Tuning

An ultrasonic stack is designed to operate in a specified mode (e.g., axial). The stack will only operate effectively if it is driven at the resonant frequency of that mode. If it is driven slightly away from its resonant frequency then the amplitude will be severely reduced. This is a consequence of the high Q of the stack.

Most ultrasonic power supplies are designed to operate over a fairly limited frequency range. For example, a 20 kHz power supply might be designed to operate between 19.5 kHz and 20.5 kHz. This is done for two reasons.

An ultrasonic stack is typically designed to operate only in their primary mode (e.g., axial). However, complex resonator stacks may have many undesired (secondary) modes (sometimes called parasitic modes). To prevent the power supply from accidentally starting on one of these undesired modes, it is designed to operate only within an allowed frequency window, even though it might otherwise operate acceptably outside this window.
An ultrasonic stack only operates effectively if it is driven at the resonant frequency of the primary mode. If it is driven slightly away from that frequency then the amplitude will be severely reduced. This is a consequence of the high Q of the stack.

Tuned dimension

A resonator's tuned dimension will depend on five factors —

Material properties. These include the modulus of elasticity (Young's modulus) and density which, together, determine the thin-wire wave speed. Also, Poisson's ratio becomes important for resonators with large lateral dimensions. (Note that the material properties generally depend on the operating temperature.)
Resonator geometry. These include exterior and interior geometries, slots, chamfers, radii, wrench flats, etc.
Vibration mode. Vibration modes include longitudinal (axial), radial, flexural, torsional, and combinations thereof.
Number of half-waves. In order to be resonant the resonator must have at least one half-wavelength. (exception - folded transducer). However, multiple half-waves are possible.
Added elements. These include studs and wear surfaces (e.g., carbide).

Simplest resonator — longitudinal thin wire

For a thin (slender) wire or rod, the following equation gives the longitudinal half-wavelength \( \Gamma \) in terms of the thin-wire wave speed \( C_O \) and the resonant frequency \( C_O \) —

\begin{align} \label{eq:10100a} f_r\propto \sqrt{\frac{k}{m} } \end{align}

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A resonator's dimensions may need to be adjusted in order to tune the resonator to the desired (primary) frequency. These dimensional adjustments are needed in the following circumstances.

In computer simulations (FEA) the initial estimate of the resonator's dimensions will probably not be correct.
Even if a resonator has previously been tuned by computer simulation, the equivalent machined resonator may not be correctly tuned because:
1. The material properties that were used in the simulation may not have been correct. For example, the simulation may have used generic handbook material properties which may not account for such factors as processing, grain direction, stock size, heat treatment, anisotropy, etc. For instance, see titanium.
2. The simulation may not be accurate because of assumptions. (for example, when the multiple components in a transducer are assumed to be rigidly attached to each other).
Tuning is required because of variability in the material's properties (see titanium)

For a lumped spring-mass system with negligible energy dissipation, the resonant frequency is given by equation \eqref{eq:10101a}.

\begin{align} \label{eq:10101a} f_r\propto \sqrt{\frac{k}{m} } \end{align}

where —

\( f_r \)	= Resonant frequency
\( k \)	= Stiffness
\( m \)	= Mass

Although equation \eqref{eq:10101a} strictly applies only to a lumped spring-mass system, it can also be applied in conceptually to vibrations of continuous-media systems such as ultrasonic resonators. Unlike the illustrated lumped spring-mass system, an ultrasonic resonator can have many vibration modes. Then equation \eqref{eq:10101a} can be written separately for each mode \( i \).

\begin{align} \label{eq:10102a} f_{ri}\propto \sqrt{\frac{k_i}{m_i} } \end{align}

For the axial mode or if the useage of equation \eqref{eq:10102a} is unambiguous, the \( i \) subscript may be omitted.

From the energy method by which equation \eqref{eq:10101a} was derived, \( k_i \) is associated with regions of the resonator that have relatively high potential energy (i.e., regions with significant stretch-strain); hence, these regions are conveniently called "strain regions". Similarly \( m_i \) is associated with regions that have relatively high kinetic energy (i.e., regions with significant velocity); hence, these regions are conveniently called "velocity regions".

Generally the strain regions will be centered at a node. However, for flexure these regions will be centered at antinodes where the strain is highest.

Methods

The following will discuss how tuning is affected by removing material as with typical machining. The opposite results would apply if material were added (for instance, using FEA, by adding a carbide face insert, etc.). For the time being the following will neglect any effect from any attached resonators. For example, any discussion of tuning a horn will neglect the effect of an attached booster or transducer.

Equation \eqref{eq:10102a} shows that removing material from a strain region will decrease the frequency (i.e., the stiffness \( k \) in the numerator decreases so the frequency decreases). Conversely, removing material from a velocity region will increase the frequency (i.e., the vibrating mass \( m \) in the denominator decreases so the frequency increases). If material is simultaneously removed from regions that have both substantial strain and velocity then the frequency may either increase, decrease, or remain unchanged, depending on whether the strain or velocity predominates.

Allowance for hardening

Steel horns are often hardened in order to increase wear resistance. However, these horns are generally tuned while soft (unhardened) which allows easier machining, especially if the horn has any face features. Then some allowance must be made for any frequency shift \( \delta{f} \) that occurs after hardening.

For example, when a 20 kHz D2 tool steel horn is hardened to R_c 55‑60 the frequency drops 100 ‑ 300 Hz, depending on the particular horn geometry. Then, rather than tuning to 20 kHz (nominal) the tuning specification might state something like, "Tune to 20200 Hz prior to hardening."

Allowance for fixed face features

Medical probes. Spool horns. Contoured horns.

Allowance for adjacent face features

Features close to the face: Slots, flutes --> Put flutes in afterwards.

Considerations if also adjusting for performance (uniformity) --> affected by slot length

Allowance for added entities

Allowance must be made for heavy entities such as carbide inserts. The compensation is exactly proportional to the mass of the entities. For example zzz

Some minor compensation may be needed for wear coatings like D‑gun. No compensation is needed for thin coatings like chrome and nickel.

Tuning with a fixed resonator length

Tuning from the resonator's face is usually easiest. However, in some cases a resonator's length must be fixed.

Medical probes with fixed heads

If must fit into pre-existing enclosure where no positioning adjustment is permitted

Tip horns on composite horns for plastic welding (tip horn's length must be fixed with respect to adjacent tip horns in order to properly contact the plastic part).

Tuning when the initial frequency is too high

Cut in nodal region - banding, pushing nodal radius

Batch tuning

Boosters. E variation along bar length. Non-critical resontors (boosters)

Composite horns

Tip horns --> tight tuning spec to avoid double axial resonances

Initial prototypes vs production resonators

Tuning rate

For a given mode the tuning rate (rate of frequency change) (Hz/mm) - axial, flexure, torsion, shear

Tuning rate depends on gain. Depends on attached components.

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zzz Flexure modes induced from axial vibration - if cut OD or thick, will depend on where these cuts occur.

Fatigue crack -- no material removed. Stress is relieved. Therefore, f must drop.