Week - 1 - Consistency of Units

AIM:

To calculate the unit conversion from SI unit to 

• 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

Theory:

Different industries use different system of units as per their convenience. The units for the basic measurement quantities, mass distance and time - dictate how the other derived quantities will be expressed. For example, automotive industries use kg,mm,ms as their standard unit system. Construction industries are likely to use tonne,m,s as their unit system. Similarly, PCB simulation in electronic industry uses gm, micrometer,ms as their system of units.

Example: Young's Modulus of steel is 210GPa which is input as 210e+9 in LS-DYNA as per SI unit system. However, if you use kg/mm/ms unit system, the input quantity will be 210. Thus the conversion factor is 10^-9.

But, how?

The basic S.I system or M.K.S system:

Length: Meter

Mass: Kilogram

Time: Second

Youngs modulus have same units of stress as strain has no units.

Stress = Force/Area

Force = Mass*Acceleration

          = kg*m / s^2 = NEWTON

Stress = N / m^2                                                                    1 MPA = 10^6 N/m^2 = N/mm^2

                                                                                                 1 GPA = 10^9 N/m^2 = 10^3 N/mm^2

Given, 

Young's Modulus of steel is 210GPa.

If we use kg/mm/ms 

Force = kg*mm / ms^2 = 1000 newton.

Stress = F/A = 1000 N/mm^2

                      = 1000* 10^6 N/m^2

                      = 1 GPA

Therefore, 

YOUNGS MODULUS in kg/mm/ms = 1 GPA

To convert in S.I system i.e., meter-kilogram-second

1 GPA = 10^9 Kg/m-s^2 = 10^9 N/m^2

The conversion factor is 10^9.

according to S.I system, E = 210* 10^9 N/m2

according to kg/mm/ms, E = 210 GPA.

Conversion of velocity:

56.33 KMPH = 15.647 mps

Detailed calculation for conversion from SI unit to gm/mm/ms unit system:

Mass: gm

Length: mm

Time: ms

Force:  gm-mm/ms^2 

                                      F = m*a = gm-mm/ms^2 

Stress: gm/mm-ms^2

                                      stress = force/area = gm/mm-ms^2

Energy: gm-mm^2/ms^2 

                                      Energy = Force * displacement = gm-mm^2/ms^2 

Density: gm / mm^3

                                     Density = mass / volume = gm / mm^3

Youngs modulus: gm/mm-ms^2

                                      Youngs modulus = force/area = gm/mm-ms^2

Velocity: mm/ms

                                      velocity = displacement/time = mm/ms

Gravity = mm/ms^2

Conversion to from S.I system to gm/mm/ms:

 Force = 1 gm-mm/ms^2

 Stress = 10^-6 gm/cm-ms^2

 Energy = 1000 gm-mm^2/ms^2 

 Density = 10^-6   gm/mm^3

 Youngs modulus = 210 *10^-6 gm/mm-ms^2

 Velocity = 15.647 mm/ms

 Gravity = 9.8*10^-3 mm/ms^2

UNITS	MASS	LENGTH	TIME	FORCE	STRESS	ENERGY	DENSITY	YOUNGS MODULUS	VELOCITY	GRAVITY
S.I UNITS kg-m-s	KG	METER	SECOND	N = kg-m / s^2	kg/m-s^2	kg-m^2 / s^2	Kg/m^3	kg/m-s^2	m/s	m/s^2
Kg cm ms	KG	100 cm	10^3 ms	10^-4   kg-cm/ms^2	10^-8 kg/cm-ms^2	10^-2 kg-mm^2/ms^2	10^-6   kg/cm^3	210*10^-8    kg/cm-ms^2	1.5647 cm/ms	9.8*10^-4     cm/ms^2
conversion factor	1	100	10^3	10^-4	10^-8	10^-2	10^-6	10^-8	0.1	10^-4
Kg cm s	KG	100 cm	s	100     kg-cm/s^2	10^-2     kg/cm-s^2	10^4   kg-cm^2/s^2	10^-6      Kg/cm^3	210*10^-2      kg/cm-s^2	1564.7 cm/sec	980 cm/s^2
conversion factor	1	100	1	100	10^-2	10^4	10^-6	10^-2	100	100
g cm s	1000 gm	100 cm	s	10^5    g-cm / s^2	10     g/cm-s^2	10^7   g-cm^2 / s^2	10^-3     g/cm^3	210*10     g/cm-s^2	1564.7 cm/s	980 cm/s^2
conversion factor	10^3	10^2	1	10^5	10	10^7	10^-3	10	100	100
g cm us	1000 gm	100 cm	10^6 us	10^-7    g-cm/us^2	10^-11     g/cm-us^2	10^-5     g-cm^2/us^2	10^-3       g/cm^3	210*10^-11       g/cm-us^2	15.647*10^-4        cm/us	9.8*10^-10       cm/us^2
conversion factor	1000	100	10^6	10^-7	10^-11	10^-5	10^-3	10^-11	10^-4	10^-10
g mm s	1000 g	1000 mm	s	10^6   g-mm/s^2	1     g/mm-s^2	10^9    g-mm^2 / s^2	10^-6      g/mm^3	210     g/mm-s^2	15647        mm/s	9800      mm/s^2
conversion factor	1000	1000	1	10^-6	1	10^9	10^-6	1	1000	1000
ton mm s	10^-3 ton	1000 mm	s	1   ton-mm/s^2	10^-6    ton/mm-s^2	1000 ton-mm^2 / s^2	10^-12     Ton/mm^3	210*10^-6    ton/mm-s^2	15647        mm/s	9800       mm/s^2
conversion factor	10^-3	1000	1	1	10^-6	1000	10^-12	10^-6	1000	1000
slug ft s	0.0685218           slug	3.28 foot	s	0.2247515    slug-ft / s^2	0.02108363     slug/ft-s^2	0.73718493  slug-ft^2 / s^2	1.942*10^-3       slug/ft^3	4.4275623     slug/ft-s^2	51.32          ft/s	32.144       ft/s^2
conversion factor	0.0685218	3.28	1	0.2247515	0.02108363	0.73718493	1.942*10^-3	0.02108363	3.28	3.28
kg mm s	kg	1000 mm	s	1000    kg-mm/s^2	10^-3    kg/mm-s^2	10^6 kg-mm^2 / s^2	10^-9      Kg/mm^3	210*10^-3     kg/mm-s^2	15647         mm/s	9800       mm/s^2
conversion factor	1	1000	1	1000	10^-3	10^6	10^-9	10^-3	1000	1000
g cm ms	1000g	100 cm	1000 ms	0.1   g-cm/ms^2	10^-5    g/cm-ms^2	10    g-cm^2/ms^2	10^-3       g/cm^3	210*10^-5      g/cm-ms^2	1.5647       cm/ms	9.81* 10^-4       cm/ms^2
conversion factor	1000	100	1000	0.1	10^-5	10	10^-3	10^-5	0.1	10^-4
kg mm ms	kg	1000 mm	1000 ms	10^-3 kg-mm/ms^2	10^-9    kg/mm-ms^2	1 kg-mm^2 /ms^2	10^-9       Kg/mm^3	210*10^-9      kg/mm-ms^2	15.647        mm/ms	9.81*10^-3        mm/ms^2
conversion factor	1	1000	1000	10^-3	10^-9	1	10^-9	10^-9	1	10^-3

           

Conclusion:

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.