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engineering
mechanical vibration analysis

Questions and Answers of Mechanical Vibration Analysis

Prove that the constant \(a\) in Eqs. (8.18) and (8.19) is negative for common boundary conditions.Equation 8.18 and 8.19:- dw dx2 - W = 0 (8.18)
Find the free-vibration solution of a cord fixed at both ends when its initial conditions are given by\[w(x, 0)=0, \quad \frac{\partial w}{\partial t}(x, 0)=\frac{2 a x}{l} \quad \text { for } \quad
What is the main difference in the nature of the frequency equations of a discrete system and a continuous system?
Fill in the Blank.When a beam is subjected to an axial force (tension), it ____________ the natural frequency.
Pinned enda. Bending moment \(=0\); shear force equals theb. Deflection \(=0\); slope \(=0\)c. Deflection \(=0\); bending moment \(=0\)d. Bending moment \(=0\); shear force \(=0\)
True or False.For a discrete system, the boundary conditions are to be applied explicitly.
What is the effect of a tensile force on the natural frequencies of a beam?
The cable between two electric transmission towers has a length of \(2000 \mathrm{~m}\). It is clamped at its ends under a tension \(P\) (Fig. 8.25). The density of the cable material is \(8890
Fill in the Blank.The Timoshenko beam theory can be considered as __________ beam theory.
Fixed end spring forcea. Bending moment \(=0\); shear force equals theb. Deflection \(=0\); slope \(=0\)c. Deflection \(=0\); bending moment \(=0\)d. Bending moment \(=0\); shear force \(=0\)
True or False.The Euler-Bernoulli beam theory is more accurate than the Timoshenko theory.
Under what circumstances does the frequency of vibration of a beam subjected to an axial load become zero?
Fill in the Blank.A drumhead can be considered as \(\mathrm{a}(\mathrm{n})\) ____________ .
Elastically restrained enda. Bending moment \(=0\); shear force equals theb. Deflection \(=0\); slope \(=0\)c. Deflection \(=0\); bending moment \(=0\)d. Bending moment \(=0\); shear force \(=0\)
If a string of length \(l\), fixed at both ends, is given an initial transverse displacement of \(h\) at \(x=l / 3\) and then released, determine its subsequent motion. Compare the deflection shapes
Why does the natural frequency of a beam become lower if the effects of shear deformation and rotary inertia are considered?
Fill in the Blank.A string has the same relationship to a beam as a membrane bears to a(n) ____________.
A cord of length \(l\) is made to vibrate in a viscous medium. Derive the equation of motion considering the viscous damping force.
a. Zero bending momentb. Zero transverse displacementc. Zero shear forced. Zero slope\(W=0\)
Give two practical examples of the vibration of membranes.
Fill in the Blank.Rayleigh's method can be used to estimate the system _____________ natural frequency of a continuous system.
a. Zero bending momentb. Zero transverse displacementc. Zero shear forced. Zero slope\(W^{\prime}=0\)
Determine the free-vibration solution of a string fixed at both ends under the initial conditions \(w(x, 0)=w_{0} \sin (\pi x / l)\) and \((\partial w / \partial t)(x, 0)=0\).
The strings of a guitar (Fig. 8.26) are made of music wire with diameter \(0.05 \mathrm{~mm}\), weight density \(76.5 \mathrm{kN} / \mathrm{m}^{3}\), and Young's modulus \(207 \mathrm{GPa}\). If the
What is the basic principle used in Rayleigh's method?
Fill in the Blank.\(E I \frac{\partial^{2} w}{\partial x^{2}}\) denotes the _____________ in a beam.
a. Zero bending momentb. Zero transverse displacementc. Zero shear forced. Zero slope\(W^{\prime \prime}=0\)
Why is the natural frequency given by Rayleigh's method always larger than the true value of \(\omega_{1}\) ?
Fill in the Blank.For a discrete system, the governing equations are _____________differential equations.
a. Zero bending momentb. Zero transverse displacementc. Zero shear forced. Zero slope\(W^{\prime \prime \prime}=0\)
The vertical and horizontal forces (reactions) at joints \(A\) and \(B\) of a typical cable of the suspension bridge shown in Fig. 8.27 are given by \(F_{x}=2.8 \times 10^{6} \mathrm{~N}\) and
What is the difference between Rayleigh's method and the Rayleigh-Ritz method?
Fill in the Blank.An axial tensile load increases the bending ___________ of a beam.
a. Longitudinal vibration of a barb. Torsional vibration of a shaftc. Transverse vibration of a string\(c=\left(\frac{P}{ho}\right)^{1 / 2}\)Data:-\(c^{2} \frac{\partial^{2} w}{\partial
Derive an equation for the principal modes of longitudinal vibration of a uniform bar having both ends free.
Fill in the Blank.The ____________ energy of a beam is denoted by \(\frac{1}{2} \int_{0}^{l} ho A\left(\frac{\partial w}{\partial t}\right)^{2} d x\).
a. Longitudinal vibration of a barb. Torsional vibration of a shaftc. Transverse vibration of a string\(c=\left(\frac{E}{ho}\right)^{1 / 2}\)Data:-\(c^{2} \frac{\partial^{2} w}{\partial
Derive the frequency equation for the longitudinal vibration of the systems shown in Fig. 8.28. M p. A, E.1 p.A. E,I M k p. A. E,I ell M k (a) (b) FIGURE 8.28 Bar with different end conditions. (c)
Fill in the Blank.The ____________ energy of a beam is denoted by \(\frac{1}{2} \int_{0}^{l} E I\left(\frac{\partial^{2} w}{\partial x^{2}}\right)^{2} d x\).
a. Longitudinal vibration of a barb. Torsional vibration of a shaftc. Transverse vibration of a string\(c=\left(\frac{G}{ho}\right)^{1 / 2}\)Data:-\(c^{2} \frac{\partial^{2} w}{\partial
A thin bar of length \(l\) and mass \(m\) is clamped at one end and free at the other. What mass \(M\) must be attached to the free end in order to decrease the fundamental frequency of longitudinal
Show that the normal functions corresponding to the longitudinal vibration of the bar shown in Fig. 8.29 are orthogonal. x=0 1000 k x=/ FIGURE 8.29 Bar fixed at one end and connected to a spring at
Derive the frequency equation for the longitudinal vibration of a stepped bar having two different cross-sectional areas \(A_{1}\) and \(A_{2}\) over lengths \(l_{1}\) and \(l_{2}\), respectively.
A steel shaft of diameter \(d\) and length \(l\) is fixed at one end and carries a propeller of mass \(m\) and mass moment of inertia \(J_{0}\) at the other end (Fig. 8.30). Determine the fundamental
A torsional system consists of a shaft with a disc of mass moment of inertia \(I_{0}\) mounted at its center. If both ends of the shaft are fixed, find the response of the system in free torsional
Find the natural frequencies for torsional vibration of a fixed-fixed shaft.
A uniform shaft of length \(l\) and torsional stiffness \(G J\) is connected at both ends by torsional springs, torsional dampers, and discs with inertias, as shown in Fig. 8.31. State the boundary
Solve Problem 8.23 if one end of the shaft is fixed and the other free.Data From Problem 8.23:-Find the natural frequencies for torsional vibration of a fixed-fixed shaft.
Derive the frequency equation for the torsional vibration of a uniform shaft carrying rotors of mass moment of inertia \(I_{01}\) and \(I_{02}\) one at each end.
An external torque \(M_{t}(t)=M_{t 0} \cos \omega t\) is applied at the free end of a fixed-free uniform shaft. Find the steady-state vibration of the shaft.
Find the fundamental frequency for torsional vibration of a shaft of length \(2 \mathrm{~m}\) and diameter \(50 \mathrm{~mm}\) when both the ends are fixed. The density of the material is \(7800
A uniform shaft, supported at \(x=0\) and rotating at an angular velocity \(\omega\), is suddenly stopped at the end \(x=0\). If the end \(x=l\) is free, determine the subsequent angular displacement
Compute the first three natural frequencies and the corresponding mode shapes of the transverse vibrations of a uniform beam of rectangular cross section \((100 \mathrm{~mm} \times 300
Derive an expression for the natural frequencies for the lateral vibration of a uniform fixedfree beam.
Prove that the normal functions of a uniform beam, whose ends are connected by springs as shown in Fig. 8.32, are orthogonal. x=0 FIGURE 8.32 x=1 k Beam connected to rotational and linear springs at
Derive an expression for the natural frequencies for the transverse vibration of a uniform beam with both ends simply supported.
Derive the expression for the natural frequencies for the lateral vibration of a uniform beam suspended as a pendulum, neglecting the effect of dead weight.
Find the cross-sectional area \((A)\) and the area moment of inertia (I) of a simply supported steel beam of length \(1 \mathrm{~m}\) for which the first three natural frequencies lie in the range
Derive the frequency equation for the transverse vibration of a uniform beam resting on springs at both ends, as shown in Fig. 8.33. The springs can deflect vertically only, and the beam is
A uniform beam, simply supported at both ends, is found to vibrate in its first mode with an amplitude of \(10 \mathrm{~mm}\) at its center. If \(A=120 \mathrm{~mm}^{2}, I=1000 \mathrm{~mm}^{4},
A simply supported uniform beam of length \(l\) carries a mass \(M\) at the center of the beam. Assuming \(M\) to be a point mass, obtain the frequency equation of the system.
A uniform fixed-fixed beam of length \(2 l\) is simply supported at the middle point. Derive the frequency equation for the transverse vibration of the beam.
A simply supported beam carries initially a uniformly distributed load of intensity \(f_{0}\). Find the vibration response of the beam if the load is suddenly removed.
Estimate the fundamental frequency of a cantilever beam whose cross-sectional area and moment of inertia vary as\[A(x)=A_{0} \frac{x}{l} \quad \text { and } \quad I(x)=\bar{I} \frac{x}{l}\]where
Derive Eqs. (E.5) and (E.6) of Example 8.10.Data From Example 8.10:-Equation E.5 and E.6:- Determine the effects of rotary inertia and shear deformation on the natural frequencies of a simply
(a) Derive a general expression for the response of a uniform beam subjected to an arbitrary force.(b) Use the result of part (a) to find the response of a uniform simply supported beam under the
Derive Eqs. (E.7) and (E.8) of Example 8.10.Data From Example 8.10:-Equation E.7 and E.8:- Determine the effects of rotary inertia and shear deformation on the natural frequencies of a simply
Prove that the constant \(a\) in Eq. (8.82) is positive for common boundary conditions. c dW(x) W(x) dx4 = 1 dT(t) T(t) dt = a = w (8.82)
A fixed-fixed beam carries an electric motor of mass \(100 \mathrm{~kg}\) and operational speed \(3000 \mathrm{rpm}\) at its midspan, as shown in Fig. 8.34. If the motor has a rotational unbalance of
Find the response of a simply supported beam subject to a uniformly distributed harmonically varying load.
A steel cantilever beam of diameter \(2 \mathrm{~cm}\) and length \(1 \mathrm{~m}\) is subjected to an exponentially decaying force \(100 e^{-0.1 t} \mathrm{~N}\) at the free end, as shown in Fig.
Find the steady-state response of a cantilever beam that is subjected to a suddenly applied step bending moment of magnitude \(M_{0}\) at its free end.
Consider a railway car moving on a railroad track as shown in Fig. 8.36(a). The track can be modeled as an infinite beam resting on an elastic foundation and the car can be idealized as a moving load
A cantilever beam of length \(l\), density \(ho\), Young's modulus \(E\), area of cross section \(A\), and area moment of inertia \(I\) carries a concentrated mass \(M\) at its free end. Derive the
Find the first two natural frequencies of vibration in the vertical direction of the floor of the suspension bridge shown in Fig. 8.27 under the following assumptions:1. The floor can be considered
A uniform beam of length \(2 l\) is fixed at the left end, supported on a simple support at the middle, and free at the right end as shown in Fig. 8.37. Derive the frequency equation for determining
A uniform fixed-fixed beam of length \(2 l\) is supported on a pin joint at the midpoint as shown in Fig. 8.38. Derive the frequency equation for determining the natural frequencies of vibration of
The L-shaped frame shown in Fig. 8.39 is fixed at the end \(A\) and free at end \(C\). The two segments of the frame, \(A B\) and \(B C\), are made of the same material with identical square cross
Consider a simply supported uniform beam resting on an elastic foundation, with a foundation modulus \(k \mathrm{~N} / \mathrm{m}\).a. Derive the equation of motion of the beam when the applied
Consider a fixed-fixed uniform beam resting on an elastic foundation, with a foundation modulus \(k \mathrm{~N} / \mathrm{m}\).a. Derive the equation of motion of the beam when the applied
Consider a simply supported uniform beam of length \(l\) subjected to a concentrated transverse harmonic force \(F(t)=F_{0} \sin \omega t\) at \(x=x_{0}\) from the left end of the beam. Determine the
Starting from fundamentals, show that the equation for the lateral vibration of a circular membrane is given by\[\frac{\partial^{2} w}{\partial r^{2}}+\frac{1}{r} \frac{\partial w}{\partial
Consider a rectangular membrane of sides \(a\) and \(b\) supported along all the edges.(a) Derive an expression for the deflection \(w(x, y, t)\) under an arbitrary pressure \(f(x, y, t)\).(b) Find
Find the free-vibration solution and the natural frequencies of a rectangular membrane that is clamped along all the sides. The membrane has dimensions \(a\) and \(b\) along the \(x\) and \(y\)
Find the free-vibration response of a rectangular membrane of sides \(a\) and \(b\) subject to the following initial conditions:\[\begin{aligned}& w(x, y, 0)=w_{0} \sin \frac{\pi x}{a} \sin \frac{\pi
Find the free-vibration response of a rectangular membrane of sides \(a\) and \(b\) subjected to the following initial conditions:\[\left.\begin{array}{c}w(x, y, 0)=0 \\\frac{\partial w}{\partial
Compare the fundamental natural frequencies of transverse vibration of membranes of the following shapes:(a) square;(b) circular; and(c) rectangular with sides in the ratio of 2:1. Assume that all
Using the equation of motion given in Problem 8.59, find the natural frequencies of a circular membrane of radius \(R\) clamped around the boundary at \(r=R\).Data From Problem 8.59:-Starting from
Find the fundamental natural frequency of a fixed-fixed beam using the static deflection curve\[W(x)=\frac{c_{0} x^{2}}{24 E I}(l-x)^{2}\]where \(c_{0}\) is a constant.
Solve Problem 8.66 using the deflection shape \(W(x)=c_{0}\left(1-\cos \frac{2 \pi x}{l}\right)\), where \(c_{0}\) is a constant.Data From Problem 8.66:-Find the fundamental natural frequency of a
Find the fundamental natural frequency of vibration of a uniform beam of length \(l\) that is fixed at one end and simply supported at the other end. Assume the deflection shape of the beam to be
Determine the fundamental frequency of a uniform fixed-fixed beam carrying a mass \(M\) at the middle by applying Rayleigh's method. Use the static deflection curve for \(W(x)\).
Applying Rayleigh's method, determine the fundamental frequency of a cantilever beam (fixed at \(x=l\) ) whose cross-sectional area \(A(x)\) and moment of inertia \(I(x)\) vary as \(A(x)=A_{0} x /
Using Rayleigh's method, estimate the fundamental frequency for the lateral vibration of a uniform beam fixed at both the ends. Assume the deflection curve to be\[W(x)=c_{1}\left(1-\cos \frac{2 \pi
Find the fundamental frequency of longitudinal vibration of the tapered bar shown in Fig. 8.41, using Rayleigh's method with the mode shape\[U(x)=c_{1} \sin \frac{\pi x}{2 l}\]The mass per unit
Approximate the fundamental frequency of a rectangular membrane supported along all the edges by using Rayleigh's method with\[W(x, y)=c_{1} x y(x-a)(y-b)\]\[V=\frac{P}{2}
The root mean square value of a signal \(x(t), x_{\mathrm{rms}}\), is defined as\[x_{\mathrm{rms}}=\left\{\lim _{T \rightarrow \infty} \frac{1}{T} \int_{0}^{T} x^{2}(t) d t\right\}^{1 / 2}\]Using
What are the various methods available for vibration control?
Fill in the Blank.The presence of unbalanced mass in a rotating disc causes ____________ .
An electronic instrument, of mass \(20 \mathrm{~kg}\), is to be isolated to achieve a natural frequency of \(15 \mathrm{rad} / \mathrm{s}\) and a damping ratio of 0.95. The available dashpots can

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