Considerations on the numerical and experimental investigation of attached-mass effects on a conical steel ultrasonic horn. A finite-difference, 2D/3D FEM and experimental FRF study

2.1. Mathematical description of ultrasonic wave propagation

An ultrasonic horn generally has two end faces with different cross-sectional areas. The one-dimensional model used to describe longitudinal-wave propagation through an ultrasonic horn is given in [5]:

where $S\left(x\right)$ is the cross-sectional area variation, $c$ is the longitudinal wave velocity in the material, and $u\left(x,t\right)$ is the axial displacement field, that depends on position $x$ and time $t$.

2.2. Types of ultrasonic horns

For most industrial horns, the cross-section is circular and the diameter follows a prescribed variation law along the horn axis. This profile affects the displacement-amplification ratio, stress distribution, resonant behavior, and manufacturability. These profiles remain common because they combine predictable dynamic behavior with relatively straightforward manufacturing. The most commonly used ultrasonic horns are [6]-[8]:

1) Stepped ultrasonic horn – consists of two axial sections with different constant diameters.

2) Conical horn – provides smooth stress distribution and efficient force transmission useful for high-force machining and welding.

3) Exponential ultrasonic horn – the diameter profile decreases according to an exponential.

4) Catenoidal horn – smooth hyperbolic-cosine profile that enables high-force transmission, moderate amplitude gain, and improved fatigue life.

Horn selection and shape optimization must balance requirements such as displacement gain, force transmission, stress level, and fatigue life [9-11].

3.1. Finite difference method

For the finite-difference model, a centered-difference scheme was applied to the propagation equation defined by Eq. (1), written for the spatial component of the solution to Webster’s equation. Applying the centered finite-difference method yields the following spatial equation:

where $\beta$ is the coefficient describing the geometric variation of the circular cross-section along the longitudinal axis.

Applying Eq. (2) at all axial nodes requires two boundary conditions: an input amplitude at the horn’s maximum cross-section and dynamic equilibrium at its output end, given by the following equation:

Assuming a harmonic ultrasonic signal, $u(x,t)=u\left(x\right)\cdot e^{j\omega t}$, and applying the centered finite-difference method, Eq. (3) becomes:

Eq. (3) and (4), combined with the prescribed input amplitude $u_0$, yield a linear system of equations. In matrix form, the tridiagonal coefficient matrix contains the terms associated with the neighboring amplitudes $u_{i+1}$, $u_i$, and $u_{i-1}$, while the right-hand-side is zero except for the component corresponding to $u_0$.

A MATLAB® 2023b [14] script implementing the centered finite-difference method simulated the horn response both without and with the experimentally tested attached mass (Fig. 1). The axial amplitudes $A$ were normalized by the input amplitude $u_0$ to determine the amplification factor.

Fig. 1Variation in signal magnification along the longitudinal axis: a) without the attached mass; b) with the attached mass

a)

b)

3.2. The 2D FEM model

A computationally efficient 2D axisymmetric PLANE183 model captured the longitudinal and radial modes [15], with local refinement and mesh convergence assessed in Table 1. Harmonic analysis results are presented in Table 2. The calibrated PLANE183 2D model of the horn–screw–mass assembly showed attenuation at 20 kHz and approximately 1.77× amplification at its natural frequency (Fig. 2).

In general, the attached mass reduces the amplification ratio.

3.3. The 3D finite element model

The 3D horn model was generated by revolving the 2D geometry and joining the volumes using VADD. SOLID187 elements were employed, with a refined 1 mm mesh near the holes and variable element sizes elsewhere. The 3D model was analyzed under free-free conditions using four, mesh sizes, while harmonic simulations at the first longitudinal natural frequency determined the signal amplification (Table 3).

A 4 mm element size provides nearly the same natural frequency as the 2 mm mesh, with a difference of only 0.0008 % and approximately 61 % fewer elements. A 6 mm mesh offers the best compromise between accuracy and computational cost, whereas coarser meshes of 8 mm produce increasingly noticeable deviations. Under free-free conditions, the horn exhibited an amplification ratio of approximately 4.14, with negligible mesh dependence. The 3D model was then extended to include the screw, contact interfaces, and attached mass, using calibrated component masses and locally refined meshes to determine the loaded natural frequencies (ultrasonic horn $m_{uh}=$ 1.231 kg, screw $m_s=$ 0.002 kg, and attached mass $m_{att}=$ 0.193 kg) (Table 4). The density assigned in ANSYS was calculated from the mass and volume of each component. All components were assumed to be made of steel, with a Young’s modulus of 210 GPa and a Poisson’s ratio of 0.3.

Fig. 2Mode shapes of the 2D models with the same element size (2 mm): a) modal analysis of the single ultrasonic horn at a natural frequency of 23,582.8 Hz; b) harmonic analysis of the assembly at a frequency of 27,622.6 Hz

a)

b)

As shown in Table 4, the harmonic analysis indicated that the horn designed for 20 kHz attenuated the output signal, whereas adjusting its length to tune the horn to the assembly’s natural frequency produced nearly mesh-independent amplification.