# Helmholtz T.F. III

## Masterpiece design x=2a
Restoring force f1(x) and driving force f2(x)

Electromagnetic attractive force as a function of AC current I、where μ=infinity is assumed for iron for simplicity.1

1) Shota Miyairi, Electro-mechanical energy conversion, Maruzen, p-25, in Japanese. Bending beam approximation Conceptual drawing Vibration of the diffuser calculated by Runge-Kutta method2

2) W. Thomson, The theory of vibration, Nelson Thornes, p-479.  Current (A)
Applied current and tine vibration amplitude

Equation of motion for the tine of tuning forks

• Given tine position 0<x<d, whereｄis the neutral gap length between the tine and the magnet taken at 1mm, the attractive force on the tine f2(x) is proportional to the squared applied AC current on the coil divided by x.
• Restoring force on the tine f1(x) is given by s(d-x) using the stiffness s calculated from the bending beam model Setting parameters N=800(T), I=0.65(A),and S=b2, a plot of f2(x) is shown in right, where the restoring force f1(x) is also shown as a straight line with s= 75,600 (N/m) calculated from E=20×1010 Pa, t=3mm, b=7mm, l=50mm.

With such proper choice of parameters, since f1(x) and f2(x) have cross point at d-x=2a, where restoring force f1(x)< driving force f2(x) always holds at any d-2a<x<d, the nonlinear equation of motion is linearized for this small amplitude 2a,

Temporal evolution of x, where f1(t)=(1-cos2ωt)/2<1
Starting at x=d, when Mdv/dt+Rv v=f1(x)-f2(x)ft(t)<0, x tends to decrease with time until f1(x)=f2(x) at x=d-2a.

Note: At x=2a=150 μm. where f1(d-2a)=10N~1kgf which is the maximum force on the sound post.