To enjoy the tuning of roll rigidity ... Thorough explanation of the basic mechanism.

With the release of the new Roadster, Mazda's Jinba-Ittai concept has been further refined.Not only economic efficiency and environmental performance, but also the performance to enjoy driving, which is the basis of automobiles, is being reviewed, and I am happy to have been involved in the development of sports cars for many years.
After all, the situation where you can fully enjoy the sense of unity between humans and horses is cornering.I think that the moment when you enter the corner comfortably and run through the line as you imagined is the true value.
On the contrary, it must be unpleasant, not to mention the dynamic sensibility, to loosen the accelerator by feeling the unstable behavior against the driver's will.Rolls (technically called rolling) are the main cause of such discomfort.
Rolls are easy for everyone to feel and are said to be the most influential exercise for human sensibilities.Motion sickness is often caused by rolls, and research results have shown that it is due to the difference between the tilt captured by the inner ear of the semicircular canal and the tilt captured by the eye.Even when tuning from mass-produced cars, there are many functional products that claim the roll reduction effect, probably because it is easy for humans to experience.
From this time, I will try to unravel the relationship between rolls and dynamic sensibilities.
Roll is a movement that combines various elements in terms of automobile engineering and kansei engineering.Lectures will proceed from multiple angles as usual, but in order to understand the whole picture, knowledge of the basic mechanism is first required.This time, I will explain the basics of roll movement, that is, the mechanism.As usual, some esoteric formulas will appear, but please do your best to get along.


■ What is a roll?What kind of exercise is it?
Both humans and vehicles try to balance when turning so that they are not pushed outward by centrifugal force.The Shinkansen has a cant on the track so that the resultant force of centrifugal force and gravity is at right angles to the track.

Therefore, passengers can move comfortably without feeling centrifugal force.Speed ​​skating athletes also lean inward to move their center of gravity, directing the resultant force of centrifugal force and gravity toward the edge of the skate and the point of contact of the ice.Both of them balance the resultant force by tilting, but you may have noticed that they are tilted in the opposite direction to the roll of the car.This so-called reverse roll attitude (although some new Mercedes-Benz models use the technology to make this possible), generally speaking, cannot be achieved in a car.

The reason is that the car is equipped with suspension to handle various road conditions and maintain a comfortable ride.When centrifugal force is applied to the car body during cornering, the suspension on the outer ring side sinks and the inner ring side stretches, balancing the centrifugal force so that it rotates around the X axis of the car.This rotational movement is the roll.

However, this explanation is not XNUMX% accurate.Let's consider it in a little more detail.Roll is the rotation of the car around the X axis (See §9), But in reality, centrifugal force is applied to push the center of gravity of the car, and the center of gravity revolves around a point called the roll center.As a result, when considering the horizon as a reference, the vehicle body tilts due to the revolution and rotates around the center of gravity.The process by which the revolution movement produces the rotation movement is the same as that at the extremely low speed of the turning movement.Think of it as a geometric result rather than a movement problem.
The roll angle (rotation angle) of the car body = the revolution angle of the center of gravity.So where is this revolution center (roll center)?The suspension moves geometrically with the cross member and suspension arm mounting points as fulcrums.Therefore, the position of the roll center depends on the suspension design as shown below, and when roll occurs, the position always moves as the suspension geometry changes.

 

Also, since the car body is a highly rigid solid, the roll center originally exists on the axis connecting the roll centers of the front and rear suspensions, but this lecture is intended to promote understanding of the basic mechanism. For the time being, let's talk about the geometric roll center of either the front or rear suspension alone.


■ Calculation of force to generate roll
It seems a little difficult to think about the force to rotate the car body, but there is a good way.The source of the force that generated the roll was centrifugal force.Since the center of gravity and the roll center are separated from each other, the centrifugal force is converted into a rotational force (roll moment) that tries to revolve the center of gravity around the roll center.

In this case, the rotation and revolution are basically the same movement just by changing the viewpoint, so "rotation moment = revolution moment".So, if it is the moment on the revolution side, the calculation can be done easily.

This revolution moment is the same as the "principle of the lever". If the distance between the center of gravity and the roll center is long, it will be large, and if it is short, it will be small.As an extreme example, if the positions of both are the same, the centrifugal force will not be a roll moment, so the car will not roll.
Let's take a closer look at the principle of roll moment generation.The figure on the right is a simplification of the relationship between the centrifugal force applied to the "roll center" and "center of gravity" of a simple suspension such as a swing arm.I think you can imagine that the "principle of the lever" works depending on the distance between the roll center and the center of gravity.The formula for calculating the roll moment is shown below.In past lectures, the unit of acceleration was "m / sec²", but when calculating rolls, G (9.8G = XNUMXm / sec²) is common, so according to that convention, this lecture uses G. I will continue to use it.

・ Roll moment = (center of gravity height-roll center height) x spring weight (N) x lateral acceleration (G)

What is important in this formula is the term (center of gravity height-roll center height).When the roll angle is reached, the roll center moves, which complicates the process, but in this case, the center of gravity and the roll center are on the vertical line because it is stationary.Therefore, the "distance between the center of gravity and the roll center" is simply "the height of the center of gravity-the height of the roll center".
Now, let's actually calculate the roll moment under the following conditions.

・ Spring weight (body weight): 9500N　
・ Lateral acceleration: 0.5G (... For example, lateral acceleration when turning a stationary circle with a radius of 50m at a constant speed of about 56.5km / h)
・ High center of gravity-Roll center height: 0.3m

Therefore, when each condition is substituted, it becomes as follows.

・ 0.3m × 9500N × 0.5G ＝ 1425N ・ m

Now you can calculate the roll moment at the moment when the roll starts.Since the above calculation is stationary (not rolled), the roll starts from here.

In other words, the suspension geometry changes from this state, and the "center of gravity height-roll center height" changes, and as a result, the roll moment also changes.This is because the direction of the centrifugal force acting in the horizontal direction and the force that rotates the center of gravity around the roll center are different by the roll angle.

In this case, the length that works effectively as a lever arm is the horizontal line passing through the center of gravity and the vertical distance to the roll center, but since the roll center itself also moves, this value is drawn while actually changing the roll angle. There is no other way to calculate it than to do it.

"Center of gravity height-roll center height" is not the distance connecting the center of gravity and the roll center, but the vertical distance between the rotation axis (center of gravity) and the revolution axis (roll center). We will call it "distance between vertical axes".In the case of the above figure, the distance between the vertical axes obtained from the drawing is 0.29m, so substitute it for the calculation.

・ 0.29m × 9500N × 0.5G ≒ 1377N ･ m (-48N ･ m)

It was found that the roll moment changes as the distance between the vertical axes changes.However, at the stage when the roll occurs in this way, the center of gravity is not directly above the roll center, so the weight of the car itself adds a roll moment according to the angle of deviation. The impact will be described later, but for the time being, you should have an overview of the force that causes rolls.

■ Calculation of force to suppress roll
Next is the calculation of the force that suppresses the roll (roll rigidity).
Since the car is in contact with the road surface, the revolution movement around the roll center is blocked by the road surface, and the reaction force causes the suspension to contract.
Therefore, if the spring constant of the suspension spring is increased, it can rotate only a little.In other words, it produces a moment in the opposite direction to the roll moment, and is obtained by multiplying the spring force of the suspension of the inner and outer wheels by the lever length equivalent to half of the tread.This moment is called "roll rigidity".To put the roll rigidity correctly, it means "the roll moment required to change the roll angle by one unit".If you increase the roll rigidity, you can reduce the roll.Now, let's look at the formula for calculating the roll rigidity.

・ Roll rigidity =Wheel position spring constant × (tread) ² ÷ 2

The force generated by the spring depends on the amount of deflection of the spring, and the amount of deflection is obtained as a string of circular motion with half the radius of the tread.Since the chord is an arc at a small angle, it can be calculated by multiplying the radius by the angle rad.And since it disappears from the calculation with the setting of 1 rad, the force generated by the spring is the spring constant x tread ÷ XNUMX.
In the calculation of the moment around the center of gravity, the force is further multiplied by half of the tread as the lever length, so the spring constant x (tread ÷ XNUMX) x (tread ÷ XNUMX). It becomes the power ÷ XNUMX.

Since this moment is generated on both the inner and outer wheels, the expansion and contraction are opposite, but the direction in which the force works is the same, so the total is doubled. The first formula, "Square ÷ XNUMX", is born. You can see that expanding the "tread" that works in the square is more effective in increasing the roll rigidity than increasing the "spring constant".
Now let's calculate the roll stiffness in a real car.The specifications of the car to be calculated are shown on the right.Basically, based on the Roadster (NC), the front and rear treads are the same and the suspension layout has been changed for easier calculation.

<Calculation formula>
・ Front wheel roll rigidity = (9506 + 11378) x 1.49² ÷ 2 ≒ 23183N ・ m / rad
・ Rear wheel roll rigidity = (13527 + 2689) × 1.49² ÷ 2 ≒ 18001N ・ m / rad

The unit is "Nm / rad". Since it is difficult to imagine with "rad", when converted to "deg", it is "1rad ≒ 57.3deg", so the force required per 1 ° roll angle is calculated as follows.

・ Front wheel: Force required per 23183 ° roll angle = 57.3 ÷ 404.5 ≒ XNUMXN ・ m / deg
・ Rear wheel: Force required per 18001 ° roll angle = 57.3 ÷ 314.1 ≒ XNUMXN ・ m / deg

This does not apply to this example of fixing the tread, but for the car as a whole, it is advantageous in terms of roll rigidity to have a wide "tread that works with squares" at the time of basic design, as in the recent ND. Also keep in mind that.


■ Calculation of roll angle during steady circular rotation
Now let's calculate the actual roll angle.The force that tries to roll the car body is the "roll moment", and the force that tries to suppress it is the "roll rigidity", so when it is stable, it is in a balanced relationship with each other.See the balance formula below.

It can be read from this formula that the roll angle is the balance point between the "roll rigidity component" and the "roll moment component", and each term is related to φ (roll angle).

Since the roll rigidity is the "roll moment that changes the roll angle by one unit", the roll rigidity component changes according to the roll angle.

On the other hand, the roll moment is caused by centrifugal force and by gravity mentioned in the previous section. The gravity effect related to φ is the roll moment when the center of gravity shifts by φ from directly above the roll center, and gravity tries to push down the center of gravity.
The moment in this case is the gravity of Ws toward the road surface, so it is multiplied by the arm (lateral deviation) perpendicular to it, but the length is ≒ hs ・ φ at a minute angle, so Ws ・ hs ・Obtained with φ. (The right is a conceptual diagram) If this formula is converted into an expression to find the roll angle, it will be as follows.

Now, let's actually substitute the numerical values ​​and calculate.The vehicle is the same as before, and the suspension arm layout is an outward opening type that opens from the inside (cross member side) to the outside (tire side) with upper and lower equal length arms.

Also, to simplify the calculation, it is assumed that the roll angle does not change during steady circular constant speed turning.In the case of this car, as you roll, the position of the roll center moves as shown in the figure on the right, and the center of gravity also moves.

The roll center moves diagonally downward to the right.
On the other hand, the height of the center of gravity decreases in the direction of rotation of the vehicle body, but the roll center decreases further, so the distance between the vertical axes becomes longer and the roll moment increases, which can be read from the drawing.Let's assume that the distance between the vertical axes at this time is 0.308m and calculate the lateral acceleration as 0.5G.


・ 0.5G × 9500N × 0.308m ÷ ((23183N ･ m / rad ＋ 18001N ･ m / rad)-9500N × 0.308m) × 57.3 ° ≒ 2.19 °

It was found that this car rolls about 0.5 ° at a lateral acceleration of 2.19G.By the way, the roll angle at 0.5G set here is called the "roll rate" of the car, and it is used when comparing the roll tendency of multiple cars easily.


■ What happens if the roll rigidity is changed or the roll is lowered?
Next, if you change the "roll rigidity", how will the "roll angle" change?
Assuming the previous condition as 100%, I simulated a change in roll rigidity from -50% to 200% and calculated the relationship with the roll angle.Think of the "roll angle in simple calculation" in the item as a rough number calculated assuming that the roll angle is simply inversely proportional to the roll rigidity, ignoring the effect of hs.It seems that this simple calculation is often used in general manuals. The calculation is φ = g (Ws ・ hs) ÷ (Kφf + Kφr). hs is a fixed value before rolling.

Rate of change in roll rigidity		50%	75%	100%	125%	150%	175%	200%
Distance between vertical axes	m	0.324	0.312	0.308	0.307	0.307	0.306	0.306
Roll angle	you	5.035	3.041	2.191	1.721	1.420	1.204	1.048
Simple calculation Roll angle	you	3.965	2.643	1.983	1.586	1.322	1.133	0.991
Condition: Lateral acceleration 0.5G

In the graph on the right, you can compare how the roll angle changes in the rigidity up area and the rigidity down area based on 100% roll rigidity.It was found that the actual roll angle approaches the curve of the roll angle in simple calculation as the rigidity increases, and conversely, the rigidity down range moves away.Although it is a slight difference, it can be read that the roll angle does not simply increase or decrease in inverse proportion to the roll rigidity.
Next, what happens if only lowdown is performed without changing the roll rigidity?A general low-down spring maintains a balance by increasing the spring constant according to the amount of down, but in order to confirm the effect of only low-down, each condition is the same as before, and the amount of down is from 0 to -40 mm. I made it step by step.

Amount of down	mm	0	-5	-10	-15	-20	-25	-30	-35	-40
垂直 Distance between axes	m	0.308	0.312	0.315	0.318	0.322	0.329	0.337	0.345	0.351
Roll angle	you	2.191	2.222	2.245	2.268	2.299	2.353	2.415	2.479	2.531
Condition: Lateral acceleration 0.5G

The lowdown lowers the mounting position of the suspension arm, which changes the arm angle.With this suspension, the position of the roll center moves more diagonally downward as the roll progresses, as shown in the figure on the right, compared to a standard ride height vehicle.In other words, the change in the distance between the vertical axes becomes large.
So, after setting the distance between each vertical axis based on the results of experiments etc., I calculated the roll angle according to the above formula.

The figure on the right is a graph of the relationship.As the amount of down increases, the distance between the vertical axes increases, the roll moment increases, and the roll angle also increases.Keep in mind that even if you lower the center of gravity, the roll angle tends to increase depending on the suspension type and layout.


■ Verification of the effect of realistic tuning
Next, low down (vehicle height -20mm) and spring constant increase are applied to this car at the same time as in normal tuning, and the effect is calculated.The items to be installed are a low-down spring with an increased spring constant and a stabilizer (front and rear).The basic calculation only reflects each setting value in the above formula, so we will compare only the results.

<Tuning details / Spring constant increase> * Both wheel positions

Item	Low down spring (Vehicle height -20mm)	stabilizer	Total
front wheel	13308N / m (140%)	19344N / m (170%)	32652N / m (156%)
Rear wheel	16232N / m (120%)	14115N / m (200%)	30347N / m (187%)
Total	29540N / m (128%)	33459N / m (237%)	62999N / m (169.8)

* The% in parentheses of the spring constant is the ratio before tuning.

<Comparison of roll rigidity and roll angle before and after tuning>

-	After tuning	Before tuning	Rate of change
Front wheel roll rigidity	36246 N ･ m / rad	23183 N ･ m / rad	156.3%
Rear wheel roll rigidity	33687 N ･ m / rad	18001 N ･ m / rad	187.1%
Total roll stiffness	69933 N ･ m / rad	41184 N ･ m / rad	169.8%
Distance between vertical axes	0.311m	0.308m	100.9%
Roll moment	1477N ・ m	1463N ・ m	100.9%
Roll angle	1.264deg	2.191deg	57.7%
Condition: Lateral acceleration: 0.5G

* The% in parentheses of the spring constant is the ratio before tuning.

From the calculation results, it was found that the roll angle decreased to 156.3% while the spring constant increased by 187.1% to 169.8% (roll rigidity increased by 57.7% in total).In this way, roll tuning is greatly influenced by the basic design of the base car, such as the layout of the suspension arm that determines the movement trajectory of the roll center, and cannot be easily changed later.In particular, increasing the spring constant of a spring does not reduce the roll angle as much as the increase rate, and there is a concern that it will have the adverse effect of deteriorating ride quality and trackability to the road surface.In order to enjoy tuning, you can understand how important it is to select the base car as the material through the above simulation.


■ Verification of tuning effect by changing G
Finally, graph the roll angle at that time to see what happens if the lateral acceleration under each of the above conditions is changed.In the case of only "roll rigidity up" simulated in the previous section, and the case of only "low down" are also described, the change of each roll is obvious.All of them look like graphs that are almost proportional to the lateral acceleration, but in reality, due to the effect of the movement of the roll center explained in the previous section, there is an increase of about several percent at the maximum roll angle.

For reference, the actual roadster (NC) is 0.5deg at 2.20G, so it has almost the same tendency as the standard car handled here.Also, in the case of the car set here, the adverse effect of the increase in hs due to lowdown is stronger due to the suspension layout.
To make it easier for you to get used to, we also graphed the roll angle, which is the conversion of G on the horizontal axis into velocity (when rotating in a steady circle with a radius of 2 m).Lateral acceleration is proportional to the square of velocity, so you can imagine it more intuitively.

The calculation of the roll angle is a little rough, but the value of about 0.5 ° at XNUMXG is within the normal level range that you can experience in an actual car.

Perhaps you might think that you are rolling a lot bigger, but it is physically like this, including the effect of increasing the roll rigidity.It goes without saying that the theme of this seminar as a whole is whether or not it feels like a slight tilt in numerical terms ... it is a matter of human dynamic sensibility, so it is important. There is no such thing.
As mentioned above, this lecture is also difficult for humanities people to understand, but it is said that mathematical formulas are "the most reliable language form that conveys the idea of ​​things".Regardless of the absolute value, I hope that you will understand the theory of "why it will happen" and that it will be useful for "intelligent tuning" that goes beyond the level of mere car lovers.

By the way, in the next lecture, I will dig deeper into the roll rigidity.
Roll rigidity can be rephrased as the amount of force that presses the tire against the road surface.When suspension springs with different spring constants are attached to the front and rear wheels, the one with the higher spring constant bears a larger load, ignoring detailed conditions.In other words, roll rigidity affects the cornering power characteristics (CP) of the tire and is closely related to the steering characteristics.This time, some of you may have wondered, but we plan to explain the benefits of lowering the center of gravity.Please look forward to it.