evaluating sdbd. crown & bridge downbearings in a new piano

[Richard Brekne]

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Ron or somebody... would you please tell me where the two numbers I have
italisized and in bold text below ?? The rest is easy enough to follow.  :)

Richard Brekne
I.C.P.T.G.  N.P.T.F.
Bergen, Norway

Ron Nossaman wrote:

>
> * As long as the assembly is in the press, this is true. As for the Voodoo
> engineering, let's conjure up a little math and see what the ancient Gods
> have to say about all this and try to ascertain who's shaking the beads
> here. Let's talk Engineer.
>
> When the rib (or panel) is bent, there is a neutral stress line running
> roughly through the center of it's height. That part is obvious, so let's
> use the rib centerline as a reference for computation. Let's use Del's
> example rib and panel as a standard and start with a rib 1000mm long, 25mm
> high, and 25mm wide, with a panel 8mm thick. Let's also assume the 18000mm
> final radius result after assembly, because we need some sort of figure as a
> benchmark and the 60' radius seems to be the most widely accepted. So what
> are the final arc lengths of the assembled surfaces? Here's where the math
> comes in. First, let's figure the segment angle of the formed arc. Angle =
> 57.29578*(riblength/radius). That's 57.29578 times 1000/18000, or 3.1831
> degrees. The formed arc segment of the top of the panel is computed as
> PanelTop = 0.017453*(radius+(ribheight/2)+PanelThickness)*angle. In the
> final assembly, that comes to 1001.1221. the formula
> Radius-sqrt(sqr(radius)-sqr(riblength/2)) gives us a crown height of
> 6.9458mm. I figured four decimal places ought to be adequate overkill. Now,
> in order to have compression in the top of the panel after assembly, and
> before string load, We have to form the assembly in a press with a curve
> radius such that the arc segment defined by the neutral stress centerline of
> the panel when it's in the press is the same, or greater, than the arc
> segment defined by the top of the panel after removal from the press and
> spring back. As it turns out, a press caul of 14487mm radius produces an arc
> segment angle of 3.955, and using
> 0.017453*(radius+(RibHeight/2)+(PanelThickness/2)*angle gives us a panel
> centerline of 1001.1222, with a crown height of 8.631mm. Therefor, if the
> spring back after taking the assembly out of the press accounts for a
> reduction in crown of more than 1.6854mm to arrive at that 18000mm radius,
> the top of the panel will be under compression. That is the case with this
> particular set of rib and panel dimensions, but the principal can be put to
> any set of dimensions you may have to test the premise.
>
> Now, in your example, if we knew the caul radius of the press used, the rib
> and panel dimensions, and the resultant crown radius of the assembly upon
> removal from the press, we could find out what we really have.
>
> If my math is incorrect, please correct me. I need all the help I can get
> with math. Also, I realise that this isn't exact to the four decimal places
> shown, primarily because bending the rib in an arc shortens the chord
> measurement from which the computations are taken. Taking this into account,
> the actual figures would be minutely different, but I assumed that a
> hundredth of a millimeter or so wouldn't invalidate the illustration.
>
>  Ron N

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