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334 lines
12 KiB
334 lines
12 KiB
2 years ago
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\chapter{Units Conversion System}\label{ch:units}
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\section{Introduction}
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The units conversion system as implemented in \aprepro{} defines
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several variables that are abbreviations for unit quantities. For
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example, if the output format for the current unit system was inches,
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the variable \cmd{foot} would have the value \cmd{12}. Therefore, an
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expression such as \cmd{8*foot} would be equal to \cmd{96} which is the
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number of inches in 8 feet\footnote{This can also be written as 8\~{}foot since \~{} has been defined to be the same as $*$ (multiplication).}.
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Seven consistent units systems have been defined including four metric
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based systems: \cmd{si}, \cmd{cgs}, \cmd{cgs-ev}, and \cmd{shock}; and
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three english-based systems: \cmd{in-lbf-s}, \cmd{ft-lbf-s}, and
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\cmd{ft-lbm-s}. The output units for these unit systems are shown in
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Table 8 (metric) and Table 9 (english). A list of the defined units
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abbreviations is given in Table 10.
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In addition to the definition of the conversion factors, several
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string variables are also defined which describe the output format of
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the current units system. For example, the string variable \cmd{dout}
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defines the output format for density units. For the \cmd{in-lbf-sec}
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units system, \cmd{dout} =
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\cmd{\texttt{"}lbf-sec\^{}2/in\^{}4}\texttt{"}
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which is the output format for densities in this system. The string
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variables can be used to document the \aprepro{} output. The
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string variable names are listed in the last column of Table 8 and Table 9.
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\begin{longtable}{l|lllll}
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\caption{Units Systems and Corresponding Output Format--Metric} \\
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\hline
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Quantity &si& cgs & cgs-ev & shock & string \\
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\hline
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Length & metre & centimetre & centimetre & centimetre & lout \\
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Mass & kilogram & gram & gram & gram & mout \\
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Time & second & second & second & micro-sec & tout \\
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Temperature & kelvin & kelvin & eV & kelvin & Tout \\
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Velocity & metre/sec& cm/sec & cm/sec & cm/usec & vout \\
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Acceleration & metre/sec$^2$ & cm/sec$^2$ & cm/sec$^2$ & cm/usec$^2$ & aout \\
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Force & newton & dyne & dyne & g-cm/usec$^2$ & fout \\
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Volume & metre$^3$ & cm$^3$ & cm$^3$ & cm$^3$ & Vout \\
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Density & kg/m$^3$ & g/cc & g/cc & g/cc & dout \\
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Energy & joule & erg & erg & g-cm$^2$/usec$^3$ & eout \\
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Power & watt & erg/sec & erg/sec & g-cm$^2$/usec$^4$ & Pout \\
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Pressure & pascal & dyne/cm$^2$ & dyne/cm$^2$ & Mbar & pout \\
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\hline
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\end{longtable}
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\begin{longtable}{l|llll}
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\caption{Units Systems and Corresponding Output Format--English} \\
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\hline
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Quantity & in-lbf-s & ft-lbf-s & ft-lbm-s & string \\
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\hline
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Length & inch & foot & foot & lout \\
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Mass & lbf-sec$^2$/in & slug & pound-mass & mout \\
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Time & second & second & second & tout \\
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Temperature & rankine & rankine & rankine & Tout \\
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Velocity & inch/sec & foot/sec & foot/sec & vout \\
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Acceleration & inch/sec$^2$ & foot/sec$^2$ & foot/sec$^2$ & aout \\
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Force & pound-force & pound-force & poundal & fout \\
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Volume & inch$^3$ & foot$^3$ & foot$^3$ & Vout \\
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Density & lbf-sec$^2$/in$^4$ & slug/ft$^3$ & lbm/ft$^3$ & dout \\
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Energy & inch-lbf & foot-lbf & ft-poundal & eout \\
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Power & inch-lbf/sec & foot-lbf/sec & ft-poundal/sec & Pout\\
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Pressure & lbf/in$^2$ & lbf/ft$^2$ & poundal/ft$^2$ & pout \\
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\hline
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\end{longtable}
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The units definitions are accessed through the \afunc{Units} function in \aprepro{}:
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\begin{apinp}
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\{Units("\var{unit\_system}")\}
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\end{apinp}
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where \var{unit\_system} is one of the strings listed in the first row
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of the previous two tables.
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\section{Defined Units Variables}
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In the following table, the first column lists the variables
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that are defined in the \aprepro{} unit system and the second column
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is a short description of the unit. All units variables are defined in
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terms of the five SI Base Units metre (length), second (time),
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kilogram (mass), temperature (kelvin), and radian (angle)\footnote{The
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radian is actually a SI Supplementary Unit since it has not been
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decided whether it is a Base Unit or a Derived Unit. There are three
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other SI Base Units, the candela, ampere, and mole, but they are not
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yet used in the Aprepro units system.}. The bolded rows
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delineate the type of unit variable and the base quantities used to
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define it where $L$ is length, $T$ is time, $M$ is mass, and $t$ is
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temperature. For example density is defined in terms of $M/L^3$ which
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is mass/ length$^3$.
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\begin{longtable}{l|l}
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\caption{Defined Units Variables} \\
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\hline\hline
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\multicolumn{2}{c}{\bf Length \boldmath{[L]}} \\
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\hline
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m, meter, metre & \cmd{Metre (base unit)} \\
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cm, centimeter, centimetre & {Metre / 100} \\
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mm, millimeter, millimetre & {Metre / 1,000} \\
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um, micrometer, micrometre & {Metre / 1,000,000} \\
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km, kilometer, kilometre & {Metre * 1,000} \\
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in, inch & Inch \\
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ft, foot & Foot \\
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yd, yard & Yard \\
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mi, mile & Mile \\
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mil & Mil (inch/1000) \\
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\multicolumn{2}{c}{} \\
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\hline \multicolumn{2}{c}{\bf Time \boldmath{[T]}} \\
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\hline
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second, sec & \cmd{Second (base unit)} \\
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usec, microsecond & Second / 1,000,000 \\
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msec, millisecond & Second / 1,000 \\
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minute & Minute \\
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hr, hour & Hour \\
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day & Day \\
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yr, year & Year = 365.25 days \\
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decade & 10 Years \\
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century & 100 Years \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Velocity \boldmath{[L/T]}} \\
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\hline
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mph & Miles per hour \\
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kph & Kilometres per hour \\
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mps & Metre per second \\
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kps & Kilometre per second \\
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fps & Foot per second \\
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ips & Inch per second \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Acceleration \boldmath{[$L/T^2$]}} \\
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\hline
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ga & Gravitational acceleration \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Mass \boldmath{[$M$]}} \\
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\hline
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kg & \cmd{Kilogram (base unit)} \\
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g, gram & Gram \\
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lbm & Pound (mass) \\
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slug & Slug \\
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lbfs2pin & Lbf-sec$^2$/in \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Density \boldmath{[$M/L^3]$}} \\
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\hline
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gpcc & Gram / cm$^3$ \\
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kgpm3 & Kilogram / m$^3$ \\
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lbfs2pin4 & Lbf-sec$^2$ / in$^4$ \\
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lbmpin3 & Lbm / in \\
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lbmpft3 & Lbm / ft$^3$ \\
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slugpft3 & Slug / ft$^3$ \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Force \boldmath{[$ML/T^2$]}} \\
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\hline
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N, newton & Newton = 1 kg-m/sec$^2$ \\
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dyne & Dyne = newton/10,000 \\
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gf & Gram (force) \\
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kgf & Kilogram (force) \\
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lbf & Pound (force) \\
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kip & Kilopound (force) \\
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pdl, poundal & Poundal \\
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ounce & Ounce = lbf / 16 \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Energy \cmd{[$ML^2/T^2$]}} \\
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\hline
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J, joule & Joule = 1 newton-metre \\
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ftlbf & Foot-lbf \\
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erg & Erg = 1e-7 joule \\
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calorie & International Table Calorie \\
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Btu & International Table Btu \\
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therm & EEC therm \\
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tonTNT & Energy in 1 ton TNT \\
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kwh & Kilowatt hour \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Power \boldmath{[$ML^2/T^3$]}} \\
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\hline
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W, watt & Watt = 1 joule / second \\
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Hp & Elec. Horsepower (746 W) \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Temperature \boldmath{[t]}} \\
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\hline
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degK, kelvin & \cmd{Kelvin (Base Unit)} \\
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degC & Degree Celsius \\
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degF & Degree Fahrenheit \\
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degR, rankine & Degree Rankine \\
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eV & Electron Volt \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Pressure \boldmath{[$M/L/T^2$]}} \\
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\hline
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Pa, pascal & Pascal = 1 newton / metre$^2$ \\
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MPa & Megapascal \\
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GPa & Gigapascal \\
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bar & Bar \\
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kbar & Kilobar \\
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Mbar & Megabar \\
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atm & Standard atmosphere \\
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torr & Torr = 1 mmHg \\
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mHg & Metre of mercury \\
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mmHg & Millimetre of mercury \\
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inHg & Inch of mercury \\
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inH2O & Inch of water \\
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ftH2O & Foot of water \\
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psi & Pound per square inch \\
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ksi & Kilo-pound per square inch \\
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psf & Pound per square foot \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Volume \boldmath{[L$^3$]}} \\
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\hline
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liter & Metre$^3$ / 1000 \\
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gal, gallon & Gallon (U.S.) \\
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\multicolumn{2}{c}{} \\
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\hline
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\multicolumn{2}{c}{\bf Angular } \\
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\hline
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rad & \cmd{Radian (base unit)} \\
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rev & Full circle = 360 degree \\
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deg, degree & Degree \\
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arcmin & Arc minute = 1/60 degree \\
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arcsec & Arc second = 1/360 degree \\
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grade & Grade = 0.9 degree \\
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\end{longtable}
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The conversion expressions were obtained from
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References~\cite{bib:isotopes}, \cite{bib:jaeger}, \cite{bib:lambe}, and~\cite{bib:simpson}.
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\section{Usage}
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The following example illustrates the basic usage of the \aprepro{} units conversion utility.
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\begin{apinp}
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\$ Aprepro Units Utility Example
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\$ \{ECHO(OFF)\} \textit{\ldots{}Turn off echoing of the conversion factors}
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\$ \{Units(``shock'')\} \textit{\ldots{}Select the shock units system}
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\$ NOTE: Dimensions - \{lout\}, \{mout\}, \{dout\}, \{pout\}
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\textit{\ldots{}This will document what quantities are used in the file after it is run through Aprepro}
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\{len1 = 10.0 * inch\} \textit{\ldots{}Define a length in an english unit (inches)}
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$ \{len2 = 12.0\textasciitilde{}inch\} \textit{\ldots{}\textasciitilde{} is a synonym for * (multiplication)}
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Material 1, Elastic Plastic, \{1890\textasciitilde{}kgpm3\} \$ \{dout\}
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Youngs Modulus = \{28.3e6\textasciitilde{}psi\} \$ {pout}
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Yield Stress = \{30\textasciitilde{}ksi\}
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Initial Veclocity = \{10\textasciitilde{}mph\} \$ {vout}
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\textit{\ldots{}Define the density and material parameters in whatever units they are available}
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End
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Point 100 \{0.0\} \{0.0\}
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Point 110 \{len1\} \{0.0\}
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Point 120 \{len1\} \{len2\}
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Point 130 \{0.0\} \{len1\}
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\end{apinp}
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The output from this example input file is:
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\begin{apout}
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\$ Aprepro Units Utility Example
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\$ NOTE: Dimensions - cm, gram, g/cc, Mbar \textit{\ldots{}The documentation of what quantities this file uses}
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\$ 25.4
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\$ 30.48
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Material 1, Elastic Plastic, 1.89 \$ g/cc
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Youngs Modulus = 1.951216314 \$ Mbar
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Yield Stress = 0.002068427188 \textit{\ldots{}All material parameters are now in consistent units}
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Initial Velocity = 0.00044704 \$ cm/usec
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End
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Point 100 0 0
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Point 110 25.4 0
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Point 120 25.4 30.48
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Point 130 0 25.4 \textit{\ldots{}Lengths have all been converted to centimetres}
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\end{apout}
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The same input file can be used to output in SI units simply by changing Units
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command from \var{shock} to \var{si}. The output in SI units is:
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\begin{apout}
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\$ Aprepro Units Utility Example
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\$ NOTE: Dimensions - meter, kilogram, kg/m^3, Pa
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\textit{\ldots{}Quantities are now output in standard SI units}
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\$ 0.254
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\$ 0.3048
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Material 1, Elastic Plastic, 1890 \$ kg/m^3
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Youngs Modulus = 1.951216314e+11 \$ Pa
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Yield Stress = 206842718.8
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Initial Velocity = 4.4704 \$ meter/sec
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End
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Point 100 0 0
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Point 110 0.254 0
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Point 120 0.254 0.3048
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Point 130 0 0.254 \textit{\ldots{}Lengths have all been converted to metres}
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\end{apout}
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\section{Additional Comments}
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A few additional comments and warnings on the use of the units system are detailed
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below.
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Omitting the \{ECHO(OFF)\} line prior to the \{Units(``unit\_system'')\}
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function will print out the contents of the units header and conversion files.
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Each line in the output will be preceded by the current comment character which
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is \$ by default.
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The comment character can be changed by invoking \aprepro{} with the
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\cmd{-c} option. For example \cmd{aprepro -c\# input\_file output\_file}
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will change the comment character at the beginning of the lines to \#.
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The temperature conversions are only valid for relative temperatures, for
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example, 100\textasciitilde{}degC is equal to 180\textasciitilde{}degF, not 212\textasciitilde{}degF.
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The functions \cmd{CtoF(x)} and \cmd{FtoC(x)} can be used to convert absolute temperatures. For example,
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\begin{apinp}
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100 C = \{CtoF(100)\} F
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98.6 F = \{FtoC(98.6)\} C
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\end{apinp}
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produces the output:
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\begin{apout}
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100 C = 212 F
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|
98.6 F = 37 C
|
||
|
|
\end{apout}
|
||
|
|
|
||
|
|
Several variables are defined in the units system; they are all
|
||
|
|
\textit{immutable} variables, so it is not possible to
|
||
|
|
redefine their values. However, if you inadvertently are using a
|
||
|
|
variable with the same name as a variable defined in the units system,
|
||
|
|
you may get the incorrect value of that variable since it cannot be
|
||
|
|
redefined from the value set by the units system. You can enter the
|
||
|
|
command \cmd{DUMP()} to see a list of all defined variables and their
|
||
|
|
current value.
|