Undaped free vibration

A vibration is the periodic motion of a body or system of connected bodies displaced from an equilibrium position. In general, there are two types of vibration Free and Forced. Free vibrations are vibrations that occur when no external force acts on the system. These vibrations are maintained by gravitational or elastic restoring forces, such as the swinging motion of a pendulum or the vibration of an elastic rod.  Forced vibration on the other hand are caused by an external period or intermittent force applied to the system. Free and forced vibrations can be damped or undamped. From idealistic point of view undamped vibrating systems can continue indefinitely. The reason why they go indefinitely is that the frictional effects are neglected in the analysis. Since in reality both internal and external friction forces are present, the motion of all vibrating bodies is actually damped.

The simplest type of vibrating motion is undamped free vibration, is represented by a model which consists of block and a spring. The model is shown in next figure.

FIGURE 1 - Undamped free vibrational system

Vibration motion occurs when the block is released from a displaced position x so that the spring pulls on the block. We say that block is oscillating around its equilibrium position which means that the block will attain a velocity such that it will proceed to move out of equilibrium when x= 0. One thing is very important in this theoretical analysis and that is the assumption that the surface is smooth. Smooth surface means that there is no friction between block and the surface. So block will oscillate without any friction.

The time dependent path of motion of the block can be determined by applying the equation of motion to the block when it is in the displaced position x. But before you derive the equation that describes the motion we need to make the Free Body Diagram. This method is used to draw the external and effective forces which acts on the body and in this case it’s the block of mass m. when you applied the method of free body diagram you can then apply the Newton second law from which we will derive the equation that describes the motion or in this case the vibration of the analyzed system.

The elastic restoring force F=kx is always directed toward the equilibrium position, whereas the acceleration a is assumed to act in the direction of a positive displacement. Since

We have

$$\begin{align} & \xrightarrow{+}\sum{{{F}_{x}}}=m{{a}_{x}} \\ & -F=ma \\ & -kx=m\ddot{x} \\ & m\ddot{x}+kx=0 \\ \end{align}$$

The previous equation describes the simple harmonic motion. The acceleration is proportional to the block displacement. Now we need to derive the expression for natural frequency. If we divide the previous equation with m we will get.

$$\begin{align} & m\ddot{x}+kx=0/:m \\ & \ddot{x}+\frac{k}{m}x=0, \\ & \ddot{x}+\omega _{n}^{2}x=0, \\ \end{align}$$

The Constant in the previous equation is called the natural frequency of the system, and in this case:

$${{\omega }_{n}}=\sqrt{\frac{k}{m}}$$

Now the solution to the differential equation which describes the motion of the analyzed system can be written in the following form:

$$x=A\sin {{\omega }_{n}}t+B\cos {{\omega }_{n}}t$$

If we look at the equation which is the solution of the system we will see that A are B are the unknowns and they represent two constants of the integration.

To prove that the previous equation is the solution of the system we need to insert it into the differential equation but before we do that we need to derive the first and the second derivation of the solution.

$$\begin{align} & \dot{x}=A{{\omega }_{n}}\cos {{\omega }_{n}}t-B{{\omega }_{n}}\sin {{\omega }_{n}}t, \\ & \ddot{x}=-A\omega _{n}^{2}\sin {{\omega }_{n}}t-B\omega _{n}^{2}\cos {{\omega }_{n}}t. \\ \end{align}$$

By substituting the second derivation of solution and solution of the system into the differential equation of the system we get:

$$\begin{align} & \ddot{x}+\omega _{n}^{2}x=0, \\ & -\omega _{n}^{2}\left( A\sin {{\omega }_{n}}t+B\omega _{n}^{2}\cos {{\omega }_{n}}t \right)+\omega _{n}^{2}\left( A\sin {{\omega }_{n}}t+B\cos {{\omega }_{n}}t \right)=0 \\ & 0=0 \\ \end{align}$$

From the previous expression we can see that the differential equation is satisfied for the proposed solution.

All that is left is to determine the constants of the integration. To determine the constants of the integration we need to apply the boundary condition.

For t=0, the x=x1 which means that at the beginning the system is at rest. By applying this boundary condition to the solution of the differential equation, we get:

$$\begin{align} & x=A\sin {{\omega }_{n}}t+B\cos {{\omega }_{n}}t \\ & {{x}_{1}}=A\sin 0+B\cos 0 \\ & B={{x}_{1}} \\ \end{align}$$

For t=0, the v=v1 and by applying that to the first derivation of solution we get:

$$\begin{align} & \dot{x}=A{{\omega }_{n}}\cos {{\omega }_{n}}t-B{{\omega }_{n}}\sin {{\omega }_{n}}t, \\ & {{v}_{1}}=A{{\omega }_{n}}\cos 0-B{{\omega }_{n}}\sin 0, \\ & A=\frac{{{v}_{1}}}{{{\omega }_{n}}} \\ \end{align}$$

So now we have determined the constants of integration. We can write the complete solution of the system in the following form.

$$x=\frac{{{x}_{1}}}{{{\omega }_{n}}}\sin {{\omega }_{n}}t+{{x}_{1}}\cos {{\omega }_{n}}t$$

The solution can be written in shorter form if we express the constants of the integration in following form:

$$\begin{align} & A=C\cos \phi , \\ & B=C\sin \phi . \\ \end{align}$$

The C and ϕ are constants of integration and they also need to be determined from boundary conditions. If we substitute the A and B into the solution we will get:

$$x=C\cos \phi \sin {{\omega }_{n}}t+C\sin \phi \cos {{\omega }_{n}}t$$

By applying trigonometric transformation:

$$\sin \left( \theta +\phi \right)=\sin \theta \cos \phi +\cos \theta \sin \phi $$

We will get:

$$x=C\sin \left( {{\omega }_{n}}t+\phi \right)$$

If this equation is plotted on an x versus the axis, the graph is obtained. The maximum displacement of the block from its equilibrium position is defined as the amplitude of vibration. From the figure the amplitude is C. The angle is called phase angle since it represents the amount by which the curve is displaced from the origin when t=0.

$$C=\sqrt{{{A}^{2}}+{{B}^{2}}}$$

Phase angle can be determined from the following expression:

$$\phi ={{\tan }^{-1}}\left( \frac{B}{A} \right)$$

Note that the sine curve completes one cycle in time or:

$$\tau =\frac{2\pi }{{{\omega }_{n}}}$$

Finally the frequency f is defined as the number of cycles completed per unit of time, which is the reciprocal of the period; that is

$$f=\frac{1}{\tau }=\frac{{{\omega }_{n}}}{2\pi }$$

The frequency is expressed in cycles/s. This ratio of units is called a hertz where 1 Hz = 1 cycle/s or 2PI rad/s. When the body or system of connected bodies is given initial displacement from its equilibrium position and released, it will vibrate at natural frequency. Provided the system has a single degree of freedom, that is, it requires only one coordinate to specify completely the position of the system at any time, then the vibration motion will have the same characteristics as the simple harmonic motion of the block and the spring just presented.