NationMaster - Encyclopedia: Polar moment of inertia

Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects) with an invariant circular cross-section and no significant warping or out-of-plane deformation.^[1] It is used to calculate the twist of an object subjected to a torque. It is analogous to the area moment of inertia, which characterizes an object's ability to resist bending and is required to calculate displacement. Look up torsion in Wiktionary, the free dictionary. ... Cross section may refer to the following In geometry, Cross section is the intersection of a 3-dimensional body with a plane. ... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ... The second moment of area, also known as the second moment of inertia and the area moment of inertia, is a property of a shape that is used to predict its resistance to bending and deflection. ... Figure 1. ... Look up displacement in Wiktionary, the free dictionary. ...

The larger the polar moment of inertia, the less the beam will twist, when subjected to a given torque.

The polar moment of inertia must not be confused with the moment of inertia, which characterizes an object's angular acceleration due to a torque. See moment (physics). Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg mÂ², Former British units slug ft2), is the rotational analog of mass. ... Ð»Insert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non-formatted text hereInsert non... Torque applied via an adjustable end wrench Relationship between force, torque, and momentum vectors in a rotating system In physics, torque (or often called a moment) can informally be thought of as rotational force or angular force which causes a change in rotational motion. ... It has been suggested that this article or section be merged with torque. ...

1 Limitations
2 Definition
3 Unit
4 Application
5 Sample calculation
6 Comparing various moments of inertias for a cylinder
7 See also
8 References
9 External links

Limitations

In objects with significant cross-sectional variation, which cannot be analyzed in segments, and/or a non-circular cross-section a more complex approach has to be used. See 3-D elasticity. 3-D elasticity is one of three methods of structural analysis. ...

Definition

A schematic showing how the polar moment of inertia is calculated for an arbitrary shape about an axis o. ρ is the radial distance to the element dA.

$J_x = int r^2, dA$

J_x = the polar moment of inertia about the axis x
dA = an elemental area
r = the radial distance to the element dA from the axis x

Or for an element of a circular section: Image File history File linksMetadata PolarMomentOfInertia. ... Image File history File linksMetadata PolarMomentOfInertia. ...

$J = int_0^r 2 pi r^3 dr$

Unit

The SI unit for polar moment of inertia, like the area moment of inertia, is metre to the fourth power (m⁴) Look up si, Si, SI in Wiktionary, the free dictionary. ... This article is about the unit of length. ...

Application

The polar moment of area appears in the formulae that describes torsional stress and angular displacement. Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. ...

Torsional stress:

$tau = frac{T r}{J_{x}}$

where $T$ is the torque, $r$ is the radius and $J x$ is the polar moment of area.

In a circular shaft, the shear stress is maximal at the surface of the shaft (as that is where the torque is maximal): Stress is a measure of force per unit area within a body. ...

$T_max = frac{{tau}_max J_{x}}{r}$

Most frequently the inverse problem is solved, in which one solves for the radius.

Sample calculation

The rotor of a modern steam turbine.

Calculation of the steam turbine shaft radius for a turboset: Image File history File linksMetadata Download high resolution version (2126x2126, 1707 KB) Steam turbine rotor produced by Siemens, Germany Photo taken from [1] with the friendly permission of Siemens Germany by Christian Kuhna, E-Mail: christian. ... Image File history File linksMetadata Download high resolution version (2126x2126, 1707 KB) Steam turbine rotor produced by Siemens, Germany Photo taken from [1] with the friendly permission of Siemens Germany by Christian Kuhna, E-Mail: christian. ... A rotor of a modern steam turbine, used in a power plant A steam turbine is a mechanical device that extracts thermal energy from pressurized steam, and converts it into useful mechanical work. ... A rotor of a modern steam turbine, used in a power plant A steam turbine is a mechanical device that extracts thermal energy from pressurized steam, and converts it into useful mechanical work. ...

Assumptions:

Power carried by the shaft is 1000 MW; this is typical for a large nuclear power plant.
Yield stress of the steel used to make the shaft is (τ_yield) 250 x 10⁶ N/m².
Electricity has a frequency of 50 Hz; this is the typical frequency in Europe. In North America the frequency is 60 Hz.

The angular frequency can be calculated with the following formula: The megawatt (symbol: MW) is a unit for measuring power corresponding to one million (106) watts. ... This article is about applications of nuclear fission reactors as power sources. ... This article is about the SI unit of frequency. ... It has been suggested that this article or section be merged into Angular velocity. ...

ω = 2π f

The torque carried by the shaft is related to the power by the following equation: In physics, power (symbol: P) is the rate at which work is performed or energy is transferred. ...

P = T ω

The angular frequency is therefore 314.16 rad/s and the torque 3.1831 x 10⁶ Nm. Some common angles, measured in radians. ... This article is about the unit of time. ... Newton metre is the unit of moment (torque) in the SI system. ...

The maximal torque is:

$T_max = frac{ { tau }_max J_{x} }{r}$

After substitution of the polar moment of inertia the following expression is obtained:

$r=sqrt[3]{frac{2 T_max}{pi {tau}_max}}$

The radius is 0.200 m. If one adds a factor of safety of 5 and re-calculates the radius with the maximal stress equal to the yield stress/5 the result is a radius of 0.343 m, or a diameter of 69 cm, the approximate size of a turboset shaft in a nuclear power plant. This article is about an authentication, authorization, and accounting protocol. ... Factor of safety (FoS) can mean either the fraction of structural capability over that required, or a multiplier applied to the maximum expected load (force, torque, bending moment or a combination) to which a component or assembly will be subjected. ... DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ... A centimetre (American spelling centimeter, symbol cm) is a unit of length that is equal to one hundredth of a metre, the current SI base unit of length. ...

Comparing various moments of inertias for a cylinder

Polar moment of inertia

$J_{x} = frac{pi r^4}{2}$

$J_{x} = frac{pi d^4}{32}$

where:

d is the diameter

DIAMETER is a computer networking protocol for AAA (Authentication, Authorization and Accounting). ...

Area moment of inertia

$I_0 = frac{pi r^4}{4}$

$I_0 = frac{pi d^4}{64}$

Moment of inertia

$I_{x} = frac{1}{2} mr^2$

$I_{x} = frac{1}{8} md^2$

References

^ Ugural AC, Fenster SK. Advanced Strength and Applied Elasticity. 3rd Ed. Prentice-Hall Inc. Englewood Cliffs, NJ. 1995. ISBN 0-13-137589-X.

External links

Torsion of Shafts - engineeringtoolbox.com
Elastic Properties and Young Modulus for some Materials - engineeringtoolbox.com
http://mywebsite.bigpond.com/npajkic/solid_mechanics/polar_moments_and_products_of_inertia/index.html - Polar moments and products of inertia and the accompanying parallel-axis theorems

Categories: Continuum mechanics | Mechanics

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