Mechanism of generation of the flexural vibrations in an ultrasonic waveguide

2.1. Parameterization of the state

Function $x\left(t,z\right)$, which define a shape of the rod at the instant of time $t$, should be determined by means of the elasticity theory equations solving ($t$ – time, $z$ – longitudinal coordinate, $x\left(t,z\right)$ – transverse deflection of the rod points). One of the commonly used numerical methods of the solution consists in the piecewise-constant or piecewise-linear approximation of $x\left(t,z\right)$, that leads to the corresponding finite-difference computational scheme. Nevertheless, in practice, this approach proved to be ineffective when applied to the problem considered, because it requires the use of very dense computational grids to achieve the appropriate accuracy (to recover the effect of interest) that makes the computational time to become too expensive.

It has been appear to be more efficient the approach based on the Fourier expansion of $x\left(z\right)$, the coefficients of which as functions of time being considered as generalized coordinates representing the state of the system. Taking into account the appropriate boundary conditions [4]:

This expansion reduces to:

2.2. Dynamical equation

After expression of Lagrangian function of the system in terms of the generalized coordinates $A_n\left(t\right)$ and appropriate transformations, we derive a system of equations for $A_n\left(t\right)$ which determine the dynamics of the system. When restricting the Eq. (1) by the first term only, the equation for $A\left(t\right)$ takes the form:

where ${\omega }_0^2=\pi h{k}^4/4\rho \beta$, $\rho$ – linear density, $k=\pi /2L$, $L$ – length of the rod, $\beta =3\pi /4-2$, $h=EI$, $E$ – elastic modulus, $I=\pi R^4/4$, $R$ – radius of the rod, $f\left(t\right)=\mathrm{\Delta }z\left(t\right)$ – longitudinal displacement of the rod caused by an external source action, that we consider to be a given function of time in this section.

Initial conditions:

The Eq. (2) differs from the conventional equation of forced oscillations in that the given function of time (source function) enters into the equation not as a free term, but as an additional summand in the expression for coefficient at the $A$ (parametric excitation [5]).

2.3. Dependence of the amplitude of flexural oscillations on the frequency of longitudinal oscillations. Low-frequency resonance

The modelling has been performed for:

where $\omega$ and $a$ are, respectively, a frequency and an amplitude of the longitudinal vibrations.

a) A case of $a=$ 0 (external excitation is absent);

In this case Eq. (2) takes a form of the free oscillations equation with frequency ${\omega }_0$. For ${\rho }_V=$ 7850 kg/m³, $E=$ 204 GPa (steel), $R=$ 0.5 mm, and $L=$ 210 mm, the value of the free flexural oscillation frequency is ${\nu }_0=$ 19.5 Hz (${\omega }_0=2\pi {\nu }_0$).

b) A case of $A^0=$ 0 (no initial deflection of the rod from the equilibrium position). Then, $A\equiv$ 0 and $x\equiv$ 0;

c) General case.

The Eq. (2) with function f defined by Eq. (3) has been integrated numerically for various values of the excitation frequency $\nu$ ($\omega =2\pi \nu$, $a=$ 1 μm, $A^0=$ 0.1 μm, the other parameters values – as in a case a). The results are presented in Fig. 2 in a form of dependence of the amplitude of flexural oscillations on the frequency of longitudinal oscillations.

Fig. 2Dependence of the amplitude of flexural oscillations on the frequency of longitudinal oscillations

A specific feature of this amplitude-frequency dependence is a sharp pick at $\nu =2{\nu }_0=$ 39 Hz, that is in accordance with the parametric resonance theory [5]. For all other frequencies (except that of the near-resonance region) the amplitude of flexural oscillations is negligible. So, the ultrasonic longitudinal vibrations per se can’t be a reason of significant flexural oscillations.