CVT - Continuously Variable Transmission


What are the steps to select a CVT ?

Note: I would like to receive your helpful comments to correct/improve this approach.
(Instructions: input your own and click on the buttons successively)

We'll need to determine the ratio spread (highest ratio and lowest ratio).

1. Determination of the "highest ratio"
Usually the "highest ratio" is selected so that it will allow the achievement of the
maximum possible speed. Nevertheless, an higher ratio is also used
to benefit highway fuel economy. (But it would not achieve the top speed.)
Thus, we need to know the vehicle's maximum speed, using the given power.

1.1. Available power?
Considering that the engine power is Pe = [hp],
and the transmission efficiency (incl. rolling resistance) is Eff = [%],
and the available power (Pt) of the transmission will be:
Pt =
Pe × Eff [hp] = [w].

1.2. Thrust force?
At a speed v (m/s), the Thrust_force will be Tf = Pt/v = /v [N] 

1.3. Drag force?
To determine the maximum speed we must also calculate the Drag_force, Df.
The Df = Cd × ½ × r × v² × A
: drag force coefficient:
  r: air density (=1.2 [kg/m³])
  v: vehicle velocity (m/s)
  A :   -Vehicle's projected frontal area: [m²]   

1.4. Maximum speed?
Therefore the maximum velocity will be calculated by equating the
Thrust_force to the Drag_force (Tf = Df), and solving for the velocity:
³ = Pt / (Cd × ½ × r × A) thus
v.max [m/s] (=[km/h] =[mph]);

1.5. Highest ratio?

Considering that the maximum power is at rpm (rpm.P),
the wheel diameter is [inch] (w.diam, use the calculator at the right → ),
and the differential ratio is :1 ,
the highest ratio (high.ratio) will be calculated by the equation:
v.max = (rpm.P ×2×3.1416/60) × ((w.diam×0.0254)/2) / (high.ratio×differential)
high.ratio = (rpm.P ×2×3.1416/60) × ((w.diam×0.0254)/2) / (v.max×differential)
and therefore the high.ratio
Note: this “highest ratio” value was selected in order to optimize the maximum speed. 


2. Determination of the lowest ratio

2.1. Force due to the slope?
Consider that the car must be able to start moving uphill, fully loaded.
The vehicle's mass is [kg] and the maximum load is [kg] (incl. passengers).
The slope inclination, (percent.incl), is [%] and the safety factor is .
The force due to the slope will be:
F.slope = (mass+load) × 9.81 × (percent.incl/100) × safety / Eff [N];
(note: Eff is the transmission efficiency ).

2.2. Available tangential force in the tyres:

The maximum engine torque is [Nm] .
However when starting to move the clutch slip results in an efficiency of % (clutch.eff).
So the tangential force in the tyres will be:
F.tyres = torque × clutch.eff × low.ratio × differential / ((w.diam×0.0254)/2)

2.3. lowest ratio?

Equating F.slope = F.tyres and isolating low.ratio results:
low.ratio = F.slope × ((w.diam×0.0254)/2) / (torque×clutch.eff×differential)
low.ratio :1
So, at this point we already know the required “lowest ratio” and “highest ratio”.

3. Addition of an external fixed ratio to obtain symmetric ratios:
The previous calculations resulted in: low.ratio:   high.ratio :.
However a V-Belt CVT usually generates (almost*) symmetric extreme ratios:
        [ low.ratio × cf = 1 / high.ratio ]
    because both pulleys move apart by equal amounts (alternatively).
 *Note: cf = is a correction factor. Lower cf values are used to lower the
whole ratio span within the CVT (cf = 0.91 .. 0.99) which is beneficial.
Therefore, we must add a fixed reduction after the V-Belt CVT.
This way the V-Belt CVT will be able to have a
(almost) symmetric ratio span.
The fixed reduction ensures that the final ratio span will be exactly from
low.ratio to high.ratio.  
Therefore, we must add (in series) a final (fixed) reduction with a ratio of :
k = SquareRootOf ( low.ratio×high.ratio ×cf ), thus k .
Now we must calculate the new CVT's high and low ratios. These will result from
dividing each of initial low.ratio and high.ratio by k.
 in this case the CVT's symmetric ratios will be
=s.l.ratio, and =s.h.ratio.

4. CVT dimensions:

4.1. Pulley
Considering that the smaller diameter (D1) is [mm]
and that the bigger is (D2) will be so that D1/D2=s.h.ratio.
Thus D2 = D1/s.h.ratio [mm].
Specify the pulley groove angle: betha = deg.
4.2. Belt:
Knowing that the pulleys centre-to-centre distance is a=[mm]
  (note: it must be greater than [mm]),
...the belt_length will be 2×a+(PI/2)×(D2+D1)+(D2-D1)×(D2-D1)/(4×a).
Thus the belt_length [mm].

5. Velocities
(considering the maximum power regime)

5.1. Velocities in the lowest ratio:
Rotational speed driver pulley = [rpm]; (n1);
Rotational speed driven pulley [rpm]; (n2=n1/s.l.ratio);
Belt_speed [m/s]; ((n1×2×PI/60)×(D1/2))

5.2. Velocities in the highest ratio:
Rotational speed driver pulley = [rpm]; (n1);
Rotational speed driven pulley [rpm]; (n2=n1/s.h.ratio;);

6. Forces:

6.1. Friction coefficient:
Consider the coefficient of friction is mu=.
Therefore, the effective friction coefficient will be calculated by
mu.e = mu/sin(betha/2) .

6.2. Belt tensions in the lowest ratio:
The lowest ratio is .
In the following, we will use D1=[mm] for the driver pulley,
and D2=[mm] for the driven pulley.
The smallest belt wrap angle is calculated by
2×acos((D2-D1)/(2×a)), thus alpha [deg];
(Note: the following is not valid for pushbelt CVTs)
The ratio of the belt forces will be exp(mu.e×alpha), so T12 = ;
Belt tension load T1  will be (Pe/Belt_speed)/(1-1/T12) = [N];
(Note:Pe is the engine power).
Belt tension load T2 will be T1/T12 = [N];
And the shaft load can be calculated by:
  (shaft.load)² = T1×T1+T2×T2-2×T1×T2×cos(alpha);
Thus, shaft.load = [N];
Axial clamping.force = ( Pe/Belt_speed )/ mu.e = [N]

6.3. Belt tensions in the highest ratio:
The highest ratio is .
Now the driver pulley is D1=[mm], and the driven pulley is D2=[mm].
The smallest belt wrap angle is alpha [deg];  
(Note: the following is not valid for pushbelt CVTs)
The ratio of the belt forces will be T12 = ;
Belt tension load T1: [N];
Belt tension load T2 = [N];
And the shaft.load = [N];
Axial clamping.force = ( Pe/Belt_speed )/ mu.e = [N].

[ this site is under  construction ]  [created: 21 Apr 2004, updated: 19 Set 2008 ]  [ designed by ]