The OEM can contract with the two suppliers either simultaneously or sequentially. We consider both cases in which the The manufacturers produce partially substitutable products, which are stocked by the retailer that sells them in the market Published Online: October 1, First Page PDF KB References Permissions Preview Abstract Abstract How does deleveraging affect the market liquidity of high-embedded-leverage securities issued by financial institutions and the funding constraints of these institutions?
We use the forced deleveraging of structured mutual funds during the Chinese First Page PDF KB References Permissions Preview Abstract Abstract This paper broadens the focus of empirical research on salesforce management to include multitasking settings with multidimensional incentives , where salespeople have private information about customers.
This allows us to ask novel substantive questions A central managerial concern is how to screen the population for only the most capable people when the capability of the population is not known. If the manager sets the The relation between currency excess returns and sovereign risk is Published Online: September 28, First Page PDF KB References Permissions Preview Abstract Abstract Driving is an integral component of many operational systems, and any small improvement in driving quality can have a significant effect on accidents, traffic, pollution, and the economy in general.
However, making improvements is challenging given the Although market-entry games in the laboratory consistently produce equilibrium outcomes, we show that a lack of market knowledge hinders the markets from consistently reaching Published Online: September 27, We show that steeper securities provide more insurance because Published Online: September 21, Three quarters of hedge funds shifted their investment styles at least once over the period from January to December Published Online: September 20, This paper uses multiple laboratory experiments to test when and why thank you gifts Consumers may focus on the symbols and pay limited attention to the ratings.
This bias leads to discontinuously increased demand at the Empirically, we exploit the staggered implementation of Schedule UTP, a discrete policy change expected to increase Published Online: September 17, First Page PDF KB References Permissions Preview Abstract Abstract We examine the impact of ambiguity, or Knightian uncertainty, on the capital structure decision, using a static tradeoff theory model in which agents are both ambiguity and risk averse.
The model confirms the well-known result that greater risk—the Published Online: September 16, I obtained data from a recruiting platform where individuals can self-report their computer programming skills and recruiters can message First Page PDF KB References Permissions Preview Abstract Abstract A wide body of empirical research has revealed the descriptive shortcomings of expected value and expected utility models of risky decision making.
Published Online: September 14, Same-day fulfillment in e-commerce has seen substantial growth in recent years, and the underlying management of such services is complex. Although the literature includes Published Online: September 13, The viability of robo-advisors crucially depends on their ability to offer personalized financial advice.
We introduce a novel framework in This problem arises in precision medicine, targeted marketing, personalized education, and many other environments.
Published Online: September 10, First Page PDF KB References Permissions Preview Abstract Abstract Conventional optimization algorithms that prescribe order packing instructions which items to pack in which sequence in which box focus on box volume utilization yet tend to overlook human behavioral deviations.
We observe that packing workers at the This result is also robust at the establishment level at which we find local analysts have a more profound impact than distant analysts. Published Online: September 9, We compile a new comprehensive data set of state We consider a practical formulation in which the unknown demand parameters for each product come from an unknown distribution prior However, managers of public firms who obtain stock-based incentives may alter inventory operations to influence the stock price.
We develop a Published Online: September 8, We consider a buyer who repeatedly demands a certain good and is unable to commit to long-term contracts.
In each time period, the buyer makes a The quality of these rosters is of utmost importance. High-quality rosters should be attractive on an individual level, but also divide the work fairly over the employees. We develop novel We evaluate the First Page PDF KB References Permissions Preview Abstract Abstract To stay competitive, high-technology manufacturers not only frequently source new technologies from their suppliers, but also financially support the development of these new technologies into component products or production tools.
We consider a When PCPs concentrate their specialist referrals—for instance, by sending their cardiology patients to fewer distinct cardiologists—repeat interactions Many newspapers have implemented paywalls in an attempt to boost subscription revenue. We study whether and how paywalls can help newspapers boost Published Online: September 7, Price run-ups coincide with increases in demand fundamentals but are followed by predictable busts.
Predictable changes account for about half of the variance of Published Online: September 2, First Page PDF KB References Permissions Preview Abstract Abstract Decades of research suggest that information exchange in groups and organizations can reliably improve judgment accuracy in tasks such as financial forecasting, market research, and medical decision making.
However, we show that improving the accuracy of Published Online: September 1, This criticism is particularly pronounced for health products, where large asymmetries in information may exist between consumers and firms. This paper Published Online: August 31, One of the main drivers for increasing profits and market share is service quality. When consumers encounter a bad experience, or a frustration , they may be tempted to stop using the service. First Page PDF KB References Permissions Preview Abstract Abstract Although extant literature investigates the role of sell-side equity analysts in capital markets, most studies do not consider that sell-side equity analysts often work in hierarchical teams.
Lead analysts manage a team of associate and junior analysts At every point in time, the consumer then decides between searching among alternatives of which he is already aware and discovering more products.
I show Published Online: August 30, Traditional algorithms for learning SVMs scale superlinearly with the training set size, which becomes infeasible Published Online: August 27, We use a product recall as our measure of quality failure because it is an objective metric that is applicable to a broad cross-section of industries. Our analysis employs a This paper adds to the debate by showing that institutional differences in setting up defined contribution personal schemes have an economically and Published Online: August 26, Using a hand-collected data set on the prevalent dialects for 2, cities counties in China, I identify the effects of Published Online: August 25, Using more than 11 million health encounters over two years in Published Online: August 24, Published Online: August 20, We determine restrictions on preferences for clear-cut results.
Risk increases of a benefit are compensated by Published Online: August 18, We hypothesize that compared with noneponymous firms, eponymous firms have higher FRQ because of reputation concerns. Using a First Page PDF KB References Permissions Preview Abstract Abstract We study the impact of using informational messaging aimed at encouraging women to seek a career in technology and the resulting trade-offs for organizations pursuing recruitment strategies of this kind. Our treatment, implemented through two field Published Online: August 16, First Page PDF KB References Permissions Preview Abstract Abstract To increase revenue or improve customer service, companies are increasingly personalizing their product or service offerings based on their customers' history of interactions.
In this paper, we show how call centers can improve customer service by Administering ESR programs throughout a complex supply network, however, is challenging. In this paper, we apply a Published Online: August 13, This facilitates personalized pricing but may also induce consumer suspicions of overpaying. Once employed, a risk-averse manager acquires imperfectly portable skills whose value is stochastic because of industry-wide demand shocks.
Published Online: August 12, First Page PDF KB References Permissions Preview Abstract Abstract Many financial arrangements reference market prices that are yet to be realized at the time of contracting and consequently susceptible to manipulation.
Two of the most common such arrangements are as follows: i guaranteed volume-weighted average price Influenza imposes heavy societal costs through healthcare expenditures, missed days of work, and numerous hospitalizations each year. Considering these costs, the healthcare and behavioral science literature offers suggestions on increasing First Page PDF KB References Permissions Preview Abstract Abstract Cash flows to hedge funds are highly sensitive to performance streaks, a streak being defined as subsequent quarters during which a fund performs above or below a benchmark, even after controlling for a wide range of common performance measures.
At the Published Online: August 11, We find that, when a firm is exposed to We reopen this debate in the context of security circumvention by employees, currently a leading cause of information security Published Online: August 10, In our laboratory experiment, senders are required to report their private information truthfully but can choose how complex to make To better understand decision making in these contexts, this paper measures and investigates the economic relevance of heterogeneity in moral universalism: the Published Online: August 9, We study driver-side payment mechanisms for such marketplaces, presenting the theoretical Published Online: August 5, Exploiting a regulatory change to credit card minimum payments in Mexico, we find that a 1-percentage point change in minimum payments leads to a 0.
Published Online: August 4, We show that trust mitigates incomplete contracting and enhances innovation by acting as an informal contracting mechanism. Trust plays an especially important role Published Online: August 2, We evaluate whether economic shocks that increase the supply of bank deposits alter the cost—benefit calculations of bank managers concerning Published Online: July 29, Market-wide price shocks are Published Online: July 28, This work has established that people often egocentrically discount advice.
If they were to place more weight on advice, their point estimates would be more Published Online: July 26, This paper develops a simple economic model to examine how leadership styles depend on the prevailing labor market conditions for workers. We show Published Online: July 13, First Page PDF KB References Permissions Preview Abstract Abstract We exploit the staggered passage of state-level fair-employment laws in the post-World War II period to examine how stronger worker protection against racial discrimination affects firm profitability and financing decisions.
We find that firms experience Published Online: July 9, First Page PDF KB References Permissions Preview Abstract Abstract We use a proprietary data set from a national department store chain and evaluate the competitive impact of the launch of a buy-online-and-pick-up-in-store service by a major competitor. The transaction-level data set includes nearly 50 million First Page PDF KB References Preview Abstract Abstract We experimentally investigate, in an unstructured bargaining environment with commonly known money payoffs, the attraction effect and compromise effect AE and CE in bargaining, namely, a tendency for bargainers to agree to an intermediate option CE or On the one hand, the more features the prototype incorporates, the larger the potential for I propose a new measure, brokerage degree , that determines the extent to which a scientist depends on a coauthor to provide him unique access to Such a polarized coexistence of dire scarcity and massive wastefulness has been mainly driven by insufficient pooling of Published Online: July 6, Employing mergers of large, multimarket banks as an exogenous credit Published Online: July 2, The main intriguing result is that previous generation valence has a positive impact on current generation sales; however, Published Online: July 1, First Page PDF KB References Preview Abstract Abstract We examine how the introduction of monetary incentives by a knowledge-sharing platform affects the nonrewarded knowledge activity on the platform.
Our setting is a question-and-answer platform that provides monetary incentives for holding live talks. Published Online: June 30, The model is based on an incomplete information game and permits individual-specific parameters of consumers. It is challenging to apply this Published Online: June 16, First Page PDF KB References Permissions Preview Abstract Abstract An important challenge for many firms is to identify the life transitions of its customers, such as job searching, expecting a child, or purchasing a home.
Inferring such transitions, which are generally unobserved to the firm, can offer the firms Published Online: June 15, Worse funding liquidity higher repo haircuts and rates leads to larger bid-ask spreads and transaction costs in corporate bonds.
Published Online: June 4, Yet, limited attention has been paid to examine the potentially disparate impact of privacy protection on underprivileged subpopulations. This study is First Page PDF KB References Permissions Preview Abstract Abstract Private and public organizations are interested in finding effective ways to reduce crime and promote ethical behavior without investing heavy resources into monitoring and compliance.
In this paper, we experimentally study how revealing different Published Online: June 3, In this paper, we use a randomized control trial to test the efficacy of crowdsourcing on enhancing user engagement and retention in the context of mobile gaming However, such automated data entry can lead to increased reimbursement requests by Published Online: June 2, First Page PDF KB References Preview Abstract Abstract In mechanism design theory it is common to assume that agents can perfectly report their preferences, even in complex settings in which this assumption strains reality.
We experimentally test whether real market participants can report their real Published Online: June 1, The sPCA improves the traditional principal component analysis PCA by scaling each predictor with its predictive slope on the target Buck Solution Manual [6th Edition] [31].
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Solution manual of modern quantum J. The total energy being constant. Let 2 be the time corresponding to the maximum displacement of the mass. We then have 2. We will choose the static equilibrium position to be the reference of zero potential energy.
Due to displacement x from this reference the increase in the potential energy of the spring which is eq,tal to the area under the force-displacement diagram of Fig.
Adding to the kinetic energy of m, we have I. Solution: Assume that the system is vibrating harmonically with amplitude 0 from its static equilibrium position. Determine its differential equation of motion for small oscillations about the lowest point. Solution: In determining the kinetic energy of the cylinder, it must be noted that both translation and rotation take place. The kinetic energy may now be written as 2 1 w. Substituting into Eq.
The energy method can be used for multimass systems or for distributed mass systems, provided the motion of every point in the system is known. In systems in which masses are joined by rigid links, levers, or gears the motion of the various masses can be ll Free Vibration expressed in terms of the motion x of some specific point and the system is simply one of single degree of freedom since only one coordinate is! If the stiffness at that point is also known, the natural frequency can be calculated from the simple equation w.
The following examples illustrate the use of both of these methods. Solution: With x equal to the velocity of the lumped mass m, we will assume the velocity of a spring element located a distance y from the fixed end to vary linearly withy as follows.
I, and the effective mass is found to be one-third the mass of the spring. New York: Dover Publications, , pp. The engine valve system of Fig.
The reduction of such a system to a simpler equivalent system is generally desirable. The rocker arm of moment of inertia J, the valve of mass mv, and the spring of mass m. Engine valve system. Determine the effective mass of the system at midspan and find its fundamental frequency.
See Fig. Although the actual description of the damping force is difficult, ideal damping models can be assumed that will often result in satisfactory prediction of the response.
Of these models, the viscous damping force, proportional to the velocity, leads to the simplest mathematical treatment. Viscous damping force is expressed by the equation 2. Symbolically it is designated by a dashpot as shown in Fig. From the. We will first examine the homogeneous equation that will give us some understanding of the role of damping. With the homogeneous equation m. Upon substitution into t! We refer to this case as overdamped.
We refer to this case as underdamped. As a limiting case between the oscillatory and nonoscillatory motion, we define critical damping as the value of c which reduces the radical to zero. It is now advisable to examine these three cases in detail, and in terms of quantities used in practice. We begin with the critical damping.
Critical Damping. For critical damping cc, the radical in Eq. We now express the roots of Eq. Figure 2. With this diagram in mind, we are now ready to examine the solution given by Eq. Oscillatory Motion. With initial conditions x O and x O , Eq. Critically Damped Motion. The solution for the initial conditions x O and. The moving parts of many electncal meters and instruments are critically damped to avoid overshoot and oscillation.
Critically damped motion 2. The larger the damping, the greater will be the rate of decay. Consider a damped vibration expressed by the general equation 2. We introduce here a term called logarithmic decrement which is defined as the natural logarithm of the ratio of any two successive amplitudes. Rate of decay of oscillation measured by the logarithmic decrement.
Determine the logarithmic decrement and the ratio of any two successive amplitudes. Plot a curve giving the number of cycles elapsed against for the amplitude to diminish by 50 per cent.
Solution: Figure 2. VI XI 2! The damping force is equal to the product of the normal force and the coefficient of friction 11 and is assumed to be independent of the velocity, once the motion is initiated. Since the sign of the damping force is always opposite to that of the velocity.
To determine the decay of amplitude, we resort to the work-energy principle of equating the work done to the change in kinetic energy. Choosing a half cycle starting at the extreme position with velocity equal to zero and the amplitude equal to X 1, the change in the kinetic energy is zero and the work done on m is also zero. I Free vibration with coulomb. It can also be shown that the frequency of oscillation is w,..
It should be noted that the amplitudes decay linearly with time. Table of Spring Stiffness. Determine the natural frequency of the system. Determine the natural period when a 2. When a 0. Determine the unknown mass m and the spring constant k Njm. A second mass m2 drops through a height h and sticks to m 1 without rebound.
Determine the subsequent motion. If the measured period of oscillation was 1. A connecting rod weighing Determine its moment of inertia about its center of gravity, which is located 0. If the period of oscillation about a vertical axis throl'gh the center of the wheel is 2. P2- Determine the frequency of oscillation due to a small unbalance weight w lb at a distance a in.
Flpre P:Z Determine the natural frequency of oscillation. A platinum wire attached to the bob completes the electric timing circuit through a drc 1 of mercury as it swings through the lowest point. Problems 39 the amplitude to limit the duration of contact to 0.
Assume that the velocity during contact is constant and that the amplitude of oscillation is small. P2-l3, is used to measure the specific gravity of liquids. The mass of the float is 0. Determine the period of vibration when the float is allowed to bob up and down in a fluid of specific gravity 1. Figure PZ The center of gravity of the buoy is 8 in. Determine the moment of inertia of the buoy about its rotational axis.
The metacenter M represents the point of intersection of the line of action of the buoyant force and the center line of the ship, and its ::! P2-l 5. Show that the period of the Figure PZ In general, the position of the roll axis is unknown and J is obtained from the period of oscillation determined from a model test. Determine its period of oscillation if it is allowed to rock on a horizontal surface. Set up the differential equation of motion for small angular oscillations of tht: bar about the vertical axis , and determine its period.
Prove that the bar will oscillate about the vertical line through the mass center, and determine the frequency of oscillation. A steel shaft 50 in. Determine the natural frequency of the system if the weight of the wheel and tire assembly is 38 lb and its radius of gyration about its axle is 9. Discuss the difference in the natural frequency with the wheel locked and unlocked to the arm. The columns are assumed to be rigidly imbedded at the ends.
Determine its natural period 'T. Refer to the table of stiffness at the end of the chapter. P and its natural frequency.
Determine the effective mass of the beam to be added to M assuming the deflection to be that of a massless beam with a concentrated force at the end, and write the equation for its fundamental frequency.
P and calculate its natural period. Flpre Pl P in terms of x. Determine the stiffness at mo and write the expression for the natural frequency. Figure Pl How large a weight must be placed on the end of a reed made of spring steel 0. Determine the critical damping coefficient. If a i lb weight produced a constant velocity of 1. Plot non-dimensional curves for the three cases with w"t as abscissa and xwnf v0 as ordinate. Determine a the natural frequency of the damped system, b the logarithmic decrement, c the damping factor, and d the damping coefficient.
Find a the dampmg factor, b the logarithmic decrement, and c the ratio of any two consecutive amplitudes. Determine a the damping factor, b the natural frequency of damped oscillation, c the logarithmic decrement, and d the ratio of any two consecutive amplitudes. Determine the expression for a the critical damping coefficient, and b the natural frequency of damped oscillation.
P and determine the natural frequency of damped oscillation and the critical damping coefficient. I r' ' Flpre Pl If T 1 is the natural period of undamped oscillation that is, with the system oscillating in air , and T2 the damped period with the plate immersed in the fluid, show that Problems where the damping force on the plate is Fd area of the plate, and vis its velocity. If the barrel recoils 4 ft on firing, determine a the initial recoil velocity of the barrel, b the critical damping coefficient of a dashpot which is engaged at the end of the recoil stroke, and c the time required for the barrel to return to a position 2 in.
Determine the maximum displacement of the piston after engaging the spring-damper. How many seconds does it take? Fipre Pl P, in terms of the displacement x. Plot In X vs. When will the two amplitudes be equal? Z Determine the differential equation of motion and establish the critical damping for the system shown in Fig. Common sources of harmonic excitation are unbalance in rotating machines, forces produced by reciprocating machines, or the motion of the machine itself.
These excitations may be undesirable to equipment whose operation may be disturbed or to the safety of the structure if large vibration amplitudes develop. Resonance is to be avoided in most cases, and to prevent large amplitudes from developing, dampers and absorbers are often used. Discussion of their behavior is of importance for their intelligent use. Finally, the theory of vibration measuring instruments is presented as a tool for vibration analysis.
It is commonly produced by the unbalance in rotating machinery. Although pure harmonic excitation is less likely to occur than periodic or other types of excitation, understanding the behavior of a system undergoing harmonic excitation is essential in order to comprehend how the system will respond to more general types of excitation. Harmonic excitation may be in the form of a force or displacement of some point in the system. We will first consider a single degree of freedom system with viscous damping, excited by a harmonic force F0 sin wt as shown in Fig.
Its 48 Forced Harmonic Vibration Flpre 3. The complimentary function, in this case, is a damped free vibration that was discussed in Chapter 2. The particular solution to the above equation is a steady-state oscillation of the same frequency w as that of the excitation. The amplitude and phase in the above equation are found by substituting Eq.
It is easily seen from this diagram that 3. Vector relationship for forced vibration with damping. Dividing the numerator and denominator of Eqs. I ;;- The above equations may be further expressed in terms of the following quantities: w,. These curves show that the damping factor has a large influence on the amplitude and phase angle in the frequency region near resonance. Further understanding. Frequency ratio 1, 1. Plot of Equations 3. The magnitude of the impressed force is then nearly equal to the spring force as shown in Fig.
The inertia force, which is now larger, is balanced by the spring force; whereas the impressed force overcomes the damping force. The amplitude at resonance can be found, either from Eqs. In summary, we can write the differential equation and its complete solution, including the transient term as 3. We consider here a spring mass system constrained to move in the vertical direction and excited by a rotating machine that is unbalanced, as shown in Fig.
Harmonic disturbing force resulting from rotating unbalance. By varying the speed of rotation, a resonant amplitude of 0.
When the speed of rotation was increased considerably beyond the resonant frequency, the amplitude appeared to approach a fixed value of 0. Determine the damping factor of the system. Solution: From Eq. Solving the two equations simultaneously, the damping factor of the system is r3. It is more likely that the unbalance in a rotating wheel or rotor is distributed in several planes.
We wish now to distinguish between two types of rotating unbalance and show how they may be corrected. Static Unbalance.
When the unbalanced masses all lie in a single plane, as in the case of a thin rotor disk, the resultant unbalance is a single radial force. The wheel will roll to a position where the heavy point is directly below the axle. Since such unbalance can be detected without spinning the wheel, it is called static unbalance. Ftpre 3. System with static unbalance. Dynamic Unbalance. When the unbalance appears in more than one plane, the resultant is a force and a rocking moment which is referred to as dynamic unbalance.
As previously described, a static test may detect the! For example, consider a shaft with two disks as shown in Fig. However, when the rotor is spinning, each unbalanced disk would set up a rotating centrifugal force, tending to rock the shaft on its bearings.
I Flpre 3. System with dynamic unbalance. Flpre 3. A rotor balancing machine. In general, a long rotor, such as a motor armature or an automobile engine crankshaft, can be considered to be a series of thin disks, each with some unbalance.
Such rotors must be spun in order to detect the unbalance. Essentially the balancing machine consists of supporting bearings which are spring mounted so as to detect the unbalanced forces by their motion, as shown in Fig. Knowing the amplitude of each bearing and their relative phase, it is possible to determine the unbalance of the rotor and correct for them. We describe one such test which can be simply performed.
The disk is supported on spring restrained bearings that can move horizontally as shown in Fig. Running at any predetermined speed, the amplitude X 0 and the wheel position "a" at maximum excursion are noted. An accelerometer on the bearing and a stroboscope can be used for this observation. The amplitude X 0 , due to the original unbalance m 0 , is drawn to scale on the wheel in the direction from o to a.
Next, a trial mass m 1 is added at any point on the wheel and the procedure is repeated at the same speed. The new amplitude X 1 and f. Balancing of Rotors 57 b Flpre 3. Experimental balancing of thin disk. The difference vector ab is then the effect of the trial mass m 1 alone. If the position of m 1 is now advanced by the angle cp shown in the vector diagram, and the magnitude of m 1 is increased to m 1 oaf ab , the vector ab will become equal and opposite to the vector oa.
The wheel is now balanced since X 1 is zero. Generally, the correction is made by drilling holes in the two end planes; i. With several unbalanced masses treated similarly, the correction to be made is found from their resultant in the two end planes.
Consider the balancing of a 4 in. The 3 oz in. The 2 oz in. Whirling is defined as the rotation of the plane made by the bent shaft and the line of centers of the bearings. The phenomenon results from such various causes as mass unbalance, hysteresis damping in the shaft, gyroscopic forces, fluid friction in bearings, etc. The whirling of the shaft may take place in the same or Qpposite direction as that of the rotation of the shaft, and the whirling speed may or may not be equal to the rotation speed.
We will consider here a single disk of mass m symmetrically located on a shaft supported by two bearings as shown in Fig. The center of mass G of the disk is at a distance e eccentricity from the geometric center S of the disk. The center line of the bearings intersects the plane of the disk at 0, and the shaft center is deflected by r - OS.
We will always assume the shaft i. For the equation of motion, we can develop the acceleration of the mass center as follows; l I t ,' 3. The latter term is directed from G to S since w is constant.
Some aspects of this motion will be treated in Sec. Syncluwwu Wlrirl. For the synchronous whirl, the whirling speed 9 is equal to the rotation speed w, which we have assumed to be constant. When the rotation ;nc liJ Figure 3. Figure 3. At very high speeds w :» w,. Assuming the critical speed w,. Assume zero damping. However r and ; terms must be retained unless shown to be zero.
Amplitude and phase relationship of synchronous whirl with viacous dampiDJ. We let y be the harmonic displacement of the support point and measure the displacement x of the mass m from an inertial reference.
System excited by motion of support point. The form of this equation is identical to that of E. It should be observed that the amplitude curves for different damping all have the same value of IX I Yl - 1.
In Fig. Thus the problem of isolating a mass from the motion of the support point is identical to that of isolating disturbing forces. Each of these ratios is referred to as transmissibility, and the ordinate of Fig. As seen from Fig. Some damping is desiraqle when it is necessary for w to vary through the resonant region, although the large amplitude at resonance can be limited by stops.
It is possible to reduce the amplitude of vibration by supporting the machine on a large mass Mas shown in Fig. However, since the amplitude X is reduced by the increased value of k. This discussion has been limited to bodies with translation along a single coordinate.
In general, a rigid body has six degrees of freedom; namely. For these more advanced cases the reader is referred to the excellent text on vibration isolation by C. Crede, Vibration and Siwek l. FTR 3. Its effect is to remove energy from the system. Energy in a vibrating system is either dissipated into heat or radiated away. Dissipation of energy into heat can be experienced simply by bending a piece of metal back and forth a number of times.
We are all aware of the sound which is radiated from an object given a sharp blow. When a buoy is made to bob up and down in the water, waves radiate out and away from it, thereby resulting in its loss of energy. In vibration analysis, we are generally concerned with damping in terms of system response.
The loss of energy from the oscillatory system results in the decay of amplitude of free vibration. A vibrating system may encounter many different types of damping forces, from internal molecular friction to sliding friction and fluid resistance. Generally their mathematical description is quite complicated and not suitable for vibration analysis. Thus simplified damping models have been developed that in many cases are found to be adequate in evaluating the system response.
For example, we have already used the viscous damping model, designated by the dashpot, which leads to manageable mathematical solutions. Energy dissipation is usually determine4 under conditions of cyclic oscillations. Depending on the type of damping present, the force-displacement relationship when plotted may differ greatly. In all cases, however, the force-displacement curve will enclose an area, referred to as the hysteresis loop, that is proportional to the energy lost per cycle.
The energy lost per cycle due to a damping force Fd is computed from the general equation 3. We will consider in this section the simplest case of energy dissipation, that of a spring-mass system with viscous damping. The energy dissipated per cycle is then given by the area enclosed by the ellipse.
If we add to Fd the force kx of the lossless spring, the hysteresis loop is rotated as shown in Fig. This representation then conforms to the Voigt model, which consists of a dashpot in parallel with a spring.
Energy dissipated by viscous damping. Damping properties of materials are listed in many different ways depending on the technical areas to which they are applied.
Of these we list two relative energy units which have wide usage. First of these is specific damping capacity defined as the energy loss per cycle Wd divided by the peak potential energy U. The toss coefficient for most materials varies between 0. When the damping loss is not a quadratic function of the strain or amplitude, the hysteresis curve is no longer an ellipse. Power is the rate of doing work which is the product of the force and velocity. The second term is a sine wave of twice the frequency that represents the fluctuating component of power, the average value of which is zero over any interval of time that is a multiple of the period.
Rewriting Eq. As seen from the response curves of Fig. In the case of viscous damping, the amplitude at resonance, Eq. It is possible, however, to approximate the resonant amplitude by substituting an equivalent damping ceq in the above equation.
The equivalent damping ceq is found by equating the energy dissipated by the viscous damping to that of the nonviscous damping force with assumed harmonic motion. From Eq. Determine the equivalent damping for such forces acting on an oscillatory system, and find its resonant amplitude.
X2 where the negative sign must be used when x is positive, and vice versa. Assuming harmonic motion with the time measured from the position of extreme negative displacement l x- -Xcoswt! Develop the equation for the equivalent damping and indicate the procedure for determining the amplitude at resonance.
Its substitution into Eq. Internal damping fitting this classification is called solid damping or structural damping. With the energy dissipation per cycle proportional to the square of the vibration amplitude, the loss coefficient is a constant and the shape of the hysteresis curve remains unchanged with amplitude and independent of the strain rate.
Using the concept of equivalent viscous damping, Eq. In the calculation of the flutter speeds of airplane wings and tail surfaces, the concept of complex stiffness is used.
It is arrived at by assuming the oscillations to be harmonic, which enables Eq. The method is justified, however, only for harmonic oscillations. With the solution x Xe;"", the steady state amplitude from Eq. To determine this quantity, we will assume viscous damping and start with Eq.
We now seek the two frequencies on either side of resonance often referred to as sidebands , where X is 0. These points are also referred to as the half-power points and are shown in Fig. FICUft 3. Thus, for structural damping, Q is equal to Q 3. In place of the Fourier series of Sec. Solution: Pis The Fourier series for the rectangular wave of amplitude which contains only odd harmonics.
Depending on the frequency range utilized, displacement, velocity, or acceleration is indicated by the relative motion of the suspended mass with respect to the case. Frequency ra t 10 Flpre 3. When the natural frequency w,. The mass m then remains stationary while the supporting case moves with the vibrating body. Such instruments are called seismometers. One of the disadvantages of the seismometer is its large size. Since Z - Y, the relative motion of the seismic mass must be of the same order of magnitude as that of the vibration to be measured.
The relative motion z is usually converted to an electric voltage by making the seismic mass a magnet moving relative to coils fixed in the case as shown in Fig. Since the voltage generated is proportional to the rate of cutting of the magnetic field, the output of the instrument will be proportional to the velocity of the vibrating body.
Such instruments are called velometers. A typical instrument of this kind may have a natural frequency between I Hz to 5 Hz and a useful frequency range of 10 Hz to Hz. Both the displacement and acceleration are available from the velocity-type transducer by means of the integrator or the differentiator provided in most signal conditioner units.
The Ranger seismometer incorporates a velocity-type transducer with the permanent magnet as the seismic mass. Its size is 15 em in diameter and it weighs II lb. Courtesy of Kinemetrics, Inc. Accelerometer-Instrument with High Natural Frequency.
When the natural frequency of the instrument is high compared to that of the vibration to be measured, the instrument indicates acceleration. Examina tion of Eq. The useful range of the accelerometer can be 8l Harmonically Excited Vibration 1. Thus an instrument with a natural frequency of Hz has a useful frequency range between 0 Hz to 20 Hz with negligible error. On the other hand, very high natural frequency instruments, such as the piezoelectric crystal accelerometers, have almost zero damping and operate without distortion up t0 frequencies of 0.
The seismic mass accelerometer is often used for low frequency vibration, and the supporting springs may be four electric strain gage wires connected in a bridge circuit. A more accurate variation of this accelerometer is one in which the seismic mass is servo-controlled to have zero relative displacement; the force necessary to accomplish this becomes a measure of the acceleration.
Both of these instruments require an external source of electric power. The piezoelectric propecties of crystals like quartz or barium titanate are utilized in accelerometers for higher frequency measurements. The r Vibration Measuring Instruments Figure 3. The natural frequency of such accelerometers can be made very high, in the 50, Hz range, which enables acceleration measurements to be made up to Hz.
The size of the crystal accelerometer is very small, approximately l em in diameter and height, and it is remarkably rugged and can stand shocks as high as 10, g's. Typical sensitivity for a crystal accelerometer is 25 pCI g with crystal capacitance of pF picofarads. To reproduce a complex wave such as the one shown in Fig.
This requires that the phase angle be zero or that all the harmonic components must be shifted equally. Thus the instrument faithfully reproduces the acceleration y without distortion. Determine the damping coefficient when a harmonic exciti11g force of When the weight is displaced and released, the period of vibration is found to be 1. Determine the amplitude and phase when a force F At resonance the amplitude is measured to be 0. P, set up the equation of motion and solve for the steady-state amplitude and phase angle by using complex algebra.
Determine the amplitude of the cylinder motion and its phase with respect to the piston. P is used to determine the vibrational characteristics of a structure of mass At a speed of rpm, a stroboscope shows the eccentric masses to be at the top at the instant the structure is moving upward through its static equilibrium posttion, and the corresponding amplitude is If the unbalance of each wheel of the exciter is 0.
Determine the diameter and position of a third hole at 10 em radius that will balance the disk. P3-ll is equivalent to an eccentric weight of w lb at a radius of r in. Determine the counterweights necessary at the two flywheels if they are also placed at a radial distance of r in. Figure PJ Determine the lowest critical speed.
Assume shaft to be simply supported at the bearings. The rotor is known to have an unbalance of 0. Determine the forces exerted on the bearings at a speed of rpm if the diameter of the steel shaft is 2.
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