sec	4.00		2.00		2.50		6.50		3.10

 

■ Calculation of centrifugal force　
Calculate the centrifugal force using the above formula.Since we want to know the centrifugal force acting on the front and rear tires, we calculate the mass of the car separately for the front and rear axles.In this example, the front and rear weight distribution is set to 50:50, so each is 550 kg.In addition, in order to simplify the conditions, we will set each section to the same radius and keep the front-back acceleration within the section constant, and obtain the instantaneous numerical value when passing through each point.I will rewrite the calculation formula just in case.

・ Angular velocity (rad / sec) = peripheral velocity (m / sec) ÷ radius of gyration (m)
・ Centrifugal acceleration (m / sec)²) = Turning radius (r) x angular velocity (rad / sec)²
・ Centrifugal force (N: Newton) = car mass (kg) x centrifugal acceleration (m / sec)²)

It is the result calculated based on the driving conditions of each point.

Item	unit	location
		a	b	c	d	e
Angular velocity	rad / sec	0.11	0.11	0.09	0.04	0.03
Centrifugal acceleration	m / sec2	0.62	0.62	1.03	0.69	0.77
Front axis centrifugal force	N	339.51	339.51	565.84	381.94	424.38
Rear axle centrifugal force	N	339.51	339.51	565.84	381.94	424.38

Now that we know the centrifugal force applied to the front and rear axles, the next step is to calculate the loads on the four wheels. This is because, as lectured in §4, the CF characteristics of a tire change depending on the wheel load.

 

■ Calculation of wheel load.
The formula for calculating the amount of movement between the inner and outer rings is calculated from the relationship of centrifugal force acting on the center of gravity point x height of the center of gravity = load movement amount x tread.

・ Inner and outer ring load transfer amount = centrifugal force x center of gravity height ÷ tread

Since it is acceleration while stepping on the accelerator, a moment due to acceleration is also generated in the front-back direction, but the idea is the same, only the "tread" changes to "wheelbase".

・ Front and rear wheel load transfer amount = inertial force (reaction force of front and rear acceleration force) x center of gravity height ÷ wheelbase

 

As an example of specific calculation, let's calculate the amount of load transfer at point C.The front-rear acceleration is a number obtained by dividing the change in vehicle speed by the elapsed time.

・ Weight transfer amount in the left-right direction = 1.03 (centrifugal acceleration) x 1100 (mass) x 0.45 (center of gravity height) ÷ 1.49 (tread) ≒ 342N
・ Load transfer amount in the front-back direction = 2.22 (front-back acceleration) x 1100 (mass) x 0.45 (center of gravity height) ÷ 2.33 (wheelbase) ≒ 472N

Next, add or subtract from the stationary state to obtain the wheel load after weight transfer.However, since we are dealing with the relationship between the tires and the road surface, we must convert the mass (kg) of the stationary vehicle into the weight (N) and adjust the unit."Gravitational acceleration = 9.8m / sec" due to the attractive force of the earth²"I will multiply.The wheel load of each wheel is shown on the right.

・ 1 wheel load in stationary state = 10780N (vehicle weight) ÷ 4 wheels = 2695N
・ Left-right direction: 342N for both inner and outer wheels
　　　　　(171N per wheel) Move.
・ Front-rear direction: Front and rear wheels → 472N on both rear wheels
　　　　　(236N per wheel) Move.

 

■ CF calculation and SA reading.
Similarly, calculate the wheel load at each point and move on to the required CF calculation.The size of the CF that the same SA exerts on the tire changes depending on the wheel load.Originally, it is not possible to simply calculate how much CF each tire earns, but in the case of a minute SA region, it is considered that the wheel load and CF are proportional, and it is calculated according to the load ratio of each wheel. can.As an example of a concrete calculation, let's calculate the CF at point C as well.

Front wheel CF is calculated by multiplying the front axle centrifugal force by the load ratio between the front wheels.
・ CF of front right tire = 565.84 (centrifugal force on front axle) x 0.535 (load ratio) ≒ 302.72N
・ CF of front left tire = 565.84 (centrifugal force on front axle) x 0.465 (load ratio) ≒ 263.11N

The rear wheel CF is calculated by multiplying the rear axle centrifugal force by the load ratio between the rear wheels.
・ CF of rear right tire = 565.84 (rear axle centrifugal force) x 0.531 (load ratio) ≒ 300.46N
・ CF of rear left tire = 565.84 (centrifugal force on rear axle) x 0.469 (load ratio) ≒ 265.37N

Below, the CF of each tire at each point is calculated in the same way.

タ イ ヤ		location
		a	b	c	d	e
front wheel	right	176.21	177.01	302.72	199.74	222.28
	left	163.29	162.50	263.11	182.21	202.10
Rear wheel	right	176.21	175.57	300.46	198.63	222.28
	left	163.29	163.93	265.37	183.31	202.10

 

Well, it's finally the end.
The figure on the right is a sample for explanation, but using point C as an example, the SA that generates each CF is read from the performance curve according to the wheel load (number in the upper margin of the graph) of each tire.
Since it is a very analog method, I would like to avoid strict numerical evaluation, but the SA of the front and rear wheels is about 0.3 °, and the difference between the front and rear wheels is within 0.05 °.In other words, it is a proof that the SA of the front wheels has a slightly large and weak under-steering characteristic.

Read the SA at each point in the same way.The left and right wheels are slightly different, but the table below shows the average values.

 

タ イ ヤ	location
	a	b	c	d	e
front wheel	0.17	0.17	0.31	0.19	0.21
Rear wheel	0.17	0.16	0.27	0.18	0.21
SA difference between front and rear wheels (Rear wheel SA-Front wheel SA)	0	-0.01	-0.04	-0.01	0

At points a and e, "SA for front wheels = SA for rear wheels", which shows that the NS characteristics are perfect.On the other hand, when acceleration is applied to the front of the vehicle body as in points b to d, "SA for front wheels> SA for rear wheels", and a slight US (understeer) characteristic can be seen.

 

■ Impact of weight distribution
For comparison, let's calculate the same turning motion for cars with different front and rear weight distributions.
Only the weight distribution is different from the NS characteristic car, and the other specifications and the speed and acceleration of each section are the same.

・ Turning in a car with a weight distribution of "front axle: rear axle = 60:40"　
Calculate by changing the mass / weight values ​​to front axle: rear axle = 60:40, and read the corresponding SA to see the table below.

タ イ ヤ	location
	a	b	c	d	e
front wheel	0.18	0.19	0.33	0.22	0.23
Rear wheel	0.15	0.14	0.24	0.16	0.20
SA difference between front and rear wheels (Rear wheel SA-Front wheel SA)	-0.03	-0.05	-0.09	-0.06	-0.03

"SA for front wheels> SA for rear wheels" for all sections.This is a feature of US characteristic cars.

・ Weight distribution "Front axle: Rear axle = 40:60 turning in a car
This time, change the values ​​of mass and weight to front axis: rear axis = 40:60 and calculate, and read the corresponding SA, the table below will be obtained.

タ イ ヤ	location
	a	b	c	d	e
front wheel	0.15	0.17	0.29	0.18	0.19
Rear wheel	0.18	0.18	0.30	0.20	0.23
SA difference between front and rear wheels (Rear wheel SA-Front wheel SA)	0.03	0.01	0.01	0.02	0.04

All sections are "SA for front wheels <SA for rear wheels".This is the characteristic of OS (oversteer) characteristic vehicles.Let us summarize the conclusions drawn through the above considerations.

The figure on the right compares the SA at point C of three vehicles with different weight distributions (resulting in different steering characteristics) in relation to the CF characteristics of the tires. It can be seen that the SA of the front and rear wheels is distributed in a distant area in the US / OS characteristic vehicle compared to the NS characteristic vehicle.The difference in SA between the front and rear wheels is a "clear indicator of the difference in steering characteristics."

・ SA of front wheels ＞ SA of rear wheels …… US characteristics
・ SA of front wheels = SA of rear wheels …… NS characteristics
・ SA of front wheels <SA of rear wheels …… OS characteristics　

 

 

■ Illustration of calculation results
This is the end of the principle calculation.The steering characteristics can be confirmed by the SA difference between the front and rear wheels when turning, and I think you understand in the calculation process that the basic factor that determines the characteristics is the load distribution of the front and rear wheels.

However, for those who can't get the point just by theory and calculation, I will visualize the result of the above simulation.The left side of the vertical axis is the SA difference between the front and rear wheels (rear wheel SA-SA front wheel), the right side of the vertical axis is the acceleration, and the horizontal axis is the set point that passed.I think you can imagine each steering characteristic more clearly.
The SA on the front wheels acts in the direction of reducing the steering angle, and the SA on the rear wheels has the same effect as tilting the vehicle body inward to increase the steering angle. , If it is a complete NS state, it will be 0.In other words, in this case, depending on the driving conditions at each passing point, it is necessary for OS cars to return the steering angle of the tire by about 0.04 ° at the maximum, increase it to about 0.1 ° for US cars and 0.04 ° for NS cars. Please understand that you need to.

I will show you another graph.As many of you may have already noticed, in such daily driving, the SA of each wheel is less than 0.5 °, and the SA difference between the front and rear wheels is an extremely small angle of less than 0.1 °.You may be wondering if it makes sense to compare numbers in that range.

Therefore, the actual steering angle (angle between the front wheels and the vehicle body) and the angle between the front wheel tires and the vehicle body in each traveling direction (temporarily defined as the effective steering angle. Actual steering angle + rear wheel SA-front wheel SA). Let's compare.).Let's think of it as a ratio, not an absolute value.Since it is a correction factor that returns or increases the steering angle according to the situation, this graph seems to be closer to the driver's feeling.

・ Actual steering angle (deg) = calculated as a geometric steering angle where SA does not occur (sin θ = wheelbase ÷ turning radius)
・ Effective steering angle (deg) = actual steering angle + rear wheel SA-front wheel SA
・ Effective rudder angle correction factor (%) = (effective rudder angle-actual rudder angle) ÷ actual rudder angle

It can be seen that the tendency of the graph according to the steering characteristics does not change, but the steering angle correction factor can be up to 30%.You can see the "big meaning" of the small numbers after the decimal point mentioned above.

 

■ Relationship with dynamic sensibility
From the viewpoint of dynamic sensitivity, it means whether the steering characteristic can be turned by intuitive steering angle determination for the curvature of the curve, or whether it must be consciously corrected.If you need to make corrections by turning the steering wheel a lot or turning it back on the way, you will feel a sense of discomfort and anxiety every time you steer.Therefore, it is a well-established theory that the steering characteristics tend to be weakly under.Even if correction is necessary, the direction of steering does not change and the correction factor is small, so the driver can respond with peace of mind.In addition, at corners such as highways, consideration has been given to the design of the road so that the steering angle is gradually applied from the straight-ahead state, and after passing the apex, the steering is gently returned to join the main line naturally. It has been.It is called a relaxation curve or clothoid curve.From the perspective of enjoying driving more actively, I think that the basics are NS, and it is ideal to have a weak US depending on the driving conditions.

However, even if the theory is understood, the driver who actually drives the car cannot directly control such steering angle correction of less than 1 °.Therefore, an appropriate gear ratio is set between the tire steering angle and the steering angle. It expands the world below 1 ° to make it easier to handle.If the gear ratio is 24: 1, it means that 1 ° of the tire steering angle is 24 ° on the steering.If you show it on the clock face, it is expanded to an angle of about 4 minutes of the minute hand.With this, you can fully control both visually and as the movement of your arm.How much the gear ratio should be is an important theme in dynamic Kansei engineering.If you zoom in too much, you will not be able to operate agilely, and if you are too sensitive, you will feel nervous and unstable.Of course, the premise is the optimum design of the total vehicle (or system details) that can accurately control the direction of the tire even in the world of less than 0.1 °, as if the driver is an extension of the body, and driving. The development process of reflecting the sensitivity evaluation by experiment is indispensable.

So far, I have explained the detailed numbers in a row, but I hope that readers can confirm the impression of the actual drive with such data.It is actually the mechanism of the micro world that controls the steering characteristics.That is why the alignment of the suspension, the rigidity of the body, the rigidity of the steering system, and the air pressure of the tires are required to have the corresponding micro precision.Everyone who loves cars, enjoys driving, and aspires to tune for a greater sense of unity between humans and horses will not only hone their driving skills and sensibilities, but will also acquire basic knowledge of automobile engineering (that is this course). I would like to sincerely hope that you will keep in mind the maintenance that keeps your car in the best condition on a daily basis.