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engineering
introduction mechanical engineering

Questions and Answers of Introduction Mechanical Engineering

Determine the potential energy of the system at an arbitrary instant for the systems of Figure P7.20. Put the potential energy in a quadratic form. Use the quadratic form to determine the stiffness
Derive the stiffness matrix for the systems of Figure P7.1 using the indicated generalized coordinates and stiffness influence coefficients. k x1 2k x2 |--- ww m m ww m X3 ww FIGURE P7.1
Derive the stiffness matrix for the systems of Figure P7.2 using the indicated generalized coordinates and stiffness influence coefficients. -25- FIGURE P7.2 m 2m k k Slender rod of mass m 2k x1 1x2
Derive the stiffness matrix for the systems of Figure P7.3 using the indicated generalized coordinates and stiffness influence coefficients. L/2 L/2 L/4- L/4 Slender rod of x1 FIGURE P7-3 mass m 2k
Derive the stiffness matrix for the systems of Figure P7.4 using the indicated generalized coordinates and stiffness influence coefficients. x1 FIGURE P7.4 - L/2- L/2 m X3 x2 Slender rod of mass m
Derive the stiffness matrix for the systems of Figure P7.5 using the indicated generalized coordinates and stiffness influence coefficients. 0.1L 0.4L 0.3L G X1 2k m2 FIGURE P7.5 x2 2k -0.2L Rod of
Derive the stiffness matrix for the systems of Figure P7.6 using the indicated generalized coordinates and stiffness influence coefficients. www 2k ww x1 2k ww C FIGURE P7.6 m 2c x2 2k 3m 2m X3
Derive the stiffness matrix for the systems of Figure P7.7 using the indicated generalized coordinates and stiffness influence coefficients. -2L/3- L/3- FIGURE P7-7 2k m x2 2m F(t) 2c Te
Derive the stiffness matrix for the systems of Figure P7.15 using the indicated generalized coordinates and stiffness influence coefficients. FIGURE P7.15 2k xDxc mum Thin disk of mass m and radius r
Derive the stiffness matrix for the systems of Figure P7.16 using the indicated generalized coordinates and stiffness influence coefficients. Thin disk of mass m and radius r rolls without slip
Derive the stiffness matrix for the systems of Figure P7.17 using the indicated generalized coordinates and stiffness influence coefficients. X1 m FIGURE P7.17 2k ww r/2 2m 21 TxX2
Derive the stiffness matrix for the systems of Figure P7.18 using the indicated generalized coordinates and stiffness influence coefficients. x2 2r 17 m m FIGURE P7.18 k x1
Derive the stiffness matrix for the systems of Figure P7.19 using the indicated generalized coordinates and stiffness influence coefficients. k L x1 Identical slender rods of length L and mass m.
Derive the stiffness matrix for the systems of Figure P7.20 using the indicated generalized coordinates and stiffness influence coefficients. 2L/3 Identical slender bars of mass m. L/3 FIGURE P7.20 M
Derive the stiffness matrix for the systems of Figure P7.21 using the indicated generalized coordinates and stiffness influence coefficients. FIGURE P7.21 k k3 J k4
Derive the stiffness matrix for the systems of Figure P7.22 using the indicated generalized coordinates and stiffness influence coefficients. M L/2 L/4 L/4 x1 Uniform slender rod of mass 2m www
Determine the flexibility matrix for the systems of Figure P7.1 using the indicated generalized coordinates and flexibility influence coefficients. k x2 2k k X3 1m 5m 5m5 ww ww k FIGURE P7.1
Determine the flexibility matrix for the systems of Figure P7.2 using the indicated generalized coordinates and flexibility influence coefficients. FIGURE P7.2 ++ L 23 k k Slender rod of mass m w m
Determine the flexibility matrix for the systems of Figure P7.3 using the indicated generalized coordinates and flexibility influence coefficients. x1 FIGURE P7.3 L/4| Slender rod of k mass m 2k Te
Determine the flexibility matrix for the systems of Figure P7.4 using the indicated generalized coordinates and flexibility influence coefficients. x1 L/2 L/2 Slender rod of m FIGURE P7.4 x3 k x2
Determine the flexibility matrix for the systems of Figure P7.5 using the indicated generalized coordinates and flexibility influence coefficients. 0.1L +24+0.34. 0.4 0.3L G FIGURE P7.5 x1 2k m2 x2
Determine the flexibility matrix for the systems of Figure P7.6 using the indicated generalized coordinates and flexibility influence coefficients. FIGURE P7.6 2k ww x1 2k m ww 2c x2 2k 3m ww 2m C X3
Determine the flexibility matrix for the systems of Figure P7.7 using the indicated generalized coordinates and flexibility influence coefficients. 2L/3. L/3- FIGURE P7-7 www m www 2k J2c x2 2m F(t)
Determine the flexibility matrix for the systems of Figure P7.15 using the indicated generalized coordinates and flexibility influence coefficients. 2k TxD xD xc Thin disk of FIGURE P7.15 mum - F(t)
Determine the flexibility matrix for the systems of Figure P7.16 using the indicated generalized coordinates and flexibility influence coefficients. Thin disk of mass m and radius r rolls without
Determine the flexibility matrix for the systems of Figure P7.17 using the indicated generalized coordinates and flexibility influence coefficients. 2k www r/2 k 2m m 17 x1 FIGURE P7.17 21 x2
Determine the flexibility matrix for the systems of Figure P7.18 using the indicated generalized coordinates and flexibility influence coefficients. 2r 0 x2 www k www m x1 12 12 m k WE FIGURE P7.18
Determine the flexibility matrix for the systems of Figure P7.19 using the indicated generalized coordinates and flexibility influence coefficients. L XIV www Identical slender rods of length L and
Determine the flexibility matrix for the systems of Figure P7.20 using the indicated generalized coordinates and flexibility influence coefficients. 2L/3 Identical slender bars of mass m. L/3 mu
Determine the flexibility matrix for the systems of Figure P7.21 using the indicated generalized coordinates and flexibility influence coefficients. k FIGURE P7.21 k2 03 J k4 www
Determine the flexibility matrix for the systems of Figure P7.22 using the indicated generalized coordinates and flexibility influence coefficients. M L/2 | L/4 L/4 X1 Uniform slender rod of mass
Determine the mass matrix for the systems of Figure P7.1 using the indicated generalized coordinates and flexibility influence coefficients. k x2 2k k X3 1m 5m 5m5 ww ww k FIGURE P7.1
Determine the mass matrix for the systems of Figure P7.2 using the indicated generalized coordinates and flexibility influence coefficients. FIGURE P7.2 ++ L 23 k k Slender rod of mass m w m 12 www
Determine the mass matrix for the systems of Figure P7.3 using the indicated generalized coordinates and flexibility influence coefficients. x1 FIGURE P7.3 L/4| Slender rod of k mass m 2k Te m 2m x2
Determine the mass matrix for the systems of Figure P7.4 using the indicated generalized coordinates and flexibility influence coefficients. x1 L/2 L/2 Slender rod of m FIGURE P7.4 x3 k x2 mass m
Determine the mass matrix for the systems of Figure P7.5 using the indicated generalized coordinates and flexibility influence coefficients. 0.1L +24+0.34. 0.4 0.3L G FIGURE P7.5 x1 2k m2 x2 0.2L 2k-
Determine the mass matrix for the systems of Figure P7.6 using the indicated generalized coordinates and flexibility influence coefficients. FIGURE P7.6 2k ww x1 2k m ww 2c x2 2k 3m ww 2m C X3
Determine the mass matrix for the systems of Figure P7.7 using the indicated generalized coordinates and flexibility influence coefficients. 2L/3. L/3- FIGURE P7-7 www m www 2k J2c x2 2m F(t) Te
Determine the mass matrix for the systems of Figure P7.15 using the indicated generalized coordinates and flexibility influence coefficients. 2k TxD xD xc Thin disk of FIGURE P7.15 mum - F(t) mass m
Determine the mass matrix for the systems of Figure P7.16 using the indicated generalized coordinates and flexibility influence coefficients. Thin disk of mass m and radius r rolls without slip
Determine the mass matrix for the systems of Figure P7.17 using the indicated generalized coordinates and flexibility influence coefficients. 2k www r/2 k 2m m 17 x1 FIGURE P7.17 21 x2
Determine the mass matrix for the systems of Figure P7.18 using the indicated generalized coordinates and flexibility influence coefficients. 2r 0 x2 www k www m x1 12 12 m k WE FIGURE P7.18
Determine the mass matrix for the systems of Figure P7.19 using the indicated generalized coordinates and flexibility influence coefficients. L XIV www Identical slender rods of length L and mass m.
Determine the mass matrix for the systems of Figure P7.20 using the indicated generalized coordinates and flexibility influence coefficients. 2L/3 Identical slender bars of mass m. L/3 mu FIGURE
Determine the mass matrix for the systems of Figure P7.21 using the indicated generalized coordinates and flexibility influence coefficients. k FIGURE P7.21 k2 03 J k4 www
Determine the mass matrix for the systems of Figure P7.22 using the indicated generalized coordinates and flexibility influence coefficients. M L/2 | L/4 L/4 X1 Uniform slender rod of mass 2m
Derive the differential equations governing the torsional oscillations of the turbomotor of Figure P7.81. The motor operates at \(800 \mathrm{rpm}\) and the turbine shaft turns at \(3200
Derive the differential equations governing the torsional oscillations of the system of Figure P7.82. 03 04 J, G J, G J, G J, G L L 1/2+ L 12 13 FIGURE P7.82
A rotor of mass \(m\) is mounted on an elastic shaft with journal bearings at both ends. A three degree-of-freedom model of the system is shown in Figure P7.83. Each journal bearing is modeled as a
A three degree-of-freedom model of a railroad bridge is shown in Figure P7.84. The bridge is composed of three rigid spans. Each span is pinned at its base. Using the angular displacements of the
A five-degree of model of a railroad bridge is shown in Figure P7.85. The bridge is composed of five rigid spans. The connection between each span and its base is modeled as a torsional spring. Using
A four degree-of-freedom model of an aircraft wing is shown in Figure P7.86.Derive the flexibility matrix for the model. E. 14 E, E. 12 E, L FIGURE P7.86
Figure P7.87 illustrates a three degree-of-freedom model of an aircraft. A rigid fuselage is attached to two thin flexible wings. An engine is attached to each wing, but the wings themselves are of
An airplane is modeled as two flexible wings attached to a rigid fuselage (Figure P7.88). Use two degrees of freedom to model each wing and derive the differential equations governing the motion of
A drum of mass \(m\) is being hoisted by an overhead crane as illustrated in Figure P7.89. The crane is modeled as a simply supported beam with a winch at its midspan. The cable connecting the crane
The beams shown in Figures P7.90 are made of an elastic material of elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and have a cross-sectional moment of inertia \(1.3 \times
The beams shown in Figures P7.91 are made of an elastic material of elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and have a cross-sectional moment of inertia \(1.3 \times
The beams shown in Figures P7.92 are made of an elastic material of elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and have a cross-sectional moment of inertia \(1.3 \times
The beams shown in Figures P7.93 are made of an elastic material of elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and have a cross-sectional moment of inertia \(1.3 \times
Determine the stiffness matrix for the three degree-of-freedom model of the freefree beam of Figure P7.94. L/4 FIGURE P7.94 L/4. L/4 - Li4
Using a two degree-of-freedom model, derive the differential equations governing the forced vibration of the system of Figure P7.95. L/3 L/3- L/3 1/3- F(t) FIGURE P7.95 EI
Use a two degree-of-freedom model to derive the differential equations governing the motion of the system of Figure P7.96. A thin disk of mass moment of inertia \(I_{D}\) is attached to the end of
The natural frequencies of a MDOF system are the eigenvalues of \(\mathbf{M}^{-1} \mathbf{K}\).Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
An \(n\) degree-of-freedom system has \(n+1\) natural frequencies.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
The mode-shape vector is the solution of \(\left(\mathbf{A M}-\frac{1}{\omega^{2}} \mathbf{I}\right) \mathbf{X}=0\).Indicate whether the statement presented is true or false. If true, state why. If
A node for a mode is a particle that has zero displacement when the vibrations are solely at that frequency.Indicate whether the statement presented is true or false. If true, state why. If false,
The mode-shape vectors are orthogonal with respect to the standard inner product. That is, \(\mathbf{X}_{j}^{T} \mathbf{X}_{i}=0\).Indicate whether the statement presented is true or false. If true,
The mode-shape vector corresponding to a natural frequency \(\omega\) for a MDOF system is unique.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
The eigenvectors are normalized by requiring that the kinetic-energy inner product of a mode-shape vector with itself is one.Indicate whether the statement presented is true or false. If true, state
The modal matrix is the transpose of the matrix whose columns are the normalized mode-shape vectors.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite
Proportional damping occurs when the damping matrix is proportional to the flexibility matrix.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
The natural frequencies of a \(n \mathrm{DOF}\) system are the roots of a \(n\) th-order polynomial.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite
\(\quad \mathbf{P}^{\mathbf{T}} \mathbf{M P}=\mathbf{I}\) where \(\mathbf{P}\) is the modal matrix and \(\mathbf{I}\) is the identity matrix.Indicate whether the statement presented is true or false.
If \(\mathbf{X}_{i}\) is a normalized mode shape corresponding to a natural frequency \(\omega_{i}\), then \(\left(\mathbf{X}_{i}, \mathbf{X}_{i}\right)_{K}=\omega_{i}^{2}\).Indicate whether the
The lowest natural frequency when \(\operatorname{det} \mathbf{K}=0\) is zero.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it
The flexibility matrix does not exist for an unrestrained system.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
Rayleigh's quotient can be applied to obtain a lower-bound on the lowest natural frequency.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
The damping ratio for a proportionally damped system where the proportional damping is proportional to the stiffness matrix is inversely proportional to the natural frequency.Indicate whether the
Matrix iteration is a method used to determine natural frequencies of a MDOF system iteratively.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
If [ 12\(]^{T}\) is a mode-shape vector corresponding to a natural frequency of \(100 \mathrm{rad} / \mathrm{s}\) for a two non-degenerate system, then \([26]^{T}\) is also a mode-shape vector
What is the normal mode solution?
What is the dynamical matrix?
The natural frequencies of an \(n \mathrm{DOF}\) system are the of the eigenvalues of \(\mathbf{A M}\).
The natural frequencies and mode-shape vectors for a \(n \mathrm{DOF}\) system have been determined. How is the free response of the system determined?
What is the name for the mode corresponding to a natural frequency equal to zero?
How many linearly independent mode-shape vectors correspond to a natural frequency that is a double root of the characteristic equation?
Define the potential-energy scalar product.
What does the term "kinetic energy" refer to in the kinetic-energy scalar product?
How is the property of commutivity of scalar products satisfied for the kineticenergy scalar product?
What is meant by mode-shape orthogonality?
What is a normalized mode-shape vector?
Define Rayleigh's quotient for an arbitrary \(n\)-dimensional vector.
When is Rayleigh's quotient stationary?
Why is the modal matrix nonsingular?
State the expansion theorem.
What are the principal coordinates for an undamped, linear MDOF system?
How is matrix iteration used to approximate the lowest natural frequency of a MDOF system?
What is the modal damping ratio?
Why can the principal coordinates of an undamped system be used as principal coordinates for a viscously damped system with proportional damping?
If the lowest natural frequency of a system is zero, what is \(\operatorname{det} \mathbf{M}^{-1} \mathbf{K}\) ?

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