OldTony

I am wondering why you think the motor will have the full supply voltage connected to it. In other words why you think the volts drop across the capacitor plus the volts drop across the transformer primary winding will equate to zero.

Take it or leave it One over two Pi root L C equals resonant frequency

Now you have me worried. My birthday is the 23rd of next month and perhaps me and anybody who responds to this thread might drop dead two days before Christmas. More seriously I think that if there was no such thing as a coincidence the word "coincidence" would not exist. BTW Although the odds against it are huge the possibility that I and anyone who responds to this thread will die on my birthday does exist

If you google capacitance multiplier circuit you will see that your idea is sound although you need to make a few changes to your circuit. To make the output variable you could add a potentiometer.

To get to grips with this circuit you need to understand the terms reactance, resistance and impedance. You then need to estimate how the properties of each circuit component fits together to find the current flowing around the circuit and its phase relationship with the generated voltage. Knowing there will be voltage drops only across the resistance component you can then find the voltage drop across each part of the circuit. Unfortunately unless the motor is running at a constant speed attached to a constant load all these parameters will be shifting. Unfortunately I feel if you can get someone to make some analysis of the circuit I feel you will just tell them they are wrong.

Looking at your diagram ( Agust 3rd 2024) it appears to me that you are trying to use the inductance of the transformer primary and the capacitor as a series tuned circuit at resonance. In such a circuit the voltage across those two components can easily exceed the input voltage. You then assume you can tap off some of that voltage to do useful work. However transformers can easily exceed 95% efficiency so the transformer with a resistive load will act as an almost pure resistor with a small amount of inductance. The net result of this is that any power delivered to the circuit by the generator will be shared between the resistor connected to the transformer secondary winding and that required to drive the motor. In other words I think that the addition of a transformer would reduce the circuit efficiency even if you achieved the circuits resonant frequency. Note power factor correction is usually achieved with capacitance connected in parallel with the inductive load. There are reasons for that, but that's another story.

Electronics is full of examples of feedback. However feedback cannot take you back in time. So the first signal round a feedback loop ignores the feedback loop and so that first time around can be very different to the very quickly established stable effect. This known as a transient condition. I'm not a philosopher but this seems relevant to me. It seems to indicate to me that feedback is in its way cause and effect.

It's a Cat's life – Perhaps 

 It's dark inside this little box
 in which they say a fatal ray
 has perhaps ended my life;
 but it's not for me to say.

 So do I live or am I dead?
 Strange as it seems it's up to you.
 I accept you have no choice
 although you know what you must do.

 “Bring some light into this place
 to look and see, Schrödinger said.
 "Because until you do as bid
 this cat's alive, but also dead.

You can expect a motor designed to provide more power than you need to be physically larger than one designed for a lower power. The main losses for an eletric motor are friction, windage and iron losses. In general these losses increase with the physical size of a motor. I would suggest you choose a motor that can provide the power you need plus, perhaps, a little more for longevity and reliability.

It so happens that I am a moderator on a completely different forum not to do with science. Assuming the arrangement here is much the same perhaps members should realise that moderators are carefully chosen for their knowledge and the benefit that they can give to their forum. They do not receive any reward for their time spent helping the forum to flow as harmoniously as possible except the occasional word of thanks. In short: it can sometimes be a thankless job. Just saying........

Mrs Tilley likes prose but not poetry

I'm having difficulty visualising a practical application as the two connecting rods seem to need to pass through the walls of their respective cylinders. Also although the horizontal piston moves faster than the vertical piston you still only get one revolution of the crank per stroke. I presume you are relying on momentum to carry the horizontal piston through its mid point. However perhaps, however rarely, due to a fault such as an engine misfire it could bounce back and reverse at the mid point which I can imagine would be catastrophic for an internal combustion engine.