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engineering
introduction mechanical engineering

Mechanical Vibrations Theory And Applications 1st Edition S. GRAHAM KELLY - Solutions

Determine an integral expression for the period of oscillation of the system of Figure P12.6. FIGURE P12.6 w Parabola y=px rotates at constant w Particle of mass m moves along parabola x= generalized coordinate
Use the method of renormalization to determine a two-term approximation for the frequency-amplitude relation for the system of Figure P12.4. If the bar is rotated \(4{ }^{\circ}\) from equilibrium and released, what is the period for \(L=4 \mathrm{~m}\), \(k=1000 \mathrm{~N} / \mathrm{m}\), and
A \(25-\mathrm{kg}\) mass is attached to a hardening spring with \(k_{1}=1000 \mathrm{~N} / \mathrm{m}\) and \(k_{3}=4,000 \mathrm{~N} / \mathrm{m}^{3}\). The mass is displaced \(15 \mathrm{~mm}\) from equilibrium and released from rest. What is the period of the ensuing oscillations?
Suppose the mass of Chapter Problem 12.20 is subject to an impulse which imparts a velocity of \(3.1 \mathrm{~m} / \mathrm{s}\) to the mass when the mass is in equilibrium. What is the period of the ensuing oscillations?Data From Chapter Problem 12.20:A \(25-\mathrm{kg}\) mass is attached to a
Suppose the mass of Chapter Problem 12.20 is attached to the same spring when a 50-N force is statically applied and suddenly removed. What is the period of the ensuing oscillations?Data From Chapter Problem 12.20:A \(25-\mathrm{kg}\) mass is attached to a hardening spring with \(k_{1}=1000
Use the method of renormalization to determine a two-term frequency-amplitude relationship for the particle on the rotating parabola of Figure P12.6, assuming the amplitude is small. FIGURE P12.6 w Parabola y=px rotates at constant w Particle of mass m moves along parabola x= generalized coordinate
Use the method of renormalization to determine a two-term frequency-amplitude relationship for a block of mass \(m\) attached to a spring with a quadratic nonlinearity. When nondimensionalized the differential equation governing free vibrations of the system is\[ \ddot{x}+\omega^{2} x+\epsilon
If \(F(t)=F_{0} \sin \omega t\), what values of \(\omega\) will lead to the presence of the following?(a) A primary resonance(b) A superharmonic resonance(c) A subharmonic resonance k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
When \(F(t)=5 \sin 8 t \mathrm{~N}\), a primary resonance condition occurs. Determine the amplitude of the forced response. k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
When \(F(t)=150 \sin 2.5 t \mathrm{~N}\) a superharmonic resonance condition occurs. Determine the amplitude of the forced response. k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
If \(F(t)=F_{0} \sin \omega t \mathrm{~N}\), for what value of \(\omega\) will a jump in amplitude occur when \(\omega\) is increased slightly beyond this value when(a) \(F_{0}=5 \mathrm{~N}\) and a primary resonance occurs.(b) \(F_{0}=150 \mathrm{~N}\) and a superharmonic resonance occurs. k =
If \(F(t)=25 \sin 22 t \mathrm{~N}\), will a nontrivial subharmonic response exist? k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
If \(F(t)=30 \sin 15 t+25 \sin \omega t \mathrm{~N}\), what values of \(\omega\) lead to a combination resonance? k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
If \(F(t)=30 \sin 2.5 t+25 \sin \omega t \mathrm{~N}\), what values of \(\omega\) lead to simultaneous resonances? k = 1000 N/m k = 950 N/m w 20 N s/m FIGURE P12.25 17.8 kg F(t)
If \(m_{2}=10 \mathrm{~kg}\), for what values of \(k_{2}\) will internal resonances exist?Refer to the systems of Figure P12.32. The spring of stiffness \(k_{2}\) is a linear spring. W 1000x + 950x FIGURE P12.32 k 10 kg w my F(t)
For what values of \(m_{2}\) are internal resonances possible? If an internal resonance is possible in terms of \(m_{2}\), for what values of \(k_{2}\) will they exist?Refer to the systems of Figure P12.32. The spring of stiffness \(k_{2}\) is a linear spring. W 1000x + 950x FIGURE P12.32 k 10 kg w
Consider the system with \(m_{2}=10 \mathrm{~kg}\) and \(k_{2}=2000 \mathrm{~N} / \mathrm{m}\). The right mass is displaced \(10 \mathrm{~mm}\) from equilibrium while the left mass is held in place. The system is released from rest from this configuration.(a) Determine the natural frequencies, mode
If \(m_{2}=10 \mathrm{~kg}, k_{2}=1000 \mathrm{~N} / \mathrm{m}\), and \(F(t)=150 \sin \omega t \mathrm{~N}\), for what values of \(\omega\) will the following resonances exist?(a) Primary resonance(b) Superharmonic resonance(c) Subharmonic resonance(d) Combination resonance Refer to the systems of
Consider the system of Figure P12.36.(a) Derive the nonlinear differential equations governing the motion of the system using the generalized coordinates shown.(b) Expand trigonometric functions of the generalized coordinates using Taylor series expansions. Rewrite the differential equations
Show that the coefficient multiplying \(p_{2}^{2} p_{1}\) for a pinned-pinned beam is zero in Equation (11.50).\(\delta=A-\epsilon \frac{A^3}{32} \tag{12.50}\)
A fixed-free rectangular steel beam ( \(\left.ho=7850 \mathrm{~kg} / \mathrm{m}^{3}, E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) with a length of \(1 \mathrm{~m}\), base of \(2 \mathrm{~cm}\), and height of \(5 \mathrm{~cm}\) is subject to a single frequency harmonic excitation. List
If the beam of Chapter Problem 12.38 is fixed-fixed, which of the following excitation frequencies should be avoided and why?(a) \(180 \mathrm{rad} / \mathrm{s}\)(b) \(1530 \mathrm{rad} / \mathrm{s}\)(c) \(2200 \mathrm{rad} / \mathrm{s}\)(d) \(7940 \mathrm{rad} / \mathrm{s}\)Data From Chapter
The Rayleigh distribution can be applied to random variables with positive values.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A stationary process is one in which a representative sample of ensemble measurements can be used for the entire process.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The Weiner-Khintchine equations imply that the autocorrelation function is the Fourier transform of the power spectral density.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
If \(P(0)=0.5\) for a normalized random variable, the probability distribution follows the Gaussian distribution.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The probability distribution function is the derivative of the probability density function.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The autocorrelation function is an even function of \(\tau\) for a stationary process.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The transfer function is defined as the Fourier transform of the output of a system divided by the Fourier transform of the input to a system is equal to the sinusoidal transfer function for the system.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite
If \(x(t)=A \sin 5 t\), then \(p(x)=\frac{1}{A}\) for \(|x|
The mean of a random function can be calculated by \(\mu=\int_{-\infty}^{\infty} x p(x) d x\) for a stationary ergodic process.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The variance is the positive square root of the standard deviation.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A narrowband process has a power spectral density defined over a narrow band of frequencies.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
For a stationary process \(R(0)=1\).Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
What is an ensemble?
What is a stationary process?
What is an ergodic process?
Which is more likely to be a random process, the wind induced vibrations of a bridge or the rotating unbalance of a machine?
What is the total area under the curve of a probability density function?
What does the Central Limit theorem imply?
What is the power spectral density function for ideal white noise?
What is the autocorrelation function for ideal white noise?
What is the Fourier transform of \(\delta(t)\) ?
If the probability density function \(p(x)\) is known for a random variable, what is the probability distribution \(P(x)\) ?
What is \(P(0)\) for the normalized Gaussian distribution?
The probability of the maximum value of the response of a system follows what probability distribution?
The probability that the absolute value of the response of a system follows what probability distribution?
A random variable has a probability distribution, \(P(x)\). What is \(p(\infty)\) ?
A random variable has a probability distribution, \(P(x)\). What is the probability that \(x>b\) ?
A random variable has a probability distribution, \(P(x)\). What is the probability that \(-1 \leq x
If the power spectral density of an input force is \(S_{F}(\omega)\) and the transfer function for the system the force is applied to is \(H(\omega)\), what is the power spectral density of the output?
The spectral density of a random process is \(S(\omega)\). How is the mean square value of the process determined?
For the normalized Gaussian distribution \(P(z)\), determine the following.(a) What is the probability that \(z
A random variable has a Gaussian distribution with \(\mu=-1.3\) and \(\sigma=2.8\). Determine the following.(a) What is the probability that \(x3.3\) ?(c) What is the probability that \(0
A random variable has a Rayleigh distribution with \(\mu=3.1\). Determine the following.(a) What is the probability that \(x>3.1\) ?(b) What is the probability that \(x
The probability density function for the standard Cauchy distribution is\(p(x)=\frac{1}{\pi\left(1+x^{2}\right)}\)(a) What is the probability distribution function for the Cauchy distribution?(b) What is the mean of the Cauchy distribution?(c) What is the mean square value of the Cauchy
The probability distribution function for the standard Weibull distribution is\(P(x)=1-e^{-x y}\)What is its probability density function \(p(x)\) ?
Consider the system shown in Figure SP13.36.(a) What is the transfer function for the system \(H(\omega)=\frac{X(\omega)}{F(\omega)}\) ?(b) Determine \(|H(\omega)|\).(c) The system is subject to ideal white noise. What is the power spectral density of the input?(d) What is the power spectral
It is desired to approximate the random displacement of a machine due to a random force \(F(t)\). What are the SI units of the following.(a) The power spectral density of the displacement \(S_{x}(\omega)\)(b) The power spectral density of the force \(S_{F}(\omega)\)(c) The Fourier transform of the
Determine the autocorrelation function for \(x(t)=A \cos 2 t\).
Determine the autocorrelation function for the rectangular wave shown in Figure P13.2. A F -2T -15T -T -7T T T 9T 2T 17T t 8 8 8 8 8 FIGURE P13.2
Determine the autocorrelation function for the rectangular wave shown in Figure P13.3. A -5T -2T -3T T T 3T 2T 7T t 2 2 2 2 2 FIGURE P13-3
Determine the autocorrelation function for the triangular wave shown in Figure P13.4. 3T N F FIGURE P13-4 IN T 72 31 2
Determine the autocorrelation function for the triangular wave shown in Figure P13.5. A 1444. -T FIGURE P13.5 0 T T 2 t
Determine the autocorrelation function for the triangular wave shown in Figure P13.6. A 717 HIN 0 FIGURE P13.6 -A T 72
A sine wave has the form\(x(t)=3-2 \sin 4 t\)Determine the expected value of \(x\) and \(x^{2}\).
Assume that \(t\) is uniformly distributed.(a) Determine the probability density function \(p(x)\) for the function in Chapter Problem 13.7.(b) Determine the probability distribution function \(P(x)\) for the function in Chapter Problem 13.7.Data From Chapter Problem 13.7:A sine wave has the
Determine the probability density function for the periodic function, one period of which is shown in Figure P13.9 P(t) A -T 0 FIGURE P13.9 T14 T
Determine the probability density function for the half-period cosine wave of Figure P13.10. A COS 72 FIGURE P13.10 N
Determine the Fourier transform of the rectangular pulse of Figure P13.11. P(t) Fo -T FIGURE P13.11 T 72
Determine the Fourier transform for the triangular pulse of Figure P13.12 Fo -T FIGURE P13.12 T 2
Determine the Fourier transform of the half-period cosine wave of Figure P13.10. FIGURE P13.10 cos T 22
Determine the power spectral density of the wave shown in Figure P13.2. A F -2T -15T -T -77 T 27 177 8 8 8 8 FIGURE P13.2
Determine the power spectral density of the wave shown in Figure P13.3. -27 FIGURE P13-3 F He T 37 22 2 27 22 77 Ela
Determine the power spectral density of the wave shown in Figure P13.4. -37 FIGURE P13-4 F T 72 3T 2
Determine the power spectral density of the wave shown in Figure P13.6. FIGURE P13.6 +14- 0
A force has band limited white noise with frequency bounds of \(\omega_{1}=100 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}=500 \mathrm{rad} / \mathrm{s}\) and magnitude \(S_{0}=2 \times 10^{2} \mathrm{~N}^{2} \cdot \mathrm{s} / \mathrm{rad}\) determine the following.(a) The autocorrelation
A SDOF system with a mass of \(20 \mathrm{~kg}, \zeta=0.1\) and \(\omega_{n}=100 \mathrm{rad} / \mathrm{s}\) is subject to white noise with \(S_{0}=1 \times 10^{-2} \mathrm{~N}^{2} \cdot \mathrm{s} / \mathrm{rad}\). What is the power spectral density of the response \(S_{x}(\omega)\) ?
A SDOF system with a mass of \(30 \mathrm{~kg}, \zeta=0.05\), and \(\omega_{n}=200 \mathrm{rad} / \mathrm{s}\) is subject to white noise with \(S_{0}=1 \times 10^{-2} \mathrm{~N}^{2} / \mathrm{Hz}\). What is the power spectral density of the response \(W_{x}(f)\) ?
A SDOF system with a mass of \(20 \mathrm{~kg}, \zeta=0.1\), and \(\omega_{n}=100 \mathrm{rad} / \mathrm{s}\) is subject to white noise with \(S_{0}=1 \times 10^{-2} \mathrm{~N}^{2} \cdot \mathrm{s} / \mathrm{rad}\).(a) What is the mean square value of the response of the system?(b) What is the
The SDOF system of Figure P13.22 is subject to a white noise with \(S_{0}=1 \times 10^{-2} \mathrm{~m}^{2} / \mathrm{rad} \cdot \mathrm{s}\) (the power spectral density of the acceleration of the base). Calculate the mean square value of the acceleration of the \(20 \mathrm{~kg}\) block. 20 kg 1
The SDOF system of Figure P13.23 is subject to a white noise with \(S_{0}=1 \times\) \(10^{-3} \mathrm{~N}^{2} \cdot \mathrm{s} / \mathrm{rad}\). What is the mean square value of the response of the \(300 \mathrm{~kg}\) mass. www 1 x 10 N/m 300 kg Txo x(t) 1 105 N/m 2000 N.s/m y(t) FIGURE P13.23
Solve Chapter Problem 13.22, assuming the acceleration is band limited with \(\omega_{1}=10 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}=30 \mathrm{rad} / \mathrm{s}\).Data From Chapter Problem 13.22:The SDOF system of Figure P13.22 is subject to a white noise with \(S_{0}=1 \times 10^{-2}
A two SDOF has governing differential equations\(\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{l}\dot{x}_{1} \\ \dot{x}_{2}\end{array}\right]+\left[\begin{array}{rr}5 & -2 \\ -2 & 2\end{array}\right]\left[\begin{array}{l}\dot{x}_{1} \\
A piecewise continuous function that satisfies the boundary conditions is an admissible function for approximation of the natural frequencies of a beam.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The boundary condition at a free end for a bar is a geometric boundary condition.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The boundary conditions at a free end for a beam are natural boundary conditions.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A beam element has four degrees of freedom.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A finite-element model of a bar with \(\mathrm{n}\) elements predicts \(\mathrm{n}\) natural frequencies of the bar.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
Natural frequency approximations using the finite element method are determined as the square roots of the eigenvalues of \(\mathbf{M}^{-1} \mathbf{K}\) where \(\mathbf{M}\) is the global mass matrix and \(\mathbf{K}\) is the global stiffness matrix.Indicate whether the statement presented is true
The finite-element method can be used to approximate the displacement of a system subject to initial conditions.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The global generalized coordinates for a pinned-pinned beam are an accumulation of the local generalized coordinates.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The stiffness matrix for an interior element of length \(\ell\) for a variable area bar is\[ k=\frac{E A}{\ell}\left[\begin{array}{rr} 1 & -1 \\ -1 & 1 \end{array}\right] \]Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it
The functions \(w_{1}(x)=x+1\) and \(w_{2}(x)=x^{2}+1\) can be used as trial functions using the assumed-mode method to predict the lowest natural frequencies of a fixed-free bar.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make
What is an admissible function?
What are natural boundary conditions?
Give a summary of the assumed-modes method.
A finite-element model of a bar fixed at \(x=0\) at one end and having a mass \(m\) rigidly attached at \(x=L\) must satisfy what boundary condition?
A finite-element model of a torsional shaft that is attached to a spring of torsional stiffness \(k_{t 1}\) at \(x=0\) and a spring of torsional stiffness \(k_{t 2}\) at \(x=L\) must satisfy what boundary conditons?
A torsional bar element has two degrees of freedom. What are the generalized coordinates associated with these degrees of freedom?

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