Nonlinear Interactions

Chapter
Part of the Soft and Biological Matter book series (SOBIMA)

Abstract

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.

Notes

Glossary

Variable

Definition (Comments)

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

An average

0

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

a

Instantaneous amplitude (Eq. 13.3.6)

A

(Effective) area of the resonator plate

f

Frequency

fbody

A body force (Eq. 13.2.1)

fd

Damped resonance frequency

f0

Resonance frequency at the fundamental (f0 = Zq/(2mq) = Zq/(2ρqdq))

F

Tangential force

F+,F

Force at interface while uS increases (+) and decreases (−)

FN

Normal force

Fmax

Maximum force in a force-displacement loop

Fsam

Force exerted onto a resonator by a sample

Fx

Tangential force

mq

Mass per unit area of the plate (mq = ρqdq = Zq/(2f0))

MR

Mass of a resonator

NP

Number of particles per unit area

RP

Particle radius

p

Pressure

ref

As an index: reference state

S

As an index: Surface

st

As an index: steady

t

Time

T

Period of oscillation

Tg

Glass temperature

us

As an index: unsteady

u(t)

Displacement

ûS, uS

Tangential displacement

umax

Maximum displacement in a force-displacement loop

uN

Normalized displacement (uN = uS/u0)

u0

Amplitude of oscillation (equal to the absolute value of ûS)

v

Velocity

x, y, z

Spatial coordinates

x

Displacement of a simple harmonic resonator

y

Auxiliary variable (Eq. 13.3.3)

Zq

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

α

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

γD

Damping factor

Γ

Imaginary part of a resonance frequency

δ

Penetration depth of a shear wave (Newtonian liquids: δ = (2ηliq/(ρliqω))1/2)

Δ

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

φpv

Phases (Eq. 13.2.3)

η

Viscosity

κ

Spring constant

κ1, κ2, κ3

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

ϕ

Instantaneous phase (Eq. 13.3.6)

μ

Friction coefficient

μD

Dynamic friction coefficient

ρ

Density

ω

Angular frequency

ωd

Damped angular resonance frequency

ω0

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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