Basic notions:

SIMPLE HARMONIC MOTION:

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TERMINOLOGIES:

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NEWTON’S LAW

Practically all motion based problems are directly derived from Newton’s Law’s of Motion, and Mechanical Vibration is no exception. In fact, after applying these equations under basic differential equations we have the basis of all that this text book will cover; albeit with increasing complexity and focus on specifics. Based on observation of experiments, in a vibrational model we notice that the acceleration in the body experiences is always in the direction of the applied force. We can then surmise in accord with a standard convention of sign an equation relating force and acceleration vectorally. This is none of than Newton’s famous second law of motion:

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Bold notion indicates that the variable is a vector. This equation correlates to the physical embodiment of a force of 1 N equivalent to a mass of 1 kg accelerating at 1 m/sec^2

By manipulating this equation we can bring forth other useful vibrational relationships. Earlier in this chapter Inertia was briefly mentioned as the property of matter which acts as a resistance to a change in the bodies momentum; this is explained as a body which has a mass m having a change in velocity. The change in momentum per unit time is related to the acceleration of the body and is represented equationally by:

This can be seen as a direct relation to Newton’s Second Law of Motion:

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This can be seen as a direct relation to Newton’s Second Law of Motion:

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We will now move along to understand Hooke’s Law which is defined as the amount which a material body is deformed (or in the case of our supposed springs, deformation of the spring; strain) is linearly related to the force that is causing the deformation (the stress). It is an approximation of the bodies elasticity. The stiffness constant is expressed in units of force per unit length and assumes spring displacement is measured from the initial resting length of the spring. This is shown in the following equation as:

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D ALEMBERT’S THEOREM:

The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the moment of the system itself along any virtual displacement consistent with the constraints of the system is zero.

Thus, in symbols d’Alembert’s principle is written as following,

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where

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is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,

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is the total applied force (excluding constraint forces) on the  image -th particle,

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is the mass of the  image -th particle,

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is the acceleration of the  image -th particle,

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together as product represents the time derivative of the momentum of the  image -th particle, and

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is the virtual displacement of the  image -th particle, consistent with the constraints.

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