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

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.

Oxide Corporation

Speckle contrast Measurement System