CVT Calculation for a Baja car

CVT in a BAJA car
As none of us are going to design and fabricate this complicate design, this section deals with the tuning of CVT rather than its design. In this discussion we focus on a particular type of CVT and try to tune it, taking into account all the required parameters. Though other methods are also available, the following one will ensure minimum time and low cost.
While looking on a CVT one should be able to differentiate from the other types from the color of the springs. The colors are provided to indicate the stiffness level of the springs. By tuning, we refer to the optimum selection of these springs and the flyweights. Since the kinematic equation of any CVT operating in similar fashion will be the same, this method could be used for all types.
Model and working:
The type of CVT considered here is V-Belt rubber type. The input (Engine) is to the primary sheave while the output is from the secondary sheave. When the speed increases, the flyweights pull the movable sheave in the primary sheave closer. The force provided by the flyweight should be enough to overcome the spring force and the frictional force between belt and the pulley. The secondary pulley consists of a similar arrangement with the movable sheaves actuated by a shift in the primary pulley.
Tuning Technique:
Here the energy balance is used as the main criteria for tuning. The energies that should be balanced are the potential energy stored in spring, energy lost due to friction, energy corresponding to the velocity of the sheaves, and angular velocity of the flyweights. The corresponding equations are provided in the table below
Potential Energy stored in springs 1/2*K*(∆x)^2
K= stiffness of spring(lbs/in)
∆x=Compression
Loss due to friction 2*N*μ_k*d N=Normal force by spring
μk=Coeff of friction between sheave and belt
d= Radial distance from belt.(dia of belt location)
Kinetic Energy stored on sheaves 1/2*m*v^2
M= mass of sheave + flyweight
V= velocity of sheave
Kinetic Energy of flyweights m*(y_cm*ω)^2
M=mass of flyweight
Ycm= Height of CG.
ω=Angular velocity
of flyweight

While calculating the dia ‘d’ in frictional loss calculation, the linear relationship between ‘x’ and d should be determined where x is the distance through which the sheave is moved. ‘d’ calculated will be different for primary and secondary sheaves as x is different.
The relationship will be of the form

Ax+B=d for primary
Cx+D=d for secondary
The coefficients A and B can be found by substitution of x and d for 2 cases. The height of the belt for different x is taken as the radius.
The sheave velocity ‘V’ can be determined by analysing the motion produced by the cam surface on the moving flyweight. By measuring the linear follower displacement as a function of the rotation of the cam, a displacement plot can be constructed. After determining a curve fit model to give an analytical displacement function, it is possible to take derivatives to find the velocity and acceleration with respect to rotation. To get true velocity, we must multiply the result by dw/dt, w being the rotational engine speed.
Finally the KE of flyweight is found by a curve fit. The accurate mathematical model of the cam surface is made.
The ideal tool for taking these measurements would be a dial indicator with a knife-edge tip. A ball end dial indicator shall also be used but it introduces a small error which could be considered negligible. Finally the location of CG is found.

The final energy balance for primary sheave is
(PE spring -E friction +KE sheave +KEflyweights )primary= 0
From the available springs the mass of flyweight to be added can be obtained from the equation
Mass of flyweight mfw = 〖(E〗_friction-〖PE〗_spring)/(1/2*m_s*v^2*ω+(y_cm*w)^2)
Now the energy balance for the whole system is
PE spring -E friction +KE sheave +KEflyweights =0
Where all the parameters are for both primary and secondary sheaves. From the above equation the list of optimum spring rates for secondary sheaves can be found for different primary spring rates and flyweights. These secondary spring rates are then compared to a list of the available secondary spring rates. The closest match is then found.
The iterations can be performed by using various computer programs. Else a trial and error method which would be very time consuming is the alternative.