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engineering
mechanical vibration analysis

Mechanical Vibrations 6th Edition Singiresu S Rao - Solutions

A damped single-degree-of-freedom system has a mass \((m)\) of \(5 \mathrm{~kg}\), stiffness \((k)\) of \(20 \mathrm{~N} / \mathrm{m}\), and a damping constant \((c)\) of \(5 \mathrm{~N}-\mathrm{s} / \mathrm{m}\). Design an active controller to achieve a settling time less than \(15 \mathrm{~s}\)
A stepped bar, fixed at \(x=0\) and free at \(x=l\), has a cross-sectional area of \(2 A\) for \(0 \leq x
Estimate the fundamental frequency for the longitudinal vibration of a uniform bar fixed at \(x=0\) and free at \(x=l\) by assuming the mode shapes as(a) \(U(x)=c_{1}(x / l)\) and(b) \(U(x)=c_{1}(x / l)+c_{2}(x / l)^{2}\).
Estimate the fundamental frequency of a fixed-fixed string, assuming the mode shape(a) \(W(x)=c_{1} x(l-x)\) and(b) \(W(x)=c_{1} x(l-x)+c_{2} x^{2}(l-x)^{2}\).
Using Rayeigh's method, determine the fundamental natural frequency of the system shown in Fig. 8.42. Disc, mass moment of inertia 5 kg-m Steel shaft, dia. 5 cm + O 0.8 m -0.2 m- FIGURE 8.42 Fixed-fixed shaft carrying a disc.
Solve Problem 8.73 using the Rayleigh-Ritz method with the mode shape\[U(x)=c_{1} \sin \frac{\pi x}{2 l}+c_{2} \sin \frac{3 \pi x}{2 l}\]
Find the first two natural frequencies of a fixed-fixed uniform string of mass density \(ho\) per unit length stretched between \(x=0\) and \(x=l\) with an initial tension \(P\). Assume the deflection functions\[\begin{aligned}& w_{1}(x)=x(l-x) \\& w_{2}(x)=x^{2}(l-x)^{2}\end{aligned}\]
Using Program \(12 . \mathrm{m}\), solve Example 8.4.Data From Example 8.4:- Find the natural frequencies of a bar with one end fixed and a mass attached at the other end, as in Fig. 8.9.
Using Program12.m, find the first five natural frequencies of a thin fixed-fixed beam.Data From Example 8.4:- Find the natural frequencies of a bar with one end fixed and a mass attached at the other end, as in Fig. 8.9.
Using MATLAB, plot the dynamic response of the plucked string, Eq. (E.6) of Example 8.1, at \(x=l / 2\). Data: \(h=0.1 \mathrm{~m}, l=1.0 \mathrm{~m}, c=100 \mathrm{~m} / \mathrm{s}\).Data From Example 8.1:-Figure 8.5:- If a string of length 1, fixed at both ends, is plucked at its midpoint as
Write a computer program for finding numerically the mode shapes of thin fixed-simply supported beams by using the known values of the natural frequencies.
A vehicle, of weight \(F_{0}\), moving at a constant speed on a bridge (Fig. 8.43(a)) can be modeled as a concentrated load traveling on a simply supported beam as shown in Fig. 8.43(b). The concentrated load \(F_{0}\) can be considered as a uniformly distributed load over an infinitesimal length
True or False.Vibration can cause structural and mechanical failures.
Fill in the Blank.Even a small excitation force can cause an undesirably large response near ____________ .
An example of a source of vibration that cannot be altered is:a. atmospheric turbulenceb. hammer blowc. tire stiffness of a car
A car moving on a rough road, in the form of a sinusoidal surface, is modeled as a springmass system, as shown in Fig. 9.41. The sinusoidal surface has a wave length of \(5 \mathrm{~m}\) and an amplitude of \(Y=1 \mathrm{~mm}\). If the mass of the car, including the passengers, is \(1500
Name some sources of industrial vibration.
True or False.The response of a system can be reduced by the use of isolators and absorbers.
Fill in the Blank.The use of close tolerances and better surface finish for machine parts tends to make a machine ____________ susceptible to vibration.
The two-plane balancing is also known as:a. static balancingb. dynamic balancingc. proper balancing
True or False.Vibration control means the elimination or reduction of vibration.
The unbalanced force caused by an eccentric mass \(m\) rotating at an angular speed \(\omega\) and located at a distance \(r\) from the axis of rotation isa. \(m r^{2} \omega^{2}\)b. \(m g \omega^{2}\)c. \(m r \omega^{2}\)
Two identical discs are connected by four bolts of different sizes and mounted on a shaft, as shown in Fig. 9.42. The masses and locations of three bolts are as follows:\(m_{1}=35 \mathrm{~g}, r_{1}=110 \mathrm{~mm}\), and \(\theta_{1}=40^{\circ} ; m_{2}=15 \mathrm{~g}, r_{2}=90 \mathrm{~mm}\), and
What is single-plane balancing?
True or False.The vibration caused by a rotating unbalanced disc can be eliminated by adding a suitable mass to the disc.
Fill in the Blank.When the speed of rotation of a shaft equals one of the natural frequencies of the shaft, it is called ____________ speed.
The following material has high internal damping:a. cast ironb. copperc. brass
Four holes are drilled in a uniform circular disc at a radius of \(100 \mathrm{~mm}\) and angles of \(0^{\circ}, 60^{\circ}\), \(120^{\circ}\), and \(180^{\circ}\). The weight removed at holes 1 and 2 is \(100 \mathrm{~g}\) each and the weight removed at holes 3 and 4 is \(150 \mathrm{~g}\) each.
Describe the two-plane balancing procedure.
True or False.Any unbalanced mass can be replaced by two equivalent unbalanced masses in the end planes of the rotor.
Fill in the Blank.The moving elements of a reciprocating engine are the crank, the connecting rod, and the ____________.
Transmissibility is the ratio ofa. force transmitted and exciting forceb. force applied and the resulting displacementc. input displacement and output displacement
Three masses, weighing \(225 \mathrm{~g}, 320 \mathrm{~g}\), and \(550 \mathrm{~g}\), are attached around the rim, of diameter \(750 \mathrm{~mm}\), of a flywheel at the angular locations \(\theta=10^{\circ}, 100^{\circ}\), and \(190^{\circ}\), respectively. Find the weight and the angular location
What is whirling?
What is the difference between stationary damping and rotary damping?
True or False.The oil whip in the bearings can cause instability in a rotor system.
Fill in the Blank.The vertical component of the inertia force of a reciprocating engine has primary and ____________ parts.
Mechanical impedance is the ratio ofa. force transmitted and exciting forceb. force applied and force transmittedc. applied force and displacement
The amplitude and phase angle due to original unbalance in a grinding wheel operating at \(1200 \mathrm{rpm}\) are found to be \(0.25 \mathrm{~mm}\) and \(40^{\circ}\) counterclockwise from the phase mark. When a trial mass \(m=170 \mathrm{~g}\) is added at \(65^{\circ}\) clockwise from the phase
How is the critical speed of a shaft determined?
True or False.The natural frequency of a system can be changed by varying its damping.
Fill in the Blank.Laminated structures have ____________ damping.
Vibration can be eliminated on the basis of theoretical analysisa. sometimesb. alwaysc. never
An unbalanced flywheel shows an amplitude of \(0.165 \mathrm{~mm}\) and a phase angle of \(15^{\circ}\) clockwise from the phase mark. When a trial weight of magnitude \(50 \mathrm{~g}\) is added at an angular position \(45^{\circ}\) counterclockwise from the phase mark, the amplitude and the phase
What causes instability in a rotor system?
True or False.The stiffness of a rotating shaft can be altered by changing the location of its bearings.
Fill in the Blank.Materials with a large value of the loss factor are subject to ____________ stress.
A long rotor can be balanced by adding weights ina. a single planeb. any two planesc. two specific planes
In order to determine the unbalance in a grinding wheel, rotating clockwise at \(2400 \mathrm{rpm}\), a vibration analyzer is used and an amplitude of \(0.1 \mathrm{~mm}\) and a phase angle of \(45^{\circ}\) are observed with the original unbalance. When a trial mass \(m=100 \mathrm{~g}\) is added
What considerations are to be taken into account for the balancing of a reciprocating engine?
True or False.All practical systems have damping.
Fill in the Blank.Vibration isolation involves insertion of a resilient member between the vibrating mass and the of ____________ vibration.
The damping caused by the internal friction of a shaft material is calleda. stationary dampingb. external dampingc. rotary damping
What is the function of a vibration isolator?
True or False.High loss factor of a material implies less damping.
A rotor, having three eccentric masses in different planes, is shown in Fig. 9.43. The axial, radial, and angular locations of mass \(m_{i}\) are given by \(l_{i}, r_{i}\), and \(\theta_{i}\), respectively, for \(i=1,2,3\). If the rotor is to be dynamically balanced by locating two masses \(m_{b
Fill in the Blank.Cork is a(n) ____________ isolator.
The damping caused by the bearing support structure of a rotating shaft is calleda. stationary dampingb. internal dampingc. rotary damping
What is a vibration absorber?
True or False.Passive isolation systems require external power to function.
Fill in the Blank.An active isolator consists of a sensor, a signal processor, and a(n) ____________ .
An undamped vibration absorber removes the original resonance peak but introducesa. one new peaka. two new peaksb. several new peaks
The rotor shown in Fig. 9.44 (a) is balanced temporarily in a balancing machine by adding the masses \(m_{1}=m_{2}=90 \mathrm{~g}\) in the plane \(A\) and \(m_{3}=m_{4}=90 \mathrm{~g}\) in the plane \(D\) at a radius of \(75 \mathrm{~mm}\), as shown in Fig. 9.44(b). If the rotor is permanently
A turbine rotor is run at the natural frequency of the system. A stroboscope indicates that the maximum displacement of the rotor occurs at an angle \(229^{\circ}\) in the direction of rotation. At what angular position must mass be removed from the rotor in order to improve its balancing?
The reciprocating mass, crank radius, and connecting-rod length of each of the cylinders in a two-cylinder in-line engine are given by \(m, r\), and \(l\), respectively. The crank angles of the two cylinders are separated by \(180^{\circ}\). Find the unbalanced forces and moments in the engine.
The cylinders of a four-cylinder in-line engine are placed at intervals of \(300 \mathrm{~mm}\) in the axial direction. The cranks have the same length, \(100 \mathrm{~mm}\), and their angular positions are given by \(0^{\circ}, 180^{\circ}, 180^{\circ}\), and \(0^{\circ}\). If the length of the
The force transmitted by an internal combustion engine of mass \(500 \mathrm{~kg}\), when placed directly on a rigid floor, is given by\[F_{t}(t)=(18,000 \cos 300 t+3600 \cos 600 t) \mathrm{N}\]Design an undamped isolator so that the maximum magnitude of the force transmitted to the floor does not
Is it always advantageous to include a damper in the secondary system of a dynamic vibration absorber?
A shaft, having a stiffness of \(3.75 \mathrm{MN} / \mathrm{m}\), rotates at \(3600 \mathrm{rpm}\). A rotor, having a mass of \(60 \mathrm{~kg}\) and an eccentricity of 2000 microns, is mounted on the shaft. Determine (a) the steady-state whirl amplitude of the rotor and (b) the maximum whirl
Find the bearing reactions and the maximum bending stress induced in the shaft at (a) operating speed, (b) critical speed, and (c) 1.5 times the critical speed for the shaft-rotor system described in Problem 9.17.Data From Problem 9.17:-A steel shaft of diameter \(2.5 \mathrm{~cm}\) and length \(1
Fill in the Blank.Oil whip in bearings can cause ____________ in a flexible rotor system.
The data obtained in a two-plane balancing procedure are given in the table below. Determine the magnitude and angular position of the balancing masses, assuming that all angles are measured from an arbitrary phase mark and all masses are added at the same radius. Amplitude (mm) Phase Angle
Fill in the Blank.The single-plane balancing is also known as ____________ balancing.
Is it better to use a soft spring in the flexible mounting of a machine? Why?
True or False.Internal and external friction can cause instability in a rotating shaft at speeds above the first critical speed.
Why does a rotating shaft always vibrate? What is the source of the shaking force?
Fill in the Blank.The combustion instabilities are a source of ____________ in engines.
Fill in the Blank.The deflection of a rotating shaft becomes very large at the ____________ speed.
A steel shaft of diameter \(2.5 \mathrm{~cm}\) and length \(1 \mathrm{~m}\) is supported at the two ends in bearings. It carries a turbine disc, of mass \(20 \mathrm{~kg}\) and eccentricity \(0.005 \mathrm{~m}\), at the middle and operates at \(6000 \mathrm{rpm}\). The damping in the system is
A washing machine of mass \(50 \mathrm{~kg}\) operates at \(1200 \mathrm{rpm}\). Find the maximum stiffness of an isolator that provides 75 percent isolation. Assume that the damping ratio of the isolator is 7 percent.
Explain why dynamic balancing can never be achieved by a static test alone.
What is the difference between a vibration isolator and a vibration absorber?
The arrangement of cranks in a six-cylinder in-line engine is shown in Fig. 9.48. The cylinders are separated by a distance \(a\) in the axial direction, and the angular positions of the cranks are given by \(\alpha_{1}=\alpha_{6}=0^{\circ}, \alpha_{2}=\alpha_{5}=120^{\circ}\), and
at an engine speed of \(1500 \mathrm{rpm}\). 1 1 Reference plane 10 cm- -10 cm- 5 cm 5 cm FIGURE 9.47 Four cylinder in-line engine.
Does spring mounting always reduce the vibration of the foundation of a machine?
Fill in the Blank.Although an undamped vibration absorber removes the original resonance peak of the response, it introduces ____________ new peaks.
True or False.The force transmitted to the foundation of an isolator with rigid foundation can never be infinity.
Avoid excessive response at resonancea. Introduce dampingb. Use vibration isolatorc. Add vibration absorberd. Avoid resonance
Why does dynamic balancing imply static balancing?
What is active vibration isolation?
Solve Problem 9.17 by assuming that the material of the shaft is aluminum rather than steel.Data From Problem 9.17:-A steel shaft of diameter \(2.5 \mathrm{~cm}\) and length \(1 \mathrm{~m}\) is supported at the two ends in bearings. It carries a turbine disc, of mass \(20 \mathrm{~kg}\) and
Solve Problem 9.18 by assuming that the material of the shaft is aluminum rather than steel.Data From Problem 9.18:-Find the bearing reactions and the maximum bending stress induced in the shaft at (a) operating speed, (b) critical speed, and (c) 1.5 times the critical speed for the shaft-rotor
A damped single-degree-of-freedom system has an undamped natural frequency of \(20 \mathrm{rad} / \mathrm{s}\) and a damping ratio of 0.20. Design an active control system which achieves an undamped natural frequency of \(100 \mathrm{rad} / \mathrm{s}\) and a damping ratio of 0.8. Assume that the
Explain the difference between passive and active isolation.
A four-cylinder in-line engine has a reciprocating weight of \(1.5 \mathrm{~kg}\), a stroke of \(15 \mathrm{~cm}\), and a connecting-rod length of \(25 \mathrm{~cm}\) in each cylinder. The cranks are separated by \(10 \mathrm{~cm}\) axially and \(90^{\circ}\) radially, as shown in Fig. 9.47. Find
A single-cylinder engine has a total mass of \(150 \mathrm{~kg}\). Its reciprocating mass is \(5 \mathrm{~kg}\), and the rotating mass is \(2.5 \mathrm{~kg}\). The stroke ( \(2 \mathrm{r})\) is \(15 \mathrm{~cm}\), and the speed is \(600 \mathrm{rpm}\).(a) If the engine is mounted floating on very
A diesel engine of mass \(300 \mathrm{~kg}\) and operating speed \(1800 \mathrm{rpm}\) is found to have a rotating unbalance of \(1 \mathrm{~kg}-\mathrm{m}\). It is to be installed on the floor of an industrial plant for purposes of emergency power generation. The maximum permissible force that can
A sensitive electronic system, of mass \(30 \mathrm{~kg}\), is supported by a spring-damper system on the floor of a building that is subject to a harmonic motion in the frequency range \(10 \mathrm{~Hz}\) to \(75 \mathrm{~Hz}\). If the damping ratio of the suspension is 0.25, determine the
A machine of mass \(1150 \mathrm{~kg}\) is mounted on springs. A piston of mass \(m=25 \mathrm{~kg}\) moves up and down in the machine at a speed of \(600 \mathrm{rpm}\) with a stroke of \(350 \mathrm{~mm}\). Considering the motion to be harmonic, determine the maximum force transmitted to the
True or False.The transmissibility is also called the transmission ratio.

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