Mechanical Vibration/Second Law of Motion

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 gravitational model we notice that the acceleration 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 vector-ally. This is none other than Newton’s famous second law of motion:

$\mathbf {F} =m\mathbf {a}$

Bold notation 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 gravitational 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:

$\mathbf {p} =m\mathbf {v}$

$\ \sum {\mathbf {F} }={\mathrm {d} \mathbf {p} \over \mathrm {d} t}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}=m\mathbf {a}$

This can be seen as a direct relation to Newton's Second Law of Motion in the experiment of Hooke's Law;

$\mathbf {F} =m\mathbf {a}$

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:

$\mathbf {F} =-k\mathbf {x} \$

There do exist more complex nonlinear relations for the spring model under Hooke's Law but they will not be used for our vibrational modeling purposes.