Modeling a Rotating Link
Reduction, Transformation, or Refer Masses and Spring-Rates
Reduction is the procedure of creating a one degree-of-freedom model. The magnitude of every mass moment-of-inertia and torsional-rigidity, and/or mass and linear stiffness, is transformed to a single magnitude of mass and a single spring-stiffness at one point in the mechanism - usually the point of interest is on the output member.
In the reduced model, the magnitudes of the same kind can be compared with each other, because transmission ratios are absent.
You must transform the values in such as a way that the dynamic properties of the model do not change.
Mass Moment-of-Inertia and Torque
To calculate the Force to accelerate a body along a straight-line, you must know the Mass of the body.
To calculate the Torque to accelerate ta body about an axis-of-rotation, you must know the Mass Moment of Inertia of the body about the axis of rotation.
Mass Moment-of-Inertia of a Point-Mass
The Mass Moment-of-Inertia of a Point-Mass about an axis-of-rotation is equal to the product of the Mass and the square of the Radius from the axis-of-rotation.
 Element Mass at Radius |
Note A Point-Mass is an idealized concept of a finite mass at a known position, but without any physical size - is it a Black Hole? |
Mass Moment of Inertia of a Body with Distributed mass
The Mass Moment-of-Inertia includes the distribution of the mass about a rotational axis.
In three dimensions, the Mass Moment-of-Inertia is described by six inertia functions as an Inertia tensor. In the case of planar motion that is parallel to the X-Y plane, only one Inertia function occurs in the analysis, which is the Mass Moment of Inertia about the axis that is normal to the plane of motion.
If we consider a solid body as a distribution of elemental masses about a chosen rotational-axis, then its Mass Moment-of-Inertia is the sum of the product of each elemental mass and the square of its radius from the rotational-axis.
To calculate the Mass Moment-of-Inertia of a solid body that has a general shape, we must choose a rotational-axis and we must know how the mass is distributed about that axis.
Mass Moment of Inertia of simple bodies rotating on a Plane.
For simple geometric shapes there are standard formulas that you can use to evaluate the Mass Moment-of-Inertia, usually about an axis that is through the Center-of-Mass and perpendicular to the rotational plane.
The standard formula for Mass Moment-of-Inertia of a rectangular, prismatic body, that rotates about an axis-of-rotation that is perpendicular to a plane, and through the Center-of-Mass*, is: |
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 Extruded Rectangle |
The mass is obviously:
* Center-of-Mass = Center-of-Gravity in a constant gravitational field. |
Parallel Axis Theorem, Mass Moment of Inertia about a parallel axis.
We know the Mass Moment-of-Inertia, , of the rectangular link, about the axis - see image above. We can use the Parallel Axis Theorem to calculate the Mass Moment-of-Inertia about any other axis that is parallel to The image below shows the rectangular link as above, but now, it rotates about axis that is parallel to . We use the Parallel Axis Theorem to calculate the Mass Moment of Inertia of the link about . |
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 Parallel Axis Theorem |
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The Concept of Rotation-of Gyration.
It is sometimes useful to redefine the Mass Moment of Inertia of a solid body as a Point-Mass at a particular radius from the axis of rotation. When the mass of the Point-Mass is equal to the actual Mass of the Solid-Body, we can calculate the radius at which to position the Point-Mass such that its Mass Moment of Inertia is equal to that of the solid body, about the same axis of rotation. |
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To calculate the Radius-of-Gyration, . , we need to know the Mass Moment of Inertia of the Solid-Body about an axis. In this case we know the value of .
Note: If the Mass Moment of Inertia is about a different rotational-axis, e.g. , the Mass Moment of Inertia changes. The Radius-of-Gyration, , will also change.
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The Concept of Equivalent-Mass.
It is sometimes useful to specify a particular Radius for a Point-Mass from an axis of rotation, such that the mass of the Point-Mass is different to that of the solid body, but its Mass Moment of Inertia is equal to that of the solid-body about the same axis of rotation. We assume we know the Radius at which we want to put the Point-Mass from the axis-of-rotation. For example, we want to refer the Mass Moment of Inertia as a mass at the end of a lever, such that Mass Moment of Inertia of the lever does not change. |
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Transforming Masses across a Lever-Ratio to give equal Mass Moment of Inertia
Referring Masses across a Lever
 Reduction of a Mass 'across' a lever ratio Use the Kinetic-Energy of a Mass to equate the Mass at points A and B.
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Referring Springs across a Lever
 Reduction of a Spring across a Lever Ratio Use the Potential-Energy of a Spring to equate the Spring-Rate at points A and B.
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