Considerations on the connectivity of magnetic springs to spring constants

0. References

[1] https://www.jstage.jst.go.jp/article/trbane/2003/48/2003_48_39/_pdf/-char/ja "Vibration Isolation Mechanisms Using Magnetic Springs and Their Applications."

This is the same as the one I pointed out earlier (late July 2021) in the discussion of the issue.

[2] The URL to the DMM.make website is pasted after.

(Update: August 25, 2021, 21:32　

Magnetic spring constant measurement device, bottom.stl - DMM.make Creators' Market https://make.dmm.com/item/1347763/)

[3] https://ryebourbon.xsrv.jp/spring-gousei/ What are synthetic spring constants? I'll show you how to deal with series and parallel springs!

1. About spring constants of magnetic springs

According to reference [1], the equation for the load characteristics of a magnetic spring is as follows.

(Equation 1.1)

2. Spring constant measuring instruments and their usage

We made the following spring constant measuring device with a 3D printer. It is not available to the public at the moment, but we will sell the measuring instrument on DMM.make later. This product, according to an important theorem in physics (see below), is a necessity for determining the spring constant!

As shown in the figure, two magnets can be made to repel each other to make a platform float. When a certain load is applied to the platform, it moves closer by a certain distance. Equation (1.1) expresses this property.

How to use

First, prepare small objects (e.g. writing utensils) of a size that can be placed on the table to apply two different loads.

Next, measure their masses with an electronic weighing scale. You can use an electronic scale to measure the mass of salt, sugar, etc., as you would in your kitchen.

Now, apply the load. You will see that it takes different displacements for different masses. Let's measure the two displacements that have changed. (For example, you can measure them from the px of the image.)

3. How to organize data

The following equations (3.1) and (3.2) are valid for load 1 and load 2, respectively. Here, m_1 and m_2 represent the mass measurement results of Load 1 and Load 2, g represents the gravitational acceleration of 9.81, and x_1 and x_2 represent the displacement when Load 1 and Load 2 are applied respectively.

By solving the above simultaneous equations, we can find the unknowns k_1 and k_3. By the way, when solving such simultaneous equations, I recommend using a computer. It can be done in a surprisingly short time.

4. Displacement as a function of spring constant

In high school physics, we learned that when making series or parallel connections, it is easier to understand if we rewrite the equation as (displacement) = (function of spring constant). If you are like me and have completely forgotten this, please check out reference [3]. I've been looking at it too, and thanks to it, I've been able to apply it here.

The scary thing about equation (3.1) and equation (3.2) is that they are cubic equations for x. When we solve it, we get the following However, x is a real number. (i=1,2)

The above results reveal a surprising fact!

"Even if the behavior of a series-connected magnetic spring coincides with that of a magnetic spring with a different spring constant, each spring constant cannot be determined from the spring constant of another original magnetic spring before the connection, since there are two different spring constants." (Theorem 4.1)

This means that the spring constant of a series-connected magnetic spring cannot be determined without experimental values.

This is why a spring constant measuring device is useful.