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engineering
introduction mechanical engineering

Questions and Answers of Introduction Mechanical Engineering

A schematic of part of a power transmission system is shown in Figure P6.46. A motor of moment of inertia \(I=100 \mathrm{~kg} \cdot \mathrm{m}^{2}\) is mounted on a shaft of shear modulus \(G=80
Determine the natural frequencies and modal fractions for the two degree-offreedom system of Figure P6.47. 2m 2k 2r: FIGURE P6.47 m k
Determine the frequency response of the system of Figure P6.1 and Chapter Problem 6.10 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the block whose displacement is \(x_{1}\).Data From
Determine the frequency response of the system of Figure P6.1 and Chapter Problem 6.10 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the block whose displacement is \(x_{2}\).Data From
Determine the frequency response of the system of Figure P6.2 and Chapter Problem 6.11 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the particle.Data From Chapter Problem
Determine the frequency response of the system of Figure P6.7 and Chapter Problem 6.42 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the block whose displacement is \(x_{1}\).Data From
Determine the frequency response of the system of Figure P6.8 and Chapter Problem 6.43 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the mass center of the machine tool.Data From
Determine the frequency response of the system of Figure P6.8 and Chapter Problem 6.43 due to a sinusoidal force \(F_{0} \sin \omega t\) applied to the right end of the machine tool.Data From Chapter
A \(50 \mathrm{~kg}\) lathe mounted on an elastic foundation of stiffness \(4 \times 10^{5} \mathrm{~N} / \mathrm{m}\) has a vibration amplitude of \(35 \mathrm{~cm}\) when the motor speed is \(95
What is the required stiffness of an undamped dynamic vibration absorber whose mass is \(5 \mathrm{~kg}\) to eliminate vibrations of a \(25 \mathrm{~kg}\) machine of natural frequency \(125
A \(35 \mathrm{~kg}\) machine is attached to the end of a cantilever beam of length \(2 \mathrm{~m}\), elastic modulus \(210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\), and moment of inertia \(1.3
A \(150 \mathrm{~kg}\) pump experiences large-amplitude vibrations when operating at \(1500 \mathrm{rpm}\). Assuming this is the natural frequency of a SDOF system, design a dynamic vibration
A solid disk of diameter \(30 \mathrm{~cm}\) and mass \(10 \mathrm{~kg}\) is attached to the end of a solid 3-cm-diameter, 1-m-long steel shaft \(\left(G=80 \times 10^{9} \mathrm{~N} /
A \(200 \mathrm{~kg}\) machine is placed on a massless simply supported beam as shown in Figure P6.59. The machine has a rotating unbalance of \(1.41 \mathrm{~kg} \cdot \mathrm{m}\) and operates at
The disk in Figure P6.60 rolls without slip and the pulley is massless. What is the mass of the block that should be hung from the cable such that steady-state vibrations of the cylinder are
Vibration absorbers are used in boxcars to protect sensitive cargo from large accelerations due to periodic excitations provided by rail joints. For a particular railway, joints are spaced \(5
A \(500 \mathrm{~kg}\) reciprocating machine is mounted on a foundation of equivalent stiffness \(5 \times 10^{6} \mathrm{~N} / \mathrm{m}\). When operating at \(800 \mathrm{rpm}\), the machine
A \(100 \mathrm{~kg}\) machine is placed at the midspan of a 2-m-long cantilever beam \(\left(E=210 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}, I=2.3 \times 10^{-6} \mathrm{~m}^{4}\right)\). The
Repeat Chapter Problem 6.63 if the excitation is due to a rotating unbalance of magnitude \(0.33 \mathrm{~kg} \cdot \mathrm{m}\).Data From Chapter Problem 6.63:A \(100 \mathrm{~kg}\) machine is
For the absorber designed in Chapter Problem 6.63, what is the minimum steady-state amplitude of the machine and at what speed does it occur?Data From Chapter Problem 6.63:A \(100 \mathrm{~kg}\)
Determine values of \(k\) and \(c\) such that the steady-state amplitude of the center of the cylinder in Figure \(\mathrm{P} 6.66\) is less than \(4 \mathrm{~mm}\) for \(60 \mathrm{rad} / \mathrm{s}
Use the Laplace transform method to analyze the situation of an undamped absorber attached to a viscously damped system, as shown in Figure P6.67.(a) Determine the steady-state amplitude of the mass
Design an undamped absorber such that the steady-state motion of the \(25 \mathrm{~kg}\) machine component in Figure P6.68 ceases when the absorber is added. What is the steady-state amplitude of the
A \(300 \mathrm{~kg}\) compressor is placed at the end of a cantilever beam of length \(1.8 \mathrm{~m}\), elastic modulus \(200 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\), and moment of inertia
An engine has a moment of inertia of \(7.5 \mathrm{~kg} \cdot \mathrm{m}^{2}\) and a natural frequency of \(125 \mathrm{~Hz}\). Design a Houdaille damper such that the engine's maximum magnification
A \(200 \mathrm{~kg}\) machine is subjected to an excitation of magnitude \(1500 \mathrm{~N}\). The machine is mounted on a foundation of stiffness \(2.8 \times 10^{6} \mathrm{~N} / \mathrm{m}\).
The differential equations for a linear MDOF system can be written in a matrix form.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to
Lagrange's equations can be used to derive the differential equations governing the motion only for linear systems.Indicate whether the statement presented is true or false. If true, state why. If
Lagrange's equations can be used for conservative systems and nonconservative systems.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to
The FBD method, when applied to a MDOF linear system, always leads to symmetric mass, stiffness, and damping matrices.Indicate whether the statement presented is true or false. If true, state why. If
Lagrange's equations, when applied to a MDOF linear system, always leads to symmetric mass, stiffness, and damping matrices.Indicate whether the statement presented is true or false. If true, state
The quadratic form of the potential energy can be used to determine the stiffness matrix for a linear MDOF system.Indicate whether the statement presented is true or false. If true, state why. If
A system is dynamically coupled if the mass matrix for the system is not symmetric.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to
The choice of generalized coordinates is irrelevant in deciding whether a system is dynamically coupled.Indicate whether the statement presented is true or false. If true, state why. If false,
The flexibility matrix is the transpose of the stiffness matrix.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it true.
A diagonal stiffness matrix means that \(k_{i j}=k_{i j}\) for all \(i, j=1,2, \ldots, n\).Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
Elements of the mass matrix for a MDOF system may have different dimensions.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to make it
The formulation of the stiffness influence coefficient method to determine the stiffness matrix for a linear MDOF system relies on the concept that potential energy is a function of position.Indicate
When flexibility influence coefficients are used to calculate the flexibility matrix for a MDOF system, the flexibility matrix is calculated one column at a time.Indicate whether the statement
The stiffness matrix for a system always exists but the flexibility matrix does not always exist.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the
A system is not statically coupled if its flexibility matrix is a diagonal matrix.Indicate whether the statement presented is true or false. If true, state why. If false, rewrite the statement to
Lagrange's equations can be used to derive the equations governing the vibrations of three masses along the span of a beam ignoring the inertia of the beam and using three degrees of freedom in the
Write the general matrix form of the differential equations governing the undamped and forced vibrations of a linear \(n\) DOF system.
State Lagrange's equations for a conservative system.
What defines whether a system is dynamically coupled?
How is Rayleigh's dissipation function used?
What is a variation?
How is the method of virtual work applied in the application of Lagrange's equations for a MDOF system?
What is Maxwell's reciprocity relation and how is it applied?
Write the differential equations governing a MDOF system in matrix form when the mass matrix, damping matrix, and flexibility matrix are known.
Describe the calculation of the stiffness influence coefficient \(k_{13}\).The generalized coordinates for modeling a system have been selected as \(x_{1}, x_{2}\), and \(\theta\) where \(x_{1}\) and
Describe the calculation of the flexibility influence coefficient \(a_{13}\).The generalized coordinates for modeling a system have been selected as \(x_{1}, x_{2}\), and \(\theta\) where \(x_{1}\)
Describe the calculation of the inertia influence coefficient \(m_{12}\).The generalized coordinates for modeling a system have been selected as \(x_{1}, x_{2}\), and \(\theta\) where \(x_{1}\) and
Describe the calculation of the inertia influence coefficient \(m_{31}\).The generalized coordinates for modeling a system have been selected as \(x_{1}, x_{2}\), and \(\theta\) where \(x_{1}\) and
What is the kinetic energy of the system of Figure SP7.29 at an arbitrary instant? 3000 N/m ww ] 10 kg x 2000 N/m www C 120 N. s/mm/300 N. s/m FIGURE SP7.29 4 kg 600 N/m ww E 8 kg 1100 N. s/mmmmmmm
What is the potential energy in the system of Figure SP7.29 at an arbitrary instant? 3000 N/m www + 120 N s/m FIGURE SP7:29 2000 N/m ww 10 kg 4 kg ++ 300 N s/m 600 N/m ww +- 100 Ns/m 8 kg 150 N s/m
What is Rayleigh's dissipation function for the system of Figure SP7.29 at an arbitrary instant? 3000 N/m + 120 N s/m FIGURE SP7:29 10 kg 2000 N/m www + 300 N s/m 4 kg 600 N/m ww +- 100 N-s/m 8 kg
What is the result of\[ \frac{d}{d t}\left[\frac{\partial}{\partial \dot{x}}(2 \dot{x}-\dot{y})^{2}\right] \]
What is virtual work done by the external forces in Figure SP7.33, assuming virtual displacements \(\delta x\) and \(\delta y\) ? M(t) 2r F(t) FIGURE SP7.33
What are the generalized forces for the system of Figure SP7.34 using \(x\) and \(\theta\) as generalized coordinates? L/2 1 -x -F2(t) A FIGURE SP7.34 F(t)
The quadratic form of the potential energy for a three degree-of-freedom system is\[ V=5 x_{1}^{2}+4 x_{1} x_{2}+2 x_{1} x_{3}+8 x_{2}^{2}+3 x_{2} x_{2}+6 x_{3}^{2} \]Determine the stiffness matrix
The kinetic energy for a three degree-of-freedom system is\[ T=3\left(\dot{x}_{2}-\frac{1}{2} \dot{x}_{1}\right)^{2}+12\left(\dot{x}_{2}+\frac{1}{3} \dot{x}_{1}\right)^{2}+4 \dot{x}_{3}^{2}
When a load of \(50 \mathrm{~N}\) is applied to the \(250 \mathrm{~kg}\) mass in the system of Figure SP7.37, the displacements of the masses are \(x_{1}=3 \mathrm{~mm}, x_{2}=5 \mathrm{~mm}\), and
When the block of mass \(10 \mathrm{~kg}\) is given a displacement of \(3 \mathrm{~mm}\) in the system of Figure SP7.38 and all other blocks are held in their equilibrium positions, it is found that
What is the determinant of the stiffness matrix of the system of Figure SP7.39? m FIGURE SP7.39 k 2m X 3k -X3 2k H m 2m X4
When block \(A\) of Figure SP7. 40 is given a velocity of \(15 \mathrm{~m} / \mathrm{s}\) and the velocities of blocks \(B\) and \(C\) remain at rest, an impulse of \(3 \mathrm{~N} \cdot \mathrm{s}\)
When the right end of the bar of the system of Figure SP7.41 is given a velocity of \(3 \mathrm{~m} / \mathrm{s}\) but the angular velocity of the bar is zero, an impulse of magnitude \(6 \mathrm{~N}
Lagrange's equations are used to derive the differential equations for a three degree-of-freedom system resulting in\[ \left[\begin{array}{lll} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.1 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.2 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.3 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.4 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.5 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.6 using the indicated generalized coordinates. Make linearizing
Use the free-body diagram method to derive the differential equations governing the motion of the systems shown in Figure P7.7 using the indicated generalized coordinates. Make linearizing
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.1. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.2. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.3. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.4. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.5. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.6. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to derive the differential equations governing the motion of the systems shown in Figures P7.7. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.15. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.16. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.17. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.18. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.19. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.20. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.21. Use the indicated generalized coordinates. Make linearizing assumptions, and
Use Lagrange's equations to drive the differential equations governing the motion of the systems shown in Figures P7.22. Use the indicated generalized coordinates. Make linearizing assumptions, and
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.1. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.3. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.4. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.5. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.15. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.19. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the kinetic energy of the system at an arbitrary instant for the systems of Figure P7.22. Put the kinetic energy in a quadratic form. Use the quadratic form to determine the mass matrix for
Determine the potential energy of the system at an arbitrary instant for the systems of Figure P7.1. Put the potential energy in a quadratic form. Use the quadratic form to determine the stiffness
Determine the potential energy of the system at an arbitrary instant for the systems of Figure P7.2. Put the potential energy in a quadratic form. Use the quadratic form to determine the stiffness
Determine the potential energy of the system at an arbitrary instant for the systems of Figure P7.15. Put the potential energy in a quadratic form. Use the quadratic form to determine the stiffness

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