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engineering
introduction mechanical engineering

Questions and Answers of Introduction Mechanical Engineering

A bar with a length of \(L\) and cross-sectional area \(A\) is made of a material with an elastic modulus \(E\) and mass density \(ho\) is fixed at \(x=0\) and has a rigid mass \(m\) attached at
A bar with a length of \(L\) and cross-sectional area \(A\) is made of a material with an elastic modulus \(E\) and mass density \(ho\) is fixed at \(x=0\) and is attached to a spring with a
The differential equation for the vibrations of a beam is\[ ho A \frac{\partial^{2} w}{\partial t^{2}}+E I \frac{\partial^{4} w}{\partial x^{4}}=f(x, t) \]Explain the physical meaning of each term
The characteristic equation for a fixed-free beam is \(\cos \lambda^{1 / 4} \cosh \lambda^{1 / 4}=-1\). This is an example of a ______________ equation to solve for \(\lambda\).
What are the boundary conditions for the free vibrations of a fixed-free beam?
What are the boundary conditions for the free vibrations of a free-free beam?
What are the boundary conditions for the free vibrations of a beam that is fixed at \(x=0\) and has a rigid mass \(m\) attached at \(x=L\) ?
The characteristic equation for the fixed-pinned beam is the same as the characteristic equation for the pinned-free beam, yet their lowest natural frequency is different. How is this possible?
A bar with a length of \(L\) and cross-sectional area \(A\) is made of a material with an elastic modulus \(E\) and mass density \(ho\) and has a normalized longitudinal mode shape \(X_{k}(x)\) which
For Short Answer Problem 10.29 what is the value of \(R(w)\) ?Data From Short Problem 10.29:A bar with a length of \(L\) and cross-sectional area \(A\) is made of a material with an elastic modulus
A beam with a length \(L\) cross-sectional area \(A\), and moment of inertia \(I\) is made of a material with an elastic modulus \(E\) and mass density \(ho\) and has a normalized transverse mode
For Short Answer Problem 10.31 what is the value of \(R(w)\) ?Data From Short Problem 10.31:A beam with a length \(L\) cross-sectional area \(A\), and moment of inertia \(I\) is made of a material
What is the wave speed for torsional oscillations in a circular shaft made from steel? The shaft is of length \(60 \mathrm{~cm}\) and has a radius of \(3 \mathrm{~cm}\).
Calculate the wave speed of longitudinal waves in a \(3-\mathrm{m}\) long steel bar \(\left(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, ho=7580 \mathrm{~kg} / \mathrm{m}^{3}\right)\) with a
Calculate the three lowest natural frequencies of a solid \(20-\mathrm{cm}\) radius steel shaft \(\left(G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, ho=7500 \mathrm{~kg} / \mathrm{m}^{3}\right)\)
The characteristic equation for a fourth-order continuous system is \(\cos \lambda=0\). What is the lowest natural frequency of the system?
What are the three lowest positive values of \(\lambda\) that satisfy the equation \(\tan \lambda=6 / \lambda\) ?
What are the three lowest positive values of \(\lambda\) that satisfy the equation \(\tan \lambda=4 \lambda\) ?
The nondimensional mode shape of a uniform bar is \(\sin 5 \pi \mathrm{x}\).(a) Determine the potential energy of this mode.(b) Determine the kinetic energy of this mode.(c) What is the
The nondimensional mode shape of a beam is \(\sqrt{2} \sin 3 \pi x\).(a) Determine the potential energy of this mode.(b) Determine the kinetic energy of this mode.(c) What is the nondimensional
A circular bar with a length of \(80 \mathrm{~cm}\) and radius of \(3 \mathrm{~cm}\) is made of steel which has an elastic modulus \(200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and mass density
A carbon nanotube ( \(E=1 G P a, ho=2.3 \mathrm{~g} / \mathrm{cm}^{3}\) ) has a length of \(200 \mathrm{~nm}\) and radius of \(5 \mathrm{~nm}\). Using a fixed-free beam model for the nanotube,
Each of the beams of Figures SP10.43 is made from a material of \(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and \(ho=7580 \mathrm{~kg} / \mathrm{m}^{3}\) with A \(=1.2 \times 10^{-2}
Each of the beams of Figures SP10.44 is made from a material of \(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and \(ho=7580 \mathrm{~kg} / \mathrm{m}^{3}\) with A \(=1.2 \times 10^{-2}
Each of the beams of Figures SP10.45 is made from a material of \(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and \(ho=7580 \mathrm{~kg} / \mathrm{m}^{3}\) with A \(=1.2 \times 10^{-2}
Find all non-trivial solutions to the boundary value problem\[ \frac{d^{2} X}{d x^{2}}+\lambda X=0 \quad X^{\prime}(0)=0 \quad X^{\prime}(1)=0 \]
Find all non-trivial solutions to the boundary value problem\[ \frac{d^{4} X}{d x^{4}}-\lambda X=0 \quad X(0)=0 \quad X^{\prime \prime}(0)=0 \quad X(1)=0 \quad X^{\prime \prime}(1)=0 \]
Specify the SI units of the given quantity.(a) Wave speed of longitudinal vibrations in a bar, \(c\)(b) Flexural rigidity of a beam, \(E l\)(c) Natural frequency of sixth mode, \(\omega_{6}\)(d)
A \(5000 \mathrm{~N} \cdot \mathrm{m}\) torque is statically applied to the free end of a solid \(20-\mathrm{cm}\) radius steel shaft ( \(\left.G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2},
A \(5000 \mathrm{~N} \cdot \mathrm{m}\) torque is statically applied to a the midspan of a solid \(20-\mathrm{cm}\) radius steel shaft \(\left(G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, ho=7500
A steel shaft \(\left(ho=7850 \mathrm{~kg} / \mathrm{m}^{3}, G=85 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right)\) with a inner radius of \(30 \mathrm{~mm}\), outer radius of \(50 \mathrm{~mm}\),
A \(10,000-\mathrm{N} \cdot \mathrm{m}\) torque is applied to the midspan of the shaft of Chapter Problem 10.3 and suddenly removed. Determine the time-dependent angular displacement of the midspan
A motor of mass moment of inertia \(85 \mathrm{~kg} \cdot \mathrm{m}^{2}\) is attached to the end of the shaft of Chapter Problem 10.1. Determine the three lowest natural frequencies of the shaft and
Show the orthogonality of the two lowest mode shapes of the system in Chapter Problem 10.5.Data From Chapter Problem 10.5:A motor of mass moment of inertia \(85 \mathrm{~kg} \cdot \mathrm{m}^{2}\) is
Operation of the motor attached to the shaft of Chapter Problem 10.5 produces a harmonic torque of amplitude \(2000 \mathrm{~N} \cdot \mathrm{m}\) at a frequency of \(110 \mathrm{~Hz}\). Determine
A 20-cm-diameter, 2-m-long steel shaft ( \(ho=7600 \mathrm{~kg} / \mathrm{m}^{3}, G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) ) has rotors of mass moment of inertia \(110 \mathrm{~kg} \cdot
Determine an expression for the natural frequencies of the shaft of Figure P10.9. FIGURE P10.9 J, G, p
An oil well drilling tool is modeled as a bit attached to the end of a long shaft, unrestrained from rotation at its fixed end.(a) Determine the equation defining the natural frequencies of the
The shaft of Chapter Problem 10.1 is at rest in equilibrium when the time dependent moment of Figure P10.11 is applied to the end of the shaft. Determine the time-dependent form of the resulting
The shaft of Chapter Problem 10.1 is at rest in equilibrium when it is subject to the uniform time-dependent torque loading per unit length of Figure P10.12. Determine the time-dependent form of the
The elastic bar of Figure P10.13 is undergoing longitudinal vibrations. Let \(u(x, t)\) be the time-dependent displacement of a particle along the centroidal axis of the bar, initially a distance
Using the results of Chapter Problem 10.13, determine the natural frequencies of longitudinal vibrations of a bar fixed at one end and free at the other.Data From Chapter Problem 10.13:The elastic
Show the orthogonality of mode shapes of longitudinal vibration of a bar fixed at one end and free at its other end.
A large industrial piston operates at \(1000 \mathrm{~Hz}\). The piston head has a mass of \(20 \mathrm{~kg}\). The shaft is made from steel \(\left(ho=7500 \mathrm{~kg} / \mathrm{m}^{3}, E=210
The free end of the piston of Chapter Problem 10.16 is subject to a force \(1000 \sin \omega t \mathrm{~N}\), where \(\omega=100 \mathrm{~Hz}\). If the diameter of the shaft is \(8 \mathrm{~cm}\),
Determine the five lowest natural frequencies of the system of Figure P10.18. k FIGURE P10.18 L P.E, A p=7500 kg/m E 200 x 109 N/m A = 1.5 10-5 m L=3m k = 1 x 10 N/m k-1.5 10 N/m
Determine the steady-state response of the system of Figure P10.19.\[ \begin{aligned} & ho=7500 \mathrm{~kg} / \mathrm{m}^{3} \\ & E=200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2} \\
Determine the steady-state response of the system of Figure P10.20. FIGURE P10.20 p, E, A m Fo sin ot k p=7500 kg/m E 200 x 10 N/m A-4.5 105 m k = 9 10 N/m m = 2.5 kg L = 3.5 m = Fo 600 N w=450
Draw frequency response curves for the response of the disk at the end of the shaft in Example 10.3. Plot the curves for \(\beta=0.5, \beta=2\), and \(\beta=20.0\).Example 10.3:The thin disk of
Determine the steady-state response of a circular shaft subject to a uniform torque per unit length \(T_{0} \sin \omega t\) applied over its entire length.
Determine the steady-state response of the system of Figure P10.23. p. J. G FIGURE P10.23 L To sin cot
Propeller blades totaling \(1200 \mathrm{~kg}\) with a total mass moment of inertia of \(155 \mathrm{~kg} \cdot \mathrm{m}^{2}\) are attached to a solid circular shaft ( \(ho=5000 \mathrm{~kg} /
A pipe used to convey fluid is cantilevered from a wall. The steel pipe \(\left(ho=7500 \mathrm{~kg} / \mathrm{m}^{3}, G=80 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, E=200 \times 10^{9} \mathrm{~N}
Verify the characteristic equation given in Table 10.4 for a pinned-free beam. TABLE 10.4 Natural frequencies and mode shapes for beams Characteristic Five Lowest Natural Frequencies Kinetic Energy
Verify the characteristic equation given in Table 10.4 for a fixed-fixed beam. TABLE 10.4 Natural frequencies and mode shapes for beams Characteristic Five Lowest Natural Frequencies Kinetic Energy
Verify the orthogonality of the eigenfunctions given in Table 10.4 for a pinnedfree beam. TABLE 10.4 Natural frequencies and mode shapes for beams Characteristic Five Lowest Natural Frequencies
Verify the orthogonality of the eigenfunctions given in Table 10.4 for a fixed-attached mass beam. TABLE 10.4 Natural frequencies and mode shapes for beams Characteristic Five Lowest Natural
Determine the time-dependent displacement for the beam shown in Figures P10.30. + FIGURE P10.30 Fo sin cor P, A, E, I
Determine the time-dependent displacement for the beam shown in Figures P10.31. FIGURE P10.31 T. 2 Fo sin cot P, A, E, I
Determine the time-dependent displacement for the beam shown in Figures P10.32. Foe-at L L FIGURE P10.32 2 p. A, E, I
Determine the time-dependent displacement for the beam shown in Figures P10.33. Fo sin cot FIGURE P10.33 L
Determine the time-dependent displacement for the beam shown in Figures P10.34. FIGURE P10.34 EI = 12, 2DALA - Fosin ot 102 13 m 0.35 PAL
A root manipulator is \(60 \mathrm{~cm}\) long, made of steel \(\left(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\right.\), \(ho=7500 \mathrm{~kg} / \mathrm{m}^{3}\) ) and has the cross section
The steam pipe of Figure P10.36 is suspended from the ceiling in an industrial plant. A heavy machine with a rotating unbalance is placed on the floor above the machine causing vibrations of the
A simplified model of the rocket of Figure P10.37 is a free-free beam.(a) Calculate the five lowest natural frequencies for longitudinal vibration.(b) Calculate the five lowest natural frequencies
Longitudinal vibrations are initiated in the rocket of Figure P10.38 when thrust is developed. Determine the Laplace transform of the transient response \(\mathrm{U}(x, s)\) when the thrust of Figure
Determine the response of a cantilever beam when the fixed support is subject to a displacement \(f(t)=A \sin \omega t\). Use the Laplace transform method and determine the transform \(W(x, s)\). Do
The tail rotor blades of a helicopter have a rotating unbalance of magnitude \(0.5 \mathrm{~kg} \cdot \mathrm{m}\) and operate at a speed of \(1200 \mathrm{rpm}\). Modeling the tail section as a
Determine the steady-state amplitude of the engine of Figure P10.41. FIGURE P10.41 4.1 m Rotating unbalance E p = 7800 kg/m E 200 x 109 N/m I= 4.5 x 10 m A 1.6 103 m m=55 kg k = 5 10 N/m moe 1.8 x
Show that the differential equation governing free vibration of a uniform beam subject to a constant axial load, \(P\), is\[ E I \frac{\partial^{4} w}{\partial x^{4}}-P \frac{\partial^{2}
Determine the frequency equation for a simply supported beam subject to an axial load.
Determine the frequency equation for a fixed-pinned beam subject to an axial load.
A fixed-fixed beam is made of a material with a coefficient of thermal expansion \(\alpha\). After installed, the temperature is decreased by \(\Delta T\). Determine the beam's frequency equation.
Show orthogonality of the mode shapes for a simply supported beam subject to an axial load.
Use Rayleigh's quotient to approximate the lowest natural frequency of a torsional shaft fixed at both ends.
Use Rayleigh's quotient to approximate the lowest natural frequency of a torsional shaft with a disk of mass moment of inertia \(I\) placed at its midspan. The shaft is fixed at both ends.
Use Rayleigh's quotient to approximate the lowest natural frequency of a fixedfree beam.
Use Rayleigh's quotient to approximate the lowest natural frequency of a simply supported beam with a mass \(m\) at its midspan. Use \(w(x)=\sin (\pi x / L)\) as the trial function.
Use the Rayleigh-Ritz method to approximate the two lowest natural frequencies of a fixed-free beam.
Use the Rayleigh-Ritz method to approximate the two lowest natural frequencies of the system of Figure P10.52. p=6000 kg/m E 200 x 10 N/m k = 1 10 N/m Im E FIGURE P10.52 2 m -20 mm 35 mm
Use the Rayleigh-Ritz method to approximate the two lowest natural frequencies for the system of Figure P10.53. FIGURE P10.53 I-7.1 kg m p=4000 kg/m G = 60 109 N/m r = 35 mn 60 cm -40 cm
Use the Rayleigh-Ritz method to approximate the three lowest natural frequencies of a fixed-pinned beam. Use polynomial of order six or less as trial functions.
Use the Rayleigh-Ritz method to approximate the three lowest natural frequencies and their corresponding mode shapes of a fixed-free beam. Use polynomials of order six or less as trial functions.
Use the Rayleigh-Ritz method to approximate the two lowest frequencies of transverse vibration of the system of Figure P10.56. 1 70 cm FIGURE P10.56 80 kg E 200 x 10" N/m 1 5.6 x 106m4 A 2.4 x 103 m
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.1 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.2 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.3 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.4 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.5 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.6 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Calculate the natural frequencies and mode shapes for the system shown in Figures P8.7 by calculating the eigenvalues and eigenvectors of \(\mathbf{M}^{-1} \mathbf{K}\). Graphically illustrate the
Two machines are placed on the massless fixed-pinned beam of Figure P8.8.Determine the natural frequencies for the system. 20 kg 30 kg -1m- 0.5 m FIGURE P8.8 E 210 x 109 N/m 1-5.6 104 m
Determine the natural frequencies and mode shapes for the system of Figure P7.2 if \(k=3.4 \times 10^{5} \mathrm{~N} / \mathrm{m}, L=1.5 \mathrm{~m}\) and \(m=4.6 \mathrm{~kg}\). 13 -25- + www FIGURE
Determine the natural frequencies of the system of Figure P7.5 if \(k=2500 \mathrm{~N} / \mathrm{m}\), \(m_{1}=2.4 \mathrm{~kg}, m_{2}=1.6 \mathrm{~kg}, I=0.65 \mathrm{~kg} \cdot \mathrm{m}^{2}\),
Determine the natural frequencies and mode shapes for the system of Figure P7.17 if \(k=10,000 \mathrm{~N} / \mathrm{m}, m=3 \mathrm{~kg}, I=0.6 \mathrm{~kg} \cdot \mathrm{m}^{2}\), and \(r=80
Determine the natural frequencies and mode shapes of the system of Figure P7.19 if \(k=12,000 \mathrm{~N} / \mathrm{m}\) and each bar is of mass \(12 \mathrm{~kg}\) and length \(4 \mathrm{~m}\). x1
A \(400 \mathrm{~kg}\) machine is placed at the midspan of a 3-m-long, 200-kg simply supported beam. The beam is made of a material of elastic modulus \(200 \times 10^{9} \mathrm{~N} /
A \(500 \mathrm{~kg}\) machine is placed at the end of a \(3.8-\mathrm{m}-\mathrm{long}, 190-\mathrm{kg}\) fixed-free beam. The beam is made of a material of elastic modulus \(200 \times 10^{9}
Determine the two lowest natural frequencies of the railroad bridge of Chapter Problem 7.84, if \(k_{1}=5.5 \times 10^{7} \mathrm{~N} / \mathrm{m}, k_{2}=1.2 \times 10^{7} \mathrm{~N} / \mathrm{m},
Determine the natural frequencies of the system of Chapter Problem 7.89. The beam is of length \(5 \mathrm{~m}\), made of a material of elastic modulus \(200 \times 10^{9} \mathrm{~N} /

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