Stretch rule

Last updated November 11, 2019

In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis.^[1] This operation leaves cylinders oriented parallel to the axis unchanged in radius.

Classical mechanics sub-field of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars and galaxies.

Edward John Routh, was an English mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the nineteenth century. He also did much to systematise the mathematical theory of mechanics and created several ideas critical to the development of modern control systems theory.

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rotation rate. It is an extensive (additive) property: for a point mass the moment of inertia is just the mass times the square of the perpendicular distance to the rotation axis. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems. Its simplest definition is the second moment of mass with respect to distance from an axis. For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3 × 3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.

This rule can be applied with the parallel axis theorem and the perpendicular axes rule to find moments of inertia for a variety of shapes.

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem, named after Christiaan Huygens and Jakob Steiner, can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

Derivation

The (scalar) moment of inertia of a rigid body around the z-axis is given by:

I_{z}=\int _{V}d^{3}r\,\rho (\mathbf {r} )\,r^{2}

Where $r$ is the distance of a point from the z-axis. We can expand as follows, since we are dealing with stretching over the z-axis only:

I_{z}=\int _{0}^{L}dz\int _{x,y}dxdy\,\rho (x,y,z)\,r^{2}

Here, $L$ is the body's height. Stretching the object by a factor of $a$ along the z-axis is equivalent to dividing the mass density by $a$ (meaning $\rho '(x,y,z)=\rho (x,y,z/a)/a$ ), as well as integrating over new limits $0$ and $aL$ (the new height of the object), thus leaving the total mass unchanged. This means the new moment of inertia will be:

I_{z}'=\int _{0}^{aL}dz\int _{x,y}dxdy\,\rho '(x,y,z)\,r^{2}

=\int _{0}^{L}adz'\int _{x,y}dxdy\,{\frac {\rho (x,y,z/a)}{a}}\,r^{2}

=\int _{0}^{L}dz'\int _{x,y}dxdy\,\rho (x,y,z')\,r^{2}=I_{z}

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References

↑ Smith, J.O. (2010). "Stretch Rule". Physical Audio Signal Processing. W3K Publishing. Retrieved 26 November 2012.

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[1] Smith, J.O. (2010). "Stretch Rule". Physical Audio Signal Processing. W3K Publishing. Retrieved 26 November 2012.