Week - 1 - Consistency of Units

UNIT CONSISTENCY

Objective: To convert the system of SI units into given system of units as per the data given.

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.

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 has to be conversant with converting quantities from one system of units to another.

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

Now, here are a few system of units:

1. Kg cm ms
2. Kg cm s
3. gm mm ms
4. kg m s
5. g cm s
6. g cm us
7. g mm s
8. ton mm s
9. slug ft s
10. kg mm s
11. g cm ms
12. kg mm ms

Write the given measurements in the above different system of units. Also, mention the conversion factors.

MASS	LENGTH	TIME	FORCE	STRESS	ENERGY	DENSITY	YOUNG’S	VELOCITY	GRAVITY
Kg	100cm	1000ms	1kg *100cm/(1000)2 = 10e-05	10e-10	10e-1	10e-6	210x10e-10	1.565	9.81 x 10e-04
Kg	100cm	s	10e+02	10e-4	10e+5	10e-6	210x 10e-04	1.565 x10e+3	9.81 x10e+02
kg	1000mm	s	10e+03	10e-03	10e+6	10e-9	210x10e-03	1.5650x10e+4	9.81x 10e+03
gm	100cm	s	10e+05	10e-01	10e+8	10e-3	210x10e-01	1.565x10e+3	9.81 x10e+02
gm	100cm	100000us	10e-05	10e-05	10e-2	10e-3	210x10e-05	1.565 x10e-2	9.81x10e-08
gm	1000mm	s	10e+06	1	10e+9	10e-9	210	1.5650x 10e+4	9.81x 10e+03
Ton=1000kg	1000mm	s	10e+03	10e+03	10e+12	10e-3	210x10e+03	1.5650x 10e+4	9.81x 10e+03
Slug=14.5kg	ft=3048mm	s	10e+9	10e+05	1.347	0.407	210x10e+05	47.69	2.982
kg	1000mm	s	10e+03	10e-09	10e+6	10e-09	210x10e-09	1.5650x 10e+4	9.81x 10e+03
gm	100cm	1000ms	10e-1	10e-07	10e+3	10e-03	210x10e-07	1.565	9.81 x 10e-04
kg	1000mm	1000ms	10e-3	10e-06	1	10e-09	210x10e-06	15.659	9.81 x 10e-03

Fundamental quantities:

Fundamental quantiles are independent quantities which are mass (M), Length (L), Time (s).

Derived quantities: Derived quantiles are dependent on fundamental quantities.

Unit conversion Calculations

Mass 1kg = 1000 gm, LENGTH 1M=1000 mm, TIME s= 1000 ms.

1. = m*a = kg *m/ s² =(1000gm * 1000 mm)/1000 ms * 1000 ms = 1 ( gm mm/ ms˄2)

= (1kg * 100 cm)/ s² = 10 ˄2 (kg cm s)

2. STRESS 1 pa = N/m² = (kg m/s² ) / (m) ² = {(1000gm *1000mm/1000 ms * 1000ms )}/ (1000)² = 10˄-7
3. ENERGY 1NM =( kg m/s-² ) m = {1000gm * 1000mm * (1000)^-2 } 1000mm = 10˄3

4. DENSITY 1kg/m³ = (1000gm/( 1000mm)³ = 10˄-6

5. YOUNG’S MODULES 210GPa = 210 × 10⁹ X N/m² = 210 x10⁹ x (kg m/s² )/ m² = 210 x10 ⁹ x {( 1000gm x 1000 mm)}/(1000)³ =210 x10˄3

6. VELOCITY 56.33KMPH ( 1kmph= 1000 x 1000 mm /(3600 *1000ms) = 10⁶ /(36x 10⁵ ) = 10/36 = 0.278 →33x0.278= 15.659

7. GRAVITY 9.81 m/s² = 1000 mm /(1000ms)² = 10³ * 10 ^-6 = 10˄ -3 = 9.81 x 10^-3

Conclusion: Unit conversion is done as per the given data.