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engineering
mechanical vibration analysis

Mechanical Vibration Analysis, Uncertainties, And Control 4th Edition Haym Benaroya, Mark L Nagurka, Seon Mi Han - Solutions

What difficulties arise if a system parameter such as \(k\) or \(m\) is known only approximately?
If the structure responds in a way that is unacceptable for a particular application, what options exist for the designer?
If the analyst decides that a structure requires control in order to fulfill the needs of an application, describe the procedure by which a control force can be derived and then verified that it performs as expected.
A simple system is represented by the governing equation\[ m \ddot{x}+k x=F(t) \]where \(m=1 \mathrm{~kg}, k=10 \mathrm{~N} / \mathrm{m}\), and \(F(t)=\cos t\) \(\mathrm{N}\), with zero initial conditions. Find the steadystate response \(x_{s}(t)\). Now suppose that the system is constrained
Consider the simple system, \(m \ddot{x}+k x=F(t)\), where \(F(t)\) is known and \(m\) and \(k\) are unknown. Suppose the initial conditions are both equal to zero. What can we determine about the values of \(m\) and \(k\) ?
Consider the simple system, \(m \ddot{x}+k x=F(t)\), where \(F(t)\) and steady-state solution \(x_{s}(t)\) are known and \(m\) and \(k\) are unknown. Suppose the initial conditions are both equal to zero. What can we determine about the values of \(m\) and \(k\) ?
Consider the simple system, \(m \ddot{x}+k x=F(t)\), where steady-state solution \(x_{s}(t), m\), and \(k\) are known and \(F(t)\) is unknown. Suppose the initial conditions are both equal to zero. What can we determine about \(F(t)\) ?
Consider a damped oscillator where the force is a function of velocity,\[ m \ddot{x}+c \dot{x}+k x=F_{0} \dot{x} \]Discuss the stability of this system in terms of the parameters \(m,c, k, F_{0}\).
Consider a damped oscillator where the force is a function of displacement,\[ m \ddot{x}+c \dot{x}+k x=F_{0} x \]Discuss the stability of this system in terms of the parameters \(m,c, k, F_{0}\).
For an \(m-c-k\) oscillator subjected to harmonic loading, \(A k \cos \omega t\), conceive a design so that the displacement response \(|x(t)|
For an oscillator governed by \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=\) \(\cos \omega t\), conceive a design so that the displacement response \(|x(t)|
Solve Problem 37 for the load shown in Figure 4.36. A F(t) 0 Figure 4.36: Ramp loading.
Solve Problem 37 for the load as shown in Figure 4.37. A F(t) 0 Figure 4.37: Simplified representation of a shock loading.
Consider the system shown in Figure 5.21. Let \(x_{1}\) and \(\theta\) be the displacement and angle of rotation of the pulley measured from their original locations when the spring is undeformed. Let \(x_{2}\) be the displacement of the cylinder, also measured from its original location when the
Consider the point mass supported by two linear springs shown in Figure 5.22. The springs are undeformed when the mass and the springs are along a straight line, where the coordinate \(y\) defines the displacement of the mass. Do not assume small angles. Derive (a) the equilibrium equation using
Consider again the system shown in Figure 2.58, reproduced in Figure 5.23. The system consists of massless pulleys, a block of mass \(m\), and two linear springs. The coordinate, \(\theta\), is the angle of rotation of the moving pulley, \(x_{1}\) is the displacement of the pulley center of mass,
Consider the system shown in Figure 5.24. Let \(x\) be the displacement of the mass measured from its original location where the springs are undeformed. Assume that the pulleys are massless and the springs are linear. (a) Find the static equilibrium position of the system using the principle of
Consider again the system shown in Figure 2.46, reproduced in Figure 5.25. The bar is horizontal when the massless springs are undeformed. Assume that the rod mass is \(m\), and the mass moment of inertia about the torsional spring is \(I_{O}\).(a) Find the static equilibrium position of the system
Consider again the system shown in Figure 2.56, reproduced in Figure 5.26. A disk is on an inclined plane supported by a spring. The coordinates \(x\) and \(\theta\) are zero when the spring is undeformed. The disk has mass moment of inertia \(I_{G}\) about its mass center and has radius \(r\).
Consider the system comprised of the disk, pulley, and block shown in Figure 5.27. The spring is connected to the disk of mass \(M\) at its mass center. The disk has radius \(r\) and mass moment of inertia \(I_{G}\) about its mass center. A cable is connected to and wrapped around the disk and
Consider the two degree-of-freedom system shown in Figure 5.28. The springs are undeformed when the rods are horizontal. The mass of each rod is \(m\), with length \(L\), and mass moment of inertia about each center of gravity is \(I_{G}\). (a) Find the static equilibrium position of the system
Derive the equation of motion for the system in Figure 5.21 for generalized coordinate \(x_{2}\) using(a) Lagrange's equation, and(b) Hamilton's principle. k M, R. IG. x1 m X2 Figure 5.21: Pulley-cylinder coupled system.
Derive the equation of motion for the system in Figure 5.22 for generalized coordinate \(y\) using(a) Lagrange's equation, and(b) Hamilton's principle. m ww www Figure 5.22: Point mass supported by two identical springs.
Derive the equation of motion for the system in Figure 5.23 for generalized coordinate \(x_{2}\) using(a) Lagrange's equation, and(b) Hamilton's principle. eeeee ellee m T Figure 5.23: A mass suspended by springs and pulleys.
Derive the equation of motion for the system in Figure 5.24 for generalized coordinate \(x\) using(a) Lagrange's equation, and(b) Hamilton's principle. k -0000 x m ellel k Figure 5.24: A mass suspended by springs and pulleys.
Derive the equation of motion for the system in Figure 5.25 for generalized coordinate \(\theta\) using(a) Lagrange's equation, and(b) Hamilton's principle. K 6000 Figure 5.25: Restrained rigid rod. k
Derive the equation of motion for the system in Figure 5.26 for generalized coordinate \(x\) using(a) Lagrange's equation, and(b) Hamilton's principle. eeeee IG 8 Position of the disk when the spring is unstretched x Figure 5.26: A rolling disk on an incline restrained by a spring.
Derive the equation of motion for the system in Figure 5.27 for generalized coordinate \(x\) using(a) Lagrange's equation, and(b) Hamilton's principle. k mmm M. r. IG x m Figure 5.27: Spring-disk-pulley mass system.
Derive the equations of motion for the system in Figure 5.28 for generalized coordinates \(\theta_{1}\) and \(\theta_{2}\) using(a) Lagrange's equation and(b) Hamilton's principle. m. L. IG eeeee m. L. IG B Figure 5.28: Spring-constrained double pendulum. eeeee
Consider again the system of Figure 2.52, reproduced in Figure 5.29. A disk of mass \(m\) and mass moment of inertia \(I_{G}\) is supported by a massless spring and a rope. Derive the equation of motion of the system using (a) Lagrange's equation, and (b) Hamilton's principle, for generalized
Consider again the system of Figure 2.53, reproduced in Figure 5.30. A mass \(m\) is suspended by a pulley system with a spring. Derive the equation of motion of the system using(a) Lagrange's equation, and(b) Hamilton's principle, for generalized coordinate \(x\). position of m when the spring is
Consider again the system of Figure 2.54, reproduced in Figure 5.31. Derive the equation of motion of the system using (a) Lagrange's equation, and (b) Hamilton's principle, for generalized coordinate \(x_{1}\). static equilibrium position for m Xeal 0 static equilibrium X2 Y2 position for m m Xeq2
Consider again the system of Figure 2.57, reproduced in Figure 5.32. Derive the equation of motion using (a) Lagrange's equation, and (b) Hamilton's principle, for generalized coordinate \(x\). eeeee k position of m when the spring is unstretched m Figure 5.32: Two masses suspended in a pulley
For the simple pendulum of Figure 5.33, derive the governing equation of motion assuming (a) that \(m\) is a point mass, and (b) that the mass is a sphere with small but finite mass moment of inertia. Given \(l=10 \mathrm{~cm}\) and \(m=5 \mathrm{~kg}\), how significant is the effect of the mass
The lightweight bar in Figure 5.34 is released from rest when the spring is undeformed at \(\theta=0\). The two masses at \(A\) and \(B\) slide in frictionless guides that are horizontal at the left and vertical at the right. Formulate the equation of motion using(a) Newton's second law,(b)
The spring supporting the rod-sphere system in Figure 5.35 is undeformed when the rod is horizontal. Assume the rod mass is negligible. The roller from which the spring is suspended permits the spring to maintain a vertical configuration. If the system is in this position when it is released from
Verify Equation 5.20,\[ m_{i} \ddot{\mathbf{r}}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{k}}=\left[\frac{d}{d t}\left(\frac{\partial}{\partial \dot{q}_{k}}\right)-\frac{\partial}{\partial q_{k}}\right]\left(\frac{1}{2} m_{i} \dot{\mathbf{r}}_{i} \cdot \dot{\mathbf{r}}_{i}\right) \]
For Example 5.10, use Newton's second law of motion to derive the equations of motion. Example 5.10 The Generalized Force Consider the simple pendulum suspended from a block that can translate on a smooth surface, as shown in Fig- ure 5.15. Gravity acts in the y direction. Derive the equations of
Derive Equation 5.28, assuming a linear model with \(m_{i j}=\) constant and \(k_{i j}=\) constant ,\[ [m]\{\ddot{q}\}+[k]\{q\}=\{0\} \]
Formulate the problem of Example 5.13 where the string is elastic with stiffness \(k\) and can stretch. Derive the equations of motion. Example 5.13 Two Mass System Equations from Lagrange's Equation Formulate the problem of Example 5.7 using Lagrange's equation and derive the equation of motion
For the elastic pendulum discussed in Example 5.14, and shown in Figure 5.19, derive the governing equations of motion using Lagrange's equation. Example 5.14 Hamilton's Principle for the Derivation of the Equations of Motion of an Elas- tic Pendulum Derive the equation of motion of the
In the pendulum shown in Figure 5.36 the length \(r\) is variable. Derive the equations of motion. Discuss the possibility and implications if \(\omega=\dot{\theta}\). x = A sin oot m Figure 5.36: Variable length pendulum.
(a) For the elastically supported mass on a vertically moving base shown in Figure 5.37 derive the governing equation of motion first using d'Alembert's principle, and then using Lagrange's equation. Discuss the effect of base motion and show that it is equivalent to a force on mass \(m\) equal to
Consider the motion of a pendulum that is supported by springs that are elastically restrained to horizontal motion, as depicted in Figure 5.38. Assume that the springs are massless and remain horizontal, that \(\theta\) is small, and \(r\) is a constant. Formulate the equations of motion using (a)
For a pendulum supported from a horizontally restrained mass, as shown in Figure 5.39, derive the governing equations of motion using (a) Lagrange's equation and (b) Hamilton's principle. Identify the constraint equations. Discuss the effect of setting \(m_{1}=0\). Finally, simplify the governing
Fill in the missing steps in the derivation of the equations of motion for Example 5.8, that is, Equations 5.15 and 5.16 . Example 5.8 Double Compound Pendulum Equa- tions from Newton's Second Law Consider a double compound pendulum attached to a pin at point O as depicted in Figure 5.12. Using the
Use Lagrange's equation to derive the equations of motion governing the displacements \(x_{1}\) and \(x_{2}\) for the two degree-of-freedom system depicted in Figure 5.40 . 12 -6000 6000 Figure 5.40: Undamped two degree-of-freedom system.
Use Lagrange's equation to derive the equations governing the rotations \(\theta_{1}, \theta_{2}\), and \(\theta_{3}\) for the springconnected triple pendulum system shown in Figure 5.41. 6000 k 6000 m m m Figure 5.41: Triple pendulum coupled by springs.
A pendulum is suspended from a torsionally restrained disk, as in Figure 5.42, where \(\alpha\) is the angle of rotation of the disk, and \(\theta\) is the relative angle of the pendulum measured from the disk. Derive the governing equation of motion using (a) Lagrange's equation, and (b)
A rigid beam acts as a compound pendulum, suspended from an elastically restrained block that can undergo horizontal motion, as drawn in Figure 5.43. Derive the equations of motion using Lagrange's equation. Consider the following cases: (a) no friction between the block and surface, and (b)
For the inverted simple pendulum of length \(l\) and mass \(m_{2}\), supported on a cart of mass \(m_{1}\), as shown in Figure 5.44, derive the equations of motion if the cart is forced by \(F(t)\). m) m F Figure 5.44: Inverted pendulum on a moving base.
The system in Figure 5.45 oscillates about the equilibrium position denoted by the horizontal lines. The coil torsional springs are each undeformed when \(\theta_{1}=\theta_{2}=0\). If the link masses are neglected, derive the equations of motion to be\[ \begin{aligned} \left(m_{1}+m_{2}\right)
Figure 5.46 shows a pulley system that is acted on by an applied moment \(M(t)\), resulting in oscillatory motion. Use Lagrange's equation to derive the equations of motion. Identify the generalized forces \(Q_{\theta}\) and \(Q_{x}\). celle k M(1) m x Figure 5.46: A pulley system subjected to a
The system sketched in Figure 5.47 depicts two shafts coupled through meshing gears. Assume that gears 2 and 4 have gear ratio \(n=r_{2} / r_{4}\), and that external moment \(M(t)\) is applied to gear 1. Identify the constraint equation and the generalized coordinates, and derive the governing
Solve Equation 3.6, \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=0\) for \(\zeta>1\).\
Derive Equation 3.9, \(x(t)=C e^{-\zeta \omega_{n} t} \cos \left(\omega_{d} t-\phi\right)\).
(a) The response of an oscillator with mass \(m=1\) kg is shown in Figure 3.51. What can you say about the properties of the oscillator and the response? Determine the equation of motion and the initial conditions.(b) One property is changed, with all others remaining the same. The resulting time
(a) Derive the equation of motion for a mass-springdamper system in free vibration. Solve for the transient response \(x(t)\) that is driven by the initial conditions \(x(0)=x_{0}\) and \(\dot{x}(0)=v_{0}\). (b) Assume parameter values: \(k=1 \mathrm{lb} /\) in, weight \(m g=100 \mathrm{lb}\),
A mass-spring-damper system is tested to determine the value of \(c\). Assume \(k=10 \mathrm{lb} /\) in and \(m=2\) slug. (a) If the vibrational amplitude is observed to decrease to \(33 \%\) of its initial value after 2 consecutive cycles, what is the value of \(c\) ? (b) If \(k\), rather than
A periodic torque with a maximum value of \(1.0 \mathrm{~N}-\mathrm{m}\) at a frequency of \(5.0 \mathrm{rad} / \mathrm{s}\) is applied to a flywheel suspended from a wire. The flywheel has a moment of inertia of \(0.10 \mathrm{~kg}-\mathrm{m}^{2}\) and the wire has a stiffness of \(1
A periodic force \(F(t)=A \cos \omega t\) is applied to the damped system shown in Figure 3.53. Calculate the work done in one cycle during the steady-state response where \(A=50 \mathrm{~N}, \omega=10 \mathrm{rad} / \mathrm{s}, k=6 \mathrm{kN} / \mathrm{m}\), \(c=200 \mathrm{~N}-\mathrm{s} /
This problem investigates the number of cycles \(n\) re-quired for a structural oscillation amplitude to decay to \(x \%\) of its maximum. (a) What is the expression for the logarithmic decrement in terms of \(n\) and \(x\) ? (b) Solve for \(n\) in terms of \(\zeta\) and \(x\). (c) Plot \(n\) as a
Solve for \(t_{\max }\) in Equation 3.17 in general for arbitrary variables. Then evaluate \(t_{\max }\) numerically for the cases in Table 3.1. Plot each case and discuss trends. Table 3.1: Cases for tmax in Problem 9. Case (0) (in) (0) (in/s) w (rad/s) 1 2 0 5 0.1 2 5 0 0.1 2 3 5 5 0.1 2 4
Coulomb used the following method to determine the viscosity of liquids. A thin plate of weight \(W\) is suspended vertically and set into motion, first in air and then fully submerged in the particular liquid as shown in Figure 3.54. The time duration, \(t_{1}\) in air and \(t_{2}\) in the liquid,
The bell-crank mechanism depicted in Figure 3.55 is rotated slightly and released to oscillate in free vibration. Derive the damped frequency of oscillation \(\omega_{d}\) and the critical damping constant \(c_{c r}\). The mass of the frame can be ignored compared to the mass attached to it at the
An automobile's suspension system determines the vehicle's response to inputs from uneven road surfaces. Figure 3.56 shows a drawing of a suspension system that includes a coil spring, a shock absorber, and connecting links to the car frame and the wheel. Next to this artist's rendering is a simple
For the system in Figure 3.57, the forcing per unit mass is \(F(t) / m=10 \sin 15 t \mathrm{ft} / \mathrm{s}^{2}\). Solve for the transient response, the steady-state response, and the constants of integration for the following initial conditions.(a) \(x(0)=0\) in, \(\dot{x}(0)=5 \mathrm{in} /
For a mass-spring-damper system under harmonic loading per unit mass \(F(t) / m=\cos \omega t \mathrm{ft} / \mathrm{s}^{2}\), solve for the response amplitude for the case where \(k=\) \(20 \mathrm{lb} / \mathrm{in}, W=40 \mathrm{lb}\), and then plot \(\beta\) for a broad range of driving
Solve Problem 14 for \(k=40 \mathrm{lb} /\) in and \(W=20 \mathrm{lb}\). 14. For a mass-spring-damper system under harmonic loading per unit mass F(t)/mcos ut ft/s, solve wt for the response amplitude for the case where k = 20 lb/in, W40 lb, and then plot 3 for a broad range of driving frequencies
Solve Problem 14 for \(k=20 \mathrm{lb} /\) in and \(W=20 \mathrm{lb}\). 14. For a mass-spring-damper system under harmonic loading per unit mass F(t)/mcos ut ft/s, solve wt for the response amplitude for the case where k = 20 lb/in, W40 lb, and then plot 3 for a broad range of driving frequencies
For Problem 14 where \(k=20 \mathrm{lb} /\) in and \(W=40 \mathrm{lb}\), solve for the steady-state responses in \(\mathrm{ft} / \mathrm{s}^{2}\) to the following excitation forces per unit mass:(a) \(F(t) / m=100 \sin \left(0.9 \omega_{n} t\right)\)(b) \(F(t) / m=100 \sin \left(\omega_{n}
For what range of frequency ratios \(\omega / \omega_{n}\) will the magnification factor \(\beta\) be greater than 1 for the idealized model of Figure 3.58? For what range of frequency ratios will the magnification factor be greater than \(86 \%\) of the peak response? Let \(m=\) 1 slug, \(k=9
For the system of Figure 3.58, what do the initial conditions have to be so that both integration constants \(C\) and \(\phi\) equal zero in Equation 3.37? x(t) = Cent cos(wato) + B coswt + B sin wt, (3.37)
Derive \(C\) and \(\phi\) to verify Equations 3.38 and 3.39 . C = x(0) - B COS (3.38) [(0) + [x(0) B]wn - Bw = tan (3.39) [x(0) - B]wd
For each case in Table 3.2, for the governing equation \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=A \cos \omega t\), evaluate \(C\) and \(\phi\), and \(B_{1}\) and \(B_{2}\) in Equation 3.37. Then plot the free and forced responses separately and then in sum. Let \(A=1 \mathrm{~cm} /
Derive Equations 3.44 and 3.45:\[ \begin{aligned} |G(i \omega)| & =\frac{1}{k \sqrt{\left(1-\left(\omega / \omega_{n}\right)^{2}\right)^{2}+\left(2 \zeta \omega / \omega_{n}\right)^{2}}} \\ \phi & =\tan ^{-1}\left[\frac{2 \zeta \omega / \omega_{n}}{1-\omega^{2} / \omega_{n}^{2}}\right]
Derive \(B_{1}\) and \(B_{2}\) in Equations 3.48 and 3.49:\[ \begin{aligned} & -\omega_{b}^{2} B_{1}+2 \zeta \omega_{n} \omega_{b} B_{2}+\omega_{n}^{2} B_{1}=A_{1} \\ & -\omega_{b}^{2} B_{2}-2 \zeta \omega_{n} \omega_{b} B_{1}+\omega_{n}^{2} B_{2}=A_{2} \end{aligned} \] Table 3.2: Cases for
Plot \(F_{T} / k Y\), Equation 3.52, vs. \(\omega_{b} / \omega_{n}\) for \(\zeta=\) \(0.05,0.10\), and 0.25 . 1/2 kY =()[ 1+ (20wo/wn) - [(1 w/w) + (2(wo/wn) ] (3.52)
The base-excited system of Figure 3.59 is driven by a force with frequencies in the range \(1.0 \leq \omega_{b} / \omega_{n} \leq\) 2.0. For parameter values \(m=1 \mathrm{~kg}, k=9 \mathrm{~N} / \mathrm{cm}\), and \(Y=10 \mathrm{~cm}\), select a value of \(c\) so that the average magnitude of the
Solve for the response of the base-excited system governed by \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=2 \zeta \omega_{n} \dot{y}+\omega_{n}^{2} y\), where \(y(t)=A \exp \left(i \omega_{b} t\right)\) for the response \(x(t)=\) \(X(i \omega) \exp (i \omega t)\). Plot the response for
The response of a system excited at its base by \(y(t)=Y \sin \omega t\) is given by\[ x(t)=X \sin \left(\omega_{b} t-\phi\right) \]with\[ \frac{X}{Y}=\left[\frac{1+\left(2 \zeta \omega_{b} / \omega_{n}\right)^{2}}{\left(1-\omega_{b}^{2} / \omega_{n}^{2}\right)^{2}+\left(2 \zeta \omega_{b} /
A vibration sensor has a damping ratio \(\zeta=0.55\) and a natural frequency \(f_{n}=10 \mathrm{~Hz}\). Estimate the lowest frequency that can be measured with \(1 \%\) error.
An engine of mass \(200 \mathrm{~kg}\) is to be supported on four helical springs. When the engine speed is \(1000 \mathrm{rpm}\) there is a primary vertical periodic force of maximum value \(350 \mathrm{~N}\) due to the unbalanced reciprocating masses. Assume that the engine vibrates in the
A rotating machine component has an eccentricity of approximately \(e=0.1 \mathrm{~mm}\) in its center of mass resulting in a harmonic load on the structure. Assume, using the notation of the chapter, that \(m_{0} / m=0.05\), operating frequency \(\omega_{r}=100 \mathrm{rad} / \mathrm{s}\),
Derive the general equations of motion, Equations 3.55 and 3.56 ,\[ \begin{aligned} & m\left(\ddot{x}-\varepsilon \dot{\Omega}^{2} \cos \Omega-\varepsilon \ddot{\Omega} \sin \Omega\right)+k_{x} x=0 \\ & m\left(\ddot{y}-\varepsilon \dot{\Omega}^{2} \sin \Omega+\varepsilon \ddot{\Omega} \cos
A simple model of a seismometer is shown in Figure 3.60. The top view shows how a pen attached to the seismometer mass traces out a pattern on a paper grid that is on a roller. The side view provides more details of the model. The absolute motion ofthe ground is defined by \(y(t)\), the relative
A machine is loaded by a periodic "sawtooth" shaped force, as depicted in Figure 3.61. The load is assumed to have existed for a very long time. Model this force using: (a) a one-term Fourier series, (b) a three-term Fourier series, and (c) a five-term Fourier series. In each case solve for the
Solve Problem 33 with (a) \(T=2 \mathrm{~s}\), and (b) \(T=\) \(0.5 \mathrm{~s}\).
Solve Problem 33 with \(A=2 \mathrm{~cm}\).
Solve Problem 33 with \(\zeta=1.0\).
A body is subjected to the two harmonic motions,\[ \begin{aligned} & x_{1}(t)=100 \sin \left(\omega t+\frac{\pi}{3}\right) \mathrm{mm} \\ & x_{2}(t)=50 \sin \left(\omega t+\frac{\pi}{6}\right) \mathrm{mm} \end{aligned} \]What additional harmonic motion should be applied to the body to bring it
An undamped system is forced by the periodic function shown in Figure 3.62. Derive the Fourier series representation for this function, and solve for the response assuming zero initial conditions. FO 1/2 3/2 2 Figure 3.62: Triangle foreing function.
For Example 3.20, plot \(F(t)\) and \(x(t)\) for the case \(\omega_{n}=6 \omega_{T}\). Compare the results with those of Figure 3.50 and discuss. Assume \(T=1 \mathrm{~s}, F_{0}=1 \mathrm{~N}\), \(k=4 \mathrm{~N} / \mathrm{m}\), and \(\zeta=0.1\). Example 3.20 Response to a Square Wave Using
Solve Example 3.20 where the structure has viscous damping, that is, the governing equation of motion is \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2}=F(t)\), the forcing \(F(t)\) is the same square wave, and \(\omega_{n}=2 \omega_{T} \mathrm{rad} / \mathrm{s}\). Example 3.20 Response to a
Solve Example 3.20, as in Problem 40, with \(\omega_{n}=\) \(4 \omega_{T} \mathrm{rad} / \mathrm{s}\). Example 3.20 Response to a Square Wave Using Fourier Series 47 Consider the simple undamped oscillator + wx F(t)/m, where F(t) is the force represented by the square wave function in Figure 3.49.
A base-excited structure is governed by the equation \(\ddot{x}+2 \zeta \omega_{n} \dot{x}+\omega_{n}^{2} x=2 \zeta \omega_{n} \dot{y}+\omega_{n}^{2} y\), where \(y(t)\) is the base motion. Solve for the response if the base motion is given by the sawtooth function shown in Figure 3.61. F(t) T
Solve Problem 42 where the input is given by the square wave of Figure 3.63. y(t) B - B T Figure 3.63: Square wave base motion.
Solve Problem 42 where the input is given by the function \(\sin t, 0 \leq t \leq \pi\), shown in Figure 3.64, repeated periodically. C y(t) 0 Figure 3.64: Base motion driven by sint.
The beam in Figure 2.30 vibrates as a result of loading not shown. State the necessary assumptions to reduce this problem to a one degree-of-freedom oscillator. Then derive the equation of motion.Figure 2.30: Vibrating beam of mass \(m\) supported by springs of stiffness \(k\) separated by length
If a beam is supported continuously on a foundation, as shown in Figure 2.31, damping must be added to an idealized model to represent the viscous effects of the mat foundation. How would you idealize this system as a one degree-of-freedom oscillator? Derive the equation of motion. beam mat

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