Passive mechanical element that can act as counterforce for a spring

An elavator is a moving cabin that gravity acts on. On the scales of the typical elevator application (a skyscraper for example), gravity is constant. So the gravitational force can be counteracted by a counterweight, and the whole elevator can be moved with comparably little force.

Is there a similar thing to counteract the force that a spring exerts? The inherent problem I see here is that springs have a position dependent force (with a negative proportionality constant), that needs to be counteracted with a position dependent force with a positive proportionality constant: To illustrate what I mean, I appended two figures:

In the first image, we can see what happens if I simply connect two springs whose equilibrium position are displaced from each other. The result is the displacement - force dependence of another spring, that simply has a bigger spring constant, and an equilibrium position between the two beforementioned springs.

Picture two shows the possible effect of the fictious element that I'm looking for, which flattens the resulting overall force curve. The combined system then still has an equilibrium position that it tends to, but the force needed to displace the system from that position are mutch smaller.

Some notes for clarification, as mechanical systems can be abitrarily complex, and some comments pointed out that the question is unclear:

The combined system of the spring and device X that I'm looking for should again have 1 degree of freedom. Displacements in this degree of freedom should result in the same displacement of the spring: There should not be any leverage. Displacing this degree of freedom should however require "less" force in some working range. Here, by "less" it is meant that the magnitude of the force is smaller than highest force in the working range of the original spring. The new combined system may or may not have a new equilibrium position. In case there is one, the force per displacement to displace the system out of that position should be smaller.

Is there such a device? Or a whole class of such devices?

To give context: I'd like to build a (Geigenwerk), and it requires the force of the player to lift the strings off the bowing wheels. Displacing the strings, they behave like a spring, and to ease the players efforts, the force needed to displace them has to be diminished. Leverage is not an option, as in that case the player will need to displace the keys more, and thus the instrument as a whole will be less responsive.

• there are such things as constant force springs, you see these quite a lot in cars and anywere where you want to lift something. But to answer your question is impossible since its so opaque. Typically you can adjust the force curve with a a suitable fourbar. Jul 18, 2023 at 20:44
• It is a bit unclear why you need to counteract a spring while changing it to one preloaded weaker spring might suffice? My point is without context this becomes a shot in the dark. Jul 18, 2023 at 23:17
• @joojaa the aim is to build a Streichklavier (Geigenwerk, Bowed Clavier). Inside, mechanisms are needed to lift strings of a bowing wheel. A string in that regard behaves exactly like a spring. I want to reduce the amount of force one needs to exert to lift the string of the wheel / put it back on as mutch as possible. Levers are not an option, because the displacements one does while exerting the force should also be kept small. Jul 18, 2023 at 23:27
• If you want to keep the lever displacement small then use compound levers. Jul 19, 2023 at 6:03
• @joojaa I gave the cams in a compound bow as an example for a mechanism that, at least in some working area, exhibits a force distance relationship different from a spring, to give the reader a better picture of what I'm looking for. I have thought about this question for some time, now I'm giving it too the internet, to get answers that I obviously couldn't come up with myself. Besides that I still think it's an interesting question (although not well posed, I give you that). What's wrong with that? Jul 19, 2023 at 10:15

Ok, what you are finally asking is bit tricky, in not going to say impossible since its probably not but just a lot of acoustics work. So your in your last addition you reveal that you want to have less force, but your not willing to trade more distance for it. This is a bit problematic. Why?

Because there is a constitutive relationship between force and energy. You want to keep the energy converted to sound same. So your energy requirement is fixed. On the other hand force and distance is energy, if you lower force you need to increase distance to get the same energy. This excess energy has to come from somewhere.

No cams wont help, they are just levers and they add damping to your system which is probably not what you need. (See the comment on counterweights below)

Now you can possibly sidestep this but you have to change the entire structure of your clavier because you need to make it do more passive sound amplification.

PS: The lift analogy is a misguided red herring since the tradeoff is that the elevator now is harder to accelerate, its only beneficial if the lift is moving mostly at constant speed. If the lift were to accelerate 50% of time and decelerate 50% of the time it would be counter productive to have a counterweight. In your case your only doing acceleration so adding inertia to the system does not give you anything. Which is why counterweight like objects wont help you. Force isn't everything.

• I know about the force distance energy relationship, but this isn't a showstopper. The combined system that I'm looking for simply has to store less energy when it's displaced. The elevator for example ( let's say it's in perfect balance) doesn't store any energy when it is moved. While moving it, the engine has to do work only against the friction, and this friction is lost. It is true that with more parts in the mechanism the inertia get's bigger, but geigenwerke are operated at slow speeds, and the inertia is not the limiting factor here. Jul 19, 2023 at 10:09
• In the case of a geigenwerk, force is the interesting quantity, because the player needs to apply it with his fingers in a healthy way, often and repeatedly. Jul 19, 2023 at 10:11
• @Quantumwhisp yes but the energy of the system needs to come form somewhere, you can probably somehow transfer the energy from the pedals into the string somehow. But not your probably not the first person to encounter this problem, im pretty sure the original designers or someone who came after them thought about this. You can just adjust pre extansion of the spring to satify your graph. So your graph no problem, but i dont think it will work in your application Jul 19, 2023 at 10:15
• Mainly because the spring is what causes the sound. So lets say you have a lever, but that lever acts as a clutch on a rotating element that provides the final string energy. Or the rotation loads a spring/counterweight/rotating mass and you take energy out of there instead. But now we are at active terretory Jul 19, 2023 at 10:22
• Bowed claviers work the following way: There is a rosined wheel turning. All strings in contact with the wheel produce sound. They gain their energy from the turning wheel. It's the job of the player to lower the strings to the wheel, (simplified) this is where the spring like reaction force of the string has to be counteracted by the player, for every note he dares to play. Reducing the force needed to lower the string to the wheel is the application I have in mind. The energy needed to displace the string doesn't take a role in soundproduction at all. Jul 19, 2023 at 12:22