Week - 1 - Consistency of Units

Week - 1 - Consistency of Units

1.Aim: To represent the given measurements/quntities in unit systems mentioned in the question along with their respective conversion factors.

2.Objective:
To convert given measurements into different system of units and finding their corresponding conversion factors.

Measurements given :

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

3.Theory:

i.Unit Systems:

• Units are most basic aspects of any measurements.
• They are arbitrarily chosen and internationally accepted.

Categories of units

1.Fundamental units

• Some units are arbitrarily chosen to be fundamental.
• They become foundation for system.
• Every other unit is derived from them.

2.Derived units

• Units derived from the fundamental units by multiplication and division are called derived units.

Example   LENGTH=SPEED*TIME

In this length is a derived unit.
There are different unit systems like SI, CGS, MKS, FPS etc.

ii.Unit Systems in LS-Dyna:

Coming to our area of interest, 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.

4.Conversions: 

The detailed conversion of the given quantities from SI unit to gm/mm/ms unit system are given below:
(Note:For other units only the answers in the form of a table are shown with the unit systems as rows and measured quantities as columns.)

1. MASS 1kg=1000g
   Conversion Factor=10^3
   
   
2. LENGTH 1m=1000mm
   Conversion Factor=10^3
   
   
3. TIME 1s=1000ms
   Conversion Factor=10^3
   
   
4. FORCE 1N= 1kgm/s^2
                    = {1000g*1000mm}/ (1000ms)^2
                    = {10^6/10^6}gmm/(ms)^2
                    = 1 g*mm/(ms)^2
   Conversion Factor=1
   
   
5. STRESS 1Pa= 1N/m^2
                      = 1(kgm/s^2)/m^2
                       = 1(g*mm/(ms)^2)/(1000mm)^2...............{from iv}
                       =  10^-6 g/{mm*(ms)^2}
   Conversion Factor=10^-6
   
   
6. ENERGY 1Nm= 1 [kg*m/s^2]*m
                         = 1 [g*mm/(ms)^2]*1000mm...............{from iv}
                         = 1000 [g*mm^2]/(ms)^2
   Conversion Factor=10^3
   
   
7. DENSITY 1Kg/m3= 1000g/(1000mm)^3
                               =  10^-6 g/mm^3
   Conversion Factor=10^-6
   
   
8. YOUNG'S Modulus  210GPa= 210*10^9 Pa
                                                = 210*10^9*10^-6 g/{mm*(ms)^2}.........(from v)
                                                = 210*10^3 g*mm/[(ms)^2)*(mm)^2]  
   Conversion Factor=10^-6 
   [Note: Since 210Gpa in SI unit will be 210*10^9Pa. Hence the coversion factor from SI unit to gm/mm/ms unit system will be 10^-6]
   
   
9. Velocity 56.33KMPH= 56.33*1000m/3600s
                                   = 56.33*(5/18) m/s
                                   = 56.33*5*1000mm/(18*1000ms)
                                   = 56.33*(5/18) mm/ms
                                   = 15.64mm/ms
   Conversion Factor= 5/18 ≈0.278
   
   
10. GRAVITY 9.8m/s2= 9.8*1000mm/(1000ms)^2
                                = 9.8*10^-3 mm/(ms)^2
    Conversion Factor=10^-3

[NOTE: The Conversion Factors are given for conversion from SI to the respective unit systems. SI unit for Stress is Pa]

Unit system	MASS	LENGTH	TIME	FORCE	STRESS	ENERGY	DENSITY	YOUNG'S Modulus	Velocity	GRAVITY
SI Unit kg m s	1kg [CF=1]	1m [CF=1]	1s [CF=1]	1N= 1kgm/s^2 [CF=1]	1Pa=1N/m^2 [CF=1]	1Nm=1[kg*m/s^2]*m  [CF=1]	1Kg/m3 [CF=1]	210GPa =210*10^9Pa [CF=10^9]	56.33KMPH =56.33*(5/18) m/s =15.64m/s [CF=5/18]	9.8m/s2 [CF=1]
Kg cm ms	1kg [CF=1]	100cm [CF=100]	1000ms [CF=1000]	1N= 10^-4kgcm/ms^2 [CF=10^-4]	CF=10^-8	CF=10^-2	CF=10^-6	CF=10^-8	CF=5/180	CF=10^-4
Kg cm s	1kg [CF=1]	100cm [CF=100]	1s [CF=1]	1N= 100kgcm/s^2 [CF=100]	CF=10^-2	CF=10^4	CF=10^-6	CF=10^-2	CF=500/18	CF=10^2
gm mm ms	CF=10^3	CF=10^3	CF=10^3	CF=1	CF=10^-6	CF=10^3	CF=10^-6	CF=10^-6	CF=5/18	CF=10^-3
g cm s	CF=10^3	CF=10^2	CF=1	CF=10^5	CF=10	CF=10^7	CF=10^-3	CF=10	CF=500/18	CF=10^2
g cm us (us=Micro second)	CF=10^3	CF=10^2	CF=10^6	CF=10^-7	CF=10^-11	CF=10^-5	CF=10^-3	CF=10^-11	CF=5/180000	CF=10^-10
g mm s	CF=10^3	CF=10^3	CF=1	CF=10^6	CF=1	CF=10^9	CF=10^-2	CF=1	CF=5000/18	CF=10^3
ton mm s	CF=10^-3	CF=10^3	CF=1	CF=1	CF=10^-6	CF=10^3	CF=10^-12	CF=10^-6	CF=5000/18	CF=10^3
slug ft s	CF=0.0685	CF=3.28	CF=1	CF=0.2246	CF=0.0208	CF=0.7369	CF=0.001941	CF=0.0208	CF=0.9111	CF=3.28
kg mm s	CF=1	CF=10^3	CF=1	CF=10^3	CF=10^-3	CF=10^6	CF=10^-9	CF=10^-3	CF=5000/18	CF=10^3
g cm ms	CF=10^3	CF=10^2	CF=10^3	CF=10^-1	CF=10^-5	CF=10	CF=10^-3	CF=10^-5	CF=5/180	CF=10^-4
kg mm ms	CF=1	CF=10^3	CF=10^3	CF=10^-3	CF=10^-9	CF=1	CF=10^-9	CF=10^-9	CF=5/18	CF=10^-3

5.Results:

The unit conversion was done for given systems of units.