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mechanical vibration analysis

Questions and Answers of Mechanical Vibration Analysis

An idealized one degree-of-freedom model is tested many times in order to estimate its natural frequency. It is relatively straightforward to measure its mass \(m\), but stiffness \(k\) can only be
The cantilever beam in Figure 2.32 undergoes harmonic oscillation, being driven by a force of amplitude \(A\) and nominal frequency \(\omega\). An examination of a long time-history of the response
For each nonlinear equation of motion, linearize the equation about the indicated equilibrium position and discuss the range of validity of the linearized equation. Plot the relative error for the
Consider the mass \(m\) suspended by a nonlinear hardening spring as shown in Figure 2.34. The spring obeys \(F=k x^{3}\) and has an original length of \(L\). The coordinate \(x\) is the stretch of
For each idealized model in Figures 2.35 to 2.38, draw a free-body diagram and derive the equation of motion using(a) Newton's second law of motion and(b) the energy method. The block in Figure 2.36
A disk of mass \(m\) is mounted between two shafts with different properties, as shown in Figure 2.39.(a) What is the natural frequency of the system?(b) If the disk is rotated \(\theta\), where
Calculate the equivalent torsional spring constant for the stepped shaft shown in Figure 2.40. The two shafts have negligible mass and are made of a material with shear modulus \(G_{1}\) and
The valve mechanism in Figure 2.41 is in equilibrium when the rocker arm is horizontal. The system is assumed to be frictionless. Use an energy method to determine the natural frequency for small
A rod is supported on two rotating grooved rollers, as depicted in Figure 2.42. The rollers rotate in opposite directions and the coefficient of friction between the rollers and the rod is
A \(25-\mathrm{kg}\) block is suspended by two cables, as depicted in Figure 2.43. Assume small displacements.(a) What is the frequency of oscillation in \(\mathrm{Hz}\) of the block in the \(x\)
Derive the equation of motion and natural frequency for a mass \(m\) on the string that is under constant tension \(T\) as shown in Figure 2.44. Assume small displacements and that \(m\) is much
Continuing Problem 13, the string is stretched to the position shown in Figure 2.45. Calculate the natural frequency of the system using the following parameter values: \(m g=2 \mathrm{lb}, T=50
Derive the equation of motion for a uniform stiff rod restrained from vertical motion by a torsional spring of stiffness \(K\) and two translational springs each of stiffness \(k\), as shown in
A uniform rigid and massless rod is pinned at one end and connected to ground via a spring at the other end. At midpoint on the rod, a spring is connected to a mass which is connected to a fixed
A component of uniform material and nonuniform cross section is pinned at one end and supported by two springs as shown in the two configurations of Figure 2.48. The component is displaced slightly
A body of mass \(m\) is suspended by a spring of constant \(k\) and attached to an elastic beam of length \(l\), as shown in Figure 2.49. When the mass is attached to the spring, measurements are
A solid cylinder floating in equilibrium in a liquid of specific gravity \(\gamma\) is depressed slightly and released. A schematic is shown in Figure 2.50. Find the equilibrium position and solve
The uniform rod is restrained by four translational springs and a torsional spring as shown in Figure 2.51. Determine the natural frequency of the system using the energy approach. M 1/4 31/4- k k k
Derive Equation 2.26,\[ x(t)=x(0) \cos \omega_{n} t+\frac{\dot{x}(0)}{\omega_{n}} \sin \omega_{n} t \]
Derive the equation of motion and natural frequency for a disk in motion that is constrained to move vertically, but can rotate, as shown in Figure 2.52. The coordinate \(y\) is the displacement of
Derive the equation of motion and natural frequency for mass \(m\) shown in Figure 2.53. Assume the pulleys are massless, the cable is inextensible, and let the spring constant be \(k\). The
For the two-mass system shown in Figure 2.54 derive the equation of motion in terms of \(x_{1}\). Assume the pulleys are massless, the cable is inextensible, and let the spring constant be \(k\). The
Derive the equation of motion and natural frequency for the disk shown in Figure 2.55. Assume the disk is rolling without slip. y k eelle 16 G k eelle Xo Figure 2.55: Disk on an inclined plane
Derive the equation of motion and natural frequency for the disk shown in Figure 2.56. Assume the disk is rolling without slip, and the cable is wrapped around the disk. y k eelle IG Position of the
Derive the equation of motion and natural frequency for mass \(m\) shown in Figure 2.57. Assume the pulleys are massless, the cable is inextensible, and let the spring constant be \(k\). eeeee k
Derive the equation of motion and natural frequency for mass \(m\) shown in Figure 2.58. Assume the pulleys are massless, and let the spring constants be \(k_{1}\) and \(k_{2}\). Let \(x_{1}\) and
Derive Equation 2.30,\[ x(t)=B_{1} \cos \omega_{n} t+B_{2} \sin \omega_{n} t \]beginning with the equation of motion.
Show that the period of free vibration of a load weighing \(W\) suspended from two parallel springs, as shown in Figure 2.59, is given by \(T\),\[ T=2 \pi \sqrt{\frac{W}{g\left(k_{1}+k_{2}\right)}}
For a body suspended between two springs as in Figure 2.60, show that the period of oscillation is\[ T=2 \pi \sqrt{\frac{W}{g\left(k_{1}+k_{2}\right)}} \] ellel k W ellel k Figure 2.60: Body
A compound pendulum in the shape of a rectangle is supported at point \(O\) and allowed to oscillate. The dimensions of the rectangle are given in Figure 2.61. Calculate the natural frequency for
Two springs in series support weight \(W\), as shown in Figure 2.62. If these springs are replaced by a single spring, find the equivalent stiffness as well as the period of oscillation. The solution
A pendulum of mass \(m\) and mass moment of inertia \(I_{O}\) is suspended from a hinge, as shown in Figure 2.63. The center of gravity is located a distance \(h\) from the hinge. For small
A bifilar pendulum of length \(2 a\) is suspended with two vertical strings, each of length \(l\), as shown in Figure 2.64. (Bifilar means fitted with or involving the use of two threads or wires.)
An inverted hinged pendulum with a mass \(m\) at the top is suspended between two springs with constants \(k\), as shown in Figure 2.65. The rod can be assumed rigid and massless, and in the vertical
Consider the inverted simple pendulum shown in Figure 2.66. Initially the pendulum is in a perfectly vertical position. If it is displaced very slightly from the vertical, what are the stability
For the system of Figure 2.67, what should the value of \(k_{3}\) be if \(k_{1}=2 k_{2}=3 k_{3}\), for a period of free vibration of \(400 \mathrm{~ms}\) for \(m=2.5 \mathrm{~kg}\) ? k k, 000 m 60000
A slender bar of mass \(m\) and length \(l\) is supported at its base by a torsional spring of stiffness \(K\), as per Figure 2.68. The bar rests in the vertical position when in equilibrium with the
Solve Problem 39 for the initial conditions \(\theta(0)=0\) and \(\dot{\theta}(0)=\dot{\theta}_{0}\) for two cases of the stiffness:\[ K>\frac{m g l}{2} \quad \text { and } \quad K
A bar supported by a hinge at its base is held in place by a spring connected to a collar, as per Figure 2.69. The spring is unstretched when the bar is vertical. As the bar is displaced from
A simple pendulum of initial length \(l_{0}\) and initial angle \(\theta_{0}\) is released from rest. If the length is a function of time according to \(l=l_{0}+\varepsilon t\), find the position
A homogeneous disk of weight \(W\) and radius \(r\) is supported by two identical cylindrical steel shafts of length \(l\), as shown in Figure 2.70. From solid mechanics, for each shaft the relation
When a manned craft is sent into space it is necessary to know the mass properties of the system, including the moment of inertia of astronauts on the flight, in order to accurately calculate its
A small pendulum is mounted in a rocket that is accelerating up at a rate of \(4 g\), as sketched in Figure 2.72. The pendulum is composed of a massless rod of length \(l=1 \mathrm{~m}\) that
For the undamped rocker arm sketched in Figure 2.73 , determine the natural frequency of the system undergoing small amplitude oscillation using Newton's second law of motion. Assume that the mass of
For the system of Figure 2.74 with spring stiffness \(k\), bar mass \(M\) and length \(l\), and end mass \(m\), derive the equation of motion for small oscillations about the horizontal equilibrium
Consider a generalization of Example 2.10, where a mass moves to the left with speed \(v\) on a platform, disconnected from two springs, as shown in Figure 2.75. Assuming that there is no friction,
A block of mass \(m\) is suspended from the ceiling by a spring of stiffness \(k\). The equation of motion is\[ m \ddot{x}+k x=0 \]where \(x\) is measured from the static equilibrium position.
Two systems with discontinuities are shown in Figure 2.76. In each system the mass oscillates and comes into contact with springs at some distance \(\pm x_{c}\). For each system, derive the equation
Derive Equation 2.37,\[ \begin{aligned} x(t) & =\frac{v_{0}}{\omega_{n}} \sin \omega_{n} t+x_{0} \cos \omega_{n} t \\ & +\frac{x_{s t}}{1-\left(\omega / \omega_{n}\right)^{2}}\left[\cos \omega
Solve the equation of motion,\[ \ddot{x}+x=\frac{1}{m} F(t) \]for: (a) \(F(t)=\cos 0.5 t\), (b) \(F(t)=\cos 0.99 t\), (c) \(F(t)=\cos t\), and (d) \(F(t)=\cos 2 t\). Plot the responses and compare
For the oscillator that is beating according to the equation\[ \ddot{x}+16 x=\frac{A}{m} \sin (4-\varepsilon) t \]where \(\varepsilon\) is small but not equal to zero, what happens as \(\varepsilon
The block shown in Figure 2.77 is acted on by the force\[ F(t)=100+25 \sin 75 t \mathrm{~N} \]and, after the transients have died out, it oscillates with an amplitude of \(0.6 \mathrm{~mm}\) about a
For Example 2.12, what range of frequencies of the motion \(y(t)\) must be excluded to keep the maximum force at \(C\) less than \(10 \mathrm{~N}\) ? Example 2.12 Driving Sinusoidal Motion A rigid
A gantry crane system is shown schematically in Figure 2.78. The crane is a simple pendulum of length \(l\) and mass \(m\). The gantry oscillates with the displacement \(x(t)=A \sin \omega t\). If
Solve for the response of the equation of motion,\[ \ddot{x}+9 x=3 \sin t+\cos 3 t \]assuming zero initial conditions.
Solve for the response of the equation of motion,\[ \ddot{x}+\omega_{n}^{2} x=\frac{A}{m} \sin \omega_{n} t \]for arbitrary initial conditions.
Derive Equation 2.41,\[ x(t)=\frac{x_{s t} \omega_{n}^{2}}{2 \varepsilon(\varepsilon+\omega)}[\sin (\varepsilon+\omega) t \sin \varepsilon t] \]
A cylinder of mass \(m\) is mounted in a water tunnel as shown in Figure 2.79 with the cylinder axis transverse to the flow direction. When there is no flow, a vertical force of \(F_{\text {static
Fill in the Blank.When finite amplitudes of motion are involved, __________ analysis becomes necessary.
Fill in the Blank.__________ principle is not applicable in nonlinear analysis.
Fill in the Blank.__________ equation involves time-dependent coefficients.
Fill in the Blank.The governing equation of a simple pendulum whose pivot is subjected to vertical vibration is called ___________ equation.
Fill in the Blank.The representation of the motion of a system in the displacement-velocity plane is known as ___________ plane representation.
Fill in the Blank.The curve traced by a typical point in the phase plane is called \(\mathrm{a}(\mathrm{n})\) _____________ .
Fill in the Blank.The velocity with which a representative point moves along a trajectory is called the ___________ velocity.
Fill in the Blank.The phenomenon of realizing two amplitudes for the same frequency is known as ____________ phenomenon.
Fill in the Blank.The forced-vibration solution of Duffing's equation has _____________ for any given amplitude \(|A|\).
Fill in the Blank.The Ritz-Galerkin method involves the solution of _____________ equations.
Fill in the Blank.Mechanical chatter is a(n) ___________ vibration.
Fill in the Blank.If time does not appear explicitly in the governing equation, the corresponding system is said to be __________ .
Fill in the Blank.The method of __________ can be used to construct the trajectories of a one-degree-of-freedom dynamical system.
Fill in the Blank.Van der Pol's equation exhibits _________ cycles.
Each term in the equation of motion of a linear system involves displacement, velocity, and acceleration of thea. first degreeb. second degreec. zero degree
A nonlinear stress-strain curve can lead to nonlinearity of thea. massb. springc. damper
If the rate of change of force with respect to displacement, \(d f / d x\), is an increasing function of \(x\), the spring is called aa. soft springb. hard springc. linear spring
If the rate of change of force with respect to displacement, \(d f / d x\), is a decreasing function of \(k\), the spring is called aa. soft springb. hard springc. linear spring
The point surrounded by closed trajectories is called aa. centerb. mid-pointc. focal point
For a system with periodic motion, the trajectory in the phase plane is a(n)a. closed curveb. open curvec. point
In subharmonic oscillations, the natural frequency \(\left(\omega_{n}\right)\) and the forcing frequency \((\omega)\) are related asa. \(\omega_{n}=\omega\)b. \(\omega_{n}=n \omega ; n=2,3,4,
In superharmonic oscillations, the natural frequency \(\left(\omega_{n}\right)\) and the forcing frequency \((\omega)\) are related asa. \(\omega_{n}=\omega\)b. \(\omega_{n}=n \omega ; n=2,3,4,
If time appears explicitly in the governing equation, the corresponding system is calleda. an autonomous systemb. a nonautonomous systemc. a linear system
Duffing's equation is given bya. \(\ddot{x}+\omega_{0}^{2} x+\alpha x^{3}=0\)b. \(\ddot{x}+\omega_{0}^{2} x=0\)c. \(\ddot{x}+\alpha x^{3}=0\)
Lindstedt's perturbation method givesa. periodic and nonperiodic solutionsb. periodic solutions onlyc. nonperiodic solutions only
\(\lambda_{1}\) and \(\lambda_{2}\) with same sign \(\left(\lambda_{1}, \lambda_{2}\right.\) : real and distinct)a. Unstable nodeb. Saddle pointc. Noded. Focus or spiral pointe. Stable node
\(\lambda_{1}\) and \(\lambda_{2}
\(\lambda_{1}\) and \(\lambda_{2}>0\left(\lambda_{1}, \lambda_{2}\right.\) : real and distinct \()\)a. Unstable nodeb. Saddle pointc. Noded. Focus or spiral pointe. Stable node
\(\lambda_{1}\) and \(\lambda_{2}\) : real with opposite signsa. Unstable nodeb. Saddle pointc. Noded. Focus or spiral pointe. Stable node
\(\lambda_{1}\) and \(\lambda_{2}\) : complex conjugatesa. Unstable nodeb. Saddle pointc. Noded. Focus or spiral pointe. Stable node
a. Nonlinearity in massb. Nonlinearity in dampingc. Linear equationd. Nonlinearity in spring\(a x \ddot{x}+k x=0\)
The equation of motion of a simple pendulum, subjected to a constant torque, \(M_{t}=m l^{2} f\), is given byIf \(\sin \theta\) is replaced by its two-term approximation, \(\theta-\left(\theta^{3} /
The equation of motion of a system is given byApproximate Eq. (E.1) using one, two, and three terms in the polynomial expansion of \(\cos \theta\) and discuss the nature of nonlinearities involved in
The free-vibration equation of a single-degree-of-freedom system with nonlinear damper and nonlinear spring is given byIf \(x_{1}(t)\) and \(x_{2}(t)\) are two different solutions of Eq. (E.1), show
Two springs, having different stiffnesses \(k_{1}\) and \(k_{2}\) with \(k_{2}>k_{1}\), are placed on either side of a mass \(m\), as shown in Fig. 13.27. When the mass is in its equilibrium
Find the equation of motion of the mass shown in Fig. 13.28. Draw the spring-force-versus- \(x\) diagram. -x(t) P sin wot k k 000 00000 KY m 00000 FIGURE 13.28 Mass-spring system of Problem 13.5.
A mass \(m\) is attached at the midpoint of a stretched wire of area of cross-section \(A\), length \(l\), and Young's modulus \(E\) as shown in Fig. 13.29. If the initial tension in the wire is
Two masses \(m_{1}\) and \(m_{2}\) are attached to a stretched wire, as shown in Fig. 13.30. If the initial tension in the wire is \(P\), derive the equations of motion for large transverse
A mass \(m\), connected to an elastic rubber band of unstretched length \(l\) and stiffness \(k\), is permitted to swing as a pendulum bob, as shown in Fig. 13.31. Derive the nonlinear equations of
A uniform bar of length \(l\) and mass \(m\) is hinged at one end \((x=0)\), supported by a spring at \(x=\frac{2 l}{3}\), and acted upon by a force at \(x=l\), as shown in Fig. 13.32. Derive the
Derive the nonlinear equation of motion of the spring-mass system shown in Fig. 13.33. 1- 00000 k 777 m k 00000 F(t) x(t) h FIGURE 13.33 Mass connected to springs in different directions.
Derive the nonlinear equations of motion of the system shown in Fig. 13.34. Also, find the linearized equations of motion for small displacements, \(x(t)\) and \(\theta(t)\). k 00000 m F(t) -x(t)

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