I want to know the relationship between the frequency applied to the piezoelectric vibrator and the exit frequency

Question

We purchased a vibrator with frequency characteristics including the desired frequency. Now, to produce a sound with a single frequency, it is difficult to know how to determine the frequency of the inlet voltage. I think that the output frequency of the vibrator varies depending on the frequency of the inlet voltage, but I would like to ask you if there are any formulas, curves, or empirical formulas for the relationship between the two.

Answer 1

A piezo device is basically a tiny loudspeaker, which means the output frequency is equal to the driving signal frequency. So, if the input frequency is a 40kHz sine wave of AC electricity, it will radiate 40kHz sound waves.

Like any loudspeaker, the point at which the acoustic output will be biggest occurs when the piezo crystal is being electrically excited at its frequency of mechanical resonance, which depends on its physical dimensions and mass. This means that when you buy a piezo transducer, the manufacturer will specify what frequency it was trimmed for and that's where you drive it, to get the maximum output power.

There are formulas you can apply to approximate the resonant frequency of the piezo "slab" same as you would use to predict the resonant frequency of a closed-end tube of length L filled with air. In the case of the piezo slab, L is its thickness and the speed of sound you plug in is not the speed of sound waves in air but instead the speed of sound in the crystal material- which you would have to look up.

Answer 2

Typically, they will have the same frequency, because the emergent stress in the crystal is linear in the potential applied across it.

Specifically, the polarization $\mathbf{P}$ in a piezoelectric crystal (which is directly proportional to the field $\mathbf{E}$, the magnitude of which is, in turn, linear in the applied potential difference across the crystal) satisfies $$\mathbf{P}=\vec{d}\sigma,\tag{A}$$ where $\vec{d}$ is a tensor of rank 3 called the "piezoelectric tensor," and $\sigma$ is the stress tensor (rank 2), changes in which can manifest as small changes in the crystal's macroscopic dimensions.

$\vec{d}$ is a constant that is determined by the structure of the piezoelectric crystal, and for many common crystals, its value has been documented through experiments (though considering the crystal's symmetry can go a long way in predicting its form). When $\mathbf{P}$ is treated as a variable through control of the electric field, $\sigma$ is not always uniquely specified (infinitely many solutions may exist; you can observe this by considering whether equation (A) is many-to-one, one-to-one, or one-to-many map), but when you observe physical deformations due to the applied $\mathbf{E}$ field, there are other constraints (e.g. external physical impediments that constrain it to uniaxial stress and strain) that lead it to take a unique form.

That being said, although it's almost certainly not observed in the system mentioned by OP in the nonlinear piezoelectric effects have been observed in some crystals. Using index notation, we have $$P^{(2)}_i=\sigma_{jk}\sigma_{lm}\vec{d}^{(2)}_{ijklm},$$ where $\vec{d}^{2)}_{ijklm}$ are elements of the crystal's fifth-rank piezoelectric tensor. In this case, the second-order polarization oscillates with double the frequency of the stress tensor (you may verify that by allowing certain elements of $\sigma$ and $\vec{P}^{(2)}$ to take forms like $\sigma_0\mathrm{e}^{\mathrm{i}\omega t}$ and $P_0\mathrm{e}^{\mathrm{i}\omega t}$ respectively).