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Consistency of units in LS-Dyna

AIM: To write the given measurements in different system of units. Also, to prepare consistent unit system table for different unit system EXPLANATION: Fundamental Quantities: These quantities are are independent of any other quantity. It includes mass, time and length. Derived Quantities: All the quantities…

Ashwen Venkatesh
updated on 28 Dec 2020

AIM:

To write the given measurements in different system of units. Also, to prepare consistent unit system table for different unit system

EXPLANATION:

Fundamental Quantities: These quantities are are independent of any other quantity. It includes mass, time and length.

Derived Quantities: All the quantities which can be derived from fundamental quantities are known as derived quantities. Example includes pressure units, force units, density units etc.

The calculation shown below establishes the conversion from SI units to mm,gm and ms. (millimeteres, grams and milliseconds)

1. Mass conversion from kg to mg

1kg = 1000 grams

⇒ 1kg = $10^{3}$ $10^3$ grams

2. Length conversion from m to mm

1m = 1000 millimetres

⇒ 1m = $10^{3}$ $10^3$ mm

3. Time conversion from s to ms

1s = 1000 milliseconds

⇒ 1s = $10^{3}$ $10^3$ ms

4. Force units conversion from SI units to mm,ms and grams

Force = 1N = 1 $k g \cdot \frac{m}{s^{2}}$ $kg*m/s^2$

⇒ $1 \cdot 10^{3} \cdot \frac{10^{3}}{{(10^{3})}^{2}}$ $1*10^3*10^3/(10^3)^2$ $g m \cdot \frac{m m}{m s^{2}}$ $gm*(mm)/(ms^2)$

⇒ 1N = 1 $g m \cdot \frac{m m}{m s^{2}}$ $gm*(mm)/(ms^2)$

5. Stress units conversion from SI units to mm,ms and grams

Stress = 1Pa = $1 \frac{N}{m^{2}}$ $1N/m^2$ = $1 \cdot k g \cdot \frac{m}{s^{2} \cdot m^{2}}$ $1*kg*m/(s^2*m^2)$ = $1 \cdot \frac{k g}{m \cdot s^{2}}$ $1*(kg)/(m*s^2)$

⇒ $1 \cdot \frac{10^{3}}{10^{3} \cdot {(10^{3})}^{2}}$ $1*10^3/(10^3*(10^3)^2)$ $\frac{g m}{m m \cdot m s^{2}}$ $(gm)/(mm*ms^2)$

⇒ 1Pa = $10^{- 6}$ $10^-6$ $\frac{g m}{m m \cdot m s^{2}}$ $(gm)/(mm*ms^2)$

6. Energy units conversion from SI units to mm,ms and grams

Energy = 1J = $1 \cdot k g \cdot \frac{m^{2}}{s^{2}}$ $1*kg*m^2/s^2$

⇒ $1 \cdot 10^{3} \cdot \frac{{(10^{3})}^{2}}{{(10^{3})}^{2}}$ $1*10^3*(10^3)^2/(10^3)^2$ $1 \cdot g m \cdot \frac{{(m m)}^{2}}{{(m s)}^{2}}$ $1*gm*(mm)^2/(ms)^2$

⇒ 1J = $1 \cdot g m \cdot \frac{{(m m)}^{2}}{{(m s)}^{2}}$ $1*gm*(mm)^2/(ms)^2$

7. Density units conversion from SI units to mm,ms and grams

Density = $1 \cdot \frac{k g}{m^{3}}$ $1*(kg)/(m^3)$

⇒ $1 \cdot \frac{10^{3}}{{(10^{3})}^{3}}$ $1*10^3/(10^3)^3$ $1 \cdot \frac{g m}{m m^{3}}$ $1*(gm)/(mm^3)$

⇒ $1 \cdot \frac{k g}{m^{3}}$ $1*(kg)/(m^3)$ = $10^{- 6}$ $10^-6$ $\frac{g m}{m m^{3}}$ $(gm)/(mm^3)$

8. YOUNG'S Modulus 210GPa units conversion mm,ms and grams

1 GPa = $10^{9}$ $10^9$ Pa

⇒ 210 GPa = $210 \cdot 10^{9}$ $210*10^9$ Pa

⇒ 1Pa = $10^{- 6}$ $10^-6$ $\frac{g m}{m m \cdot m s^{2}}$ $(gm)/(mm*ms^2)$

⇒ $210 \cdot 10^{9}$ $210*10^9$ Pa = $210 \cdot 10^{9} \cdot 10^{- 6}$ $210*10^9*10^-6$ $\frac{g m}{m m \cdot m s^{2}}$ $(gm)/(mm*ms^2)$

⇒ 210 GPa = $210 \cdot 10^{3} \frac{g m}{m m \cdot m s^{2}}$ $210*10^3 (gm)/(mm*ms^2)$

9. Velocity 56.33KMPH units conversion to mm,ms and grams

⇒ 1 hour = 3600 seconds and 1 km = 1000 metres

⇒ 56.33 KMPH = $56.33 \cdot \frac{1000}{3600} \frac{m}{s}$ $56.33*1000/3600 m/s$

⇒ 56.33 KMPH = $15.65 \frac{m}{s}$ $15.65 m/s$

⇒ $1 \cdot \frac{m}{s} = \frac{10^{3}}{10^{3}} \frac{m m}{m s} = 1 \frac{m m}{m s}$ $1*(m)/s = 10^3/10^3 (mm)/(ms) = 1 (mm)/(ms)$

⇒ 56.33 KMPH = $15.65 \frac{m m}{m s}$ $15.65 (mm)/(ms)$

10. GRAVITY 9.8m/s2 units conversion to mm,ms and grams

⇒ $9.8 \frac{m}{s^{2}} = 9.8 \cdot \frac{10^{3}}{{(10^{3})}^{2}} \frac{m m}{{(m s)}^{2}}$ $9.8 m/(s^2) = 9.8 * (10^3)/(10^3)^2 (mm)/(ms)^2$

⇒ $9.8 \frac{m}{s^{2}} = 9.8 \cdot 10^{- 3} \frac{m m}{{(m s)}^{2}}$ $9.8 m/(s^2) = 9.8 *10^-3 (mm)/(ms)^2$

Note: 1 slug = 14.5939 kg

Unit System	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus (210 GPa)	Velocity (56.33 Kmph)	Gravity
kg m s	1kg	1m	1s	1N = 1 kg*m/ $s^{2}$ $s^2$	1Pa = 1 N/ $m^{2}$ $m^2$	1J = $1 \cdot k g \cdot \frac{m^{2}}{s^{2}}$ $1kgm^2/s^2$	$1 \cdot \frac{k g}{m^{3}}$ $1*(kg)/(m^3)$	$210 \cdot 10^{9}$ $210*10^9$ N/ $m^{2}$ $m^2$	15.647 m/s	9.8 m/ $s^{2}$ $s^2$
kg cm ms	1 kg	$10^{2} c m$ $10^2 cm$	$10^{3} m s$ $10^3 ms$	$10^{- 4}$ $10^-4$ kg*cm/ $m s^{2}$ $ms^2$	$10^{- 8}$ $10^-8$ kg/ $(c m \cdot m s^{2})$ $(cm*ms^2)$	$10^{- 2}$ $10^-2$ $k g \cdot \frac{c m^{2}}{m s^{2}}$ $kg*(cm^2)/(ms^2)$	$10^{- 6} \frac{k g}{c m^{3}}$ $10^-6 (kg)/(cm^3)$	2100 $\frac{k g}{c m \cdot m s^{2}}$ $(kg)/(cm*ms^2)$	1.5647 cm/ms	$9.8 \cdot 10^{- 4} \frac{c m}{m s^{2}}$ $9.8*10^-4 (cm)/(ms^2)$
kg cm s	1 kg	$10^{2} c m$ $10^2 cm$	1s	$10^{2}$ $10^2$ kg*cm/ $s^{2}$ $s^2$	$10^{- 2}$ $10^-2$ kg/ $(c m \cdot s^{2})$ $(cm*s^2)$	$10^{4}$ $10^4$ $k g \cdot \frac{c m^{2}}{s^{2}}$ $kg*(cm^2)/s^2$	$10^{- 6} \frac{k g}{c m^{3}}$ $10^-6 (kg)/(cm^3)$	$210 \cdot 10^{7}$ $21010^7$ $\frac{k g}{c m \cdot s^{2}}$ $(kg)/(cms^2)$	1564.7 cm/s	$9.8 \cdot 10^{2} \frac{c m}{s^{2}}$ $9.8*10^2(cm)/(s^2)$
gm mm ms	$10^{3} g m$ $10^3 gm$	$10^{3} m m$ $10^3 mm$	$10^{3} m s$ $10^3 ms$	$1$ $1$ gm*mm/ $m s^{2}$ $ms^2$	$10^{- 6}$ $10^-6$ gm/ $(m m \cdot m s^{2})$ $(mm*ms^2)$	$10^{3}$ $10^3$ $g m \cdot \frac{m m^{2}}{m s^{2}}$ $gm*(mm^2)/(ms^2)$	$10^{- 6} \frac{g m}{m m^{3}}$ $10^-6 (gm)/(mm^3)$	$210 \cdot 10^{3}$ $21010^3$ $\frac{g m}{m m \cdot m s^{2}}$ $(gm)/(mmms^2)$	15.647 mm/ms	$9.8 \cdot 10^{- 3} \frac{m m}{m s^{2}}$ $9.8*10^-3 (mm)/(ms^2)$
gm cm s	$10^{3} g m$ $10^3 gm$	$10^{2} c m$ $10^2 cm$	1s	$10^{5}$ $10^5$ gm*cm/ $s^{2}$ $s^2$	$10$ $10$ gm/ $(c m \cdot s^{2})$ $(cm*s^2)$	$10^{7}$ $10^7$ $g m \cdot \frac{c m^{2}}{s^{2}}$ $gm*(cm^2)/s^2$	$10^{- 3} \frac{g m}{c m^{3}}$ $10^-3 (gm)/(cm^3)$	$210 \cdot 10^{10}$ $21010^10$ $\frac{g m}{c m \cdot s^{2}}$ $(gm)/(cms^2)$	1564.7 cm/s	$9.8 \cdot 10^{2} \frac{c m}{s^{2}}$ $9.8*10^2 (cm)/(s^2)$
gm cm us	$10^{3} g m$ $10^3 gm$	$10^{2} c m$ $10^2 cm$	$10^{6} μ s$ $10^6 mus$	$10^{- 7}$ $10^-7$ kg*cm/ $μ s^{2}$ $mus^2$	$10^{- 11}$ $10^-11$ gm/ $(c m \cdot μ s^{2})$ $(cm*mus^2)$	$10^{- 5}$ $10^-5$ $g m \cdot \frac{c m^{2}}{μ s^{2}}$ $gm*(cm^2)/(mus^2)$	$10^{- 3} \frac{g m}{c m^{3}}$ $10^-3 (gm)/(cm^3)$	$2.1$ $2.1$ $\frac{g m}{c m \cdot μ s^{2}}$ $(gm)/(cm*mus^2)$	$15.647 \cdot 10^{- 4}$ $15.647*10^-4$ cm/ $μ s$ $mus$	$9.8 \cdot 10^{- 10} \frac{c m}{μ s^{2}}$ $9.8*10^-10 (cm)/(mus^2)$
g mm s	$10^{3} g m$ $10^3 gm$	$10^{3} m m$ $10^3 mm$	1s	$10^{6}$ $10^6$ gm*mm/ $s^{2}$ $s^2$	$1$ $1$ gm/ $(m m \cdot s^{2})$ $(mm*s^2)$	$10^{9}$ $10^9$ $g m \cdot \frac{m m^{2}}{s^{2}}$ $gm*(mm^2)/s^2$	$10^{- 6} \frac{g m}{m m^{3}}$ $10^-6 (gm)/(mm^3)$	$210 \cdot 10^{11}$ $21010^11$ $\frac{g m}{m m \cdot s^{2}}$ $(gm)/(mms^2)$	15647 mm/s	$9.8 \cdot 10^{3} \frac{m m}{s^{2}}$ $9.8*10^3 (mm)/(s^2)$
ton mm s	$10^{- 3}$ $10^-3$ ton	$10^{3} m m$ $10^3 mm$	1s	$1$ $1$ ton*mm/ $s^{2}$ $s^2$	$10^{- 6}$ $10^-6$ ton/ $(m m \cdot s^{2})$ $(mm*s^2)$	$10^{3}$ $10^3$ ton* $m m^{2}$ $mm^2$ / $s^{2}$ $s^2$	$10^{- 12}$ $10^-12$ ton/ $(m m^{3})$ $(mm^3)$	$210 \cdot 10^{3}$ $21010^3$ ton/ $(m m \cdot s^{2})$ $(mms^2)$	15647 mm/s	$9.8 \cdot 10^{3} \frac{m m}{s^{2}}$ $9.8*10^3 (mm)/(s^2)$
slug ft s	0.068 slug	3.28 ft	1s	0.224 slug*ft/ $s^{2}$ $s^2$	0.0208 slug/ $(f t \cdot s^{2})$ $(ft*s^2)$	$0.736$ $0.736$ $s l u g \cdot \frac{f t^{2}}{s^{2}}$ $slug*(ft^2)/s^2$	$1.93 \cdot 10^{- 3} \frac{s l u g}{f t^{3}}$ $1.93*10^-3 (slug)/(ft^3)$	$4.368 \cdot 10^{9}$ $4.36810^9$ $\frac{s l u g}{f t \cdot s^{2}}$ $(slug)/(fts^2)$	51.322 ft/s	32.144 ft/ $s^{2}$ $s^2$
kg mm s	1 kg	$10^{3} m m$ $10^3 mm$	1s	$10^{3}$ $10^3$ kg*mm/ $s^{2}$ $s^2$	$10^{- 3}$ $10^-3$ kg/ $(m m \cdot s^{2})$ $(mm*s^2)$	$10^{6}$ $10^6$ $k g \cdot \frac{m m^{2}}{s^{2}}$ $kg*(mm^2)/s^2$	$10^{- 9} \frac{k g}{m m^{3}}$ $10^-9 (kg)/(mm^3)$	$210 \cdot 10^{6}$ $21010^6$ $\frac{k g}{m m \cdot s^{2}}$ $(kg)/(mms^2)$	15647 mm/s	$9.8 \cdot 10^{3} \frac{m m}{s^{2}}$ $9.8*10^3 (mm)/(s^2)$
g cm ms	$10^{3} g m$ $10^3 gm$	$10^{2} c m$ $10^2 cm$	$10^{3} m s$ $10^3 ms$	$10^{- 1}$ $10^-1$ g*cm/ $m s^{2}$ $ms^2$	$10^{- 5}$ $10^-5$ gm/ $(c m \cdot m s^{2})$ $(cm*ms^2)$	$10$ $10$ $g \cdot \frac{c m^{2}}{m s^{2}}$ $g*(cm^2)/(ms^2)$	$10^{- 3} \frac{g m}{c m^{3}}$ $10^-3 (gm)/(cm^3)$	$210 \cdot 10^{4}$ $21010^4$ $\frac{g m}{c m \cdot m s^{2}}$ $(gm)/(cmms^2)$	1.5647 cm/ms	$9.8 \cdot 10^{- 4} \frac{c m}{m s^{2}}$ $9.8*10^-4 (cm)/(ms^2)$
kg mm ms	1 kg	$10^{3} m m$ $10^3 mm$	$10^{3} m s$ $10^3 ms$	$10^{- 3}$ $10^-3$ kg*mm/ $m s^{2}$ $ms^2$	$10^{- 9}$ $10^-9$ kg/ $(m m \cdot m s^{2})$ $(mm*ms^2)$	$1$ $1$ $k g \cdot \frac{m m^{2}}{m s^{2}}$ $kg*(mm^2)/(ms^2)$	$10^{- 9} \frac{k g}{m m^{3}}$ $10^-9 (kg)/(mm^3)$	$210$ $210$ $\frac{k g}{m m \cdot m s^{2}}$ $(kg)/(mm*ms^2)$	15.647 mm/ms	$9.8 \cdot 10^{- 3} \frac{m m}{m s^{2}}$ $9.8*10^-3 (mm)/(ms^2)$

INFERENCE: Since LS-Dyna does not have any unit system settings, the user must be able to maintain the consistency of the units by properly converting them so that the simulation becomes error free.