Week - 1 - Consistency of Units
Derived from the need of being flexible to accomodate analyses in different units systems, LS-Dyna requires the information to be input in consistent dimensions so the calculations display logical results.
The following challenge will help us construct a tool to calculate coherent units systems to perform our analyses.
1. Table of units conversions on sample dimensions
| Property | Quantity | Systems of units | |||||||||||||||||||||||
| Kg/cm/ms | Kg/cm/s | g/mm/ms | kg/m/s | g/cm/s | g/cm/us | g/mm/s | ton/mm/s | slug/ft/s | kg/mm/s | g/cm/ms | kg/mm/ms | ||||||||||||||
| Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Conversion factor | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | Conversion factor | Magnitude in new system of units | ||
| Mass | 1 kg = | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E-03 | 1.00E-03 | 6.85E-02 | 6.85E-02 | 1.00E+00 | 1.00E+00 | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 |
| Length | 1 m = | 1.00E+02 | 1.00E+02 | 1.00E+02 | 1.00E+02 | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 | 1.00E+02 | 1.00E+02 | 1.00E+02 | 1.00E+02 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 3.28E+00 | 3.28E+00 | 1.00E+03 | 1.00E+03 | 1.00E+02 | 1.00E+02 | 1.00E+03 | 1.00E+03 |
| Time | 1 s = | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+06 | 1.00E+06 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+03 | 1.00E+03 | 1.00E+03 | 1.00E+03 |
| Force | 1 N = | 1.00E-04 | 1.00E-04 | 1.00E+02 | 1.00E+02 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+00 | 1.00E+05 | 1.00E+05 | 1.00E-07 | 1.00E-07 | 1.00E+06 | 1.00E+06 | 1.00E+00 | 1.00E+00 | 2.25E-01 | 2.25E-01 | 1.00E+03 | 1.00E+03 | 1.00E-01 | 1.00E-01 | 1.00E-03 | 1.00E-03 |
| Stress | 1 Pa = | 1.00E-08 | 1.00E-08 | 1.00E-02 | 1.00E-02 | 1.00E-06 | 1.00E-06 | 1.00E+00 | 1.00E+00 | 1.00E+01 | 1.00E+01 | 1.00E-11 | 1.00E-11 | 1.00E+00 | 1.00E+00 | 1.00E-06 | 1.00E-06 | 2.09E-02 | 2.09E-02 | 1.00E-03 | 1.00E-03 | 1.00E-05 | 1.00E-05 | 1.00E-09 | 1.00E-09 |
| Energy | 1 N*m = | 1.00E-02 | 1.00E-02 | 1.00E+04 | 1.00E+04 | 1.00E+03 | 1.00E+03 | 1.00E+00 | 1.00E+00 | 1.00E+07 | 1.00E+07 | 1.00E-05 | 1.00E-05 | 1.00E+09 | 1.00E+09 | 1.00E+03 | 1.00E+03 | 7.38E-01 | 7.38E-01 | 1.00E+06 | 1.00E+06 | 1.00E+01 | 1.00E+01 | 1.00E+00 | 1.00E+00 |
| Density | 1 Kg/m3 = | 1.00E-06 | 1.00E-06 | 1.00E-06 | 1.00E-06 | 1.00E-06 | 1.00E-06 | 1.00E+00 | 1.00E+00 | 1.00E-03 | 1.00E-03 | 1.00E-03 | 1.00E-03 | 1.00E-06 | 1.00E-06 | 1.00E-12 | 1.00E-12 | 1.94E-03 | 1.94E-03 | 1.00E-09 | 1.00E-09 | 1.00E-03 | 1.00E-03 | 1.00E-09 | 1.00E-09 |
| Young's Modulus | 210 GPa = | 1.00E-08 | 2.10E+03 | 1.00E-02 | 2.10E+09 | 1.00E-06 | 2.10E+05 | 1.00E+00 | 2.10E+11 | 1.00E+01 | 2.10E+12 | 1.00E-11 | 2.10E+00 | 1.00E+00 | 2.10E+11 | 1.00E-06 | 2.10E+05 | 2.09E-02 | 4.39E+09 | 1.00E-03 | 2.10E+08 | 1.00E-05 | 2.10E+06 | 1.00E-09 | 2.10E+02 |
| Velocity | 56.33 Km/h = | 1.00E-01 | 1.56E+00 | 1.00E+02 | 1.56E+03 | 1.00E+00 | 1.56E+01 | 1.00E+00 | 1.56E+01 | 1.00E+02 | 1.56E+03 | 1.00E-04 | 1.56E-03 | 1.00E+03 | 1.56E+04 | 1.00E+03 | 1.56E+04 | 3.28E+00 | 5.13E+01 | 1.00E+03 | 1.56E+04 | 1.00E-01 | 1.56E+00 | 1.00E+00 | 1.56E+01 |
| Gravity | 9.8 m/s^2 = | 1.00E-04 | 9.81E-04 | 1.00E+02 | 9.81E+02 | 1.00E-03 | 9.81E-03 | 1.00E+00 | 9.81E+00 | 1.00E+02 | 9.81E+02 | 1.00E-10 | 9.81E-10 | 1.00E+03 | 9.81E+03 | 1.00E+03 | 9.81E+03 | 3.28E+00 | 3.22E+01 | 1.00E+03 | 9.81E+03 | 1.00E-04 | 9.81E-04 | 1.00E-03 | 9.81E-03 |
2. Development of g mm ms system
Mass
`1 kg = 1000 g`
Length
`1m=1000mm`
Time
`1s = 1000 ms`
Force
`1N= 1kg*frac{m}{s^2}`
`1kg*frac{m}{s^2}*frac{1000g}{1kg}*frac{1000mm}{1m}*(frac{1s}{1000ms})^2`
`1cancel(kg)*frac{cancel(m)}{cancel(s^2})*(frac{cancel(1000)g}{1cancel(kg)}*frac{cancel(1000)mm}{1cancel(m})*frac{1cancel(s^2)}{cancel(1000^2)ms^2})^`
`1N= 1kg*frac{m}{s^2}= 1g*frac{mm}{ms^2}`
Stress
`1 Pa = 1 frac{N}{m^2}`
From the previous definition we can take N with the new units. for the square meters, we take the conversion of m to mm and square it in the divisor
`1frac{N}{m^2} = 1*frac{1*frac{g*mm}{ms^2}}{1*m^2*(frac{1000mm}{1m})^2}`
If we look at the lower part of the division we get:
`1*cancel(m^2)*(frac{1000^2mm^2}{1cancel(m^2)}) = 1*10^6`
And the conversion is given by:
`1frac{N}{m^2} = 1*frac{1*frac{g*mm}{ms^2}}{1*10^6 mm^2}=1*10^-6frac{g}{mm*ms^2}`
Energy
`1 N*m=1g*frac{mm}{ms^2}*1m=1g*frac{mm}{ms^2}*1cancel(m)*frac{1000mm}{1cancelm)}`
`1 N*m=1*10^3g*frac{mm^2}{ms^2}`
Density
`1 frac{kg}{m^3} * frac{1000g}{1kg}*(frac{1m}{1000mm})^3`
`1 frac{cancel(kg)}{cancel(m^3)} * frac{1000g}{1cancel(kg)}*frac{1cancel(m^3)}{1*10^9mm^3}= 1*10^-6frac{g}{mm^3}`
Young's Modulus
`210 GPa = 210*10^9 Pa`
`210*10^9cancel(Pa)*frac{1*10^-6frac{g}{mm*ms^2}}{1cancel(Pa)}=2.1*10^5frac{g}{mm*ms^2}`
Velocity
`56.33 cancel(frac{Km}{h})*frac{1000m}{1cancel(Km)}*frac{1cancel(h)}{3600s}=15.647frac{m}{s}`
`15.647 cancel(frac{m}{s})*frac{cancel(1000)mm}{1cancel(m)}*frac{1cancel(s)}{cancel(1000)ms}=15.647frac{mm}{ms}`
Gravity
`9.806 frac{m}{s^2}*frac{1000mm}{1m}*(frac{1s}{1000ms})^2`
`9.806 cancel(frac{m}{s^2})*frac{1000mm}{1cancel(m)}*frac{1cancel(s^2)}{1*10^6ms^2}= 9.806*10^-3frac{mm}{ms^2}`
Conclusion
The development of a calculator to change dimensions from one system to another allows us to move freely around problems that may be built in a specific units system, or in cases were the solution is needed in a specific dimension. Moreover, it also gives us a better perspective to know if our solution is correct or if something is not working properly.