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Evaluation of Moments of Inertia

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Evaluation of Moments of Inertia 

? 2008 Waterloo Maple Inc 

 

Introduction 

The concept of a moment of inertia is important in many design and analysis problems encountered in mechanical and civil engineering.  It is required in the design of machines, bridges, and other engineering systems.  The computation of moments of inertia can often be cumbersome.  In the general case, analytical integrals must be evaluated.  In specific cases involving composite shapes, a large number of repetitive formulas and numbers must be manipulated. Maple is an ideal tool to perform both of these operations. 

 

 

Method of Integration 

In this example, the general expression for the moment of inertia for a triangular area is derived. 

 

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Step 

Result 

Define the differential moment of inertia. 

`:=`(dIx, `*`(`^`(y, 2), `*`(dA))) 

`*`(`^`(y, 2), `*`(dA)) (2.1)
 

 

Define the differential area element. 

`:=`(dA, `*`(L, `*`(dy))) 

`*`(L, `*`(dy)) (2.2)
 

 

Using geometric contraints (similar triangles), derive the expression for L . 

`/`(`*`(L), `*`(b)) = `/`(`*`(`+`(h, `-`(y))), `*`(h)) 

`/`(`*`(L), `*`(b)) = `/`(`*`(`+`(h, `-`(y))), `*`(h)) (2.3)
 

L = `/`(`*`(`+`(h, `-`(y)), `*`(b)), `*`(h)) 

 

Assign this value of L . 

`:=`(L, `/`(`*`(`+`(h, `-`(y)), `*`(b)), `*`(h))) 

`/`(`*`(`+`(h, `-`(y)), `*`(b)), `*`(h)) (2.4)
 

 

The corresponding expression for the differential moment of inertia is: 

dIx 

`/`(`*`(`^`(y, 2), `*`(`+`(h, `-`(y)), `*`(b, `*`(dy)))), `*`(h)) (2.5)
 

 

Integrate to derive a formula for the moment of inertia for a general triangle. Note the dy is assigned the value 1 so that the Maple integrator does not confuse it as a mathematical variable. 

`:=`(dy, 1); -1 

Ix = int(dIx, y = 0 .. h) 

Ix = `+`(`*`(`/`(1, 12), `*`(`^`(h, 3), `*`(b)))) (2.6)
 

 

 

Composite Objects (Parallel Axis Theorem) 

Maple can extend the usefulness of the well-known parallel axis theorem by facilitating parametric design. The symbolic Maple engine can keep track of important parameters and unknowns to produce expressions that are entirely general but can be conveniently manipulated to produce optimal designs.Given the following cross-sectional shape, derive the general expression for the moment of inertia about the axis Z-Z with respect to a parameter t (thickness). 

 

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If the object above was the cross-section of a beam, one could easily determine the optimal plate thickness ( t ) for various loading conditions. This is the direct result of a Maple-based symbolic approach to this problem.The parallel axis theorem solves the overall problem by considering a series of smaller, simpler problems. The general formula for the centroid position of the entire object follows. 

`:=`(ybar, `/`(`*`(sum(`*`(y[i], `*`(A[i])), i = 1 .. 3)), `*`(`#msub(mi( 

`/`(`*`(`+`(`*`(y[1], `*`(A[1])), `*`(y[2], `*`(A[2])), `*`(y[3], `*`(A[3])))), `*`(`#msub(mi( (3.1)
 

 

Step 

Result 

Define the sub-areas and total area of the object. 

`:=`(A[1], `+`(`*`(10, `*`(t)))); -1 

`:=`(A[2], `*`(`+`(3, `-`(t)), `*`(t))); -1 

`:=`(A[3], `*`(`+`(3, `-`(t)), `*`(t))); -1 

`:=`(`#msub(mi( 

`+`(`*`(10, `*`(t)), `*`(2, `*`(`+`(3, `-`(t)), `*`(t)))) (3.2)
 

 

Define the centroid heights of the sub-areas of the object. 

`:=`(y[1], `+`(3, `-`(`*`(`/`(1, 2), `*`(t))))); -1 

`:=`(y[2], `*`(`+`(3, `-`(t)), `/`(1, 2))); -1 

`:=`(y[3], `*`(`+`(3, `-`(t)), `/`(1, 2))); -1 

The variable ybar (centroid position) has kept track of the specific expressions for the various terms. Print out the current contents of ybar. 

ybar 

`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))), `*`(`+`(3, `-`(t)), `*`(t)))))), `*`(`+`(`*`(10, `*`(t)), `*`(2, `*`(`+`(... (3.3)
 

 

Simplify this expression. 

`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))), `*`(`+`(3, `-`(t)), `*`(t)))))), `*`(`+`(`*`(10, `*`(t)), `*`(2, `*`(`+`(... 

`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))), `*`(`+`(3, `-`(t)), `*`(t)))))), `*`(`+`(`*`(10, `*`(t)), `*`(2, `*`(`+`(... (3.4)
 

 

`+`(`-`(`/`(`*`(`/`(1, 2), `*`(`+`(39, `-`(`*`(11, `*`(t))), `*`(`^`(t, 2))))), `*`(`+`(`-`(8), t))))) (3.5)
 

 

Display the expression for the moment of inertia (parallel axis theorem). 

`:=`(`#msub(mi( 

`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
`+`(I0[1], `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(`*`(`/`(1, 2), `*`(t)))),...
(3.6)
 

 

 

 

Define the individual moments of inertia  `+`(`*`(`/`(1, 12), `*`(b, `*`(`^`(h, 3))))) . 

`:=`(I0[1], `*`(`+`(`*`(10, `*`(`^`(t, 3)))), `/`(1, 12))); -1 

`:=`(I0[2], `+`(`*`(`/`(1, 12), `*`(t, `*`(`^`(`+`(3, `-`(t)), 3)))))); -1 

`:=`(I0[3], `+`(`*`(`/`(1, 12), `*`(t, `*`(`^`(`+`(3, `-`(t)), 3)))))); -1 

Display the final expression for the moment of inertia. Once again, Maple has kept track of all variable changes. 

`#msub(mi( 

`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
(3.7)
 

 

Simplify the result. 

`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(... 

`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
`+`(`*`(`/`(5, 6), `*`(`^`(t, 3))), `*`(10, `*`(t, `*`(`^`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t))), `-`(`/`(`*`(`+`(`*`(10, `*`(`+`(3, `-`(`*`(`/`(1, 2), `*`(t)))), `*`(t))), `*`(2, `*`(`+`(`/`(3, 2), `-`(...
(3.8)
 

 

`+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(t, 4)), `-`(`*`(22, `*`(`^`(t, 3)))), `*`(139, `*`(`^`(t, 2))), `-`(`*`(378, `*`(t))), 621), `*`(t))), `*`(`+`(`-`(8), t))))) (3.9)
 

 

Convert the general expression to "operator" or a functional form (i.e., `#msub(mi(). The result is a convenient form for the expression that facilitates quick calculation of `#msub(mi(  for specific values of t . This is performed with the unapply() function. 

`:=`(`#msub(mi( 

proc (t) options operator, arrow; `+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(t, 4)), `-`(`*`(22, `*`(`^`(t, 3)))), `*`(139, `*`(`^`(t, 2))), `-`(`*`(378, `*`(t))), 621), `*`(t))), `*`(`+`(`-`(8), t)))... (3.10)
 

 

Calculate the moment of inertia for some values of t . 

`#msub(mi( 

2.875525032 (3.11)
 

`#msub(mi( 

11.44471154 (3.12)
 

 

The operator form also accepts non-numeric information. For example, the thickness t could be a function of another parameter e.g., t = (a-1): 

`#msub(mi( 

`+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(`+`(a, `-`(1)), 4)), `-`(`*`(22, `*`(`^`(`+`(a, `-`(1)), 3)))), `*`(139, `*`(`^`(`+`(a, `-`(1)), 2))), `-`(`*`(378, `*`(a))), 999), `*`(`+`(a, `-`(1))))), `*...
`+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(`+`(a, `-`(1)), 4)), `-`(`*`(22, `*`(`^`(`+`(a, `-`(1)), 3)))), `*`(139, `*`(`^`(`+`(a, `-`(1)), 2))), `-`(`*`(378, `*`(a))), 999), `*`(`+`(a, `-`(1))))), `*...
(3.13)
 

 

`+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(a, 4)), `-`(`*`(26, `*`(`^`(a, 3)))), `*`(211, `*`(`^`(a, 2))), `-`(`*`(726, `*`(a))), 1161), `*`(`+`(a, `-`(1))))), `*`(`+`(`-`(9), a)))))
`+`(`-`(`/`(`*`(`/`(1, 6), `*`(`+`(`*`(`^`(a, 4)), `-`(`*`(26, `*`(`^`(a, 3)))), `*`(211, `*`(`^`(a, 2))), `-`(`*`(726, `*`(a))), 1161), `*`(`+`(a, `-`(1))))), `*`(`+`(`-`(9), a)))))
(3.14)
 

 

By using a plot we can summarize the general relationship between the moment of inertia and the thickness parameter t. 

 

plot(`#msub(mi(
plot(`#msub(mi(
plot(`#msub(mi(
 

Plot_2d
 

 

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