Week - 1 - Consistency of units

Unit Consistency challenge:

Aim: To practice unit consistency and make a chart of different unit system used in CAE applications showing relation between them.

In CAE there are various applications for FEA and as per the scale of components the unit system should be adaptable. For instance the unit system for crashworthiness analysis where the time steps are of milliseconds or microseconds scale are not practical for static or modal analysis. As a result a number of solvers and numerical analysis tools don’t have a default unit system. They either have a option to change to various unit systems as per the application or rely on the user to have a unit consistency. It implies that the user has to make sure all the input parameters have units form the same unit system. For instance in a static structural analysis, if the density of the material is input in ‘kg/m³’ then the units for young’s modulus ‘E’ has to be in ‘Pa’ and nor ‘MPa’.

Given below is a chart of some of the various unit systems used in CAE and how the parameters are changed to have consistency can be seen:

Calculations example:

An example of the conversion factor calculation for unit consistency from SI units to kg-mm-ms system:

For unit system ‘kg-mm-ms’ which is commonly used in crash-worthiness analysis,

Mass is in ‘kg’, length in ‘mm’=10^-3m, and time in ‘ms’=10^-3s. Hence mass will remain the same.

As m=10³ mm,

for length conversion factor will be ‘10^-3’

As s = 10³ ms,

for time conversion factor will be ‘10^-3’

1 N of force converted as, 1N = 1*(1kg*10^-2m/10^-6s²) = 1000N or 1kN

Hence the force figures will be in ‘kN’ both input and output with a conversion factor of ’10^-3’

Similarly,

1 Pa of input stress will be, 1Pa = 1N/1m² = 1*(1000N/10^-6m²) = 10⁹Pa = 1GPa.

Stress will be in GPa and conversion factor of ’10^-9’

Density in kg/m³ = 1*(1/10^-9m³) = 10⁹ kg/m³, with conversion factor of ’10^-9’

1 joule of energy = 1*(1000N*10^-3m) = 1 Joule, conversion factor of ’1’.

Velocity of 1km/hr = 1*(1000*10^-3m/10^-3s*3600 hr/s) = 0.2777 per km/hr, conversion factor of ‘1/0.277 = 3.6’.

Acceleration of 1m/s² = 1*(0.001m/10^-6s²) = 10³ per m/s², conversion factor of ’10^-3’

The material property input will be, for example:

‘E’ for steel 210 GPa, = 210*10⁹*(10^-9) = 210Pa

‘ρ’ density 8000 kg/m³ = 8000*10^-9 = 8*10^-6 kg/m³

‘v’ velocity if 50 km/hr = 50*0.277 = 13.88 km/hr

‘g’ acceleration due to gravity = 9.81m/s² = 9.81*(10^-3) = 0.00981 m/s²

^{Unit Consistency Chart:}

Unit system in terms if SI units:
	Kg cm ms	Kg cm s	gm mm ms	Kg m s	g cm s	g cm µs	g mm s	ton mm s	slug ft s	kg mm s	g cm ms	kg mm ms
Mass 1kg = in kg's	1	1	0.001	1	0.001	0.001	0.001	1000	14.594	1	0.001	1
Length 1m = in m's	0.01	0.01	0.001	1	0.01	0.01	0.001	0.001	0.3003003	0.001	0.01	0.001
Time 1s = in s	0.001	1	0.001	1	1	0.000001	1	1	1	1	0.001	0.001
Force 1N = Kgm/s^2 in N's	10000	0.01	1	1	0.00001	10000000	0.000001	1	0.228176	0.001	10	1000
Stress 1Pa = N/m^2 in Pa	1E+08	100	1000000	1	0.1	1E+11	1	1000000	2.5302204	1000	100000	1E+09
Energy 1Nm in Nm	100	0.0001	0.001	1	1E-07	100000	1E-09	0.001	0.0685213	0.000001	0.1	1
Density 1kg/m3	1E-06	1E-06	0.000001	1	0.001	0.001	0.000001	1E-12	0.0018556	1E-09	0.001	1E-09
Young’s Modulus 210GPa=210e9 N/m^2	2100	2.1E+09	210000	2.1E+11	2.1E+12	2.1	2.1E+11	210000	8.3E+10	210000000	2E+06	210
Velocity 56.33km/hr	1.56472	1564.72	15.64722	15.6472	1564.72	0.001565	15647.222	15647.22	52.10525	15647.222	1.5647	15.647222
Gravity 9.81m/s2	0.00098	981	0.00981	9.81	981	9.81E-10	9810	9810	2.9459459	9810	0.001	0.00981
Area m^2	0.0001	0.0001	0.000001	1	0.0001	0.0001	0.000001	0.000001	0.0901803	0.000001	0.0001	0.000001
volume m^3	1E-06	1E-06	1E-09	1	1E-06	0.000001	1E-09	1E-09	0.0270812	1E-09	1E-06	1E-09