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Panel Resonance Calculation Validity

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  • Panel Resonance Calculation Validity

    Can anyone vouch for the validity of the following equation to calculate panel resonance frequencies?

    f = (pi/ 2) * SQRT{D/ p[(1/a^2) + (1/ b^2)]}

    where:
    f = natural frequency
    D = plate stiffness factor = [E(h^3)] / [12(1-u^2)]
    E = modulus of elasticity in lb/ in^2
    Note 3/4" no-void ply has an MOE of 1.8 mill psi,
    while typical MDF is 0.53 mill psi.
    h = plate thickness, inches
    u = Poisson ratio, MDF = .33

    p = mass per unit area = v * h / g
    v = material density in lb/in^3, MDF =~ 50lb/ft^3 = 0.028935 lb/in^3
    h = plate thickness, inches
    g = acceleration of gravity, 386 inch/sec^2

    a = length of the plate, in inches
    b = width of the plate, in inches.

    Source was here.

    I found E needed to be in kilo psi (so, 530 for MDF and 1800 for ply) to give what look like appropriate results (I think):

    For MDF:

    40" x 12" x .75" = 83.3Hz (153.5Hz ply)
    20" x 12" x .75" = 93.1Hz
    20" x 6" x .75" = 166.6Hz
    10" x 6" x .75" = 186.1Hz
    6" x 6" x .75" = 225.7Hz
    4" x 4" x .75" = 338.5Hz

    The values seem perhaps to be a little low and, counter to common wisdom, doubling up the panel thickness had a more significant effect that dividing the panel in 2.

    40" x 12" x 1.5" = 166.6Hz (307Hz ply)
    20" x 12" x 1.5" = 186.1Hz
    20" x 6" x 1.5" = 333.2Hz
    10" x 6" x 1.5" = 372.2Hz
    6" x 6" x 1.5" = 451.4Hz
    4" x 4" x 1.5" = 677Hz

    Seems like it could be a useful tool for designing cabinets and bracing even if the results are only ballpark figures. Can anyone comment?

  • #2
    Re: Panel Resonance Calculation Validity

    Originally posted by mobius View Post
    The values seem perhaps to be a little low and, counter to common wisdom, doubling up the panel thickness had a more significant effect that dividing the panel in 2.
    That's assuming all bracing does is divide the MDF into smaller bits. It acts to stiffens the panels and force their frequency of resonance higher. Just like stiffening the spring with a mass on it; Fs goes up and Q goes down.

    I appreciate that you are looking for a mathematical equation for bracing. I am studying C++ maybe when the equation stuff is ironed out I can attempt to make a simple little C++ program to do the math for us.

    Comment


    • #3
      Re: Panel Resonance Calculation Validity

      Originally posted by mobius View Post
      I found E needed to be in kilo psi (so, 530 for MDF and 1800 for ply) to give what look like appropriate results (I think):
      KSI is the correct term.

      As for the validity of the equations, I will have to think about it.....

      Comment


      • #4
        Re: Panel Resonance Calculation Validity

        Originally posted by mobius View Post

        The values seem perhaps to be a little low and, counter to common wisdom, doubling up the panel thickness had a more significant effect that dividing the panel in 2.
        Dividing the panel area in half should increase the resonant frequency by a factor of 4. Doubling the thickness should increase the resonant frequency by only a factor of 2. This is assuming the panel is square. Your calculations confirms this.

        Bracing the cabinet correctly is better than adding another layer.

        Comment


        • #5
          Re: Panel Resonance Calculation Validity

          Adding thickness is far less effective than panel to panel bracing, especially where strength to weight and cost ratios are concerned. In the picture below adding the single red brace connecting the middles of two panels the result is the same as doubling the panel thickness. Adding the blue braces is the equivalent of quadrupling the panel thickness. That's why my cabs are never made of anything other than 1/2 inch plywood, and they don't have audible panel vibrations, not even pro-touring subs.

          www.billfitzmaurice.com
          www.billfitzmaurice.info/forum

          Comment


          • #6
            Re: Panel Resonance Calculation Validity

            Originally posted by jcrane82 View Post
            Dividing the panel area in half should increase the resonant frequency by a factor of 4. Doubling the thickness should increase the resonant frequency by only a factor of 2. This is assuming the panel is square. Your calculations confirms this.
            Thanks, jcrane82. I can see that doubling the thickness doubles the frequency, but halving the area looks like it only increases the frequency marginally.

            Originally posted by billfitzmaurice View Post
            Adding thickness is far less effective than panel to panel bracing, especially where strength to weight and cost ratios are concerned. In the picture below adding the single red brace connecting the middles of two panels the result is the same as doubling the panel thickness. Adding the blue braces is the equivalent of quadrupling the panel thickness. That's why my cabs are never made of anything other than 1/2 inch plywood, and they don't have audible panel vibrations, not even pro-touring subs.
            Thanks billfitzmaurice. What I'm looking for is the science to back that up.

            I'm not a math wiz but I think monkish54 may have it right - in the equation, the stiffness factor varies directly with h, the panel thickness, but it doesn't look like the ratio of length to width is accounted for.

            Anyone have a better equation?

            I found this one also:

            f = {SQRT(K/m)}/2pi

            where:
            K = panel stiffness = Eh^3/alpha*b^4
            and
            E = modulus of elasticity
            h = panel thickness
            a = panel length
            b = panel width
            m = panel mass
            but
            alpha = some ratio based on a/b in a limited chart form only so not really very useful

            Source for that is on these 2 pages:
            http://www.audioholics.com/loudspeak...cal-background
            http://www.audioholics.com/loudspeak...ative-analysis

            Comment


            • #7
              Re: Panel Resonance Calculation Validity

              Originally posted by mobius View Post
              Thanks billfitzmaurice. What I'm looking for is the science to back that up.
              It's very basic civil/structural/acoustical engineering. Ask any trained architect, bridge builder or professional loudspeaker designer.
              www.billfitzmaurice.com
              www.billfitzmaurice.info/forum

              Comment


              • #8
                Re: Panel Resonance Calculation Validity

                Do these formula predict the resonant frequency of a free floating panel, or one with its edges held to something rigid? I.E. is it assumed to be a vibrating plate, or something like a speaker box panel?

                Comment


                • #9
                  Re: Panel Resonance Calculation Validity

                  In order for something to have a resonating mode, it needs to have boundary conditions to constrain the model. This means the edges are fixed.

                  Comment


                  • #10
                    Re: Panel Resonance Calculation Validity

                    Originally posted by billfitzmaurice View Post
                    That's why my cabs are never made of anything other than 1/2 inch plywood, and they don't have audible panel vibrations, not even pro-touring subs.
                    So realistically, I should be able to build a 10-ish cubic foot ported subs for an SI HT18 driver with 1/2" stock, using this bracing scheme. Do you have a rule of thumb for spacing the dowels?

                    Comment


                    • #11
                      Re: Panel Resonance Calculation Validity

                      Just keep in mind that these equations assume that the braced locations are rigidly constrained, which in fact they aren't. The braces will also have their own resonant modes that will energized the main panels. Having the braces constrained to each other will improve things.

                      Comment


                      • #12
                        Re: Panel Resonance Calculation Validity

                        Originally posted by rhodesj View Post
                        So realistically, I should be able to build a 10-ish cubic foot ported subs for an SI HT18 driver with 1/2" stock, using this bracing scheme. Do you have a rule of thumb for spacing the dowels?
                        Spaced 6 inches apart you'll be fine. That will give the same result as 12 inch spacing with one inch plywood, or 24 inch unbraced spans with two inch plywood. Bracing with thin materials is a labor intensive process, and that's why you don't see commercial cabs done this way, as thicker materials are a lot less expensive than labor.
                        One issue with an 18 loaded sub might be sub dancing, as the cab weight might not be enough to counteract the motive forces. You might want to go with 3/4" ply and nine inch brace spacing for that reason.
                        www.billfitzmaurice.com
                        www.billfitzmaurice.info/forum

                        Comment


                        • #13
                          Re: Panel Resonance Calculation Validity

                          IMHO, most speakers enclosures are complex enough that a simple formula will not accurately solve your natural frequency. You would need to use FEA I think.
                          Melby Audio - Flat Pack Speaker Kits

                          Comment


                          • #14
                            Re: Panel Resonance Calculation Validity

                            Originally posted by jcrane82 View Post
                            In order for something to have a resonating mode, it needs to have boundary conditions to constrain the model. This means the edges are fixed.
                            so if I excite a tuning fork and toss it into the air it will stop resonating because there is no longer a boundary to constrain it ? or if we were able to levitate a tuning fork into the air and play a sympathetic frequency it would not cause the tuning fork to become excited ?
                            craigk

                            " Voicing is often the term used for band aids to cover for initial design/planning errors " - Pallas

                            Comment


                            • #15
                              Re: Panel Resonance Calculation Validity

                              Originally posted by craigk View Post
                              so if I excite a tuning fork and toss it into the air it will stop resonating because there is no longer a boundary to constrain it?
                              I have not confirmed the equations above, so I may have mispoke when saying the edges are fixed. My main point was that the boundary conditions need to be taken into account. I am only assuming that the edges are fixed, because if the edges aren't fixed in the above equations....they are not representative of a speaker cabinet.

                              In order to excite the tuning fork in the first place you had to hold it (constrained boundary conditions).

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