Vibrations Fundamentals and Practice ch12 Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.
de Silva, Clarence W “Vibration Design and Control” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 12 Vibration Design and Control It has been pointed out that there are desirable and undesirable types and situations of mechanical vibration This chapter discusses ways of either eliminating or reducing the undesirable effects of vibration Undesirable vibrations are those that cause human discomfort and hazards, structural degradation and failure, performance deterioration and malfunction of machinery and processes, and various other problems General approaches to vibration mitigation can be identified from the dynamic systems point of view Consider the schematic diagram of a vibratory system shown in Figure 12.1 Forcing excitations f(t) to the mechanical system S cause the vibration responses y The objective here is to suppress y to a level that is acceptable Clearly, there are three general ways of doing this Isolation: Suppress the excitations of vibration This method primarily deals with f Design modification: Modify or redesign the mechanical system so that, for the same levels of excitation, the resulting vibrations are acceptable This method deals with S Control: Absorb or dissipate the vibrations, using external devices, through implicit or explicit sensing and control This method primarily deals with y Within each of these three categories, several approaches can be used to achieve the objective of vibration mitigation Essentially, all these approaches involve designing (either complete redesign or incremental design modification) of the system on the one hand, and controlling the vibration through external means (passive or active devices) on the other The analytical basis for many such approaches was presented in previous chapters Further analytical procedures will be given in the present chapter Note that removal of faults (e.g., misalignments and malfunctions by repair or parts replacement) can also remove vibrations This may fall into any of the three categories listed above, but primarily into the second category of modifying S The category of vibration isolation involves “isolating” a mechanical system (S) from vibration excitations (f) so that the excitation signals are “filtered” out or dissipated prior to reaching the system The use of properly designed suspension systems, mounts, and damping layers falls within this category The category of design modification involves making changes to the components and the structure of a mechanical system according to a set of specifications and design guidelines Balancing of rotating machinery, and structural modification through modal analysis and design techniques, fall into this category The category of control involves either passive devices (which not use external power) such as dynamic absorbers and dampers, or active control devices (which need external power for operation) In the passive case, the control device implicitly senses the vibration response and dissipates it (as in the case of a damper), or absorbs and stores its energy where it is slowly dissipated (as in the case of a dynamic absorber) In the active case, the vibrations y are explicitly sensed through sensors and transducers; what forces should be acted on the system to counteract and suppress vibrations are determined by a controller; and the corresponding forces or torques are applied to the system through one or more actuators Note that there may be some overlap in the three general categories of vibration mitigation mentioned above For example, the addition of a mount (category 1) can also be interpreted as a design modification (category 2) or as incorporating a passive damper (category 3) It should be noted as well that the general approach commonly known as that source alteration may fall into either category or category The purpose in this case is to alter or remove the source of vibration The source could be either external (e.g., road irregularities that result in vehicle vibrations), a category problem; or internal (imbalance or misalignment in rotating devices that results in periodic ©2000 CRC Press FIGURE 12.1 A vibrating mechanical system forces, moments, and vibrations), a category problem It can be more difficult for a system user to alter external vibration sources (e.g., resurfacing the roadways) than to modify the internal sources (e.g., balancing of rotating machinery) Furthermore, the external source of vibration can be quite random and also may not be accessible at all for alteration (e.g., aerodynamic forces on an aircraft) The present chapter will address some useful topics on the design for vibration suppression and the control of vibration Typically, a set of vibration specifications is given as simple threshold values (bounds) or frequency spectra, and the goal is to either design or control the system so as to meet these specifications SHOCK AND VIBRATION Sometimes, response to shock loads are considered separately from response to vibration excitations for the purpose of design and control of mechanical systems For example, shock isolation and vibration isolation are treated under different headings in some literature This is actually not necessary Although vibration analysis predominantly involves periodic excitations and responses, transient and random oscillations (vibrations) are also commonly found in practice The frequency band of the latter two types of signals is much broader than that of a simple periodic signal A shock signal is transient by definition, and has a very short duration (in comparison to the predominant time constants of the mechanical system to which the shock load is applied) Hence, it will possess a wide band of frequencies Consequently, frequency-domain techniques are still applicable Time-domain techniques are particularly suited to dealing with transient signals in general, and shock signals in particular In that context, a shock excitation can be treated as an impulse whose effect is to instantaneously change the velocity of an inertia element Then, in the time domain, a shock load can also be treated as an initial-velocity excitation of an otherwise free (unforced) system 12.1 SPECIFICATION OF VIBRATION LIMITS Design and control procedures of vibration have the primary objective of ensuring that, under normal operating conditions, the system of interest does not encounter vibration levels that exceed the specified values In this context, then, the ways of specifying vibration limits become important This section will present some common ways of vibration specification ©2000 CRC Press 12.1.1 PEAK LEVEL SPECIFICATION Vibration limits for a mechanical system can be specified either in the time domain or in the frequency domain In the time domain, the simplest specification is the peak level of vibration (typically acceleration in units of g, the acceleration due to gravity) Then, the techniques of isolation, design, or control should ensure that the peak vibration response of the system does not exceed the specified level In this case, the entire time interval of operation of the system is monitored and the peak values are checked against the specifications Note that in this case, it is the instantaneous peak value at a particular time instant that is of interest, and what is used in representing vibration is an instantaneous amplitude measure rather than an average amplitude or an energy measure 12.1.2 RMS VALUE SPECIFICATION The root-mean-square (rms) value of a vibration signal y(t) is given by the square root of the average (mean value) of the squared signal: yrms 1 = T T ∫ 2 y dt (12.1) Note that by squaring the signal, its sign is eliminated and essentially the energy level of the signal is used The period T over which the squared signal is averaged will depend on the problem and the nature of the signal For a periodic signal, one period is adequate for averaging For transient signals, several time constants (typically four times the largest time constant) of the vibrating system will be sufficient For random signals, a value that is as large as feasible should be used In the method of rms value specification, the rms value of the acceleration response (typically, acceleration in gs) is computed using equation (12.1) and is then compared with the specified value In this method, instantaneous bursts of vibration not have a significant effect because they are filtered out as a result of the integration It is the average energy or power of the response signal that is considered The duration of exposure enters into the picture indirectly and in an undesirable manner For example, a highly transient vibration signal can have a damaging effect in the beginning; but the larger the T that is used in equation (12.1), the smaller the computed rms value Hence, the use of a large value for T in this case would lead to diluting or masking the damage potential In practice, the longer the exposure to a vibration signal, the greater the harm caused by it Hence, when using specifications such as peak and rms values, they have to be adjusted according to the period of exposure Specifically, larger levels of specification should be used for longer periods of exposure 12.1.3 FREQUENCY-DOMAIN SPECIFICATION It is not quite realistic to specify the limitation to vibration exposure of a complex dynamic system by just a single threshold value Usually, the effect of vibration on a system depends on at least the following three parameters of vibration: Level of vibration (peak, rms, power, etc.) Frequency content (range) of excitation Duration of exposure to vibration This is particularly true because the excitations that generate the vibration environment may not necessarily be a single-frequency (sinusoidal) signal and may be broad-band and random; and ©2000 CRC Press FIGURE 12.2 Operating vibration specification (nomograph) for a machine furthermore, the response of the system to the vibration excitations will depend on its frequencytransfer function, which determines its resonances and damping characteristics Under these circumstances, it is desirable to provide specifications in a nomograph, where the horizontal axis gives frequency (Hz) and the vertical axis could represent a motion variable such as displacement (m), velocity (m·s–1), or acceleration (m·s–2 or g) It is not very important which of these motion variables represents the vertical axis of the nomograph This is true because, in the frequency domain, Velocity = jω × Displacement Acceleration = jω × Velocity and one form of motion can be easily converted into one of the remaining two motion representations In each of the forms, assuming that the two axes of the nomograph are graduated in a logarithmic scale, the constant displacement, constant velocity, and constant acceleration lines are straight lines Consider a simple specification of machinery vibration limits as given by the following values: Displacement limit (peak) = 0.001 m Velocity limit = 0.01 m ⋅ s −1 Acceleration limit = 1.0 g This specification can be represented in a velocity vs frequency nomograph (log–log) as in Figure 12.2 Usually, such simple specifications in the frequency domain are not adequate As noted before, the system behavior will vary, depending on the excitation frequency range For example, motion sickness in humans might be predominant in low frequencies in the range of 0.1 Hz to 0.6 Hz, and passenger discomfort in ground transit vehicles might be most serious in the frequency range of ©2000 CRC Press FIGURE 12.3 A severe-discomfort vibration specification for ground transit vehicles Hz to Hz for vertical motion and Hz to Hz for lateral motion Also, for any dynamic system, particularly at low damping levels, the neighborhoods of resonant frequencies should be avoided and, hence, should be specified by low vibration limits in the resonant regions Furthermore, the duration of vibration exposure should be explicitly accounted for in specifications For example, Figure 12.3 presents a ride comfort specification for a ground transit vehicle, where lower vibration levels are specified for longer trips Before leaving this section, it should be noted that the specifications of concern in the present context of design and control are upper bounds of vibration The system should perform below (within) these specifications under normal operating conditions Test specifications, as discussed in Chapter 10, are lower bounds The test should be conducted at or above these vibration levels so that the system will meet the test specifications Some considerations of vibration engineering are summarized in Box 12.1 12.2 VIBRATION ISOLATION The purpose of vibration isolation is to “isolate” the system of interest from vibration excitations by introducing an isolator in between them Examples of isolators are machine mounts and vehicle suspension systems Two general types of isolation can be identified: Force isolation (related to force transmissibility) Motion isolation (related to motion transmissibility) ©2000 CRC Press BOX 12.1 Vibration Engineering Vibration Mitigation Approaches: • Isolation (buffers system from excitation) • Design modification (modifies the system) • Control (senses vibration and applies a counteracting force: passive/active) Vibration Specification: • Peak and rms values • Frequency-domain specs on a nomograph —Vibration levels —Frequency content —Exposure duration Note: Velocity = ω × Displacement Acceleration = ω × Velocity Limiting Specifications: • Operation (design) specs: Specify upper bounds • Testing specs: Specify lower bounds In force isolation, vibration forces that would be ordinarily transmitted directly from a source to a supporting structure (isolated system) are filtered out by an isolator through its flexibility (spring) and dissipation (damping) so that part of the force is routed through an inertial path Clearly, the concepts of force transmissibility are applicable here In motion isolation, vibration motions that are applied at a moving platform of a mechanical system (isolated system) are absorbed by an isolator through its flexibility and dissipation so that the motion transmitted to the system of interest is weakened The concepts of motion transmissibility are applicable in this case The design problem in both cases is to select applicable parameters for the isolator so that the vibrations entering the system are below specified values within a frequency band of interest (the operating frequency range) Force transmissibility and motion transmissibility were studied in Chapter 3, but the main concepts are revisited here Figure 12.4(a) gives a schematic model of force transmissibility through an isolator Vibration force at the source is f(t) In view of the isolator, the source system (with impedance Zm) is made to move at the same speed as the isolator (with impedance Zs) This is a parallel connection of impedances, as noticed in Chapter Hence, the force f(t) is split so that part of it is taken up by the inertial path (broken line) of Zm, and only the remainder (fs) is transmitted through Zs to the supporting structure, which is the isolated system As derived in Chapter 3, force transmissibility is Tf = fs Zs = f Zm + Zs (12.2) Figure 12.4(b) gives a schematic model of motion transmissibility through an isolator Vibration motion v(t) of the source is applied through an isolator (with impedance Zs and mobility Ms) to the isolated system (with impedance Zm and mobility Mm) The resulting force is assumed to transmit ©2000 CRC Press FIGURE 12.4 (a) Force isolation; (b) motion isolation; (c) force isolation example; and (d) motion isolation example directly from the isolator to the isolated system and, hence, these two units are connected in series (see Chapter 3) Consequently, one obtains the motion transmissibility: Tm = Zs vm Mm = = v Mm + Ms Zs + Z m (12.3) It is noticed, according to these two models, Tf = Tm ©2000 CRC Press (12.4) As a result, the concepts of force transmissibility and motion transmissibility can usually be studied using just one common transmissibility function T Simple examples of force isolation and motion isolation are shown in Figure 12.4(c) and (d) As derived in Chapter 3, for both cases, the transmissibility function is given by T= ( k + bjω k − mω + bjω (12.5) ) where ω is the frequency of vibration excitation Note that the model (12.5) is not restricted to sinusoidal vibrations Any general vibration excitation can be represented by a Fourier spectrum, which is a function of frequency ω Then, the response vibration spectrum is obtained by multiplying the excitation spectrum by the transmissibility function T The associated design problem is to select the isolator parameters k and b to meet the specifications of isolation Equation (12.5) can be expressed as T= (ω ω 2n + jζω n ω n − ω + jζω n ω ) (12.6) where ω n = k m = undamped natural frequency of the system ζ= b = damping ratio of the system km Equation (12.6) can be written in the nondimensional form T= + jζr − r + jζr (12.7) where the nondimensional excitation frequency is defined as r = ω ωn The transmissibility function has a phase angle as well as magnitude In practical applications, it is the level of attenuation of the vibration excitation that is of primary importance, rather than the phase difference between the vibration excitation and the response Accordingly, the transmissibility magnitude T = + 4ζ r (1 − r ) 2 is of interest It can be shown that ΈTΈ < for r > + 4ζ r (12.8) , which corresponds to the isolation region Hence, the isolator should be designed such that the operating frequencies ω are greater than ωn Furthermore, a threshold value for ΈTΈ would be specified, and the parameters k and b of the isolator should be chosen so that ΈTΈ is less than the specified threshold in the operating frequency range (which should be given) This procedure can be illustrated using an example ©2000 CRC Press FIGURE 12.5 A simplified model of a machine tool and its supporting structure EXAMPLE 12.1 A machine tool and its supporting structure are modeled as the simple mass–spring–damper system shown in Figure 12.5 a Draw a mechanical-impedance circuit for this system in terms of the impedances of the three elements: mass (m), spring (k), and viscous damper (b) b Determine the exact value of the frequency ratio r in terms of the damping ratio ζ, at which the force transmissibility magnitude will peak Show that for small ζ, this value is r = c Plot ΈTfΈ versus r for the interval r = [0, 5], with one curve for each of the five ζ values 0.0, 0.3, 0.7, 1.0, and 2.0 on the same plane Discuss the behavior of these transmissibility curves d From part (c), determine for each of the five ζ values, the excitation frequency range with respect to ωn, for which the transmissibility magnitude is i Less than 1.05 ii Less than 0.5 e Suppose that the device in Figure 12.5 has a primary, undamped natural frequency of Hz and a damping ratio of 0.2 It is required that the system has a force transmissibility magnitude of less than 0.5 for operating frequency values greater than 12 Hz Does the existing system meet this requirement? If not, explain how one should modify the system to meet this requirement SOLUTION a Here, the elements m, b, and k are in parallel, with a common velocity v across them, k as shown in Figure 12.6 In the circuit, Zm = mjω, Zb = b, and Zk = jω The force transmissibility is Tf = Zb + Zk Fs Fs V Zs = = = F FV Zs + Z0 Zm + Zb + Zk Substituting the element impedances, one obtains ©2000 CRC Press (i) TABLE 12.3 Eigenvalues of the Open-Loop (Uncontrolled) Beam Eigenvalue (rad·s–1) (Multiply by 26.27) Mode –0.000126 –0.000776 –0.002765 –0.007453 –0.016741 –0.033.75 ( ) ( ± ± ± ± ± ± j j j j j j 1.0 4.0 9.0 16.0 25.0 36.0 ) E * ω j = g2 ω j + g2 (12.172) in which ωj is the jth undamped natural frequency given ω j = ( jπ l ) EI ρA (12.173) The numerical values used for the damping parameters are g1 = 88 × 104 kPa (12.5 × 104 psi) and g2 = 3.4 × 104 kPa·s (5 × 103 psi·s) For the present problem, Yi(x) = sin(jπx/l) and αj = ρAl for all j First, ωj and γj are computed using equations (12.173) and (12.153), respectively, along with (12.172) Next, the open-loop system matrix A is formed according to equation (12.156) and its eigenvalues are computed These are listed in Table 12.3, scaled to the first undamped natural frequency (ω1) Note that in view of the very low levels of internal material damping of the beam, the actual natural frequencies, as given by the imaginary parts of the eigenvalues, are almost identical to the undamped natural frequencies Next, attempt to place the real parts of all the (scaled) eigenvalues at –0.20, while exercising no constraint on the imaginary parts (i.e., damped natural frequencies) by using (1) single damper, and (2) two dampers In the cost function (12.171), the first three modes are more heavily weighted than the remaining three Initial values of the damper parameters are b1 = b2 = 0.1 lbf·s·in–1 (17.6 N⋅s⋅m−1 ) and the initial locations l1/l = 0.0 and l2/l = 0.5 At the end of the numerical optimization, using a modified gradient algorithm, the following optimized values are obtained: (1) Single-damper control: ( b1 = 36.4 lbf ⋅ s ⋅ in.−1 6.4 × 10 N ⋅ s ⋅ m −1 (b) ) l1 l = 0.3 The corresponding normalized eigenvalues (of the closed-loop system) are given in Table 12.4 Two-damper control: ( ) ( ) b1 = 22.8 lbf ⋅ s ⋅ in.−1 4.0 × 10 N ⋅ s ⋅ m −1 b2 = 12.1 lbf ⋅ s ⋅ in.−1 2.1 × 10 N ⋅ s ⋅ m −1 l1 l = 0.25, l2 l = 0.43 The corresponding normalized eigenvalues are given in Table 12.5 ©2000 CRC Press TABLE 12.4 Eigenvalues of the Beam with an Optimal Single Damper Mode Eigenvalue (rad·s–1) (Multiply by 26.27) –0.225 –0.307 –0.037 –0.119 –0.355 –0.158 ± ± ± ± ± ± j j j j j j 0.985 3.955 8.996 15.995 24.980 35.990 TABLE 12.5 Eigenvalues of the Beam with Optimized Two Dampers Mode Eigenvalue (rad·s–1) (Multiply by 26.27) –0.216 –0.233 –0.174 –0.079 –0.145 –0.354 ± ± ± ± ± ± j j j j j j 0.982 3.974 8.997 15.998 24.999 35.989 It would be highly optimistic to expect perfect assignment of all real parts at –0.2 However, note that good levels of damping have been achieved for all modes except for Mode in the singledamper control and for Mode in the two-damper control In any event, since the contribution of the higher modes toward the overall response is relatively smaller, it is found that the total response (say, at point x = l/12) is well damped in both cases of control PROBLEMS 12.1 In general terms, outline the procedures of: a System design for vibration b Vibration control in a system What are the differences and similarities of the two procedures? 12.2 On a velocity versus frequency nomograph, to log–log scale, mark suitable operating (design) vibration regions for the following applications: a Ground transportation (30-minute trips) b Ground transportation (8-hour trips) c Tool-workpiece region of milling machines d Automobile transmissions e Lateral vibration of a building tower f Pile drivers in civil engineering constructions (bridges) g Forging machines h Concrete drilling machines i Delicate robotic experimentation in a space station j Compact-disk players ©2000 CRC Press 12.3 Indicate similarities and differences in the specifications of vibration environment for: a Product operation b Product testing 12.4 The following are procedures for achieving a desired vibration performance from a mechanical system Categorize them into vibration isolation, vibration design modification, and vibration control a Shock absorbers in an automobile b Stiffening crossbars added to a structure c Rotary damper placed on the shaft of a rotating machine d Dynamic absorber mounted on the casing of a delicate instrument e Elastomeric mounts of an exhaust fan f Spring mounts of a heavy engine g Inertia block at a machine base h Stabilizer suspensions of power transmission lines i Helical spoilers of tall incinerators and chimneys j Vibration sensor-actuator combinations for distributed systems k Active suspension systems of transit vehicles l Balancing of rotating machines through the addition and removal of mass 12.5 Consider the transmissibility function magnitude of a simple-oscillator mechanical system, as given by: + 4ζ r T = − r + 4ζ r ( 12 ) Using straightforward differentiation of this function, it can be shown that the peak transmissibility occurs at [ + 8ζ − 1] = rpeak 12.6 12 2ζ In particular, for ζ = 0, rpeak = By computing rpeak for a range of ζ values from 0.001 to 1.9, show that rpeak decreases with ζ A landlord rents a room in his basement to two university students The room is just beneath the kitchen Two weeks later, the students complain about the shaking of their ceiling when the dishwasher is operating The landlord decides to install the dishwasher on four spring mounts in order to achieve a vibration isolation level of 80% The following data are known: Mass of the dishwasher = 50 kg Normal operating speed = 300 rpm 12.7 Determine the required stiffness for each of the four spring mounts What will be the static deflection of the springs? Under normal conditions, a washing machine operates at a steady speed in the range of 1200 to 1800 rpm The weight of the washing machine is 75 kg It is required to achieve a vibration isolation level of at least 80%, and preferably about 90% Also, during starting and stopping conditions of the washing machine, the peak (resonant) transmissibility should be about 2.5, but not exceed Design a damped spring mount to achieve these operating requirements Specifically, determine the stiffness k and damping constant b ©2000 CRC Press FIGURE P12.8 A milling machine with damped flexible mounts 12.8 of each of the four identical mounts to be incorporated at the base of the washing machine What is the (undamped) natural frequency of the system? Hint: Use the approximate design relation (assuming small or damping ratio) and then check adherence to the specifications by using more accurate relations (for a sufficiently large damping ratio) A milling machine weighing 500 kg is rigidly mounted on a concrete floor Loadcells were placed on the base and measurements were made to determine the vertical forces generated by the machine that are transmitted to the floor during operation, in the frequency range of 10 Hz to 60 Hz The worst-case amplitude of the transmitted force was found to be 2000 N and the vibrations were nearly sinusoidal Also, large-amplitude vibratory motion was noticed during start-up and shut-down procedures To reduce floor vibrations that affect adjoining operations and offices, vibration isolation was found to be required Furthermore, in order to maintain the machining accuracy during normal operation, the vibratory motion during these steady operating conditions needed to be reduced The following specifications are given: Amplitude of vibratory motion at resonace = 1.0 cm or less Level of vibration isolation under normal operation = 80% (approx.) Amplitude of vibratory motion under normal operation = 2.0 mm or less 12.9 Design a mounting system to achieve these specifications A schematic representation of the system is shown in Figure P12.8 Consider the flexible vibrating system with a vibration isolator as shown in Figure 12.8 For the case of negligible damping (B and b are neglected) and a unity mass ratio (m/M = 1), show that the transmissibility ratio Tflexible /Tinertial of the flexible system to the inertial system (where B, k, and m are absent as in Figure 12.4(c)) is given by Tflexible r2 − = Tinertial − r r − rω2 + 2r − ( ©2000 CRC Press ) FIGURE P12.10 Static load-deflection characteristics of four spring mounts where the nondimensional frequency variable: r= ω k M and the natural frequency ratio: rω = K M = k M K k The excitation frequency is ω Plot this transmissibility ratio against r in the range r = to 10, for a frequency ratio value of rω = 10.0 For the inertial system with an undamped isolator, what is the minimum operating frequency ratio rop for achieving an isolation level of 90%? What is the isolation level of the flexible system at the same operating frequency? ©2000 CRC Press 12.10 A high-speed punch press operates at a steady speed in the range of 2400 to 3600 rpm under normal conditions In order to stop vibration effects of the punch press from affecting other processes and environments, a 75% vibration isolation is sought Four flexible mounts are available for this purpose and their static characteristics are shown in Figure P12.10 Which of these four mounts would you choose for this application? Perform an analysis to justify your choice Assume that the damping in the mounts is very small 12.11 The following data are obtained from an experiment carried out on a disk that is mounted very close to a bearing, in a single-plane balancing problem Magnitude and location of the trial mass with respect to a body reference line: v Mt = 13.5∠0° gm Magnitude and phase angle (with respect to a body reference) of the accelerometer signal at the bearing, in the absence of the trial mass: v Vu = 356∠242.2° Magnitude and phase angle (with respect to the same body reference) of the accelerometer signal at the bearing, in the presence of the trial mass: v Vr = 348∠75.6° Determine the magnitude and orientation of the necessary balancing mass to be mounted on the disk at the same radius as the trial mass in order to completely balance it, after removing the trial mass 12.12 a List five causes of unbalance in rotating devices What are detrimental effects of unbalance? Give some of the ways of eliminating/reducing unbalance b A pancake motor has a disk-like rotor When rotating at a fixed speed, an accelerometer mounted on the rotor bearing shows an excitation amplitude of 350 mV through a charge amplifier Also, this signal has a phase lead of 200° with respect to a body reference of the rotor, as determined with respect to a synchronized stroboscope signal A trial mass of 15 gm was placed on the rotor at a known radius and an angular location of 0° with respect to a body-reference radius that is marked on the rotor surface Then, the accelerometer signal was found to have an amplitude of 300 mV and a phase lead of 70° with respect to the same synchronized (with respect to both frequency and phase) strobe signal as before Determine the magnitude and location of the mass that must be placed at the same radius as the trial mass in order to balance the rotor, after removing the trial mass 12.13 a When is two-plane balancing preferred over single-plane balancing? Comment on the terms “static balancing” and “dynamic balancing.” b A turbine rotor is supported on two bearings at the two ends Two accelerometers are mounted on the housing of these bearings The rotor is driven at a fixed speed and the accelerometer signals obtained Their amplitudes and phase leads, with respect to a strobe signal that is synchronized to a fixed body reference, are found to be: 400 mV and 100° at bearing 700 mV and 120° at bearing ©2000 CRC Press FIGURE P12.14 A dynamic balancing problem Next, a trial mass of 20 gm is placed at a known radius and an angular position of 90° with respect to a known body reference, in the balancing plane (close to bearing 1) The new readings of the accelerometer signals are: 350 mV and 140° at bearing 600 mV and 130° at bearing Subsequently, this trial mass is removed and a second trial mass of 25 gm is placed at a known radius and an angular position of 30° with respect to a known body reference, in the balancing plane (close to bearing 2) The resulting readings of the accelerometers are: 10 mV and 150° at bearing 750 mV and 170° at bearing Determine the magnitudes and locations of the balancing masses that should be placed on planes and 2, at the same radii as the trial masses placed on these planes, after removing the trial masses 12.14 a Give four causes of dynamic imbalance in rotating machinery b What is static balancing and what is dynamic balancing? Assuming that any location/plane of a rotating machine is available for placing a balancing mass, is it possible to completely balance the machine using the static balancing (single-plane) method? Fully justify your answer c Consider a completely balanced rigid shaft that is supported horizontally on two bearings at distance l apart A point mass m1 is attached to the shaft at a distance l1 from the left bearing using a light, rigid radial arm of length r1 measured from the rotating axis of the shaft Similarly, a point mass m2 is attached to the shaft at a distance l2 from the right bearing using a light, radial arm of length r2 The mass m2, however, is placed in a radially opposite configuration with respect to m1, as shown in Figure P12.14 The two masses securely rotate with the shaft as a single rigid body, without any deformation Note that when m1 is vertically above the shaft axis, m2 will be vertically below the axis The angular speed of rotation of the shaft is ω Suppose that at time t = 0, the mass m1 is vertically above the shaft axis ©2000 CRC Press FIGURE P12.15 Angular coordinates for locating the masses in a single-plane balancing problem i Giving all the necessary steps, derive expressions for the horizontal (x-axis) and vertical (z-axis) components of the reactions on the shaft at the left bearing (1) and the right bearing (2) Specifically, obtain Rx1, Rz1, Rx2, and Rz2 in terms of the given parameters (m1, m2, r1, r2, l1, l2, and l) of the system, for a general time instant t ii Using the planes at the two bearings as the balancing planes, determine the magnitude and orientation of the balancing masses mb1 and mb2 that should be placed on these planes at a specified radius r, in order to dynamically balance the system Give the orientation of the balancing masses with respect to the orientation of the mass m1 iii If m1 = kg, m2 = kg, l = 1.0 m, l1 = 0.2 m, l2 = 0.3 m, r1 = 0.1 m, r2 = 0.2 m, and r = 0.1 m, compute the balancing masses and their orientations d Would it be possible to balance this problem by the single-plane (static) method? Explain your answer Also, consider the special case where m1 = m2 = m and r1 = r2 = r 12.15 Consider the problem of single-plane balancing It should be clear that the angular position for locating the trial and balancing masses should be measured in the same direction as the angular velocity of the rotating disk, in using the equation v v v Vu M b = v v Mt Vr − Vu ( ) v v where the phase angles of the accelerometer signals V u and V r are taken as phase leads in the usual notation a In a laboratory-experimental setup, the disk was found to be graduated as shown in Figure P12.15, while the angular speed ω was in the indicated direction (counterclockwise) What interpretations must be made on the experimental data in this case, in using the above equation for computing the balancing mass? b With an experimental setup of the above type, the following data were obtained: Without a trial mass, the amplitude and the phase lead (with respect to the strobe signal) of the accelerometer signal were 36.1 mV and 209.3° ©2000 CRC Press FIGURE P12.17 A possible crank arrangement of a four-cylinder engine FIGURE P12.18 A single-cylinder model used in load analysis With a trial mass of 10.4 gm placed at the location of 130°, the amplitude and the phase lead of the accelerometer signal were 38.7 mV and 247.5° Determine the magnitude and the location of the balancing mass, with the trial mass removed 12.16 Although it is possible to accurately compute the magnitude and the location of the necessary balancing mass for a rotating component, in practice it may not be possible to achieve a perfect balance, particularly in a single trial Give reasons for this situation 12.17 Examine the statement “the most difficult part of balancing the inertial loading of a reciprocating engine is the removal of effects due to the equivalent reciprocating mass.” Consider a four-cylinder engine where the engines are placed in parallel (in-line) and equally spaced (z0) with their cranks phased at the angles 0°, 90°, 270°, and 180°, in sequence, with respect to a rotating reference Show that, for this engine, the inertial loading on the crankshaft, due to the reciprocating masses, has the following characteristics: i Primary components (frequency ω) of the forces are balanced ii Primary components of the bending moments are not balanced iii Secondary components (frequency 2ω) of the forces are balanced iv Secondary components of the bending moments are balanced Here, ω is the angular speed of the crankshaft The crank arrangement is shown in Figure P12.17 12.18 Clearly justify the assumption of massless crank and connecting rod in the balancing analysis of a reciprocating engine Also, justify the assumption that the resultant end force at each end of a connecting rod acts along the length of the rod Suppose that a force f acts on the piston of a single-cylinder engine, as shown in Figure P12.18 Note that f can represent either the inertia force of the equivalent reciprocating mass, or the force due to gas pressure in the cylinder As a result, a torque τ is ©2000 CRC Press FIGURE P12.19 An alternative crank configuration of a four-cylinder engine applied on the crankshaft in the direction of its rotation, and an equal reaction torque τ is applied by the crankshaft on the crank, in the opposite direction Use the principle of virtual work, with justification of its use, to show that τ = (rcosθ + lcosφ)f tanφ, where r and l are the lengths of the crank and the connecting rod, respectively; θ is the inclination of the crank; and φ is the inclination of the connecting rod, to the line connecting the crankshaft and the piston, as shown 12.19 In the analysis of inertial load balancing in reciprocating engines, the inertia of the connecting rod is frequently represented by two lumped masses at its two ends, joined by a massless rod The end masses are chosen such that the vector sum of their inertia forces is equal to the inertia force of the mass of the rod assumed to be concentrated at the centroid What are the limitations of this model? A four-cylinder in-line engine where the cylinders are equally spaced and the cranks placed at the angles 0°, 180°, 180°, and 0°, sequentially, is schematically shown in Figure P12.19 Show that in this case the inertial loading on the crankshaft, due to the reciprocating masses, is such that i primary components (with frequency ω) of the forces and bending moments are completely balanced ii secondary components (with frequency 2ω) of the forces and bending moments are not balanced Note that ω is the angular speed of the crankshaft 12.20 In a multicylinder engine, the inertia force of each reciprocating mass causes on the crankshaft a lateral reaction force in the direction of reciprocation and a torque in the direction of rotation of the shaft For proper operation of the engine, all these reaction forces, bending moments, and torques should be balanced Similarly, the piston force due to gas pressure in the cylinder causes a reaction force, a bending moment, and a torque in the crankshaft It is desirable to balance the reaction forces and bending moments, but not the torques, in this case Explain why: i Balancing of the gas-pressure load is much more difficult than that of the inertial load ii Torque on the crank shaft due to the gas-pressure load should not be balanced Consider a six-cylinder in-line engine with the crank orientations 0°, 120°, 240°, 240°, 120°, and 0°, in sequence Check whether the torques generated on the crankshaft due to the inertia forces of the reciprocating masses are balanced 12.21 Consider a light shaft that is supported by bearings at its ends and carries a rotor in its mid-span The shaft is driven at an angular speed of ω The magnitude of the shaft deflection at the rotor, in whirling motion, at steady state is given by ©2000 CRC Press r= [(ω eω 2 n − ω2 ) + (2ζω ω )] 2 12 n where e = eccentricity of the rotor centroid from the axis of rotation of the shaft ωn = undamped natural frequency of bending vibration of the shaft-rotor system ζ = equivalent damping ratio for whirling motion Show that the peak value of r occurs at a shaft speed of ω= ωn − 2ζ 12.22 A light shaft that is supported on two bearings carries a fly wheel at its mid-span The centroid of the flywheel has an eccentricity e with respect to the axis of rotation of the shaft The effective damping ratio in whirling motion of the shaft is ζ The bending stiffness of the shaft at its midspan is k What is the reaction at each bearing when the shaft system rotates at its critical speed? 12.23 An experimental procedure for determining the equivalent damping ratio that is provided by a pair of bearings on a shaft in whirling motion is as follows A radial arm with a lumped mass of 1.0 kg is rigidly attached at the mid-span so that the eccentricity of the mass from the axis of rotation of the shaft is 10.0 cm The shaft is driven at the normal operating speed of 2400 rpm, and the average reaction at the two bearings is measured using load cells It was found to be 4.56 × 103 N Using strobe lighting with manually adjustable frequency, directed axially from one end of the shaft, the mid-span deflection of the shaft is measured approximately using a background scale, while the strobe frequency is synchronized with the operating speed of the shaft This reading was found to be 4.7 cm Also, using similar means, the angle between the radial arm and the direction of bending (bowing) of the shaft was measured approximately This was found to be 20° The bending stiffness of the shaft at mid-span was measured while the system was stationary, by applying a known load and measuring the deflection It was determined to be k = 2.0 × 105 N·m–1 Estimate the damping ratio 12.24 A turbine rotor has a mass of 50 kg and is supported on a light shaft with end bearings The bending stiffness of the shaft at the rotor location is 3.0 × 106 N·m–1 The centroid of the rotor has an eccentricity of 2.0 cm from the axis of rotation of the shaft The normal operating speed of the turbine is 3600 rpm The equivalent damping ratio of the system in whirling is 0.1 a What is the critical speed of rotation of the system? b What is the shaft deflection at the rotor during normal operation? c A mass of kg is added to the rotor in order to achieve a better balance By what factor should the eccentricity be reduced by this means in order to reduce the shaft deflection by a factor of 10 during normal operation? 12.25 A student proceeds to determine the growth of the deflection of a whirling shaft that carries a rotor in its mid-span and is supported on bearings at its ends as follows: Assume synchronous whirl (steady whirling rotation at the same speed as the shaft spin) so that θ˙ = ω and θ = ωt – φ, where φ is the phase lag between the whirl and the spin The equations of motion [see equations (12.77) and (12.78)] become ©2000 CRC Press ( ) ˙˙ r + 2ζ v ω n r˙ + ω 2n − ω r = eω cos φ r˙ + ζω n r = eω sin φ (i) (ii) Solve equation (ii) with r = at t = to get r= [ eω sin φ − e −ζω nt ζω n ] (iii) Substitute equation (iii) in (i) Set the overall coefficient of e −ζω nt to 0, as e −ζω nt ≠ in general Then set the rest of the terms to In this manner, obtain φ and hence the time variation of r Do you agree with this approach? If so, provide justification If not, give reasons why the approach might fail 12.26 Explain how experimental modal analysis can be used in the design of a mechanical system for proper performance under vibration What are limitations of its use? In substructuring, if the linkages of the subsystem have inertial elements that are not negligible, what additional issues should be addressed in a vibration design procedure? 12.27 Consider two single-degree-of freedom subsystems (k1, m1) and (k2, m2) that are interconnected by a spring element of stiffness kc , as shown in Figure 12.22 Use ω = k1 m1 in nondimensionalizing the frequencies according to ri = ωi /ω0 Suppose that k2 /k1 = 7.0 and m2 /m1 = 1.0 Design the interconnection element kc so that the two natural frequencies satisfy the condition ri2 ≥ 2.0 for i = and 12.28 Determine an expression for the separation interval between the two dominant resonant frequencies of a vibrating (primary) system once a vibration absorber is added Show that this expression, with respect to the original resonance of the primary system, can be expressed as [ r2 − r1 = α (1 + µ ) + − 2α where ri ωi ωp ωa mp ma = = = = = = ] 12 ωi /ωp, α = ωa /ωp, and µ = ma /mp, and a newly created resonant frequency original resonant frequency of the primary system resonant frequency of the vibration absorber mass of the primary system mass of the vibration absorber For a system with α = 1.0, µ = 0.1, and ωp = 120π rad·s–1, compute the frequency interval of the two resonances 12.29 An induction motor weighing 10 kg is mounted on a relatively light structure and is used to drive a conveyor at a steady speed A schematic diagram is given in Figure P12.29 The normal operating speed of the motor is 2400 rpm, as required for driving the conveyor When the motor speed was slowly increased, a significant vertical resonance was found at 3000 rpm With the intention of mitigating this problem and further reducing its spill-over effect at the normal operating speed, a technician installs a vibration absorber ©2000 CRC Press FIGURE P12.29 A conveyor motor on the support frame of the motor The absorber is tuned to 2700 rpm (a value in between the operating speed and the original resonance) The absorber mass is 1.0 kg Determine whether the addition of the vibration absorber mitigates the problem In particular, answer the following questions: a What are the main resonant frequencies of the modified system when the vibration absorber is added? b What is the effective speed range of operation of the modified system? c What was the magnitude of vibration amplification of the original system at the operating speed, and what is it after the modification? Neglect damping in this analysis 12.30 An undamped vibration absorber generates two resonances for which separation is equal to β times the tuned frequency (resonant frequency) of the absorber Obtain an expression for the fractional mass µ of the absorber in terms of β and the nondimensional resonant frequency α of the absorber (with respect to the primary frequency) A machine of mass 100 kg has a significant resonance at 2400 rpm The normal operating speed is 2200 rpm Design an undamped vibration absorber, tuned to the operating speed, such that the two generated resonances are at least 20% apart with respect to the operating frequency 12.31 The tubes of a steam generator in a nuclear power plant facility exhibited significant wear and tear due to vibration Vibration monitoring and signal analysis showed that under normal operation, the tube vibration was narrow-band, and limited to a very small interval near 30 Hz Furthermore, vibration testing indicated that the primary significant resonance of the steam generator occurs at 32 Hz The mass of the steam generator is 50 kg Design a damped vibration absorber for the system Check the magnitude of the ©2000 CRC Press FIGURE P12.34 A wood-cutting saw mounted on a beam operating vibration in the modified system and compare it with the performance before modification 12.32 Compare and contrast a simple linear damper and a dynamic absorber as vibration control devices The exhaust fan of a pulp and paper mill operates at 450 rpm At the time of installation of the fan on its support structure, a static deflection of 1.0 cm was experienced by the structure During normal operation, the amplitude of vertical vibrations of the fan was found to be 3.0 mm During start-up and shut-down of the fan, it exhibited a vertical resonance with vibration amplitude 2.0 cm Estimate the damping ratio of the fan-supportstructure system 12.33 Using the damped simple oscillator model of a system, justify that at low frequencies of excitation, system dynamics are primarily determined by its stiffness characteristics; and at high frequencies of excitation it is primarily determined by its inertia characteristics Furthermore, justify that near the resonance, it is the damping that primarily determines the dynamic characteristics of the system It is also known that a dynamic absorber can be quite effective in vibration control in the neighborhood of a system resonance But, typically, a dynamic absorber is a lowdamping device that can function properly even without any damping — at least in theory Is this a contradiction in view of the previous observation about the dominance of damping in determining dynamics near a resonance? 12.34 a In comparison with lumped-parameter systems, what are some of the difficulties that arise in the vibration control of a distributed-parameter (i.e., continuous) system? Also, give several reasons for considering active control to be more difficult than passive control in the vibration reduction of distributed-parameter systems b A scragg saw commonly used in cutting relatively small diameter logs of wood is firmly supported at the mid-span of a beam of length l = m, as schematically shown in Figure P12.34 The saw weighs 20 kg and normally operates at a steady speed of 600 rpm The beam is simply supported at its ends and has the following parameter values: I (2nd moment of area of cross section about the neutral axis) ρA (mass per unit length) E (Young’s modulus) ©2000 CRC Press = 1.0 × 10–7 m4 = 5.0 kg⋅m-1 = 2.0 × 1011 N⋅m-2 (Note: N·m–2 = Pa = × 10–3 kPa) Estimate the fundamental natural frequency of the overall system (the saw and the supporting beam) by first determining the equivalent mass of the beam at the midspan in the first mode of vibration Design an optimal (and damped) vibration absorber, to be tuned to the normal operating speed of the saw and mounted at the mid-span of the supporting beam Note: The equivalent stiffness (force/displacement) at the mid-span of a simply sup48 EI ported beam is given by l3 12.35 The shock absorbers of an automobile are primarily damping devices, with springs provided by the suspension system Explain why conventional dynamic absorbers are not suitable in this application Consider the use of dynamic absorbers to control vibration of a beam Formulate a state-space model for this problem, as shown in Figure P12.35 Specifically, determine the model for the case of a single dynamic absorber Note that x = lj is the location of the jth absorber along the beam, and sj is the displacement of the mass of this absorber Also, mj, kj, and bj are the mass, stiffness, and damping constant, respectively, of the jth absorber FIGURE P12.35 Use of dynamic absorbers for vibration control of a beam ©2000 CRC Press ... Undesirable vibrations are those that cause human discomfort and hazards, structural degradation and failure, performance deterioration and malfunction of machinery and processes, and various... involves periodic excitations and responses, transient and random oscillations (vibrations) are also commonly found in practice The frequency band of the latter two types of signals is much broader... vibration environment may not necessarily be a single-frequency (sinusoidal) signal and may be broad-band and random; and ©2000 CRC Press FIGURE 12.2 Operating vibration specification (nomograph) for