APKJ

VIBRATION CONTROL AND MEASUREMENT

Transmissibility

Assuming that the forcing function is harmonic in nature, we shall consider two cases of vibration transmission - one in which force is transmitted to the supporting structure, and one in which the motion of the supporting structure is transmitted to the machine.

            (a) Force excitation

Consider the system shown in Figure 1, where f(t) is the harmonic force acting on the system and fT(t) is the force transmitted to the supporting structure or base. The force transmitted through the spring and damper to the supporting structure is :

(1)

Figure 1 Force Excitation Model

The magnitude of this force as a function of frequency is :

(2)

The oscillation magnitude as a function of frequency is :

(3)

Substituting equation (3) into (2) :

(4)

T is defined as the transmissibility and represents the ratio of the amplitude of the force transmitted to the supporting structure to that of the exciting force.

(b) Motion excitation

The system that illustrates motion excitation is shown in Figure 2. The motion of the dynamic system is represented by the variable x and the harmonic displacement of the supporting base is represented by the variable y. The equation that describes the dynamics of the system is :

(5)

Figure 2 Motion Excitation Model

Then the ratio of the magnitudes of the displacements as a function of frequency, which is the transmissibility, is given by the expression

(6)

Note that the transmissibility expressions for both force and motion excitation are identical. Therefore, it would appear that the engineering principles employed to protect the supporting structure under force excitation are the same as those used to protect the dynamic system from motion excitation.

 

Design Curves

(a) Transmissibility vs. damping ratio

The curve in Figure 3 demonstrates the effectiveness of an isolator to reduce vibration. Figure 3 also indicates a number of important concepts: (i) isolators should be chosen so as not to excite the natural frequencies of the system; (ii) damping is important in the range of resonance whether the dynamic system is operating near resonance or must pass through resonance during start-up; (iii) in the isolation region, the larger the ratio (i.e., the smaller the value of ), the smaller the transmissibility will be.

(b) Isolation efficiency vs. w and

Another graphical method of illustrating the regions of isolation and amplitude as a function of the disturbing frequency and the natural frequency of the system is shown in Figure 4. In using this figure we must note that percent isolation is defined by the expression :

(7)

Figure 3 Design Curves for the Transmissibility vs. the Frequency ratio as a Function of the Damping Ratio z for a Linear Single-Degree-of-Freedom System

The forcing frequency on the ordinate and the percent isolation lines in the graph locate a point, the abscissa of which is the natural frequency of the system necessary to achieve the required isolation. The system parameters may then be selected or adjusted to obtain this desired natural frequency.

(c) Static deflection vs. natural frequency

The static deflection is the deflection of an isolator that occurs due to the dead weight load of the mounted equipment. Since the static deflection is given by the expression , and since the undamped natural frequency of a single-degree-of-freedom system is determined by the equation :

(8)

where is in centimeters.

Figure 4 Design Curves for Isolation Efficiency vs. Frequency

(Damping Ratio, z = 0)

The graphical presentation of this equation is given in Figure 5. Thus, we can determine the natural frequency of a system by measuring the static deflection. This statement is correct provided that the spring is linear and that the isolator material possesses the same type of elasticity under both static and dynamic conditions. As mentioned previously, however, we are assuming a single-degree-of-freedom linear system throughout our analysis, and thus all the design curves presented above are applicable.

Figure 5 Design Curves for the Static Deflection vs. Natural Frequency for a Linear Single-Degree-of-Freedom System

The examples that follow demonstrate the use of these design curves.

EXAMPLE 1

A pump in an industrial plant is mounted rigidly to a massive base plate. The base plate rests on four springs, one at each corner. If the static deflection of each spring is two centimeters, then the natural frequency of the system is given by :

 

Control Techniques

In the control of noise we basically considered three areas: the source, the path, and the receiver. Vibration control may involve one or a combination of the following techniques.

(a) Source alteration

In the control of vibration it is important to first check and see if the noise or vibration level can be reduced by altering the source. This may be accomplished by making the source more rigid from a structural standpoint, changing certain parts, balancing, or improving dimensional tolerances. The system mass and stiffness may be adjusted in such a way so that resonant frequencies of the system do not coincide with the forcing frequency. This process is called detuning. Sometimes it is also possible to reduce the number of coupled resonators that exist between the vibration source and the receiver of interest. This technique is called decoupling. Although these techniques can be applied during design or construction, they are perhaps more often used as a correction scheme. However, it is also important to ensure that the application of these schemes does not produce other problems elsewhere.

(b) Isolation

In general, vibration isolators can be broken down into three categories: (i) metal springs, (ii) elastomeric mounts, and (iii) resilient pads. Before examining each of these areas, a few general comments can be made which are pertinent to all categories. We must always remember that we are assuming a single-degree-of-freedom system, and therefore our analysis will not be exact in every case. However, practical systems are normally reduced to this model because it is the only one that we understand thoroughly.

When building or correcting a design, always ensure that the machine under investigation and the element that drives it both rest on a common base. Always design the isolators to protect against the lowest frequency that can be generated by the machine. Design the system so that its natural frequency will be less than one-third of the lowest forcing frequency present. The isolation device should also reduce the transmissibility at every frequency contained in the Fourier spectrum of the forcing function.

(i) Metal springs

Metal springs are widely used in industry for vibration isolation. Their use spans the spectrum from light, delicate instruments to very heavy industrial machinery. The advantages of metal springs are: (a) they are resistant to environmental factors such as temperature, corrosion, solvents, and the like; (b) they do not drift or creep; (c) they permit maximum deflection; and (d) they are good for low-frequency isolation. The disadvantages of springs are (a) they possess almost no damping and hence the transmissibility at resonance can be very high; (b) springs act like a short circuit for high-frequency vibration; and (c) care must be taken to ensure that a rocking motion doe not exist.

Careful engineering design will minimize the effect of some of these disadvantages. For example, the damping lacked by springs can be obtained by placing dampers in parallel with the springs. Rocking motions can be minimized by selecting springs in such a way that each spring used will deflect the same amount. In addition, the use of an inertia block that weighs from one to two times the amount of the supported machinery minimizes rocking lowers the center of gravity of the system, and helps to uniformly distribute the load. High-frequency transmission through springs caused by the low damping ratio can be blocked by using rubber pads in series with the springs. A typical damping ratio for steel springs is 0.005.

The design procedure for selecting springs for vibration isolation is outlined below:

EXAMPLE 2

A machine set operating at 2400 rpm is mounted on an inertia block. The total system weighs 907 N. The weight is essentially evenly distributed. We want to select four steel springs upon which to mount the machine. The isolation required is 90%.

(ii) Elastomeric mounts

Elastomeric mounts consist primarily of natural rubber and synthetic rubber materials such as neoprene. In general, elastomeric mounts are used to isolate small electrical and mechanical devices from relatively high forcing frequencies. They are also useful in the protection of delicate electronic equipment. In a controlled environment, natural rubber is perhaps the best and most economical isolator. Natural rubber contains inherent damping, which is very useful if the machine operates near resonance or passes through resonance during "startup" or "shutdown." Synthetic rubber is more desirable when the environment is somewhat hazardous.

Rubber can be used in either tension, compression or shear; however, it is normally used in compression or shear and rarely used in tension. In compression it possesses the capacity for high-energy storage; however, its useful life is longer when used in shear. Rubber is classified by a durometer number. Rubber employed in isolation mounts normally ranges from 30-durometer rubber, which is soft, to 80-durometer rubber, which is hard. The typical damping ratio for natural rubber and neoprene is z = 0.05.

One word of caution when dealing with rubber: it possesses different characteristics depending upon whether the material is used in strips or bulk, and whether it is used under static or dynamic conditions. The steps for selecting an elastomeric mount are essentially those enumerated in the previous section on metal springs. The following examples will illustrate the procedure.

EXAMPLE 3

A drum weighing 120 N and operating at 3600 rpm induces vibration in adjacent equipment. Four vertical mounting points support the drum. Choose one of the isolators shown in Figure 6 so as to achieve 90°/ vibration isolation.

Figure 6 Typical Load vs. Deflection Curve for an Elastomeric Mount

(iii) Isolation pads

The materials in this particular classification include such things as cork, felt, and fiberglass. In general, these items are easy to use and install. They are purchased in sheets and cut to fit the particular application, and can be stacked to produce varying degrees of isolation. Cork, for example, can be obtained in squares (like floor tile) 1 to 2.5 cm in thickness or in slabs up to 15 cm thick for large deflection applications. Cork is very resistive to corrosion and solvents and is relatively insensitive to a wide range of temperatures. Some of the felt pads are constructed of organic material and hence should not be employed in an industrial environment where solvents are used. Fiberglass pads, on the other hand, are very resistant to industrial solvents. A typical damping ratio for felt and cork is z = 0.05 to 0.06.

EXAMPLE 4

A large machine is mounted on a concrete slab. The lowest expected forcing frequency is 60 Hz. If the isolator will be loaded at 7 N/cm2, choose the proper fiberglass isolator from the manufacturer's data shown in Figure 7 to produce 80% isolation. Assume that the damping ratio of the material is z = 0.05.

Figure 7 Typical Natural Frequency vs. Static Load Curves

for Fiberglass

(iv) Inertia blocks

Isolated concrete inertia blocks play an important part in the control of vibration transmission. Large-inertia forces at low frequencies caused by equipment such as reciprocating compressors may cause motion that is unacceptable for proper machine operation and transmit large forces to the supporting structure. One method of limiting motion is to mount the equipment on an inertia base. This heavy concrete or steel mass limits motion by overcoming the inertia forces generated by the mounted equipment.

Low natural frequency isolation requires a large deflection isolator such as a soft spring. However, the use of soft springs to control vibration can lead to rocking motions which are unacceptable. Hence, an inertia block mounted on the proper isolators can be effectively used to limit the motion and provide the needed isolation.

Inertia blocks are also useful in applications where a system composed of a number of pieces of equipment must be continuously supported. An example of such equipment is a system employing calibrated optics.

Thus, inertia blocks are important because they lower the center of gravity and thus offer an added degree of stability; they increase the mass and thus decrease vibration amplitudes and minimize rocking; they minimize alignment errors because of the inherent stiffness of the base; and they act as a noise barrier between the floor on which they are mounted and the equipment that is mounted on them. One must always keep in mind, however, that to be effective, inertia blocks must be mounted on isolators

Consider the system shown in Figure 8. The equations of motion that describe the systems are :

(9)

Figure 8 Model for the Analysis of Vibration Absorber

The magnitude of the frequency response is obtained from the following equations :

(10)

Now note what happen to the equations above if the forcing frequency w is equal to the natural frequency of the vibration absorber (i.e. ). Under this condition :

(11)

Therefore, the motion of the main mass is ideally zero and the spring force of the absorber is at all times equal and opposite to the applied force, . Hence no force is transmitted to the supporting structure.

 

Vibration Measurements

Measurements should be made to produce the data needed to draw meaningful conclusions from the system under test. These data can be used to minimize or eliminate the vibration and thus the resultant noise. There are also examples where the noise is not the controlling parameter, but rather the quality of the product produced by the system. For example, in process control equipment, excessive vibration can damage the product, limit pro-cessing speeds, or even cause catastrophic machine failure. The basic measurement system used for diagnostic analyses of vibrations consists of the three system components shown in Figure 9.

Figure 9 Basic Vibration Measurement System

(i) Transducers

In general, the transducers employed in vibration analyses convert mechanical energy into electrical energy; that is, they produce an electrical signal which is a function of mechanical vibration. In the following section, both velocity pickups and accelerometers mounted or attached to the vibrating surface will be studied.

(a) Velocity Pickups

The electrical output signal of a velocity pickup is proportional to the velocity of the vibrating mechanism. Since the velocity of a vibrating mechanism is cyclic in nature, the sensitivity of the pickup is expressed in peak milli-volts/cm/s and thus is a measure of the voltage produced at the point of maximum velocity. The devices have very low natural frequencies and are designed to measure vibration frequencies that are greater than the natural frequency of the pickup.

Velocity pickups can be mounted in a number of ways; for example, they can be stud-mounted or held magnetically to the vibrating surface. However, the mounting technique can vastly affect the pickup's performance. For example, the stud-mounting technique shown in Figure 10(a), in which the pickup is mounted flush with the surface and silicone grease is applied to the contact surfaces, is a good reliable method. The magnetically mounted pick-up, as shown in Figure 10(b), on the other hand, in general has a smaller usable frequency range than the stud-mounted pickup. In addition, it is important to note that the magnetic mount, which has both mass and spring like properties, is located between the velocity pickup and the vibrating surface and thus will affect the measurements. This mounting technique is viable, but caution must be employed when it is used.

Figure 10 Two Transducer Mounting Technique

(a) Stud-Mount Pickup; (b) Magnetically Held Velocity Pickup

The velocity pickup is a useful transducer because it is sensitive and yet rugged enough to withstand extreme industrial environments. In addition, velocity is perhaps the most frequently employed measure of vibration severity. However, the device is relatively large and bulky, is adversely affected by magnetic fields generated by large ac machines or ac current carrying cables, and has somewhat limited amplitude and frequency characteristic.

(b) Accelerometers

The accelerometer generates an output signal that is proportional to the acceleration of the vibrating mechanism. This device is, perhaps, preferred over the velocity pickup, for a number of reasons. For example, accelerometers have good sensitivity characteristics and a wide useful frequency range; they are small in size and light in weight and thus are capable of measuring the vibration at a specific point without, in general, loading the vibrating structure. In addition, the devices can be used easily with electronic integrating networks to obtain a voltage proportional to velocity or displacement. However, the accelerometer mounting, the interconnection cable, and the instrumentation connections are critical factors in measurements employing an accelerometer. The general comments made earlier concerning the mounting of a velocity pickup also apply to accelerometers.

Some additional suggestions for eliminating measurement errors when employing accelerometers for vibration measurements are shown in Figure 11. Note that the accelerometer mounting employs an isolation stud and an isolation washer. This is done so that the measurement system can be grounded at only one point, preferably at the analyzer. An additional ground at the accelerometer will provide a closed (ground) loop which may induce a noise signal that affects the accelerometer output. The sealing compound applied at the cable entry into the accelerometer protects the system from errors caused by moisture.

Figure 11 Mounting Technique for Eliminating Selected Measurement Errors

The cable itself should be glued or strapped to the vibrating mechanism immediately upon leaving the accelerometer, and the other end of the cable, which is connected to the preamplifier, should leave the mechanism under test at a point of minimum vibration. This procedure will eliminate or at least minimize cable noise caused by dynamic bending, compression, or tension in the cable.

(ii) Preamplifiers

The second element in the vibration measurement system is the preamplifier. This device, which may consist of one or more stages, serves two very useful purposes: it amplifies the vibration pickup signal, which is in general very weak, and it acts as an impedance transformer or isolation device between the vibration pickup and the processing and display equipment.

Recall that the manufacturer provides both charge and voltage sensitivities for accelerometers. Likewise, the preamplifier may be designed as a voltage amplifier in which the output voltage is proportional to the input voltage, or a charge amplifier in which the output voltage is proportional to the input charge. The difference between these two types of preamplifiers is important for a number of reasons. For example, changes in cable length (i.e., cable capacitance) between the accelerometer and preamplifier are negligible when a charge amplifier is employed. When a voltage amplifier is used however, the system is very sensitive to changes in cable capacitance. In addition, because the input resistance of a voltage amplifier cannot in general be neglected, the very low frequency response of the system may be affected. Voltage amplifiers, on the other hand, are often less expensive and more reliable because they contain fewer components and thus are easier to construct.

(iii) Processing and display equipment

The instruments used for the processing and display of vibration data are, with minor modifications, the same as those described earlier for noise analyses. The processing equipment is typically some type of spectrum analyzer. The analyzer may range from a very simple device which yields, for example, the rms value of the vibration displacement, to one that yields an essentially instantaneous analysis of the entire vibration frequency spectrum. As discussed earlier, these analyzers, which are perhaps the most valuable tool in a vibration study, are typically either a constant-bandwidth or constant-percentage-bandwidth type of device. They normally come equipped with some form of graphical display, such as a cathode ray tube, which provides detailed frequency data.

  Source : http://personal.cityu.edu.hk
Practical Methods for Vibration Control of Industrial Equipment, can download in here.   
Reference standards for Vibration Monitoring and Analysis, here.

THEORY OF VIBRATION

Introduction

There are two general classes of vibrations - free and forced. Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution.

Vibration that takes place under the excitation of external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of major structures such as bridges, buildings, or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies of major importance in the study of vibrations.

Vibrating systems are all subject to damping to some degree because energy is dissipated by friction and other resistances. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculation for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance.

The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. Thus, a free particle undergoing general motion in space will have three degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three components of position and three angles defining its orientation. Furthermore, a continuous elastic body will require an infinite number of coordinates (three for each point on the body) to describe its motion; hence, its degrees of freedom must be infinite. However, in many cases, parts of such bodies may be assumed to be rigid, and the system may be considered to be dynamically equivalent to one having finite degrees of freedom. In fact, a surprisingly large number of vibration problems can be treated with sufficient accuracy by reducing the system to one having a few degrees of freedom.

 

Harmonic Motion

Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal, ,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship .

Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w , the displacement x can be written as

(1)

Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle

The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship

(2)

where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively.

The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain

(3)

(4)

 

Vibration Model

The basic vibration model of a simple oscillatory system consists of a mass, a massless spring, and a damper. The spring supporting the mass is assumed to be of negligible mass. Its force-deflection relationship is considered to be linear, following Hooke's law, , where the stiffness k is measured in newtons/meter.

The viscous damping, generally represented by a dashpot, is described by a force proportional to the velocity, or . The damping coefficient c is measured in newtons/meter/second.

Figure 2 Spring-Mass System and Free-Body Diagram

 

Equation of Motion : Natural Frequency

Figure 2 shows a simple undamped spring-mass system, which is assumed to move only along the vertical direction. It has one degree of freedom (DOF), because its motion is described by a single coordinate x.

When placed into motion, oscillation will take place at the natural frequency fn which is a property of the system. We now examine some of the basic concepts associated with the free vibration of systems with one degree of freedom.

Figure 2 Spring-Mass System and Free-Body Diagram

Newton's second law is the first basis for examining the motion of the system. As shown in Fig. 2 the deformation of the spring in the static equilibrium position is D , and the spring force kD is equal to the gravitational force w acting on mass m

(5)

By measuring the displacement x from the static equilibrium position, the forces acting on m are and w. With x chosen to be positive in the downward direction, all quantities - force, velocity, and acceleration are also positive in the downward direction.

We now apply Newton's second law of motion to the mass m :

and because kD = w, we obtain :

(6)

It is evident that the choice of the static equilibrium position as reference for x has eliminated w, the force due to gravity, and the static spring force kD from the equation of motion, and the resultant force on m is simply the spring force due to the displacement x.

By defining the circular frequency w n by the equation

(7)

Eq. 6 can be written as

(8)

and we conclude that the motion is harmonic. Equation (8), a homogeneous second order linear differential equation, has the following general solution :

(9)

where A and B are the two necessary constants. These constants are evaluated from initial conditions , and Eq. (9) can be shown to reduce to

(10)

The natural period of the oscillation is established from , or

(11)

and the natural frequency is

(12)

These quantities can be expressed in terms of the static deflection D by observing Eq. (5), . Thus, Eq. (12) can be expressed in terms of the static deflection D as

(13)

Note that , depend only on the mass and stiffness of the system, which are properties of the system.

 

Viscously Damped Free Vibration

Viscous damping force is expressed by the equation

(14)

where c is a constant of proportionality.

Symbolically. it is designated by a dashpot, as shown in Fig. 3. From the free body diagram, the equation of motion is .seen to be

(15)

The solution of this equation has two parts. If F(t) = 0, we have the homogeneous differential equation whose solution corresponds physically to that of free-damped vibration. With F(t) ¹ 0, we obtain the particular solution that is due to the excitation irrespective of the homogeneous solution. We will first examine the homogeneous equation that will give us some understanding of the role of damping.

Figure 3 Viscously Damped Free Vibration

With the homogeneous equation :

(16)

the traditional approach is to assume a solution of the form :

(17)

where s is a constant. Upon substitution into the differential equation, we obtain :

which is satisfied for all values of t when

(18)

Equation (18), which is known as the characteristic equation, has two roots :

(19)

Hence, the general solution is given by the equation:

(20)

where A and B are constants to be evaluated from the initial conditions and .

Equation (19) substituted into (20) gives :

(21)

The first term, , is simply an exponentially decaying function of time. The behavior of the terms in the parentheses, however, depends on whether the numerical value within the radical is positive, zero, or negative.

When the damping term (c/2m)2 is larger than k/m, the exponents in the previous equation are real numbers and no oscillations are possible. We refer to this case as overdamped.

When the damping term (c/2m)2 is less than k/m, the exponent becomes an imaginary number, . Because

the terms of Eq. (21) within the parentheses are oscillatory. We refer to this case as underdamped.

In the limiting case between the oscillatory and non oscillatory motion , and the radical is zero. The damping corresponding to this case is called critical damping, cc.

(22)

Any damping can then be expressed in terms of the critical damping by a non dimensional number z , called the damping ratio:

(23)

and

(24)

(i) Oscillatory Motion (z < 1.0) Underdamped Case :

(25)

The frequency of damped oscillation is equal to :

(26)

Figure 4 shows the general nature of the oscillatory motion.

Figure 4 Damped Oscillation z < 1

(ii) Non oscillatory Motion (z > 1.0) Overdamped Case :

(27)

The motion is an exponentially decreasing function of time as shown in Fig. 5.

Figure 5 Aperiodic Motion z > 1

(iii) Critically Damped Motion (z = 1.0) :

(28)

Figure 6 shows three types of response with initial displacement x(0).

Figure 6 Critically Damped Motion z = 1

 

Forced Harmonic Vibration

Harmonic excitation is often encountered in engineering systems. It is commonly produced by the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system.

We will first consider a single DOF system with viscous damping, excited by a harmonic force , as shown in Fig. 7. Its differential equation of motion is found from the free-body diagram.

(29)

Figure 7 Viscously Damped System with Harmonic Excitation

The solution to this equation consists of two parts, the complementary function, which is the solution of the homogeneous equation, and the particular integral. The complementary function. in this case, is a damped free vibration.

The particular solution to the preceding equation is a steady-state oscillation of the same frequency w as that of the excitation. We can assume the particular solution to be of the form :

(30)

where X is the amplitude of oscillation and f is the phase of the displacement with respect to the exciting force.

The amplitude and phase in the previous equation are found by substituting Eqn. (30) into the differential equation (29). Remembering that in harmonic motion the phases of the velocity and acceleration are ahead of the displacement by 90° and 180°, respectively, the terms of the differential equation can also be displayed graphically, as in Fig. 8.

Figure 8 Vector Relationship for Forced Vibration with Damping

It is easily seen from this diagram that

(31)

and

(32)

We now express Eqs (31) and (32) in non-dimensional term that enables a concise graphical presentation of these results. Dividing the numerator and denominator of Eqs. (31) and (32) by k, we obtain :

(33)

and

(34)

These equations can be further expressed in terms of the following quantities:

The non-dimensional expressions for the amplitude and phase then become

(35)

and

(36)

These equations indicate that the non dimensional amplitude , and the phase f are functions only of the frequency ratio , and the damping factor z and can be plotted as shown in Fig 9.

Figure 9 Plot of Eqs. (35) and (36)

Rotating Unbalance

Unbalance in rotating machines is a common source of vibration excitation. We consider here a spring-mass system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig. 10. The unbalance is represented by an eccentric mass m with eccentricity e that is rotating with angular velocity w . By letting x be the displacement of the non rotating mass (M - m) from the static equilibrium position, the displacement of m is :

Figure 10 Harmonic Disturbing Force Resulting from Rotating Unbalance

The equation of motion is then :

which can be rearranged to :

(37)

It is evident that this equation is identical to Eq. (29), where is replaced by , and hence the steady-state solution of the previous section can be replaced by :

(38)

and

(39)

These can be further reduced to non dimensional form :

(40)

and

(41)

Example

A counter rotating eccentric weight exciter is used to produce the forced oscillation of a spring-supported mass as shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of 0.60 cm was recorded. When the speed of rotation was increase considerably beyond the resonant frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the damping factor of the system.

Figure 11

Solution :

From Eqn. (40), the resonant amplitude is :

When w is very much greater than , the same equation becomes

By solving the two equations simultaneously, the damping factor of the system is

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