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Unit Conversion for Consistency of Units in LS Dyna

Objective: To write the given measurements according to the given system of units and to mention the conversion factors for the same. Value of the following measurements are given as follows: MASS - 1kg LENGTH - 1m TIME - 1s FORCE -1N STRESS - 1Pa ENERGY - 1Nm DENSITY - 1Kg/m3 YOUNG'S Modulus - 210GPa Velocity -…

Sam Cherian
updated on 02 Nov 2020

Objective:

To write the given measurements according to the given system of units and to mention the conversion factors for the same.

Value of the following measurements are given as follows:

MASS - 1kg
LENGTH - 1m
TIME - 1s
FORCE -1N
STRESS - 1Pa
ENERGY - 1Nm
DENSITY - 1Kg/m3
YOUNG'S Modulus - 210GPa
Velocity - 56.33KMPH
GRAVITY - 9.8m/s2

System of units:

Kg cm ms
Kg cm s
gm 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

Detailed calculation for conversion from SI unit to gm/mm/ms unit system needs to be shown and a table for the remaining units need to be created with the unit systems as rows and measured quantities as columns.

Procedure:

1. As asked in the question a detailed calculation for conversion from SI unit to gm/mm/ms unit system is shown below.

Mass

1 kg = 1000 grams

Length

1 m = 1000 mm

Time

1 second = 1000 milli-seconds

Force

$F = m \cdot a = k g \cdot \frac{m}{s^{2}}$ $F =m*a = kg*m/s^2$

`1kg*m/s^2 = 10^3g * (10^3mm)/(10^6ms^2) $=$ $=$ 1g*(mm)/(ms^2)`

Therefore,

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

Stress

Stress = Force / Unit Area

1 Pa = 1 N/m^2

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

Therefore,

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

Energy

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

Therefore

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

Density

$1 \frac{k g}{m^{3}} = 10^{3} \frac{g}{10^{9} m m^{3}}$ $1(kg)/m^3 = 10^3 g / (10^9mm^3)$

Therefore:

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

Young's Modulus

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

Therefore:

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

Velocity

$56.33 \frac{k m}{h} = \frac{56.33 \cdot 10^{6} m m}{3.6 \cdot 10^{6} m s}$ $56.33 (km)/h = (56.33* 10 ^6 mm) / (3.6 * 10^6 ms)$

Therefore

$56.33 \frac{k m}{h} = 15.64722 \frac{m m}{m} s$ $56.33 (km)/h = 15.64722 (mm)/ms$

Gravity

$9.8 \frac{m}{s^{2}} = 9.8 \cdot 10^{3} \frac{m m}{10^{6} m s^{2}}$ $9.8m/s^2 = 9.8 * 10^3 (mm) / (10^6 ms^2)$

Therefore:

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

The quantities in their corrresponding unit systems are shown bellow:

Units	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus	Velocity	Gravity
Kg cm ms	1 kg	100 cm	1000 ms	$10^{- 4} k g \cdot \frac{c m}{m s^{2}}$ $10^-4 kg*(cm)/(ms^2)$	$10^{- 8} \frac{k g}{c m \cdot m s^{2}}$ $10^-8 (kg)/(cm*ms^2)$	$10^{- 2} (k g) \cdot \frac{c m^{2}}{m s^{2}}$ $10^-2 (kg)*(cm^2)/(ms^2)$	$10^{- 6} \frac{k g}{{(c m)}^{3}}$ $10^-6 (kg)/(cm)^3$	$2100 k \frac{g}{c m \cdot m s^{2}}$ $2100 kg/(cm*ms^2)$	$156.4 \cdot 10^{- 2} \frac{c m}{m s}$ $156.4*10^-2 (cm)/(ms)$	$9.8 \cdot 10^{- 4} (c \frac{m}{m} s^{2})$ $9.8 * 10^-4 (cm/ms^2)$
Kg cm s	1 kg	100 cm	1 s	$10^{2} k g \cdot \frac{c m}{s^{2}}$ $10^2 kg*(cm)/(s^2)$	$10^{- 2} \frac{k g}{c m \cdot s^{2}}$ $10^-2 (kg)/(cm*s^2)$	$10^{4} (k g) \cdot \frac{c m^{2}}{s^{2}}$ $10^4 (kg)*(cm^2)/(s^2)$	$10^{- 6} \frac{k g}{{(c m)}^{3}}$ $10^-6 (kg)/(cm)^3$	$210 \cdot 10^{7} k \frac{g}{c m \cdot s^{2}}$ $21010^7 kg/(cms^2)$	$1564.722 \frac{c m}{s}$ $1564.722 (cm)/s$	$980 \frac{c m}{s^{2}}$ $980 (cm)/(s^2)$
g mm ms	1000 g	1000 mm	1000 ms	$1 g \cdot \frac{m m}{m s^{2}}$ $1 g*(mm)/(ms^2)$	$10^{- 6} \frac{g}{m m \cdot m s^{2}}$ $10^-6 g/(mm*ms^2)$	$10^{3} g \cdot \frac{m m^{2}}{m s^{2}}$ $10^3 g*(mm^2)/(ms^2)$	$10^{- 6} \frac{g}{{(m m)}^{3}}$ $10^-6 (g)/(mm)^3$	21010^3 g/(mmms^2)`	$15.647 \frac{m m}{m s}$ $15.647 (mm)/(ms)$	$9.8 \cdot 10^{- 3} \frac{m m}{m s^{2}}$ $9.8*10^-3 (mm)/(ms^2)$
kg m s	1 kg	1 m	1 s	$1 k g \cdot \frac{m}{s^{2}}$ $1kg*(m)/(s^2)$	$1 \frac{k g}{m \cdot s^{2}}$ $1 (kg)/(m*s^2)$	$1 (k g) \cdot \frac{m^{2}}{s^{2}}$ $1 (kg)*(m^2)/(s^2)$	$1 \frac{k g}{m^{3}}$ $1 (kg)/m^3$	$210 \cdot 10^{9} k \frac{g}{m \cdot s^{2}}$ $210* 10^9 kg/(m*s^2)$	$15.64 \frac{m}{s}$ $15.64 m/s$	$9.8 \frac{m}{s^{2}}$ $9.8 m/(s^2)$
g cm s	1000 g	100 cm	1 s	$10^{5} g \cdot \frac{c m}{s^{2}}$ $10^5 g*(cm)/(s^2)$	$10 \frac{g}{c m \cdot s^{2}}$ $10 g/(cm*s^2)$	$10^{7} g \cdot \frac{c m^{2}}{s^{2}}$ $10^7 g*(cm^2)/(s^2)$	$10^{- 3} \frac{g}{{(c m)}^{3}}$ $10^-3 (g)/(cm)^3$	$210 \cdot 10^{10} \frac{g}{c m \cdot s^{2}}$ $21010^10 g/(cms^2)$	$1564.7 \frac{c m}{s}$ $1564.7 (cm)/s$	$980 \frac{c m}{s^{2}}$ $980 (cm)/s^2$
g cm $μ$ $mu$ s	1000 g	100 cm	$10^{6}$ $10^6$ $μ$ $mu$ s	$10^{- 7} g \cdot \frac{c m}{s^{2}}$ $10^-7 g*(cm)/(s^2)$	$10^{- 11} \frac{g}{c m \cdot μ s^{2}}$ $10^-11 g/(cm*mus^2)$	$10^{- 5} g \cdot \frac{c m^{2}}{μ s^{2}}$ $10^-5 g*(cm^2)/(mus^2)$	$10^{- 3} \frac{g}{c} m^{3}$ $10^-3 (g)/cm^3$	$210 \cdot 10^{- 2} \frac{g}{c m \cdot μ s^{2}}$ $21010^-2 g/(cmmus^2)$	$15.647 \cdot 10^{- 4} \frac{c m}{μ s}$ $15.647 * 10^-4 (cm)/(mus)$	$9.8 \cdot (10^{- 10}) \frac{c m}{μ s^{2}}$ $9.8 *(10^-10) (cm)/(mus^2)$
g mm s	1000 g	1000 mm	1 s	$10^{6} g \cdot \frac{m m}{s^{2}}$ $10^6 g*(mm)/(s^2)$	$1 \frac{g}{m m \cdot s^{2}}$ $1 g/(mm*s^2)$	$10^{9} g \cdot \frac{m m^{2}}{s^{2}}$ $10^9 g*(mm^2)/(s^2)$	$10^{- 6} \frac{g}{m} m^{3}$ $10^-6 (g)/mm^3$	$210 \cdot 10^{9} \frac{g}{m m \cdot s^{2}}$ $21010^9 g/(mms^2)$	$15647.2 \frac{m m}{s}$ $15647.2 (mm)/s$	$9.8 \cdot 10^{3} \frac{m}{s^{2}}$ $9.8 *10^3 m/(s^2)$
ton mm s	0.001 ton	1000 mm	1 s	$1 ton \cdot \frac{m m}{s^{2}}$ $1"ton"*(mm)/s^2$	$10^{- 6} \frac{ton}{m m \cdot s^{2}}$ $10^-6 "ton"/(mm*s^2)$	$10^{3} ton \cdot \frac{m m^{2}}{s^{2}}$ $10^3 "ton"*(mm^2)/(s^2)$	$10^{- 12} \frac{ton}{m} m^{3}$ $10^-12 ("ton")/mm^3$	$210 \cdot 10^{3} \frac{ton}{m m \cdot s^{2}}$ $21010^3 "ton"/(mms^2)$	$51.33 \frac{f t}{s}$ $51.33 (ft)/s$	$9.8 \cdot 10^{3} \frac{m m}{s^{2}}$ $9.8 *10^3 (mm)/(s^2)$
slug ft s	0.068 slug	3.28 ft	1 s	$0.224 s l u g \cdot \frac{f t}{s^{2}}$ $0.224 slug*(ft)/s^2$	$0.020 \frac{s l u g}{f t \cdot s^{2}}$ $0.020 (slug)/(ft*s^2)$	$0.731 slug \cdot \frac{f t^{2}}{s^{2}}$ $0.731 "slug"*(ft^2)/(s^2)$	$1.92 \cdot 10^{- 3} \frac{s l u g}{f} t^{3}$ $1.92*10^-3 (slug)/ft^3$	$4.35 \cdot 10^{9} \frac{s l u g}{f t \cdot s^{2}}$ $4.3510^9 (slug)/(fts^2)$	$15647.2 \frac{m m}{s}$ $15647.2 (mm)/s$	$32.144 \frac{f t}{s^{2}}$ $32.144 (ft)/(s^2)$
kg mm s	1 kg	1000 mm	1 s	$1000 k g \cdot \frac{m m}{s^{2}}$ $1000 kg*(mm)/(s^2)$	$10^{- 3} \frac{k g}{m m \cdot s^{2}}$ $10^-3 (kg)/(mm*s^2)$	$10^{6} (k g) \cdot \frac{m m^{2}}{s^{2}}$ $10^6 (kg)*(mm^2)/(s^2)$	$10^{- 9} \frac{k g}{m} m^{3}$ $10^-9 (kg)/mm^3$	$210 \cdot 10^{6} k \frac{g}{m m \cdot s^{2}}$ $21010^6 kg/(mms^2)$	$15647.22 \frac{m m}{s}$ $15647.22 (mm)/s$	$9.8 \cdot 10^{3} \frac{m}{s^{2}}$ $9.8*10^3 (m)/s^2$
g cm ms	1000 g	100 cm	1000 ms	$0.1 g \cdot \frac{c m}{m s^{2}}$ $0.1 g*(cm)/(ms^2)$	$10^{- 5} \frac{g}{c m \cdot m s^{2}}$ $10^-5 g/(cm*ms^2)$	$10 g \cdot \frac{c m^{2}}{m s^{2}}$ $10 g*(cm^2)/(ms^2)$	$10^{- 3} \frac{g}{c} m^{3}$ $10^-3 (g)/cm^3$	$210 \cdot 10^{4} \frac{g}{c m \cdot m s^{2}}$ $21010^4 g/(cmms^2)$	$1.564 \frac{c m}{s^{2}}$ $1.564 (cm)/(s^2)$	$9.8 \cdot 10^{- 4} \frac{c m}{s^{2}}$ $9.8*10^-4 (cm)/s^2$
kg mm ms	1 kg	1000 mm	1000 ms	$10^{- 3} k g \cdot \frac{m m}{m s^{2}}$ $10^-3 kg*(mm)/(ms^2)$	$10^{- 9} \frac{k g}{m m \cdot m s^{2}}$ $10^-9 (kg)/(mm*ms^2)$	$1 (k g) \cdot \frac{m m^{2}}{m s^{2}}$ $1 (kg)*(mm^2)/(ms^2)$	$10^{- 9} \frac{k g}{m} m^{3}$ $10^-9 (kg)/mm^3$	$210 k \frac{g}{m m \cdot m s^{2}}$ $210 kg/(mm*ms^2)$	$15.647 \frac{m m}{m} s$ $15.647 (mm)/ms$	$9.8 \cdot 10^{- 3} \frac{m m}{m s^{2}}$ $9.8*10^-3 (mm)/(ms^2)$

Conversion units:

To convert the given value in (kg m s) to the required units, the conversion values are shown below:

Units	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus	Velocity	Gravity
Kg cm ms	1	100	1000	0.0001	0.00000001	0.01	0.000001	0.00000001	0.027765	0.0001
Kg cm s	1	100	1	100	0.01	10000	0.000001	0.01	27.77777	100
g mm ms	1000	1000	1000	1	0.000001	1000	0.000001	0.000001	0.277774	0.001
kg m s	1	1	1	1	1	1	1	1	0.27765	1
g cm s	1000	100	1	100000	10	10000000	0.001	10	27.77738	100
g cm $μ$ $mu$ s	1000	100	1000000	1E-07	1E-11	0.00001	0.001	1E-11	2.78E-05	1E-10
g mm s	1000	1000	1	1000000	1	1000000000	0.000001	1	277.7774	1000
ton mm s	0.001	1000	1	1	0.000001	1000	1E-12	0.000001	0.911237	1000
slug ft s	0.068	3.28	1	0.224	0.02	0.731	0.00192	0.020714286	277.7774	3.28
kg mm s	1	1000	1	1000	0.001	1000000	1E-09	0.001	277.7777	1000
g cm ms	1000	100	1000	0.1	0.00001	10	0.001	0.00001	0.027765	0.0001
kg mm ms	1	1000	1000	0.001	1E-09	1	1E-09	0.000000001	0.277774	0.001

Result:

The given measurements were converted according to the given system of units and the mention the conversion factors for the same are mentioned.
Calculation for one conversion is shown extensively.

Learning outcomes: