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engineering
mechanical vibration analysis

Questions and Answers of Mechanical Vibration Analysis

Fill in the Blank.Vibration neutralizer is also known as dynamic vibration ____________.
Control natural frequencya. Introduce dampingb. Use vibration isolatorc. Add vibration absorberd. Avoid resonance
Masses of \(1 \mathrm{~kg}, 3 \mathrm{~kg}\), and \(2 \mathrm{~kg}\) are located at radii \(50 \mathrm{~mm}, 75 \mathrm{~mm}\), and \(25 \mathrm{~mm}\) in the planes \(C, D\), and \(E\),
Figure 9.46 shows a rotating system in which the shaft is supported in bearings at \(A\) and \(B\). The three masses \(m_{1}, m_{2}\), and \(m_{3}\) are connected to the shaft as indicated in the
Reduce transmission of excitation force from one part to anothera. Introduce dampingb. Use vibration isolatorc. Add vibration absorberd. Avoid resonance
Is the shaking force proportional to the square of the speed of a machine? Does the vibratory force transmitted to the foundation increase with the speed of the machine?
Fill in the Blank.Phase marks are used in ____________ plane balancing using a vibration analyzer.
Reduce response of the systema. Introduce dampingb. Use vibration isolatorc. Add vibration absorberd. Avoid resonance
A flywheel, with a mass of \(50 \mathrm{~kg}\) and an eccentricity of \(12 \mathrm{~mm}\), is mounted at the center of a steel shaft of diameter \(25 \mathrm{~mm}\). If the length of the shaft
Derive the expression for the stress induced in a shaft with an unbalanced concentrated mass located midway between two bearings.
Fill in the Blank.Machine errors can cause ____________ in rotating machines.
A water tank of mass \(10^{5} \mathrm{~kg}\) is supported on a reinforced cement concrete column, as shown in Fig. 9.49(a). When a projectile hits the tank, it causes a shock, in the form of a step
An electronic instrument of mass \(20 \mathrm{~kg}\) is to be isolated from engine vibrations with frequencies ranging from \(1000 \mathrm{rpm}\) to \(3000 \mathrm{rpm}\). Find the stiffness of the
In the PCB described in Problem 9.63, it is desired to reduce the displacement transmissibility to a value of 0.25. If the chassis mass is 50 percent of the mass of the PCB, determine the necessary
A machine with a natural frequency of \(4.2 \mathrm{~Hz}\) is subjected to a rotating unbalance force of amplitude \((F)\) of \(20 \mathrm{~N}\) at a frequency of \(4 \mathrm{~Hz}\). Design a
Derive an expression for the displacement transmissibility of a damped single-degree-offreedom system whose base is subjected to a general periodic displacement.
An electronic instrument is to be isolated from a panel that vibrates at frequencies ranging from \(25 \mathrm{~Hz}\) to \(35 \mathrm{~Hz}\). It is estimated that at least 80 percent vibration
An exhaust fan, having a small unbalance, weights \(800 \mathrm{~N}\) and operates at a speed of \(600 \mathrm{rpm}\). It is desired to limit the response to a transmissibility of 2.5 as the fan
An air compressor of mass \(500 \mathrm{~kg}\) has an eccentricity of \(50 \mathrm{~kg}-\mathrm{cm}\) and operates at a speed of \(300 \mathrm{rpm}\). The compressor is to be mounted on one of the
The armature of a variable-speed electric motor, of mass \(200 \mathrm{~kg}\), has an unbalance due to manufacturing errors. The motor is mounted on an isolator having a stiffness of \(10 \mathrm{kN}
A dishwashing machine weighing \(75 \mathrm{~kg}\) operates at \(300 \mathrm{rpm}\). Find the minimum static deflection of an isolator that provides 60 percent isolation. Assume that the damping in
It is found that an exhaust fan, of mass \(80 \mathrm{~kg}\) and operating speed \(1000 \mathrm{rpm}\), produces a repeating force of \(10,000 \mathrm{~N}\) on its rigid base. If the maximum force
A compressor of mass \(120 \mathrm{~kg}\) has a rotating unbalance of \(0.2 \mathrm{~kg}-\mathrm{m}\). If an isolator of stiffness \(0.5 \mathrm{MN} / \mathrm{m}\) and damping ratio 0.06 is used,
An internal combustion engine has a rotating unbalance of \(1.0 \mathrm{~kg}-\mathrm{m}\) and operates between \(800 \mathrm{rpm}\) and \(2000 \mathrm{rpm}\). When attached directly to the floor, it
A small machine tool of mass \(100 \mathrm{~kg}\) operates at \(600 \mathrm{rpm}\). Find the static deflection of an undamped isolator that provides 90 percent isolation.
Design the suspension of a car such that the maximum vertical acceleration felt by the driver is less than \(2 g\) at all speeds between \(70 \mathrm{~km} / \mathrm{h}\) and \(140 \mathrm{~km} /
Consider a single-degree-of-freedom system with Coulomb damping (which offers a constant friction force, \(F_{c}\) ). Derive an expression for the force transmissibility when the mass is subjected to
Consider a single-degree-of-freedom system with Coulomb damping (which offers a constant friction force, \(F_{c}\) ). Derive expressions for the absolute and relative displacement transmissibilities
When a washing machine, of mass \(200 \mathrm{~kg}\) and an unbalance \(0.02 \mathrm{~kg}\) - \(\mathrm{m}\), is mounted on an isolator, the isolator deflects by \(5 \mathrm{~mm}\) under the static
An electric motor, of mass \(60 \mathrm{~kg}\), rated speed \(3000 \mathrm{rpm}\), and an unbalance \(0.002 \mathrm{~kg}\)-m, is to be mounted on an isolator to achieve a force transmissibility of
An engine is mounted on a rigid foundation through four springs. During operation, the engine produces an excitation force at a frequency of \(3000 \mathrm{rpm}\). If the weight of the engine causes
A printed circuit board of mass \(1 \mathrm{~kg}\) is supported to the base through an undamped isolator. During shipping, the base is subjected to a harmonic disturbance (motion) of amplitude \(2
An electronic instrument of mass \(10 \mathrm{~kg}\) is mounted on an isolation pad. If the base of the isolation pad is subjected to a shock in the form of a step velocity of \(10 \mathrm{~mm} /
A viscously damped single-degree-of-freedom system has a body of mass \(25 \mathrm{~kg}\) with a spring constant of \(70 \mathrm{kN} / \mathrm{m}\). Its base is subjected to harmonic vibration.(a)
A single-degree-of-freedom system is used to represent an automobile, of mass \(m\), damping constant \(c\), and stiffness \(k\), which travels on a rough road that is in the form of a sinusoidal
A sensitive instrument of mass \(100 \mathrm{~kg}\) is installed at a location that is subjected to harmonic motion with frequency \(20 \mathrm{~Hz}\) and acceleration \(0.5 \mathrm{~m} /
A delicate instrument weighing \(200 \mathrm{~N}\) is suspended by four identical springs, each with stiffness \(50,000 \mathrm{~N} / \mathrm{m}\), in a rigid box as shown in Fig. 9.50. The box is
A damped torsional system is composed of a shaft and a rotor (disk). The torsional stiffness and the torsional damping constant of the shaft are given by \(k_{t}=6000 \mathrm{~N}-\mathrm{m} /
The force transmissibility of a damped single-degree-of-freedom system with base motion is given by Eq. (9.106):\[T_{f}=\frac{F_{t}}{k Y}=r^{2}\left\{\frac{1+(2 \zeta
Derive an expression for the relative displacement transmissibility, \(\frac{Z}{Y}\), where \(Z=X-Y\), for a damped single-degree-of-freedom system subjected to the base motion, \(y(t)=Y \sin \omega
A printed circuit board (PCB), made of fiber reinforced plastic composite material, is attached to a chassis that is attached to a motor vibrating at a speed of \(3000 \mathrm{rpm}\). The PCB can be
An air compressor of mass \(200 \mathrm{~kg}\), with an unbalance of \(0.01 \mathrm{~kg}-\mathrm{m}\), is found to have a large amplitude of vibration while running at \(1200 \mathrm{rpm}\).
Derive the equations of motion, using Newton's second law of motion, for each of the systems shown in Figs. 6.18. F(t) 5k 00000 mi k F3(1) F2(t) 00000 m2 m3 00000 k k X2(1) k x(1) (1)Ex FIGURE 6.18
Derive the equations of motion, using Newton's second law of motion, for each of the systems shown in Figs. 6.20. 000 x(t) F(t) 21 31 Rigid bar, mass - 2m A G k X3(1) F3(1) 5m x2(1) F2(t) FIGURE
Derive the equations of motion, using Newton's second law of motion, for each of the systems shown in Figs. 6.19. 8(t) 2k 2m k 4 M,(t) 000 3k Free Rigid bar, mass = 2m T x(1) F(t) m I X2(1) t F(1)
Derive the equations of motion, using Newton's second law of motion, for each of the systems shown in Figs. 6.21. Pulley, mass M, mass moment of inertia Jo 3r 3m T x(1) F(t) 2k 000 T x2(1) m T X3(1)
Derive the equations of motion, using Newton's second law of motion, for each of the systems shown in Figs. 6.22. M, cos wt G 15 G3 13 ka 03 4 kn 12 G k3 16 G6 G Number of teeth on gear G, n, (i = 1
A car is modeled as shown in Fig. 6.23. Derive the equations of motion using Newton's second law of motion.Figure 6.23:- x(t) F(t) 21 31 Rigid bar, mass- 2m A G X3(1) F3(t) 5m x2(1) F(t) FIGURE 6.20
The equations of motion derived using the displacements of the masses, \(x_{1}, x_{2}\), and \(x_{3}\) as degrees of freedom in Fig. 6.12 (Example 6.10) lead to symmetric mass and stiffness matrices
A simplified vibration analysis of an airplane considers bounce and pitch motions (Fig. 6.24(a)). For this, a model consisting of a rigid bar (corresponding to the body of the airplane) supported on
Consider the two-degree-of-freedom system shown in Fig. 6.25 with \(m_{1}=m_{2}=1\) and \(k_{1}=k_{2}=4\). The masses \(m_{1}\) and \(m_{2}\) move on a rough surface for which the equivalent viscous
A simplified model of the main landing gear system of a small airplane is shown in Fig. 6.27 with \(m_{1}=100 \mathrm{~kg}, m_{2}=5000 \mathrm{~kg}, k_{1}=10^{4} \mathrm{~N} / \mathrm{m}\), and
For a simplified analysis of the vibration of an airplane in the vertical direction, a threedegree-of-freedom model, as shown in Fig. 6.26, can be used. The three masses indicate the masses of the
Derive the stiffness matrix of each of the systems shown in Figs. 6.18 using the indicated coordinates. F(t) 5k 0000 m2 F2(t) m1 k k x1(t) F3(t) 00000 m3 k k x(1) X3(1) FIGURE 6.18
Derive the stiffness matrix of each of the systems shown in Figs. 6.19 using the indicated coordinates. 31 4 4 3k O 8(t) 2k M,(t) Rigid bar, mass= 2m C 2m x1(t) F(t) m T x2(t) F2(t) FIGURE 6.19 Rigid
Derive the stiffness matrix of each of the systems shown in Figs. 6.20 using the indicated coordinates. k x1(t) -21- 31 Rigid bar, mass = 2m A G k F(t) k F3(t) 5m T x2(t) F2(t) FIGURE 6.20
Derive the stiffness matrix of each of the systems shown in Figs.6.21 using the indicated coordinates. Pulley, mass M, mass moment of inertia Jo 2k k 3r 000 0 3m x1(t) F(t) I x2(t) m T x3(t) F2(t) 3k
Derive the stiffness matrix of each of the systems shown in Figs. 6.22 using the indicated coordinates. G5 04 15 16 G3 G6 13 k12 03. 14 02 M cos wot G4 11 12 G G2 Number of teeth on gear G = n; (i
Derive the stiffness matrix of each of the systems shown in Figs. 6.23 using the indicated coordinates. 4 G X3 Mass = M, mass moment of inertia = JG m1 k2 F1 X1 11 k F2 X2 11 m2 k C1 C1 k FIGURE 6.23
Derive the flexibility matrix of the system shown in Fig. 5.39. mo e(t) ellee k1 k2 lllll m x(t) FIGURE 5.39 Mass hanging from a pulley.
Derive the stiffness matrix of the system shown in Fig. 5.39. mo e(t) ellee k1 k2 lllll m x(t) FIGURE 5.39 Mass hanging from a pulley.
Derive the flexibility matrix of the system shown in Fig. 5.42. 00000 2m 8(t) * 2m ellee x(t) m 00000 FIGURE 5.42 Rigid bar connected to masses and springs.
Derive the stiffness matrix of the system shown in Fig. 5.42. 00000 2m 8(t) * 2m ellee x(t) m 00000 FIGURE 5.42 Rigid bar connected to masses and springs.
Derive the mass matrix of the system shown in Fig. 5.42. 00000 2m 8(t) * 2m ellee x(t) m 00000 FIGURE 5.42 Rigid bar connected to masses and springs.
Find the flexibility and stiffness influence coefficients of the torsional system shown in Fig. 6.28. Also write the equations of motion of the system. (GJ)1 01 Compressor (GJ)2 (Jan) Turbine (142)
Find the flexibility and stiffness influence coefficients of the system shown in Fig. 6.29. Also, derive the equations of motion of the system. k1 k3 k4 000 /711 000 m2 000 my 000 FIGURE 6.29
An airplane wing, Fig. 6.30(a), is modeled as a three-degree-of-freedom lumped-mass system, as shown in Fig. 6.30(b). Derive the flexibility matrix and the equations of motion of the wing by assuming
Determine the flexibility matrix of the uniform beam shown in Fig. 6.31. Disregard the mass of the beam compared to the concentrated masses placed on the beam and assume all \(l_{i}=l\). m m m3
Derive the flexibility and stiffness matrices of the spring-mass system shown in Fig. 6.32 assuming that all the contacting surfaces are frictionless. m m2 m3 00000 m3 m2 00000 m1 k k FIGURE 6.31
Derive the equations of motion of the system shown in Fig. 6.34. 000 m 00000 m 2m k 000000 2k 000 3k FIGURE 6.34 Three-degree-of-freedom spring-mass system.
Derive the equations of motion for the tightly stretched string carrying three masses, as shown in Fig. 6.33. Assume the ends of the string to be fixed. P m1 m2 x1(1) X2(1) m3 X3(1) P FIGURE 6.33
Four identical springs, each having a stiffness \(k\), are arranged symmetrically at \(90^{\circ}\) from each other, as shown in Fig. 2.65. Find the influence coefficient of the junction point in an
Show that the stiffness matrix of the spring-mass system shown in Fig. 6.3(a) is a band matrix along the diagonal.Figure 6.3(a):- F(t) F2(t) Fi(t) kj F(t) kn Fn(t) kn+1 000 000 000 000 000 000 m m m
Derive the mass matrix of each of the systems shown in Figs. 6.18 using the indicated coordinates. F(1) 5k F2(t) 0.0 m 00000 m2 00000 k k F3(1) m3 00000 m 00000 k k -x(1) -x(1) -X3(1) FIGURE 6.18
Derive the mass matrix of each of the systems shown in Figs. 6.19 using the indicated coordinates. 31 774 3k Freee 8(t) 2k C M,(t) Rigid bar, mass = 2m k 2m T x(1) F(t) m X(1) F2(t) FIGURE 6.19
Derive the mass matrix of each of the systems shown in Figs. 6.20 using the indicated coordinates. R x(1) -21- 31- Rigid bar, mass = 2m A G k F(t) 5m X2(1) F(t) X3(1) F3(1) FIGURE 6.20
Derive the mass matrix of each of the systems shown in Figs. 6.21 using the indicated coordinates. k Pulley, mass M, mass moment of inertia Jo 2k 3r 3m T X1(1) F(t) T x2(1) m T X3(1) F2(t) 3k F3(t)
Derive the mass matrix of each of the systems shown in Figs. 6.22 using the indicated coordinates. G3 G5 15 13 k2 04 Mcos wt 01 02 4 kn 12 G G4 16 G6 G Number of teeth on gear G = n, (i = 1 to 6)
The inverse mass influence coefficient \(b_{i j}\) is defined as the velocity induced at point \(i\) due to a unit impulse applied at point \(j\). Using this definition, derive the inverse mass
For the four-story shear building shown in Fig. 6.35, there is no rotation of the horizontal section at the level of floors. Assuming that the floors are rigid and the total mass is concentrated at
Derive the equations of motion of the system shown in Fig. 6.36 by using Lagrange's equations with \(x\) and \(\theta\) as generalized coordinates.Figure 6.36:- m k 00000 m T x FIGURE 6.36 Mass and
Derive the equations of motion of the system shown in Fig. 5.12(a), using Lagrange's equations with (a) \(x_{1}\) and \(x_{2}\) as generalized coordinates and (b) \(x\) and \(\theta\) as generalized
Derive the equations of motion of the system shown in Fig. 6.29 using Lagrange's equations. k k k3 k4 000 m 000 m2 000 m3 000 FIGURE 6.29 Spring-mass system.
When an airplane (see Fig. 6.37(a)) undergoes symmetric vibrations, the fuselage can be idealized as a concentrated central mass \(M_{0}\) and the wings can be modeled as rigid bars carrying end
Derive the equations of motion of the triple pendulum shown in Fig. 6.10 using Lagrange's equations.Figure 6.10:- F(1) m4 F3(1) F(1) F(1) k/2 kz/2 k/2 k/2. m3 1112 m1 k/2 k/2 m 00000 m FIGURE 6.35
Use Lagrange's equations to derive the equations of motion of each of the systems shown in Figs. 6.18. F(t) 5k F(1) F3(1) m m 00000 m 00000 k k k k x2(1) -x1 (1) X3(1) FIGURE 6.18
Use Lagrange's equations to derive the equations of motion of each of the systems shown in Figs. 6.19. 7+ + CE 0(1) 2k 000 2m 3k Mt) Rigid bar, mass = 2m T x1(1) F(t) m x2(1) F2(t) FIGURE 6.19 Rigid
Use Lagrange's equations to derive the equations of motion of each of the systems shown in Figs. 6.20. 000 k Free -21- -31- Rigid bar, mass 2m A G T x(1) F(t) X3(1) F3(1) 5m x2(1) F(1) FIGURE 6.20
Use Lagrange's equations to derive the equations of motion of each of the systems shown in Figs. 6.21. Pulley, mass M, mass moment of inertia Jo 3r 2k 3m m T x1(1) F(t) I x2(1) 7 X3(1) F2(t) 3k F3(1)
Use Lagrange's equations to derive the equations of motion of each of the systems shown in Figs. 6.22. G5 15 16 G3 G6 13 kp 03 14 02 G4 Mcos at 4 kn G 12 G Number of teeth on gear G = n, (i = 1 to 6)
Set up the eigenvalue problem of Example 6.11 in terms of the coordinates \(q_{1}=x_{1}, q_{2}=\) \(x_{2}-x_{1}\), and \(q_{3}=x_{3}-x_{2}\), and solve the resulting problem. Compare the results
Derive the frequency equation of the system shown in Fig. 6.29. k k k3 KA 000 m 000 m2 000 m3 000 FIGURE 6.29 Spring-mass system.
Find the natural frequencies and mode shapes of the system shown in Fig. 6.6 (a) when \(k_{1}=k, k_{2}=2 k, k_{3}=3 k, m_{1}=m, m_{2}=2 m\), and \(m_{3}=3 m\). Plot the mode shapes.Figure 6.6(a):- k
Set up the matrix equation of motion and determine the three principal modes of vibration for the system shown in Fig. 6.6(a) with \(k_{1}=3 k, k_{2}=k_{3}=k, m_{1}=3 m\), and \(m_{2}=m_{3}=m\).
Find the natural frequencies of the system shown in Fig. 6.10 with \(l_{1}=20 \mathrm{~cm}, l_{2}=30 \mathrm{~cm}\), \(l_{3}=40 \mathrm{~cm}, m_{1}=1 \mathrm{~kg}, m_{2}=2 \mathrm{~kg}\), and
(a) Find the natural frequencies of the system shown in Fig. 6.31 with \(m_{1}=m_{2}=m_{3}=m\) and \(l_{1}=l_{2}=l_{3}=l_{4}=l / 4\).(b) Find the natural frequencies of the beam when \(m=10
Determine the eigenvalues and eigenvectors of the system shown in Fig. 6.29, taking \(k_{1}=k_{2}=k_{3}=k_{4}=k\) and \(m_{1}=m_{2}=m_{3}=m\). k k k3 KA 000 m 000 mmy 000 FIGURE 6.29 Spring-mass
The frequency equation of a three-degree-of-freedom system is given by\[\left|\begin{array}{rrr}\lambda-5 & -3 & -2 \\-3 & \lambda-6 & -4 \\-1 & -2 & \lambda-6\end{array}\right|=0\]Find the roots of
Find the natural frequencies and principal modes of the triple pendulum shown in Fig. 6.10, assuming that \(l_{1}=l_{2}=l_{3}=l\) and \(m_{1}=m_{2}=m_{3}=m\). 4 m y2 102 m2 3 03 -x2 -X3- m3 FIGURE
Find the natural frequencies and mode shapes of the system shown in Fig. 6.29 for \(k_{1}=k_{2}=k_{3}=k_{4}=k, m_{1}=2 m, m_{2}=3 m\), and \(m_{3}=2 m\). k k k3 KA 000 m 000 mmy 000 FIGURE 6.29

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