Week - 1 - Consistency of Units

CONSISTENCY OF UNITS

INTRODUCTION

Depending on what is most convenient for each industry, multiple systems of units are used. The other derived numbers must be stated in terms of the units for the basic measuring quantities, mass, distance, and time. For instance, the kg, mm, ms system is used as the standard unit system in the automotive industry. Tone,m,s is most likely the unit system used by the construction industries. Similar to this, the electronic industry employs the gm, micrometer, and ms systems of units for PCB simulation.

There is no built-in unit system in LS-DYNA. The user must provide material characteristics and physical dimensions using their preferred, standardized system of units. This implies that users must be familiar with converting quantities from one system of units to another in order to use LS-DYNA in various situations.

AIM

• Write the given measurements in the above different system of units. Also, mention the 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

Now, here are a few systems of units:

1. Kg cm ms
2. Kg cm s
3. gm mm ms
4. kg m s
5. g cm s
6. g cm us
7. g mm s
8. ton mm s
9. slug ft s
10. kg mm s
11. g cm ms
12. kg mm ms

• The assignment should show the detailed calculations for conversion from SI unit to gm/mm/ms unit system. For others, only the answers in the form of a table are sufficient with the unit systems as rows and measured quantities as columns.

EXPLANATION

DETAILED CALCULATION FOR CONVERSION FROM SI UNIT TO GM/MM/MS UNIT SYSTEM

• Mass 1Kg

1Kg = 1000 grams = 10³ gm

Conversion Factor or C.F= 10³

• Length 1m

1m = 1000 millimeters = 10³mm

C.F = 10³

• Time 1s

1s = 1000 milliseconds = 10³ ms

C.F = 10³

• Force 1N

1N = 1 Kg m/ s² = (1000gm _* 1000mm)/ (1000)²ms²

= 1 gm mm ms^-2

C.F= 1

• Stress 1Pa

1 Pa = 1N / m² = 1gm mm ms^-2 / (1000) mm²

= 10^-6 gm mm^-1 ms^-2

C.F = 10^-6

• Energy N/m

1 Joule or 1J = N/m = 1gm mm ms^-2 / 1000 mm

= 10^-3 gm ms^-2

C.F = 10^-3

• Density 1Kg/m³

1 Kg/m³ = 1000 gm / (1000)³ mm³

= 10^-6 gm mm^-3

C.F = 10^-6

• Young’s Modulus 210 Gpa

Since, 1 GPa = 10⁹ N/m² = 10⁹ _* 10^-6 gm mm^-1 ms^-2

= 10³ gm mm^-1 ms^-2

Therefore 210GPa = 210 _* 10³ gm mm^-1 ms^-2

C.F = 10³

• Velocity 56.33 KmpH

Since 1 KmpH = 1000000 mm/ (60 _* 60) _* 1000 ms

= 0.277 mm/ms

Therefore 56.33 KmpH = 56.33 _* 0.277 mm/ms = 15.64 mm/ms

C.F = 0.277

• Gravity 9.8 m/s²

1 m/s² = 1000 mm / (1000)² ms² = 10^-3 mm ms^-2

Therefore 9.8 m/s² = 9.8 * 10^-3 mm ms^-2 = 9800 mm ms^-2

C.F = 10^-3

TABLE WITH THE UNIT SYSTEMS AS ROWS AND MEASURED QUANTITIES AS COLUMNS.

• The table compiled below is a collaborated data of standard unit conversion both column-wise and row-wise. The fundamental units and derived units used in this data are confined by a conversion factor and a prescribed factor. For example, the value of stress we found earlier for 1N/m² is 10^-6 gm mm^-1 ms^-2 but here we have converted the units pertaining to other system units like kg cm ms, kg m s so on and so forth. Therefore after conversion for afore mentioned other system units we obtain 10^-8 Kg cm^-1 ms^-2 and 1 Kg m^-1 s^-2 likewise. Here millisecond or 1ms = 10^-3 seconds, 1 gm or gram = 10^-3 kg, and 1 centimeter or 1 cm = 100 mm. Henceforth we can convert other derived or base units by the same following method.

S.No

SI units

Mass

CF

Length

CF

Time

CF

Force

CF

Stress

CF

Energy

CF

Density

CF

Youngs modulus

CF

Velocity

CF

Gravity

CF

kg

m

sec

1 N

N /m²

Nm

1kg / m³

210*10⁹ N/m²

56.33 kmph

9.8 m/s^2

Other unit systems

1

Kg cm ms

1 kg

1

10² cm

10²

10³ms

10³

10⁻⁴ Kg.cm / ms²

10⁻⁴

10⁻⁸ kg. / cm ms²

10⁻⁸

10⁻² kg cm² / ms²

10⁻²

10⁻⁶ kg / cm³

10⁻⁶

2100 kg. / cm ms²

10⁻⁸

1.564 cm /ms

0.027

9.8 *10⁻⁴cm /ms²

10⁻⁴

2

Kg cm s

1 kg

1

10² cm

10²

1 sec

1

10² kg cm / s²

10²

10⁻² kg / cm s²

10⁻²

10⁴ kg cm² / s²

10⁴

10⁻⁶ kg / cm³

10⁻⁶

210*10⁷ kg / cm s²

10⁷

1564.7 cm /s

27.77

9.8 *10² cm /s²

10²

3

Gm mm ms

10³ gm

10³

10³ mm

10³

10³ ms

10³

1 gm mm / ms²

1

10⁻⁶ gm / mm ms²

10⁻⁶

10³ gm mm² /ms²

10³

10⁻⁶ gm /mm³

10⁻⁶

210*10³ gm / mm ms²

10³

15.64 mm/ms

0.277

9.8 *10⁻³ mm / ms²

10⁻³

4

Kg m s

1 kg

1

1m

1

1 sec

1

1 Kg m / s²

1

1 kg / m s²

1

1 kg m² / s²

1

1kg / m³

1

210*10⁹ kg / m s²

1

15.64 m /s

0.277

9.8 m / s²

1

5

Gm cm s

10³ gm

10³

10² cm

10²

1 sec

1

10⁵ gm cm / s²

10⁵

10 gm / cm s²

10

10⁷ gm cm² / s²

10⁷

10⁻³ gm / cm³

10⁻³

210*10¹⁰gm / cm s²

10

1564.7 cm /s

27.77

9.8 *10² cm /s²

10²

6

Gm cm µs

10³ gm

10³

10² cm

10²

10⁶ sec

10⁶

10⁻⁷ gm cm / µs²

10⁻⁷

10⁻¹¹ gm / cm µs²

10⁻¹¹

10⁻⁵ gm cm² / µs²

10⁻⁵

10⁻³ gm / cm³

10⁻³

210*10² gm / cm µs²

10⁻¹¹

1.564*10⁻³ cm /µs

2.7 *10⁻⁵

9.8 *10⁻¹⁰ cm/ µs²

10⁻¹⁰

7

Gm mm s

10³ gm

10³

10³ mm

10³

1 sec

1

10⁶ gm mm / s²

10⁶

1 gm / mm s²

1

10⁹ gm mm² / s²

10⁹

10⁻⁶ gm /mm³

10⁻⁶

210*10⁹ gm / mm s²

1

15647.22 mm /s

277.77

9.8 *10³ mm / s²

10³

8

Ton mm s

10⁻³ ton

10⁻³

10³ mm

10³

1 sec

1

1 ton mm / s²

1

10⁻⁶ ton / mm s²

10⁻⁶

10³ ton mm² / s²

10³

10⁻¹² ton / mm³

10⁻¹²

210*10³ ton / mm s²

10⁻⁶

15647.22 mm /s

277.77

9.8 *10³ mm / s²

10³

9

Slug ft s

0.068 slug

0.07

3.28 ft

3.3

1 sec

1

0.22 sl. ft / s²

0.22

0.020 sl / ft. s²

0.02

0.72 sl .ft² / s²

0.72

1.9 *10⁻³ sl / ft³

1.9 *10⁻³

4.2*10⁻⁹ sl / ft. s²

0.02

51.32 ft/sec

0.91

32.144 ft /s²

3.28

10

Kg mm s

1 kg

1

10³ mm

10³

1 sec

1

10³ kg mm / s²

10³

10⁻³ kg / mm s²

10⁻³

10³ kg mm² / s²

10³

10⁻⁹ kg / mm³

10⁻⁹

210*10⁶ kg / mm s²

10⁻³

15647.22 mm /s

277.77

9.8 *10³ mm / s²

10³

11

Gm cm ms

10³ gm

10³

10² cm

10²

10³ ms

10³

0.1 gm cm / ms²

0.1

10⁻⁵ gm/cm ms²

10⁻⁵

10 gmcm² / ms²

10

10⁻³ gm /mm³

10⁻³

210*10⁴ gm/cm ms²

10⁻⁵

1.5647 cm/ms

0.027

9.8 *10⁻⁴ cm / ms²

10⁻⁴

12

Kg mm ms

1 kg

1

10³ mm

10³

10³ ms

10³

10⁻³ kg mm / ms²

10⁻³

10⁻⁹ kg / mm ms²

10⁻⁹

1 kg mm² /ms²

1

10⁻⁹ kg / mm³

10⁻⁹

210kg / mm ms²

10⁻⁹

15.64 mm/ms

0.277

9.8 *10⁻³ mm / ms²

10⁻³