Week - 1 - Consistency of Units

AIM :

To convert the given system of units into different unit system and finding the respective unit conversion factor.

OBJECTIVE :

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.
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.

PROCEDURE :

Given Measurements : (To be converted to gm/mm/ms unit system and find its conversion factor)

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

Conversion of units are as follows :

1. Mass 1Kg : 1Kg = 1000grams so `10^3`grams. The conversion factor is `10^3`

2. Length 1m : 1m = 100cm (10mm = 1cm) so, 100cm = (100*10) mm = 1m. So the conversion factor is `10^3`

3. Time 1s : 1s to 1ms. 1s = 1000milliseconds (ms). So the conversion factor is `10^3`

4. Force 1N : 1 N = 1 Kg * 1 m/`s^2` (According to F=ma) now we know that 1Kg = `10^3`grams and 1m = `10^3`mm and 1s = `10^3` ms
Substituting these values, we get :

1N = `10^3`gm. `10^3` mm / `(10^3 ms)^2`

so, 1 N = `10^3` * `10^3`* `10^-6` `gm*(mm) / (ms^2)` = 1 `gm* (mm) / (ms^2)`
The conversion factor is 1.

5. Stress 1Pa : 1Pa = 1 `N/m^2` We know that 1N = 1 `gm*(mm) / (ms)^2` & 1`m^2`= `10^6` `mm^2`

Substituting those value we get, 1 Pa = `10^-6` `(gm) / (mm. ms^2)`

The conversion factor is `10^-6`

6. Energy 1Nm : We know that 1Nm = 1 Joule and 1N = 1 `gm* (mm)/(ms)^2` and 1m = `10^3`mm

Substituting all these values, we get 1Nm = `10^3` `gm* (mm)^2 / (ms)^2`

The conversion factor is `10^3`

7. Density 1 Kg/`m^3` : We know that 1Kg=`10^3`gm and `1m^3` = `10^9 mm^3`

Substituting these values we get 1 Kg/`m^3` = `10^-6` `(gm) / (mm^3)`

The conversion factor is `10^-6`

8. Young's modulus 210 Gpa : We know that, 1Gpa = `10^3` N /`(mm^2)` and 1N = 1 `gm*(mm)/(ms^2)`

Substituting these values we get 1Gpa = `10^3 gm*(mm)/(ms^2) / (mm^2)` and we will get `10^3` `(gm)/ (mm*ms^2)`

So, 210 Gpa = `210*10^3 (gm)/ (mm*ms^2)`

The conversion factor is `10^3`

9. Velocity 56.33 Km/hr : We know that 1 Km = `10^6`mm and 1Hr = `3600*10^3` ms. Substituting these values

We get 56.33 Km/hr = `56.33*(10^6 mm) / (3600*10^3 ms)`

So, we get 56.33 Km/hr = 15.647 mm/ms

The conversion factor is `5/18`

10. Gravity 9.8 m/`s^2` : We know that 1m = `10^3` mm and 1s = `10^3` ms. Substituting these values

We get `9.8* 10^3*(mm)/ (10^3ms)^2` and so the value would be 9.8*`10^-3` `(mm)/(ms^2)`

The conversion factor is `10^-3`


Now, here are the few unit systems which needs to be converted with respect to SI unit systems :

Unit system	Mass	Length	Time	Force	Stress	Energy
SI Unit System (Kg/m/s)	1Kg	1m	1s	1N =`1Kg*m/s^2`	1Pa=`(1Kg)/(m*s^2)`	1J=1Nm=`1Kg*(m^2/s^2)`
Kg/cm/ms	1Kg	`10^2`cm	`10^3`ms	`10^-4``Kg*(cm)/(ms^2)`	`10^-8``(Kg)/(cm*ms^2)`	`10^-2` `Kg*(cm^2)/(ms^2)`
Kg/cm/s	1Kg	`10^2`cm	1s	`10^2``Kg*(cm)/(s^2)`	`10^-2``(Kg)/(cm*s^2)`	`10^4` `Kg*(cm^2)/(s^2)`
g/mm/ms	`10^3`g	`10^3`mm	`10^3`ms	1 `g*(mm)/(ms^2)`	`10^-6``(g)/(mm*ms^2)`	`10^3` `g*(mm^2)/(ms^2)`
g/cm/s	`10^3`g	`10^2`cm	1s	`10^5``g*(cm)/(s^2)`	`10^1``(g)/(cm*s^2)`	`10^7` `g*(cm^2)/(s^2)`
g/cm/`mus`	`10^3`g	`10^2`cm	`10^6` `mus`	`10^-7``g*(cm)/ (mus^2)`	`10^-11``g/(cm*mus^2)`	`10^-5``g*(cm^2)/(mus^2)`
g/mm/s	`10^3`g	`10^3`mm	1s	`10^6``g*(mm)/s^2`	1 `(g)/(mm*s^2)`	`10^9` `g*(mm^2)/s^2`
Tonne/mm/s	`10^-3`Tonne	`10^3` mm	1s	1tonne`*(mm)/(s^2)`	`10^-6`tonne`/(mm*s^2)`	`10^3`tonne`(mm^2)/(s^2)`
Slug/ft/s	0.068 slug	3.28 ft	1s	0.2246 `slug*(ft)/(s^2)`	0.0208`(slug)/(ft*s^2)`	0.7366`slug*(ft^2)/(s^2)`
Kg/mm/s	1Kg	`10^3`mm	1s	`10^3``Kg*(mm)/(s^2)`	`10^-3``(Kg)/(mm*s^2)`	`10^6``Kg*(mm^2)/(s^2)`
g/cm/ms	`10^3` g	`10^2`cm	`10^3`ms	`10^-1``g*(cm)/(ms^2)`	`10^-5``(g)/(cm*ms^2)`	`10^1``(g)*(cm^2)/(ms^2)`
Kg/mm/ms	1Kg	`10^3`mm	`10^3`ms	`10^-3``Kg*(mm)/(ms^2)`	`10^-9``(Kg)/(mm*ms^2)`	1`Kg*(mm^2)/(ms^2)`

RESULT :

Thus, all the given unit systems were converted to the respective unit system and its conversion factor has been identified.