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engineering
mechanical vibration analysis

Mechanical Vibrations 6th Edition Singiresu S Rao - Solutions

a. Nonlinearity in massb. Nonlinearity in dampingc. Linear equationd. Nonlinearity in spring\(\ddot{x}+c \dot{x}+k x=a x^{3}\)
The equation of motion of a simple pendulum subject to viscous damping can be expressed as\[\ddot{\theta}+c \dot{\theta}+\sin \theta=0\]If the initial conditions are \(\theta(0)=\theta_{0}\) and \(\dot{\theta}(0)=0\), show that the origin in the phase-plane diagram represents (a) a stable focus for
The phase-plane equation of a single-degree-of-freedom system is given by\[\frac{d y}{d x}=\frac{-c y-\left(x-0.1 x^{3}\right)}{y}\]Investigate the nature of singularity at \((x, y)=(0,0)\) for \(c>0\).
Identify the singularity and find the nature of solution near the singularity for van der Pol's equation:\[\ddot{x}-\alpha\left(1-x^{2}\right) \dot{x}+x=0\]
Identify the singularity and investigate the nature of solution near the singularity for an undamped system with a hard spring:\[\ddot{x}+\omega_{n}^{2}\left(1+k^{2} x^{2}\right) x=0\]
Solve Problem 13.30 for an undamped system with a soft spring:\[\ddot{x}+\omega_{n}^{2}\left(1-k^{2} x^{2}\right) x=0\]Data From Problem 13.30:-Identify the singularity and investigate the nature of solution near the singularity for an undamped system with a hard
Solve Problem 13.30 for a simple pendulum:\[\ddot{\theta}+\omega_{n}^{2} \sin \theta=0\]Data From Problem 13.30:-Identify the singularity and investigate the nature of solution near the singularity for an undamped system with a hard spring:\[\ddot{x}+\omega_{n}^{2}\left(1+k^{2} x^{2}\right) x=0\]
Verify that the following equation exhibits chaotic behavior:\[x_{n+1}=k x_{n}\left(1-x_{n}\right)\]
Using MATLAB, solve the simple pendulum equations, Eqs. (E.1)-(E.3), given in Example 13.6, for the following data:\[\omega_{0}=0.1, \quad \theta(0)=0.01, \quad \dot{\theta}(0)=0\]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 a nonlinear single-degree-of-freedom system governed by Eq. (E.1) of Example 13.8 under a pulse load for the following data: \(m=10, k_{1}=4000\), \(k_{2}=1000, F_{0}=1000, t_{0}=5, x(0)=0.05, \dot{x}(0)=5\).Data From Example 13.8:- Using MATLAB, find the solution
Solve the equation of motion \(\ddot{x}+0.5 \dot{x}+x+1.2 x^{3}=1.8 \cos 0.4 t\), using the Runge-Kutta method with \(\Delta t=0.05, t_{\max }=5.0\), and \(x_{0}=\dot{x}_{0}=0\). Plot the variation of \(x\) with \(t\). Use Program18.m for the solution.
Find the time variation of the angular displacement of a simple pendulum (i.e., the solution of Eq. (13.5)) for \(g / l=0.5\), using the initial conditions \(\theta_{0}=45^{\circ}\) and \(\dot{\theta}_{0}=0\). Use the RungeKutta method given in Program18.m.Equation 13.5:- + (0-10) = 0 (13.5)
In the static firing test of a rocket, the rocket is anchored to a rigid wall by a nonlinear spring-damper system and fuel is burnt to develop a thrust, as shown in Fig. 13.35. The thrust acting on the rocket during the time period \(0 \leq t \leq t_{0}\) is given by \(F=m_{0} v\), where \(m_{0}\)
Write a computer program for finding the period of vibration corresponding to Eq. (13.14). Use a suitable numerical integration procedure. Using this program, find the solution of Problem 13.45.Data From Problem 13.45:-Find the time variation of the angular displacement of a simple pendulum (i.e.,
A single-degree-of-freedom system has a softening spring and is subjected to a harmonic force with the equation of motion given byFind the response of the system numerically using the fourth-order Runge-Kutta method for the following data for two cases-one by neglecting the nonlinear spring term
Solve Problem 13.48 using a forcing frequency \(\omega=10 \mathrm{rad} / \mathrm{s}\) instead of \(5 \mathrm{rad} / \mathrm{s}\).Data From Problem 13.48:-A single-degree-of-freedom system has a softening spring and is subjected to a harmonic force with the equation of motion given byFind the
In some periodic vibratory systems, external energy is supplied to the system over part of a period and dissipated within the system in another part of the period. Such periodic oscillations are known as relaxation oscillations. Van der Pol [13.22] indicated several instances of occurrence of
A machine tool is mounted on two nonlinear elastic mounts, as shown in Fig. 13.36. The equations of motion, in terms of the coordinates \(x(t)\) and \(\theta(t)\), are given bywhere \(m\) is the mass and \(J_{0}\) is the mass moment of inertia about \(G\) of the machine tool. Using the Runge-Kutta
What is the difference between a sample space and an ensemble?
True or False.A deterministic system requires deterministic system properties and loading.
The strength of the foundation of a reciprocating machine \((x)\) has been found to vary between \(1 \mathrm{MPa}\) and \(1.5 \mathrm{MPa}\) according to the probability density function:\[p(x)= \begin{cases}k\left(1-\frac{x}{1.5}\right), & 1 \leq x \leq 1.5 \\ 0, & \text { elsewhere
Fill in the Blank.When the vibrational response of a system is known precisely, the vibration is called ___________ vibration.
True or False.Most phenomena in real life are deterministic.
Each outcome of an experiment, in the case of a random process, is called aa. sample pointb. sample spacec. sample function
How are the mean value and variance of a random variable defined?
True or False.A random variable is a quantity whose magnitude cannot be predicated precisely.
The probability density function of a random variable \(x\) is given by\[p(x)= \begin{cases}0 & \text { for } x2\end{cases}\]Determine \(E[x], E\left[x^{2}\right]\), and \(\sigma_{x}\).
Fill in the Blank.The pressure fluctuation at a point on the surface of an aircraft flying in the air is a(n) _____________ process.
The probability density function, \(p(\widetilde{x})\), denotesa. \(P(x \leq \tilde{x})\)b. \(P(x>\tilde{x})\)c. \(P(\tilde{x} \leq x \leq \tilde{x}+\Delta x)\)
The variance of \(x\) is given bya. \(\overline{x^{2}}\)b. \(\left(\overline{x^{2}}\right)-(\bar{x})^{2}\)c. \((\bar{x})^{2}\)
Fill in the Blank.The standard deviation is the positive square root of ___________ .
True or False.The correlation coefficient \(ho_{X Y}\) satisfies the relation \(\left|ho_{X Y}\right| \leq 1\).
Normalization of probability distribution impliesa. \(P(\infty)=1\)b. \(\int_{-\infty}^{\infty} p(x)=1\)c. \(\int_{-\infty}^{\infty} p(x)=0\)
The life \(T\) in hours of a vibration transducer is found to follow exponential distribution\[p_{T}(t)= \begin{cases}\lambda e^{-\lambda t}, & t \geq 0 \\ 0, & t
Fill in the Blank.If any parameter of a vibrating system is not known precisely, the resulting vibration is called ____________ vibration.
The probability distribution function, \(P(\widetilde{x})\), denotesa. \(P(x \leq \tilde{x})\)b. \(P(x>\widetilde{x})\)c. \(P(\tilde{x} \leq x \leq \tilde{x}+\Delta x)\)
What is a bivariate distribution function?
True or False.The expected value of \(x\), in terms of its probability density function, \(p(x)\), is given by \(\int_{-\infty}^{\infty} x p(x) d x\).
Find the temporal mean value and the mean square value of the function \(x(t)=x_{0} \sin (\pi t / 2)\).
Fill in the Blank.In a random process, the outcome of an experiment will be a function of some _________ such as time.
What is the covariance between two random variables \(X\) and \(Y\) ?
The joint density function of two random variables \(X\) and \(Y\) is given by\[p_{X, Y}(x, y)= \begin{cases}\frac{x y}{9}, & 0 \leq x \leq 2,0 \leq y \leq 3 \\ 0, & \text { elsewhere }\end{cases}\](a) Find the marginal density functions of \(X\) and \(Y\).(b) Find the means and standard deviations
Define the correlation coefficient, \(ho_{X Y}\).
True or False.The autocorrelation function \(R\left(t_{1}, t_{2}\right)\) is the same as \(E\left[x\left(t_{1}\right) x\left(t_{2}\right)\right]\).
If \(x\) and \(y\) are statistically independent, then \(E[x y]=E[x] E[y]\). That is, the expected value of the product \(x y\) is equal to the product of the separate mean values. If \(z=x+y\), where \(x\) and \(y\) are statistically independent, show that
What are the bounds on the correlation coefficient?
True or False.The mean square value of \(x(t)\) can be determined as \(E\left[x^{2}\right]=R(0)\).
Find the autocorrelation functions of the periodic functions shown in Fig. 14.24. x(t) 0 -xo (a) x(1) x0 (b) FIGURE 14.24 Periodic function of Problem 14.7.
The standard normal variable, \(z\), corresponding to the normal variable \(x\), is defined asa. \(z=\frac{\bar{x}}{\sigma_{x}}\)b. \(z=\frac{x-\bar{x}}{\sigma_{x}}\)c. \(z=\frac{x}{\sigma_{x}}\)
Fill in the Blank.The distribution of several random variables is called ____________ distribution.
The spectral density of a random signal is given by\[S(f)= \begin{cases}0.0001 \mathrm{~m}^{2} / \text { cycle } / \mathrm{s}, & 10 \mathrm{~Hz} \leq f \leq 1000 \mathrm{~Hz} \\ 0, & \text { elsewhere }\end{cases}\]Find the standard deviation and the root mean square value of the signal by assuming
The marginal density function of \(x\) can be determined form the bivariate density function \(p(x, y)\) asa. \(p(x)=\int_{-\infty}^{\infty} p(x, y) d y\)b. \(p(x)=\int_{-\infty}^{\infty} p(x, y) d x\)c. \(p(x)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} p(x, y) d x d y\)
Fill in the Blank.Univariate distributions describe the probability distributions of __________ random variables.
What is a marginal density function?
True or False.If \(x(t)\) is stationary, its mean will be independent of \(t\).
Compute the autocorrelation function of a periodic square wave with zero mean value and compare this result with that of a sinusoidal wave of the same period. Assume the amplitudes to be the same for both waves.
The correlation coefficient of \(x\) and \(y\) is given bya. \(\sigma_{x y}\)b. \(\sigma_{x y} /\left(\sigma_{x} \sigma_{y}\right)\)c. \(\sigma_{x} \sigma_{y}\)
Fill in the Blank.The distribution of two is known as _________ distribution
What is an autocorrelation function?
True or False.The autocorrelation function \(R(\tau)\) is an even function of \(\tau\).
The autocorrelation function of a random process \(x(t)\) is given by\[R_{x}(\tau)=20+\frac{5}{1+3 \tau^{2}}\]Find the mean square value of \(x(t)\).
True or False.The Wiener-Khintchine formulas relate the power spectral density to the autocorrelation function.
If the excitation of a linear system is a Gaussian process, the response will bea. a different random processb. a Gaussian processc. an ergodic process
Fill in the Blank.The standard deviation of a stationary random process \(x(t)\) will be independent of __________ .
What are the bounds on the autocorrelation function of a stationary random process?
True or False.The ideal white noise is a physically realizable concept.
An air compressor of mass \(100 \mathrm{~kg}\) is mounted on an undamped isolator and operates at an angular speed of \(1800 \mathrm{rpm}\). The stiffness of the isolator is found to be a random variable with mean value \(\bar{k}=2.25 \times 10^{6} \mathrm{~N} / \mathrm{m}\) and standard deviation
For a normal probability density function, \(\operatorname{Prob}[-3 \sigma \leq x(t) \leq 3 \sigma]\) isa. 0.6827b. 0.999937c. 0.9973
Fill in the Blank.If all the probability information of a stationary random process can be obtained from a single sample function, the process is said to be ___________.
Define an ergodic process.
Find the complex form of the Fourier series for the wave shown in Fig. 14.24(b).Figure 14.24(b):- 1 (b) Ox- 0 (1).x
The mean square response of a stationary random process can be determined from the:a. autocorrelation function onlyb. power spectral density onlyc. autocorrelation function or power spectral density
Fill in the Blank.The Gaussian density function is a symmetric _____________ -shaped curve about the mean value.
Find the Fourier transform of the functions shown in Figs. 14.25 and plot the corresponding spectrum.Figure 14.25:- x(t) A t a -a FIGURE 14.25 Function considered in Problem 14.13.
What are temporal averages?
All possible outcomes of a random variablea. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.The standard normal variable has mean of _____________ and standard deviation of ______________ .
Find the Fourier transform of the functions shown in Figs. 14.26 and plot the corresponding spectrum.Figure 14.26:- x(t) A Ae-at FIGURE 14.26 Function considered in Problem 14.14.
What is a Gaussian random process? Why is it frequently used in vibration analysis?
All possible outcomes of a random processa. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.A nonperiodic function can be treated as a periodic function having a(n) __________ period.
What is Parseval's formula?
Find the Fourier transform of the functions shown in Figs. 14.27 and plot the corresponding spectrum.Figure 14.27:- x(t) A Ae-aln FIGURE 14.27 Function considered in Problem 14.15.
Statistical connections between the valuesa. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.The ___________ spectral density function is an even function of \(\omega\).
Define the following terms: power spectral density function, white noise, band-limited white noise, wide-band process, and narrow-band process.
Find the Fourier transform of the functions shown in Figs. 14.28 and plot the corresponding spectrum.Figure 14.28:- x(t) 8(t - a) a FIGURE 14.28 Function considered in Problem 14.16. t
A random process invariant under a shifta. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.If \(S(\overline{\omega)}\) has significant values over a wide range of frequencies, the process is called \(\mathrm{a}(\mathrm{n})\) _____________ process.
Derive Eq. (14.46) from Eq. (14.45).Equation 14.46 and 14.45:- 00 T/2 Cn * [cos(n - m)wt + i sin (n - m)w] dt (14.45) T/2
How are the mean square value, autocorrelation function, and the power spectral density function of a stationary random process related?
Mean and standard deviations of \(x(t)\) vary with \(t\)a. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.If \(S(\overline{\omega) \text { has }}\) significant values only over a small range of frequencies, the process is called \(\mathrm{a}(\mathrm{n})\) _____________ process.
What is an impulse-response function?
The autocorrelation function of a random process is given by\[R_{x}(\tau)=A \cos \omega \tau ; \quad-\frac{\tau}{2 \omega} \leq \tau \leq \frac{\pi}{2 \omega}\]where \(A\) and \(\omega\) are constants. Find the power spectral density of the random process.
Power spectral density is constant over a frequency rangea. Correlation functions in an experimentb. Nonstationary processc. Sample space of \(x(t)\) at times \(t_{1}, t_{2}, \ldots\)d. White noise of the time scalee. Stationary processf. Ensemble
Fill in the Blank.The power spectral density \(\mathrm{S}(\omega)\) of a stationary random process is defined as the transform of \(R(\tau) / 2 \pi\).
Express the response of a single-degree-of-freedom system using the Duhamel integral.
A periodic function \(F(t)\) is shown in Fig. 14.29. Use the values of the function \(F(t)\) at ten equally spaced time stations \(t_{i}\) to find(a) the spectrum of \(F(t)\) and(b) the mean square value of \(F(t)\). 500 -500 F(1), N [10 1 - t, s S 11 12 13 14 ts tg FIGURE 14.29 Discretized
Fill in the Blank.If the band of frequencies has finite cut-off frequencies for a white noise, it is called ____________ white noise.

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