37 Replies Latest reply: Mar 28, 2007 12:00 AM by GR_Aero-disabled

# max value of function

Better post 11.2a or lower .

We have a work sheet for functions with multiple peaks [�).
You seem confused: max of a col and Maximize/Minimize .

jmG
• ###### max value of function and optimization
This sheet seems to be your original sheet, with all the problems I noted there.

To us maximize or minimize you must create a function of the variables on which you wish to optimize. Unless you wish to play with the symbolic processor, which allows undefined variables as free variables, you have to make everything that depends on any of the unkonwns an explicit function of such unkowns.

Also note that while you choice of discrete points for calculating forces limits your calculations to integer values of i, the forces are in face continuous functions of i, and the true maximum could easily be (and likely is) between the tabulated values. Personally, I would suggest getting rid of i altogether and using the angle that it represents directly and consistently as a function argument.

� � � � Tom Gutman
• ###### max value of function and optimization
Well, the diagram clarifies the geometry. But I have trouble relating the diagram to your worksheet. I assume that IFS is the fixed hinge. You seem to give alternative coordinates for it, based on possible positions along a circle. Yet in the diagram the only available reference point, hence the likely origin for a coordinate system is that fixed hinge. Further you describe the coordinates as station and radius. What are those in the diagram? The diagram looks like you would be using rectangular coordinates, but the term "radius" is not normally used for one of two rectangular coordinates.

There are some questions about the relationships between the lengths in the diagram. Is the length of the door (a+b) equal to the height of the opening (T), so that the door closes at 90° to the horizontal? Or is it larger, so that the door closes at a smaller angle? For the nonce I'll assume T=a+b. I also assume that L is somewhat arbitrary, but larger than T (so that the drag link is still not vertical when the door is completely open.

I see nothing that calls for i, or specific configurations, in that diagram. The values of X, α, β, and the height of the bottom of the door (not in the diagram, I'll call it h if I need to refer to it) are all interelated, given one the others are fixed. Any of them can be used as a parameter for the configuration of the system. β looks like a likely candidate for the parameter, as it has a fixed range of zero to π/2. h, with a range of zero to T is another possibility. And calculating eithe β or T from the other is close to trivial. α and X are also easily calculated from β. You should be able to set up the geometry as just a simple set of equations, without needing a solve block.

The pressure is obviously more complicated. I don't understand why you choose to divide the door into segments for the calculations, nor why you do the pressure only for discrete postions, rather than for the whole door and for arbitrary position. What is your model for the pressure? I gather that you are looking a dynamic pressure from some sort of flow. Clearly you need to make major simplifications to have any hope of a solution. Even if all you have is empirical measurements at specific points on the door and specific angles you can interpolate to get an approximation to the continuous pressure distribution. And what is the shape of the door? Is it rectangular or some other shape?

If course your optimization failed. You hadn't even put in the fixes to the calculation of the quantity to be optimized. Nor had you made a function out of that quantity, a function of the parameters to be otimized. And you have entirely too many parameters being optimized. Given T as a constant (that sets the scale) you have only two (L and a) parameters if the door closes vertically and three (L, a, and b) if it closes obliquely. Those completely determine the configuration. You cannot solve for more than two or three variables (although they need not be those specific variables). The maximize/minimize functions have as arguments specifically the function to be optimized and the variables whose values are to be determined so as to achieve that optimum. When using any built in function (or, for that matter, a user function) you must follow the rules for that function exactly. No guessing, no fudging.

� � � � Tom Gutman
• ###### max value of function and optimization
As is common, the diagrams answer some questions and raise others. According to the diagram a+b>T, so that the door would close in an oblique position. Correct? But it also looks like the door never actually closes, and the maximum stroke is limited by other factors, perhaps just arbitrary. Also, the door is not limited to horizontal, but can slope upward. That would indicate that either there is no roof, or the roof is above the hinge. The latter is also implied by your diagram showing the hinges below the top of the door. But with the door sloping upward the top of the door has to go below the hinge. That would require either that the hinge drive is below the line of hing motion, or that the hinge is set back from the plane of the door. The latter affects the geometric calculations. And is the minimum stroke arbitrary, or is it constrained by the geometry?

I see the door is trapezoidal. And contrary to the position diagram the upper hinge is not at the edge but below the edge. One implication of the trapozoidal shape is that, except possibly at maximum deployment, there will be gaps, hence airflow, around all the sides of the door. That means that the drag from turbulence at the edges occurs not just at the bottom edge, but on all the edges. Hence pressure will not be just a function of height on the door, but also of distance from the sides. So exactly what are the pressures in your table? If they average across a height, why does the table not go all the way and average across all heights, giving a net force and moment? Also, the primary variable for the pressure will be the angle. But with different heights for the upper hinge the same angle will have different gaps, hence the pressures will be different. If you are going to be changing the positions of the hinges, including the upper hinge, you may need a more comprehensive pressure model, and perhaps more measurements. In addition, since at minimum stroke (i=0) the door is not horizontal the pressure cannot really be assumed to be zero.

While calculating the other position parameters from X is not as straightforward as the other calculations, it can be done directly. Draw a perpendicular from A to the top line. Connect A and B. You now have a right triangle with two known sides (T and X) and can solve for the angles and the remaining side. With that you know the three sides of the triangle ABC and so can solve that triangle. With both those triangles solved the remaining calculations are trivial.

You can also solve for the forces without resorting to a solve block. Only the pressure force and the drag link contribute to the moments on the door. And you know the direction of the drag link force. Hence you can easily solve for the magnitude of that force. That gives you two out of the three forces on the door, hence you can easily get the force on the upper hinge. And the force on the ifs hinge is just the force of the drag link on the lower hinge, there being no other forces on the drag link. Presumably all parts are light enough that ignoring gravity is suitable.

However i is still odd. It is not the percentage of the stroke, nor even linearly related to that. You have a messy conditional function to calculate the X that goes with each i. That suggests that i is in fact reasonably related to some other variable in the system. What is it?

There are also a lot of oddities about your input data. Your diagram is in terms of he construction, T, L, a, b, etc. But your input seems to be mostly the positions of the links at the two ends of the stroke, and you calculate the various parameters from that. But even then, not consistently. While the stroke length is easily calculated from the position of the upper hinge, you input that separately (and hope it's consistent). And your value for alpha0 is unbelievably low.

If you are to do any optimization you must start by being very clear on what are the independent variables that you might want to vary to get an optimum. And then you must keep everything that depends on those as explicit functions of thos variables. Mathcad does not do implicit functions (except in a limited way by the symbolic processor).

� � � � Tom Gutman
• ###### max value of function and optimization
Thinking a bit more about this I find I have a further problem. The door is trapezoidal, with the maximum width at the top. I have been assuming that the door is deployed in a channel that is also trapezoidal, and that the maximal stroke (fully deployed) is determined by where the door fits snugly in the channel, with contact along both sides (and likely the top). However for this to be so the width of the door at the top must equal the width of the channel at the height of the top of the door at full deployment. But that would mean that the top of the door can never go below its position at full deployment (the channel would not be wide enough to accomodate it). And for that to be the case would require that the upper hinge be at the top edge of the door, contrary to your diagram, and in conflict with your desire to "optimize" the hinge location. So I am in a complete quandary as to what defines the limits of the stroke, and the minimally and fully deployed positions.

� � � � Tom Gutman
• ###### max value of function and optimization
I am getting more confused. Your diagram for the door motion looks rectangular. Are the upper and lower lines, the ones that appear to be parallel to a conventional x axis, actualy circles? That results in a quite different geometry. You would have to have a very small fraction of a circle as the relevant arc to use straight lines as reasonable approximations. If they are arcs, is the actual stroke parallel to those arcs, or is it straight, a section of a chord?

The doors are next to each other. In which direction? In the direction of the stroke? Or perpendicular to the paper in the diagram as drawn? That would be side by side. If side by side, how close to each other are they? Close enough to interact, perhaps touch at an extreme? Or are there side walls to the channels, so that the doors are isolated from each other?

Is the length of the stroke fixed? The strike itself is not a significant physical variable. The position of the door corresponding to the stroke is. But if you change the configuration of the hinges and the drag link, you change the relationship betwee the stroke position and the door position. Your pressure data can apply to stroke position only for the particular set of geometry variables. Change them (as required for optimization) and you change the relationship between stroke position and door position.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

The drawing for the door distribution is attached. The plane shown is perpendicular to the analysis plane. It shows 8 doors (actually we used 12 doors for the analysis) attached to the outer surface panel with hinges and the doors are at fully deployed positions. Just a single door needs analysis and optimization. Hope this helps you to get clearer.

GR
• ###### max value of function and optimization
"The purpose is to find the max hinge load transmitted from a four-bar linkage consisting of a drag link, blocker door, and a slider".
_______________________

Then you must express the "hinge load" as a function of these 4 parameters.

jmG
• ###### max value of function and optimization
On 1/26/2007 4:40:02 PM, jmG wrote:
>"The purpose is to find the
>blocker door, and a slider".
>_______________________
>
>Then you must express the
>"hinge load" as a function of
>these 4 parameters.
>
>jmG
_________________________

The attached example is most interesting as you Minimize an SSE . Therefore the SSE is a numerical value, i.e: an indexed task related to a data set .

jmG

• ###### max value of function and optimization
... I don't fully understand yet !

V1 is a number and not a function.

Wouldn't it be simpler to examine all the case as a data table and Regress for bivariate ? In fact the problem is like a "non-linear from far away items", i.e: a very difficult one to solve . Two work sheets attached

1. The Given/Minimize "technique"

2. an NL difficult

jmG
• ###### max value of function
Max, and match if you need the index.

� � � � Tom Gutman
• ###### max value of function and optimization
Let's see if I can relate this to your other diagrams. It looks like this is a view from above the doors, looking down on the doors. I assume that the various gaps are just artifacts of the way the BMP was created and processed (BTW, BMPs are unecessarily large, I recommend converting them to GIF, PNG, or JPG files before uploading them).

Presumably the three dots on door 1 represent the three hinge locations. That indicates that your geometry diagram is oriented with the outside (outer cylinder) to the left and the inside (inner cylinder) to the right. IIRC your force diagram implied that the airflow was from left to right, hence in this diagram it would be from outside to inside.

You show the upper and lower edges of the doors as circular arcs (or, more correctly, elliptical arcs that project to circles at full deployment). If the doors actually have straight tops and bottoms that would replace your circles with polygons. Not much of a change.

You show the doors meeting edge to edge at full deployment. That means that the door angle, β, for full deployment is fixed, and can be calculated from the dimensions of the door and the angular spread between doors (wich can be gotten from the number of doors and the size of the slot, where there is no door). That, together with the position of the three hinges (relative to the top and bottom of the door for the top and bottom hinges, relative to the position of the door for the ifs hinge), determines one end of the stroke. It is not clear if the position of the doors at full deployment (the clearances, if any, between the top and bottom of the doors and the top and bottom plates of the channel). This affects whether T is fixed by the position of the top hinge or whether it is an independent variable.

It is still not clear what determines the lower limit of the stroke. I would have expected it to be where β is zero, but you calculate it as lower than that, with a negative β. Is that minimum β fixed (i.e., does it define the lower limit of the stroke)? Further, that non-zero β at zero deployment is inconsistent with your zero pressure at zero deployment. I would expect, with any simple model, zero pressure at zero β.

Note that with changing hinge positions the relationship between the stroke position, and even the fraction of the stroke, and the position of the door will change. Hence you cannot use your pressure table (there are other questions and issues relating to that table, I'll bring them up when we get that far) to relate pressure to stroke position. You must relate pressure to something that has an invariant relationship to door position, perhaps β.

I am a bit surprised at your choice of data to determine the layout. You are taking the absolute hinge positions in two states, rather than using what should be the known secifications used to built the system. I would expect things like the length of the drag link and the distance between the top and bottom hinges to be available from the blueprints. You have redundant measurements, so one can check whether the data you have is self-consistent.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

The diagram last attached should be viewed from the side (not from top or bottom). The view plane is perpendicular to the analysis plane. Concerning the optimization, the reference point should be the top edge of the door at start of the translation (0% deployed) and the upper hinge coordinates (B0) can be considered as stroke lower limit point (far left or start point) which moves in translation direction to another end (B1) as the door deploys. All coordinates including for upper and IFS hinges except for the upper edge are optimal for minimizing the load at IFS hinge (V1). Some of the initial and ending angles of the door and drag link may be considered as design constraints depending geometric and equilibrium conditions. It is assumed that the given pressure loads on the door correlate to the percentage of the stroke distance. For the model, we just apply 8 stroke steps (20%, 40%, 60%, 70%, 80%, 90%, 99%, and 100%) for analysis and optimization. If the stroke is contant, then the pressure pattern on the door at different stroke step is known.

Tom- as you reviewed the template for ANALYSIS part, have you found any mistakes made in addition to the comments you gave earlier, based on many communications we had on this topic? Thank you.

Another question: Is it possible to generate �header/footer� in a mathcad document that completely matches that in Word document? If yes, how? I�m now trying to draft a report using mathcad, but could not make it due to the problem with entering title and logo to the header/footer. Thank you again for sincere help.

GR

• ###### max value of function and optimization
You have not answered my questions, and your statements leave me completely confused.

I don't know what you mean when you say that the diagram should be viewed from the side. The diagram is a 2D projection and can only be viewed from one direction -- along an axis perpendicular to the paper. What I need to know is how this diagram relates to your previous diagram for the geometry. The only way I can make sense of this is if the view axis on this diagram (I'll call it the cylinder diagram) corresponds to the vertical axis in the geometry diagram. And the horizontal axis in the geometry diagram corresponds to a radius in the cylinder diagram, with the left side of the geometry diagram corresponding to the outside of the cylinder diagram. Is this correct or not? If not, how do the axes of the two diagrams relate?

You cannot use the stroke with the existing parameters to define the stroke in general. The stroke will change with the position of the three hinges. You need to define, logically, what determines the ends of the strokes. I asked if the doors meet side to side when fully deployed. If yes, then given the dimensions of the doors and the size of the slot the fully deployed positioned is fixed and can provide a reference point. If not, what is meant by the fully deployed position? The position of the door when fully deployed with the current parameters? If so, both T and the position of the end of the stroke will vary depending on the position of the top hinge.

And what defines the zero percent deployment. It cannot be the existing stroke length. With some positions of the hinges it will be physically impossible to achieve that stroke length. Is it that strange negative door angle (which is not consistent with your pressure calculations)? You say the position of the top of the door. But, without further qualification that seems unlikely. When the door is near horizontal the top edge of the door will be near T above the floor. But T will vary with the position of the top hinge. Fixing T means fixing the position of the top hinge. That removes the possibility of optimizing the location of that hinge.

It is physically unrealistic to assume that the pressure loads depend directly and invariantly on the percent of stroke. The only variable I see for which that might be true (and there are questions even about that) is the door angle. Tghe door angle is well defined, regardless of the hinge positions, and can be expected to be the major determinant of the pressure load. But the horizontal and vertical (referenced to the geometric diagram) positions for any angle will vary, depending on the hinge positions. So there is a question of how much such variation in position affects the pressure loads.

I have not noticed any errors that I have not commented on. That doesn't mean there aren't any. I have not looked over the sheet like a teacher looking for all errors. I am only looking for enough information to be able to show you how to do the optimization. This might also be a good point to add that if you want help you really need to be more responsive to the questions asked. I find that I am repeating myself, asking the same questions and making the same points repeatedly. That is no fun, and the only reason I participate in this forum is for fun.

There is also a thread in the Electrical Engineering section, max and min of array, that has a considerable similarity to your problem. You might want to read through that and see the problems caused by estimating maxima and minima by sampling.

Tom Gutman
• ###### max value of function and optimization
Now we're getting somewhere.

You say there are two slots. In you worksheet you have an entry of 30° for the slot. Is that the size of each slot? So that the two slots together take up 60° of the circle and the 12 doors take up the remaining 300°? Or are the slots the same size as the doors, and the 12 doors are actually ten real doors and the two slots?

Why is the term cylinder not appropriate for the aggregate of the inner and outer panels? Naively I would expect that the panels are either flat, with each door (or slot) fitting between flat plates, and the plates fitting together so that the cross section is a polygonal approximation of a circle; or that the plates are ars of a circle on a common center, so that they fit together to yield a circular cross section. So what is the actual shape of the plates? With the door top/bottoms being curved I would expect arcs of a circle.

You have the doors meeting each other and the outer plate when fully deployed. With the sleeve at a fixed distance from the outer plate your upper hinge position is fixed. You cannot vary it. You have, at most, two degrees of freedom, the lower hinge and the drag bar length. If any aspect of the door postion in the stowed position is fixed, that will reduce you to one degree of freedom. I don't know whether that is best handled by doing an optimization on two variables with an equality constraing or whether the constraint can be solved to reduce the problem to a single variable. That will require further specification of the actual constraint, and some analysis of the geometry.

Assuming a straight line instead of a curve certainly simplifies the area calculations. But it also makes them wrong. The point of using a system like Mathcad is so that you can do more complicated calculations and don't have to approximate the system by something you can calculate on a slide rule. So what do the dimensions given for the doors really mean? Are they the dimensions of the trapezoid joining the corners of the door? Or are the ficticious numbers that are in some way supposed to be "equivalent" to the actual door? The problem is that you need accurate areas for the force calculations, accurate area distribution for the moment calculations, and accurate positioning of key points (like the corners) for the geometry calculations.

� � � � Tom Gutman
• ###### max value of function and optimization
The slot at the bottom is much smaller than the top one so it was negligent.

The attached is diagram for outer panel and the door at stowed position. Drag links shown are towards inner surface panel hinge locations. Overall outer panel is arc but with upper and lower edge portions (the red lines) being machined to flat to fit with the tracks of the box structures and move horizontally (z).
• ###### max value of function and optimization
Just as a note, avoid posting BMP files -- they are overly large. Convert them to GIF, PNG, or JPG files before posting.

Ah, so the entire outer panel moves. Presumably there's a short fixed rod (or similar) connecting the outer panel to the upper door hinges. So the upper and lower hinges are actually on opposite faces of the door. That doesn't really affect the geometry, except that the eliminates a possible constraint, a horizontal driving rod.

Just to be sure, the upper edge of the door is flush with the outer panel when full deployed. And the sides of the doors meet along their whole length when fully deployed. It seems to me that to accomplish this the upper hinges must be on the line joining the two upper corners. It will be interesting to see if the geometric calculations are consistent with this.

Just how "negligible" is that lower slot? You geometric information is given with three decimal places. Is that lower slot smaller than .001 radians?

I have started looking some more at your sheet. There are questions and problems.

Why the odd choice of given data? Why the positions of the hinges at the two extremes? For the plate you have one actual measurement, from the lower hinge to the bottom edge. Why that one measurement, and not the other measurements of the plate? The hinge positions appear to be too consistent to be the results of actual measurements, they seem to have been calculated. Why can you not just use the data used for those calculations?

For the IFS hinge position you have one regular setting (in yellow) and a set of alternatives (in green). What are those alternatives? I don't think any of them are compatible with your other hinge positions. But in your sheet you did have one of them enabled rather than the base value.

You have a problem with some of your angle calculations. The inverse trigonometric functions are not really functions (the trigonometric functions being periodic, and generally non-monotonic). To make a function rather arbitrary principle values are used. For the arc tangent the principle values are in the range ±π/2. You cannot use the arc tangent to find an angle that is not already known to lie within those limits. And at the stowed position α exceeds π/2. In general it is best to use atan2 or angle for determining angles, depending on the desired range. Both provide a full 2π range and so can be used for any angle.

What is the "Blocker Door upper edge coordinates"? I would think, from your previous statements and your usage that it would be the position of the center of the top edge of the door in the stowed position. But it is not in line with the two door hinges in that position -- not even close. Is it something else? So far calculation have bee done assuming that the hinges are actually in the door. Are they significantly separated from the door? If so, the geometrice calculations need adjustment for that difference. But the direction of the discrepancy seems opposite to what would be expected from offset hinges.

Your calculations of the top and bottom widths are problematical. You are calculating the length of the arc that is the orthogonal projection of the edge of the plate (at full deployment). This is not the length of the actual edge, nor the distance between corners.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-
Each door upper hinge is connected to the outer panel with a bracket (fittings) and the design was made to enough space for the door to rotate without interference of the lip of door with the panel.

I think you are now getting clear about overall relationship between components and kinematics mechanism involved. We return to the original question- analysis and optimization of a SINGLE door.

For ANALYSIS, all the coordinates of the hinges and the door upper edge are known; stroke, etc are given in the blueprint so we just need to apply them without making arguments about them. Some parameter like distance between door lower hinge to the bottom edge which is given also is used to calculate the door area and forces only.

The work sheet you reviewed flows like this:

Angles calculated for drag link and the door are consistent with what I originally designated for and angles on the blueprint.

Alternatives in green for IFS hinge positions were used to test influence of change of IFS hinge locations on the reactions. It is for test only. Only one either yellow or green one is needed to be active.

The blocker door upper edge coordinate is for the extension line of the door's upper hinge which is not shown in the diagram (but it was explained early, I recalled)

For OPTIMIZATION, I'd set up the following as knowns:
Door upper edge (stowed)position, T (distance bwt door upper hinge and IFS hinge, valued as in ANALYSIS), Stroke, initial door angle (stowed, valued as in analysis) AND/OR drag-link angle (fully deployed, valued as calculated in analysis)

The hinge positions are design variables which would affect the door area and reactions at IFS hinge. Set the minimum IFS hinge as optimization objective which is subject to geometric and equilibrium constraints.

GR

• ###### max value of function and optimization
Yes, the overall configuration now seems clear. Clear enough that I thought I could start on some calculations. So I went back to the sheet and started adding some calculations, checking the data for consistency.

And ran into the fact that the data are not consistent, at least not with the assumptions that have been being made. The upper edge of the door is not in a line with the two hinges. So either the door is not a flat plate (but curves), or the hinges are not on the plate but significantly distanced from the plate. Or that position is not actually the position of something on the door plate. Which is it? I hope, and assume, that it is the hinges off the door. In which case I need information about these hinges. First, I would like to confirm my previous conclusion (from your last diagram) that the lower and upper hinges are on opposite sides of the door, with the lower hinge on the front side and the upper hinge on the back side. I stated this assumption earlier, but you did not confirm nor contradict it. Are the upper and lower hinges identical (other than the side on which they are mounted)? If so, there is enough information to calculate their offset from the plate. If not, you need to know the distances of the hinges from the plate.

Ah, so there are blueprints. Shouldn't the blueprints include the shape and dimensions of the doors? And the positions of the hinges relative to the doors? Why try to calculate those? And as long as we're on the subject, exactly what is that distance between the lower hinge and the bottom edge? Is it the distance in space between the hinge and the center of the bottom edge? Or is it the distance on the door plate between the center of the bottom edge and the perpendicular projection of the hinge on the plane of the plate?

You talk about the extension line of the upper hinge. I don't remember that term and don't know what is meant by it. If you mean the extension of the line between the hinges, the upper edge of the door is not on that line.

I have some problems with the fixed and variable values for optimization. I understand that the stroke is fixed, presumably being created by an external mechanism which is not subject to modification. Fixing T is reasonable, and means that the mounting bracket for the upper hinge is fixed and not subject to modification. Fixing T means that the position of the upper hinge on the door can vary only as the door angle at full deployment varies. But why fix the upper edge position? That is just floating in the middle of nowhere, and seems completely unimportant. And why is the initial door angle fixed? I can see some limit on the absolute value of β, as the door needs to be sufficiently horizontal so as not to interfere with the air flow. I can see a lower limit on β, as the bottom of the door must not go through the roof (but note that in the actual configuration, as the stowed value for α is over π/2, β goes through a minimum on its way to the stowed position). But why fix the current angle, which seems by and large arbitrary? And why fix the drag link angle at full deployment? Why is that not simply determined by the hinge positions (including the IFS hinge)? You state that the door area will vary with the hinge positions. That implies that the door angle at full deployment is not fixed, but is allowed to vary. That means a different size and shape for the door.

Overall it seems that you are constructing completely new doors with new hinge positions, and repositioning the IFS hinge (and constructing a new drag link to match), with the rest of the mechanism being left alone. That opens the question of what determines the lower door edge -- I would assume that the distance from that edge to the inner cylinder, when fully deployed, is fixed.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

Suppose we call the diagram last attached as cylinder diagram as you called and the one prior to that called drag-link diagram. In these two diagrams I did not give axis notation which may cause some misunderstanding. For the drag-link diagram, now we can define the horizontal line (dish) as x, vertical line as y, and line perpendicular to x-y plane as z. Now consider the cylinder diagram. Suppose one door is separated for analysis purpose and rotates its central vertical line to be at 12'clock position. Then, horizontal is defined as x', vertical as y' and z' perpendicular to x'-y' plane. so the two diagrams correlate to each other with

x=z'
y=y'
z=x'

so the cylinder diagram (fully deployed)is viewed from x or z' direction.

The stroke is supposed to be a constant for this analysis. Actually, the door is attached to another component (not shown) called translating sleeve with the fittings. The translating sleeve moves horizontally and the door follows. The door also has a rotation relative to the sleeve through upper hinges. So the stroke is actually for the moving distance of the translating sleeve from its original position (stowed). Since the door, particularly the upper hinge position is completely follows the sleeve, you can select those locations as reference points for analysis.

GR
• ###### max value of function and optimization
It's getting a bit clearer. The doors are arrayed around a circle. When fully deployed the upper and lower edges of the doors form, approximately, outer and inner circles. The airflow, and the movement of the stroke, are parallel to the axis of the circle (perpendicular to the plane of the circle). For absoute orientation you say that the sleeve moves horizontally. So the circle is vertical, with a horizontal axis. The geometry diagram shows a side view parallel to the circle, showing a door at the top of the circle. The cylinder diagram is a view perpendicular to the circle (along the direction of airflow and stroke movement). It show the slot at the top, and would need to be rotated to put a door at the top in order to match the geometry diagram. Is that finally correct?

Now to repeat a few questions. Do the doors meet when fully deployed (the cylinder diagram seems to indicate that they do)? You mention inner and outer cylinders. Are these physical cylinders, with the doors and their mechanisms between these teo cylinders? Do the doors touch any part of these cylinders in either the fully deployed or full retracted positions? If so, exactly what contacts occur? Are the cylinders truly cylinders (with circular cross sections)? Or are the actually polygonal, with a pair of flat faces bounding each door? Are the top and bottom of the doors actually straight (as shown in the door diagram)? Or are they curved (as shown in the cylinder diagram)?

You say that the stroke is fixed by the mechanism, so that the positions of the upper hinge at zero and full deployment are fixed. That severely limits the possible geometries. What are the constraints on the position of the doors at the two limits? I assume that the zero deployment needs to be close to horizontal. How close? Does it need to match the angle of the existing geometry? Is the stroke position completely fixed, or only fixed in the direction of the stroke (i.e., can the distance of the upper hinge from the inner and outer cylinders be varied or is that also fixed)?

I assume that the radii of the inner and outer cylinders are fixed (the radii of the inscribed cylinders if they are actually prismatic). If the doors meet when fully deployed then the dimensions of the doors fix the inner and outer radii of the doors, which fixes the position of the doors relative to the inner and outer cylinders. If the distance of the sleeve from the inner and outer cylinders is fixed, then the position of the upper hinge is also fixed. There is not much point in trying to optimize a variable whose value is fixed. I am trying to find out what the limiting conditions are to determine what (if anything, it is possible that you have enough constraints to fix all the parameters) can be varied for optimization.

� � � � Tom Gutman
• ###### max value of function and optimization
You are right about the orientations. There are two slots in the structures, one is at the top as indicated in the diagram and the other is located at the bottom in y� dir (not shown). Each slot is filled with a box shape structure incorporated with tracks at each side of the box to support the translating sleeve panel (a big and approximate arc panel or called outer surface panel). The sleeve slides on the tracks and moves horizontally. The mention of rotating the door to the top was to just want you to understand the orientations clearly. Since the analysis is for a single door (all doors are assumed identical), any central line passing drag-link/door hinge (or mid points of 2 upper hinge points can be defined as y axis and the plane passing the y axis and perpendicular to door surface forms an analysis plane, as in drag-link diagram.

You are right. The doors will meet when fully deployed, but there is a small gap clearance present between the door bottom edge and the inner surface panel. Use of the term cylinder is not accurate. It is appropriate to use outer surface panel to stand for the movable translating sleeve and inner surface panel for the fixed part. The outer surface panel and inner panel form the passage to let the air go through at 0% deployed (stowed phase) and block air flow during the deployment and redirect the air out from the exposed vents (not shown). The top and bottom of the edge is in curve shape, but for simplification of area calculation we assumed that they are straight lines

The motion pairs among the components are as follows:

box structure's track (fixed) /sleeve (moveable) = sliding pair
sleeve/door = hinge pair (rotation only)
drag link/inner surface panel (fixed) = hinge pair

Yes you can assume that the radii of the inner and outer panels are fixed, together with given door's upper edge coordinates, stroke distance, and some constraints of start and ending angles of the door to optimize the best hinge locations and capture the minimum the load reacted at on the inner surface panel/drag link hinge.

GR
• ###### max value of function and optimization
"For ANALYSIS, all the coordinates of the hinges and the door upper edge are known; stroke, etc are given in the blueprint so we just need to apply them without making arguments about them".

_______________________

Well ! if it would be that simple and clear in the work sheet ... it would have been done already.

Was just a comment.

"apply them without making arguments about them"

... Jean, install my 48 inches valve in the 4 inches pipe .

It looks like you have a data table of values, that the table may not be square and you want some optimization applied ... ?

Just trying to help, not bother .

Re-post your latest work sheet, never know ?

jmG
• ###### max value of function and optimization
Yes, I understand about the central plane for analysis. The issue is the position of the upper hinges relative to that line. With the doors being curved, moving the hinges out towards the edge also moves the forward, possibly beyond the analysys front surface. I'm already assuming that the hinge mountings take care of the slope involved in the curvature, so that the two hinges can be considered as a single hinge.

You cannot simply consider the doors as flat. That overstates the force. Based on curvature in only one direction, and symmetry, you could probably project the door onto an equivalent flat plate. But at this point it's not clear if doing that projection is any simpler than simply doing the integration on the actual surface. But at this point the position of the hinges with respect to the door surface is still open. And since the information you have is mainly for the hinges, the positions of the door itself is still open.

And I still don't understand the constraints for optimization. What is special about the particular angle of the stowed door? I don't see that it minimizes the interference with the air flow (I would expect that to happen at a β of zero). And it doesn't even stop at the first occurance of that angle. The stroke continues past that point, with the angle continuing to decrease, until it eventually starts increasing again and again reaches the final angle. It seems that the stowed position of the stroke was chosen for reasons other than the angle of the door. Note that fixing T and fixing the angle of the stowed door means fixing the position of the upper hinge.

Also, it is not clear if the IFS hinge position is changeable. I thought from previous messages that it was, at least as to station (the radius is fixed by its attachment to the inner cylinder). Now it seems it might be fixed. If so, and if the angle of the drag bar a full deployment is fixed, you might have no degrees of freedom left for any optimization. The fixed position of the stowed door will result in a relationship between the lower hinge position and the drag link length. The fixed angle of the drag bar will result in a, probably different, relationship between the lower hinge position and the length of the drag link. The expected result is a unique solution for the position of the lower hinge and the length of the drag bar.

BTW, I specifically asked about the relationship of the upper hinge and the front surface, since the choice of mounting could make that anything. And I never did hear back about exactly how "negligible" that second slot really is.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

The door thickness is assumed as 1.1 in. The offset of the door hinge plane (analysis plane) from the wind loaded door surface side is less than that, it is 0.8 in. Whether the door is curve or flat is not a big deal in terms of calculation of force worked on the surface, since the pressure load normal to the surface will be eventually projected to the analysis plane to gain a resultant force and a moment. The resultant force on the door is dictated by the door measurements only and the loads transferred to the hinges are determined by how the hinges were distributed on the doors and on the inner surface.

My presumption was to place the door's wind loaded surface onto the door hinge surface (analysis plane) for simplification, not including offset influence, otherwise it may be too much involved. In this way, both door hinges are moveable along the central line as design variables. The door's upper edge coordinates are the base for determination of the door dimensions, it is 199.733in (x) for station and 58.533in for radial (y) (I'm not if they were given on the previous sheet).

From the analysis sheet, the stowed angle for the door is about -8.75 deg, which would led the door's surface to smoothly merge with the inner surface of the outer panel to form good air flow passage, as explained before. (The passage is not exactly in the cylindrical shape, it is a cowl shape). When in stowed position, there is no movement for any parts. As the door (outer panel) deploys, the door starts rotation clockwise only, and the drag link follows and rotates clockwise too, until the fully deployment occurs. The door angle starts as -8.75 deg, then -6.04 deg, -4.48deg, 22,53deg, ... , and 68.11 deg at full deployment. The rotation is in consistence. Similarly for the drag link, it is rotating from 91.91 deg to 14.6 deg. All the angles are obtained relative to the horizontal line.

The upper hinge position can be moved horizontally in terms of design consideration, even constrained by T value.

The IFS hinge is changeable, giving upper and lower bound of about 3in relative to the original coordinates.

The bottom slot angle is about one-thirds of the upper slot angle. It was not included because of conservative considerations with regard to the resultant force derived in that way.

GR
• ###### max value of function and optimization
OK, I think that is enough information to get the dimensions. So both hinges are .8" from the front surface, as measured by the projection of the hinge into the central plane (the plane through the center line of the door and normal to the surface at that centerline) perpendicular to the center line. The distance between the upper hinges and the front surface at the point of attachement will be considerably greater, due to the curvature of the door. I assume that the hinge mounts provide for that offset, but you might want to verify that.

You can't just ignore the curvature of the doors. That defines the geometry of the doors. You could project the doors onto a suitable plane and do the calculations on that projection. But you first have to do the projection, you can't just unroll the doors. And the projection ends up with elliptical edges. Ellipses don't integrate very well, and I still have hopes for integrals that work out. We'll see about that ...

Because α starts out over 90° the door will originally rotate counterclockwise as it starts deployment. Not by much (the maximum α seems to be 91.999°) but by some amount.

I don't understand about the upper hinge being moved horizontally. With respect to what? The actual horizontal position (station) has not been an issue (but see below), only the relative horizontal positions of the upper and IFS hinges. Or do you mean horizontally with respect to the door (whose vertical is set by the upper and lower edges). That would mean the distance between the front surface and the hinge.

I had thought that the possible adjustment was the distance from the hinge to the top edge of the door. Changing that will cause the stowed door to move, changing the distance between the door lower edge and the outer panel. Is that acceptable? You latest statement suggests that in the stowed position the bottom edge of the door is in contact, or close thereto, with the outer panel.

There is a further problem with changing that hinge position. It will change the angle of the deployed door, and so the point at which the upper edge meets the outer panel. With the assumption of a cylindrical outer panel that did not really matter, the radius of the meeting point was the same, no matter the station. With a non-cylindrical outer panel the radius will vary with the station, and we need to know the shape of the panel to determine that radius.

There might be a similar problem at the other end. The radius of the lower edge has been assumed fixed. With a cylindrical inner surface that would keep a fixed clearance. If the inner surface is not cylindrical keeping a fixed clearance requires adjusting the radius based on the station.

There is another potential problem. The upper hinge has to be behind the line of the upper edges (in the deployed position). That means that the entire upper edge will, at least initially, move down (smaller radius) as the doors start moving towards the stowed position. But the ends of the edges, the corners, are in contact with the adjacent doors. And at a smaller radius there is simply not enough room for the chords (lines joining the corners). I have no idea how that is handled (or if there is something wrong with my analysis), so I intend to pretty much ignore this.

� � � � Tom Gutman
• ###### max value of function and optimization
Here's part of the work. Note how the hinge geometry is encapsulated prior to any values being specified (you have some global assignments, which really should be just ordinary assignments, but I'm ignoring that) so that the calculations can be used with any parameters.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

As the curve projects onto the plane, seems to me the projection would look like a symmetric trapezoid, as shown in the previous diagram. The force accumulated either by projection or integration at the central line of the door in the direction perpendicular to the line as shown in previous diagram) should be close.

You�re right. As the door starts moving from the stowed position, there is a very much little rotation of the door counterclockwise because initial link angle is larger than 90 deg. The door rotates clockwise afterward.

T is given as a constant, relying on the distance between the upper hinge center (y value- vertical) and INITIALLY given IFS hinge center (y value-vertical). To my concern, upper hinge position (x value-horizontal only) and IFS hinge (both x, y) are designable parameters. Since the upper hinge moves horizontally (x-dir), the variation of the upper hinge can only be made along this direction. It is not relative to the door itself.

As the door is in stowed position, the door was designed to seat on the pads supported by the outer panel. So the door stowed angle design should be made to be consistent with this to form suitable channel for air flow. Keeping the stowed angle of the door as required is a must constraint to avoid any impediment to air flow movement in the stowed phase.

It is assumed that the design allows for enough clearance between the door upper edge and the outer panel to avoid the interference as the door deploys. The inner surface is assumed cylindrical, with gap of 0.75in to the full deployed door lower edge.

I could not quiet follow your last point. Each individual door is separated each other. There may be design tolerances available between the adjacent doors to keep them no interference at all operation phases. Need to mention there is a plate spring designed at the lower hinge area working with the drag link as preloading component to secure the door firmly. it was not considered in this analysis work.

GR
• ###### max value of function and optimization
Tom-

Points in red are for hinge centers and upper edge of the door where the coordinates are given. Solid lines are for stowed position while dash lines for fully deployed. The door upper edge is not exactly on extension line of the door two hinges and there is an offset between them. Door hinges are on the same of the door, towards the outer panel side. The drag link goes through the door to connect the door with the lower hinge. The door is in minor curve, not flat.

The blueprint does not offer the area which is needed for calculation. The distance btw the lower hng and bottom edge is for the distance on the door plate between the center of the bottom edge and the perpendicular projection of the hinge on the plane of the plat, as you guessed.

What I meant for fixing the upper edge position is to select it as reference point for calculation of door dimensions/area. The designable door hinge locations are also relevant to this point. Fixing the initial door angle is to allow for placing an appropriate angle of the door to fit the outer surface panel so that the channel btw the inner and outer surface can receive better air flow at the stowed position. Fixing drag link angle at fully deployment is to avoid the interference of the link with inner surface panel with assumption that that the inner panel may not be in cylinder shape (bumpers may exist somewhere after the IFS hinge). The values for above constraints may not exactly same as those used in analysis step, but it is more effective to get an easy comparison between old design (analyzed one) and one with optimization approach.

You�re right. We need to construct a new door with new hinge positions, but applying as many as constraints used in the previous analysis process. The distance from lower edge of the door the inner surface is a gap (assumed as 0.75 in). If you review the sheet, you can see that in the analysis, I used the door fully deployed angle (beta1) combined with entire height of the door to calculate the perimeter of the circle made of all door�s bottom edges. As the door is fully deployed, you can consider the doors patterned (distributed) as a truncated cone, with all upper edges of the doors being as the front end of the cone, and all lower edges of the doors as aft end of the cone.

GR
• ###### max value of function and optimization
... maybe you should use CATIA .
My friend has a CATIA business, but I don't know much about the "maths" behind . However, it will open/closed the door(s) on the screen and plot all the gaps vs tolerances +++.

jmG
• ###### max value of function and optimization
So the doors are curved (approximately) around an axis parallel to the line from top to bottom. This curvature is such that the doors fit together as a truncated cone. The top and bottom edges are therefore actually segements of a circle. It's not clear exactly how that affects the pressure calculations. It might be possible to project the door onto a plane paralled to the central axis (and symmetric sideways) and calculate pressures on that plate. Or it might be easier to just do the integration directly on the curved door.

Your picture raises a number of new questions. The door appears to have appreciable thickness, with the hinges embedded within it. For pressure calculations it is presumably the front surface that matters. What is parallel, or in line with, what? The back surface appears not be be parallel with the front surface. The hinges seem not be both on the back surface (the upper one is, the lower one seems to be depressed below that surface. From the diagram it looks like the line between the hinges might be parallel to the front surface. Is it? The top edge seems not to be in line with either the hinges nor the main part of the front face. Where is it located relative to these?

I am inclined to simplify the geometry a bit by extending the front face to the point where it meets the outer cylinder (when fully deployed), treating the actual displacement of the upper edge as an insignificant beveling. We need to know the position of the actual top edge with respect to the rest of the door to calculate the position of this virtual top edge.

The geometry of the doors collectively (in the fully deployed state) can be defined by three parameters. The radius of the upper edge is the radius of the outer cylinder, and should be known, or can be calculated. The radius of the lower edge is the radius of the inner cylinder plus the clearance. We don't know the height of the cone, as that depends on the position of the upper edge with respect to the main part of the front surface. If we assume that the two hinges are equidistant from the front edge then we would have the angle of the cone, and so know the cone. The geomety of an individual door is then gotten by considering the angle subtended by the door. The position of the door with respect to the hinges is not known, but the constraints allow only motion parallel to the cylinder axis. And AFAICT the position along this axis is not germane to the calculations.

There is still a problem with the upper hinge. That is, according to the early diagrams, not a single hinge on the center line but a pair of hinges off to the side. Presumably the mounting takes care of the slope of the sides so that the hinges have common rotation axis. But the sides are also displaced towards the front surface. Unless the mounting offsets that the upper hinge can be expected to be significantly foward of the lower hinge -- possibly even in front of the nominal (center line) front surface. Do the mountings provide the necessary offset to align the upper hinge with the lower hinge?

� � � � Tom Gutman
• ###### max value of function and optimization
Tom-

This is a 3-dimension problem, but we have to simplify it to 2-D for analysis and optimization. The analysis plane is x-y, formed by the door central line passing lower hinge center and the midst point of the two upper hinge centers. So you can take a virtual hinge at the upper which functions or is equivalent as the upper two hinges. This will make calculation much easy. The door is curve, but when calculating door area and length and distribute pressure load on it, you just should consider it as a plate for simplification.

GR
• ###### max value of function and optimization
There is indeed a plane where the projection of the door is a trapazoid. But this plane is the horizontal plane (based on your hinge diagrams). The lengths of the upper and lower edges of the trapezoid are the cords of the arcs formed by the edges of the door (not the arc length) and the height of the trapezoid is the horizontal projection of the height of the door. And the net force is definitely not perpendicular to this plane. In any other plane the projection will have top and bottom edges which are sections of ellipses.

The doors are sectors of the frustum of a cone. Orienting the cone on the hinge diagram the axis of the cone is horizontal, along the x axis. The door is a sector centered along the top of the base circle, centered on the y axis. The base and top of the frustum are circles parallel to the y-z plane, perpendicular to the x axis. Projected on the x-z plane the door becomes a trapezoid. Projected on the y-z plane the door becomes the sector of an annulus. Unrolled the door is again a sector of an annulus, but with different inner and outer radii and different central angle.

So the IFS hinge position is adjustable in two dimensions -- you can raise it off of the central cylinder by an arbitrary amount (and possibly sink it into that cylinder). But I am unclear about the upper hinge. There are two mountings for the upper hinge, one on the door and one on the outer plate. What are the designable values for each? I assume that the position on the door can be varied along the center line of the door, represented by the distance from the top edge. What about the distance from the hinge to the front surface (currently .8")? Is that fixed or designable? By a the horizontal position of the upper hinge I assume you mean the postion relative to the outer plate. Adjusting that means putting the mounting bracket at a different location. But the outer plate is not cylindrical, so moving the horizontal position of the bracket means changing the radial position of the base of the bracket. If the size of the bracket is fixed, then the radial position of the hinge will vary. To keep the radial position of the upper hinge constant you have to change the size of the bracket. But if you can adjust the size of the bracket, why presuppose that the radial position of the upper hinge is fixed, rather than designable? Also, unless there is some other relevant reference point on the outer panel, why move the mounting bracket at all? The relative horizontal position of the upper hinge can be adjusted by moving the IFS hinge.

The stowed position of the door is noe being limited by those pads. What kind of flexibility is in those pads? The pads contact the back surface of the door, I presume fairly low down. Is that contact position fixed, or can it vary? Are the pads stiff enough that the angle of the door is fixed, or is a reasonable variation possible? Even a few degrees of tolerance might allow considerable freedom to the rest of the parameters. And is the position of the pads on the outer surface fixed, or is that adjustable? If adjustable, the curvature of the outer panel should result in changes to both the radial position and the angle of the stowed door. Depending, the position of the stowed door might be completely fixed, allowed to vary only by translation along its axis, or might only require that it be at a proper radial distance at the point of contact.

If the door position can be adjusted there is the matter of the relationship between the front and back surfaces. The diagram of the door cross section showed thes as non-parallel. Constraints based on the back surface would have to take that into account. Either that, or the door design would need to vary that relationship to match ther requirements.

Do you have the designed value of the inner cylinder? My calculations seem to indicate a value very near 38".

My understanding is that a full deployment the doors meet side to side to form a continuous surface, the frustum of a cone (ignore the slots). In particular, the upper corners of adjacent doors are in contact. The chords joining those corners then form a polygon inscribed in the circle that is the base of the cone. If you move the doors so that the corners move inward, towards the center of the circle the radius of a circle through those points decreases. There is then not enough room in the circle to accomodate the original chords.

� � � � Tom Gutman
• ###### max value of function and optimization
The load analysis is to be on a single door. That is well and good. And if we had a full description of the door we could simply start from there and proceed with the analysis. But for some reason you are unable or unwilling to get the door information from the blueprints. You are trying to calculate the door parameters from other data. This calculation depends on the relationship of the door to the other components, including the other doors.

The doors are not all that flat. I calculate that in the deployed position at the top (relative to door) edge the sides of the door are about 1.8" below the center of the door. That does not seem negligible.

I am always leery of simplistic assumptions that a particular approximation is "close enough". I have seen too many serious mistakes (including my own) with such assumptions. A nice example of a situation where intuition as to the size of an effect is likely to fail can be found here. This collaboratory is about how to use Mathcad, and I consider that one of the important reasons for using a powerful tool like Mathcad, rather than just a slide rule, is that you can avoid most of these assumptions. You can afford to do an accurate calculation on the actual geometry and see what is actually important or not.

The spring may not be part of the analysis. But it is a part of the stowed geometry. If it does what I think it does, allow the lower hinge to move relative to the door face, then far from the stowed door angle being a fixed reference point you don't even know the angle of the stowed door. You know the angle of the hinge line but unless you have accurate information on the relationship between the hinges and the door you know little about the door. There is not much point to knowing the hinge positions to three decimal places if the spring allows the hinge to move an inch or two from its nominal location (relative to the door surface).

You still seem to miss my concerns about the upper hinge. I fully understand that the two hinges work on a common axis, and that for analysis purposes are treated as a single hinge on that axis at the door centerline. But the actual points of attachement are off to the side, not on that centerline. If the hinges are mounte .8" from the front surface at the point of attachement, the projection to the centerline could easily be as much as a half inch or so in front of the door face, rather than the .8" in back of the face assumed. This make a difference in the relationship of the door angle to the hinge angle.

As to movement of the upper hinge, my assumption was that the .8" (which a distance perpendicular to the center line, not a position along it) is fixed and that it is possible to move the hinge position along the center line while keeping the perpendicular distance fixed. I don't see why changing the position of the upper hinge on the door would change the bracket design at the hinge position. What happens to the bracket depends on how the stowed door position is assumed to be affected. If the stowed door position (not just the angle) is assumed to be fixed then changing the upper hinge position will change the position of the bracket on the outer panel. That will affect the length of the bracket (distance from hinge to outer panel) and the mounting of the bracket on the panel (different radius and curvature at the new attachment point).

I don't quite understand the fixing of the stowed door angle as matching the outer plate. Since the outer plate is curved (not cylindrical or conical) the angle is a matter of what you use as the reference point on the outer panel. This could vary if the door position is allowed to vary. Making a fixed angle for the door implies having a fixed reference point on the outer panel.

Which inner cylinder dimension? The value that the computed value approximates (38")? Or the actual computed value (assuming a .75" clearance) of 37.98"? And apply how? Which other value should be recalculated or ignored based on this value?

A big gap between the doors, rather than their assumed meeting, would mean a large overestimate of the door widths and areas. How big a gap? And how will that affect the pressure when different deployment angles are used?

BTW, I worked a bit more with the integrals of the conical sections, fixing up the area element and doing some additional integrals. Somewhat to my surprise the direction of the net pressure force is not perpendicular to a plane through the center line. It approaches that as the subtended angle approaches zero, but is noticeably different.

� � � � Tom Gutman
• ###### max value of function and optimization
While there are still questions about the actual geometry, here is the sheet with the force calculations done exactly, for the assumed frustum of a cone geometry. The resulting hinge forces are similar to those you calculated, although noticeably different. There seems to be some question about the sign convention you are using.

� � � � Tom Gutman
• ###### max value of function and optimization
Tom: just to thank you for your valuable input and effort to try to help me out with the problem. Glad to get back to this forum and hope to learn more from your experts in mathcad.
• ###### max value of function and optimization
I forgot about that spring. When the spring is active (deformed) there is presumably some change in the geometry. That makes trying to calculate the door parameters based on the geometry somewhat iffy. I don't see the effect of that spring in the data. The key distances appear to be the same in the stowed and deployed position. That leads me to think that the spring is installed such that it allows the distance of the lower hinge from the front surface to vary, along a line perpendicular to the front surface. Such motion would not affect the length of the drag bar, and unless fairly large would be unoticeable in the interhinge distance. But it would affect the relationship between the hinge positions and the door positions. That's why I would prefer getting the information on the door parameters directly from the blueprints, using whatever measurements are available there and, if necessary, deducing the parameters needed.

The attached is some preliminary analysis of normal forces in a cone. It looks like the offset of the hinge from the door face does not nicely drop out.

� � � � Tom Gutman