Vibration Of Mechanical Systems 1st Edition Alok Sinha - Solutions

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Consider the model of a bladed disk (Sinha, 1986) shown in Figure P5.1 where each blade is represented by a single mass. Furthermore, it should be noted that \(i+1=1\) when \(i=N\) and \(i-1=N\) when \(i=1\), where \(N\) is the number of blades. Model parameters are as follows: \(m_{t}=0.0114
Consider the model of a turbine blade (Griffin and Hoosac, 1984) shown in Figure P5.2. Model parameters (SI units) are as follows: \(m_{1}=0.0114, m_{2}=0.0427, m_{3}=0.0299, k_{1}=430,300, k_{2}=\) \(17,350,000\), and \(k_{3}=7,521,000\).a. Compute the natural frequencies and the mode shapes.
Consider the half car model in Figure P5.3. The vehicle is traveling with a velocity \(V\) on a sinusoidal road surface with an amplitude of \(0.011 \mathrm{~m}\) and a wavelength of \(5.3 \mathrm{~m}\).Parameters of the system are as follows: \(\ell_{1}=1.35 \mathrm{~m}, \ell_{2}=\) \(1.05
The steel wire of length \(0.9 \mathrm{~m}\) and cross-sectional area of \(1.3 \mathrm{~mm}\) is fixed at both ends in a musical instrument. The tension in the string is \(220 \mathrm{~N}\). A musician plucks the string while adjusting the lengths of the strings in the following sequence:
Find the natural frequencies and the mode shapes of a fixedfixed longitudinal bar.
Find the natural frequencies and the mode shapes of a fixedfixed torsional shaft.
Find the natural frequencies and the mode shapes of a cantilever beam attached to a spring at its end (Figure P5.7).Figure P5.7 E, Ia, P, A, l
A sinusoidal force is applied at the midpoint of a fixed-fixed elastic beam. Determine the response of the system using zero initial conditions.Figure P5.8 fo sin wot E, Ia, P, A, l
Consider a fixed-fixed steel bar with the length \(=0.1 \mathrm{~m}\) and the cross-sectional area \(=4 \times 10^{-4} \mathrm{~m}^{2}\). Using the finite element method, determine the first three natural frequencies and compare them to the theoretical values.
Consider a fixed-fixed steel beam with the length \(=0.1 \mathrm{~m}\), the cross-sectional area \(=4 \times 10^{-4} \mathrm{~m}^{2}\) and the area moment of inertia \(=\) \(0.5 \times 10^{-8} \mathrm{~m}^{4}\). Using the finite element method, determine the first three natural frequencies and
Consider the SDOF system shown in Figure P1.1. All the shafts and the connections among them are massless. The material of the shaft is steel. Also,\[ m_{1}=1.1 \mathrm{~kg}, m_{2}=1.4 \mathrm{~kg}, m_{b}=2.1 \mathrm{~kg} \]The stiffness \(k\) is half of the cantilever beam stiffness.Find the
Find the equivalent mass of a spring under the assumption that the velocity distribution along the length of the spring is parabolic.
Consider the cantilever beam with mass \(m\) and length \(\ell\) (Figure P1.3). Obtain the equivalent mass of the cantilever beam under the assumption that the beam deflection is \(y(z)=\frac{x}{2}\left[1-\cos \left(\frac{\pi z}{\ell}\right)\right]\).Figure P1.3 z=0 N z = l (7)x
An object with mass \(m\) and rectangular cross section \(A\) is floating in a liquid with mass density \(ho\) (Figure P1.4). Derive the governing differential equation of motion, and obtain the natural frequency of the system.Figure P1.4 Floating object x Liquid level
An L-shaped bracket hinged at point A is supported by two springs with stiffnesses \(k_{1}\) and \(k_{2}\) (Figure P1.5). The mass of the bracket is \(m\) and is uniformly distributed.Figure P1.5 k2 FWW a2 12 Hinge A 15 - ww 1 k
The mass of a complex-shaped object is \(3 \mathrm{~kg}\). When this object is suspended like a pendulum (Figure P1.6), its frequency of oscillation is 30 cycles \(/ \mathrm{min}\). The center of mass is at a distance of \(0.2 \mathrm{~m}\) from the pivot point \(A\).Find the mass-moment of inertia
An uniform rigid bar of length \(\ell=50 \mathrm{~cm}\) and mass \(m=7 \mathrm{~kg}\) is hinged at one end (Figure P1.7). At the other end, this bar is suddenly attached to a massless spring with stiffness \(k=1,000 \mathrm{~N} / \mathrm{m}\). Derive an expression for the angular oscillation of the
A cylinder of mass \(m_{2}\) rolls without slipping inside the box with mass \(m_{1}\) (Figure P1.8). Derive the equivalent mass and the stiffness of the system.Figure P1.8 Box with mass m Cylinder with mass m k x(t)
Consider the gear shaft system in Figure P1.9. The length and the diameter of shaft \(A\) are \(50 \mathrm{~cm}\) and \(4 \mathrm{~cm}\), respectively. Similarly, the length and the diameter of shaft \(B\) are \(60 \mathrm{~cm}\) and \(3 \mathrm{~cm}\), respectively. Masses of gears A and B are
An object with mass \(=500 \mathrm{~kg}\) is attached to a table with four steel legs of diameter \(=0.015 \mathrm{~m}\) and length \(=0.1 \mathrm{~m}\) (Figure P1.10).a. Derive the differential equation for vibration of the object in vertical direction.b. Determine the natural frequency of
A tank (Figure P1.11) with mass \(m_{1}=2,000 \mathrm{~kg}\) fires a cannon with mass \(m_{2}=2 \mathrm{~kg}\) and velocity \(=10 \mathrm{~m} / \mathrm{sec}\). The recoil mechanism consists of a spring with stiffness \(=11,000 \mathrm{~N} / \mathrm{m}\), and a damper. There are three possible
Consider the system shown in Figure P1.12a, where \(m=10 \mathrm{~kg}\), \(\ell_{1}=35 \mathrm{~cm}\), and \(\ell=50 \mathrm{~cm}\).A record of free vibration is shown in Figure 1.12b. Find the values of the spring constant \(k\) and the damping coefficient \(c\).Figure P1.12(a,b) 41 l HWW (a) k
Consider a spring-mass system (Figure P1.13) where the mass is on a surface with the coefficient of friction \(\mu\). Assuming that the initial displacement of the mass is \(x(0)\), determine the expression for the displacement \(x(t)\).Figure P1.13 k www. m X Coulomb friction
Consider a simple electromagnetic suspension system shown in Figure P1.14.Figure P1.14 The electromagnetic force \(f_{m}\) is given by\[ f_{m}=\alpha \frac{I^{2}}{h^{2}} \]where \(I\) and \(h\) are the coil current and the air gap, respectively. The constant \(\alpha=\mu_{0} N^{2} A_{p}\) where
Consider the system in Figure P1.15. Determine the natural frequencies when the mass \(m\) is constrained to move along \(x\) and \(y\) directions, respectively.Figure P1.15 k2 m X
Consider the system shown in Figure P2.1 a where \(a=25 \mathrm{~cm}\), \(\ell_{1}=50 \mathrm{~cm}\), and \(\ell_{2}=30 \mathrm{~cm}\). When the force \(f(t)\) is a step function of magnitude \(1 \mathrm{~N}\), the response is as shown in Figure P2.1b. Determine the mass \(m\), the stiffness \(k\),
Consider a spring-mass-damper system (Figure P2.2) with \(m_{e q}=100 \mathrm{~kg}, k_{e q}=10,000 \mathrm{~N} / \mathrm{m}\), and \(c_{e q}=20 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\).Figure P2.2 It is required to place a \(10 \mathrm{~kg}\) mass on the main mass \(m_{e q}\) such that the new
Consider the system shown in Figure P2.1 a where \(a=25 \mathrm{~cm}\), \(\ell_{1}=50 \mathrm{~cm}\), and \(\ell_{2}=30 \mathrm{~cm}\). Here, \(k=1,100 \mathrm{~N} / \mathrm{m}\) and \(0.5 \mathrm{~kg}Find a value of damping constant \(c\) such that the steady state is reached without any
Consider the system shown in Figure P1.12.a. First the damper is detached and the mass is excited by a force \(f(t)=20 \sin \omega t \mathrm{~N}\). Find and plot responses when \(\omega=0.8 \omega_{n}, \omega_{n}\), and \(1.5 \omega_{n}\). Assume that the initial conditions are zero. Compare your
Consider a rotor on a massless and rigid shaft, which is supported by ball bearings at the ends (Figure P2.5).Figure P2.5 The mass of the rotor is \(10 \mathrm{~kg}\) and the eccentricity is \(0.5 \mathrm{~cm}\). If the operating speed of the rotor is \(4,200 \mathrm{rpm}\), what should be the
Consider a rotor on a massless and flexible steel shaft, which is simply supported at the ends (Figure P2.6).Figure P2.6 The mass of the rotor is \(10 \mathrm{~kg}\) and the eccentricity is \(0.5 \mathrm{~cm}\). The length and the diameter of the shaft are \(50 \mathrm{~cm}\) and \(5
An instrument with mass \(=13 \mathrm{~kg}\) is to be isolated from aircraft engine vibrations ranging from 18,00 to \(2,300 \mathrm{cpm}\). What should be the stiffness of an isolator for at least \(65 \%\) isolation? Assume that the damping ratio is 0.045 .Assuming that the initial conditions are
A spring-mass-damper system with sinusoidal base displacement P2.8 Determine the amplitude and the phase of the steady state response of the mass \(m\) in Figure P2.8.Figure P2.8 a Massless and rigid bar x(t) C l1 k 2 y(t) = yo sin wot T m
A vehicle with mass \(m_{e q}=1,050 \mathrm{~kg}\) and suspension stiffness \(k_{e q}=435,000 \mathrm{~N} / \mathrm{m}\) is traveling with a velocity \(V\) on a sinusoidal road surface with amplitude \(=0.011 \mathrm{~m}\) and a wavelength of \(5.3 \mathrm{~m}\) (Figure P2.9).a. Determine the
The natural frequency and the damping ratio of a vibrometer are \(6 \mathrm{~Hz}\) and \(0.22 \mathrm{~Hz}\), respectively. What is the range of frequencies for the measurement error to be below \(3 \%\) ?Corroborate the validity of your design using MATLAB ODE23 or ODE45 for a signal frequency.
An accelerometer with mass \(=0.01 \mathrm{~kg}\) and a damping ratio \(=\) 0.707 is to be designed. What should be the undamped natural frequency of the system so that the measurement error never exceeds \(2 \%\) ? The vibration signal, which is to be measured, can have a frequency as high as
The force-deflection curve for a structure is experimentally obtained (Figure P2.12). What is the equivalent viscous damping if the frequency of oscillation is \(100 \mathrm{~Hz}\) ?Figure P2.12 -0.53 Force (N) 794 -0.2 0.2 0.53 Deflection (mm) -794
Find the Fourier series expansions of the periodic functions shown in Figures P3.1a-c.Figure P3.1(a, b, c) -T P f (t) Half sine wave 0 T 2T Figure P3.1a Periodic function with half sine waves f(t)^ P ADA -T T 2 -P T T 2 -P Figure P3.1b Periodic function with triangular waves f(t)^ P -2T -T 0 0.3T T
Consider a spring-mass-damper system with mass \(=1.2 \mathrm{~kg}\) and damping ratio \(=0.05\), which is subjected to the periodic force shown in Figures P3.1a-c. Let the natural frequency of the system be \(\pi / T\).a. Determine the steady-state response for each forcing function.b. Verify your
In Example 3.1.4, the second nozzle is inadvertently closed. Find the Fourier coefficients of the force experienced by each blade. Also, find the maximum steady-state amplitude using the natural frequency and the damping ratio provided in Example 3.1.5.Example 3.1.4Example 3.1.5 Consider the rotor
A turbine blade in a rotor of a gas turbine experiences the force (Table P3.4) during each rotation. An SDOF model of the turbine blade is described in Figure P3.4a where \(m=1 \mathrm{~kg}, k=80,000\) \(\mathrm{N} / \mathrm{m}\), and \(c=3 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). The clearance is
Consider an SDOF spring-mass-damper system subjected to the step forcing function (Figures 2.1.1 and 2.1.2).Using the convolution integral, derive Equations 2.1.21, 2.1.28, and 2.1.38 for underdamped, critically damped, and overdamped system, respectively.Figure 2.1.1Figure 2.1.2Equation
Using the convolution integral, derive the expression for the response of an undamped \((c=0)\) spring-mass system subjected to the forcing function \(f(t)\) shown in Figure P3.6. Assume that all initial conditions are zero.Figure P3.6 f(t) m x(t) f(t) P a Q
Consider the cantilever of the atomic force microscope (Binning and Quate, 1986) with length \(=100 \mu \mathrm{m}\), thickness \(=0.8 \mu \mathrm{m}\), and width \(=20 \mu \mathrm{m}\). The material of the cantilever is silicon nitride having \(E=310 \times 10^{9} \mathrm{~Pa}\) and density
The base displacement \(y(t)\) in Figure P3.8a is described in Figure P3.8b. Using the convolution integral, determine the response \(x(t)\).Figure P3.8(a, b) x(t) Static equilibrium y(t) (a) meq www keg x(0)0 x(0) = 0 y(t) yo (b) b 1
Consider an SDOF spring-mass-damper system subjected to the step forcing function (Figures 2.1.1 and 2.1.2).Using the Laplace transformation technique, derive Equations 2.1.21, 2.1.28, and 2.1.38 for underdamped, critically damped, and overdamped systems, respectively.Figure 2.1.1Figure
Solve the problem P3.7 using the Laplace transformation technique.Problem P3.7Consider the cantilever of the atomic force microscope (Binning and Quate, 1986) with length \(=100 \mu \mathrm{m}\), thickness \(=0.8 \mu \mathrm{m}\), and width \(=20 \mu \mathrm{m}\). The material of the cantilever is
Solve the problem P3.8 using the Laplace transformation technique.Problem P3.8The base displacement \(y(t)\) in Figure P3.8a is described in Figure P3.8b. Using the convolution integral, determine the response \(x(t)\).Figure P3.8(a, b) x(t) Static equilibrium y(t) (a) meq www keg x(0)0 x(0) = 0
Consider the system shown in Figure P3.12, where \(a=25 \mathrm{~cm}\), \(\ell_{1}=50 \mathrm{~cm}\), and \(\ell_{2}=30 \mathrm{~cm}\). Here, \(k=1,100 \mathrm{~N} / \mathrm{m}, m=2 \mathrm{~kg}\), and the damping ratio is 0.1 .Figure P3.12 a. Derive the transfer function \(\frac{x(s)}{y(s)}\).b.
The dynamics of pedestrian-bridge interaction is given (Newland, 2004) by\[ M \ddot{y}+K y(t)+m \alpha \beta \ddot{y}(t-\Delta)+2 \xi \sqrt{K M} \dot{y}=-m \beta \ddot{x}(t) \]where \(M\) : Mass of the bridge, \(K\) : Stiffness of the bridge, \(\xi\) : damping ratio, \(Y\) : displacement of the
Consider the system in Figure P4.1.Figure P4.1a. Derive the differential equations of motion and obtain the mass and stiffness matrices.b. Calculate the natural frequencies and the mode shapes.c. Find the initial conditions such that the free vibration is sinusoidal with each natural frequency.
Consider the system in Figure P4.2.Figure P4.2 a. Derive the differential equations of motion and obtain the mass and stiffness matrices.b. Calculate the natural frequencies and the mode shapes.c. Find the initial conditions such that the free vibration is sinusoidal with each natural frequency.
Consider the system in Figure P4.3.Figure P4.3 a. Derive the differential equations of motion and obtain the mass and stiffness matrices.b. Calculate the natural frequencies and the mode shapes.c. Find the initial conditions such that the free vibration is sinusoidal with each natural frequency.
Consider the system in FigureP4.4.FigureP4.4 a. Derive the differential equations of motion and obtain the mass and stiffness matrices.b. Calculate the natural frequencies and the mode shapes.c. Find the initial conditions such that the free vibration is sinusoidal with each natural frequency.
Obtain and plot free vibration response for the system shown in Figure P4.1 when \(m=1 \mathrm{~kg}, k=530 \mathrm{~N} / \mathrm{m}, k_{c}=130 \mathrm{~N} / \mathrm{m}\). Assume that \(x_{1}(0)=0.01 \mathrm{~m}, \dot{x}_{1}(0)=1 \mathrm{~m} / \mathrm{sec}, x_{2}(0)=0\), and \(\dot{x}_{2}(0)=0\).
Consider the system in Figure P4.6.Figure P4.6 a. Derive the differential equations of motion and obtain mass, stiffness, and damping matrices.b. Assume that \(m=11 \mathrm{~kg}, k=4,511 \mathrm{~N} / \mathrm{m}\), and \(c=20 \mathrm{~N}-\mathrm{sec} / \mathrm{m}\). Determine the damping ratio and
A quarter car model of an automobile is shown in Figure P4.7. The vehicle is traveling with a velocity \(V\) on a sinusoidal road surface with amplitude \(=0.011 \mathrm{~m}\) and a wavelength of \(5.3 \mathrm{~m}\).Figure P4.7a. Derive the differential equations of motion and obtain mass,
A rotor-shaft system (Figure P4.8) consisting of torsional stiffness \(k_{1}\) and mass-moment of inertia \(J_{1}\) is subjected to a sinusoidal torque with magnitude \(1.3 \mathrm{kN}\)-meter. When the excitation frequency equals \(80 \mathrm{~Hz}\), there is a large amount of vibration.Figure
A rotor-shaft system consisting (Figure P4.9) of torsional stiffness \(k_{1}\) and mass-moment of inertia \(J_{1}\) is subjected to a sinusoidal torque with magnitude \(1.3 \mathrm{kN}\)-meter. When the excitation frequency equals \(80 \mathrm{~Hz}\), there is a large amount of vibration.Figure
Determine the response of the two-mass system in Figure P4.10 via modal decomposition when the force \(f(t)\) is a step function of magnitude \(5 \mathrm{~N}\) and the modal damping ratio in the vibratory modeFigure P4.10 Assume that \(m=1.5 \mathrm{~kg}, k=1,250 \mathrm{~N} / \mathrm{m},

Vibration Of Mechanical Systems 1st Edition Alok Sinha - Solutions

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