Week - 1 - Consistency of Units

Aim-

To convert given system of measurement to different measurement system (for ex.-SI to CGS) and to provide its conversion factors.

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

Conversion System-

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-

Unit conversion is a multi-step process that involves multiplication or division by a numerical factor, selection of the correct number of significant digits, and rounding.

Unit Conversion Process-

This process uses the fact that any number or expression can be multiplied by "one" without changing its value. This allows the conversion of units by multiplying the initial measurement by one (or more) forms of the number 1. While the multiplication by 1 does not change the value of the measurement, it does change the measurement units.

It’s very easy to systematically apply unit conversion process to solve conversions within or between measurement systems. It may be necessary to multiply by more than one conversion ratio in more complex conversions. Use these steps to construct a unit conversion problem so one (or more) of the units cancel until only the desired unit remains:

Step 1. Identify the unit you have. These are the Starting Units.

Step 2. Identify the unit you want. These are the Desired Units.

Step 3. Identify appropriate unit conversion factor(s). These are the Linking (or Ratio) Unit(s). Use EXACT conversion factors whenever available.

Step 4. Cancel units and perform the math calculations (e.g., multiply, divide). Repeat the calculation (double check).

Step 5. Evaluate the result. Does the answer make sense?

 

Purpose-

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.

Procedure-

Conversion-

MASS-1kg- 1000 gm;

                -Conversion factor-10³

LENGTH-1m-1000 mm; 

                    -Conversion factor-10³

TIME-1s-1000ms; 

             -Conversion factor-10³

FORCE-1N-1kg.m/s²

               -1000gm.1000mm/1000000ms²

               -10⁶ /10⁶ gm.mm/ms²

               - 1 gm.mm/ms²

               - Convesion factor-1

STRESS-1Pa-1N/m²

                   -1kg.m/s²m²     

                   -1kg/ms²

                   -1000gm/1000mm*1000000ms²

                   -10^-6 gm/mm.ms²

                       -Conversion factor-10^-6

ENERGY-1Nm-1kgm²/s²

                     -10³ gm*10⁶ mm²/10⁶ ms²

                      -10³ gm.mm²/ms²

                      -Conversion factor-10³

DENSITY-1Kg/m3

              -10³ gm/10⁹ mm³

              -10^-6 gm/mm³

              -Conversion factor-10^-6

YOUNG'S Modulus -210GPa-210*10⁹Pa

                                           -210*10⁹N/m²

                                          -210*10⁹kg.m/s²m²     

                                           -210*10⁹kg/ms²

                                           -210*10⁹*1000gm/1000mm*1000000ms²

                                           -210 * 10³ gm/mm.ms²

                                                   -Conversion factor-10³

Velocity-56.33KMPH-56.33 (10³m/3600s)

                               -56.33 (10⁶mm/3600*10³ms)

                               -15.647 mm/ms

                              - Conversion factor-5/18

GRAVITY 9.8m/s2-9.8*1000mm/1000000ms²

                            -9.8 * 10^-3 mm/ms²

                                   -Conversion factor-10^-3

 

Table for the unit systems as rows and measured quantities as columns-

Sr. No.	Unit System	Mass (1kg)	Length (1m)	Time (1s)	Force (1N)	Stress (1Pa)	Energy (1J)	Density (1kg/m3)	Young's Modulus(210GPa)	Velocity (53.66kmph)	Gravity(9.8m/s2)
1	kg m s	1kg	1m	1s	1N=1kg.m/s2	1Pa=1kg/m. s2	1Nm=1kg.m2/s2	1kg/m3	210*109 kg/m. s2	15.6m/s	9.81m/s2
2	Kg cm ms	1kg	102cm	1s	10-4kg.cm/ms2	10-8 kg/cm.ms2	10-2kg.cm2/ms2	10-6kg/cm3	210*101 kg/cm.ms2	1.56cm/ms	9.81*10-4cm/ms2
3	Kg cm s	1kg	102cm	1s	102kg.cm/s2	10-2 kg/cm. s2	104kg.cm2/s2	10-6kg/cm3	210*107 kg/cm. s2	1.56*102 cm/s	9.81*102cm/s2
4	gm mm ms	103 gm	103mm	103ms	1gm.mm/ms2	10-6 gm/mm.ms2	103gm.mm2/ms2	10-6gm/mm3	210*103 gm/mm.ms2	15.6mm/ms	9.81*10-3mm/ms2
5	gm cm s	103gm	102cm	1s	105gm.cm/s2	10 gm/cm. s2	107gm.cm2/s2	10-3gm/cm3	210*1010 gm/cm. s2	1.56*102 cm/s	9.81*102cm/s2
6	gm cm µs	103gm	102cm	106µs	10-7gm.cm/µs2	10-11 gm/cm.µs2	10-5gm.cm2/ µs2	10-3gm/cm3	210*10-2 gm/cm.µs2	1.56*10-4cm/µs	9.81*10-10cm/µs2
7	gm mm s	103gm	103mm	1s	106gm.mm/s2	1 gm/mm. s2	109gm.mm2/s2	10-6gm/mm3	210*109 gm/mm. s2	1.56*103mm/s	9.81*103mm/s2
8	ton mm s	10-3ton	103mm	1s	1ton.mm/s2	10-6 ton/mm. s2	103ton.mm2/s2	10-12ton/mm3	210*103 ton/mm. s2	1.56*103mm/s	9.81*103mm/s2
9	slug ft s	0.068slug	3.28ft.	1s	0.2246slug.ft/s2	0.0208slug/ft. s2	0.7366slug.ft2/s2	0.00194slug/ft3	4.305*109 slug/ft s2	4.77ft./s	32.17 ft/s2
10	kg mm s	1kg	103mm	1s	103kg.mm/s2	10-3 kg/mm. s2	106kg.mm2/s2	10-9kg/mm3	210*106 kg/mm. s2	1.56*103mm/s	9.81*103mm/s2
11	gm cm ms	103gm	102cm	103ms	10-1gm.cm/ms2	10-5 gm/cm. ms2	10gm.cm2/ms2	10-3gm/cm3	210*103 gm/mm.ms2	1.56cm/ms	9.81*10-4cm/ms2
12	kg mm ms	1kg	103mm	103ms	10-3kg.mm/ms2	10-9 kg/mm. ms2	1kg.mm2/ms2	10-9kg/mm3	210kg/mm.ms2	15.6mm/ms	9.81*10-3mm/ms2

 

Conclusion

• We have successfully revised the concept of unit conversion and its importance.
• Also learnt how this method will be useful in LS-Dyna for inouting correct values.