Slot optimization - Under development
Need
As the lateral dimensions of horns become large (relative to the thin-wire half wavelength), Poisson coupling near the node distorts the face amplitude. If the lateral dimensions become sufficiently large then the longitudinal resonance may disappear entirely — nodes will appear on the face and/or stud surfaces.
To aleviate this problem, wide horns have slots that are aligned parallel to the direction of longitudinal vibration. The slot length is a considerable portion of the half wavelength.
Unfortunately, these slots introduce stress concentrations which may cause fatigue failures toward slot ends.
Standard slots
At 20 kHz the slots are typically 10 mm wide with a full 5 mm radius at each end. (Wider slots 12–13 mm can be used but these may cause performance problems.) Toward the horn's output face the slot web length is typically between 12 mm and 20 mm. Toward the horn's input surface the slot web may be considerably larger if the horn is designed with gain.
Figure 1 shows the slot stress distribution in a 20 kHz 75 mm wide x 13 mm thick bar horn. (The slot is 10 mm wide with a full 5 mm radius at the end. The web length is 15 mm.) Note the gradual stress rise from the horn's face and then the sudden stress increase at the slot. The stress peaks at 4.1 mm just before the slot arc becomes tangent to the slot wall.
Important — Unless noted otherwise —
- All stresses are normalized with respect to the highest (nodal) stress in a longitudinally resonant thin wire. This thin wire is made of the same material and has the same output amplitude as the actual horn. Figure 1 shows a normalized stress curve. For this curve the maximum value of the relative stress curve is 1.07. This means that the maximum slot stress is 7% higher than the highest stress in a longitudinally resonant thin wire. This allows for comparison of various slot designs that are independent of the horn's material, amplitude, and frequency.
- All dimensions are for 20 kHz horns. However, these can easily be scaled to other frequencies.
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Reducing the slot stress may have several advangates —
- The horn life may be extended at the current amplitude.
- The existing horn can be run at a higher amplitude without failure.
- The slot web can be increased which may, in some circumstances, improve the horn's performance (better frequency separation, better amplitude uniformity, reduced transverse amplitude).
Advanced slots
In order to overcome the stress limitations of standard slots, slots with different end geometries have been optimized with finite element analysis. The investigated geometries include compound arcs, various ellipses, catenaries, and keyholes. Other geometries are possible but may not yield significant improvements. In some cases (e.g., keyholes) the stresses were significantly worse.
Design considerations
The slot design was constrained by machining considerations.
- The slots should be machinable using conventional rotary tools. (CNC capability was assumed but not absolutely required.) Then these slots could be machined by most small shops. This criterion eliminated some potential slot geometries (e.g., a slot with a pointed end) that could have been machined by wire EDM or broaching.
- The machining should not depend on the horn material. This eliminated wire EDM (which isn't particularly suited to softer materials like aliminum) and broaching (which isn't particularly suited to titanium).
- The slots should be machinable to considerable depths such as in large slotted block horns. This requires that the machine tool (e.g., end mill) should be as rigid as possible (i.e., large diameter).
Based on the above criteria, the main slot length (between the optimized ends) should be machined with a Ø10.0 mm tool. A Ø8.0 mm tool was chosen to machine the optimized slot ends. (A Ø6.0 mm was investigated but didn't give better results.)
Compound radius
The single 5 mm end arc can be replaced with two tangential arcs — a smaller arc with radius \( R_{1} \) at the slot end and a larger arc with radius \( R_{2} \) that transitions from the smaller arc to a tangential termination at the slot wall. The two arcs are tangent to each other where they meet.
The required value of \( R_{2} \) is given by —
where —
| \( R_{1} \) | = radius of small arc at slot end |
| \( R_{2} \) | = radius of large transition arc |
| \( x \) | = distance to the centerline of the large arc, measured parallel to the slot axis |
If \( R_{1} \) is specified and if the distance \( X \) from the slot end to the centerline of the larger radius is specified, then the size of the larger radius is given by the following equation —
The intersection of the smaller and larger arcs occurs at —
Figure 2 shows the results with \( R_{1} = 4 \) mm and several \( R_{2} \). \( R_{2} = 12.5 \) mm gives the best overall results.
Ellipse
Ellipses of varying degrees are described by the following equation —
\begin{align} \label{eq:11501a} {{\huge\lgroup}{\frac{x - x_{0}}{a}}{\huge\rgroup}}^p + {{\huge\lgroup}{\frac{y - y_{0}}{b}}{\huge\rgroup}}^q = 1\end{align}
where —
| \( x \) | = curve coordinate along major ellipse axis |
| \( y \) | = curve coordinate along minor ellipse axis |
| \( x_{0}, y_{0} \) | = coordinates of the ellipse center |
| \( a \) | = radius of major ellipse span |
| \( b \) | = radius of minor ellipse span |
| \( p, q \) | = determines the curvatures in the \( x \) and \( y \) directions |
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