Hi All,
This is basically an engineering / math question.
Please review the enclosed worksheet & plot.
1 - I have 2 contra rotating spirals with their starting degree position 180 deg to each other.
2 - Both spirals share the same 360 rotation radius value (k2)
3 - The value k2 is derived from the time balance value function between V1 & V2 (including B values).
4 - The slope V1 is Fixed shown in dotted black. (not including B values)
5 - The slope V2 is shown in red and can have its slope adjusted by changing the value of miss in blue top left of spiral. (not including B values)
6 - <<= See plot & try changing the value of miss to any value for example try (i - 2.25 and (ii) 2.25, note positive & negative values here...
After changing values notice the position of the (blue circle marker) on the outer circle moves its position above or below the fixed slope of V1 (black dotted line). Also note the Volume values of V2 & V1 in the yellow and orange boxes, (respectively), on right side of the spiral plot.
The yellow box retains the value of 110592 even though the slope of V2 has changed... But...
The orange box value has change from 110592 to some other value.
Also note that the spacing between the spiral rotations (k2) changes...
7 - The GOAL of this worksheet & my question to you is :- Are you able to setup the CORRECT configuration here so that when the miss value = 0.25 then the 2 volume values in the yellow & orange boxes = 110592 AND the blue circle marker falls exactly on the V1 slope.
Further explanation of the goal :-
1 - The GOAL of the configuration.
when miss = 0.25 then the 3 equalities are satisfied. ( I will explain 3 equalities later.)
when miss does not = 0.25 then the 3 equalities are not satisfied.
It's all about configuring to find the value of miss = 0.25.
2 - We have 2 slopes V1 = 1/6 which is a known Fixed slope AND V2 = 1/(2.75+miss)
In this example for the purpose of setting up the CORRECT configuration let's say we have the value of miss should be = 0.25.
3 - The purpose of the spiral configuration is :-
(i ) - Create & use a common time balance value between V1 & V2 = k2
(ii) - Within this spiral configuration there should be a trigonometric setup, ( not done yet), that defines the k4 angle in such a way that the required angle of 8.9...deg is isolated.
( It needs the Trigonometry 3 sides trig approach. We know RadiusV1 & RadiusV2 & symbolically can work out the 3rd side, I think. Therefore we should be able to find the angle 8.9...deg.... I think, but I don't know how to do this.)
(iii) -There will be multiple answers for k4 degree for values of miss value. Therefore we use the spiral AND repetitions of time / degree and KNOWN volumes from the V1 slope for further equalities to complete the overall configuration.
4 - To complete the configuration & achieve the goal of "extracting" the value of miss = 0.25 we need to introduce a 2nd pair of spirals with similar configuration with the only difference being the slopes.
Still using the known V1 = 1/6 but changing V2 to V2odd = 1/(2.75+miss)*squareRoot of 2.
By using the new V2odd slope another time balance value will created instead of the value for k2 ( we will obviously have to rename k2 to avoid confusion.) But now we have another spiral configuration that produces similar V1 volumes on the V1 slope line using the SAME miss value = 0.25.
If we have found a way to use the trigonometry 3 side approach to "lock" the k4 angle then the miss value = 0.25 can be isolated by setting up 3 equalities as follows:-
Equality No.1 - spiral configuration of V1 & V2 = known VolumeV1 at Time 1.
Equality No.2 - spiral configuration of V1 & V2odd = known VolumeV1 at Time 2.
Equality No.3 - spiral configuration of V1 & V2 known VolumeV1 = spiral configuration of V1 & V2odd known VolumeV1 at Time 3.The 3 values produced from this system of 3 equalities are :-
miss = 0.25 (ii) k2 from spirals V1 & V2 and (iii) k2(renamed) from spirals V1 & V2odd.
Obviously the miss value = 0.25 is a critical factor in determining the slopes of V2 & V2odd, therefore it is imperative to incorporate a trigonometric "locking" mechanism into the overall configuration.
Therefore, please could you review your worksheet to include the above.
8 - On first inspection of this worksheet it would appear that I have already achieved this result... BUT I have cheated or incorrectly configured this to demonstrate what I am trying to do. Most of this configuration is correct, but the key area or place is at k4. See the function relating to k4 marked in Green... I have simply taken the degree value difference between V2 - V1 = deg.
9 - Therefore I was wondering if anyone could provide the solution to achieving the required goal of this worksheet. My feeling is that the blue connecting line between the V1 & V2 circle markers should be defined & somehow restricted in some way so that the blue circle meets the fixed slope of V1 only when the miss value = 0.25
10 - Additionally I think that there is a relationship(s) between the FIXED RadiusV1 & the dynamic Radius & k2 that could be exploited further and would lead to the desired result of this configuration. But I have tried, in vain, for some time & haven't found it.
Many thanks for your help & attention.
Any help or advice would be greatly appreciated shown with working example please.
Best regards,
Lea...
PS - I am using Mathcad version 15
