Week - 1 - Consistency of Units

Aim : To Convert the following measurements into the given system of units and determine their corresponding conversion factors.

Measurements:

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

Theory : 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.

Conversion Factors: To convert the given measurements into different unit systems, conversion factors are determined based on the relationships between the units in each system.

Conversion from SI Unit to gm/mm/ms unit system

MASS: 1kg = 1000 gm (conversion factor = 10³)

LENGTH: 1m = 1000 mm (conversion factor = 10³)

TIME: 1s = 1000 ms (conversion factor = 10³)
FORCE: 1N = (1kg * 1m)/ 1s²

= (1000 gm * 1000 mm)/ (1000 ms)²

= (1000000 gm*mm) / (1000000 ms² )

= 1 gm*mm/ms²

(conversion factor = 1)

STRESS: 1Pa = 1N/m²

= 10^-4 kg*cm/ (ms² * (1000 ms)²)

= 10^-4 x 1000 gm *10 mm/ (ms² * (1000 ms)²)

= 10^-6 gm*mm / ms⁴

(conversion factor = 10^-6)

ENERGY: 1Nm = 1 kg *m² /s²

= 1000 gm * (1000 mm)² / (1000 ms)²

= 10³ gm * mm² /ms²

(conversion factor = 10³)

DENSITY: 1Kg/m3 = 1kg/ (100 cm)3

= 1kg / 1000000 cm³

= 10^-6 kg/cm³

(conversion factor = 10^-2)

YOUNG'S Modulus: 210GPa = 210 x 10⁹ Pa

= 210 x 10⁹ Nm

= 210 x 10⁹ x 10^-10 kg*cm / ms⁴

= 210 x 10^-1 kg*cm / ms⁴

(conversion factor = 10^-1)

Velocity: 56.33KMPH = 56.33 * 100000 cm / 3600000 ms

= 56.33 cm/ 36 ms

= 56.33 * 0.0278 cm/ms

(conversion factor = 0.0278)
GRAVITY: 9.8m/s² = 9.8 * 100 cm / (1000 ms)²

= 9.8 * 100 cm / 1000000 ms2

= 9.8 * 10⁴ cm/ ms²

(conversion factor = 10⁴)

Results (for other unit systems): (Table format)

Unit System	Mass	Length	Time	Force	Stress	Energy	Density	Young's Modulus	Velocity	Gravity
Kg m s	1 kg	1 m	1 s	1 N = 1 Kg*m/s2	1 Pa = 1 kg/(m* s2)	1J = 1 Nm = 1 kg*m2/s2	1Kg /m3	210 GPa = 210 x 109 Pa = 210 x 109 Nm	56.33 Km/hr = 56.33 * 1000 m /3600 s = 56.33 * 0.28 m/s	9.8 m/s2
Conversion Factor (C.F.)	1	1	1	1	1	1	1	109	0.28	1
Kg cm ms	1kg	100 cm	1000 ms	10-4 kg*cm/ms2	10-10 kg*cm / ms4	10-2 kg*cm2/ms2	10-6 kg/cm3	210 x 10-1 kg*cm / ms4	56.33 * 0.0278 cm/ms	9.8 * 104 cm/ ms2
C.F.	1	102	103	10-4	10-10	10-2	10-6	10-1	0.0278	104
Kg cm s	1	102	1	102 kg * cm /s2	10-2 kg / cm * s2	104 kg * cm2/s2	10-6 kg/ cm3	210 x 107 kg/ cm * s2	1577.24 cm/s	980 cm /s2
C.F.	1	102	1	102	10-2	104	10-6	107	28	102
gm mm ms	103 gm	103 mm	103 ms	1 gm * mm /ms2	10-6 gm/ mm* ms2	103 gm * mm2 /ms2	10-6 gm/mm3	210 x 103 gm / mm * ms2	15.77 mm/ms	9.8 x 10-3 mm/ms2
C.F.	103	103	103	1	10-6	103	10-6	103	0.28	10-3
g cm s	1000 gm	1000 cm	1 s	105 gm * cm /s2	10 gm / cm * s2	107 gm * cm2 /s2	10-3 gm/cm3	210 x 1010 gm/ cm * s2	15772 cm/s	980 cm /s2
C.F.	103	103	1	105	10	107	10-3	1010	280	102
g cm us	1000 gm	1000 cm	106 us	10-7 gm * cm / us2	10-11 gm/ cm* us2	10-5 gm*cm2 /us2	10-3 gm/cm3	210 x 10-2 gm / cm * us2	0.0157 cm/us2	9.8 x 10-10 cm/us2
C.F.	103	103	106	10-7	10-11	10-5	10-3	10-2	2.8 x 10-4	9.8 x 10-10
g mm s	1000 gm	105 mm	1 s	106 gm * mm /s2	102 gm / mm * s2	109 gm * mm2 /s2	10-6 gm/mm3	210 x 109 gm/ mm * s2	157724 mm/s	9800 mm /s2
C.F.	103	105	1	106	102	109	10-6	109	2800	103
ton mm s	10-3 ton	105 mm	1 s	102 ton * mm/s2	10-8 ton / mm * s2	107 ton * mm2/s2	10-18 ton/mm3	210 x 10 ton / mm * s2	157724 mm/s	9800 mm /s2
C.F.	10-3	105	1	102	10-8	107	10-18	10	2800	103
slug ft s	6.8 x 10-2 slug	3.28 ft	1 s	2.2 x 10-1 slug * ft /s2	2.07 x10-3 slug / ft * s2	7.32 x 10-1 slug * ft2 / s2	1.7 x 10-3 slug / ft3	4.3 x109 slug / ft * s2	0.518 ft/s	0.32 ft/s2
C.F.	6.8 x 10-2	3.28	1	2.2 x 10-1	2.07 x10-3	7.3 x 10-1	1.7 x 10-3	2.07 x 10-2	9.2 x 10-3	3.28 x10-2
kg mm s	1 kg	103 mm	1 s	103 kg * mm /s2	10-3 kg / mm * s2	106 kg * mm2 /s2	10-9 kg /mm3	210 x106 kg /mm * s2	1.57 x 104 mm/s	9.8 x 103 mm/s2
C.F.	1	103	1	103	10-3	106	10-9	106	280	103
g cm ms	103 gm	102 cm	103 ms	10-1 gm * cm / ms2	10-5 gm/cm * ms2	10 gm * cm2 / ms2	10-3 gm /cm3	210 x 104 gm/ cm * ms2	1.57 cm/ms	9.8 x 104 cm / ms2
C.F.	103	102	103	10-1	10-5	10	10-3	104	0.028	104
kg mm ms	1 kg	103 mm	103 ms	10-3 kg * mm / ms2	10-9 kg / mm * ms2	1 Kg * mm2 / ms2	10-6 kg / mm3	210 kg / mm * ms2	15.77 mm/ms	9.8 x 10-3 mm/ms2
C.F.	1	103	103	10-3	10-9	1	10-6	1	0.28	10-3

Conclusion : Thus we have converted different mesurements in different units.