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Computer Aided Resonator Design (CARD)

Description, Equipment Requirements, Support


Copyright 1991 .. 2002
Donald R. Culp
All rights reserved



If this document is reproduced for further
distribution, this title page must be included.


Table of Contents

bulletMethod of analysis
bulletTuning methods
bulletAcceptable resonator shapes
bulletInstruction manual
bulletCopy protection
bulletNewest beta version
bulletDemonstration version
bulletEquipment requirements
bulletAdditional information
bulletPrice and delivery
bulletReturn policy
bulletFrequently asked questions
bulletKnown issues
bulletFigure 1 -- Amplitude graph
bulletFigure 2 -- Stress graph
bulletFigure 3 -- Loss graph
bulletFigure 4 -- Spreadsheet data entry
bulletFigure 5 -- Slicing the resonator




Computer Aided Resonator Design (CARD) is software that applies quantitative techniques to the design of ultrasonic resonators (horns, boosters, and transducers) that vibrate in a longitudinal mode. CARD provides assistance in the design of resonators having low-to-moderate complexity.

With CARD, alternative resonator designs can be quickly evaluated without machining and testing. The effects of proposed resonator modifications can be easily determined. CARD is especially useful for designing low-stress resonators, resonators with a specified gain, and resonators with a specified node location.

CARD automatically tunes the horn to the desired frequency by adjusting the resonator dimensions. The adjustable dimensions include the length, thickness or diameter, and location of a transition radius. In addition, CARD can automatically adjust the gain and minimize the stress. (Note: see below for limitations for block horns.)

CARD calculates numerous acoustic parameters, including tuned length, tuned frequency, gain, node location, maximum stress, stored energy, loss, overall quality factor (Q), and weight. When calculating the stress, CARD considers the effect of stress concentrations at radii and slot ends. CARD graphically displays the calculated amplitude, stress, and strain-loss distributions at each point along the length of the resonator. (See figures 1, 2 and 3 for typical results.) Analysis results can be viewed, printed, and saved to a file.

CARD figure 1 (graph): Amplitude distribution in a slotted bar horn

Figure 1.  Amplitude distribution for a slotted bar horn

CARD figure (graph) 2: stress distribution in a slotted bar horn

Figure 2. Stress distribution in a slotted bar horn.

CARD figure 3 (graph): loss distribution in a slotted bar horn

Figure 3. Loss distribution in a slotted bar horn.

CARD uses a spreadsheet format for easy input of the resonator shape. (See figure 4.) The resonator shape can consist of any combination of straight, curved, exponential, and catenoidal surfaces, arranged in any order.

CARD figure 4:  spreadsheet for input of resonator shape

Figure 4. Spreadsheet for input of resonator shape.

The resonator shape can be graphically displayed to verify its correctness. The resonator can be composed of multiple, user-defined materials. The resonator can have a cavity in the face and can have studs, wrench flats, and spanner wrench holes.

CARD allows up to 10 different user-defined materials and ultrasonic equipment configurations. These defaults can be saved to disk.

CARD is very easy to learn and use, so that even those with minimum computer experience should have little difficulty. All user input is from menus; there are no commands to memorize. From any menu within CARD, a single keypress will change between metric and English units. Extensive hypertext help is available for each menu option. Also included with the help is a glossary of over 300 acoustic terms. (Note: although CARD is very easy to use, the user must have some understanding of resonator design in order to evaluate the computer-generated output.)

In addition to CARD's main analysis, CARD also has a section devoted to theoretical and empirical equations for longitudinal, flexural, torsional, and radial vibration. CARD also includes a database of metric and English bolt dimensions. Other sections include calculations for press-fits, bolt stresses, and piezoelectric ceramics.

Method of analysis

CARD analyzes the resonator by dividing it into a large number of thin slices. Each slice is perpendicular to the resonator's axis and extends entirely across the thickness (or diameter) of the resonator. (Figure 5 shows a typical resonator.) Starting from the back of the resonator, each successive slice is analyzed until the opposite end of the resonator is reached. This process is repeated several times until a specified convergence criterion is achieved.

CARD figure 5:  Sliced Resonator

Figure 5.  Slicing the resonator
   Top view:  resonator shape. 
   Bottom view:  sliced resonator.

All acoustic parameters are assumed to be constant within each slice. Hence, such parameters as stress and amplitude cannot vary across the resonator thickness but can only vary along the resonator length. Thus, CARD gives a one-dimensional analysis of what is actually a three-dimensional acoustic phenomenon. In this respect, CARD is somewhat like one-dimensional finite element analysis.

CARD uses an equation to account for the decrease in wave speed that occurs as the resonator cross-sectional dimensions increase. Thus, "fat" resonators will have shorter tuned lengths than the equivalent "thin" resonators. This is consistent with theoretical and empirical studies.

Tuning methods

CARD operates in either of two modes:

  1. Auto-tuning. If the resonator frequency is specified, then CARD will automatically change the dimensions of a user-specified resonator surface until the specified frequency is achieved. CARD can adjust a length dimension, a thickness dimension, or the location of a radius or step.

    While the resonator is being tuned, the gain can be automatically adjusted to a desired value. The stress can also be minimized automatically.
  2. Manual tuning. If all of the resonator dimensions are specified, then CARD will calculate the resonant frequency. If desired, the resonator dimensions can then be manually adjusted to change the frequency. Auto-tuning is used most often. Manual tuning is used to adjust certain surfaces where auto-tuning is not appropriate.

Acceptable resonator shapes

Although all acoustic phenomena are actually three-dimensional, there are many resonators for which a one-dimensional analysis gives very satisfactory results. These include most unslotted cylindrical horns, bar horns (slotted or unslotted) up to about 250 mm wide by 75 mm thick at 20 kHz, boosters, and transducers (converters). Cylindrical horns can have a cavity, as long as the cavity is not excessive. CARD can analyze flatted-cylindrical horns (horns that have a cylindrical cross-section which is flatted toward the output end.)

CARD can analyze cross-slotted cylindrical or block horns, where the slots intersect at right angles. However, only the tuned length or frequency will be determined. Automatic gain and stress adjustments are not possible for these horns.

Resonators can be designed to any frequency between 5 kHz and 99 kHz. Resonators may be up to three half-wavelengths long. (5/99: This has been increased to 99 half-waves in the latest beta (test) version.)


Because CARD uses a one-dimensional analysis, there are limitations on the parameters that can be calculated. Amplitude uniformity across the input and output surfaces cannot be calculated since the amplitude is not permitted to vary across each slice. Similarly, the frequencies of nonaxial resonances cannot be determined. However, this is not a severe limitation, since the majority of resonators for which CARD is suitable do not have problems with nonaxial resonances.

CARD designs resonators by looking at a side view of the resonator. (For slotted bar horns, this is the side view where the slots cannot be seen.) When looking at the resonator from the side, the resonator cross-section must be symmetric about the resonator axis (i.e., the cross-sectional shape on one side of the resonator axis must be the same as the shape on the other side of the resonator axis). Thus, CARD can not analyze resonators with asymmetric face contours, resonators with asymmetric face bevels, etc. unless these can be reduced to an equivalent symmetric design.

Except for flatted-cylindrical horns (above), CARD cannot be used for resonators whose cross-sectional shape changes along the length of the resonator. Thus, for example, if a resonator has a rectangular cross section, then this cross-sectional shape must continue along the entire resonator length. Although the cross-sectional shape cannot change, the cross-sectional dimensions can be adjusted as desired. (See the attached figures.) Note that certain nonsymmetric entities (e.g., wrench flats and spanner holes) are permitted.

CARD cannot be used on bar horns with risers on the back.


If CARD is run without prior calibration, resonators of certain geometries or dimensions will give less accurate results. The following are listed approximately in order of decreasing accuracy:

  1. Large thickness. Shaped resonators with large thickness (relative to the half-wavelength) generally give somewhat less reliable results than equivalent resonators with small thickness.
  2. Large, abrupt area changes. Resonators with large, sharp steps in the area of the node give somewhat unreliable results. However, such designs would not normally be used because they have high stresses at the step.
  3. Large cavities. Resonators with large cavities, such as bell resonators, usually have significant radial motion which tends to invalidate a one-dimensional analysis.

To correct for these limitations, a calibration mode has been provided. This mode allows CARD to be calibrated against a resonator of known design. For resonators that fall into the above categories, such a calibration can improve CARD's accuracy. Thus, CARD is intended to provide assistance with resonator design. It is not intended to replace the good judgment of the resonator designer. Neither is it intended to replace finite element analysis (or similar methods) as a final analysis tool.


The following tree diagram shows the equations section of CARD.

          - Cylindrical Horns
             -- Unshaped (*)
             -- Conical (*)
             -- Exponential (*)
             -- Catenoidal (*)
             --- Spool (*)
          -- Tip force (*)
          -- Radial disk resonance (*)
          --- Radial ring resonance
          -- Rod --- (typical) --- Clamped-fixed plate (*)
          -- Beam                       - Clamped-free plate (*)
          -- Plate                        -- Infinite plate (*)
          --- Disk
          -- Temperature effects (*)
          -- Thin-wire parameters
          --- Q
      --- OTHER
           -- Bolt dimensions
           -- Bolt calculations
           -- Wrench flat equivalent diameter
           -- Spanner hole equivalent diameter
           -- Press-fit (*)
           -- Von Mises stress
           -- Piezo-ceramic calculations
               -- Static performance
               --- Dynamic performance

      (*) indicates that calculations can be graphed for the specified item.

Instruction manual

CARD is supplied with a comprehensive 300 page manual, with over 40 figures. The manual gives step-by-step instructions for using CARD. The manual includes numerous examples, with corresponding resonator designs on the distribution disk.

Copy protection

CARD is copy protected. The copy protection method allows CARD to be installed on multiple computers. However, CARD can only be operated from one computer at a time.

Newest beta version

The newest beta (test) version provides additional capabilities.

Demonstration version

A functional demonstration version is available for downloading.


Equipment requirements

Required equipment


IBM micro-computer (or true compatible) based on the Intel 80286 (or higher) microprocessor.

Operating system

Version 3.3 (or later) of Microsoft DOS. Version 5.0 (or later) is recommended for printing VGA graphics. (See below.)

Although CARD is a DOS program, it will run acceptably under Microsoft Windows. (Contact Krell Engineering for additional information.) Also, see known issues


The computer must have approximately 480 KB of free conventional memory after any other memory-resident programs have been loaded.


A VGA color display (640 x 480, or higher) is recommended. (The values in parentheses give the graphics resolution.) However, CARD will also run with any of the following displays: any video adapter and monitor capable of displaying IBM graphics (CGA, 320 x 200; MCGA, 320 x 200; EGA, 640 x 350).

Disk drives

A hard (fixed) disk with approximately 5 MBytes of free space is required. One 1.44 Mbyte floppy drive is required to load CARD onto the hard disk (not required for the demo version).

Optional equipment


A two or three button mouse may be used. The mouse is only active when the on-line help is invoked.


CARD is configured to support any printer that is compatible with the IBM graphics printer (dot matrix) or Hewlett-Packard laser. CARD expects the printer to be connected to parallel port #1.

CARD does not support plotters.

Additional Information


Krell Engineering will provide free support to registered users for one year from the date of the initial license agreement. During this time, Krell Engineering will answer questions about the use of CARD and will endeavor to fix any bugs that significantly affect the use of CARD.

Prices and delivery

Full version .......................................................


Upgrade from version 8.xx or earlier ...................
(includes several chapters from the CARD manual)


Copy protection key, if required ..........................
(see note 1 below)


Shipping and handling

With full manual .........................
Without full manual ....................
   (typically for upgrades)



  1. All prices are US dollars.
  2. Payment may be made by corporate check, personal check, purchase order, or any guaranteed negotiable instrument (cashiers check, money order, etc.). Purchase orders are subject to approval. Personal checks must clear before CARD is shipped.

Shipment -- two weeks (approximately) after receipt of payment.


  1. CARD requires a diskette drive in order to access the copy protection diskette. If you intend to use CARD on a computer that does not have a diskette drive, then you will have to purchase a copy protection key that plugs into the computer's parallel printer port. Note that the copy protection key will not work under Windows 2000 or Windows XP.
  2. The copy protection diskette or the copy protection key is your evidence of authorized use. If you have lost either of these, then you cannot upgrade directly. Instead, you must purchase a full version.

Return policy

Starting from the shipment date, CARD may be returned within sixty days.  The purchase price will be refunded, less an evaluation fee of $150 and the shipping charge (above).  (Prior to purchase, you may wish to download a functional demonstration version to insure that CARD meets your requirements.)

Defective diskettes or copy protection keys will be replaced at no charge within the first sixty days; after sixty days, a nominal fee will be charged.  A lost copy protection diskette or copy protection key will not be replaced; a complete CARD purchase will be required.


Check out the current upgrades to older versions of CARD.


Frequently asked questions

See frequently asked questions.

Known Issues

The following are known issues as of 6/02.  If you feel that these may be relevant, contact Krell Engineering.

  1. Printing over a network may not work.  Since network issues can be very complex, Krell Engineering can only give limited support for this issue.
  2. The copy protection diskette can not be accessed over a network.  It must be installed and accessed from the local computer (the computer on which the work is being performed).
  3. For Windows NT:
    a. During printing, DOS programs (including CARD) may eject an extra sheet of paper before the actual printing begins.
    b. The print screen key may not send the graphics output to the printer.  However, the graphics can be copied to the Windows clipboard and then pasted into a Windows application (e.g., Microsoft Word), from which they can be printed.  This procedure is explained by CARD's online help.
  4. CARD's manual has been written for CARD version 6.xx. It has only been partially updated for the newest version of CARD. However, CARD's extensive on-line hypertext help is up-to-date.



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