- Refosco Davide – firstname.lastname@example.org – matr. 1063000 – Laurea in Ingegneria Meccanica
- Nardon Mattia – email@example.com – matr. 1057329 – Laurea in Ingegneria Meccanica
The aim of the project is to realize a support system and a rotational system for a bike wheel to put inside a wind channel. In the real simulations, the tests into the wind channel are made to obtain information like the power lost during the rotation and the drag coefficient changing these parameters:
- velocity and direction of the wind
- velocity of the wheel
- type of wheel
Into the wind channel the wind is created using an electric fan. Changing its velocity, the wind velocity changes. To change the orientation of the flow, the support system is moved around the vertical axis and not the fan. To evaluate the power lost and the drag coefficient, forces and moments acting on the wheel must be known. To do this, the support system must be connected to a dynamometric sensor. In this project, the layout of the structure will be defined and the forces acting at the upper part of the support will be evaluated. To do this, the drag coefficient must be known as a preliminary information: it is found in literature as the result of previous experimental tests.
The project could be divided into two parts. The aim of the first part is to choose the rotational system, to define the geometry of the support system and to define the global layout. To do this, the Catia environment of the software Virtual Lab has been used. In the second part a simulation is done, with the software Virtual lab, in order to realize a structural analysis of the support system by evaluating some forces (forces and moments) which act at the upper part of the structure:
- contact force between the wheel and the rotational system
- aerodynamic forces with different conditions:
- different velocities and directions of the wind
- different velocities of the wheel
- different types of wheels
- total forces: contact and aerodynamic forces
THE MODELLING PROBLEM
The structure is composed by two main parts: the support system and the rotational system.
The support system has to sustain the wheel, to change the relative direction between the wheel and the wind direction and to connect the wheel to the acquisition system, a dynamometric platform which can evaluate the forces and the moments. The dimensions and the geometry of each component have been chosen following these criteria:
- to not modify the wind flux, it has been chosen an appropriate distance between the forks;
- to minimize the disturbs during the acquisition, it has been reduced the distance between the dynamometric platform and the frame;
- to make tests with different types of wheel, it has been chosen some solutions that will be described better later.
The main components of the support system are these:
system helical screw-crown wheel
engine for the movement of the screw
platelets to support the engine and the screw
All the components are made of structural steel S235. Now let’s describe better the components.
- 2 squared profiles 40×40 mm, thickness 2 mm and length 440 mm
- 2 squared profiles 40×40 mm, thickness 2 mm and length 270 mm
- Rectangular plate with dimensions 440x350x10 mm
- Circular plate with diameter 64 mm, thickness 16 mm and with 4 holes M6
- Total mass 15,23 kg; inertia around the vertical axis 0,456 kgxm2.
DESCRIPTION: The squared profiles are welded together; all the profiles are welded with the upper plate, which give rigidity to the structure. The circular plate connects the frame with the upper part and gives a support for the crown wheel.
- Threated part M30x1,5 with length 126 mm
- Hollow tubular profile with diameter 26,9 mm, thickness 3,2 mm and length 329,5 mm
- Fixing system for the wheel pivot with diameter 30 mm
- Mass of each fork 1,74 kg
DESCRIPTION: Each fork is fixed to the frame using two nuts fixed to the threatened part, which can be used to change the distance of the rotation axis of the wheel from the frame, in order to change the diameter of the wheel tested and also give a certain preload to the contact force between the wheel and the rotational system. The fixing system supports the wheel pivot and reduce the inflexion of the structure. To not modify the wind flux, the distance between the two forks is set to 400 mm.
3. Wheel pivot
- an inner part with diameter 9 mm and length 460 mm
- an outside part with two profiles with diameter 20 mm and length 165 mm
- mass 0,907 kg ; inertia around the vertical axis 0,01 kgxm2
DESCRIPTION: the two extremities of the inner part are threaded and fixed with nuts to the forks; the profiles with the bigger diameter give the right position to the wheel and also give rigidity to the structure.
- Inferior part with diameter 50 and length 16 mm with 4 holes M6x1,5
- Middle part with diameter 60 and length 66,6 mm
- Upper part with diameter 55 and length 18,5mm
- Total mass 2,2 kg
DESCRIPTION: the shaft connects the frame to the upper part of the structure; to do this 4 screws M6x1,5 are used. The diameters and also the lengths are chosen to give enough rigidity to the structure. On the middle part there is double raw angular contact ball bearing SKF Explorer 3212 A-2RS1, which can give the rotation to the structure, accommodate radial loads as well as axial loads acting in both directions, provide stiff bearing arrangements and accommodate tilting moments.
5. Upper part
In the upper part the crown wheel is connected with a forced rim to the shaft, and its movement is impressed using an helical screw. The movement of the screw is imposed using an engine. Engine and helical screw are supported by two platelets, which are fixed by using screws to the dynamometric platform, which has been chosen to evaluate forces and moments. To sustain the structure, there is an hollow tubular profile in which the ball bearing connected to the shaft is supported.
They support the engine and the helical screw; to do this in the inferior part of each platelet there is a deep groove ball bearing SKF-Explorer 61802-2RS1 (there is only the weight of the helical screw to sustain).
5.2. Helical screw- crown wheel
- crown wheel :
external diameter = 91 mm
- helical screw:
Threated part : modulus: m = 0,8
length 20 mm
DESCRIPTION: the crown wheel is connected with a forced rim to the shaft; the helical screw is supported by the ball bearings and can rotate thanks to the engine (the helical screw is connected with a forced rim to the engine).
5.3. Hollow tubular profile and dynamometric platform
- hollow tubular profile: external diameter 160 mm, thickness 36 mm and length 64 mm
- dynamometric platform: 250x260x15 mm with 5 holes M7
DESCRIPTION: Into the hollow tubular profile there is the double raw angular contact ball bearing; so it sustains the structure. The dynamometric platform is fixed with five screws to the ground. Hollow tubular profile and dynamometric platform are welded together.
The rotational system has to give the rotational movement to the wheel, to not disturb the wind flux and to simulate the right contact with a real road. The wheel also changes its orientation, rotating around the vertical axis, in order to simulate different directions of the wind flow, so the rotational system must be always align to the wheel to give the right velocity and to not transmit forces in the transversal direction. To move simply the rotational system around the vertical axis it has been chosen a cylinder; to simulate the straight road and to not disturb the wind flow the diameter is set to 1000 mm. To have a better grip with the wheel, the cylinder is covered with a film of rubber.
In this simulation it has been considered two different types of wheel, which differ for geometry; masses and inertias are considered the same: mass = 2 kg; radius = 350 mm; inertia around the rotational axis = 0,35 kgxm2
TYPE 1 TYPE 2
The fork and the wheel pivot are fixed together with nuts in the reality; for this reason this joint is simulated by using a bracket joint. For the same reason there is a bracket between the frame and the shaft, between the forks and the frame and between the platelets and the dynamometric platform. The crown wheel is connected with a forced rim to the shaft; it is reasonable to connect them with another bracket joint. To allow the rotation of the wheel, the wheel is connected with a revolute joint to the wheel pivot (rotation allowed around the wheel axis). About the system helical screw-crown wheel, the screw is connected with a forced rim to the engine which is also fixed with nuts to the platelet. Therefore there is a bracket joint between the engine and the screw and another bracket joint between the engine and platelet. The helical screw is also supported by the two deep groove ball bearings put inside the inferior part of the platelets; to simulate this there are two revolute joints between the helical screw and the platelets in the position in which there are the ball bearings. The crown wheel is fixed to the shaft with a bracket joint. The shaft can rotate around the vertical axis thanks to the double raw angular contact ball bearing: to simulate this condition, there is a revolute joint between the shaft and the hollow tubular profile. To simulate the relative movement of the system helical screw-crown wheel, a gear joint has been used. To work, the gear joint must use joints which have a common body. To do this the two revolute joints screw-platelets are replaced by two other revolute joints in the same position of the previous ones but between the helical screw and the hollow support system. In this way the gear joint is defined by a revolute joint between the helical screw and the hollow tubular profile and a revolute joint between the shaft and the hollow tubular profile; having a common body (the hollow tubular profile) the gear joint can work. The constrains between the dynamometric platform and the ground are imposed in a different way for the first analysis (contact forces) and the second analysis (aerodynamic forces). As a matter of fact, in the first analysis the platform is constrained with a translational joint to the ground with vertical direction of movement; in the second analysis is fixed to the ground using a bracket joint. These conditions will be described better later during the analysis.
The rotational system must rotate around its transversal axis to give the rotation to the wheel and also around the vertical axis to follow the wheel during the movement. To simulate this behavior, first the cylinder is constrained to a virtual body with a revolute joint (rotation allowed: rotation that give the movement to the wheel); then the virtual body is attached to the ground with a revolute joint (rotation allowed: around the vertical axis). The system of movement isn’t modeled because it isn’t the aim of the project.
SIMULATIONS AND ANALYSIS OF RESULTS
1. CONTACT FORCES
In this first analysis the aim is to evaluate the forces and the moments which act at the upper part of the support system due to the contact between the wheel and the cylinder. A first idea was to fix the upper part of the support system to the ground with a bracket joint, create some roads with a circular profile attached to the rotational cylinder and then apply tire forces between the tire and every road (using one of the two forks as chassis). Then using a joint velocity driver the movement was imposed to the cylinder. Using this approach unfortunately, the velocity profile obtained is quite good, but the contact force in the direction of the rotation of the wheel was too much big and not regular. This is due to the fact that the vertical load is not imposed (both the wheel and the cylinder are constrained to the ground) and the contact force between the tire and the road acts only when the z axis of the road and the z axis of the chassis have the same direction. These are the results of this analysis:
To avoid this problem, it has been used a straight road with the z axis always in the same direction of the z axis of the chassis. Now the road isn’t attached to the rotational cylinder, but to a virtual body which is constrained with a translational joint (direction of movement: in the direction of rotation of the wheel) to the ground in order to simulate the rotation of the cylinder and of the wheel. In this way the z axis of the road preserves the right orientation and the road can move and put into rotation the wheel. To consider the right value of the vertical load of the force acting between the tire and the road, it has been constrained the upper part of structure with a translational joint to the ground in the vertical direction. This isn’t the real condition of work of the system, but with this simulation we can consider the right value of the normal force to the road and consequently obtain the right longitudinal force. In the real conditions it is assumed a total preload of 40 kg; to simulate this the total mass of the system is increased by 21,4 kg (total mass of the system:18,6 kg ; to consider the preload, the mass of the upper part is increased by 21,4 kg). It is reasonable to approximate the circular profile of the rotational cylinder with a straight road because the radius of the cylinder is big enough to do this. Moreover to simulate the real contact the tire is modeled in this way:
The force between the road and tire is imposed by using the “Simple Tire force model” with these parameters:
tire body: the wheel (z axis as rotation axis)
chassis body: the left fork (z axis: vertical axis)
road: the virtual body constrained to the ground (z axis with upward direction)
radius: 350 mm
damping constant: 120 kg/s
friction coefficient: 1,16 (typical value for contact between rubber and rubber)
cornering stiffness: 20000 m Kg/( s2 rad)
vertical stiffness: 100000 N/m
The simulation is done waiting the system reaches a position of equilibrium due to the gravity force; after reached the equilibrium the road is moved with a joint velocity driver. In this simulation the aim is to compare forces and moments due to different velocity laws imposed to the joint velocity driver, in order to simulate different type of control of the system. In the second simulation (aerodynamic simulation), the target velocities of the wheel are 30 km/h and 40 km/h. For this reason, in this first simulation these laws are chosen:
- linear increase of the velocity from 0 to 30 km/ in 3s
- linear increase of the velocity from 0 to 30 km/ in 6s
- linear increase of the velocity from 0 to 40 km/ in 4s
- linear increase of the velocity from 0 to 40 km/ in 7s
The second and the third law are chosen to have the same gradient. In this simulation the wheel doesn’t rotate around the vertical axis (a null law is given to the helical screw in order to fix the position of the wheel) so it has always the same orientation respect to the road. It is reasonable to do this because in the real condition the cylinder rotates around the vertical axis with the same angle of the wheel, so they are always aligned each other and once reached the target velocity the wheel doesn’t change its velocity (in fact the velocity changes for the aerodynamic force, but with a good control the velocity is quite constant). Here there is a video of the simulation: contact simulation
Now let’s analyze the results of the simulation. The next longitudinal forces and moments are evaluated at the translational joint between the upper part of the structure and the ground. It is reasonable to do this because using a translational joint there are the same constraints about the longitudinal direction and about the rotation around the vertical axis for the rotation of the wheel than using a bracket joint (which is used indeed in the real conditions). Other forces and moments are not plotted because they are null. Being masses and inertias the same for the two wheels, the results are the same in both cases.
Linear increase to 30 km/h Linear increase to 40 km/h
Increase in 3s: velocity and longitudinal force Increase in 4s: velocity and longitudinal force
Increase in 6s: velocity and longitudinal force Increase in 7s: velocity and longitudinal force
Comparison: longitudinal forces Comparison: longitudinal forces
Comparison: moments Comparison: moments
Total comparison of Longitudinal force: increase to 30 and increase to 40 km/h
Total comparison of moment (the longitudinal force produces a moment): increase to 30 and increase to 40 km/h
As it is possible to see from the previous graphs, the longitudinal force is quite constant when the velocity increases, being the increase of velocity linear in time, and then it is quite null when the velocity is constant (it is quite null for the vertical oscillations of the wheel). The first oscillation is due to the fact that there is translational joint along the vertical direction, and so the wheel reaches the equilibrium after a certain period. However the mean value is good. Moreover the longitudinal forces have the same gradient (the parameters of the tire force are the same). Being the gradient of the linear velocity law the same for the increase to 40 km/ in 4 s and for the increase to 30 km/h in 3 s, the maximum value of the longitudinal force is the same (they differ only for the time). In the other two cases, the maxim value is different (the velocity laws don’t have the same gradient). From this comparison, it is possible to say that an increase in time slower due to a longitudinal force that is smaller than a faster increase.
The same thing happens for the moment.
2. AERODYNAMIC FORCES
In this second analysis the aim is to evaluate the forces and the moments which act at the upper part of the support system due to the wind flux.
In the real test into the wind channel, the direction of the wind is fixed, but the orientation of the wheel changes. To simulate this it has been used a “wind body” with a fixed orientation and other virtual body attached to the tire (which will be better described later); tire which can rotate using the gear joint. As a matter of fact, if the aerodynamic force is applied to the global tire, the expression will be:
because the tire doesn’t move forward, but in this way the rotation of the wheel isn’t considered. To avoid this problem the global aerodynamic force is divided into different parts which act at different distances from the center of the wheel; thus for each wheel we can identify three significant values of distances in which apply the force:
ri = 58,333 mm; rm = 175 mm; re = 291,66 mm
In this way it is possible to identify three different forces:
In which (vt)i , (vt)m , (vt)e are the velocity in the direction of the wind at the different radius. If we consider the wheel perfectly aligned to the wind direction or not aligned :
In which ω is the angular velocity of the wheel. Moreover, each force Fi, Fm, Fe is divided in a simplified way into the number of spokes of the wheel. In this project, the simulation is made using two different types of wheel. For the first wheel , in literature it is possible to find the values of CDxA at different values of α angle , as it is rapresented in the diagram below:
Thus the final force has the following expression:
In this way there are eight internal forces, eight medial forces and eight external forces. Regarding the different values of (CDxA)i , (CDxA)m and (CDxA)e , it is possible to assume that (CDxA)i = (CDxA)m = (CDxA)e = (CDxA)’ (the variation is less than the variation of the speed). In this way it is possible to calculate (CDxA)’ with a simplified approach:
The simulation is done with the following combinations of values:
Now let’s describe how the simulation has been done in Virtual Lab. We define a wind body, and for each wheel we define some virtual bodies constrained with spherical joints to the wheel. Let’s consider the first tire. First we define a number of bodies equal to the number of subdivision of the global force, in this case twenty-four bodies. Then we put them into the right orientation: with the x axis with the same sign of the x axis of the wind body. In the end each body is constrained with a spherical joint to the wheel in the point in which each aerodynamic force Fi’, Fm’ and Fe’ acts (at ri, rm and re ). In this way each body preserves its orientation with respect to the wind body during the rotation of the wheel and so it is possible to simulate a constant direction of the wind flow. The same thing is done for the second wheel (in this case there are nine bodies). (in the picture there are shown only 3 virtual bodies to be clear).
Body: each virtual body attached to the wheel
Wind body: the wind body
Fluid density: 1,3 kg/ m3
Cd x Char area: 1 m2
Velocity in x : variable with respect to the position of the wind body (x axis) with this expression:
To consider the different values of (CDxA)’ and the different values of the velocity with respect to α angle , each law is defined for a different position of the wind body. Thus each position of the wind body represents an angle. For example if = 0° for 10 seconds and for the next 10 seconds it becomes 10°, for the first 10 seconds the wind body could have a distance of 450 mm from the center of the wheel and for the other 10 seconds 750mm. In this way, for a different position of the wind body there is a particular law. The diagram below describes is an example of this:
To change the position of the wind body, the wind body has been constrained with a translational joint to the ground along the x axis and it is moved with a joint position driver. To obtain the right value of the velocity, the velocity of the wheel is controlled using a joint velocity driver connected to the revolute joint between the wheel and the wheel pivot. In this way only the aerodynamic forces are evaluated and not also the contact force between the tire and the cylinder. To not consider the contact force, the simulation is done also fixing the upper part of the structure with a bracket joint to the ground (in the real condition the structure is fixed with five screw, but using only one bracket joint the constrains are the same) and not considering the tire force between the wheel and the road defined in the previous analysis. About the angle, this is controlled using the gear joint between the helical screw and the crown wheel and controlling it using a joint position driver connected to the engine. In the reality, the movement is impressed by the cylinder, which rotates around the vertical axis with the same angle of the wheel. To simulate this, the cylinder is constrained with a revolute joint with a virtual body (this give the possibility to rotate around the longitudinal axis and so simulate the transmission of the movement to the wheel), which is also constrained with another revolute joint to the ground (in this way it can rotate along the vertical axis). However in the simulation the movement of the wheel is impressed by the joint velocity driver. Here there is a video that shows this second analysis: one virtual body connected to the wheel and the wind body (white axis on the right) are shown: aerosimulation
Here there are the results of the analysis. Forces and moments are evaluated at the bracket joint between the upper part of the system and the ground. The convention used is this: Y axis: in the direction of the flow (direction of the aerodynamic forces) ; Z axis: points upward ; X axis: complete the system.
The oscillations are due to the fact that each force is applied to the wheel in a different position and the law is a sinusoidal law; the total force is a sum of these laws. The results show that the most important parameter for the aerodynamic force is the velocity of the wheel; as a matter of fact the maximum value of the force, for the same angle, is higher when the wheel velocity is at the maximum target value, 40 km/h.
In the chart below it is shown the longitudinal force at a different α angle :
The results show that the maximum moment is the tx; as a matter of fact the aerodynamic force acts in the y direction. When the angle between the wheel and the wind flow changes, tx decrease its value, and the ty and the tz appears: this is due to the fact that when angle is not 0° (wheel and wind flow are not aligned) the aerodynamic force also produces a moment around the other two axis. However, tx is the principal moment also when the wheel rotates around the vertical axis.
Moment around x – axis: TX Moment around y – axis: TY
Moment around z – axis: TZ
These graphs emphasize that the most important parameter is the wheel velocity; as a matter of fact the moments for the combination Vwind= 30 km/h , Vwheel = 40 km/h and Vwind = 20 km/h , Vwheel = 40 km/h are more or less coincidence. The same thing happens for the other two combination.
For the second wheel the oscillations are higher than the first wheel when the wheel moves around the vertical axis: this is due to the fact that the aerodynamic forces act in a less number of point than the first wheel, so the different laws don’t sum each other very well. However is possible to identify a mean value. In the graph below it is shown the longitudinal force to vary α angle:
In this next picture the moments tx, ty and tz for only one condition are shown in order to compare their values and profiles.
Moment around x – axis: TX Moment around y – axis: TY
Moment around z – axis: TZ
COMPARISON BETWEEN THE TWO WHEELS
Moment around x – axis: TX
Moment around y – axis: TY
Moment around z – axis: TZ
As it is possible to see, the longitudinal force and the moments are higher for the first wheel than the second one.
3. TOTAL FORCES
In this third step the aim is to obtain the global forces and the global moments due to the combination of the contact force and the aerodynamic force. For a structural analysis, it is useful to know the maximum forces and moments which act; thus in this third part are combined the results of the increase of the velocity to 30 km/h in 3s from the contact analysis (maximum values of the contact analysis) and the results of the aerodynamic analysis with vwind = 30 km/h andvwheel = 40 km/h (maximum values of the aerodynamic analysis) referred to the first wheel. If we consider the condition in which the wheel reaches the target speed and then the wind flow starts, it is reasonable to combine the results of the previous two analysis in the following way without any new analysis:
The longitudinal force first increases for the contact force, decreases when the velocity is constant and then starts to increase for the aerodynamic force.
A more interesting situation is when the wheel increases its velocity after the wind flow has already started. To do this, the results from the previous aerodynamic analysis can’t be used because they refer to a constant velocity of the wind. A first idea was to use the same conditions of the first analysis (translational joint at the upper part and contact with a straight road moved with a joint velocity driver) and add an aerodynamic force, in order to have the contact and both the aerodynamic forces. In this way unfortunately the wind body and the virtual bodies attached to the wheel don’t have the x axis aligned (due to the fact that the wheel moves downward for the gravity) and if the wheel moves around the vertical axis, other forces and moments appear. To avoid these problems the results of the first analysis have been used and combined with another new aerodynamic analysis, in which the simulation is done by constraining the upper part with the bracket joint and by using the virtual bodies attached with spherical joints to the wheel, like in the second analysis , but changing the expression of the wind velocity: in this situation a linear law which increases from 30 km/h (only wind) to 70 km/ (vwind + vwheel ) is used to simulate the increase of velocity. The global force is divided in the same way of the second analysis (into 24 parts) but not considering the variation due to the term (in a simplified way) and only for = 0°, because the aim is to combine the results of the first and the second analysis, but the first analysis has been done only for = 0°. Here there is the result of the analysis for the longitudinal force:
Thus the global longitudinal force is the sum of the two previous forces:
As it is reasonable, there is an offset for the previous results; thus it is possible to use the results of the second analysis with different angles and sum them:
The same thing should be done for the moments.
First the support system and the rotaional system had been modeled. After that, some analysis have been done by using two different types of wheel. In the first analysis, forces and moments, due to the contact between the wheel and the rotational system, at the upper part of the support system have been evaluated. Being masses and iertias the same for both the wheels, the results are the same in both situations. This analysis has been done by simulating a different type of control of the incease of the velocity of the wheel in order to compare the different profiles of forces and moments due to the different accelerations in time. The results show the different profiles of forces and moments by using the different laws. However, in all the situations, the maximum values are not very high. In second analysis, forces and moments, due to the aerodynamic force, at the upper part of the support system have been evaluated. The simulation has been done by considering different combinations between typical velocities of the wind and typical velocities of the wheel. The results have been expressed in a compact way in function of the angle between the wheel and the wind direction and they can be used to compare the two types of wheel. From the comparison it is possible to say that the first wheel leads to bigger values of forces and moments than the second one. In the last analysis, total forces and total moments, at the upper part of the structure, have been evaluated. In the situation in which the wheel reaches the target speed and then the wind flow starts, the results of the two previous analysis have been combined together. In the situation in which the wheel increases its velocity after the wind flow has already started , in order to have a better evaluation the results of the first analysis have been combined with the results of a new aerodynamic simulation. The limit of this approach is not consider the contact and the aerodynamic force in an only simulation; however the results are reasonable.