Mechanical Dr. B.M. El-Souhily References: Vibrations +. بس#وني 'لس$#لي 1-! Mechanical Vibrations Singiresu S. Rao Addison_Wesley publishing company. 2-! Theory of Vibration with Applications William T. Thomson Prentice hall, Englewood Cliffs, New Jersey. 3-! Vibration of Mechanical and Structural systems M. L. James, G. M. Smith, J. C. Wolford, and P. W. Whaley Harper & Row, publishers, New York. 4-! A course in Mechanical Vibrations Mahmoud Mostafa Faculty of Engineering, University of Alexandria. 1
CHAPTER 1 Fundamentals of vibration Vibration: It's the motion of a body or a system that is repeated after a given interval of time known as the period. Frequency: -The number of cycles of the motion per unit time (c.p.m, r.p.m, cps, Hz, rad/s). Amplitude: -! The maximum displacement (velocity, acceleration or force) of the body or some parts of the system from the equilibrium position is the amplitude of the vibration of that point. -! Figure 1 Phase relationships among displacement, velocity, and acceleration are shown on these time history plots. Natural Frequency: If a body is suddenly disturbed in some manner it will vibrate at a definite frequency known as its natural frequency (ω). 2
Figure 2: The Main parameters of the Simple harmonic motion Importance of the study of vibration: I- Most human activities involve vibration in one from or other: 1- We hear because our eardrums vibrate. 2- We see because light waves undergo vibration. 3- Breathing is associated with the vibration of lungs. 4-Walking involves oscillatory motion of legs and hands. 5-We speak due to the oscillatory motion of larynges (tongue). II- Most prime movers have vibrational problems due to unbalance in the engines. The unbalance may be due to faulty design or poor manufacture: 1-! Imbalance in diesel engines can cause ground waves sufficiently powerful to create a nuisance in urban areas. 2-! The wheels of some locomotion can rise more than a centimeter off the track at high speeds due to unbalance. 3-! In turbines vibrations cause spectacular mechanical failures. 4-! The structures designed to support heavy centrifugal machines, like motors and turbines, or reciprocating machines, like steam and gas engines and reciprocating pumps, are 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. 5-! The vibration causes more rapid wear of machine parts such as bearings and gears and also creates excessive noise. 6-! Vibration causes looseness of fasteners, poor surface finish. III- Whenever the natural frequency of vibration of a machine or structure coincides with the frequency of the external excitation, there occurs a phenomenon known as 3
resonance, which leads to excessive defection and failure (Tacoma narrows bridge during wind induced vibration opened on july\1940,collapsed on nov,7(1940). IV- The transmission of vibration to human beings results in discomfort and loss of efficiency. One of the important purposes of vibration study is to reduce vibration through proper design of machines and their mountings, the mechanical engineer tries to design the engine or machine so as to minimize unbalance, while the structural engineer tries to design the supporting structure so as to ensure that the effect of the imbalance will not be harmful. In spite of its detrimental effects, vibration can be utilized profitably in several consumer and industrial applications. 1-! Vibration is put to work in vibratory conveyors, hoppers, sieves, compactors, washing machines electric tooth brushes, dentist's drill, clocks, and electric massaging units. 2-! Vibration is also used in pile driving, vibratory testing of materials, vibratory finishing processes, and electronic circuits to filter out the unwanted frequencies. 3-! Vibration has been found to improve the efficiency of certain machining, casting, forging, and 4-! It is employed to simulate earthquakes for geological research. Basic concepts of vibration 1- Elementary parts of vibrating systems: -Vibratory system, in general includes: a- A mean for storing kinetic energy (mass or inertia). b- A mean for storing potential energy (springs or elastic members). c- A mean by which energy is gradually lost. (dampers). -the vibration of a system involves the transfer of its potential energy to kinetic energy and kinetic energy to potential energy, alternately. If the system is damped, some energy is dissipated in each cycle of vibration and must be replaced by an external source if a state of steady vibration is to be maintained. 4
2- Degree of freedom: Degree of freedom is the minimum number of independent coordinates required to determine completely the positions of all parts of a system at any instant of time. 5
3- Discrete and Continuous System:- -Some systems, especially those involving continuous elastic members, have an infinite number of degrees of freedom. (e.g. cantilever beam). -Systems with a finite number of degree of freedom are called continuous, or distribute systems. -Most of the time, continuous systems are approximated as discrete system, (more accurate results are obtained by increasing the number of degrees of freedom. -The analysis methods available for dealing with continuous systems are limited to a narrow selection of problems, such as uniforms beams, slender rods, and thin plates. 6
Classification of Vibration - Free and forced vibration: Free vibrations: If a system, after an initial disturbance, is left to vibrate on its own, the vibration is known as free vibration no external force acts on the systems (e.g. simple pendulum). Forced vibration: If a system is subjected to an external force, the resulting vibration is known as forced vibration if the frequency of the external force coincides with one of the natural frequencies of the system, a conditions known as resonance occurs (large oscillation). Failures of such structures as building, bridges, turbines and airplane wings have been associated with the occurrence of resonance. - Undamped and damped vibration: - If no energy is lost or dissipated in friction or other resistance during oscillation, the vibration is undamped vibration. - If any energy is lost, it is called damped vibration. -consideration of damping becomes extremely important in analyzing vibration systems near resonance. Figure 2.4: Effect of damping in free vibration on the 7 amplitude of the vibration
- Linear and nonlinear vibration: - If all the basic components of a vibratory system, the spring, the mass, and the damper, behave linearly, the resulting linear vibration. -If any behave non-linearly, the vibration is called nonlinear vibration. - Deterministic and random vibration: -If the value or magnitude of the excitation (force or motion) acting on vibrating system is known at any given time, the excitation is called deterministic. The resulting vibration is known as deterministic vibration. x(t) τ t Deterministic (periodic) excitation -If the value of the excitation at a given time cannot be predicted, the resulting vibration is known as random. (e.g. wind velocity, road roughness, and ground motion during earthquakes). 8
Random excitation Vibration analysis procedure - A vibratory system is a dynamic system for which the variables such as the excitations (input) and responses (outputs) are time-dependent. The response of a vibrating system generally depends on the initial conditions as well as the external excitations. -Most practical vibrating systems are very complex, and it is impossible to consider all the details for a mathematical analysis. -The analysis of a vibration system usually involves mathematical modeling, derivation of the governing equations solution of the equations, and interpretation of the result. Step1: Mathematical Modeling: The purpose of it is to represent all the important features of the system for the purpose of deriving the mathematical (or analytical) equations governing the behavior of the system. The mathematical model is gradually improved to obtain more accurate results. 9
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Sometimes the mathematical model is gradually improved to obtain most accurate results. In this approach, first a very crude or elementary model is used get a quick insight into the overall behavior of the system. Subsequently, the model is refined by including more components and/or details so that the behavior of the system can be observed in more detail. To illustrate the procedure of refinement used in mathematical modeling, consider the forging hammer shown. The forging hammer consists of a frame, a falling weight known as the tup, an anvil and a foundation block. The anvil is a massive steel block on which material forged into desired shape by the repeated blows of the tup. The anvil is usual mounted on an elastic pad to reduce the transmission of vibration to the foundation block and the frame. For a first approximation, the frame, anvil, elastic pal foundation block, and the soil are modeled as a single degree of freedom system. For a refined approximation, the weights of the frame an anvil and the foundation block are represented separately with a two degree of freedom model as shown. Further refinement of the model can be made by considering eccentric impacts of the tup, which cause each of the masses to have both vertical and rocking (rotation) motions in the plane of the paper. 11
Modeling of a forging hammer. 12
Step 2: Derivation of governing Equations: 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 by Newton's second law, d'alembert principle, and the principle of conservation of energy. Step3: Solution of the governing Equations: The equations of motion must be solved to find the response of the vibrating system, using the following techniques. 1-Standard methods of solving differential equations. 2-Laplace transformation methods. 3-Matrix method. 4-Numerical methods. Steps4: Interpretation of the results: -The solution 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 possible design implications of the results. 13
Spring Elements The stiffness "k" of a spring element is a relation between the force F and the deflection x where; k = df d x = the slope of curve F a b c x The relation between the force and the deflection is illustrated in previous figure. If the stiffness increases with the force, curve (a), the spring is called nonlinear hard. If the stiffness decreases with the force, curve (c), the spring is called nonlinear soft. If the stiffness is constant, line (b), the spring is called linear spring. Linear springs are available in a broad range of springs and elastic elements. All springs are considered approximately linear over a certain range of deformation. The analysis in this course is concerned only with applications having linear springs. The following table lists a variety of spring elements. Element Name Stiffness 14
4 Gd Coil k = 3 64nR n = number of coils R = radius d = wire diameter k 1 k 2 Series k k = k 1 1 k 2 + k 2 k 1 k 2 E, A, L E, I, L Parallel k = k 1 + k 2 Longitudinal bar E A k = L 3EI Cantilever beam k = 3 L a b 3EI(a + b) Simply supported beam k = 2 a 2 b L a Fixed-hinged beam 24EI = a (3L + 8a) k 2 L a 3EI Hinged-hinged k = 2 (L + a)a EI Spiral spring k t = L Torsion bar GJ k t = L 15
Damping Elements The vibrational energy is gradually converted to heat or sound. Dampers are devices which are used to dissipate energy from the system in order to reduce the vibrations. A damper is assumed to have neither mass nor elasticity. Damping force exists only if there is relative velocity between the two ends of the damper. Damping is modeled as one or more of the following: 1-Viscous damping Viscous damping is the most commonly used damping mechanism in vibration analysis (fluid medium such as air, gas, water, or oil). Typical examples of viscous damping include: a-! Fluid films between sliding surfaces, b-! Fluid flow around a piston in a cylinder, c-! Fluid flow through an orifice, and d-! Fluid film around a journal bearing. F d = c x! 2-Coulomb damping (or Dry Friction) The resisting force is constant in magnitude. Its direction is opposite to the direction of the velocity. It is caused by friction between rubbing surfaces that either dry or have insufficient lubrication. 3-Internal, solid, material or hysteric damping When a material is deformed, there is an internal resistance due the sliding action of the molecules, energy dissipated by the material, due to friction between internal planes, which slip or slide. 16
SIGNAL ANALYSIS When a body vibrates, it undergoes an oscillatory motion. In order to study the nature of the vibrations of bodies we transfer the mechanical motion to an electrical signal which is easier to deal with. A signal in its broad meaning has repetitive nature. Generally, a signal is a combination of several signal components. For example, if we study the electromagnetic signals of radio stations, T V, or cellular phones, we find that these signals are combinations of discrete frequencies, each represents one particular station. Sound signals are also examples of compound signals. We hear different sounds at the same time. Sound results from the vibration of bodies. Also, hearing the sounds is a result of the vibration of the ear drum. We can hear different sounds at the same time and distinguish between them. Each body has its unique vibration characteristics. So, in general, a compound signal is the sum of many fundamental signals, each has a single frequency and certain strength. Such fundamental signal is known as the harmonic signal. In the following sections we shall discuss the different types of signals. For compound signals, we shall be able to analyze them to extract the constituents of the harmonic signals. This is known as signal analysis. Harmonic Motion It is the simplest form of a periodic motion. It is known as simple harmonic motion. Mathematically, it is represented by the sinusoidal function (a sine or a cosine function). Graphically, the sine curve is the vertical projection of a vector of length A rotating with an angular frequency (simply called frequency) ω which is measured in rad/s; it makes an angle θ = ωt with the horizontal axis; t is the time. The cosine curve is the vertical projection of a vector of length A rotating with an angular velocity ω ; it makes angle θ = ωt with the vertical axis, as shown in the figure. We conclude that the cosine function is leading the sine function by 90 o. x = A sin θ (1) y = A cos θ (2) 17
A ωt ωt A x = A sin ωt 2 π ω x = A cos ωt t 2 π ω t Where, A is known as the amplitude and θ = ωt. Both curves make a complete cycle when the vectors rotate an angle equals to 2π". This corresponds to one complete cycle of the harmonic curve. The time of one cycle is called the period "τ" such that ωτ = 2π 2 π τ = ω The circular frequency "f" is the number of cycles per second. Its unit is Hz (Hertz). Its value is given by f = 1 ω = τ 2π The relation between the circular frequency and the angular frequency is ω = 2 π f Important conclusion For the sake of graphical representation, a sine function can be considered as a vector that makes an angle with the horizontal axis. Likewise, a cosine function can be considered as a vector that makes an angle with the vertical axis as shown. 18
Combination of Two Harmonics : a-! Having the Same Frequency Consider a signal given by, x = A 1 sin ωt + A 2 sin (ωt + φ) y C A 2 φ α ωt A 1 A 1 x φ is a phase angle between the two signals. Using the vector representation, A 1 sin ωt is represented by a vector of length A 1 that makes an angle ωt with the x-axis. Similarly, A 2 sin (ωt + φ) is represented by a vector of length A 2 that makes an angle ωt + φ with the x-axis. The resultant vector is a vector of length C which makes an angle ωt + α with the x-axis, and is represented by x = C sin (ω t + α) (3) C 2 2 2 = A + 1 A + 2 A 2 1 A 2 cos φ (4) tan α= A 1 A2 sin ϕ + A cos ϕ 2 (5) b-! Having Different Frequencies Suppose we have a signal which is composed from two harmonics with frequencies ω 1 and ω 2 ; ω 2 > ω 1. The resultant signal is x = b sin ω 1 t +a sin ω 2 t 19
Periodic motion 1-! The resultant motion is not simple harmonic motion but periodic motion, 2-! The amplitude various between (a+b) when the vectors are in phase and minimum value (b-a) when they are 180 o off phase, 3-! The period of the compound periodic motion is the time interval required for one component vector to rotate a complete revolution relative to the other [i.e. 2π/(ω 1 ω 2 )], 4-! The angular velocity of the resultant is (ω 1 + ω 2 )/2, 5-! The interval between successive peaks is. 2π/[(ω 1 + ω 2 ) /2] Beats: Beating is an interesting phenomenon that occurs when a system with very little damping is subjected to an excitation source that has a frequency very close to its natural frequency, (or, when two harmonic motions, with frequencies close to one another, are added). x = A 1 sin ω 1 t +A 2 sin ω 2 t Case 1: When A 1 = A 2 = A and ω 2 is very close to ω 1 20
y A C ω 2 t ψ = (ω 2 ω 1 )t/2 A ω 1 t x The resultant vector bisects the angle between the two vectors. Then ψ = ½ (ω 2 - ω 1 )t The angle of the resultant vector is ω 1 t + ½ (ω 2 - ω 1 ) t = ½ (ω 2 + ω 1 )t The length of the resultant vector is given by C = 2 A cos ½ (ω 2 - ω 1 )t x Therefore, x = 2 A [cos ½ (ω 2 - ω 1 )t sin ½ (ω 2 + ω 1 )t] 1-! The amplitude of this motion slowly fluctuated between 0 and 2A according to the term 2 A cos ½ (ω 2 - ω 1 )t. 21
2-! The period of fluctuating is 2π/ ½ (ω 2 - ω 1 ), 3-! The time between two successive maximum and minimum values will be 2π/(ω 2 - ω 1 ), (period of beating), 4-! The frequency of the harmonic signal is ½(ω 2 + ω 1 ), 5-! The vibration period is 2π/ ½ (ω 2 + ω 1 ), 6-! The beating frequency is ω 2 - ω 1. Case 2: When A 1 A 2 and ω 2 is very close to ω 1 x ( ω2 ω1)t 2π x 1 = A 1 sin ω 1 t and x 2 = A 2 sin ω 2 t if ω = ½ (ω 2 + ω 1 ), = ½ (ω 2 - ω 1 ) x 1 = A 1 sin (ω - )t and x 2 = A 2 sin (ω + )t x = (A 1 +A 2 ) cos t sin ωt + (A 1 - A 2 ) sin t cos ωt = C sin (ωt +φ) Where, C= A 2 1 + A 2 2 + 2 A 1 A 2 cos 2 t Max. amplitude C max = A 1 + A 2 Min. amplitude C min = A 1 - A 2 Vibration period 4π/ (ω 2 + ω 1 ), Beating period 2π /(ω 2 - ω 1). 22
Periodic Motion Fourier Series When machines or structures are subjected to an excitation involving more than one frequency, the resulting vibration is said to be periodic. Consider a signal which consists of several components. x = A 1 sin ω 1 t + A 2 sin ω 2 t +. +A n sin ω n t (7) Consider the case when the frequencies of the components have definite relations such that ω 2 = 2 ω 1, ω 3 = 3 ω 1,, ω n = n ω 1 The resultant is a periodic signal with a periodic frequency equal to ω 1, ω 1 is 2 π called the fundamental frequency. The period is equal to. If the number of ω1 components is infinite, the function is still periodic. The shape of the function depends on the amplitude of the components. In general, the terms in Eq. (7) may include sine and cosine functions. This type of functions is represented by Fourier series. It is written in the form x(t) = n= ( an cos nωt + bn sin nωt) (8) 0 x(t) τ t 23
The summation of signals with frequencies which are multiple of the first frequency results in a periodic signal. Here we want to do the reverse. We have a periodic function x(t) with a period τ and need to obtain its frequency components. In the case of periodic functions Fourier analysis is used. In general, x(t) is put in the form of Eq. (8) which can be written in the form: x (t) = ½ a o + n= Expanding Eq. (9), then Where, 1 ( an cos nω1 t + bn sin nω1 t) (9) x (t) = ½ a o + a 1 cos ω 1 t + a 2 cos ω 2 t + + b 1 sin ω 1 t + b 2 sin ω 2 t + (10) ω 1 = 2 π τ ω n = n ω 1 Multiply both sides of Eq. (10) by cos ω n t and integrate over the period τ. For the right hand side it is known that: τ # $% 0 if m n cos ωnt cosωmt dt = (11) 0 τ / 2 if m = n τ # $% 0 if m n cos ωnt sinωmt dt = (12) 0 0 if m = n Similarly, multiplying both sides of Eq. (10) by sin ω n t and integrate over the period τ. τ # $% 0 if m n sin ωnt sin ωmt dt = (13) 0 τ / 2 if m = n 24
τ # $% 0 if m n sin ωnt cosωmt dt = (14) 0 0 if m = n Use Eqs. (11), (12), (13) and (14) we get 2 a n = τ x(t) cos ω τ 0 nt dt 2 b n = τ x(t) sin ω τ 0 nt dt n = 0, 1, 2, (15) n = 1, 2, (16) After obtaining the coefficients a n and b n, Eq. (9) can be written in the form x (t) = ½ a o + n= 1 c n sin (nω1 t +ψn ) (17) Where c n is the amplitude of the component with frequency ω n and ψ n is a phase angle. Their values are given by Example 1 2 2 c n = a + b n ψ n = tan -1 a b n n n Determine the Fourier series for the half sine wave signal shown in Figure. x (t) 0 τ 2τ t 25
Solution The signal is represented by πt x = 2 sin for 0 t τ τ 4 a n = τ πt 2πnt sin cos dt τ 0 τ τ 2 8(cos πn) = 2 π(4n 1) 8 = n = 0, 1, 2 π(4n 2 1) 4 b n = τ πt 2πnt sin sin dt τ 0 τ τ 4sin 2πn = π( 4n 2 1) = 0 for all values of n. The Fourier series for this signal is 4 x (t) = + = π n 1 (4n 2 8 cos 1)π 2nπt τ c n n The frequency spectrum is a plot of the amplitudes with the frequencies. They give a true picture for the whole signal. The spectrum of the signal of Example 1 is shown in the above figure. C n = (4n 8 2 1)π 26
Example 2 Determine the Fourier series for the rectangular pulses shown in figure. Plot the frequency spectrum. Solution The signal is represented by x = A for x = 0 for 0 t τ τ t T x (t) A 0 τ T t Fig. 1-10 2 π ω 1 =, ωn = T 2πn T Apply Eqs. (1-15) and (1-16), then 2A a n = τ cos ωnt dt T 0 2 A = sin ωnτ ω T n 2 A b n = τ sin ωnt T 0 dt 27
2 A = (1 cos ωnτ) ωnt The amplitude is given by c n 2 2 c n = a + b n n Substitute the values of a n, b n, and ω n. After simplifications c n = 2A nπτ sin nπ T The Fourier series is given by x (t) = n = 0 ( 2A & ' nπ sin nπτ T 2nπt sin ( T + ψ n ) % # $ The spectrum of this signal is shown in Figure. Random Motion Fourier integral/fourier Transforms As the period of a periodic function approaches infinity, the frequency spectrum approaches being continuous, rather than discrete. At the same time the Fourier series approaches an integral that is referred as the Fourier integral. The Fourier transform pair consists of two integrals, one that transforms a time function to the frequency domain and one that a frequency function to time domain. If the frequencies in Eq. (7) have different values, the resulting signal is random. It does not repeat itself and has a random shape, as shown in figure. 28
x(t) t Random signal contains components with discrete frequencies. There are several methods to analyze such signals. The most common method is Fourier transform. If a random signal is represented by x(t) its Fourier transform pair is given by X(ω) = x( t) e iω t d t (18) x(t) = X ( ω) e i ω t d ω The absolute value of X(ω) is the amplitude of the component which has a frequency ω. Example 3 Find the Fourier transform for the rectangular signal shown in Figure. Solution x(t) The random function x(t) is given by A x (t) = A 0 t τ 29 0 τ t
x (t) = 0 t > τ Apply Eq. (11), then i A X (ω) = ω (e - i ω τ - 1) A i = (cos ω τ - i sin ω τ - 1) ω The absolute value of X (ω) is given by X( ω) = A ω (cos ωτ 1) 2 + sin 2 ωτ A = 2 2cosωτ ω = 2A ωτ sin ω 2 = A τ ωτ sin 2 ωτ 2 The frequency spectrum is shown in Figure. There is a major difference between the frequency spectrum of the repeated rectangular pulses of Example 2 and the rectangular pulse of Example 3. The first has distinct frequencies due to the periodic nature of the signal, while the second contains all the values of the frequencies. In other words, the spectrum of the first signal is formed from lines while the spectrum of the second is continuous. 30
Frequency Analyzers Spectrum Analyzers -! Digital frequency analyzers, or spectrum analyzers, at use at present to experimentally determine the Fourier coefficients of a complicated vibrations of machines, such analyzers have a fast Fourier transform (FFT) processor, which transforms digitally sampled time-domain into a finite number of frequency components (Fourier coefficients). -! FFT is a computer algorithm, which is fast and efficient scheme for computing a finite number of Fourier coefficients. -! Digital frequency analyzers can also synthesize the original time-domain signal from the frequency components. -! Digital frequency analyzers are also used as part of preventive-(and predictive) maintenance procedures for machines, for example, such procedure might involve the periodic check of a machine s vibration characteristics to determine if any significant changes have occurred in them because of bearing or gear wear, loose fasteners, fractures and so on. Impact Hammers Impact hammers are used to excite small structures and machines with an impulse, the width of the impulse, and the frequency range over which the amplitudes are essentially constant, depends upon the hardness of the hammer striker tip and upon the material and stiffness of the system to which the hammer is applied. Measuring Parameters A vibration signal is measured by means of several parameters, namely! Displacement: It is a measure for the amount of the distant traveled by the vibrating part. It is measured in micrometers (10-6 m). Displacement measurement is used in the cases where the vibration frequency is below 10 Hz.! Peak value: It represents the severity of the signal. It is equal to the amplitude "A".! Peak-to- peak value: The distance from the top of the signal to its bottom is the peak-to-peak value. It is equal to 2 A. 31
! Velocity: It is equal to ω A. It is used to measure the vibration severity over a wide range of frequencies from 10 Hz to 1000 Hz. It is measured in mm/s.! Acceleration: It is equal to ω 2 A. It is usually expressed in "g"; where g is the gravitational acceleration. The acceleration is used to measure vibrations with frequencies higher than 1000 Hz.! Spike Energy: This is a fairly abstract quantity that cannot be related to a picture of vibrating weight. The spike energy measurements include very short duration, high frequency, spike-like pulses of vibration that occur in machinery, for example, faulty rolling bearings and gears.! Average value: It is the average value of the rectified harmonic signal. It is given by x av = A π π 0 sinθ dθ 2A = = 0.637 A π! Root mean square value (rms): The square of the displacement is associated with the amount of energy. The root mean square value is a measure for the energy. It is obtained from the average of the squared harmonic signal. 1 π 2 A A dθ = = 0.707 A π 0 2 x rms = ( sinθ)! Decibel (db): It is a unit of measurement which is frequently used in sound measurement. It is defined in terms of the power ratio. db = 10 log 10 & x $ % x 1 2 #! " 2 & x1 = 20 log 10! # $ % x 2 " x 2 is a reference value depends on the measured parameter. 32
! Octave: An octave is a frequency band width from f 1 to f 2 such that & f2 # $ = 2 f!. Thus the band width from 2 Hz to 4 Hz is an octave. Also, the band % 1 " width from 100 Hz to 200 Hz is an octave, and so on. So, the octave is not a fixed band width but depends on the frequency range being considered. Modulation Modulation is the variation of one parameter of a signal by the action of another signal. A common type of modulation is amplitude modulation, where the amplitude of one signal (called the "carrier") is caused to fluctuate in response to a modulating signal. This is the way AM radio transmission works; a high-frequency wave called the carrier is caused to fluctuate in level in accordance with the voice or music signal being transmitted. The radio receiver picks up the modulated carrier and performs a demodulation to extract the audio signal. Frequency modulation is another type where the frequency of the carrier is varied rather than the amplitude. Modulation of a carrier causes new components to appear in the spectrum and they are called sidebands. The frequencies of the sidebands are equal to the carrier frequency plus and minus the modulating frequency. In rotating machinery there are many fault mechanisms which can cause amplitude and frequency modulation, and vibration analysis exposes the sidebands. Demodulation can be performed to detect the modulation frequencies directly. Amplitude Demodulation Amplitude modulation is defined as the multiplication of one time-domain signal by another time-domain signal. The signals may or may not be complex in nature, i.e., either or both signals may contain harmonics components. It is impossible to have amplitude modulation unless at least two different signals are involved. The signals may be electrical in nature, or they can be vibration signals. Modulation is inherently a non-linear process, and always gives rise to frequency components that did not exist in either of the two original signals 33
Amplitude Modulated Wave Form If the amplitude-modulated signal shown here is passed through a frequency analyzer, the following spectrum is the result. The highest peak is the carrier frequency. The right-hand peak is the upper sideband, and has a frequency of the carrier frequency plus the modulating frequency. The left-hand peak or lower sideband has a frequency of the carrier minus the modulating frequency. The sidebands are sometimes called sum and difference frequencies because of their symmetrical spacing around the carrier. Amplitude modulation also occurs in sound reproducing equipment, where it is called Intermodulation Distortion. The sum and difference frequencies are not in musical harmony with the tones that cause them, making intermodulation a particularly noticeable form of sound distortion. Spectrum of Modulated Wave Form Rectified Wave Form Recovered Modulating Signal This process of demodulation is exactly what happens in an AM radio -- the carrier is a very high frequency signal generated by the radio station, and the modulating signal is the voice or music that constitutes the program. The radio receives the modulated carrier, amplifies it, and rectifies ( detects ) it to recover the program. Modulation Effects Modulation is a non-linear effect in which several signals interact with one another to produce new signals with frequencies not present in the original signals. 34
Modulation effects are the bane of the audio engineer, for they produce "intermodulation distortion", which is annoying to the music listener. There are many forms of modulation, including frequency and amplitude modulation, and the subject is quite complex. Frequency modulation (FM) is the varying in frequency of one signal by the influence of another signal, usually of lower frequency. The frequency being modulated is called the "carrier". In the spectrum shown above, the largest component is the carrier, and the other components which look like harmonics, are called "sidebands". These sidebands are symmetrically located on either side of the carrier, and their spacing is equal to the modulating frequency. Frequency modulation occurs in machine vibration spectra, especially in gearboxes where the gear mesh frequency is modulated by the rpm of the gear. It also occurs in some sound system loudspeakers, where it is called FM distortion, although it is generally at a very low level. 35
This example shows amplitude modulation at about 50% of full modulation Notice that the frequency of the waveform seems to be constant and that it is fluctuating up and down in level at a constant rate. This test signal was produced by rapidly varying the gain control on a function generator while recording the signal. The spectrum has a peak at the frequency of the carrier, and two more components on each side. These extra components are the sidebands. Note that there are only two sidebands here compared to the great number produced by frequency modulation. The sidebands are spaced away from the carrier at the frequency of the modulating signal, in this case at the frequency at which the control knob was wiggled. In this example, the modulating frequency is much lower than the modulated or carrier frequency, but the two frequencies are often close together in practical situations. Also these frequencies are sine waves, but in practice, both the modulated and modulating signals are often complex. For instance, the transmitted signal from an AM radio station contains a high-frequency carrier, and many sidebands resulting from the carrier modulation by the voice or music signal being broadcast. Beats If two sounds, vibrations, or electrical signals have nearly the same frequency and they are linearly added together, their combined amplitude will fluctuate up and down at a rate equal to the difference frequency between them. This phenomenon is 36
called beating, and is very commonly seen in practice. For instance, a musician tunes his instrument by listening for beats between two tones that are nearly the same pitch. This waveform looks like amplitude modulation, but is actually just two sine wave signals added together to form beats. Because the signals are slightly different in frequency, their relative phase varies from zero to 360 degrees, and this means the combined amplitude varies due to reinforcement and partial cancellation. The spectrum shows the frequency and amplitude of each component, and there are no sidebands present. In this example, the amplitudes of the two beating signals are different, causing incomplete cancellation at the null points between the maxima. Beating is a linear process -- no additional frequency components are created. Electric motors often produce sound and vibration signatures that resemble beating, where the beat rate is at twice the slip frequency. This is not actually beating, but is in fact amplitude modulation of the vibration signature at twice the slip frequency. Probably it has been called beating because it sounds somewhat like the beats present in the sound of an out of tune musical instrument. The following example of beats shows the combined waveform when the two beating signals are the same amplitude. At first glance, this looks like 100% amplitude modulation, but close inspection of the minimum amplitude area shows that the phase is reversed at that point. 37
' This looks like 100% amplitude modulation! This example of beats is like the previous one, but the levels of the two signals are the same, and they cancel completely at the nulls. This complete cancellation is quite rare in actual signals encountered in rotating equipment. Earlier we learned that beats and amplitude modulation produce similar waveforms. This is true, but there is a subtle difference. These waveforms are enlarged for clarity. Note that in the case of beats, there is a phase change at the point where cancellation is complete. A beating waveform looks very much like amplitude modulation, but it is actually completely different. A spectrum analysis of beats produces only the two frequency 38
components that are combined -- there are no new frequencies such as sidebands present. It is easy to confuse beats with amplitude modulation, but a spectrum analysis will show the difference. In general, beats are benign, and do not imply faults in machines. For example, the sound of two similar machines running side by side at slightly different speeds will often produce audible beats. This is simply the sounds made by the machines combining in air to produce the amplitude fluctuations. 39
Sheet 2 Vibration Signals 40