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engineering
introduction mechanical engineering

Questions and Answers of Introduction Mechanical Engineering

Solve Chapter Problem 13.21, assuming the power spectral density is band limited with \(\omega_{1}=50 \mathrm{rad} / \mathrm{s}\) and \(\omega_{2}=200 \mathrm{rad} / \mathrm{s}\).Data From Chapter
Solve Chapter Problem 13.21, assuming the force is narrowband with a power spectral density given by \(S_{F}(\omega)=\frac{3 \times 10^{-3}}{2+5 \omega^{2}}\).Data From Chapter Problem 13.21:A SDOF
Answer the following questions.(a) All bodies which possess ____________ and ______________ are capable of vibrations.(b) Name the three fundamental categories of vibration.(c) How many degrees of
For the waveform shown below, answer the following questions.(a) Is this motion periodic?(b) If the motion is periodic, what is the period of the motion (in seconds)?Fig. P1.2 Example waveform (c) If
To explore the Fourier series representation of signals, complete the following.(a) Approximate a square wave by plotting the following function in MATLAB \({ }^{\circledR}\). Begin a new \(m\)-file
If displacement can be described as \(x=5 \cos (\omega t) \mathrm{mm}\), where \(\omega=6 \pi \mathrm{rad} / \mathrm{s}\), complete the following.(a) Plot the displacement over the time interval from
The complex exponential function, \(x=e^{i \omega t}\), can be used to describe harmonic motion (the function can be defined in MATLAB \({ }^{\circledR}\) using \(\mathrm{x}=\exp\) \(\left(1 i
A harmonic motion has an amplitude of \(0.2 \mathrm{~cm}\) and a period of \(15 \mathrm{~s}\).(a) Determine the maximum velocity \((\mathrm{m} / \mathrm{s})\) and maximum acceleration
Determine the sum of the two vectors \(x_{1}=6 e^{i \frac{\pi}{6}}\) and \(x_{2}=-1 e^{i \frac{\pi}{3}}\).
If the velocity at a particular point on a body is \(v(t)=250 \sin (100 t)\), complete the following.(a) Plot the velocity in the complex plane at \(t=0.1 \mathrm{~s}\).(b) Using the velocity
In bungee jumping, a person leaps from a tall structure while attached to a long elastic cord. Would the resulting oscillation be best described as free, forced, or self-excited vibration?
The sine function can be represented as \(\sin (\theta)=\theta-\frac{\theta^{3}}{3 !}+\frac{\theta^{5}}{5 !}-\frac{\theta^{7}}{7 !}+\frac{\theta^{9}}{9 !} \cdots\). Plot the percent error between
For a single degree of freedom spring-mass system with \(m=1 \mathrm{~kg}\) and \(k=\) \(4 \times 10^{4} \mathrm{~N} / \mathrm{m}\), complete the following for the case of free vibration.(a)
For a single degree of freedom spring-mass system, complete the following.(a) If the free vibration is described as \(x(t)=A \cos \left(\omega_{n} t+\Phi_{c}\right)\), determine expressions for \(A\)
The differential equation of motion for a cylinder rolling on a concave cylindrical surface is \(\ddot{\theta}+\frac{2}{3} \frac{g}{R-r} \theta=0\), where \(g\) is the gravitational constant.Fig.
For a single degree of freedom spring-mass-damper system with \(m=1 \mathrm{~kg}\), \(k=4 \times 10^{4} \mathrm{~N} / \mathrm{m}\), and \(c=10 \mathrm{~N}-\mathrm{s} / \mathrm{m}\), complete the
For a single degree of freedom spring-mass-damper system with \(m=0.2 \mathrm{lb}_{\mathrm{m}}\), \(k=2.5 \times 10^{3} \mathrm{lb}_{\mathrm{f}} / \mathrm{in} ., c=10.92
For a single degree of freedom spring-mass-damper system under free vibration, determine the values for the mass, \(m(\mathrm{~kg})\), viscous damping coefficient, \(c(\mathrm{~N}-\mathrm{s} /
For a single degree of freedom spring-mass-damper system under free vibration, the following information is known: \(m=2 \mathrm{~kg}, k=1 \times 10^{6} \mathrm{~N} / \mathrm{m}, c=\) \(500
The requirement for small features on small parts has led to increased demands on measuring systems. One approach for determining the size of features (such as a hole's diameter) is to use a probe to
If the free vibration of a single degree of freedom spring-mass system is described as \(x(t)=A \sin \left(\omega_{n} t+\Phi_{s}\right)\), determine expressions for \(A\) and \(\Phi_{s}\) if the
For a single degree of freedom spring-mass-damper system, the free vibration response shown in the Fig. P2.10a was obtained due to an initial displacement with no initial velocity.Fig. P2.10a Free
An apparatus known as a centrifuge is commonly used to separate solutions of different chemical compositions. It operates by rotating at high speeds to separate substances of different densities.Fig.
For a single degree of freedom spring-mass-damper system with \(m=2.5 \mathrm{~kg}\), \(k=6 \times 10^{6} \mathrm{~N} / \mathrm{m}\), and \(c=180 \mathrm{~N}-\mathrm{s} / \mathrm{m}\), complete the
A single degree of freedom lumped parameter system has mass, stiffness, and damping values of \(1.2 \mathrm{~kg}, 1 \times 10^{7} \mathrm{~N} / \mathrm{m}\), and \(364.4 \mathrm{~N}-\mathrm{s} /
For the single degree of freedom torsional system under harmonic forced vibration (see Fig. P3.4), complete parts a through c if \(J=40 \mathrm{~kg}-\mathrm{m}^{2} / \mathrm{rad}\), \(C=150
For a single degree of freedom spring-mass-damper system subject to forced harmonic vibration with \(m=1 \mathrm{~kg}, k=1 \times 10^{6} \mathrm{~N} / \mathrm{m}\), and \(c=120 \mathrm{~N}-\mathrm{s}
A single degree of freedom spring-mass-damper system which is initially at rest at its equilibrium position is excited by an impulsive force over a time interval of \(1.5 \mathrm{~ms}\); see Fig.
For a single degree of freedom spring-mass-damper system subject to forced harmonic vibration, the following FRF was measured (two figures are provided with different frequency ranges). Using the
For a single degree of freedom spring-mass-damper system with \(m=2 \mathrm{~kg}\), \(k=1 \times 10^{7} \mathrm{~N} / \mathrm{m}\), and \(c=200 \mathrm{~N}-\mathrm{s} / \mathrm{m}\), complete the
In a crank-slider setup, it is desired to maintain a constant rotational speed for driving the crank. Therefore, a flywheel was added to increase the spindle inertia and reduce the speed sensitivity
A single degree of freedom spring-mass-damper system with \(m=1.2 \mathrm{~kg}\), \(k=1 \times 10^{7} \mathrm{~N} / \mathrm{m}\), and \(c=364.4 \mathrm{~N}-\mathrm{s} / \mathrm{m}\) is subjected to a
Given the eigenvalues and eigenvectors for the two degree of freedom system shown in Fig. P4.1, determine the modal matrices \(m_{q}(\mathrm{~kg}), c_{q}(\mathrm{~N}-\mathrm{s} / \mathrm{m})\), and
Given the two degree of freedom system in Fig. P4.2, complete the following.(a) Write the equations of motion in matrix form.(b) Write the system characteristic equation using Laplace notation. Your
Given the two degree of freedom system shown in Fig. P4.3, complete the following.(a) Write the equations of motion in matrix form.(b) Write the system characteristic equation using Laplace notation.
A two degree of freedom spring-mass system is shown in Fig. P4.4. For harmonic free vibration, complete the following if \(k=5 \times 10^{6} \mathrm{~N} / \mathrm{m}\) and \(m=2 \mathrm{~kg}\).(a)
A two degree of freedom spring-mass system is displayed in Fig. P4.5. For harmonic free vibration, complete the following if \(k_{1}=2 \times 10^{6} \mathrm{~N} / \mathrm{m}\), \(m_{1}=0.8
For the same system as described in problem 5, complete the following.(a) The initial displacements for the system's free vibration are \(x_{1}(0)=0.378\) \(\mathrm{mm}\) and \(x_{2}(0)=1
A two degree of freedom spring-mass-damper system is shown in Fig. P4.7. For harmonic free vibration, complete the following if \(k_{1}=2 \times 10^{5} \mathrm{~N} / \mathrm{m}\), \(c_{1}=60
Given the two degree of freedom system in Fig. P4.8, complete the following.(a) Write the equations of motion in matrix form.(b) Verify that proportional damping exists.(c) Determine the roots of the
Determine the mass, damping, and stiffness matrices in local coordinates for the model shown in Fig. P4.9.Fig. P4.9 Three degree of freedom spring-mass-damper model. k 3c 3m C 2c + m X2 k 2m X3 2k C
Given the mass, damping, and stiffness matrices for the model shown in Fig. P4.9 determined from problem 9, can proportional damping exist for this system? Justify your answer.Problem 9.Determine the
A two degree of freedom spring-mass-damper system is shown Fig. P5.1. For harmonic forced vibration (due to the external force at coordinate \(x_{2}\) ), complete the following if \(k_{1}=2 \times
A two degree of freedom spring-mass-damper system is shown in Fig. P5.2. For harmonic forced vibration (due to the external force applied at coordinate \(x_{2}\) ), complete the following if
Consider the single degree of freedom spring-mass system shown in Fig. P5.3, where \(k=4 \times 10^{5} \mathrm{~N} / \mathrm{m}\) and \(m=8 \mathrm{~kg}\). It is being excited by a harmonic forcing
A two degree of freedom spring-mass-damper system is shown in Fig. P5.4. For harmonic forced vibration (due to the external force at coordinate \(x_{1}\) ), complete the following if \(k_{1}=2 \times
For the two degree of freedom spring-mass-damper system shown in Fig. P5.5, complete the following if \(k_{a}=2 \times 10^{5} \mathrm{~N} / \mathrm{m}, \quad k_{b}=5.5 \times 10^{4} \mathrm{~N} /
A dynamic absorber is to be designed to eliminate the vibration at coordinate \(x_{1}\) for the system shown in Fig. P5.6, where the excitation frequency is \(400 \mathrm{rad} / \mathrm{s}\) and the
Given the modal mass matrix, \(m_{q}=\left[\begin{array}{ll}2 & 0 \\ 0 & 2\end{array}\right] \mathrm{kg}\), the modal stiffness matrix, \(k_{q}=\left[\begin{array}{cc}5.858 \times 10^{6} & 0 \\ 0 &
After installation, it was found that a particular machine exhibited excessive vibration due to a harmonic excitation force with a frequency of \(100 \mathrm{~Hz}\). A dynamic absorber was designed
Given the eigenvalues and eigenvectors for the two degree of freedom system shown in Fig. P5.9, complete the following.\[\begin{array}{ll}s_{1}^{2}=-1 \times 10^{6} \mathrm{rad} / \mathrm{s}^{2}
Given the eigenvalues and eigenvectors for the two degree of freedom system shown in Fig. P5.9, determine the DC (zero frequency) compliance for the real part of the direct FRF \(X_{2} /
For a single-degree-of-freedom spring-mass-damper system subject to forced harmonic vibration, the measured FRF is displayed in Figs. P6.1a and P6.1b. Using the peak picking method, determine \(m\)
The direct and cross FRFs for the two degree of freedom system shown in Fig. P6.2a are provided in Figs. P6.2b and P6.2c.Fig. P6.2a Two degree of freedom spring-mass-damper system under forced
An FRF measurement was completed to give the two degree of freedom response shown in Fig. P6.3. Use the peak picking approach to identify the modal mass, stiffness, and damping parameters for the two
An FRF measurement was completed to give the two degree of freedom response shown in Fig. P6.4 (a limited frequency range is displayed to aid in the peak picking activity).Fig. P6.4 FRF measurement
Figures P6.5a through P6.5e show direct, \(X_{1} / F_{1}\), and cross FRFs, \(X_{2} / F_{1}\) through \(X_{5} / F_{1}\), measured on a fixed-free beam. They were measured at the beam's free end and
The measurement bandwidth was increased so that the first three modes were captured. Again, the direct, \(X_{1} / F_{1}\), and cross FRFs, \(X_{2} / F_{1}\) through \(X_{5} / F_{1}\), were measured.
Find the mass matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.7. if \(m_{1}=10 \mathrm{~kg}\) and \(m_{2}=12 \mathrm{~kg}\).Fig. P6.7 Two degree of freedom
Find the stiffness matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.7. if \(k_{1}=2 \times 10^{5} \mathrm{~N} / \mathrm{m}, k_{2}=2 \times 10^{5} \mathrm{~N} /
Find the mass matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.9. if \(m_{1}=10 \mathrm{~kg}\) and \(m_{2}=12 \mathrm{~kg}\).Fig. P6.9 Two degree of freedom
Find the stiffness matrix (in local coordinates) for the two degree of freedom system displayed in Fig. P6.9. if \(k_{1}=2 \times 10^{5} \mathrm{~N} / \mathrm{m}, k_{2}=2 \times 10^{5} \mathrm{~N} /
Complete the following statements.(a) Receptance is the frequency-domain ratio of _________________ to __________________ .(b) Mobility is the frequency-domain ratio of _____________________ to
Find three commercial suppliers of impact hammers for modal testing.
Find three commercial suppliers of dynamic signal analyzers for modal testing.
Digital data acquisition is to be used to record vibration signals for a particular system. If the highest anticipated frequency in the measurements is \(5,000 \mathrm{~Hz}\), select the minimum
An impact test was completed using an instrumented hammer to excite a structure and an accelerometer to measure the vibration response.(a) Show how to convert the acceleration-to-force frequency
As described in Sects. 7.2 and 7.4, FRFs are often measured using impact testing. In this approach, an instrumented hammer is used to excite the structure and a transducer is used to record the
For a particular measurement application, an accelerometer must be selected with a bandwidth or useful frequency range of \(5,000 \mathrm{~Hz}\). If the allowable deviation in the scaling coefficient
A single degree of freedom spring-mass-damper system which is initially at rest at its equilibrium position is excited by an impulsive force with a magnitude of \(250 \mathrm{~N}\) over a time
Determine the FRF for the system described in Problem 8 using Euler integration to calculate the time-domain displacement due to the impulsive input force. To increase the FRF frequency resolution,
The existence of modes with frequencies higher than the measurement bandwidth leads to an effect referred to as __________________ when performing a modal fit to the measured FRF.
Consider a uniform cross section fixed-free (i.e., clamped-free or cantilever) beam.(a) Sketch the first bending mode shape (lowest natural frequency).(b) Sketch the second mode shape (next lowest
In describing beam vibrations using Euler-Bernoulli beam theory, we derived the equation of motion \(\left(\partial^{4} Y / \partial x^{4}\right)-\lambda^{4} Y=0\).(a) In the equation of motion, what
Consider a fixed-free beam. The general solution to the equation of motion can be written as \(Y(x)=A \cos (\lambda x)+B \sin (\lambda x)+C \cosh (\lambda x)+D \sinh (\lambda x)\). To determine the
Consider the free-sliding beam shown in Fig. P8.4a. Direct and cross FRFs were measured at six locations and the imaginary parts are provided for the frequency interval near its second bending
Complete the following for the transverse deflection of a free-free cylindrical beam. The beam's diameter is \(15 \mathrm{~mm}\) and it is \(480 \mathrm{~mm}\) long. The beam material is 6061-T6
Complete the following for the torsion vibration of a free-free cylindrical beam. The beam's diameter is \(15 \mathrm{~mm}\) and it is \(480 \mathrm{~mm}\) long. The beam material is 6061-T6 aluminum
Consider the transverse vibration of a free-free cylindrical beam. If the diameter of a solid beam is \(d\), determine the outer diameter, \(d_{o}\), of a hollow beam with the same length and
For a \(25 \mathrm{~mm}\) diameter \(6061-\mathrm{T} 6\) aluminum \(\operatorname{rod}\left(ho=2,700 \mathrm{~kg} / \mathrm{m}^{3}\right.\) and \(E=70 \mathrm{GPa}\) ) with a nominal length of \(190
The Timoshenko beam model is more accurate than the Euler-Bernoulli beam model because it includes the effects of ___________________ and _______________________.
Determine the direct frequency response function, \(\frac{X_{2}}{F_{2}}\), for the two degree of freedom system shown in Fig. P9.1 using receptance coupling. Express your final result as a function
Determine the direct frequency-response function, \(\frac{X_{1}}{F_{1}}\), for the two degree of freedom system shown in Fig. P9.2 using receptance coupling. Express your final result as a function
Use receptance coupling to rigidly join two free-free beams and find the freefree assembly's displacement-to-force tip receptance. Both steel cylinders are described by the following parameters:
Plot the displacement-to-force tip receptance for a sintered carbide cylinder with free-free boundary conditions. The beam is described by the following parameters: \(19 \mathrm{~mm}\) diameter,
Determine the fixed-free displacement-to-force tip receptance for a sintered carbide cylinder by coupling the free-free receptances to a rigid wall (with zero receptances). The beam is described by
For a rigid coupling between two component coordinates \(x_{1 a}\) and \(x_{1 b}\), the compatibility condition is _______________________.
For a flexible coupling (spring stiffness \(k\) ) between two component coordinates \(x_{1 a}\) and \(x_{1 b}\), the compatibility condition is __________________ . An external force is applied to
What are the units for the rotation-to-couple receptance, \(p_{i j}\), used to describe the transverse vibration of beams?
What are the identical units for the displacement-to-couple, \(l_{i j}\), and rotationto-force, \(n_{i j}\), receptances used to describe the transverse vibration of beams?
Derive an expression for normal and shear stresses in a body subjected to biaxial loading.
Defined principal stresses and principal planes.
Derive and expression for principal stresses in a body subjected to complex state of stress.
Explain the procedure for drawing the Mohr’s circle under biaxial loading.
Explain the procedure for drawing Mohr’s circle under complex stresses.
Derive the relationship between elastic constants.
The biaxial stresses in a material are \(+100 \mathrm{MPa}\) and - \(50 \mathrm{MPa}\). Determine the normal shear and resultant stresses on a plane whose normal is inclined at \(30^{\circ}\) to the
At a certain point in a material the resultant stress on a plane is \(80 \mathrm{MPa}\) tensile inclined at \(30^{\circ}\) to the normal to the plane. The stress on a plane at right angles to this
The state of stress at a point is given by:\[ \sigma_{x}=90 \mathrm{MPa}, \sigma_{y}=-50 \mathrm{MPa}, \tau_{x y}=40 \mathrm{MPa} \]Determine (a) the planes of maximum shear stress, and (b)
The resultant stress at a point on plane \(A\) is \(+90 \mathrm{MPa}\) having an obliquity of \(15^{\circ}\). On another plane \(B\), the resultant stress is \(+50 \mathrm{MPa}\) having an obliquity

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