Nonlinear Interactions

  • Diethelm Johannsmann
Part of the Soft and Biological Matter book series (SOBIMA)


If the stress at the resonator surface is not proportional to the displacement (if there is nonlinear response), the frequency shift depends on the amplitude of oscillation. Nonlinearities can also be detected in the form of steady forces, steady flows, second-harmonic signals, and third-harmonic signals. The chapter gives particular emphasis to the amplitude dependence of the frequency shift. It is analyzed in the frame of a time-averaged periodic stress inserted into the SLA. An application example is the study of partial slip.


Steady Flow Frictional Heating Amplitude Dependence Partial Slip Resonator Surface 
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Definition (Comments)

\( \left\langle \ldots \right\rangle \)

An average


As an index: amplitude of a real-valued time-harmonic function


Instantaneous amplitude (Eq. 13.3.6)


(Effective) area of the resonator plate




A body force (Eq. 13.2.1)


Damped resonance frequency


Resonance frequency at the fundamental (f 0 = Z q /(2m q ) = Z q /(2ρ q d q ))


Tangential force


Force at interface while u S increases (+) and decreases (−)


Normal force


Maximum force in a force-displacement loop


Force exerted onto a resonator by a sample


Tangential force


Mass per unit area of the plate (m q  = ρ q d q  = Z q /(2f 0))


Mass of a resonator


Number of particles per unit area


Particle radius




As an index: reference state


As an index: Surface


As an index: steady




Period of oscillation


Glass temperature


As an index: unsteady



ûS, uS

Tangential displacement


Maximum displacement in a force-displacement loop


Normalized displacement (u N  = u S /u 0)


Amplitude of oscillation (equal to the absolute value of û S )



x, y, z

Spatial coordinates


Displacement of a simple harmonic resonator


Auxiliary variable (Eq. 13.3.3)


Acoustic wave impedance of AT-cut quartz (Z q  = 8.8 × 106 kg m−2 s−1)


Ratio of the normal and the tangential component of the plate’s motion (depends on position, x and y; Sect. 13.2)


Damping factor


Imaginary part of a resonance frequency


Penetration depth of a shear wave (Newtonian liquids: δ = (2η liq /(ρ liq ω))1/2)


As a prefix: A shift induced by the presence of the sample


Phases (Eq. 13.2.3)




Spring constant

κ1, κ2, κ3

Coefficients in a power series of force versus displacement (κ1 is the conventional spring constant)


Instantaneous phase (Eq. 13.3.6)


Friction coefficient


Dynamic friction coefficient




Angular frequency


Damped angular resonance frequency


Undamped angular resonance frequency


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Physical ChemistryClausthal University of TechnologyClausthal-ZellerfeldGermany

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