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. Compare this result with that of the same rotor mounted on a steel shaft of diameter 1. Assume the shaft to be simply supported at the bearings. In the turbine of Prob. J Problems 87 Figure PJ Assume that the critical speed is reached with zero amplitude.
Determine the equation for the amplitude of W as a function of the speed and determine the most unfavorable speed. Find the critical speed when the trailer is traveling over a road with a profile approximated by a sine wave of amplitude 7.
What will be the amplitude of vibration at Neglect damping. Write the differential equation of motion for small amplitude of. Determine the solution for x j x 0 and show that when '" - v'2 w,. Derive Eqs. If the unit operates at rpm, what should be the value of the spring constant k if only 10 per cent of the shaking force of the unit is to be tr nsmitted to the supporting structure?
If the machine has a rotating unbalance of 0. Assume damping to be negligible. What acceleration is Iransmitted to the instrument? Give numerical values to substantiate your solution. P , verify that the transmissibility TR lxlyl is the same as that for force.
Plot the transmissibility in decibels, 20 logl TR I vs. State under what condit1on the logarithmic decrement 8 is independent of the a mplituJe. Determine the equivalent viscous damping. Under what condition can this motion be maintained? P determine the steady-state response of the spring-mass damper system to the excitation of Prob. P is applied to a spring-mass system, determine the ratio of the response to the various harmonics compared with the fundamental.
I FJcurePJ P , undergoes harmonic torsional oscillation 80 sin wt. Determine the expression for the relative amplitude of the outer wheel with respect to a the shaft, b a fixed reference. I I r Figure PJ What is the lowest frequency that can be measured with a I percent error, b 2 percent error? I An undamped vibration pickup having a natural frequency of I cps is used to measure a harmonic vibrat1on of 4 cps. If the amplitude indicated by the pickup relative amplitude between pickup mass and frame is 0.
Sensitivity: 0. If a reading of 0. Give reasons. If I g acceleration is maintained over this frequency range, what will be the output voltage at a 10 Hz and b at Hz. Assuming that 3 mV rms is the accuracy limit of the instrument, determine the upper frequency limit of the instrument for lg excitation.
What voltage would be generated at Hz? Determine its voltage output per g. Such oscillatiOns take place at the natural frequencies of the system with the amplitude varying in a manner dependent on the type of excitation.
Such forces are called impulsive. Figure 4. As E approaches zero, such forces tend to become infinite; however, the impulse defined by its time integral is F which is considered to be finite. Under free vibratiOn we found that the undamped spring-mass system with initial conditions x O and x O behaved according to the equation Hence the response of a spring-mass system initially at rest and excited by an impulse F is x F.
For this development, we consider the arbitrary force to be a series of impulses as shown in Fig. Base Excitation. Ofter the support of the dynamical system is subjected to a sudden movement specified by its displacement, velocity, or acceleration.
For an undamped system initially at rest, the solution for the relative displacement becomes 4. Step function excitation. Considering the undamped system. Response to a unit step function. The velocity together with its time rate of change is shown in Fig.
Determine the steady-state response. If steady state is attained, the displacement and velocity after each cycle must repeat themselves.
WnT 2wnm sm-- k 1 2 The maximum spring force F, Eq. Displacement and velocity. When damping is included, a similar procedure can be applied, although the numerical work is increased considerably. For those unfamiliar with the method, a bnef presentation of the Laplace transform theory is given in the Appendix. In this section we will illustrate its use by some simple examples.
If only the forced solution is considered, we can define the impedance transform as F s. The admittance transform H s then cari also be considered as the system transfer function, defined as the ratio in the subsidiary plane of the output over the input with all initial conditions equal to zero. Block diagram. Such considerations are of paramount importance in the landing of airplanes or the cushioning of packaged articles. Consider the spring-mass system of Fig.
If x is measured from the position of m at the instant t - 0 when the spring first contacts the floor. From the inverse transformation of. Mindlin, "Dynamics of Package Cushioning. July, , pp. Response Spectrum Figure 4. The maximum value of the response is a good measure of the severity of the shock and is, of course, dependent upon the dynamic characteristics of the system.
In order to categorize all types of shock excitations, a single degree of Transient Vibration freedom undamped oscillator spring-mass system is chosen as a standard system. Engineers have found the concept of the response spectrum to be useful in design.
A response spectrum is a plot of the maximum peak response of the single degree of freedom oscillator as a function of the natural frequency of the oscillator. Different types of shock excitations will then result in different response spectra.
Since the response spectrum is determined from a single point on the time response curve, which is itself an incomplete bit of information, it does not uniquely define the shock input.
In fact, it is possible for two different shock excitations to have very similar response spectra. In spite of this limitation, the response spectrum is a useful concept that is extensively used. The response of a system to arbitrary excitation f t was expressed in terms of the impulse response h t by Eq. Figures 4. The dynamic factor of a shock is then generally less than two. Pseudo Response Spectra. In ground shock situations, it is often convenient to express the response spectra in terms of the velocity spectra.
The displacement and acceleration spectra then can be expressed in terms of the velocity spectra by dividing or multiplying by w,. Such results are called pseudo spectra since they are exact only if the peak response occurs after the shock pulse has passed, in which case the motion is harmonic. Thus the peak velocity response Sv or the velocity spectrum is given with sufficient accuracy by the peak value of the envelope 4.
Approximate relations for the peak displacement and acceleration, known as pseudo spectra, are then 4. IP must be greater than. It is evident that for this problem this results in a transcendental equation which must be solved by plotting. To avoid this numerical task, we will consider a different approach as follows.
Response Spec:trum y t Flpre 4. Impulsive doublet. For small w,, or a very soft spring, the duration of the input would be small compared to the period of the system.
Hence the input would appear as an impulsive doublet shown in Fig. Response spectrum for the base velocity input y t.. This may well be the case when the system is nonlinear or if the system is excited by a force that cannot be expressed by simple analytic functions. The solution is approximate, but with a sufficiently small time increment the solution of acceptable accuracy is obtainable. Although there are a number of different finite difference procedures available, in this chapter we consider only two methods chosen for their simplicity.
Merits of the various methods are associated with the accuracy, stability, and length of computation, which are discussed in a number of texts on numerical analysis listed at the end of the chapter. In the first method the second-order equation is integrated without change in form; in the second method the second-order equation is reduced to two first-order equations before integration.
Subtracting and ignoring higher-order terms, we obtain 4. Starting the Computation. This is supplied by the first of Taylor's series, Eq. In this development we have ignored higher-order terms that introduce what is known as truncation errNs.
Other errors;such as round-off errors. In general, better accuracy is obtained by choosing a smaller! A safe rule to use in Method 1 is to l. A flow diagram f,n the J1gital calculation is shown in Fig. Flow diagram undamped system. F t Figure 4. From the differential equation we M. The latter was obtained by the superposition of the solutions for the step function and the ramp function in the following F X 0 Initial acceleration and initial conditions zero.
X 1 will also be zero and the computation cannot be started because Eq. This condition can be rectified by developing new starting equations based on the assumption that during the first-time interval the acceleration varies linearly from. X 2 as follows:. The differential equation of motion and the initial conditions are given as The triangular force is defined in Fig. F f Figure 4. The index I controls the computation path on the diagram. The Fortran program can be written in many ways, one of which is shown in Fig.
The response x vs. When damping is present, the differential equation contains an additional term X; and Eq.
Considering again the first three terms of the Taylor senes, Eq. The procedure is thus repeated for other values of X; and X; using the Taylor series. YES -. Sfl ISil! Stl ISU! Stl ISrl! Stl ISII! A brief discussion of its basis is presented here. In the Runge-Kutta method the second-order differential equation is first reduced to two first-order equations. The Runge-Kutta method is very similar to the above computations, except that the center term of the above equation is split into two terms and four values of t, x, y, and fare computed for each point i as follows T,- II X1.
Solution: The differential equation of motion is. Q3 13 0. Q2 Table 4. It is seen that the Runge-Kutta method gives greater accuracy than the central difference method. Mathematical Methods for Digital Computers, Vols. I and II. Englewood Cliffs, N. Numerical Methods in Finite Element Analysis. Considering the pulse to be the sum of two step pulses, as shown in Fig. Fot, - - Figure P Evaluate the second term due to initial conditions by the inverse transforms.
Show that its solution is t Determine the time elapsed from first contact of the spring until it breaks contact again. Flpre P If the system is lifted so that the bottom of the springs are just free and released, determine the maximum displacement of m, and the time for maximum compression.
P4-l5 has a Coulomb damper which exerts a constant friction force f. Jmv0 as parameter. P is the response spectrum for the sine pulse. Explore Ebooks. Bestsellers Editors' Picks All Ebooks. Explore Audiobooks.
Bestsellers Editors' Picks All audiobooks. Explore Magazines. Editors' Picks All magazines. Explore Podcasts All podcasts. Difficulty Beginner Intermediate Advanced. Explore Documents. Uploaded by H B. Document Information click to expand document information Description: Solution manual for the 5th edition of theory of vibration with application.
Original Title Theory of Vibration with application 5th Solution. Did you find this document useful? Is this content inappropriate? Report this Document. Description: Solution manual for the 5th edition of theory of vibration with application.
Flag for inappropriate content. Download now. Save Save Theory of Vibration with application 5th Solutio For Later. Theory of Vibration With Application 5th Solution. Original Title: Theory of Vibration with application 5th Solution. Related titles. Carousel Previous Carousel Next. Theory of Vibration With Application 3rd Solution. Fundementals of Mechanical Vibration - S. Graham Kelly. Fundamentals of Vibrations by Leonard Meirovitch Includes a chapter on computer methods, and an accompanying disk with four basic Fortran programs covering most of the calculations encountered in vibration problems.
Author : William T. Author : William Tyrrell. Thomson Publisher: Springer ISBN: Category: Science Page: View: Read Now » This fourth edition of this volume features a new chapter on computational methods that presents the basic principles on which most modern computer programs are developed. It adds coverage of the methods of assumed modes and incorporates a new section on suspension bridges to illustrate the application of the continuos system theory to simplified models for the calculation of natural frequencies.
Author : A. The book presents in a simple and systematic manner techniques that can easily be applied to the analysis of vibration of mechanical and structural systems.
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