An Essay on Higher Order Mathematics for Rotational Moment of Inertia A Cylindrical Mapping of Boolean Shapes By Joshua Milgram

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1 An Essay on Higher Order Mathematics for Rotational Moment of Inertia A Cylindrical Mapping of Boolean Shapes By Joshua Milgram Abstract: This essay makes three main estimates: (1) Boolean Volume (2) Boolean Mass () Boolean Rotational Moment of Inertia each via a cylindrical mapping of Volume. The hypothetical method outlined herein can estimate for these measures for geometric shapes and also for complex and non geometric shapes. Illustrations are based on this non geometric Lump Hypothetical D Volume This essay assumes a knowledge of calculus, but mostly just the integral. The theory uses a Cylindrical Volume Integral, along radius. Each partial radius defines a potentially unique, iterative height domain. At any partial radius there is a calculation for 2D Area. The integral for area values, along radius, outputs Volume. page 1

2 An Essay on Higher Order Math for Rotational Moment of Inertia A Cylindrical Mapping of Boolean Shapes By Joshua Milgram This essay has three main topics: (1) Verifying the Rotational Cylindrical Moment of Inertia Rule I = ((mass*radius2)/2) (for cylinder's only) (2) Demonstration of a hypothetical Higher-order Equation, which outputs simpler equations, such as ((mass*radius2)/2) () Demonstration of an estimate-equation that, in hypothesis, is a direct-mathematical (pre-safety-factors) calculation of Rotational Moment of Inertia, about an axis of calculated-rotation. The "Volume Integral for Moment of Inertia" section is a theoretical summary of such an equation for complex and non-geometric boolean shapes. Introduction and Mathematical Context: radius,circle,height order radius,height,circle order where 'M' is the Boolean Inclusion Ratio, variable over height/radius <=1 inclusion of one 2D circle, defined by a height during height integral, at a radius during radius integral. M = 1 example What is significant about this? circumference at radius '2πx' is multiplied by less than one multiplier 'M'. If a boolean shape has variable circumference inclusion at radius, and variable height, along a '0 to radius' integral, that is: Volume (area at radius defined as the integral along height domain of arc lengths) (variable arc lengths as variable Boolean Arc Ratio 'M') Knowing this, an estimate of the volume of complex and non-geometric shapes, is possible via calculation, using a specialized volume integral method, explained here... page 2

3 What follows is a lot of dense technical stuff, best calculated with programming. It's okay to not read this whole thing at once. Read at your own pace. The Volume-Integral, Boolean Volume Estimate simplifies to output a cylinder equation: V Thus verifying the Volume Integral as a direct-estimate of volume. Variables: height: Domain Range Length of Height, range between lowest and highest points, iterative at partial radius M Boolean Inclusion Ratio (<=1), along height = (Arc Length Sum) / (partial circumference) x The Independent Variable, along radius Constants: radius (largest) 2π to convert partial radius to circumference at radius Boolean-Volume Estimate Derivation ( radius(height(circle())) order) is a fancy way to measure Arc Length value-input for Area roundabout alone, this function-form allows for functional-scalars, such as non-uniform density (P), explained on page 4 calculates Hollow-Cylinder-Boolean Area value-input for Volume = V calculates Boolean Volume height domain is from the lowest point at radius, in the calculated area shape, to highest point. M is the (Arc Length over Circumference), Maximum Circle=1.0 iterative during calculation for L page

4 Boolean-Volume Estimate Derivation ( radius(circle(height())) order) The two volume integrals are essentially equivalent, simply in a different-arrangement. M is not-calculated-for during a radius,circle,height order calc, to assume that there is height to measure all-around the circle at partial-radius. The main difference from radius,circle,height order (previous page) is that the rudimentary-inner calc is for height. Similarities to Arc-Method: The first 2 steps still output an area. That area is calculated-with along radius just as with the arc-method. The volumes output are equivalent, roughly equal for complex shapes. Estimating Mass via Volume-Integral Applying this method to the complex-integrals, output boolean-ready estimates for mass: for 'D', uniform density OR 'P', non uniform density along iterative height page 4

5 Hypothesis: That 'Boolean-Mass', and thereby 'Boolean Rotational Moment-of-Inertia', can be estimated via a Cylindrically-Mapped Volume-Integral. (explained above) Furthermore, that multiplying the above equation for Mass, by another x, results-in Cylindrical Moment of Inertia. 2 Proof using calculated-cylinder: Using simpler equation, for cylinder only:...multiplying by x outputs x, thus: 2 Hypothetical "Cylindrical Moment of Inertia" estimate (for a cylinder) is: Solved for via calculator: (lump-sum cylinder) dividing by mass, as: outputs: Knowing that we just divided by mass, the Verified Answer, for a cylinder only is: Thus verifying, for a cylinder only: page 5

6 "Volume Integral for Rotational Moment of Inertia" With the supporting equation verified, I extend the hypothesis to estimate that: AND where: radius, max axial distance height, iterative length P, non uniform point density x, independent variable So... Why x?, you might ask: The integral for volume involves one x calculation-term. The x can either be part of: 2 π x * M for any iterative circle during radius,height,circle OR x as a main-term in the integral over radius. during radius,circle,height 2 Thus multiplying by another x, ( thus x ) scales the function along radius, such that the new equation estimates Rotational Moment-of-Inertia for Boolean Shapes. Rephrased: One ' x ' term is from the integral for volume or mass. 2 The other ' x ' term converts mass to rotational-moment-of-inertia. Combined, the simplified estimate-equation has an ' x ' term. Illustrated is a 2D hollow-cylinder-calculation example, with 'example height calculation' along circumference. Profundity: The significance of the hypothesis is that, if so, Rotational Moment-of-Inertia is: an x-to-the-third ( x ) function (cubic with offset) of partial-radius from axis-of-rotation, at 0D in-volume FEA example-points. Note: This estimate is not a measure of rigidity, nor structural-integrity, nor pivot-balance, only difficulty of axial-rotation. Furthermore, the calculation is a pre-safety-factor estimate. For a cylinder only, at constant P, the above simplifies to: page 6

7 Summary (Please Critique) Boolean-Shape-Ready Estimates: Rotational Moment-of-Inertia (RMOI) with Non-Uniform-Density: Estimate via "Volume Integral for Rotational Moment of Inertia" differing as : x where 'P' is iterative non uniform density at point, along height height is iterative length at rotation, along 2π 2π is a circular mapping at partial radius, along radius (the x from 2πx, has been simplified to the x term) radius is the main integral domain range, a length from zero at axis, to furthest point, in boolean shape 'x' is the independent variable along the radius range, representing partial radius. A Cylindrical mapping of arbitrary shapes around an axis, via a modified integral for volume, can estimate for mass ( linear x -value ), thus to estimate rotational moment-of-inertia (difficulty of turning, pre-safety-factor)...even for complex and non-geometric shapes. the inner-equation is third-power-x-value over radius ( x ), outputs: 4 fourth-power-x-value total-radius ( x ), over 4, multiplied by iterative area. Partial-equation, neglecting area shows the fourth-power term: Summary: Though the inner calculation is for x, multiplied by iterative area-calc at partial-radius. 4 ignoring area to examine power-equation, the outer-function is fourth-power x, over 4. This theory is organized for all tangent-rotation vectors, along height-calcs, to be parallel. page 7

8 Hypothetical Estimation of Rotational Kinetic Energy - Lump Sum where ω is the angular velocity about the rotational axis. Vector Bundle for Rotational Kinetic Energy Estimate Using the ω2 in the equation, allows for a calc at 0D point along iterative-height Useless as-is, the equation allows for angular-velocity to be calculated-for using tangential-velocity, which is vector ready. Thus the vector ready equation can be calculated: Simplifies, interestingly, to a lower-power equation ( x ): KE Programmer's Note: If using python, remember the float() function for all integer-values, to convert to decimals. Extraneous integer-formatted numbers cause calculation-precision issues in python. page 8