In our days in automotive world there is a new trend for engine policy: reduce the specific fuel consumption. According to this actual philosophy, each carmakers propose their systems that allow a variation of valves motion and consequently a variation of the engine working.
One of this system, designed by BMW, is called Valvetronic and it is actually used in many engine built by this carmaker. The Valvetronic system guarantees a continuous variation of valves lift and valves timing of the intake system. Thus, the Valvetronic engine no longer requires a throttle butterfly, which causes fluid dynamics losses at low speed. The obtained results are an increase of the efficiency and consequently a reduction of fuel consumption (up to 10% throughout the entire operating range) and exhaust emissions. However, the mechanism used is not so complex and represents an artful solution.
The objectives of the project are:
- Evaluation of the valve lift profile in different situations
- Preliminary evaluation of the spring characteristics
- Spring sizing
- Evaluation of the camshaft torque necessary to move the mechanism in the worst work condition and in the maximum acceleration condition.
Instead of the twin-cam engine, Valvetronic use an additional eccentric shaft (3) moved by an electric stepper motor (4) and a traditional camshaft (2) that has the half speed of the engine shaft. Moreover the cam shaft (2) is connected to the motor with another BMW system called Vanos. It permits the variation of the angular position of the maximum valve lift but it is not considered in this script. These two components move the lever or intermediate arm (5) that is the most important component of the system and for this reason, it is finished to a tolerance of 0,008 mm. The intermediate arm impress an oscillating movement to a rocker arm (6) that permits the valve (8) displacement. The hydraulic compensating element (7) helps to obtain the desired maximum and minimum valve lift and in this script it is considered fix.
The virtual implementation of Valvetronic system is made starting from non-official drawings, so without knowing the real dimensions of the mechanism. Hence, all dimensions are referred to the diameter of the valve disk.
The first element modeled was the ground, which supports all the system. The important features of this element are:
- the upper profile (circumference arc) that is the rail that guide the intermediate arm.
- the position of the right support of the traditional cam
- the position of the left support of the eccentric cam
- the position of the highest point of the hydraulic compensating element
- the position and the inclination of the valve guide.
The traditional cam is connected to the right support of the fixed ground. By its profile the cam define the valve lift, in particular the time and the crank angle of the maximum lift. Its angular velocity is variable from 450 to 3500 rpm.
The eccentric cam is connected to the left support of the fixed ground. It can rotate from the initial position (0 degree) to the last position (210 degree) in just 0,3 seconds.
As already said, the most important component of Valvetronic system is surely the intermediate arm. It is composed of a principal body and two rollers. The upper roller is dug in the middle part, so the external profile of the roller rolls on the fixed ground guide and the inner profile is always in contact with the eccentric cam. The bottom roller is in contact with the traditional cam. Both rollers are maintained in contact with the respective cams by a torsion spring, fixed to the ground and acting on the front end of the lever. The lower part of the lever is the guide of the rocker arm and its geometric definition required small tolerance both in virtual mechanism and in reality.
One of the main problems of this system is the difficulty to recover the real dimensions of every bodies. Particularly, the intermediate arm in reality is made with low tolerance and this permits to obtain the desired lift profile. Starting from the non-official draws it is possible to define an approximate lower profile that is the support of the spline. The joint that connect the intermediate arm with the roller of the rocker arm is called “Roll-Curve joint”. In LMS it was defined starting from two close splines:
- the roller spline that is the roller’s sketch
- the intermediate arm spline that is supported from the geometry of the body (it need to be close)
The device must vary the lift profile of the valve, but also ensure the closing of the valve during the remaining engine stages. Especially, when the traditional cam is maintained in the minimum position (valve closed), the valve must be tight for each position of the eccentric cam. To satisfy both the demand of a regular lift, compatible with minimum and maximum values declared by the manufacturer, and the necessity to maintain the valve closed, the following results was reached:
The rocker arm, moved by the lever, oscillates in the vertical plane. It is connected with the fixed ground at the right end and with the valve at the left side. It is composed of a principal body and a roller that rolls on the lower profile of the intermediate arm. The contact between the intermediate arm and the roller of the rocker arm is guaranteed by the valve spring.
The valve is moved by the rocker arm and its minimum lift is 0,3 mm, while the maximum lift is 9,5 mm.
First, it is important to specify that the mechanism can be assumed plan with good approximation. For this reason were define the following constraints in order to avoid redundancies:
- Cylindrical joint between fixed ground and eccentric cam
- Cylindrical joint between fixed ground and traditional cam
- Spherical joint between fixed ground and rocker arm
- Cylindrical joint between fixed ground and valve
- Planar joint between fixed ground and intermediate arm
- Point-Curve joint between fixed ground and the center of the upper roller of intermediate arm
Moreover, two rollers are constrained on the intermediate arm with two cylindrical joints. The interactions between the rollers and cams, frame and lever were modelled with contact constraints between curves. For each body that requires a contact constraints was defined a curve on the symmetry plane, having the same shape of the contact surfaces. Hence, were defined the contact joints between the curves, particularly Roll-Curve Joint between the rollers and the two cams and between the active surface of the lever and the roller on the rocker arm.
For the contact between the valve and the rocker arm was used a Point-Surface Joint. In this way the vertex of the valve stem is constrained to move on the left end plane surface of the rocker arm.
Finally, was defined a circumference concentric to the upper fixed ground guide and with a radius equal to the guide radius minus the radius of the upper lever roller. So, with a Point-Curve Joint, the centre of the upper roller was forced to move on this circumference.
Below is the complete mechanism assembled:
Valve lift profile evaluation
The aim of the Valvetronic System is the variation of the intake valves lift profile to meet different regimes of engine. The azimuthal position of maximum opening level is not changed.
With this geometry of the intermediate arm active surface, when the eccentric cam rotates form 0° to 210°, a displacement of the valve is generated, as represented in the following graph:
It is a negative lift. In reality, the valve movement from closed position to negative direction is blocked from its seat, but in this model the valve stem end and the rocker arm are constrained with a Point-Surface Joint. Therefore, the contact between the valve and the rocker arm is always guaranteed and the valve is dragged upwards. However, this aspect generates a small error in the valve lift: in fact at 210° the valve negative lift is 0.044 mm against a positive lift of about 9.3 mm, therefore lift due to the eccentric cam rotation can be considered neglectable.
In the following graph are shown the valve lift profile obtained at four different position of the eccentric cam: 0°, 70°, 140°, 210° while the traditional cam was rotating:
These results permit the comparison between the minimum and maximum valve lifts obtained and the reference ones:
These values together with the neglectable lift due to the eccentric cam rotation are the results of many iteration of the active profile of the intermediate arm. In the following graph are represented the maximum valve lift values for each position of eccentric cam:
The eccentric cam geometry is such that the relation between the angular rotation of the cam and the lift is not linear, but parabolic with good approximation.
Togehter with the lift profiles is also necessary the evaluation of the velocity profiles of the valve but, unlike the position analysis, they depend to the cam speed. The cam velocity can vary from a minimum (450 turn/min) to a maximum (3500 turn/min). In the following graphs are shown these profiles as a function of the camshaft angle for the same four position of the eccentric cam (0°, 70°, 140°, 210°), the first for 450 turn/min and the second for 3500 turn/min:
This two graphs have the same shape and the curves differ only in amplitude. For this reason every following analysis regard only the maximum cam rotation speed, because it represent the worst operating condition. The velocity profiles obtained are closely related to the intermediate arm geometry. Discontinuities in these profiles have an impact on the acceleration of the valve and could generate unwanted vibrations.
The return spring works between the frame and the valve and is used to:
- close the valve
- maintain the contact between the valve stem and the rocker arm
As shown in the picture, the spring works between the two points indicated by the white cross. The distance between these points is a function of the eccentric cam rotation:
The graph below shows the forces values in global coordinates acting on the upper end of the valve in the worst case (210° eccentric cam position). Pmag is the resultant of the forces acting on the valve, directed along the valve axis:
Since the valve is always guided by the rocker arm, the contact force Pmag must always be negative in order that the valve works properly. Hence, considering the maximum positive value of the Pmag, is possible the evaluation of the minimum elastic constant of the spring (with the hypothesis of linear elastic spring):
where l (mm) is the free length of the spring and it must be greater than lw .
In the following graph are reported the Kmin values calculated by the previous equation:
Thus, different couples of l and K values are possible. The domain of this graph can be restricted because the values of l less than 45,5 mm are meaningless. Moreover, is possible to estimate the rate of global spring compression on spring free length:
This two function are plotted below:
In order to maintain the value of r ratio around 20%, the values selected for the spring are:
The value of Kmin is the minimum value that allows the valve to remain in contact with the rocker arm. Therefore, it’s necessary to assume a value of the spring constant K greater than Kmin. Assuming the maximum contact force Fmag approximately equal to -50 N, the value of K become:
Hence, the contact forces assume the following values:
The resultant Pmag is always negative, so the task of bringing the valve in closed position is given only to the spring. In this manner, the modelling of roller-cam contact joint can be considered valid.
Valve spring sizing
Once determined the spring constant K and its free length l is possible to define approximately the spring geometrical dimensions. First, it is necessary to define the maximum force T acting on the spring for a position of the eccentric cam equal to 210°:
The maximum value of the force T is approximately equal to 380 N. It has been assumed to realize the spring with tempered spring steel (for instance C85S or C100S). Following are typical values of E, G and Rm for this material:
From the ultimate tensile strength is possible to obtain the maximum allowable stress and the maximum allowable shear stress for this material:
For the purpose it was used a helical compression cylindrical spring mainly stressed to torsion. A geometric constraint is the spring housing hole diameter D’ therefore, assuming D’ equal to 20 mm, the spring outer diameter De must be less than D’. Indicating with D the spring average diameter and with d the spring wire diameter, the following relation can be written:
The following approximated results were obtained assuming a linear spring and ignoring the axial and flexional stresses and the effects of the shear stress and of the curvature. The lower the helix angle α, the better the approximation. The results are:
Using this values the spring constant K becomes:
Were also defined the parameters:
where c is the spring index, s is the spring pitch and l is the total length of the spring.
It is also possible the determination of the spring helix angle α with this relation:
therefore, these calculations can be considered valid since the helix angle is small.
The stress state τ of the spring can be evaluated in the most exterior point of the wire using the data calculate above:
where τMt is the shear stress given by the twisting moment, τt is the shear stress due to the shear force and k1 is a coefficient that consider the effect of the shear stress:
Hence the expression of τ becomes:
and the spring is verified.
Intermediate arm spring
As can be seen in the real mechanism image, to maintain the contact between the intermediate arm and the cams has been inserted a torsion spring at the top of the mechanism, fixed to the frame and acting on the front ends of the intermediate arm. To determine the main features of this spring, first of all, must be evaluated the forces acting between the cam and the lower roller of the intermediate arm. Referring to the roller, are shown in the graph below the behaviour of the horizontal force Fy in the global coordinates system in function of the camshaft angle for four different values of the eccentric cam position:
Since the contact between the cam and the roller must be guaranteed, necessary the force Fy1 must be negative. As can be seen in the previous graph, the worst condition occurs when the eccentric cam is in 0° position:
To insert the torsion spring in the model was added a new massless body, constrained to the frame by a cylindrical joint with the longitudinal axis normal to the mechanism plane and coincident with the spring helix. Finally to simulate the behaviour of the real spring, on this joint was added a torsional spring RSDA.
It was necessary to define an orientation angle for the spring body. Has been assumed an orientation angle δ such that, with the valve closed and with eccentric cam at 0° position, the rotation α of the spring was zero. The orientation angle δ is equal to 87.27° that is the maximum orientation angle.
The spring orientation angle in function of the camshaft angle is shown in the following graph:
The maximum horizontal force Fy1 at the contact point between the lower roller and the cam is:
The fundamental relation for the spring is:
where M is the torque express in Nm, k is the elastic constant of the spring express in Nm/rad and ϕ is evaluated with this relation:
The relationship that connect the spring properties to the contact force is complex so to simplify the problem it’s possible to assume the torque M as a product of the only horizontal contact force Fy1 and a middle arm l that is de distance between the spring circumference and the contact point intermediate arm-spring:
With good approximation, the force arm can be assumed equal to the following average value:
Hence, the elastic constant of the spring is:
The following graph shows the values of K in function of the orientation angle δ:
The difference δmax – δ can also be called β and it represents the preload angle of the spring. With the hypothesis that β is equal to 10°, the values of K and δ chosen for the torsion spring are:
With this value of δ the torsion spring works with a minimum preload of 3,4 Nm, which corresponds to a rotation of about 10°.
Thus, with the spring the contact force Fy1 between the roller and the cam is always lower than 0 N:
Looking at this graph it is possible to verify that the maximum value of Fy1 is about -30 N. For this reason is not necessary to increase the value of K to have a safety margin.
Intermediate arm spring sizing
Once known the spring constant and its working angle ϕ is possible to determine the main sizes of the helix spring that is solicited by bending moment Mf:
From this graph it’s possible to obtain the worst operating condition of the spring that is for a position of eccentric cam equal to 210°:
As done before, below are reported the final calculated values for the average diameter D of the spring, the diameter d of the wire, and the number of coils n:
With these values and with the material characteristics defined before is possible to estimate the elastic spring constant:
Moreover the spring index c is:
The stress value is variable along the wire diameter so it is possible to define two stress intensity factors (one internal kwi and one external kwe ):
The internal stress value and the external stress values are:
Camshaft torque in stationary conditions
At this point the springs are defined for the worst work condition (maximum camshaft velocity) and so it’s possible to evaluate the camshaft torque at the different engine velocity in function of the camshaft angle. In particular, in the graphs below, are shown the camshaft torques for the minimum (450 turn/min) and the maximum (3500 turn/min) camshaft velocities and for different positions of the eccentric cam:
At the maximum camshaft velocity, in our model, there are two symmetric discontinuities in the camshaft torque profile. They are due to an approximate lower intermediate arm profile and to the way in which the surfaces contacts were defined. These discontinuities were already present in the contact forces.
When the real mechanism works, the position of the eccentric cam and the velocity of the camshaft are straightly connected: in fact to obtain power from the engine are necessary high valve lifts. If the engine configuration is a six cylinder in line with two intake valves for cylinder (as N52) it is possible to define the camshaft torque profile at the maximum engine velocity (3500 turn/min for camshaft) with 210° of eccentric cam position in function of camshaft angle:
Camshaft torque in variable conditions
As time-dependent condition, there are these two cases:
- Abrupt change in position of the eccentric cam at the maximum camshaft velocity.
- Change in position of the eccentric cam with variable camshaft velocity.
The camshaft velocity is constant and equal to the maximum velocity (3500 turn/min). The eccentric cam position, on the contrary, depends by the time as is described in this graph:
The eccentric cam rotation from zero to 210° is accomplished in 0,3 seconds as reported in technical characteristics of the system. In the graph below is defined the camshaft torque profile that is necessary to maintain the chosen engine’s velocity (the printing interval is reduced from 0.5 s to 0.8 s).
As already said the camshaft torque is continuously variable and the curves that compose this graph are only different in amplitude (with the exception of the discontinuities). The amplification is linear as the variation of the eccentric cam position. As for the discontinuity, they are valuable for camshaft angle values equals to 140° that correspond a valve’s lift equal to 4mm.
In this case, the camshaft assume a variable velocity from the minimum to the maximum in 1.5 s. At the same time, the eccentric cam can rotate from 0° to 210° in 0.3s, as represented in Fig.30. The starting time is 0.5 s for both. With this simulation we want to recreate the condition of a sharp acceleration of the engine when the vehicle is stopped. The two cam profiles are reported below:
With this profile it is possible to obtain the necessary camshaft torque:
As this graph shows, referring to the maximum values, a good fitting curve for this values is a parabolic curve. Hence it’s possible to say that with a linear variation of camshaft velocity and with a linear variation of a position of the eccentric cam there is a parabolic variation in amplitude of the homologous point in camshaft torque.
With this project has been possible to determinate, first of all, the Valvetronic geometry and understanding how this compact mechanism works. Without the implementation of a contact between solid bodies it has been possible to recreate the transmission, evaluate the valve’s behavior and estimate the principal characteristics of the elastic elements. Moreover, with these properties, were sized the structural characteristics of the two different springs. In the end, with this model was calculated the camshaft torque that is necessary to move the mechanism at the different velocities.
Future developments, starting from what has been done, can relate to the insertion of the contacts between the various organs in order to evaluate the friction forces, and also the possibility to develop a modal analysis of the system.