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engineering
mechanical vibration analysis
Mechanical Vibrations 6th Edition Singiresu S Rao - Solutions
True or False.The Ritz-Galerkin method finds the approximate solution by satisfying the nonlinear equation in the average.
True or False.Dry friction can introduce nonlinearity in the system.
True or False.Poincaré's solution of nonlinear equations is in the form of a series.
True or False.The secular term appears in the solution of the free Duffing's equation.
According to Lindstedt's perturbation method, the angular frequency is assumed to be a function of the amplitude.
True or False.An isocline is the locus of points at which the trajectories passing through them have a constant slope.
True or False.Time does not appear explicitly in a trajectory plotted in the phase plane.
True or False.The time variation of the solution can be found from the phase-plane trajectories.
True or False.A limit cycle denotes a steady-state periodic oscillation.
True or False.Approximate solutions of nonlinear vibration problems can be found using numerical methods such as the Houbolt, Wilson, and Neumark methods.
What is the importance of vibration measurement?
What is the difference between a vibrometer and a vibrograph?
What is a transducer?
Discuss the basic principle on which a strain gage works.
Define the gage factor of a strain gage.
What is the difference between a transducer and a pickup?
What is a piezoelectric material? Give two examples of such material.
What is the working principle of an electrodynamic transducer?
What is an LVDT? How does it work?
What is a seismic instrument?
What is the frequency range of a seismometer?
What is an accelerometer?
What is phase-shift error? When does it become important?
Give two examples of a mechanical vibration exciter.
What is an electromagnetic shaker?
Discuss the advantage of using operational deflection shape measurement.
What is the purpose of experimental modal analysis?
Describe the use of the frequency-response function in modal analysis.
Name two frequency-measuring instruments.
State three methods of representing the frequency-response data.
How are Bode plots used?
How is a Nyquist diagram constructed?
What is the principle of mode superposition? What is its use in modal analysis?
State the three types of maintenance schemes used for machinery.
How are orbits used in machine diagnosis?
Define the terms kurtosis and cepstrum.
True or False.A strain gage is a variable-resistance transducer.
True or False.The value of the gage factor of a strain gage is given by the manufacturer.
True or False.The voltage output of an electromagnetic transducer is proportional to the relative velocity of the coil.
True or False.The principle of the electrodynamic transducer can be used in vibration exciters.
True or False.A seismometer is also known as a vibrometer.
True or False.All vibration-measuring instruments exhibit phase lag.
True or False.The time lag is important when measuring harmonic motion of frequency \(\omega\).
True or False.The Scotch yoke mechanism can be used as a mechanical shaker.
True or False.The time response of a system gives better information on energy distribution than does the frequency response.
True or False.A spectrum analyzer is a device that analyzes a signal in the frequency domain.
True or False.The complete dynamic response of a machine can be determined through modal testing.
True or False.The damping ratio of a vibrating system can be found from the Bode diagram.
True or False.The spectrum analyzers are also known as fast Fourier transform (FFT) analyzers.
True or False.In breakdown maintenance, the machine is run until failure.
True or False.Time-domain waveforms can be used to detect discrete damages of machinery.
Fill in the Blank.A device that transforms values of physical variables into equivalent electrical signals is called a(n) _____________ .
Fill in the Blank.Piezoelectric transducers generate electrical ______________ when subjected to mechanical stress.
Fill in the Blank.A seismic instrument consists of a(n) ______________ system mounted on the vibrating body.
Fill in the Blank.The instrument that measures the acceleration of a vibrating body is called ____________ .
Fill in the Blank.____________ can be used to record earthquakes.
Fill in the Blank.The instrument that measures the velocity of a vibrating body is called a(n) _____________ .
Fill in the Blank.Most mechanical frequency-measuring instruments are based on the principle of _____________ .
Fill in the Blank.The Frahm tachometer is a device consisting of ______________ several carrying masses at free ends.
Fill in the Blank.The main advantage of a stroboscope is that it can measure the speed without making _______________ with the rotating body.
Fill in the Blank.In real-time frequency analysis, the signal is continuously analyzed over all the ___________ bands.
Fill in the Blank.Real-time analyzers are useful for machinery ___________ monitoring, since a change in the noise or vibration spectrum can be observed immediately.
Fill in the Blank.\(\mathrm{A}(\mathrm{n})\) ___________ is the interval between any two frequencies \(\left(f_{2}-f_{1}\right)\) whose frequency ratio \(\left(\frac{f_{2}}{f_{1}}\right)\) is 2.
Fill in the Blank.The dynamic testing of a machine involves finding the ______________ of the machine at a critical frequency.
Fill in the Blank.For vibration testing, the machine is supported to simulate \(a(n)\) _______________ system so that rigid body modes can also be observed. \( \qquad \) condition of the
Fill in the Blank.The excitation force is measured by a(n) _____________ cell.
Fill in the Blank.The response of a system is usually measured by _______________.
Fill in the Blank.The frequency response of a system can be measured using ____________ analyzers.
Fill in the Blank.The condition of a machine can be determined using _____________ severity charts.
Fill in the Blank.The life of a machine follows the classic ___________ curve.
Fill in the Blank.The ____________ observed in Lissajous figures can be used to identify machinery faults.
Fill in the Blank.Cepstrum can be defined as the power spectrum of the logarithm of the ______________ .
Using Rayleigh's method, find the fundamental frequency for the lateral vibration of the beam shown in Fig. 8.40. The restoring force in the spring \(k\) is proportional to the deflection, and the restoring moment in the spring \(k_{t}\) is proportional to the angular deflection. x k 0000 FIGURE
It has been found that a printing press, of mass \(300 \mathrm{~kg}\) and operating speed \(3000 \mathrm{rpm}\), produces a repeating force of \(30,000 \mathrm{~N}\) when attached to a rigid foundation. Find a suitable viscously damped isolator to satisfy the following requirements:(a) the static
During operation, the compressor unit of a refrigerator, with mass \(75 \mathrm{~kg}\) and rotational speed \(900 \mathrm{rpm}\), experiences a dynamic force of \(200 \mathrm{~N}\). The compressor unit is supported on four identical springs, each with a stiffness of \(k\) and negligible damping.
An electric motor, having an unbalance of \(2 \mathrm{~kg}-\mathrm{cm}\), is mounted at the end of a steel cantilever beam, as shown in Fig. 9.51. The beam is observed to vibrate with large amplitudes at the operating speed of \(1500 \mathrm{rpm}\) of the motor. It is proposed to add a vibration
The pipe carrying feedwater to a boiler in a thermal power plant has been found to vibrate violently at a pump speed of \(800 \mathrm{rpm}\). In order to reduce the vibrations, an absorber consisting of a spring of stiffness \(k_{2}\) and a trial mass \(m_{2}^{\prime}\) of \(1 \mathrm{~kg}\) is
A reciprocating engine is installed on the first floor of a building, which can be modeled as a rigid rectangular plate resting on four elastic columns. The equivalent mass of the engine and the floor is \(1000 \mathrm{~kg}\). At the rated speed of the engine, which is \(600 \mathrm{rpm}\), the
Find the values of \(k_{2}\) and \(m_{2}\) in Problem 9.54 in order to have the natural frequencies of the system at least 30 percent away from the forcing frequency.Data From Problem 9.54:-An electronic instrument of mass \(20 \mathrm{~kg}\) is to be isolated from engine vibrations with
A hollow steel shaft of outer diameter \(50 \mathrm{~mm}\), inner diameter \(38 \mathrm{~mm}\), and length \(750 \mathrm{~mm}\) carries a solid disc of diameter \(380 \mathrm{~mm}\) and mass \(40 \mathrm{~kg}\). Another hollow steel shaft of length \(500 \mathrm{~mm}\), carrying a solid disc of
A rotor, having a mass moment of inertia \(J_{1}=15 \mathrm{~kg}-\mathrm{m}^{2}\), is mounted at the end of a steel shaft having a torsional stiffness of \(0.6 \mathrm{MN}-\mathrm{m} / \mathrm{rad}\). The rotor is found to vibrate violently when subjected to a harmonic torque of \(300 \cos 200 t
Plot the graphs of \(\left(\Omega_{1} / \omega_{2}\right)\) against \(\left(m_{2} / m_{1}\right)\) and \(\left(\Omega_{2} / \omega_{2}\right)\) against \(\left(m_{2} / m_{1}\right)\) as \(\left(m_{2} / m_{1}\right)\) varies from 0 to 1.0 when \(\omega_{2} / \omega_{1}=0.1\) and 10.0.
Determine the operating range of the frequency ratio \(\omega / \omega_{2}\) for an undamped vibration absorber to limit the value of \(\left|X_{1} / \delta_{\text {st }}\right|\) to 0.5. Assume that \\omega_{1}=\omega_{2}\) and \(m_{2}=0.1 m_{1}\).
When an undamped vibration absorber, having a mass \(30 \mathrm{~kg}\) and a stiffness \(k\), is added to a spring-mass system, of mass \(40 \mathrm{~kg}\) and stiffness \(0.1 \mathrm{MN} / \mathrm{m}\), the main mass ( \(40 \mathrm{~kg}\) mass) is found to have zero amplitude during its
An electric motor, of mass \(20 \mathrm{~kg}\) and operating speed \(1350 \mathrm{rpm}\), is placed on a fixed-fixed steel beam of width \(15 \mathrm{~cm}\) and depth \(1.2 \mathrm{~cm}\), as shown in Fig. 9.53. The motor has a rotatingunbalance of \(0.1 \mathrm{~kg}-\mathrm{m}\). The amplitude of
A bridge is found to vibrate violently when a vehicle, producing a harmonic load of magnitude \(600 \mathrm{~N}\), crosses it. By modeling the bridge as an undamped spring-mass system with a mass \(15,000 \mathrm{~kg}\) and a stiffness \(2 \mathrm{MN} / \mathrm{m}\), design a suitable tuned damped
A small motor, of mass \(50 \mathrm{~kg}\), is found to have a natural frequency of \(100 \mathrm{rad} / \mathrm{s}\). It is proposed that an undamped vibration absorber of mass \(4 \mathrm{~kg}\) be used to suppress the vibrations when the motor operates at \(80 \mathrm{rad} / \mathrm{s}\).
Consider the system shown in Fig. 9.54 in which a harmonic force acts on the mass \(m\). Derive the condition under which the steady-state displacement of mass \(m\) will be zero. Disc, mass M 000 R Rolls without slipping Pulley FIGURE 9.54 System considered in Problem 9.80. m Cord Fo sin cot T x(t)
Show the variation of the transmission ratio, \(\frac{X_{1}}{\delta_{\mathrm{st}}}\), with \(\frac{\omega}{\omega_{1}}\) for an undamped dynamic vibration absorber for \(\omega_{2}=\omega_{1}\) and \(m_{2}=0.25 m_{1}\).
In a dynamic vibration absorber having \(\frac{\omega_{2}}{\omega_{1}}=1\) and \(\mu=\frac{m_{2}}{m_{1}}=\frac{1}{2}\), determine the frequency range over which the value of the transmission ratio, \(\frac{X_{1}}{\delta_{\mathrm{st}}}\), is less than one.
Using MATLAB, plot Eq. (9.94) for \(\zeta=0,0.25,0.5,0.75\), and 1 over the range \(0 \leq r \leq 3\).Equation 9.94:- Tf = FT Fo = = (k - - k + wc mw) + wc] 1 + (25) 212 + (25)2 1/2 (9.94)
Using MATLAB, plot Eqs. (9.140) and (9.141) for \(f=1, \zeta=0.2,0.3\), and 0.4, and \(\mu=0.2\) and 0.5 over the range \(0.6 \leq \omega / \omega_{1}\).Equation 9.140 and 9.141:- X dst st X2 k2 1 W2 - k2 DOLOR 1 + 1 1 k2 *-ei-04 8st k2 k1 T W (9.140) (9.141)
Using MATLAB, plot the ratios \(\Omega_{1} / \omega_{2}\) and \(\Omega_{2} / \omega_{2}\) given by Eq. (9.146) for \(\omega_{2} / \omega_{1}=1.5\), 3.0, and 4.5 and \(m_{2} / m_{1}=0\) to 1 .
Using Program13.m, solve Problem 9.13.Data From Problem 9.13:-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
Write a computer program to find the displacement of the main mass and the auxiliary mass of a damped dynamic vibration absorber. Use this program to generate the results of Fig. 9.39. 16 m 1 =0
a. Nonlinearity in massb. Nonlinearity in dampingc. Linear equationd. Nonlinearity in spring\(\ddot{x}+f \frac{\dot{x}}{|\dot{x}|}+\omega_{n}^{2} x=0\)
The equation of motion of a nonlinear system is given by\[\ddot{x}+c \dot{x}+k_{1} x+k_{2} x^{2}=a \cos 2 \omega t\]Investigate the subharmonic solution of order 2 for this system.
a. Nonlinearity in massb. Nonlinearity in dampingc. Linear equationd. Nonlinearity in spring\(\ddot{x}+\omega_{0}^{2}\left(x-\frac{x^{3}}{6}\right)=0\)
Prove that, for the system considered in Section 13.5.1, the minimum value of \(\omega^{2}\) for which the amplitude of subharmonic oscillations \(A\) will have a real value is given by\[\omega_{\min }=\omega_{0}+\frac{21}{2048} \frac{F^{2}}{\omega_{0}^{5}}\]Also, show that the minimum value of the
Derive Eqs. (13.113b) and (13.116b) for the Mathieu equation.Equation 13.113b and 13.116b:- a 11 + E 62 - 8 + (13.113b)
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