On the use of a nickel-tube resonator for measuring the complex shear modulus of liquids in the kHz-range

Summary An apparatus for the measurement of liquid-shear impedance in the frequency range 4–200 kHz with the aid of a thin-walled Ni-tube resonator is described. A magnetostrictive mechanism is used for setting the tube into torsional oscillation. Real and imaginary parts of the liquid-shear impedance are found from the change in the 3 dB band-width of the resonance curve and the shift of the resonance frequency, respectively, when the tube is immersed from the air into the liquid. The amount of liquid required is 20 ml. The necessary theory is given and some preliminary results are presented.

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Zusammenfassung Es wird über einen Apparat zur Messung der Scherimpedanz von Flüssigkeiten im Frequenzbereich von 4 bis 200 kHz mit Hilfe eines Resonators berichtet. Der Resonator, ein Nickelrohr mit geringer Wandstärke, wird mittels eines magnetostriktiven Mechanismus in Torsionsschwingungen versetzt. Real- und Imaginärteil der Scherimpedanz der Flüssigkeit werden aus der Änderung der Bandbreite der Resonanzkurve und der Verschiebung der Resonanzfrequenz, wenn das Rohr aus der Luft in die Flüssigkeit eingetaucht wird, berechnet. Die benötigte Flüssigkeitsmenge beträgt 20 ml. Die zugehörige Theorie wird mitgeteilt, und einige vorläufige Meßergebnisse werden vorgestellt.

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a,b outer and inner radii of the tube - 3 dB band-width of resonance curve of the loaded and unloaded tube - c,c ₀ phase velocities of torsional waves in the loaded and unloaded tube - c ^*,c ₀ ^* complex velocities of propagation of torsional waves in the loaded and unloaded tube - c ^* =c ₁ + ic₂ - c _L ^* complex velocity of propagation of shear waves in the fluid - f frequency - f _n, f _n ⁰ resonance frequency ofn-th overtone of the loaded and unloaded tube - G ^* complex shear modulus of the tube material - G _L ^* complex shear modulus of the fluid - i - I ₀ moment of inertia per cm of unloaded tube - I effective moment of inertia per cm of loaded tube - K instrument constant - l length of the tube - m =b/a - M angular momentum - n positive integer - R _L, X_L real and imaginary parts of the characteristic plane shear impedance of the fluid - Z _L characteristic plane shear impedance of the fluid (=R _L + iX_L) - Z _cyl characteristic cylindrical shear impedance of the fluid - , ₀ damping factors of torsional waves in the loaded and unloaded tube - , ₀ phase factors of torsional waves in the loaded and unloaded tube - propagation constant (= + i) - loss angle - tan cos^–1 (/ _max) - viscosity - angular displacement - wave length - displacement amplitude - , ₀ densities of the fluid and the tube material - angular frequency (= 2f) - _n ⁰ = 2f _n ⁰ - _R reduced frequency With 4 figures and 2 tables