Calculate Mass Properties and Uncertainties of Tree Structures • massProps

The massProps package extends rollupTree with functions to recursively calculate mass properties (and optionally, their uncertainties) for arbitrary composition trees. Formulas implemented are described in a technical paper published by the Society of Allied Weight Engineers (Zimmerman and Nakai 2005).

Installation

install.packages("massProps")

You can install the development version of massProps from GitHub with:

# install.packages("pak")
pak::pak("jsjuni/massProps")

Example

Suppose we have the following mass properties table:

library(massProps)
test_table
#>     id parent mass Cx Cy Cz Ixx  Ixy   Ixz Iyy   Iyz Izz POIconv Ipoint
#> 1  A.1          NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 2  A.2    A.1   NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 3  A.3    A.1   NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 4  C.1    A.1    5  0  0  0  80 -4.0 -24.0  80 -24.0  75       -  FALSE
#> 5  P.1    A.2    2  1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 6  P.2    A.2    2  1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 7  P.3    A.2    2  1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 8  P.4    A.2    2  1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 9  P.5    A.3    2 -1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 10 P.6    A.3    2 -1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 11 P.7    A.3    2 -1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 12 P.8    A.3    2 -1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE

POIconv indicates the products of inertia sign convention. In the negative convention, for example, $I_{XY} \equiv -\int{xy \rho \, dV}$ . In the positive convention, $I_{XY} \equiv \int{xy \rho \, dV}$ .

Ipoint indicates whether this object is to be considered a point mass. The same algebraic result can be achieved by setting all moments and products of inertia to zero, but rollup_mass_props() by default ensures that all leaf items in the tree have mass properties that correspond to physically-realizable objects. A zero inertia tensor will fail this check. Rather than relax the check (which is essential for trustworthy results), a TRUE value for Ipoint indicates that the inertia tensor should be excluded from computations.

We construct a graph with edges representing child-parent relations:

library(igraph)
E(test_tree)
#> + 11/11 edges from 7d8200c (vertex names):
#>  [1] A.2->A.1 A.3->A.1 C.1->A.1 P.1->A.2 P.2->A.2 P.3->A.2 P.4->A.2 P.5->A.3
#>  [9] P.6->A.3 P.7->A.3 P.8->A.3

We can roll up mass properties to non-leaf elements as follows:

rollup_mass_props(test_tree, test_table)
#>     id parent mass Cx Cy Cz Ixx  Ixy   Ixz Iyy   Iyz Izz POIconv Ipoint
#> 1  A.1          21  0  0  0 144 -4.8 -24.8 144 -23.2 139       -  FALSE
#> 2  A.2    A.1    8  1  0  0  32 -0.4  -0.4  24   0.4  24       -  FALSE
#> 3  A.3    A.1    8 -1  0  0  32 -0.4  -0.4  24   0.4  24       -  FALSE
#> 4  C.1    A.1    5  0  0  0  80 -4.0 -24.0  80 -24.0  75       -  FALSE
#> 5  P.1    A.2    2  1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 6  P.2    A.2    2  1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 7  P.3    A.2    2  1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 8  P.4    A.2    2  1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 9  P.5    A.3    2 -1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 10 P.6    A.3    2 -1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 11 P.7    A.3    2 -1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 12 P.8    A.3    2 -1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE

Note that, although the table shows the parent of each element for clarity of exposition, the child-parent relations are coneveyed only by the tree passed as the first argument.

The input data may include mass properties uncertainties as well:

test_unc_table
#>     id parent mass Cx Cy Cz Ixx  Ixy   Ixz Iyy   Iyz Izz POIconv Ipoint
#> 1  A.1          NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 2  A.2    A.1   NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 3  A.3    A.1   NA NA NA NA  NA   NA    NA  NA    NA  NA       -  FALSE
#> 4  C.1    A.1    5  0  0  0  80 -4.0 -24.0  80 -24.0  75       -  FALSE
#> 5  P.1    A.2    2  1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 6  P.2    A.2    2  1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 7  P.3    A.2    2  1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 8  P.4    A.2    2  1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 9  P.5    A.3    2 -1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 10 P.6    A.3    2 -1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 11 P.7    A.3    2 -1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 12 P.8    A.3    2 -1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#>    sigma_mass sigma_Cx sigma_Cy sigma_Cz sigma_Ixx sigma_Iyy sigma_Izz
#> 1          NA       NA       NA       NA        NA        NA        NA
#> 2          NA       NA       NA       NA        NA        NA        NA
#> 3          NA       NA       NA       NA        NA        NA        NA
#> 4         0.5      0.0      0.0      0.0       8.0       8.0       7.5
#> 5         0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 6         0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 7         0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 8         0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 9         0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 10        0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 11        0.2      0.1      0.1      0.1       0.4       0.4       0.4
#> 12        0.2      0.1      0.1      0.1       0.4       0.4       0.4
#>    sigma_Ixy sigma_Ixz sigma_Iyz
#> 1         NA        NA        NA
#> 2         NA        NA        NA
#> 3         NA        NA        NA
#> 4       0.40      2.40      2.40
#> 5       0.01      0.01      0.01
#> 6       0.01      0.01      0.01
#> 7       0.01      0.01      0.01
#> 8       0.01      0.01      0.01
#> 9       0.01      0.01      0.01
#> 10      0.01      0.01      0.01
#> 11      0.01      0.01      0.01
#> 12      0.01      0.01      0.01

Mass properties and their uncertainties can be rolled up as follows:

rollup_mass_props_and_unc(test_tree, test_unc_table)
#>     id parent mass Cx Cy Cz Ixx  Ixy   Ixz Iyy   Iyz Izz POIconv Ipoint
#> 1  A.1          21  0  0  0 144 -4.8 -24.8 144 -23.2 139       -  FALSE
#> 2  A.2    A.1    8  1  0  0  32 -0.4  -0.4  24   0.4  24       -  FALSE
#> 3  A.3    A.1    8 -1  0  0  32 -0.4  -0.4  24   0.4  24       -  FALSE
#> 4  C.1    A.1    5  0  0  0  80 -4.0 -24.0  80 -24.0  75       -  FALSE
#> 5  P.1    A.2    2  1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 6  P.2    A.2    2  1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 7  P.3    A.2    2  1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 8  P.4    A.2    2  1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 9  P.5    A.3    2 -1  1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 10 P.6    A.3    2 -1  1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 11 P.7    A.3    2 -1 -1  1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#> 12 P.8    A.3    2 -1 -1 -1   4 -0.1  -0.1   4   0.1   4       -  FALSE
#>    sigma_mass   sigma_Cx   sigma_Cy   sigma_Cz sigma_Ixx sigma_Iyy sigma_Izz
#> 1   0.7549834 0.03809524 0.03809524 0.03809524  8.313844  8.275264  7.792946
#> 2   0.4000000 0.05000000 0.07071068 0.07071068  1.600000  1.200000  1.200000
#> 3   0.4000000 0.05000000 0.07071068 0.07071068  1.600000  1.200000  1.200000
#> 4   0.5000000 0.00000000 0.00000000 0.00000000  8.000000  8.000000  7.500000
#> 5   0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 6   0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 7   0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 8   0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 9   0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 10  0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 11  0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#> 12  0.2000000 0.10000000 0.10000000 0.10000000  0.400000  0.400000  0.400000
#>    sigma_Ixy sigma_Ixz sigma_Iyz
#> 1  1.0586784 2.5924506 2.5924506
#> 2  0.4004997 0.4004997 0.6931089
#> 3  0.4004997 0.4004997 0.6931089
#> 4  0.4000000 2.4000000 2.4000000
#> 5  0.0100000 0.0100000 0.0100000
#> 6  0.0100000 0.0100000 0.0100000
#> 7  0.0100000 0.0100000 0.0100000
#> 8  0.0100000 0.0100000 0.0100000
#> 9  0.0100000 0.0100000 0.0100000
#> 10 0.0100000 0.0100000 0.0100000
#> 11 0.0100000 0.0100000 0.0100000
#> 12 0.0100000 0.0100000 0.0100000

References

Zimmerman, Robert L., and John H. Nakai. 2005. “Are You Sure? Uncertainty in Mass Properties Engineering.” In 64th Annual International Conference on Mass Properties Engineering, 123–60. Society of Allied Weight Engineers.