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engineering
mechanical vibration analysis

Questions and Answers of Mechanical Vibration Analysis

Find the natural frequencies and mode shapes of the system considered in Problem 6.27 with \(m_{1}=m, m_{2}=2 m, m_{3}=m, k_{1}=k_{2}=k\), and \(k_{3}=2 k\).Data From Problem 6.27:-Derive the
Show that the natural frequencies of the system shown in Fig. 6.6(a), with \(k_{1}=3 k\), \(k_{2}=k_{3}=k, m_{1}=4 m, m_{2}=2 m\), and \(m_{3}=m\), are given by \(\omega_{1}=0.46 \sqrt{k / m}\),
Find the natural frequencies of the system considered in Problem 6.28 with \(m_{1}=2 m, m_{2}=m, m_{3}=3 m\), and \(l_{1}=l_{2}=l_{3}=l_{4}=l\).Data From Problem 6.28:-Derive the equations of motion
Find the natural frequencies and principal modes of the torsional system shown in Fig. 6.28 for \((G J)_{i}=G J, i=1,2,3,4, J_{d 1}=J_{d 2}=J_{d 3}=J_{0}\), and \(l_{1}=l_{2}=l_{3}=l_{4}=l\).Figure
The mass matrix \([m]\) and the stiffness matrix \([k]\) of a uniform bar arewhere \(ho\) is the density, \(A\) is the cross-sectional area, \(E\) is Young's modulus, and \(l\) is the length of the
The mass matrix of a vibrating system is given by\[[m]=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 1\end{array}\right]\]and the eigenvectors by Find the
For the system shown in Fig. 6.38, (a) determine the characteristic polynomial \(\Delta\left(\omega^{2}\right)=\operatorname{det}\left|[k]-\omega^{2}[m]\right|\), (b) plot
(a) Two of the eigenvectors of a vibrating system are known to beProve that these are orthogonal with respect to the mass matrix\[[m]=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 2 & 0 \\0
Find the natural frequencies of the system shown in Fig. 6.18 for \(m_{i}=m, i=1,2,3\). F(1) 5k F3(1) F2(t) m 00000 my m3 k x(1) k k (1)x -X3(1) FIGURE 6.18 Three-degree-of-freedom spring-mass system.
Find the natural frequencies of the system shown in Fig. 6.19 with \(m=1 \mathrm{~kg}, l=1 \mathrm{~m}\), \(k=1000 \mathrm{~N} / \mathrm{m}\), and \(c=100 \mathrm{~N}-\mathrm{s} / \mathrm{m}\).Figure
Consider the eigenvalue problem\[\left[[k]-\omega^{2}[m]\right] \vec{X}=\overrightarrow{0}\]whereFind the natural frequencies by finding the roots of the characteristic
Find the eigenvalues and eigenvectors of the following matrix:\[[A]=\left[\begin{array}{cc}8 & -1 \\-4 & 4\end{array}\right]\]\[[[A]-\lambda[I]] \vec{X}=\overrightarrow{0}\]where \(\lambda\) is the
Consider the eigenvalue problem:a. Find the natural frequencies and mode shapes of the system.b. Change the coordinates in Eq. (E.1) as \(X_{1}=Y_{1}\) and \(X_{2}=3 Y_{2}\) and express the
Consider the eigenvalue problem\[\left[[k]-\omega^{2}[m]\right] \vec{X}=\overrightarrow{0}\]where\[[m]=\left[\begin{array}{cc}2 & 0 \\0 & 1\end{array}\right] \text { and }[k]=\left[\begin{array}{cc}8
Consider the eigenvalue problem:where Equation (E.1) can be expressed as\[[D] \vec{X}=\lambda \vec{X}\]where\[[D]=\left([m]^{\frac{1}{2}}\right)^{-1}[k]\left([m]^{\frac{1}{2}}\right)^{-1}\]is called
A symmetric positive definite matrix, such as the mass matrix of a multidegree-of-freedom system, \([m]\), can be expressed as the product of a lower triangular matrix, \([L]\), and an upper
Find the natural frequencies and mode shapes of the system shown in Fig. 6.14 with \(m_{1}=m, m_{2}=2 m, m_{3}=3 m\), and \(k_{1}=k_{2}=k\). k k m m2 000 m3 FIGURE 6.14 Semidefinite system.
Find the modal matrix for the semidefinite system shown in Fig. 6.39 for \(J_{1}=J_{2}=\) \(J_{3}=J_{0}, k_{t 1}=k_{t}\), and \(k_{t 2}=2 k_{t}\). k1 k2 J3,03 J1,01 J2,02 FIGURE 6.39 Semidefinte
Using Rayleigh's method, find the fundamental natural frequency of the torsional system shown in Fig. 6.11. Assume that \(J_{1}=J_{0}, J_{2}=2 J_{0}, J_{3}=3 J_{0}\), and \(k_{t 1}=k_{t 2}=k_{t
Fill in the Blank.For a shaft carrying masses \(m_{1}, m_{2}, \ldots\), Rayleigh's method gives the natural frequency as\[\omega=\left\{\frac{g\left(m_{1} w_{1}+m_{2} w_{2}+\cdots\right)}{m_{1}
True or False.In the matrix iteration method, any computational error will not yield incorrect results.
Can we use any trial vector \(\vec{X}_{1}\) in the matrix iteration method to find the largest natural frequency?
Rayleigh's methoda. Finds the natural frequencies and mode shapes of the system, one at a time, using several trial values for each frequency.b. Finds all the natural frequencies using trial vectors
A uniform simply supported beam carries two masses \(m_{1}\) and \(m_{2}\) with \(m_{2}=3 m_{1}\) as shown in Fig. 7.12. Find the fundamental natural frequency of the beam using Dunkerley's method.
Find the free-vibration response of the spring-mass system shown in Fig. 6.29 for \(k_{i}=k(i=1,2,3,4), m_{1}=2 m, m_{2}=3 m, m_{3}=2 m\) for the initial conditions \(x_{1}(0)=x_{10}\) and
Find the free-vibration response of the triple pendulum shown in Fig. 6.10 for \(l_{i}=l(i=1,2,3)\) and \(m_{i}=m(i=1,2,3)\) for the initial conditions \(\theta_{1}(0)=\theta_{2}(0)=0\),
Find the free-vibration response of the spring-mass system shown in Fig. 6.32 for \(m_{1}=m, m_{2}=2 m, m_{3}=m, k_{1}=k_{2}=k\), and \(k_{3}=2 k\) corresponding to the initial conditions
In the freight car system shown in Fig. 6.14, the first car acquires a velocity of \(\dot{x}_{0}\) due to an impact. Find the resulting free vibration of the system. Assume \(m_{i}=m(i=1,2,3)\) and
Find the free-vibration response of the three-degree-of-freedom airplane model considered in Problem 6.10 for the following data: \(m=5000 \mathrm{~kg}, l=5 \mathrm{~m}, E=7 \mathrm{GPa}, I=8 \times
Find the free-vibration response of the spring-mass system shown in Fig. 6.6 (a) for \(k_{1}=k, k_{2}=2 k, k_{3}=3 k, m_{1}=m, m_{2}=2 m\), and \(m_{3}=3 m\) corresponding to the initial conditions
Find the free-vibration response of the tightly stretched string shown in Fig. 6.33 for \(m_{1}=2 m, m_{2}=m, m_{3}=3 m\), and \(l_{i}=l(i=1,2,3,4)\). Assume the initial conditions as
Find the free-vibration response of a three-degree-of-freedom system governed by the equationAssume the initial conditions as \(x_{i}(0)=0.1\) and \(\dot{x}_{i}(0)=0 ; i=1,2,3\). 2 -1 0 2 -1 x(t) = 0
The free-vibration solution of a two-degree-of-freedom system can be determined by solving the equationswith \(\vec{x}=\left\{\begin{array}{l}x_{1}(t) \\ x_{2}(t)\end{array}\right\}\), using the
(a) Determine the natural frequencies and mode shapes of the torsional system shown in Fig. 6.11 for \(k_{t 1}=k_{t 2}=k_{t 3}=k_{t}\) and \(J_{1}=J_{2}=J_{3}=J_{0}\).(b) If a torque \(M_{t 3}(t)=\)
Determine the amplitudes of motion of the three masses in Fig. 6.40 when a harmonic force \(F(t)=F_{0} \sin \omega t\) is applied to the lower left mass with \(m=1 \mathrm{~kg}, k=1000 \mathrm{~N} /
Using the results of Problems 6.24 and 6.56, determine the modal matrix \([X]\) of the system shown in Fig. 6.29 and derive the uncoupled equations of motion.Data From Problem 6.24:-Find the
Using modal analysis, determine the free-vibration response of a two-degree-of-freedom system with equations of motion\[2\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]
Consider the free-vibration equations of an undamped two-degree-of-freedom system:\[[m] \ddot{\vec{x}}+[k] \vec{x}=\overrightarrow{0}\]with\[[m]=\left[\begin{array}{ll}1 & 0 \\0 & 4\end{array}\right]
For the two-degree-of-freedom system considered in Problem 6.85, find the free-vibration response, \(x_{1}(t)\) and \(x_{2}(t)\), using the modal equations derived in Problem 6.85 for the following
Determine the response of the system in Problem 6.51 to the initial conditions \(x_{1}(0)=1\), \(\dot{x}_{1}(0)=0, x_{2}(0)=2, \dot{x}_{2}(0)=1, x_{3}(0)=1\), and \(\dot{x}_{3}(0)=-1\). Assume \(k /
An approximate solution of a multi-degree-of-freedom system can be obtained using the mode acceleration method. According to this method, the equations of motion of an undamped system, for example,
A simplified model of a bicycle with its rider is shown in Fig. 6.41. Find the vertical motion of the rider when the bicycle hits an elevated road, as shown in the figure.Figure 6.41:- x2 L m = 80 kg
Find the response of the triple pendulum shown in Fig. 6.10 for \(l_{i}=0.5 \mathrm{~m}(i=1,2,3)\) and \(m_{i}=1 \mathrm{~kg}(i=1,2,3)\) when a moment, in the form of a rectangular pulse of magnitude
Consider a two-degree-of-freedom system with the equations of motion \([m] \ddot{\vec{x}}+[k] \vec{x}=\vec{f}(t)\) witha. Derive the modal equations for the forced-vibration response of the system.b.
Find the response of the spring-mass system shown in Fig. 6.6 (a) for \(k_{1}=k, k_{2}=2 k\), \(k_{3}=3 k, m_{1}=m, m_{2}=2 m\), and \(m_{3}=3 m\) with \(k=10^{4} \mathrm{~N} / \mathrm{m}\) and \(m=2
An airplane wing, Fig. 6.42(a), is modeled as a twelve-degree-of-freedom lumped-mass system as shown in Fig. 6.42(b). The first three natural mode shapes, obtained experimentally, are given below.The
Using MATLAB, find the eigenvalues and eigenvectors of a system with mass and stiffness matrices given in Example 6.13.Data From Example 6.13:- Determine the eigenvalues and eigenvectors of a
Show that the initial conditions of the generalized coordinates \(q_{i}(t)\) can be expressed in terms of those of the physical coordinates \(x_{i}(t)\) in modal analysis as\[\vec{q}(0)=[X]^{T}[m]
Find the steady-state response of the system shown in Fig. 6.17 with \(k_{1}=k_{2}=k_{3}=k_{4}\) \(=100 \mathrm{~N} / \mathrm{m}, c_{1}=c_{2}=c_{3}=c_{4}=1 \mathrm{~N}-\mathrm{s} / \mathrm{m},
Using MATLAB, find and plot the free-vibration response of the system described in Problem 6.79 for the following data: \(x_{20}=0.5, P=100, l=5, m=2\).Data From Problem 6.79:-Find the free-vibration
Using the MATLAB function ode23, find and plot the forced-vibration response of the system described in Problem 6.89.Data From Problem 6.89:-Determine the amplitudes of motion of the three masses in
Find the forced-vibration response of a viscously damped three-degree-of-freedom system with equations of motion:Assume zero initial conditions. (1) 0(t) +5 0 10 0 2 0 0 3 5 cos 2t 3 T - 7 -3 0 4 -3
Using the MATLAB function ode23, solve Problem 6.99 and plot \(x_{1}(t), x_{2}(t)\), and \(x_{3}(t)\).Data From Problem 6.99:-Find the steady-state response of the system shown in Fig. 6.17 with
Using the MATLAB function roots, find the roots of the following equation:\[f(x)=x^{12}-2=0\]
Using Program8 . m, find the steady-state response of a three-degree-of-freedom system with the following data: W == 25.076 rad/s, Si = 0.001, i = = w2 53.578 rad/s, 1,2,3 = w3 110.907 rad/s
Using Program 7 .m, generate the characteristic polynomial corresponding to the matrix\[ [A]=\left[\begin{array}{lll}5 & 3 & 2 \\3 & 6 & 4 \\1 & 2 & 6\end{array}\right]\]
Find and plot the response, \(x_{1}(t)\) and \(x_{2}(t)\), of a system with the following equations of motion:using the initial conditions:Solve the differential equations, numerically using a
Write a computer program for finding the eigenvectors using the known eigenvalues in \(\mathrm{Eq}\). (6.61). Find the mode shapes of Problem 6.57 using this program.Data From Problem 6.57:-Find the
Write a computer program for generating the \([m]\)-orthonormal modal matrix \([X]\). The program should accept the number of degrees of freedom, the normal modes, and the mass matrix as input. Solve
The equations of motion of an undamped system in SI units are given byUsing subroutine MODAL, find the steady-state response of the system when \(\omega=5 \mathrm{rad} / \mathrm{s}\). 10 sin wot 2 0
Find the response of the system in Problem 6.112 by varying \(\omega\) between \(1 \mathrm{rad} / \mathrm{s}\) and \(10 \mathrm{rad} / \mathrm{s}\) in increments of \(1 \mathrm{rad} / \mathrm{s}\).
Find the natural frequencies of vibration and the corresponding mode shapes of the beam shown in Fig. 6.9 using the mass matrixand the flexibility matrix given by Eq. (E.4) of Example 6.6.Equation
A heavy machine tool mounted on the first floor of a building, Fig. 6.43(a), has been modeled as a three-degree-of-freedom system as indicated in Fig. 6.43(b).(a) For \(k_{1}=875 \mathrm{kN} /
Figure 6.44 shows a model of an automobile including the masses of the automobile, seat and body, and head and neck of the passenger. The suspension system of the automobile is modeled by two linear
It is proposed to transport a precision electronic equipment of weight \(\mathrm{W}_{1}=5000 \mathrm{~N}\) by a trailer. The electronic equipment is placed in the trailer on a rubber mount of
When the trial vector\[\vec{X}^{(1)}=\left\{\begin{array}{l}1 \\1 \\1\end{array}\right\}\]is used for the solution of the eigenvalue problem,\[\left[\begin{array}{lll}1 & 1 & 2 \\1 & 2 & 2 \\1 & 2 &
Estimate the fundamental frequency of the beam shown in Fig. 6.9 using Dunkerley's formula for the following data:(a) \(m_{1}=m_{3}=5 m, m_{2}=m\) and(b) \(m_{1}=m_{3}=m, m_{2}=5 m\).Figure 6.9:-
Name a few methods for finding the fundamental natural frequency of a multidegree-offreedom system.
True or False.The fundamental frequency given by Durkerley's formula will always be larger than the exact value.
Fill in the Blank.Any symmetric positive definite matrix \([A]\) can be decomposed as \([A]=[U]^{T}[U]\), where \([U]\) is ___________ triangular matrix.
For a semidefinite system, the final equation in Holzer's method denotes thea. amplitude at the end as zerob. sum of inertia forces as zeroc. equation of motion
Find the fundamental frequency of the torsional system shown in Fig. 6.11, using Dunkerley's formula for the following data:(a) \(J_{1}=J_{2}=J_{3}=J_{0} ; k_{t 1}=k_{t 2}=k_{t 3}=k_{t}\); and(b)
What is the basic assumption made in deriving Dunkerley's formula?
True or False.The fundamental frequency given by Rayleigh's method will always be larger than the exact value.
Fill in the Blank.The method of decomposing a symmetric positive definite matrix \([A]\) as \([A]=[U]^{T}[U]\) is known as ___________ method.
Estimate the fundamental frequency of the shaft shown in Fig. 7.3, using Dunkerley's formula for the following data: \(m_{1}=m, m_{2}=2 m, m_{3}=3 m, l_{1}=l_{2}=l_{3}=l_{4}=l / 4\).Figure 7.3:- m
Dunkerley's formula is given bya. \(\omega_{1}^{2} \approx a_{11} m_{1}+a_{22} m_{2}+\cdots+a_{n n} m_{n}\)b. \(\frac{1}{\omega_{1}^{2}} \approx a_{11} m_{1}+a_{22} m_{2}+\cdots+a_{n n} m_{n}\)c.
What is Rayleigh's principle?
True or False.\([A] \vec{X}=\lambda[B] \vec{X}\) is a standard eigenvalue problem.
Fill in the Blank.Each step of Jacobi's method reduces a pair of off-diagonal elements to ___________ .
Rayleigh's quotient is given bya. \(\frac{\vec{X}^{T}[k] \vec{X}}{\vec{X}^{T}[m] \vec{X}}\)b. \(\frac{\vec{X}^{T}[m] \vec{X}}{\vec{X}^{T}[k] \vec{X}}\)c. \(\frac{\vec{X}^{T}[k]
The natural frequency of vibration, in bending, of the wing of a military aircraft is found to be \(20 \mathrm{~Hz}\). Find the new frequency of bending vibration of the wing when a weapon of mass
State whether we get a lower bound or an upper bound to the fundamental natural frequency if we use(a) Dunkerley's formula and(b) Rayleigh's method.
True or False.\([A] \vec{X}=\lambda[I][B] \vec{X}\) is a standard eigenvalue problem.
Fill in the Blank.The ___________ theorem permits the representation of any vector as a linear combination of the eigenvectors of the system.
Rayleigh's quotient satisfies the following relation:a. \(R(\vec{X}) \leq \omega_{1}^{2}\)b. \(R(\vec{X}) \geq \omega_{n}^{2}\)c. \(R(\vec{X}) \geq \omega_{1}^{2}\)
In an overhead crane (see Fig. 7.11) the trolley weighs ten times the weight of the girder. Using Dunkerley's formula, estimate the fundamental frequency of the system. 12 -Trolley Girder FIGURE 7.11
What is Rayleigh's quotient?
True or False.Jacobi's method can find the eigenvalues of only symmetric matrices.
Fill in the Blank.If the matrix iteration method converges to the smallest eigenvalue with \([D] \vec{X}=\lambda \vec{X}\), the method converges to the ___________ eigenvalue with \([D]^{-1}
Using Dunkerley's formula, determine the fundamental natural frequency of the stretched string system shown in Fig. 5.33 with \(m_{1}=m_{2}=m\) and \(l_{1}=l_{2}=l_{3}=l\). T m m2 FIGURE 5.33 Two
Design a minimum-weight tubular section for the shaft shown in Fig. 7.3 to achieve a fundamental frequency of vibration of \(0.5 \mathrm{~Hz}\). Assume \(m_{1}=20 \mathrm{~kg}, m_{2}=50 \mathrm{~kg},
For a vibrating system with \([k]=\left[\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right]\) and \([m]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\), the mode shape closest to the
A uniform fixed-fixed beam carries two masses \(m_{1}\) and \(m_{2}\) with \(m_{2}=m_{1}\) as shown in Fig. 7.13. Find the fundamental natural frequency of the beam using Dunkerley's method. TITTI m
What is the matrix iteration method?
Fill in the Blank.Rayleigh's quotient provides ___________ bound to \(\omega_{1}^{2}\) and bound to \(\omega_{n}^{2}\)
True or False.Jacobi's method uses rotation matrices.
Dunkerley's formulaa. Finds the natural frequencies and mode shapes of the system, one at a time, using several trial values for each frequency.b. Finds all the natural frequencies using trial
What is the basic principle used in Holzer's method?
Holzer's methoda. Finds the natural frequencies and mode shapes of the system, one at a time, using several trial values for each frequency.b. Finds all the natural frequencies using trial vectors

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