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So, for a school project, I decided to investigate the SHM of a trampoline like system of springs. Basically, I took an ring, affixed eight springs (at equal angle from each other) to a central circular mass, and oscillated the central mass vertically and calculated the spring constant of these vertical oscillations (the effective force constant of the whole system) at different diameters. To my utter shock, I found that the effective force constant seems to vary at a quadratic rate with the radius. Initially, it decreases with diameter, hits a minimum value, then increases with diameter. According to the formula I derived, the variation should be hyperbolic, but my derived formula seems to fail.

Any formulae/help with explaining my experimental trend would be greatly appreciated!

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I assume from the context that "SHM" stands for Simple Harmonic Motion.

Let's look at your system:

Trampoline.

I'm assuming 8 identical springs of unstressed length $L$ with spring constant $k$ and the central material to be non-elastic.

To the right is the force diagram for one spring for a displacement $y$ of the centre of the trampoline.

$$L=R_0-R_1$$

$$(L(y)+R_1)^2=R_0^2+y^2$$

$$L(y)=\sqrt{R_0^2+y^2}-R_1$$

$$F=k(L(y)-L)$$

$$F=k\big(\sqrt{R_0^2+y^2}-R_0\big)$$

The $y$-component of $F$ is:

$$F_y=F \cos\theta$$ $$\cos\theta=\frac{y}{R_0^2+y^2}$$ $$F_y=k\big(\sqrt{R_0^2+y^2}-R_0\big)\frac{y}{R_0^2+y^2}$$

As there are springs, for a central displacement $y$ the trampoline delivers an upward force of:

$$F_{total}=8F_y$$

Now if we place a mass $m$ centrally on the trampoline we can write the Newtonian equation of motion as:

$$ma=m\frac{d^2y}{dt^2}=8F_y(y)$$

But due to the complicated nature of $F_y(y)$ this is not the equation of motion of a simple harmonic oscillator. Such a system would 'oscillate' but not in a simple sine or cosine mode.

I don't believe the $ma=8F_y$ equation has analytical solutions (Mathematica's DSolve yielded nothing useful, for instance). Perhaps numerical solutions for various radii might confirm your empirical conclusions.

Of course you could evaluate the influence of $R_0$ on $4F_y(y)$ but as the latter is also strongly dependent on $y$ in a complicated way that would not by itself allow to make pronouncements about the period of oscillation of the system.

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