I seem to remember that a wrong side failure of tilt (like you mention) was one of the problems the APT systems had to resolve. A wrong side tilt failure would take the APT out of gauge.
It wouldn’t. Fully tilted either way it must be in gauge.
They do it with cant deficiency
cant is the amount the outer rail is higher than the inner rail on a curve measured in degrees.
Passengers in a train going round the curve at the right speed will feel no lateral acceleration
If you go a little faster, the passengers will start to feel lateral acceleration. you could measure that with a plumb line and measure the angle between the uprights in the coach and the pendulum. or use an accelerometer.
That angle is cant deficiency. the amount the track isn't canted to cope with the curve at that speed.
You can remove that cant deficiency by tilting. The APT ran into trouble as passengers felt sick when their perception of up differed from what they saw outside.
You cant tilt too much because of slow trains (esp freight) that cant cope with large amounts of cant.
I think the original rule for cant deficiency was 8 degrees. But when the LM wanted to do 110mph trains they started to use 'relaxed curving rules' and 9 degrees.
The ER had a different tactic, got out bulldozers and straightened out curves.
Cant (or superelevation) is the amount the track is elevated on the high rail. For a given curve radius, and a given amount of cant, there will be a balancing speed where a passenger would feel no lateral acceleration. The equilibrium cant (E) is 11.82 x the square of the speed (in km/h) / the curve radius in metres.
Maximum cant applied (in this country) is 150mm, roughly six inches, which by coincidence is also roughly 6 degrees. Exceptionally you can go to 180mm.
Cant deficiency is the theoretical amount of additional cant that would need to be applied to achieve an equilibrium for a given speed. It is usual practice to have less cant deficiency than cant; I was taught that good practice was 2/3 cant 1/3 deficiency, but it’s a lot more complicated than that. You can go up to 50:50.
However there are also maximum levels of cant deficiency, this is 110mm (4.5inches), exceptionally 150mm (6 inches). This is what the passenger experiences as lateral force; if you go round a curve with 150mm deficiency you will know it. I referred earlier to research in BR days that enabled the exceptional values to be used.
For tilting trains, maximum cant deficiency is 265mm. The difference between that and the normal cant deficiency values is compensated for by the tilt system.
All the above apply to CWR only - there’s lower values for jointed tracks, and there are variations for short radius curves and other factors.