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introduction mechanical engineering

Questions and Answers of Introduction Mechanical Engineering

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
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 \(15 \mathrm{~mm}\)
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
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
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
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
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 =
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
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
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
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.
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
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
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
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} /
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
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.
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,
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
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
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
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
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,
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
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
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)
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
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
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
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}
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} /
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 \(S_{0}=1 \times 10^{-2} \mathrm{~N}^{2} / \mathrm{Hz}\).
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}
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
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
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
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 &
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
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
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
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
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
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
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
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
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
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
A torsional bar element has two degrees of freedom. What are the generalized coordinates associated with these degrees of freedom?
What are the local generalized coordinates associated with a beam element.

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