updated on June, 2020

## Introduction

The distribution ot the 2007 Honda CBR 600 RR engine is studied, with the particular modifications implemented by the Race UP Team, the Formula Student team of the University of Padua. The international regulation binds 3 fundamental aspects on the choice and development of the engine: 4 stroke, petrol or ethanol and a 20 mm diameter restrictor at the start of the intake system; however, the study and design of the various external and internal components of the engine are left free. The students then deal with professional engineers to present and motivate their choices in the Engineering Design test. The focus of this report is on the changes to the intake cam profile (according to the “Polidyne” procedure described in the book “Motori a quattro tempi” by Giuseppe Bocchi) and the reduction of the limiter to 13500 rpm due to the restrictor.

## Objectives

The system will be simplified and stylized, to focus more on kinematic and dynamic analysis; in particular, the aspects that will be analyzed are:

- Check the real lift law imposed by the cam on the valve and check there is no flickering (the lifter must continue to follow the imposed law) at the limiter (13500 rpm)
- Check that the speed of the valve is consistent with the data obtained from Polidyne’s Matlab script
- Get the required torque from the camshaft to move the valve group
- Subsequently, you would like to modify the coaxial springs as they are suitable for the original engine that rotates up to 17000 rpm and see if you can decrease the torque required by the camshaft

## The modelling problem

THE REAL MODEL

The system you want to model is made up of:

- Intake valve group (valve, valve lifter, retainer, cotters, 2 coaxial springs)
- Camshaft: the profile has been modified to adapt the engine to the tortuous Formula Student circuits
- Engine case: the guides for the valves and the seats where they rest are fixed; the camshaft is fixed

In the engine, the intake side distribution is made with 8 valves (2 per cylinder); on each valve at the end of the stem there is a slot for housing the cotters which in turn seal on the retainer. The two coaxial springs (one longer and more rigid outside and one shorter and less rigid inside) act between the engine casing and the plate, and allow the valve to return to the closed position after the opening ramp imposed by the cam. By means of a spacer (useful for adjusting the play) and the lifter, the cam communicates the law of motion to the valve.

The original steel valve was imported into the Adams model; cotters, retainer and valve lifter have been integrated into the valve geometry. The valve guide, where the spring acts, and the seat, where the valve rests during the closing phase, have been reproduced. The cam was designed as an interpolation with a spline of a series of points which defines the design of the cam.

The cam is moved by a motor to the limiter on the “revolutional joint” between eccentric and “ground”. Guide and seat have been assembled to the “ground”, the valve slides with a “translational joint” along the guide; on the lifter a plane has been created to allow the contact between “plane and curve” with high stiffness and dynamic friction coefficient equal to 0.1 as recommended by Bocchi. Finally the spring acts between the guide and the plate, as in the real engine; the progressive stiffness of the spring has been inserted into the model in function of load and deformation, obtained from a compression test with press and balance.

SPRINGS

First, the minimum stiffness on detachment was verified, imposing the maximum acceleration that the mobile crew undergoes in relation to the movement given to the system:

The characterization of the springs was carried out using a press – balance system to check the stiffness before the individual springs (external and internal) and then the two springs in parallel, as they are mounted in the engine. A comparison is then made with the stiffness calculated by the geometry and the material of which the springs are made, according to the formula:

The experimental data and the graphs obtained from the tests with the single springs are shown below:

The formula provides the following values; the vibration frequencies of the springs were also calculated to compare them with the engine frequency:

The parallel spring test provided the following data and graphs:

It should be noted that the results obtained are all almost compatible with each other; in particular, it can be observed that from the springs placed in parallel a progressivity is created after about 5 mm of movement. This means that the first part of the stroke is controlled by the external spring and both springs are engaged after 5 mm to obtain a higher stiffness.

SPRING – MASS MODEL

A useful model for verifying the natural frequency of the springs was created; in particular, the comparison between the formula suggested by Bocchi and the mass-spring model in Adams was performed for the external spring. In the Adams model the mass of the spring was divided into 4 equal masses connected together by springs with stiffness in series. The formula instead works with the geometry and material of the spring. Here are the results obtained:

Adams derives a first natural frequency of 561 Hz, which corresponds to about 34000 rpm, a value almost in line with the Bocchi’s formula.

To verify the vibration modes of the distribution, at mass-spring model was added the mass of the valve group, which effectively reflects the real system. In particular, the model is made up of a main mass (valve + retainer + lifter) and 4 smaller masses which represent the springs; the five elements are connected together by 6 springs in series, the stiffness of which has been previously calculated.

To verify the correctness of the model, a Matlab script from the same system was also created; this script works with mass and stiffness matrices, calculates the dynamic matrix as:

and finally look for the values of ω such that the determinant of the dynamic matrix is zero.

% Valutazioni frequenze naturali molle asp con modello a 6 molle

mv=47.7/1000; %massa gruppo valvola di asp [kg]

m1=11.65/1000; m2=11.65/1000; m3=11.65/1000; m4=11.65/1000; %massa molle [kg]

k0=311600; k1=311600; k2=311600; k3=311600; k4=311600; k5=225600; %rigidezza molle in serie [N/m]

syms w

% matrice di massa M

M=[mv 0 0 0 0 ; 0 m1 0 0 0 ; 0 0 m2 0 0 ; 0 0 0 m3 0 ; 0 0 0 0 m4 ];

% matrice di rigidezza K

K=[k0+k1 -k1 0 0 0; -k1 k1+k2 -k2 0 0; 0 -k2 k2+k3 -k3 0; 0 0 -k3 k3+k4 -k4; 0 0 0 -k4 k4+k5];

% matrice dinamica D

MW = w^2.*M;

D = K-MW;

DD = sym2poly(det(D));

w=(roots(DD)); %omega naturale in rad/s

f=w.*(30/3.14) %frequenza naturale in cicli/1′

keq=((1/k0)+(1/k1)+(1/k2)+(1/k3)+(1/k4)+(1/k5))^(-1);

%Importare matrici ABCD da Adams per calcolo autovalori

A = importdata(‘state_matrixa1′);

autovalori = eig(A)

## Simulations and analysis of results

POSITION AND VELOCITY

First of all we want to verify that the law imposed by the cam on the valve is actually the one obtained by Matlab with the “Polidyne” procedure; then the position of the center of mass of the valve is measured. With a simple difference, it can be seen that the valve actually rises 7.7 mm, as required by the profile of the cam.

From Adams model:

From the model “Polidyne” in Matlab:

Then the valve speeds given by the two calculation systems are compared; they are compatible:

VIBRATION MODES

After the mass-spring model has been built, it is linearized in Adams by simply using the “Linear Modes” command in the simulation window. A “script” simulation has also been created that allows you to extrapolate the ABCD matrices; subsequently the Matlab is able to import the matrix A and calculate the eigenvalues that are the vibration modes of the system. Another script in Adams does export the MKB matrices, useful for checking the correctness of the models.

Now let’s look at the results obtained by Adams:

The first vibration mode is at a frequency of 349 Hz, which corresponds to 20960 rpm; therefore it can be said that the system is verified since the speed at the limiter of the original engine is 17000 rpm.

From the Matlab model, a first natural frequency of 20900 rpm is obtained, a value perfectly in line with the previous value.

TORQUE AND POWER

As noted, the power spent by the camshaft to open the valve and then compress the spring is then returned when the valve is closing because the spring extends; this occurs without the friction between the cam and the lifter. So now we want to analyze the relative power of the camshaft with and without the dynamic friction coefficient 0.1 suggested by Bocchi.

In the first simulation with the Real Model and with the original progressive spring, a script is created that deactivates the friction between the cam and the lifter; the following graph is obtained:

Note that on the right side of the graph (without friction) the required power peak is lower and all energy is recovered; in the left part of the graph (with friction) from the integration curve it can be seen that the peak of power required in the opening phase is greater and in the closing phase about 17% of the inserted power is lost.

DEVELOPMENT

Since in the engine modified by Race UP the limiter has been lowered to 13500 rpm due to the sonic block created by the restrictor, one could think of also lowering the first natural frequency of the vibration system, in order to recover some energy spent by the camshaft.

A modification is proposed which consists in the variation of the diameter of the wire with which the external spring and the internal spring are produced; the diameter is reduced by 2 and 1 tenths of a millimeter respectively, maintaining the same geometry as the original springs. The stiffness is calculated with the formula seen above:

The values of the new stiffnesses have been reported in the Matlab model and in the mass-spring model in Adams; the following results are obtained:

Now in the real model the progressive spring proposed as development is inserted and the same simulation is repeated:

It can be seen that the peak power required during the opening phase is lower than the previous graph and from the integral it can be seen that approximately 15.8% of the inserted power is now lost.

The maximum force acting on the springs is about 375 N and they are usually produced with tempered carbon harmonic steel, whose characteristics are:

From the ultimate tensile strength is possible to obtain the maximum allowable stress and shear stress for this material:

One can define these parameters from the geometry of the spring:

where c is the spring index, s is the spring pitch and l is the total length of the spring.

The stress state of the spring can be evaluated using the following calculation:

where is the shear stress due to the twisting moment, is the shear stress due to the shear force and is a coefficient that consider the effect of the shear stress:

So the expression of becomes:

and the springs are verified.

## Conclusion

This project made it possible to verify the movement and speed of the valve with the new lifting law given by the “Polidyne” profile. Subsequently, the mass – spring model allows to calculate the vibrating modes of the distribution system and to compare the frequencies with those of the engine. The spring-mass model also compare the frequency of the only spring with the theorical formula given by Bocchi.

A new configuration of the coaxial springs was then proposed with which the vibration frequency of the distribution can be decreased since the limiter was brought from 17000 to 13500 rpm due to the restrictor. An improvement of 1.2% of energy lost may seem little, but considering that the engine is a 4 cylinder 180 ° out of phase and that the 8 intake valves to the limiter open two by two 6750 times per minute, in a long run it could be a good improvement for engine efficiency.

## References

[1] Giuseppe Bocchi, “Motori a quattro tempi”, Hoepli

[2] RoyMech, “Helical spring surge / Natural Frequency”

[3] Bernd Heissing – Metin Ersoy, “Chassis Handbook”, ATZ

[4] www.multibody.net