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.
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 :
The oscillation magnitude as a function of frequency is :(3)Substituting equation (3) into (2) :
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 :
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.
(a) Transmissibility vs. damping ratioThe 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 :
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 frequencyThe 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.
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 TechniquesIn 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 mountsElastomeric 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.
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.
(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.
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 Curvesfor 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 isolatorsConsider 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 :
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.
Source : http://personal.cityu.edu.hkVibration MeasurementsMeasurements 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
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 PickupsThe 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) AccelerometersThe 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.
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.
Practical Methods for Vibration Control of Industrial Equipment, can download in here.
Reference standards for Vibration Monitoring and Analysis, here.