Week - 1 - Consistency of Units

OBJECTIVE: calculation for conversion from SI unit to gm/mm/ms unit system. For others, only the answers in the form of a table are sufficient with the unit systems as rows and measured quantities as columns.

LS-DYNA doesn't have a default system of units. User needs to input material properties and physical dimensions according to a consistent system of units as per their choice. This warrants that to use LS-DYNA in different circumstances, users have to be conversant with converting quantities from one system of units to another.

Few systems of units are described as given below:

• Kg cm ms
• Kg cm s
• gm mm ms
• kg m s
• g cm s
• g cm s
• g mm s
• ton mm s
• slug ft s
• kg mm s
• g cm ms
• kg mm ms

Values of some measurements are given as below:

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

Detailed Calculation:

Here we will see the conversion of the above measurements with a conversion factor.

• The detailed calculation for conversion from SI unit to gm/mm/ms unit system can be as given below:

MASS: 1kg = 1000 gm, Conversion factor = 1 * 1000 = 10³

LENGTH: 1m = 1000 mm, Conversion factor = 1 * 1000 = 10³

TIME: 1s = 1000 ms, Conversion factor = 1 * 1000 = 10³

FORCE: 1N = 1 (Kg * m) / S²

= 1 (1000 gm * 1000 mm) / (1000 ms) ²

= (1 gm*mm)/ ms²

Conversion factor = 1

STRESS: 1Pa = 1 N / m² = {(1 Kg * m) / S²} / m²

= {(1000gm * 1000mm) /1000000 ms²} / 1000000mm²

= {(10⁶ gm mm)/ 10⁶ ms²} / 10⁶ mm²

= (1 gm mm)/ms² / 10⁶ mm²

= (10^-6 gm)/mm * ms²

Conversion factor = 10^-6

ENERGY: 1Nm = {1 (Kg * m) / S²} * m

= {1 (1000 gm * 1000 mm) / (1000 ms) ²} * 1000mm

= {1 (10⁶ gm * mm) / (10⁶ ms ²} * 10³ mm

= 1{(gm * mm) / ms ²}* 10³ mm

= 1{(gm * mm) / ms ²} * 10³ mm

= 10³(gm * mm²) / ms ²

Conversion factor =10³

DENSITY: 1Kg/m3 = 1000 gm / (1000mm)³

= 1000 gm / 10⁹ mm³

= 10³ gm / 10⁹ mm³

= 10^-6 gm /mm³

Conversion factor =10^-6

YOUNG'S Modulus: 210GPa = 210 * 1000 N / mm²

= {210 * 1000 (Kg * m) / S²}/ mm²

= {210 * 1000 (1000 gm * 1000 mm) / (1000 ms) ²} / mm²

= {210 * 1000 (1 gm * mm) /ms ²} / mm²

= (210 * 10³ gm * mm) / mm² * ms²

= 210 * 10³ gm / mm * ms²

Conversion factor =10³

VELOCITY: 56.33KMPH = (56.33 * 1000 m) / H

= (56.33 * 10⁶ mm) / 3600 * 1000 s

= (56.33 * 10⁶ mm) / 36 * 10⁵ ms

= 15.6472 mm/ ms

Form Factor, taking Round figure of 15.6472 is 15. So Prime Factors are 5 and 3.

• To get 5 in RHS, divide it by 3.12944.
• To get LHS, divide it by 3.12944.
• By division of 3.12944 , we get LHS = 18 and RHS = 5
• So, RHS = 5 / 18

Conversion factor =5/18

GRAVITY: 9.8m/s2 = 9.8 (1000mm) / (1000 ms) ²

= 9.8 (10³ mm) / 10⁶ms²

= 9.8 *10^-3 mm /ms²

Conversion factor =10^-3

• The detailed calculation for conversion from SI unit to kg/cm/s unit system can be as given below:

Mass = 1 kg

Length = 1m Conversion factor = 1*100=100 cm

Time = 1 sec

Force = 1 N

• F = mass* acceleration
• = 1 kg * 1m/s²
• = (1 kg*100cm)/(1s)²
• = 100 kg cm/s²

Conversion factor = 100

Stress = 1 Pa

• 1 Pa = 1N/m²
• As from the above derived 1N = 100 kg cm/s²
• = (100 kg cm/s² )/ (100 cm)²
• = 10^-2 kg/cm s²

Conversion factor = 10^-2 kg/cm*s²

Energy = 1 Nm

• 1N *1 m
• 100 kg cm/s² * 100 cm
• 10⁴ kg cm²/s²

Conversion factor = 10⁴

Density = 1Kg/m^3

• 1 kg / (100 cm)³
• 1 kg/10000 cm³
• 10^-4 kg/cm³

Conversion factor = 10^-4

Young’s Modulus = 210 GPa

• 1 Pa = 10^-2 kg/cm s²
• 1 GPa = 10⁹ * 10^-2 kg/cm s²
• 210 GPa = 210 * 10⁷ kg/cm*s²

Conversion factor = 10⁷

Velocity = 56.33 Kmph

• Km/h
• 1Km = 10000 cm = 10⁴ cm
• 1 hr = 60min*60sec
• = 3600
• = 3600 s
• 1 kmph
• = (10⁴ )/(3600)
• = 2.77 cm/s
• 56.33 kmph =
• = 56.33 * 2.77 cm/s
• = 156.03 cm/s

Conversion factor = `10^4`

Gravity = 9.88 m/s²

• 1 m/s2 = 100 cm/(1s)²
• = 100 cm/s2
• 9.8 m/s2 = 9.8*`10^2` cm/s2

Conversion Factor = 100

Units	Mass	Length	Time	Force	Stress	Energy	Density	Young’s Modulus	Velocity	Gravity
Kg cm ms	1 Kg	`10^2`cm	`10^3`ms	`10^-4`kg cm/ms2	`10^-8`kg /cm*ms2	`10^-2`kg*cm2/ms2	`10^-6`kg/cm3	2100kg/cm*ms2	1.56472 cm/ms	9.8*`10^-4`cm/ms2
Kg cm s	1 Kg	`10^2`cm	1s	`10^2`kg*cm/s2	`10^-2`kg/cm*s2	`10^4`kg*cm2/s2	`10^-6`kg/cm3	210*`10^7`kg/cm*s2	1564.72 cm/s	9.8*`10^2` cm/s2
gm mm ms	`10^3`gm	`10^3`mm	`10^3`ms	1gm*mm/m`10^-8`kg	`10^-6` gm/mm*ms2	`10^3`gm*mm2/ms2	`10^-6`gm/mm3	210*`10^3`gm/mm*ms2	15.6472 mm/ms	9.8*`10^-3` mm/ms2
kg m s	`10^3`gm	1m	1s	1kg*m/s2	1 kg/m*s2	1 kg*m2/s2	1kg/m3	210*`10^9`N/m2	15.6472 m/s	9.8 m/s2
gm cm s	`10^3`gm	`10^2`cm	1s	`10^5`gm*cm/s2	10 gm/cm*s2	`10^7`gm*cm2/s2	`10^-3`gm/cm3	210*`10^10`gm/cm*s2	1564.72cm/s	9.8*`10^2` cm/s2
gm cm μs	`10^3`gm	`10^2`cm	`10^6`μs	`10^-7`gm*cm/μs2	`10^-11`gm/cm*μs2	`10^-5`gm*cm2/μs2	`10^-3`gm/cm3	2.1gm/cm*μs2	15.6472 *`10^-4`cm/μs	9.8*`10^-10` cm/μs2
gm mm s	`10^3` gm	`10^3`mm	1s	`10^6`gm*mm/s2	1gm/mm*s2	`10^9`gm*mm2/s2	`10^-6`gm/mm3	210*`10^11`gm/mm*s2	156472 mm/s	9.8*`10^3` mm/s2
ton mm s	`10^-3`gm	`10^3`mm	1s	1 ton mm/s2	`10^-6`ton /mm*s2	`10^3`ton*mm2/s2	`10^-12`ton/mm3	210*`10^3`ton/mm*s2	156472 mm/s	9.8*`10^3` mm/s2
slug ft s	0.068 slug	3.28 ft	1s	0.2246 slug*ft /s2	0.0208slug/ft*s2	0.7366slug*ft2/s2	1.93*`10^-3`slug/ft3	4.368*slug/ft*s2	51.322 ft/s	32.144 ft/s2
kg mm s	1 Kg	`10^3`mm	1s	`10^3`kg mm/s2	`10^-3`kg/mm*s2	106kg*mm2/s2	10-9kg/mm3	210*106kg/mm*s2	156472 mm/s	9.8*`10^3` mm/s2
gm cm ms	`10^3`gm	`10^2`cm	`10^3`ms	`10^-1`gm cm/ms2	`10^-5`gm/cm*ms2	10gm*cm2/ms2	10-3gm/cm3	210*104gm/cm*ms2	1.56472 cm/ms	9.8*`10^-4` cm/ms2
kg mm ms	1 Kg	`10^3`mm	`10^3`ms	`10^-3`kg*mm/ms2	`10^-9`kg/mm*ms2	1kg*mm2/ms2	10-9 kg/mm3	210kg/mm*ms2	15.6472 mm/ms	9.8*`10^-3`mm/ms2

Conclusion:

• As we know that LS Dyna has no own unit system, so we have to convert each unit as per requirements.
• We can convert units with help of the conversion factor very easily and fast.
• So above conversion should always keep in mind and whenever we need, we must convert units with help of the conversion factor.