Class 21-23 -Transmissibility Phenomenon

System Modeling Coursework

P.R. VENKATESWARANFaculty, Instrumentation and Control Engineering,

Manipal Institute of Technology, ManipalKarnataka 576 104 INDIAPh: 0820 2925154, 2925152

Fax: 0820 2571071Email: [email protected], [email protected]

Web address: http://www.esnips.com/web/SystemModelingClassNotes

Class 21-23: (i) Transmissibility of Vibrations

(ii) Vibration Isolation and control

(iii) Dynamic Vibration measurement

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http://www.esnips.com/web/SystemModelingClassNotes

July – December 2008 prv/System Modeling Coursework/MIT-Manipal 2

WARNING!

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I claim no originality in all these notes. These are the compilation from various sources for the purpose of delivering lectures. I humbly acknowledge the wonderful help provided by the original sources in this compilation.

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For best results, it is always suggested you read the source material.


Contents

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Transmissibility phenomenon–

Force excitation and motion excitation model

–

Damping ratio vs. Transmissibility•

Vibration isolation–

Source isolation

–

Isolation mounts–

Isolation pads

–

Inertia Block•

Vibration measurement


Transmissibility

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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

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one in which the motion of the supporting structure is transmitted to the machine


(i) Force Excitation


Transmissibility calculations

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The magnitude of this force in terms of frequency is given by:

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The oscillation magnitude of the frequency is:


Transmissibility ratio

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Substituting the relation for x(w) in the first equation, we get transmissibility ratio

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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.


(ii) Motion excitation model


Transmissibility for motion excitation


Important concepts:

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Isolators should be chosen so as not to excite the natural frequencies of the system

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damping is important in the range of resonance whether the dynamic system is operating near resonance or must pass through resonance during start-up;

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in the isolation region, the larger the ratio (i.e., the smaller the value of ), the smaller the transmissibility will be.


Damping ratio vs

Transmissibility

Figure 3 Design Curves for the Transmissibility vs. the Frequency ratio


Control Techniques (a) Source alteration

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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.

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However, it is also important to ensure that the application of these schemes does not produce other problems elsewhere.


(b) Isolation

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In general, vibration isolators can be broken down into three categories: (i) metal springs, (ii) elastometric

mounts, and (iii) resilient pads.•

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.

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Always design the isolators to protect against the lowest frequency that can be generated by the machine.

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Design the system so that its natural frequency will be less than one-third of the lowest forcing frequency present.


(i) Metal Springs

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Their use spans the spectrum from light, delicat

e instruments to very heavy industrial machinery.

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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.

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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 does not exist.


Eliminating disadvantages

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The damping lacked by springs can be obtained by placing dampers

in parallel with the springs.

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Rocking motions can be minimized by selecting springs in such a way that each spring used will deflect the same amount.

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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.

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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.


Design procedure for springs


Design procedure for the springs


Numerical No. 1

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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%.


Solution to Numerical No.1


Solution to Numerical No.1


Typical Load vs. Deflection Curve for an Elastomeric Mount

Figure 6!


(ii) Elastomeric mounts

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Elastomeric mounts consist primarily of natural rubber and synthetic rubber materials such as neoprene.

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In general, elastomeric mounts are used to isolate small electrical and mechanical devices from relatively high forcing frequencies.

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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.

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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.


(ii) Elastomeric mounts

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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.

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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.

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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.


Numerical No. 3

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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.


Solution to Numerical No. 3


(iii) Isolation pads

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The materials in this particular classification include such things as cork, felt, and fiberglass.

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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.

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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.

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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 ζ

= 0.05 to 0.06.


Numerical No. 4

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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.


Typical Natural Frequency vs. Static Load Curves

Figure 7


Solution to Numerical No. 4


(iv) Inertia blocks

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Isolated concrete inertia blocks play an important part in the control of vibration transmission.

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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.

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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


Examples for Inertia blocks


Inertia Blocks

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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.

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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.


Advantages of inertia blocks

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they lower the center of gravity and thus offer an added degree of stability;

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they increase the mass and thus decrease vibration amplitudes and minimize rocking;

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they minimize alignment errors because of the inherent stiffness of the base; and

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they act as a noise barrier between the floor on which they are mounted and the equipment that is mounted on them.

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One must always keep in mind, however, that to be effective, inertia blocks must be mounted on isolators


Vibration measurement


How to measure vibration

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Measurements should be made to produce the data needed to draw meaningful conclusions from the system under test.

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These data can be used to minimize or eliminate the vibration and thus the resultant noise.

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There are also examples where the noise is not the controlling parameter, but rather the quality of the product produced by the system.

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For example, in process control equipment, excessive vibration can damage the product, limit processing speeds, or even cause catastrophic machine failure.


Transducers

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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.


Velocity pickups

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The electrical output signal of a velocity pickup is proportional to the velocity of the vibrating mechanism.

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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 pick ups

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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.

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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.


Transducer mounting pickup

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


Final comments

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The velocity pickup is a useful transducer because it is sensitive and yet rugged enough to withstand extreme industrial environments.

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In addition, velocity is perhaps the most frequently employed measure of vibration severity.

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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.


Accelerometers

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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.

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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.

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In addition, the devices can be used easily with electronic integrating networks to obtain a voltage proportional to velocity or displacement.


Accelerometer mounting

Figure Mounting Technique for Eliminating Selected Measurement Errors


Accelerometer pick up

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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.

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An additional ground at the accelerometer will provide a closed (ground) loop which may induce a noise signal that affects the accelerometer output.

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The sealing compound applied at the cable entry into the accelerometer protects the system from errors caused by moisture.


Preamplifiers

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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.


Preamplifiers

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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.

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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.

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When a voltage amplifier is used however, the system is very sensitive to changes in cable capacitance.

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Voltage amplifiers, on the other hand, are often less expensive and more reliable because they contain fewer components and thus are

easier to construct.


Processing and data equipment

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The instruments used for the processing and display of vibration data are, with minor modifications, the same as those described earlier for noise analyses.

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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.

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They normally come equipped with some form of graphical display, such as a cathode ray tube, which provides detailed frequency data.


And, before we break…

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“It is good to have money and the things money can buy, but it is good too, to check up once in a while and make sure you haven’t lost the things money can’t buy”–

George Horace Lorimer

Thanks for listening…

Class 21-23 -Transmissibility Phenomenon

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