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engineering
mechanical vibration analysis
Mechanical Vibrations 6th Edition Singiresu S Rao - Solutions
Find the natural time period of oscillation of the pendulum shown in Fig. 13.1(a) when it oscillates between the limits \(\theta=-\pi / 2\) and \(\theta=\pi / 2\), using Eqs. (13.1) and (13.12).Figure 13.1(a):-Equation 13.1 and 13.12:- (a) m
A simple pendulum of length \(0.8 \mathrm{~m}\) is released from the initial position of \(80^{\circ}\) from the vertical. How long does it take to reach the position \(\theta=0^{\circ}\) ?
Find the exact solution of the nonlinear pendulum equation\[\ddot{\theta}+\omega_{0}^{2}\left(\theta-\frac{\theta^{3}}{6}\right)=0\]with \(\dot{\theta}=0\) when \(\theta=\theta_{0}\), where \(\theta_{0}\) denotes the maximum angular displacement.
Find the solution of Example 13.1 using the following two-term approximation for \(x(t)\) :\[\underset{\sim}{x}(t)=A_{0} \sin \omega t+A_{3} \sin 3 \omega t\]Data From Example 13.1:- Using a one-term approximation, find the solution of the pendulum equation E[x] = x + wx = 0 6 by the Ritz-Galerkin
Using a three-term expansion in Lindstedt's perturbation method (Eq. (13.30)), find the solution of the pendulum equation, Eq. (13.20).Equation 13.20 and 13.30:- wx x + w x + ax = = 0 (13.20)
The equation of motion for the forced vibration of a single-degree-of-freedom nonlinear system can be expressed as\[\ddot{x}+c \dot{x}+k_{1} x+k_{2} x^{3}=a_{1} \cos 3 \omega t-a_{2} \sin 3 \omega t\]Derive the conditions for the existence of subharmonics of order 3 for this system.
The equation of motion of a single-degree-of-freedom system is given by\[2 \ddot{x}+0.8 \dot{x}+1.6 x=0\]with initial conditions \(x(0)=-1\) and \(\dot{x}(0)=2\).(a) Plot the graph \(x(t)\) versus \(t\) for \(0 \leq t \leq 10\).(b) Plot a trajectory in the phase plane.
Find the equilibrium position and plot the trajectories in the neighborhood of the equilibrium position corresponding to the following equation:\[\ddot{x}+0.1\left(x^{2}-1\right) \dot{x}+x=0\]
Obtain the phase trajectories for a system governed by the equation\[\ddot{x}+0.4 \dot{x}+0.8 x=0\]with the initial conditions \(x(0)=2\) and \(\dot{x}(0)=1\) using the method of isoclines.
Plot the phase-plane trajectories for the following system:\[\ddot{x}+0.1 \dot{x}+x=5\]The initial conditions are \(x(0)=\dot{x}(0)=0\).
A single-degree-of-freedom system is subjected to Coulomb friction so that the equation of motion is given by\[\ddot{x}+f \frac{\dot{x}}{|\dot{x}|}+\omega_{n}^{2} x=0\]Construct the phase-plane trajectories of the system using the initial conditions \(x(0)=10\left(f / \omega_{n}^{2}\right)\) and
The equation of motion of a simple pendulum, subjected to external force, is given by\[\ddot{\theta}+0.5 \dot{\theta}+\sin \theta=0.8\]Find the nature of singularity at \(\theta=\sin ^{-1}(0.8)\).
Determine the eigenvalues and eigenvectors of the following equations:a. \(\dot{x}=x-y, \quad \dot{y}=x+3 y\)b. \(\dot{x}=x+y, \quad \dot{y}=4 x+y\)
Find the trajectories of the system governed by the equations\[\dot{x}=x-2 y, \quad \dot{y}=4 x-5 y\]
Find the trajectories of the system governed by the equations\[\dot{x}=x-y, \quad \dot{y}=x+3 y\]
Find the trajectories of the system governed by the equations\[\dot{x}=2 x+y, \quad \dot{y}=-3 x-2 y\]
Using Lindstedt's perturbation method, find the solution of the van der Pol's equation, Eq. (13.143).Equation 13.143:- - x = a(1 x)x + x = 0, 0 (13.143)
Verify that the following equation exhibits chaotic behavior:\[x_{n+1}=2.0 x_{n}\left(x_{n}-1\right)\]
Using MATLAB, solve the simple pendulum equations, Eqs. (E.1)-(E.3), given in Example 13.6, for the following data:\[\omega_{0}=0.1, \quad \theta(0)=0.01, \quad \dot{\theta}(0)=10\]Data From Example 13.6:- Using MATLAB, find the solution of the following forms of the pendulum equation with = 0.09.
Using MATLAB, find the solution of the nonlinearly damped system, Eq. (E.1) of Example 13.7, for the following data: \(m=10, c=0.1, k=4000, F_{0}=200, \omega=20, x(0)=0.5\), \(\dot{x}(0)=1.0\).Data From Example 13.7:- Using MATLAB, find the solution of a single-degree-of-freedom system with
Define probability density function and probability distribution function.
Explain the difference between a stationary process and a nonstationary process.
What is complex frequency-response function?
How are the power spectral density functions of input and output of a single-degree-offreedom system related?
Fill in the Blank.The joint behavior of several random variables is described by the ____________ probability distribution function.
Each outcome of an experiment for a random variable is calleda. a sample pointb. a random pointc. an observed value
The autocorrelation function of a stationary random process \(x(t)\) is given by\[R_{x}(\tau)=a e^{-b|\tau|}\]where \(a\) and \(b\) are constants. Find the power spectral density of \(x(t)\).
Find the autocorrelation function of a random process whose power spectral density is given by \(S(\omega)=S_{0}=\) constant between the frequencies \(\omega_{1}\) and \(\omega_{2}\).
A machine, modeled as a single-degree-of-freedom system, has the following parameters: \(m=900 \mathrm{~kg}, k=7 \mathrm{MN} / \mathrm{m}\), and \(c=135 \mathrm{~N}-\mathrm{s} / \mathrm{m}\). It is subjected to the force shown in Fig. 14.29. Find the mean square value of the response of the machine
A mass, connected to a damper as shown in Fig. 14.30, is subjected to a force \(F(t)\). Find the frequency-response function \(H(\omega)\) for the velocity of the mass. m F(t) y(1) FIGURE 14.30 Mass-damper system subjected to force F(1).
The motion of a lifting surface about the steady flight path due to atmospheric turbulence can be represented by the equation\[\ddot{x}(t)+2 \zeta \omega_{n} \dot{x}(t)+\omega_{n}^{2} x(t)=\frac{1}{m} F(t)\]where \(\omega_{n}\) is the natural frequency, \(m\) is the mass, and \(\zeta\) is the
The mean square value of the response of a machine, \(E\left[y^{2}\right]\), subject to the force shown in Fig. 14.29, is given by (see Problem 14.26):\[E\left[y^{2}\right]=\sum_{n=0}^{N-1} \frac{\left|c_{n}\right|^{2}}{\left(k-m \omega_{n}^{2}\right)^{2}+c^{2}
When a transducer is used in conjunction with another device to measure vibration, it is called aa. vibration sensorb. vibration pickupc. vibration actuator
The instrument that measures the displacement of a vibrating body is called a(n)a. seismometerb. transducerc. accelerometer
The circuit that permits the passage of frequency components of a signal over a frequency band and rejects all other frequency components is called aa. bandpass filterb. frequency filterc. spectral filter
A decibel \((\mathrm{dB})\) is a quantity, such as power \((P)\), defined in terms of a reference value \(\left(P_{\text {ref }}\right)\), asa. \(10 \log _{10}\left(\frac{P}{P_{\text {ref }}}\right)\)b. \(\log _{10}\left(\frac{P}{P_{\text {ref }}}\right)\)c. \(\frac{1}{P_{\text {ref }}} \log
The following function plays an important role in the experimental modal analysis:a. time-response functionb. modal-response functionc. frequency-response function
The method of subjecting a system to a known force as an initial condition and then releasing is known asa. step relaxationb. excitation by electromagnetic shakerc. impactor
The process of using an electrical signal, generalized by a spectrum analyzer, for applying a mechanical force on a system is known asa. step relaxationb. excitation by electromagnetic shakerc. impactor
The procedure of using a hammer with a built-in load cell to apply load at different points of a system is known asa. step relaxationb. excitation by electromagnetic shakerc. impactor
During the initial running-in period, usually the deterioration of a machinea. decreasesb. increasesc. remains constant
During the normal operating period, the deterioration of a machine usuallya. decreasesb. increasesc. remains constant
During the aging or wearout period, the deterioration of a machine usuallya. decreasesb. increasesc. remains constant
Piezoelectric accelerometera. produces light pulses intermittentlyb. has high output and is insensitive to temperaturec. frequently used in velocity pickupsd. has high sensitivity and frequency rangee. variable-length cantilever with a mass at its free end
Electrodynamic transducera. produces light pulses intermittentlyb. has high output and is insensitive to temperaturec. frequently used in velocity pickupsd. has high sensitivity and frequency rangee. variable-length cantilever with a mass at its free end
LVDT transducera. produces light pulses intermittentlyb. has high output and is insensitive to temperaturec. frequently used in velocity pickupsd. has high sensitivity and frequency rangee. variable-length cantilever with a mass at its free end
Fullarton tachometera. produces light pulses intermittentlyb. has high output and is insensitive to temperaturec. frequently used in velocity pickupsd. has high sensitivity and frequency rangee. variable-length cantilever with a mass at its free end
A Rochelle salt crystal, having a voltage sensitivity of \(0.098 \mathrm{~V}-\mathrm{m} / \mathrm{N}\) and thickness \(2 \mathrm{~mm}\), produced an output voltage of 200 volts under pressure. Find the pressure applied to the crystal.
A spring-mass system with \(m=0.5 \mathrm{~kg}\) and \(k=10,000 \mathrm{~N} / \mathrm{m}\), with negligible damping, is used as a vibration pickup. When mounted on a structure vibrating with an amplitude of \(4 \mathrm{~mm}\), the total displacement of the mass of the pickup is observed to be \(12
The vertical motion of a machine is measured by using the arrangement shown in Fig. 10.43. The motion of the mass \(m\) relative to the machine body is recorded on a drum. If the damping constant \(c\) is equal to \(c_{\text {cri }} / \sqrt{2}\), and the vertical vibration of the machine body is
It is proposed that the vibration of the foundation of an internal combustion engine be measured over the speed range \(500 \mathrm{rpm}\) to \(1500 \mathrm{rpm}\) using a vibrometer. The vibration is composed of two harmonics, the first one caused by the primary inertia forces and the second one
Determine the maximum percent error of a vibrometer in the frequency-ratio range \(4 \leq r
Solve Problem 10.5 with a damping ratio of \(\zeta=0.67\).Data From Problem 10.5:-Determine the maximum percent error of a vibrometer in the frequency-ratio range \(4 \leq r
A vibrometer is used to measure the vibration of an engine whose operating-speed range is from \(500 \mathrm{rpm}\) to \(2000 \mathrm{rpm}\). The vibration consists of two harmonics. The amplitude distortion must be less than 3\%. Find the natural frequency of the vibrometer if (a) the damping is
A spring-mass system, having a static deflection of \(10 \mathrm{~mm}\) and negligible damping, is used as a vibrometer. When mounted on a machine operating at \(4000 \mathrm{rpm}\), the relative amplitude is recorded as \(1 \mathrm{~mm}\). Find the maximum values of displacement, velocity, and
A vibration pickup has a natural frequency of \(5 \mathrm{~Hz}\) and a damping ratio of \(\zeta=0.5\). Find the lowest frequency that can be measured with a \(1 \%\) error.
A vibration pickup has been designed for operation above a frequency level of \(100 \mathrm{~Hz}\) without exceeding an error of \(2 \%\). When mounted on a structure vibrating at a frequency of 100 \(\mathrm{Hz}\), the relative amplitude of the mass is found to be \(1 \mathrm{~mm}\). Find the
A vibrometer has an undamped natural frequency of \(10 \mathrm{~Hz}\) and a damped natural frequency of \(8 \mathrm{~Hz}\). Find the lowest frequency in the range to infinity at which the amplitude can be directly read from the vibrometer with less than \(2 \%\) error.
Determine the maximum percent error of an accelerometer in the frequency-ratio range \(0
Solve Problem 10.12 with a damping ratio of 0.75 .Data From Problem 10.12:-Determine the maximum percent error of an accelerometer in the frequency-ratio range \(0
Determine the necessary stiffness and the damping constant of an accelerometer if the maximum error is to be limited to \(3 \%\) for measurements in the frequency range of \(0 \mathrm{~Hz}\) to \(100 \mathrm{~Hz}\). Assume that the suspended mass is \(0.05 \mathrm{~kg}\).
An accelerometer is constructed by suspending a mass of \(0.1 \mathrm{~kg}\) from a spring of stiffness \(10,000 \mathrm{~N} / \mathrm{m}\) with negligible damping. When mounted on the foundation of an engine, the peak-to-peak travel of the mass of the accelerometer has been found to be \(10
A spring-mass-damper system, having an undamped natural frequency of \(100 \mathrm{~Hz}\) and a damping constant of \(20 \mathrm{~N}-\mathrm{s} / \mathrm{m}\), is used as an accelerometer to measure the vibration of a machine operating at a speed of \(3000 \mathrm{rpm}\). If the actual acceleration
A machine shop floor is subjected to the following vibration due to electric motors running at different speeds:\[x(t)=20 \sin 4 \pi t+10 \sin 8 \pi t+5 \sin 12 \pi t \mathrm{~mm}\]If a vibrometer having an undamped natural frequency of \(0.5 \mathrm{~Hz}\), and a damped natural frequency of \(0.48
A machine is subjected to the vibration\[x(t)=20 \sin 50 t+5 \sin 150 t \mathrm{~mm} \quad(t \text { in } \mathrm{s})\]An accelerometer having a damped natural frequency of \(80 \mathrm{rad} / \mathrm{s}\) and an undamped natural frequency of \(100 \mathrm{rad} / \mathrm{s}\) is mounted on the
A variable-length cantilever beam of rectangular cross section \(1.6 \mathrm{~mm} \times 25 \mathrm{~mm}\), made of spring steel, is used to measure the frequency of vibration. The length of the cantilever can be varied between \(50 \mathrm{~mm}\) and \(250 \mathrm{~mm}\). Find the range of
Show that the real component of the harmonic response of a viscously damped single-degreeof-freedom system (from \(X\) in Eq. (3.54)) attains a maximum at\[R_{1}=\frac{\omega_{1}}{\omega_{n}}=\sqrt{1-2 \zeta}\]and a minimum at\[R_{2}=\frac{\omega_{2}}{\omega_{n}}=\sqrt{1+2 \zeta}\]Equation 3.54:-
Find the value of the frequency at which the imaginary component of the harmonic response of a viscously damped single-degree-of-freedom system (from \(X\) in Eq. (3.54)) attains a minimum.Equation 3.54:- |H(iw)| == kX Fo == 1 [(12)2 + (23r) 211/2 (3.55)
Construct the Nyquist diagram for a single-degree-of-freedom system with hysteretic damping.
The Bode plot of shaft vibration of a turbine obtained during coast-down is shown in Fig. 10.44. Determine the damping ratio of the system when the static deflection of the shaft is equal to \(1.27 \mu \mathrm{m}\). Peak-to-peak displacement (um) 25.4 20.32 15.24 10.16 5.08 0 1 2 3 4 5 6 7 8 9 10
The vibratory response at the bearing of an internal combustion engine is shown in Fig. 10.45. Determine the equivalent viscous damping ratio of the system. Vibration velocity (cm/s) 17.78 15.24 12.70 10.16 7.62 5.08 2.54 0 Time (ms) 40 50 60 70 -2.54 -5.08 -7.68 -10.16 -12.70 -15.24 -17.78 FIGURE
Suggest a method of using the Bode plot of phase angle versus frequency (Fig. 3.11(b)) to identify the natural frequency and the damping ratio of the system.
Two ball bearings, each with 16 balls, are used to support the shaft of a fan that rotates at \(750 \mathrm{rpm}\). Determine the frequencies, in hertz, corresponding to the following defects: * cage, inner race, outer race, and ball. Assume that \(d=15 \mathrm{~mm}, D=100 \mathrm{~mm}\), and
Determine the defect frequencies in hertz* corresponding to roller, inner race, outer race, and cage defects for a roller bearing with 18 rollers when installed in a machine that runs at a speed of \(1000 \mathrm{rpm}\). Assume \(d=2 \mathrm{~cm}, D=15 \mathrm{~cm}\), and \(\alpha=20^{\circ}\).
An angular contact thrust bearing consists of 18 balls, each of diameter \(10 \mathrm{~mm}\), and is mounted on a shaft that rotates at \(1500 \mathrm{rpm}\). If the contact angle of the bearing is \(40^{\circ}\) with a pitch diameter \(80 \mathrm{~mm}\), find the frequencies corresponding to cage,
Find the value of kurtosis for a vibration signal that is uniformly distributed in the range \(1-5 \mathrm{~mm}\);\[f(x)=\frac{1}{4} ; \quad 1 \leq x \leq 5 \mathrm{~mm}\]
Find the value of kurtosis for a vibration amplitude that can be approximated as a discrete random variable with the following probability mass function: x (mm) f(x) 32 -8 2 3 32 5 10 9 16 36 66 36 3 16 16 32 -18 32 +
Figure 10.46 shows the experimental transfer function of a structure. Determine the approximate values of \(\omega_{\mathrm{i}}\) and \(\zeta_{\mathrm{i}}\). Magnitude (dB) 20,000 2000 200 20 20 180 2 0 250 500 750 1000 Frequency (Hz) 1250 1500 1750 0 -180 FIGURE 10.46 Experimental transfer
The experimental Nyquist circle of a structure is shown in Fig. 10.47. Estimate the modal damping ratio corresponding to this circle. -1.25 -1.25 2.5 r=1 u 1.25 FIGURE 10.47 Experimental Nyquist circle of structure.
Design a vibration exciter to satisfy the following requirements:a. Maximum weight of the test specimen \(=10 \mathrm{~N}\)b. Range of operating frequency \(=10 \mathrm{~Hz}\) to \(50 \mathrm{~Hz}\)c. Maximum acceleration level \(=20 \mathrm{~g}\)d. Maximum vibration amplitude \(=0.5 \mathrm{~cm}\)
Frahm tachometers are particularly useful to measure the speeds of engines whose rotating shafts are not easily accessible. When the tachometer is placed on the frame of a running engine, the vibration generated by the engine will cause one of the reeds to vibrate noticeably when the engine speed
A cantilever beam with an end mass \(m\) is fixed at the top of a multistory building to measure the acceleration induced at the top of the building during wind and earthquake loads (see Fig. 10.48). Design the beam (i.e., determine the material, cross-sectional dimensions, and the length of the
Describe the procedure of the finite difference method.
Using Taylor's series expansion, derive the central difference formulas for the first and the second derivatives of a function.
What is a conditionally stable method?
What is the main difference between the central difference method and the Runge-Kutta method?
Why is it necessary to introduce fictitious mesh points in the finite difference method of solution?
Define a tridiagonal matrix.
True or False.The grid points in the finite difference methods are required to be uniformly spaced.
True or False.The Runge-Kutta method is stable.
True or False.The Runge-Kutta method is self-starting.
True or False.The finite difference method is an implicit integration method.
True or False.The Newmark method is an implicit integration method.
True or False.For a beam with grid points \(-1,1,2,3, \ldots\), the central difference equivalence of the condition \(\left.\frac{d W}{d x}\right|_{1}=0\) is \(W_{-1}=W_{2}\).
Fill in the Blanks.Numerical methods are to be used when the equations of motion cannot be solved in ____________ form.
Fill in the Blanks.In finite difference methods, approximations are used for ______________ .
Fill in the Blanks.Finite difference equations can be derived using ____________ different approaches.
Fill in the Blanks.In finite difference methods, the solution domain is to be replaced by _______________ points.
Fill in the Blanks.The finite difference approximations are based on ___________ series expansion.
Fill in the Blanks.Numerical methods that require the use of a time step \((\Delta t)\) smaller than a critical value \(\left(\Delta t_{\text {cri }}\right)\) are said to be ______________ stable.
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