Question

Newton's second law of motion can always be used to derive the equations of motion of a vibrating system

Answer 1

Practically all motion based problems are directly derived from Newton’s Law’s of Motion, and Mechanical Vibration is no exception.

You can consider the following example :

The system above consists of a spring with spring constant k attached to a block of mass m resting on a frictionless surface. The origin of the coordinate system is located at the position in which the spring is unstretched.

Now imagine the block is pulled to the right and let go. Hopefully you can convince yourself that the block will oscillate back and forth. Let's apply Newton's Second Law at the instant the mass is at an arbitrary position, x. The only force acting on the mass in the x-direction is the force of the spring.

Because of our choice of coordinate system, the stretch of the spring (s) is exactly equal to the location of the block (x). Therefore,

Note that when the block is at a positibe position, the force of the spring is in the negative direction and when the block is at a negative positions, the force of the spring is in the positive direction. Thus, the force of the spring always acts to return the block to equilibrium.

Rearranging gives

and defining a constant, w2, as

Granted, it seems pretty silly to define k/m as the square of a constant, but just play along. You may also find it frustrating to learn that this "omega" is not an angular velocity. The block does not even have an angular velocity!)

yields,

Therefore, the position function for the block must have a second time derivative equal to the product of (-w2) and itself. The only functions whose second time derivative is equal to the product of a negative constant and itself ar the sine and cosine functions. Therefore, a solution to this differential equation1 can be written

or equivalently with the sine function, where A and ϕ are arbitrary constants.