Week - 1 - Consistency of Units

AIM:

To write the Conversion factors for the given measurements in different units within the International System of Units (SI).

OBJECTIVE:

• To convert the quality factor to the given conversant different system of units.
• Must represent the detailed calculation for conversion from the given SI unit to gm/mm/ms unit system.
• Create the table for all the conversion factor, the unit system as rows and measured quantities as columns.

Introduction:

Acknowledge the need of Unit system in LS-Dyna:

Studying or analyzing a phenomenon with FEM is often referred to as Finite Element Analysis(FEA). LS-DYNA is a advanced general-purpose multiphysics simulation software used to perform FEM simulations like complex real-world problems. It is particularly suited for highly nonlinear transient dynamic finite element analysis. It can carried out various kind of simulation analysis like Non-linear, Impact, Crashworthiness, Drop test, Explosion and Penetration etc.

Before performing analysis, the inputs of the simulation must have a Standard unit system to its given physical quantities. So its need to know converting the value of physical quantities to its constant unit system provides the the accuracy and consistency of the results throughout the analysis.

Understanding unit systems to carry out the simulation is vital in LS-DYNA as a finite element analysis software, it requires a consistent set of units for all parameters in the model. This ensures accurate simulations and correct interpretation of results. Users need to be conversant with converting quantities from one system of units to another to use LS-DYNA in different circumstances. Built-in unit system conversions in LS-DYNA ensure consistent units across the entire model, eliminating errors from inconsistent units.

In LS-DYNA, the unit system plays a crucial role in ensuring the accuracy and consistency of the results. The key principle is to use a consistent unit system throughout the analysis.     

Here are some important points to consider:

1. Consistency: LS-DYNA requires a consistent set of units to be used. This means all units used in the analysis must be consistent with each other. For example, if you’re using SI units, then length should be in meters, mass in kilograms, and time in seconds.
2. Derived Units: Derived units such as force, stress, and density must be derived from the base units. For example, in the SI system, force (in Newtons) is derived from mass (kg), length (m), and time (s).
3. Software: 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. Model Transfer: When transferring a model or solver file to/from somewhere else, it’s critical to understand the unit system used in the model.
5. Unit Conversion: If you need to convert units (for example, from SI to US customary units), make sure to convert all related quantities. For example, if you convert length from meters to millimetres, you should also convert other quantities like area and volume.

In conclusion, maintaining a consistent unit system is crucial in LS-DYNA to ensure accurate results. Misuse of units can lead to significant errors in tha analysis.

Definitions:

Unit System:

A unit system is a standard for expressing the measurement of a physical quantity. A set of fundamental and derived units is called a system of units. It is a collection of units of measurement and rules relating them to each other.

The International System of Units, also known as the SI system, is a widely used unit system that specifies seven base units and numerous derived units.

There are two types of unit systems:

• Fundamental Units: These are the units of basic quantities, such as length, mass, and time. Fundamental units are independent and cannot be derived from other units. For example, meters, seconds, and kilograms are fundamental units.
• Derived Units: Derived units are the units of physical quantities that can be obtained by mathematically combining the seven base units. For example, velocity is measured as (m/s), acceleration is measured as (m/s²), and pressure is measured as (N/m²) etc.

The following are the four internationally accepted systems of units:

• CGS Units: In this system, the fundamental units of length, mass, and time are taken as centimeter, gram, and second respectively.
• FPS Units: In this system, the fundamental units of length, mass, and time are taken as foot, pound, and second respectively.
• MKS Units: In this system, the fundamental units of length, mass, and time are taken as meter, kilogram, and second respectively.
• SI Units (International System of Units): It is also referred to as the metric system. This system has seven basic units, two supplementary units and 22 derived units. The seven basic types of unit system are meter (for length), kilogram (fo
• r mass), second (for time), ampere (for electric current), kelvin (for thermodynamic temperature), mole (for amount of substance), and candela (for luminous intensity). There are two supplementary units are also called as dimensionless derived units: radian (for plane angle) and steradian (for solid angle).

Conversion factor:

• A conversion factor is a number used to change one set of units to another, by multiplying or dividing. When a conversion is necessary, the appropriate conversion factor to an equal value must be used. For example, to convert kilogram to gram, the appropriate conversion value is 1Kilogram equal 1000 gram.

The conversion factors for the different units:

1. Mass: 1 kilogram = 1000 gram = 0.001 tonne = 0.0685 slug

2. Length: 1 meter = 100 centimeter = 1000 millimeter = 3.281 feet

3. Time: 1 second = 1000 millisecond(ms) = 1,000,000 microsecond(µs) = 1000,000,000 nanosecond(ns)

4. Force: 1 Newton = 1,00,000 dynes = 0.22481 pound force(lbf) = 0.102 kilogram force(kgf)

5. Stress:1 Pascal = 1 N/m² = 10 dyne/cm² = 0.000145approx. (psi) = 9.86923*`10^-6` atm = `10^-5` bar

6. Energy: 1 Nm = `10^7` erg = 1 Joule

7. Density: 1 Kg/m3 =  `10^-3` g/cm³ = `10^-6` g/mm³ = 1 gram per litre(g/L)

8. Young's modulus: 1 GPa = `10^9` Pa = `10^6` KPa = `10^3` MPa

9. Velocity: 1 KMPH =  0.27778 m/s = 27.778 cm/s  ≈ 0.62137 mph

10. Gravity: 1 m/s² = 100cm/s² =0.10194 g 

Conversion of Unit systems:

Conversion factors are used to convert a quantity expressed in one set of units into an equivalent value in another set of units. They are based on the relationships between different units of measurement.

Mass: 

Mass is the amount of matter in an object. The kilogram (kg) and gram (g) are both units of mass in the metric system. 1 kg is equal to 0.001 tonne or 1000 g, so the conversion factor from (kg) to (g) is 1000 and from kg to tonne is 0.001.

Measurement conversion:

from (kg) to (g):

 1 kg × `((1000g)/(1kg))` = 1000g            (or)              mass(g) = 1 kg × 1000 = 1000 gram

Conversion factor: `10^3`

from (kg) to (tonne):

 1 kg × `((0.001t)/(1kg))` = 0.001t            (or)               mass(t) = 1 kg × 0.001 = 0.001 tonne

Conversion factor: `10^-3`

from (kg) to (slug):

 1kg × 0.0685 = 0.0685 slug

Conversion factor: 0.0685

Length:

Length is the measurement of something from end to end. The meter (m), centimeter (cm), and millimeter (mm) are all units of length in the metric system. 1 m is equal to 100 cm or 1000 mm, so the conversion factor from m to cm is 100, and from m to mm is 1000.

Measurement conversion:

from (m) to (mm):

 1 m × `((1000 mm)/(1 m))` = 1000 m     (or)              length(mm) = 1 m × 1000 = 1000 mm

Conversion factor: `10^3`

from (m) to (cm):

 1 m × `((100 cm)/(1 m))` = 100 cm     (or)              length(cm) = 1 m × 100 = 100 cm

Conversion factor: `10^2`

Time:

The measurable period during which an action, process, or condition will start or exists or continues.The second (s), millisecond (ms),  microsecond (µs) and nanoseconds(ns) are all units of time. 1 s is equal to 1000 ms or 1000000 µs or `10^9`ns, so the conversion factor from s to ms is 1000, or from s to µs is `10^6` and from s to ns is `10^9` .

Measurement conversion:

from (s) to (ms):

 1 s × `((1000 ms)/(1 s))` = 1000 ms       (or)             time(ms) = 1 s × 1000 = 1000 ms

Conversion factor: `10^3`

from (s) to (μs):

 1 s × `((10,00,000 μs)/(1 s))` = 10,00,000 μs       (or)             time(μs) = 1 s × 10,00,000 = `10^6` μs

Conversion factor: `10^6`

Force:-

 A force is an influence that can cause an object to change its velocity. The newton (N) and dyne are both units of force. 1 N is equal to `10^5` dyne, so the conversion factor from N to dyne is `10^5`.

Newton’s second law of motion states that the force (F) acting on an object is equal to its mass (m) times its acceleration (a). This is often written as:

F = m * a

In the International System of Units (SI), mass is measured in (kg), acceleration in (m/s²), and force in Newtons (N). Therefore, if you have a mass of 1 kg and it’s accelerating at a rate of 1 m/s², the force exerted is 1 N. Hence, we say that 1 N is equivalent to 1 kg*m/s².

Measurement conversion from SI unit to gm/mm/ms unit system:

from N to dyne:

 Dyne = 1 N × `10^5` = `10^5` dyne

from N to g × mm/ms²:

Force: 

  1 N = 1 kg* 1 m/s²

        = `(1000g×1000mm)/(1000ms)^2 => (10^6×10^-6 (g×mm))/(ms^2)`

  1 N = 1 g*mm/ms²

Conversion factor: 1

Stress:-

Stress is the force that acts on a unit area of a material. Stress can deform a given body. The effect that stress has on the body is known as strain. The pascal (Pa) and dyne/cm² are both units of stress. 1 Pa is equal to 10 dyne/cm², so the conversion factor from Pa to dyne/cm² is 10.

Measurement conversion:

from Pa to dyne:

 1 Pa = `1N/m^2` = `1((10^5dyn e)/(100cm)^2)` => `10(dyn e)/(cm^2)` 

from Pa to psi:

 psi = 1 Pa × 0.000145 = 0.000145 psi

from Pa to g/(mm*ms²):

Stress:

 1 Pa = `1N/m^2 =>``({1 kg× (1 m/s^²)})/m^²` `=> ({1 kg×(1m)}/(m^2×s^2))`

      `=> (1 kg)/(m*s^2)` `=> (10^3g)/(10^3mm*(1000ms)^2`

      `=> (10^3g)/(10^3mm*10^6(ms^2)` `=> (10^-6g)/(mm*ms^2`

 1 Pa = `10^-6` g/mm*ms²

 Conversion factor: `10^-6`

Energy:-

Energy is defined as the capacity to do work. It is a physical quantity that is transferred from one object to another to perform work. The  SI unit of energy is the joule (J). The newton meter (Nm) and erg are both units of energy. 1 Nm is equal to `10^7` erg or 1 Joule, so the conversion factor from Nm to erg is `10^7` or from Nm to joule is 1.

Measurement conversion:

from Nm to erg:

 erg = 1 Nm × `10^7` =  `10^7`erg   

Energy:

from Nm to g*mm²/ms²:

 1 J = 1 Nm `=>` `(1 kg × 1 m/(s²))×m` `=> 1 kg × m^2/(s²)`

    `=> (10^3g)×(10^3mm)^2/(10^3ms)^2` `=> (10^3g)×({(10^6)×mm^2})/({(10^6)×ms^2})`

    `=> 10^3×(g×mm^2)/(ms^2)`

 1 Nm = `10^3`(g*mm²/ms²)  

Conversion factor: `10^3`

Density:-

Density is a physical property that measures how tightly a material is packed together. It is defined as the mass per unit volume. The kg/m³ and g/cm³ are both units of density. 1 kg/m³ is equal to 0.001 g/cm³, so the conversion factor from kg/m³ to g/cm³ is 0.001.

Measurement conversion:

from kg/m³ to g/cm³:

 `1(kg)/m^3 = (10^3g)/(100cm)^3 => (10^3g)/(10^6cm^3) =>(10^3×10^-6g)/(cm^3)`

 `=>(1kg)/m^3 = (10^-3g)/(cm^3)`

Density:

from kg/m³ to g/mm³:

`(1kg)/(m^3) = (10^3g)/(10^3mm)^3 => (10^3×10^-9×g)/(mm^3) => (10^-6g)/(mm^3)`

`(1kg)/(m^3) => (10^-6g)/(mm^3)`

Conversion factor: `10^-6`

Young’s Modulus:-

Young's modulus is a measure of the stiffness of an elastic material. It is a measure of elasticity, equal to the ratio of the stress acting on a substance to the strain produced. GPa and kgf/mm² are both units of Young’s Modulus. 210 GPa is equal to 210000 kgf/mm², so the conversion factor from GPa to kgf/mm² is 210000.

Measurement conversion:

from GPa to kgf/mm²:

 kgf/mm² = 1 GPa × 1000 = 1000 kgf/mm²

from 1 GPa to MPa or KPa or Pa:

 MPa =1GPa×`10^3=10^3 MPa`    (and)     KPa =1GPa×`10^6=10^6 KPa`   (and)     Pa =1GPa×`10^9=10^9 Pa`

Young’s Modulus:

from GPa to  g/(mm*ms²):

 210GPa = `210e^(+9)Pa => 210e^(+9) ×N/m^2` 

 = `210e^(+9)×({1 kg× (1 m/s^²)})/m^²` `=> ({1 kg×(1m)}/(m^2×s^2))`

   `=>210e^(+9)×(kg)/(m×s^2)` `=>210e^(+9)×(10^3g)/((10^3mm)×(10^3ms)^2)`

   `=>210e^(+9)×(10^-6*g)/(mm×ms^2) =>210e^(+3)×(g)/(mm×ms^2)`

  210GPa = `210×10^3×(g)/(mm×ms^2)`

Conversion factor: `10^3`

Velocity:

It is defined as the rate of change of displacement of a body with respect to time. km/h and m/s are both units of velocity. To convert km/h to m/s, we divide by 3.6 because there are 1000 meters in a kilometer and 3600 seconds in an hour. 1 km/h is equal to 0.27778m/s, then 56.33 km/h is equal to 15.65m/s. So the conversion factor from km/h to m/s is 0.27778.

Measurement conversion:

from km/h to m/s:                                                       {1hr = 3600s}

 m/s = 1km/h × `(10^3m)/(3600s)` `=> (1km)/h × 0.2777 => 0.2777 m/s`

from km/h to mm/ms:

 56.33 km/h × 0.2777 = 15.65m/s `=> 15.65×(10^3mm)/(10^3ms) => 15.65×(mm)/(ms)`

56.33 km/h = 15.65×(mm)/(ms)

Conversion factor: 0.2777 (or) 5/18

Gravity:-

It is also called gravitation, in mechanics, the universal force of attraction acting between all matter. m/s² is a unit of acceleration due to gravity. It remains constant as it’s a universal physical constant. 9.8 m/s² is equal to 980 cm/s² and from m/s² to cm/s² is 100.

Measurement conversion:

from m/s² to cm/s²: 

 cm/s² = 1 m/s² × 100 = 100 cm/s²            (or)            cm/s² = 9.8 m/s² × 100 = 980 cm/s²  

Gravity:

from m/s² to mm/ms²:

 `1m/s^2 = (10^3mm)/(10^3ms)^2 => (10^3mm)/(10^6×ms^2) =>10^-3(mm)/(ms^2)`

 `1m/s^2 = 10^-3(mm)/(ms^2)`

Conversion factor: `10^-3`

The given measurements were also converted to several other unit systems using the appropriate conversion factors are represented in the table will be,

Unit system	Mass    (1kg)	Length (1m)	Time (1s)	Force      (1N)	Stress (1Pa)	Energy (1Nm)	Density (kg/m³)	Young’s Modulus (210GPa)	Velocity (56.33KMPH)	Gravity (9.8m/s²)
Kg, cm, ms	1 Kg	100 cm	1000 ms	`10^-4` kg*cm/ms²	`10^-8` kg/cm*ms²	`10^-2` kg*cm²/ms²	`10^-6` kg/cm³	`210×10^1` kg/cm*ms²	`15.65×10^-1` cm/ms	`9.8×10^-4` cm/ms²
Kg, cm, s	1 Kg	100 cm	1 s	`10^2` kg*cm/s²	`10^-2` kg/cm*s²	`10^4` kg*cm²/ms²	`10^-6` kg/cm³	`210×10^7` kg/cm*s²	`15.65×10^2` cm/s	`9.8×10^2`  cm/s²
gm, mm, ms	1000 gm	1000 mm	1000 ms	1 g*mm/ms²	`10^-6` g/mm*ms²	`10^3` g*m/ms²	`10^-6` g/mm³	`210×10^3` g/mm*ms²	15.65 mm/ms	`9.8×10^-3` mm/ms²
kg, m, s	1 Kg	1 m	1 s	1 kg*m/s²	1 kg/m*s²	1 kg*m²/s²	1 kg/m³	`210×10^9` kg/m*s²	15.65 m/s	9.8 m/s²
g, cm, s	1000 g	100 cm	1 s	`10^5` g*cm/s²	`10^1` g/cm*s²	`10^7` g*cm²/s²	`10^-3` g/cm³	`210×10^10` g/cm*s²	`15.65×10^2` cm/s	`9.8×10^2` cm/s²
g, cm, µs	1000 g	100 cm	`10^6` µs	`10^-1` g*cm/µs	`10^-11` g/cm*µs²	`10^5`    cm²/µs²	`10^-3` g/cm³	`210×10^-2` g/cm*µs²	`15.65×10^-4` cm/µs	`9.8×10^-10`  cm/µs²
g, mm, s	1000 g	1000 mm	1 s	`10^6` g*mm/s²	1 g/mm*s²	`10^9` g*mm²/s²	`10^-6` g/mm³	`210×10^9` g/mm*s²	`15.65×10^3` mm/s	`9.8×10^3` mm/s²
ton, mm, s	0.001 ton	1000 mm	1 s	1 ton*mm/s²	`10^-6` ton/mm*s²	`10^3` ton*mm²/s²	`10^-12` ton/mm³	`210×10^3` ton/mm*s²	`15.65×10^3` mm/s	`9.8×10^3` mm/s²
slug, ft, s	0.0685 slug	3.281 ft	1 s	0.22474 slug*ft/s²	0.02087 slug/ft*s²	0.737 slug*ft²/s²	0.00194 slug/ft³	4.3827*`10^9` slug/ft*s²	51.347ft/s	32.1538 ft/s²
kg, mm, s	1 kg	1000 mm	1 s	`10^3` kg*mm/s²	`10^-3` kg/mm*s²	`10^6` kg*mm²/s²	`10^-9` kg/mm³	210×`10^6` kg/mm*s²	`15.65×10^3` mm/s	`9.8×10^3` mm/s²
g, cm, ms	1000 g	100 cm	1000 ms	`10^-1` g*cm/ms²	`10^-5` g/cm*ms²	`10^1` g*cm²/ms²	`10^-3` g/cm³	210×`10^4` g/cm*ms²	`15.65×10^-1` cm/ms	`9.8×10^-4` cm/ms²
kg, mm, ms	1 kg	1000 mm	1000 ms	`10^-3` kg*mm/ms²	`10^-9` kg/mm*ms²	1 kg*mm²/ms²	`10^-9` kg/mm³	210 kg/mm*ms²	15.65 mm/ms	`9.8×10^-3` mm/ms²

Conclusion:

The Unit conversion from the SI units to the different system of units has been done.