Questions and Answers of Dynamic Response Of Linear Mechanical Systems

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
Consider the model of a turbine blade (Griffin and Hoosac, 1984) shown in Figure P5.2. Model parameters (SI units) are as follows: \(m_{1}=0.0114, m_{2}=0.0427, m_{3}=0.0299, k_{1}=430,300, k_{2}=\)
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
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
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
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}
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},
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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},
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
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
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
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
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
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 \(=100 \mu \mathrm{m}\),
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
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, \(k=1,100 \mathrm{~N} / \mathrm{m}, m=2 \mathrm{~kg}\),
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
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.
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.
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.
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.
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
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}
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
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
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
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