First and foremost, I want to mention that this isn't the holy grail of an infinitely variable transmission- the gearing ratio is restricted to the range between two limits set by geometric parameters. Additionally, it does involve a friction joint as opposed to an occluding gear type mechanism, but as opposed to alternative systems, it does not require any sliding frictional couplings- the frictional element involves only revolute contact points. That is, there's no scraping in the coupling as you change the gear ratio.

I'll start the description with where I began, conceptually: a truncated cone. A cone is the limiting case of a series of increasingly small gears stacked together, so it seems, I think, natural to drift towards this sort of construction. With a single truncated cone and a friction joint it's fairly simple to construct a continuously variable transmission (CVT), albeit with a clutch. The mechanism would look something like this:

I'll start the description with where I began, conceptually: a truncated cone. A cone is the limiting case of a series of increasingly small gears stacked together, so it seems, I think, natural to drift towards this sort of construction. With a single truncated cone and a friction joint it's fairly simple to construct a continuously variable transmission (CVT), albeit with a clutch. The mechanism would look something like this:

On the left, an initial placement , with the red and blue In and Out representing the delivery of the rotation to and from the device. As the truncated cone rotates, it turns the thin disk at rate set by the ratio of the radius of the thin disk to the radius of the conic at the contact point. Manipulation of this ratio is accomplished by shifting the contact point up and down the cone, as illustrated on the right above.

There are a couple problems with this machine. First, if the thin disk contacts the cone at multiple points, the different points will want to establish turn rates that vary along that edge, so there will always be some grinding. You could make the contact edge smaller or round it to reduce this, but those tactics come at a cost of reduced contact friction, and thus lowered performance. Also, the mechanism has the rotating central shaft of the output disk changing horizontal position. With a sliding coupling link, vertical position can be fixed, but it still needs to shift left and right. This could be controlled for using a belt or chain type linkage for the output, but that introduces a number of additional problems, practically speaking.

Instead, I added another cone, as below, with slope parallel to the first one, so that the distance between the two cones is constant, but the in/out ratio changes as you slide the transmitting disk up and down. Because the transmitter is not coupled to output, it can rotate freely around a canted bearing, such as the one illustrated on the right.

There are a couple problems with this machine. First, if the thin disk contacts the cone at multiple points, the different points will want to establish turn rates that vary along that edge, so there will always be some grinding. You could make the contact edge smaller or round it to reduce this, but those tactics come at a cost of reduced contact friction, and thus lowered performance. Also, the mechanism has the rotating central shaft of the output disk changing horizontal position. With a sliding coupling link, vertical position can be fixed, but it still needs to shift left and right. This could be controlled for using a belt or chain type linkage for the output, but that introduces a number of additional problems, practically speaking.

Instead, I added another cone, as below, with slope parallel to the first one, so that the distance between the two cones is constant, but the in/out ratio changes as you slide the transmitting disk up and down. Because the transmitter is not coupled to output, it can rotate freely around a canted bearing, such as the one illustrated on the right.

This eliminates the issues with the misplacement of the output link, but retains the problems with the slipping and grinding, so how to eliminate that differential rate problem? Also, this configuration still requires a clutch, which is something we'd really like to eliminate. Though the fluid variability of ratios is desirable in its own right, it would appear to stress the 'continuous' part if we have to decouple the linkage like a normal transmission.

On the other hand, though, this design does have some nice elegance to it- the input and output axis are fixed by careful choice of cones and the transmission link is free-rotating, which seems intuitively correct. In looking at it, it really seems like this is the correct kind of template for a positive contact CVT (as it opposed to hydraulic, magnetic or other non-interface contact drives), but it feels almost like the dimensional qualities are off.

What I'm getting at here is that there is a root problem with the fact that the disk rotation axis for transmission is close to parallel with the movement that changes the gearing ratio. If those directions were perpendicular, movement could be achieved in two different dimensions, allowing for the removal of the clutch action. We need a sphere!

Here's the idea: A sphere wedged between two disks, the rotation of one transferred to the other via the rotating sphere, which is held fixed with an armature:

On the other hand, though, this design does have some nice elegance to it- the input and output axis are fixed by careful choice of cones and the transmission link is free-rotating, which seems intuitively correct. In looking at it, it really seems like this is the correct kind of template for a positive contact CVT (as it opposed to hydraulic, magnetic or other non-interface contact drives), but it feels almost like the dimensional qualities are off.

What I'm getting at here is that there is a root problem with the fact that the disk rotation axis for transmission is close to parallel with the movement that changes the gearing ratio. If those directions were perpendicular, movement could be achieved in two different dimensions, allowing for the removal of the clutch action. We need a sphere!

Here's the idea: A sphere wedged between two disks, the rotation of one transferred to the other via the rotating sphere, which is held fixed with an armature:

Now, when we want to change the gearing ratio, the two disks move together or apart at (this is the important part) the exact same rate. If they move in tandem, then the sphere can rotate both in the transmission direction (axis perpendicular to the axis of the rotating disks) and to compensate for the movement of the disks (perpendicular to both those previous axis), meaning no scraping. Additionally, since the sphere is in point contact with the disks, no grinding occurs during the rotation.

Of course, in doing this, if the distance from the center of the disks to the sphere is the same, then the input/output ratio will not change when we move them. However, if there is some initial difference in distances, then the ratio will change as the two move apart, hence the drawing of the disks offset as they are.

Note that there is an issue of setting up a mechanism to properly align this equal movement (an active controller is not really in the proper spirit, you know), as well as the fact that the movement of the output rotation gear has resurfaced, and now the input gear moves as well. These problems have solutions, but for the moment I'm going to set them aside to discuss the mathematics for the gear ratio. I'll get back to the practicals shortly.

So, we have the two disks, which are set at some initial distances from the sphere contact point. These distances are offset from one another, because the ratio of the two distances sets the gearing ratio, so if they start out at the same distance to their centers, the transmission ratio will be 1:1. And, since they pull apart at the same rate (by design), the mirrored movement will keep the two distance equal, and thus fix the ratio at 1:1. Obviously this is pretty much useless, so we need to have the initial distances offset.

So then, if we have one set at an initial radius r1, and the other r2, the transmission ratio will be tr = r2/r1. Let's say you then move the disks, so that each one is displaced a distance Δd from their initial locations relative to the contact points with the sphere. Then the ratio will be tr = (r2+Δd)/(r1+Δd), which gives a changing ratio as the relative positions change, making it a useful transmission.

For a moment, let's look at this function. Lets start by replacing the two radii with ratios of one term, r, so that r1 = r, and r2 = αr. The new expression is then (αr+Δd)/(r+Δd), which can be further recast as (α-1)/(1 + q) + 1, where q = Δd/r. This expression gives the transmission ratio as a function of the ratio of initial radii and the change as a proportion of the lower of those radii. If we pin r1 and r2 as the actual radii of the disks (basically saying we 'start' with the sphere at the edges of the two disks), and note that then Δd is limited (in theory, depending on if the width of the sphere-holding bracket is less than the small radius) by the radius of the smaller disk:

Note that there is an issue of setting up a mechanism to properly align this equal movement (an active controller is not really in the proper spirit, you know), as well as the fact that the movement of the output rotation gear has resurfaced, and now the input gear moves as well. These problems have solutions, but for the moment I'm going to set them aside to discuss the mathematics for the gear ratio. I'll get back to the practicals shortly.

So, we have the two disks, which are set at some initial distances from the sphere contact point. These distances are offset from one another, because the ratio of the two distances sets the gearing ratio, so if they start out at the same distance to their centers, the transmission ratio will be 1:1. And, since they pull apart at the same rate (by design), the mirrored movement will keep the two distance equal, and thus fix the ratio at 1:1. Obviously this is pretty much useless, so we need to have the initial distances offset.

So then, if we have one set at an initial radius r1, and the other r2, the transmission ratio will be tr = r2/r1. Let's say you then move the disks, so that each one is displaced a distance Δd from their initial locations relative to the contact points with the sphere. Then the ratio will be tr = (r2+Δd)/(r1+Δd), which gives a changing ratio as the relative positions change, making it a useful transmission.

For a moment, let's look at this function. Lets start by replacing the two radii with ratios of one term, r, so that r1 = r, and r2 = αr. The new expression is then (αr+Δd)/(r+Δd), which can be further recast as (α-1)/(1 + q) + 1, where q = Δd/r. This expression gives the transmission ratio as a function of the ratio of initial radii and the change as a proportion of the lower of those radii. If we pin r1 and r2 as the actual radii of the disks (basically saying we 'start' with the sphere at the edges of the two disks), and note that then Δd is limited (in theory, depending on if the width of the sphere-holding bracket is less than the small radius) by the radius of the smaller disk:

Which gives us the range of values Δd can occupy: (0,r1/2); with the actual distances between the rotational axis of the disks given by (r1+r2,r2). Because the function for tr is continuous as Δd changes (note that the denominator 1+q is never zero), we can assure a continuous range of ratios between the values given by substituting the limits of Δd into the tr function. For Δd of 0, this gives tr = α, the ratio of disk radii. For Δd = r1/2, tr = (1+2∙α)/3. So, for instance, if we have α = 2, we can adopt transmission ratios in the range (5/3,2). If α = 10, then we have the range (7,10). We can express the dynamic range (tdr) between the limits as a ratio to α: [α - (1+2∙α)/3 ]/α; or more compactly:

tdr = (α-1)/3α. Consequently, if we need to design for a dynamic range that is a proportion k of the maximum tr, we can find an α which satisfied that constraint easily by equating, which works out to be α = 1/(1-3k). After which selection, additional gearing may be applied to adjust the net output range.

One of the things to notice is that this transmission has asymptotic limits on the net displacement ratio- tdr can range from 0 (at the limit of α = 1) to 1/3 as α goes to infinity. If we are designing for a dynamic range less than 33%, then we can use output gearing instead to shift the set points of the transmission. For instance, say we need a 15% dynamic range. This gives us an α of ~1.82, for a tr range of (1.55,1.82). Then, we can apply fixed gearing which adjusts the net output of the whole system by a factor of G, for output range (1.55∙G,1.82∙G). So if we want a range of (2.55,3), we can take G = 1.65, which keeps our dynamic range at 15%, and shifts the actual ratios to the desired values.

To ameliorate this limitation (which is really an expression of the constraint that Δd be mirrored on each disk), we may consider modifying the mechanism so that the armature support is no longer blocking the movement of the smaller disk (other stabilizing mechanism do exist that could make up for this). Then the maximum value Δd can take is r1, instead of r1/2. If we recalculate the range, then tr can range between α and (α/2) + 1/2, which admits a tdr of

(α- 1)/2α, which, naturally, has an asymptotic limit of 50%. This upper limit to the dynamic range is the expense of the elimination of the clutch, because the smoothness requirement imposes the mirror movement on the position of the transmission disks. It does seem intuitively satisfying that splitting the movements equally would cause a 50% loss in dynamic range. The same logic applying to fixed output gearing to adjust net range applies to this variant of the mechanism, of course.

Having analyzed the transmission properties of the device, let's now turn our attention to the practical implementation. The machine described above bears the core mechanism, but lacks well defined control of the distance between the disks, a mechanism to ensure the equal displacement of the disks, and cogent input and output terminals. In particular, we would rather the positions of the rotational axis of the terminals not actually move, so that all of the disk-shifting occurs within the transmission itself. We'll address all these parts with some combined structures.

First, we're going to look at ensuring that the disk separations Δd are equal. To achieve this, we're going to use an armature with prismatic joints on a track, fixed to rotate about the central point at which the sphere is held:

tdr = (α-1)/3α. Consequently, if we need to design for a dynamic range that is a proportion k of the maximum tr, we can find an α which satisfied that constraint easily by equating, which works out to be α = 1/(1-3k). After which selection, additional gearing may be applied to adjust the net output range.

One of the things to notice is that this transmission has asymptotic limits on the net displacement ratio- tdr can range from 0 (at the limit of α = 1) to 1/3 as α goes to infinity. If we are designing for a dynamic range less than 33%, then we can use output gearing instead to shift the set points of the transmission. For instance, say we need a 15% dynamic range. This gives us an α of ~1.82, for a tr range of (1.55,1.82). Then, we can apply fixed gearing which adjusts the net output of the whole system by a factor of G, for output range (1.55∙G,1.82∙G). So if we want a range of (2.55,3), we can take G = 1.65, which keeps our dynamic range at 15%, and shifts the actual ratios to the desired values.

To ameliorate this limitation (which is really an expression of the constraint that Δd be mirrored on each disk), we may consider modifying the mechanism so that the armature support is no longer blocking the movement of the smaller disk (other stabilizing mechanism do exist that could make up for this). Then the maximum value Δd can take is r1, instead of r1/2. If we recalculate the range, then tr can range between α and (α/2) + 1/2, which admits a tdr of

(α- 1)/2α, which, naturally, has an asymptotic limit of 50%. This upper limit to the dynamic range is the expense of the elimination of the clutch, because the smoothness requirement imposes the mirror movement on the position of the transmission disks. It does seem intuitively satisfying that splitting the movements equally would cause a 50% loss in dynamic range. The same logic applying to fixed output gearing to adjust net range applies to this variant of the mechanism, of course.

Having analyzed the transmission properties of the device, let's now turn our attention to the practical implementation. The machine described above bears the core mechanism, but lacks well defined control of the distance between the disks, a mechanism to ensure the equal displacement of the disks, and cogent input and output terminals. In particular, we would rather the positions of the rotational axis of the terminals not actually move, so that all of the disk-shifting occurs within the transmission itself. We'll address all these parts with some combined structures.

First, we're going to look at ensuring that the disk separations Δd are equal. To achieve this, we're going to use an armature with prismatic joints on a track, fixed to rotate about the central point at which the sphere is held:

Here, the red dot in the center represents the location of the sphere, the green line is the armature, the turning of which about the fixed revolute joint at the red dot shifts the disks. The orange curves are circular sections, and the path of the tracks on which the prismatic joints that affix the rotation shafts of the disks to the track. When the armature (green) is rotated, the prismatic joints force the disks to displace relative to one another:

The curves are constructed by overlapping a pair of circles such that the distance at the initial point (disks closest together) corresponds to the greatest Δd. Here, note that since the curves are 180 degree rotations about the sphere center point, the movements are mirrored at the point of prismatic contact, which allows satisfaction of the Δd criterion. However, they need not be circular to do so- the circular component will actually help us out with other components.

Speaking of which, we've taken care of the issue to enforce the Δd mirroring, now we're going to see to the output. Because we forced the disks to move in circular paths when we specified the distance tracks, we now have the opportunity to place the output drivers at the center points of the circles from which the tracks were derived (purple squares on the above images). Because the axis of the transmission disks are constant distances from these centers, we can attach a fixed pair of gears- one on the transmission disk shaft, and one at the center of the track circle. As the transmission disk spins, the axis will transmit movement to these output gears, and as the distance control armature moves, the gear will simply shift position around the completely fixed output gear:

Speaking of which, we've taken care of the issue to enforce the Δd mirroring, now we're going to see to the output. Because we forced the disks to move in circular paths when we specified the distance tracks, we now have the opportunity to place the output drivers at the center points of the circles from which the tracks were derived (purple squares on the above images). Because the axis of the transmission disks are constant distances from these centers, we can attach a fixed pair of gears- one on the transmission disk shaft, and one at the center of the track circle. As the transmission disk spins, the axis will transmit movement to these output gears, and as the distance control armature moves, the gear will simply shift position around the completely fixed output gear:

These images above represent the final construction. The Green and Red gears are affixed to the shafts of the transmission disks in contact with the central sphere. The yellow and purple gears are fixed in place, and the green and red gears, respectively, transmit the motion from the disks to these fixed gears, which constitute the input and output. Furthermore, because the distances from the center of each fixed I/O gear to the corresponding shifting disk shafts are fixed, desired output gearing to change the set points, such as that described in the transmission analysis, can be applied here, within the main body of the device. The principal control element is the armature (light green in the above right) which can be coupled to a planetary gear and driven by a stepper motor or other high precision device.

The final dimensions of the device can be controlled by adjusting the radii of the disks, as all calculations were performed in terms of the ratios thereof. Additionally, The sphere securement may be coupled to the armature or the chassis, depending on clearance, and due to the layout, the alternative version which removes part of the r1 shaft to increase dynamic range is also applicable in this design. I can also imagine a number of other improvements to increase efficiency, such as rolling spherical tensioners to assure good friction between the disks and the sphere, and spherical bearings in the sphere support, to reduce friction.

The last bit of commentary I'd like to make is to repeat the assertion of the limitations of the device- it's continuously variable, not infinitely variable. Additionally, the dynamic range of transmission ratios is limited by the nature of the kinematic restraints- the device is continuously variable, but the price of that smooth operation is reduced range. The reason I end on limitations is because often little concern is payed to the costs associated with some technical inventions, and I feel it is important to make clear that the benefits are, as always, offset.

The final dimensions of the device can be controlled by adjusting the radii of the disks, as all calculations were performed in terms of the ratios thereof. Additionally, The sphere securement may be coupled to the armature or the chassis, depending on clearance, and due to the layout, the alternative version which removes part of the r1 shaft to increase dynamic range is also applicable in this design. I can also imagine a number of other improvements to increase efficiency, such as rolling spherical tensioners to assure good friction between the disks and the sphere, and spherical bearings in the sphere support, to reduce friction.

The last bit of commentary I'd like to make is to repeat the assertion of the limitations of the device- it's continuously variable, not infinitely variable. Additionally, the dynamic range of transmission ratios is limited by the nature of the kinematic restraints- the device is continuously variable, but the price of that smooth operation is reduced range. The reason I end on limitations is because often little concern is payed to the costs associated with some technical inventions, and I feel it is important to make clear that the benefits are, as always, offset.