# MECHANICAL VIBRATIONS PDF

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any means graphic, electronic, or mechanical, including but not limited to This edition of Mechanical Vibrations: Theory and Applications has been adapted to. Mechanical. Vibrations. Fifth Edition. Singiresu S. Rao. University of Miami. Prentice Hall. Upper Saddle River Boston Columbus San Francisco New York. PDF | This book is about mechanical vibration. It is not a handbook rather intended as a textbook for the present and hopefully future.

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Vibrations are oscillations in mechanical dynamic systems. The role of mechanical vibration analysis should be to use mathematical tools for modeling. Mechanical Vibrations. A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position. are ubiquitous in engineering and thus the study of vibrations is extremely important. The most basic . system that arises in the study of mechanical vibrations.

Structures designed to support heavy centrifugal machines, like motors and turbines, or reciprocating machines, like steam and gas engines and reciprocating pumps, are also subjected to vibration.

The structure or machine component subjected to vibration can fail because of material fatigue resulting from the cyclic variation of the induced stress. Vibration causes more rapid wear of machine parts such as bearings and gears and also creates excessive noise. In machines, vibration can loosen fasteners such as nuts. In metal cutting processes, vibration can cause chatter, which leads to a poor surface finish.

Whenever the natural frequency of vibration of a machine or structure coincides with the frequency of the external excitation, this results in a phenomenon known as resonance, which leads to excessive deflections and failure.

The transmission of vibration to human beings results in discomfort and loss of efficiency. The vibration and noise generated by engines causes annoyance to people and, sometimes, damage to property. Vibration of instrument panels can cause their malfunction or difficulty in reading the meters. In spite of its detrimental effects, vibration can be utilized profitably in several consumer and industrial applications.

In fact, the applications of vibratory equipment have increased considerably in recent years. Some of the benefits of vibration are as follows: Vibration is also used in pile driving, vibratory testing of materials, vibratory finishing processes, and electronic circuits to filter out the unwanted frequencies.

Vibration has been found to improve the efficiency of certain machining, casting, forging, and welding processes. Vibration is employed to simulate earthquakes for geological research and also to conduct studies in the design of nuclear reactors.

Any motion that repeats itself after an interval of time is called vibration or oscillation. The swinging of a pendulum and the motion of a plucked string are typical examples of vibration.

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The theory of vibration deals with the study of oscillatory motions of bodies and the forces associated with them. Figure 1. At position 1 the velocity of the As an example, consider the vibration of the simple pendulum shown in Fig.

## Mechanical Vibration - MV Study Materials

Since the gravitational force mg induces a bob and hence its kinetic energy is zero. This gives the bob certain angular acceleration in the clockwise direction, and by the time it reaches position 2, all of its potential energy will be converted into kinetic energy.

Hence the bob will not stop in position 2 but will continue to swing to position 3. However, as it passes the mean position 2, a counterclockwise torque due to gravity starts acting on the bob and causes the bob to decelerate. The velocity of the bob reduces to zero at the left extreme position. By this time, all the kinetic energy of the bob will be converted to potential energy.

Again due to the gravity torque, the bob continues to attain a counterclockwise velocity. Hence the bob starts swinging back with progressively increasing velocity and passes the mean position again.

This means that some energy is dissipated in each cycle of vibration due to damping by the air. Single-degree-of-freedom systems The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time defines the number of degrees of freedom of the system.

The simple pendulum shown in Fig. For example, the motion of the simple coordinates x and y. For the slider shown in Fig. In Fig. For the torsional system long bar with a heavy disk at the end shown in Fig. Three degree-of-freedom systems Some examples of two- and three-degree-of-freedom systems are shown in Figs.

The motion of the system shown in Fig. For the systems shown in Figs. In the case of the system shown in Fig. The coordinates necessary to 2 2 2 describe the motion of a system constitute a set of generalized coordinates.

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These are usually denoted as q1, q2 , A cantilever beam an infinite-number-of-degrees-of-freedom system Most of the time, continuous systems are approximated as discrete systems, and solutions are obtained in a simpler manner.

Most structural and machine systems have deformable elastic members and therefore have an infinite number of degrees of freedom.

Most of the practical systems are studied by treating them as finite lumped masses, springs, and dampers.

In general, more accurate results are obtained by increasing the number of masses, springs, and dampers that is, by increasing the number of degrees of freedom. If a system, after an initial disturbance, is left to vibrate on its own, the ensuing vibration is known as free vibration.

No external force acts on the system. The oscillation of a simple pendulum is an example of free vibration. If a system is subjected to an external force often, a repeating type of force , the resulting vibration is known as forced vibration.

The oscillation that arises in machines such as diesel engines is an example of forced vibration. If the frequency of the external force coincides with one of the natural frequencies of the system, a condition known as resonance occurs, and the system undergoes dangerously large oscillations.

Failures of such structures as buildings, bridges, turbines, and airplane wings have been associated with the occurrence of resonance. If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is known as undamped vibration. If any energy is lost in this way, however, it is called damped vibration. However, consideration of damping becomes extremely important in analyzing vibratory systems near resonance.

If all the basic components of a vibratory system the spring, the mass, and the damper behave linearly, the resulting vibration is known as linear vibration.

If, however, any of the basic components behave nonlinearly, the vibration is called nonlinear vibration. If the vibration is linear, the principle of superposition holds, and the mathematical techniques of analysis are well developed. For nonlinear vibration, the superposition principle is not valid, and techniques of analysis are less well known. Since all vibratory systems tend to behave nonlinearly with increasing amplitude of oscillation, knowledge of nonlinear vibration is desirable in dealing with practical vibratory systems.

If the value or magnitude of the excitation force or motion acting on a vibratory system is known at any given time, the excitation is called deterministic. The resulting vibration is known as deterministic vibration. In some cases, the excitation is nondeterministic or random; the value of the excitation at a given time cannot be predicted.

In these cases, a large collection of records of the excitation may exhibit some statistical regularity.

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It is possible to estimate averages such as the mean and mean square values of the excitation. Examples of random excitations are wind velocity, road roughness, and ground motion during earthquakes. If the excitation is random, the resulting vibration is called random vibration. In this case the vibratory response of the system is also random; it can be described only in terms of statistical quantities.

Deterministic and random excitations 1. The response of a vibrating system generally depends on the initial conditions as well as the external excitations. The purpose of mathematical modeling is to represent all the important features of the system for the purpose of deriving the mathematical or analytical equations governing the system s behavior. The mathematical model should include enough details to allow describing the system in terms of equations without making it too complex.

To illustrate the procedure of refinement used in mathematical modeling, consider the forging hammer shown in Fig. It consists of a frame, a falling weight known as the tup, an anvil, and a foundation block. Once the mathematical model is available, we use the principles of dynamics and derive the equations that describe the vibration of the system.

The equations of motion can be derived conveniently by drawing the free-body diagrams of all the masses involved. The free-body diagram of a mass can be obtained by isolating the mass and indicating all externally applied forces, the reactive forces, and the inertia forces. The equations of motion of a vibrating system are usually in the form of a set of ordinary differential equations for a discrete system and partial differential equations for a continuous system.

The equations may be linear or nonlinear, depending on the behavior of the components of the system. Several approaches are commonly used to derive the governing equations. Among them are Newton s second law of motion, D Alembert s principle, and the principle of conservation of energy.

The equations of motion must be solved to find the response of the vibrating system.

Depending on the nature of the problem, we can use one of the following techniques for finding the solution: If the governing equations are nonlinear, they can seldom be solved in closed form. Furthermore, the solution of partial differential equations is far more involved than that of ordinary differential equations.

Numerical methods involving computers can be used to solve the equations. However, it will be difficult to draw general conclusions about the behavior of the system using computer results. Interpretation of the Results. The solution of the governing equations gives the displacements, velocities, and accelerations of the various masses of the system.

These results must be interpreted with a clear view of the purpose of the analysis and the possible design implications of the results. Example 1.

Mathematical Model of a Motorcycle The Figure a shows a motorcycle with a rider. Develop a sequence of three mathematical models of the system for investigating vibration in the vertical direction. Consider the elasticity of the tires, elasticity and damping of the struts in the vertical direction , masses of the wheels, and elasticity, damping, and mass of the rider. If the motion is repeated after equal intervals of time, it is called periodic motion. The simplest type of periodic motion is harmonic motion.

The mass m of the spring-mass system are displaced from their middle positions by an amount x in time t given by 1. The velocity of the mass m at time t is given by 1. Such a vibration, known as simple harmonic motion. Representation of a complex number It is more convenient to represent harmonic motion using a complex-number representation. Any vector in the xy-plane can be represented as a complex number: Some examples are the oscillations of the pendulum of a grandfather clock, the vertical oscillatory motion felt by a bicyclist after hitting a road bump and the motion of a child on a swing after an initial push.

Figure 2. It is called a single-degree-of-freedom system, since one coordinate x is sufficient to specify the position of the mass at any time. There is no external force applied to the mass; hence the motion resulting from an initial disturbance will be free vibration. A spring-mass system in horizontal position Since there is no element that causes dissipation of energy during the motion of the mass, the amplitude of motion remains constant with time; it is an undamped system.

In actual practice, except in a vacuum, the amplitude of free vibration diminishes gradually over time, due to the resistance offered by the surrounding medium such as air. Select a suitable coordinate to describe the position of the mass or rigid body in the system.

Use a linear coordinate to describe the linear motion of a point mass or the centroid of a rigid body, and an angular coordinate to describe the angular motion of a rigid body. Determine the static equilibrium configuration of the system and measure the displacement of the mass or rigid body from its static equilibrium position.

Draw the free-body diagram of the mass or rigid body when a positive displacement and velocity are given to it. Indicate all the active and reactive forces acting on the mass or rigid body. Apply Newton s second law of motion to the mass or rigid body shown by the freebody diagram. Newton s second law of motion can be stated as follows: The rate of change of momentum of a mass is equal to the force acting on it.

Equation 2. The application of Eq. If no work is done on a conservative system by external forces other than gravity or other potential forces , then the total energy of the system remains constant. Since the energy of a vibrating system is partly potential and partly kinetic, the sum of these two energies remains constant. The kinetic energy T is stored in the mass by virtue of its velocity, and the potential energy U is stored in the spring by virtue of its elastic deformation.

Thus the principle of conservation of energy can be expressed as: Substitution of Eq. The two values of s given by Eq. Since both values of s satisfy Eq. By using the identities Eq.

The constants C1 and C2 or A1 and A2 can be determined from the initial conditions of the system. Two conditions are to be specified to evaluate these constants uniquely. Note that the number of conditions to be specified is the same as the order of the governing differential equation.

Harmonic Response of a Water Tank a b. In this case, the displacement of the body is measured in terms of an angular coordinate. In a torsional vibration problem, the restoring moment may be due to the torsion of an elastic member or to the unbalanced moment of a force or couple.

Torsional vibration of a disc Figure 2. From the theory of end of a solid circular shaft, the other end of which is fixed. Let the angular rotation of the disc torsion of circular shafts, we have the relation 2.

Thus the shaft acts as a torsional spring with a torsional spring constant 2. By considering the free body diagram of the disc Fig. Thus the natural circular frequency of the torsional system is 2. If the cross section of the shaft supporting the disc is not circular, an appropriate torsional spring constant is to be used. The polar mass moment of inertia of a disc is given by 2. The torsional spring-inertia system shown in Fig.

One of the most important applications of a torsional pendulum is in a mechanical clock, where a ratchet and pawl convert the regular oscillation of a small torsional pendulum into the movements of the hands. A single-degree-of- freedom system with a viscous damper is shown in Fig. Inserting this function into Eq. Thus the general solution of Eq.

The critical damping is defined as the value of the damping constant c for which the radical in Eq. Case 1: The quantity 2. The underdamped case is very important in the study of mechanical vibrations, as it is the only case that leads to an oscillatory motion. Underdamped solution Case 2: Comparison of motions with different types of damping Case 3: In this case, the solution, Eq. Since roots and are both negative, the motion diminishes exponentially with time, as shown in Fig.

Logarithmic Decrement The logarithmic decrement represents the rate at which the amplitude of a free-damped vibration decreases. It is defined as the natural logarithm of the ratio of any two successive amplitudes. Let t1 and t 2 denote the times corresponding to two consecutive amplitudes displacements , measured one cycle apart for an underdamped system, as in Fig. Using Eq. The logarithmic decrement is dimensionless and is actually another form of the dimensionless 2.

Determine the frequency of vibration of the system. Determine the damping coefficient of the damper in the system. When the damping is provided, the frequency of damped vibrations was observed to be 0. Find 1. Logarithmic decrement, and 4. Assuming that the damping force varies as the velocity, determine: External energy can be supplied through either an applied force or an imposed displacement excitation. The applied force or displacement excitation may be harmonic, nonharmonic but periodic, nonperiodic, or random in nature.

The response of a system to a harmonic excitation is called harmonic response. The nonperiodic excitation may have a long or short duration. The response of a dynamic system to suddenly applied nonperiodic excitations is called transient response.

Under a harmonic excitation, the response of the system will also be harmonic. If the frequency of excitation coincides with the natural frequency of the system, the response will be very large.

This condition, known as resonance, is to be avoided to prevent failure of the system. A spring-mass-damper system If a force F t acts on a viscously damped spring-mass system as shown in Fig.

The homogeneous solution, which is the solution of the homogeneous equation 3.

## Advanced Mechanical Vibrations.pdf

As seen in Section 2. Figure 3. Homogenous, particular, and general solutions of Eq. The steady-state motion is present as long as the forcing function is present. The variations of homogeneous, particular, and general solutions with time for a typical case are shown in Fig. The part of the motion that dies out due to damping the free-vibration part is called transient.

The rate at which the transient motion decays depends on the values of the system parameters k, c, and m. In this chapter, except in Section 3.

The physical reception of sound in any hearing organism is limited to a range of frequencies. Other species have different ranges of hearing. As a signal perceived by one of the major senses , sound is used by many species for detecting danger , navigation , predation , and communication. Earth's atmosphere , water , and virtually any physical phenomenon , such as fire, rain, wind, surf , or earthquake, produces and is characterized by its unique sounds.

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Many species, such as frogs, birds, marine and terrestrial mammals , have also developed special organs to produce sound. In some species, these produce song and speech. Furthermore, humans have developed culture and technology such as music, telephone and radio that allows them to generate, record, transmit, and broadcast sound. Noise is a term often used to refer to an unwanted sound.

In science and engineering, noise is an undesirable component that obscures a wanted signal. However, in sound perception it can often be used to identify the source of a sound and is an important component of timbre perception see above. Soundscape is the component of the acoustic environment that can be perceived by humans.

The acoustic environment is the combination of all sounds whether audible to humans or not within a given area as modified by the environment and understood by people, in context of the surrounding environment. There are, historically, six experimentally separable ways in which sound waves are analysed.

They are: pitch , duration , loudness , timbre , sonic texture and spatial location. More recent approaches have also considered temporal envelope and temporal fine structure as perceptually relevant analyses. Pitch perception Pitch is perceived as how "low" or "high" a sound is and represents the cyclic, repetitive nature of the vibrations that make up sound. For simple sounds, pitch relates to the frequency of the slowest vibration in the sound called the fundamental harmonic. In the case of complex sounds, pitch perception can vary.

Sometimes individuals identify different pitches for the same sound, based on their personal experience of particular sound patterns. Selection of a particular pitch is determined by pre-conscious examination of vibrations, including their frequencies and the balance between them.

Specific attention is given to recognising potential harmonics. For example: white noise random noise spread evenly across all frequencies sounds higher in pitch than pink noise random noise spread evenly across octaves as white noise has more high frequency content. Figure 1 shows an example of pitch recognition. During the listening process, each sound is analysed for a repeating pattern See Figure 1: orange arrows and the results forwarded to the auditory cortex as a single pitch of a certain height octave and chroma note name.

Duration[ edit ] Figure 2. Duration perception Duration is perceived as how "long" or "short" a sound is and relates to onset and offset signals created by nerve responses to sounds. The duration of a sound usually lasts from the time the sound is first noticed until the sound is identified as having changed or ceased. For example; in a noisy environment, gapped sounds sounds that stop and start can sound as if they are continuous because the offset messages are missed owing to disruptions from noises in the same general bandwidth.

Figure 2 gives an example of duration identification. When a new sound is noticed see Figure 2, Green arrows , a sound onset message is sent to the auditory cortex.In this case the vibratory response of the system is also random; it can be described only in terms of statistical quantities. Single-degree-of-freedom systems The minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time defines the number of degrees of freedom of the system.

We shall assume that the rotor is subjected to a steady-state excitation due to mass unbalance. Find 1. Hence the bob will not stop in position 2 but will continue to swing to position 3.

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