##### INTRODUCTION

*The fundamentals of Sound and Vibrations are part of the broader field of mechanics, with strong connections to classical mechanics, solid mechanics and fluid dynamics.*

**Dynamics** is the branch of physics concerned with the motion of bodies under the action of forces.

**Vibrations or oscillations** can be regarded as a subset of dynamics in which a system subjected to restoring forces swings back and forth about an equilibrium position, where a system is defined as an assemblage of parts acting together as a whole. The restoring forces are due to elasticity, or due to gravity.

The subject of Sound and Vibrations encompasses the generation of sound and vibrations, the distribution and damping of vibrations, how sound propagates in a free field, and how it interacts with a closed space, as well as its effect on man and measurement equipment. Technical applications span an even wider field, from applied mathematics and mechanics, to electrical instrumentation and analog and digital signal processing theory, to machinery and building design. Most human activities involve vibration in one form or other. For example, we hear because our eardrums vibrate and see because light waves undergo vibration. Breathing is associated with the vibration of lungs and walking involves (periodic) oscillatory motion of legs and hands. Human speak due to the oscillatory motion of larynges (tongue).

**In most of the engineering applications, vibration is signifying to and fro motion , which is undesirable. Galileo discovered the relationship between the length of a pendulum and its frequency and observed the resonance of two bodies that were connected by some energy transfer medium and tuned to the same natural frequency. Vibration may results in the failure of machines or their critical components. The effect of vibration depends on the magnitude in terms of displacement, velocity or accelerations, exciting frequency and the total duration of the vibration.**

**Free Vibration**– In Free vibration, the object is not under the influence of any kind of outside force.

In free vibration the body at first is given an initial displacement and the force is withdrawn. The body starts vibrating and continues the motion of its own accord. No external force acts on the body further to keep it in motion. The frequency of free vibration is known as free or natural frequency.

The free vibration of an elastic body can further be of three types:

a)Longitudinal vibration: when the particles of the body move parallel to the axis of the body, the vibration is known as longitudinal vibration.

b)Transverse vibration: when the particles of the body move nearly perpendicular to the axis of the body, the vibration is known as transverse vibration.

c)Torsional vibration: When the particles of the body move in a circle about the axis of the body, the vibration is known as torsional vibration.

**Forced Vibration**– In forced vibration, the object is under the influence of an outside force.

This can be understood more clearly by the following example:-

When a pendulum vibrates it is free vibration because it does not depend on any outside force to vibrate whereas when a drilling machine vibrates, it depends on a force from outside. Therefore, it is an example of forced vibration.

##### LINEAR SYSTEMS

A **linear system** is defined as one in which the relationship between the input and output signals can be described by a linear differential equation.

Often in Vibrations and Acoustics, the calculation of the effect of a certain physical quantity termed as the input signal on another physical quantity, called the output signal.

An example is that of calculating vibration velocity v(t), which is obtained in a structure when it is excited by a given force F(t). That problem can be solved by making use of the theory of linear time- invariant systems. A linear time-invariant system describes the relationship between an input signal and an output signal. For example, the input signal could be a velocity v(t), and the output signal a force F(t), or the input signal an acoustic pressure p(t) and the output signal an acoustic particle velocity u’(t). If the coefficients are, moreover, independent of time, i.e., constant, then the system is also **time invariant**.

##### SINGLE-DEGREE-OF-FREEDOM SYSTEMS

**Discrete System Components** A system is defined as an aggregation of components acting together as one entity. The components of a vibratory mechanical system are of three different types, and they relate forces to displacements, velocities, and accelerations. The component relating forces to displacements is known as a spring. For a linear spring the force F_{s} is proportional to the elongation or

where k represents the spring constant, or the spring stiffness, and x_{2} and x_{1} are the displacements of the end points.

**Viscous damper or a dashpot **The component relating forces to velocities is called a viscous damper or a dashpot. It consists of a piston fitting loosely in a cylinder filled with liquid so that the liquid can flow around the piston when it moves relative to the cylinder. The relation between the damper force and the velocity of the

piston relative to the cylinder is

in which **c** is the coefficient of viscous damping; note that dots denote derivatives with respect to time. Finally, the relation between forces and accelerations is given by Newton’s second law of motion:

*The spring constant k, coefficient of viscous damping c, and mass m represent physical properties of the components and are the system parameters.*

Note that springs and dampers are assumed to be massless and masses are assumed to be rigid.

**Equivalent spring constant **Springs can be arranged in parallel and in series. Then, the proportionality constant between the forces and the end points is known as an equivalent spring constant and is denoted by k_{eq}, as shown in Table below:

Certain elastic components, although distributed over a given line segment, can be regarded as lumped with an equivalent spring constant given by k_{eq} = F/δ, where δ is the deflection at the point of application of the force F. A similar relation can be given for springs in torsion. Table given above lists the equivalent spring constants for a variety of components.

**Equation of Motion** The dynamic behavior of many engineering systems can be approximated with good accuracy by the mass-damper spring model. Using Newton’s second law in conjunction with equations for F_{s}, F_{d} and F_{m} given above and measuring the displacement x(t) from the static equilibrium position, we obtain the differential equation of motion as below:

which is subject to the initial conditions x(0)=x_{0, }ẋ(0)=v_{0}, where x_{0 }and v_{0 }are the **initial displacement** and **initial velocity**, respectively.

*Equation given above is in terms of a single coordinate. namely x(t) is therefore said to be a single-degree-of-freedom system.*

**Free Vibration of Undamped Systems** Assuming zero damping and external forces and dividing above equation through by m, we obtain

In this case, the vibration is caused by the initial excitations alone. The solution of above equation is

which represents simple sinusoidal, or simple harmonic oscillation with **amplitude** A, **phase angle** ф, and **frequency** .

The time necessary to complete one cycle of motion defines the **period.**

The reciprocal of the period provides another definition of the **natural frequency**,namely,

where Hz denotes hertz[1 Hz = 1 cycle per second (cps)].

**Free Vibration of Damped Systems** Let F(t)=0 and divide through by m.Then, Equation of motion reduces to

**ξ** is the **damping factor**, a non-dimensional quantity. The nature of the motion depends on ξ. The most important case is that in which 0<ξ<1.

In this case, the system is said to be **underdamped** and the solution of above equation is

ω_{d }is the frequency of damped free vibration and is the **period of damped oscillation.**

The case **ξ=1, **represents** critical damping, **and** C _{c} **is the critical damping coefficient,although there is nothing critical about it. It merely represents the borderline between oscillatory decay and aperiodic decay. In fact, C

_{c }is the smallest damping coefficient for which the motion is aperiodic. When

**ξ>1**, the system is said to be

**overdamped.**

**Logarithmic Decrement** Quite often the damping factor is not known and must be determined experimentally. In the case in which the system is underdamped, this can be done conveniently by plotting x(t)

versus t, and measuring the response at two different times separated by a complete period.

**Whirling of Rotating Shafts** Many mechanical systems involve rotating shafts carrying disks. If the disk has some eccentricity, then the centrifugal forces cause the shaft to bend, as shown in Figure (a) below. The rotation of the plane containing the bent shaft about the bearing axis is called **whirling**. Figure (b) below shows a disk with the body axes *x, y* rotating about the origin O with the angular velocity ω.

The geometrical center of the disk is denoted by S and the mass center by C.The distance between the two points is the eccentricity *e*. The shaft is massless and of stiffness k_{eq} and the disk is rigid and of mass m. The *x* and *y* components of the displacement of S relative to O are independent from one another and, for no damping, satisfy the equations of motion

**Resonance** occurs when the whirling angular velocity coincides with the natural frequency. In terms of rotations per minute, it has the value

where f_{c }is called the **critical speed.**

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###### REFERENCES

- Marks’ Standard Handbook for Mechanical Engineers Eleventh Edition.
- Fundamentals of Sound and Vibrations by KTH Sweden.

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