# Helmholtz T.F. III

## Masterpiece design

x=2a

Restoring force f_{1}(x) and driving force f_{2}(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 method^{2}

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 f
_{2}(x) is proportional to the squared applied AC current on the coil divided by x. - Restoring force on the tine f
_{1}(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=b^{2}, a plot of f_{2}(x) is shown in right, where the restoring force f_{1}(x) is also shown as a straight line with s= 75,600 (N/m) calculated from E=20×10^{10} Pa, t=3mm, b=7mm, l=50mm.

With such proper choice of parameters, since f_{1}(x) and f_{2}(x) have cross point at d-x=2a, where restoring force f_{1}(x)< driving force f_{2}(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 f_{1}(t)=(1-cos2ωt)/2<1

Starting at x=d, when Mdv/dt+Rv v=f_{1}(x)-f_{2}(x)ft(t)<0, x tends to decrease with time until f_{1}(x)=f_{2}(x) at x=d-2a.

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