System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Textbook Question
Chapter 4, Problem 4.10P
Compute the equivalent torsional spring constant of the stepped shaft arrangement shown in Figure P4.10. For the shaft material,
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The mechanism shown in Figure P6.10 is in equilibriumfor an applied load P = 20 kN. Specifications for the mechanismlimit the shear stress in the steel [G = 80 GPa] shaft BC to 70 MPa,the shear stress in bolt A to 100 MPa, and the vertical deflection ofjoint D to a maximum value of 25 mm. Assume that the bearingsallow the shaft to rotate freely. Using L = 1,200 mm, a = 110 mm,and b = 210 mm, calculate(a) the minimum diameter required for shaft BC.(b) the minimum diameter required for bolt A.
Q4/ For the figure below the spring is used to stop a 10
kg package. If the maximum deflection in the spring is 70
mm, (a) the total work of the system in the figure below
is:
8m/s
H=0,25
/K=400N/m
10kg
500m
0-30
(b): The total work when
theta =0 equal to: *
3. Consider the spring-mass system in an elevator that moves vertically. The natural length of the
spring is as shown in the figure below, and at time t=0 the elevator starts moving with prescribed
displacement y(t). Suppose that the relative displacement of the mass is x(t), i.e., its displacement
with respect to the elevator. Assume mass m can only move vertically, the elevator is massless, and
gravity is included.
a. Derive the equation of motion of the mass using Newton's 2nd law.
b. When is the mass in the elevator in a condition of effective "zero gravity"?
y(t)
x(t)
Chapter 4 Solutions
System Dynamics
Ch. 4 - Prob. 4.1PCh. 4 - In the spring arrangement shown in Figure P4.2....Ch. 4 - In the arrangement shown in Figure P4.3, a cable...Ch. 4 - In the spring arrangement shown in Figure P4.4,...Ch. 4 - For the system shown in Figure P4.5, assume that...Ch. 4 - The two stepped solid cylinders in Figure P4.6...Ch. 4 - A table with four identical legs supports a...Ch. 4 - The beam shown in Figure P4.8 has been stiffened...Ch. 4 - Determine the equivalent spring constant of the...Ch. 4 - Compute the equivalent torsional spring constant...
Ch. 4 - Plot the spring force felt by the mass shown in...Ch. 4 - Calculate the expression for the natural frequency...Ch. 4 - Prob. 4.13PCh. 4 - Obtain the expression for the natural frequency of...Ch. 4 - 4.15 A connecting rod having a mass of 3.6 kg is...Ch. 4 - Calculate the expression for the natural frequency...Ch. 4 - For each of the systems shown in Figure P4.17, the...Ch. 4 - The mass m in Figure P4.18 is attached to a rigid...Ch. 4 - In the pulley system shown in Figure P4.19, the...Ch. 4 - Prob. 4.20PCh. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - In Figure P4.23, assume that the cylinder rolls...Ch. 4 - In Figure P4.24 when x1=x2=0 the springs are at...Ch. 4 - 4.25 In Figure P4.25 model the three shafts as...Ch. 4 - In Figure P4.26 when 1=2=0 the spring is at its...Ch. 4 - Prob. 4.27PCh. 4 - For the system shown in Figure P4.28, suppose that...Ch. 4 - For the system shown in Figure P4.29, suppose that...Ch. 4 - Prob. 4.30PCh. 4 - For Figure P4.31, the equilibrium position...Ch. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - 4.34 For Figure P4.34, assume that the cylinder...Ch. 4 - Use the Rayleigh method to obtain an expression...Ch. 4 - Prob. 4.36PCh. 4 - 4.37 Determine the natural frequency of the system...Ch. 4 - Determine the natural frequency of the system...Ch. 4 - Use Rayleigh's method to calculate the expression...Ch. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Prob. 4.42PCh. 4 - The vibration of a motor mounted on the end of a...Ch. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - A certain cantilever beam vibrates at a frequency...Ch. 4 - Prob. 4.47PCh. 4 - 4.48 The static deflection of a cantilever beam is...Ch. 4 - Figure P4.49 shows a winch supported by a...Ch. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - 4.53 In Figure P4.53 a motor supplies a torque T...Ch. 4 - Derive the equation of motion for the lever system...Ch. 4 - Prob. 4.55PCh. 4 - Figure P4.56a shows a Houdaille damper, which is a...Ch. 4 - 4.57 Refer to Figure P4.57. Determine the...Ch. 4 - For the system shown in Figure P4.58, obtain the...Ch. 4 - Find the transfer function ZsXs for the system...Ch. 4 - Prob. 4.60PCh. 4 - Find the transfer function YsXs for the system...Ch. 4 - Prob. 4.62PCh. 4 - 4.63 In the system shown in Figure P4.63, the...Ch. 4 - Prob. 4.64PCh. 4 - Figure P4.65 shows a rack-and-pinion gear in which...Ch. 4 - Figure P4.66 shows a drive train with a spur-gear...Ch. 4 - Prob. 4.67PCh. 4 - Prob. 4.68PCh. 4 - Prob. 4.69PCh. 4 - Figure P4.70 shows a quarter-car model that...Ch. 4 - Prob. 4.71PCh. 4 - 4.72 Derive the equation of motion for the system...Ch. 4 - A boxcar moving at 1.3 m/s hits the shock absorber...Ch. 4 - For the systems shown in Figure P4.74, assume that...Ch. 4 - Refer to Figure P4.75a, which shows a ship’s...Ch. 4 - In this problem, we make all the same assumptions...Ch. 4 - Refer to Figure P4.79a, which shows a water tank...Ch. 4 - The “sky crane” shown on the text cover was a...Ch. 4 - Prob. 4.81PCh. 4 - Prob. 4.82PCh. 4 - Suppose a mass in moving with a speed 1 becomes...Ch. 4 - Consider the system shown in Figure 4.6.3. Suppose...Ch. 4 - Prob. 4.86PCh. 4 - Figure P4.87 shows a mass m with an attached...Ch. 4 - Figure P4.88 represents a drop forging process....Ch. 4 - Refer to Figure P4.89. A mass m drops from a...Ch. 4 - Prob. 4.90PCh. 4 - (a) Obtain the equations of motion of the system...Ch. 4 - Refer to part (a) of Problem 4.90. Use MATLAB to...Ch. 4 - Refer to Problem 4.91. Use MATLAB to obtain the...Ch. 4 - 4.94 (a) Obtain the equations of motion of the...Ch. 4 -
4.95 (a) Obtain the equations of motion of the...
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- Figure Q3(b) shows a uniform bar AB of mass = 8 kg hinged at point C. Point A is connected to a spring to maintain the bar in vertical direction, and the stiffness k = 500 N/m. If point A is displaced counter-clockwise by a small angle θ = 3.5 degree and released, (i) With the free body diagram and kinetic diagram, determine the initial horizontal displacement of A.arrow_forwardPlease solve all parts send only handwrittenarrow_forwardFor the simple pendulum shown in the figure, write the governing equation in s-domain. Assume small angles and linearize your model. Initial conditions are zero. Explain each step clearly. - 0(1) L, length m, massarrow_forward
- Consider three masses m1, m2, and m3 connected by two springs as in the figure below. Suppose m1 = 3 = m3 and m2 = 2 and k12 = 12 = k23. Use a second-order system to find the general solution.arrow_forwardRefer to Figure Q2. A tray of mass mı is supported by 3 springs as shown in Figure 3(a). The natural frequency fa is 5.0Hz. An additional mass motor of m2 = 3.0kg (in OFF condition) is placed at the center on top of the mass, the natural frequency is observed to be 2.5Hz. a) Calculate the mass mı. The motor m2 is ON and it rotates at the speed of 600 rpm. Calculate: a) The transmissibility b) Attenuation c) Explain what will happen if the system run at Resonant Frequency m2 m1 Figure 2(a): Original system Figure 2(b): system with m2 addedarrow_forwardQUESTIION A. B AND Carrow_forward
- 4 = k = k k2 = 2k F X kz = 3k 1. Derive the expression for the equivalent spring constant that relates the applied force F to the resulting displacement x of the system shown in the figure above. Assume the displacement of the link to be small.arrow_forwardFigure below shows a body of mass m attached to a spring of stiffness k. Find the differential equation for the displacement, xarrow_forwardQ1: The system shown has two masses. Beam of mass (Jo#m L² kg.m²) rotates about fixed point (O) and its free end is connected to disk rotates about fixed point (O₂). Consider all connecting links are massless and rigid. Find 1- The displacements of points A, B, and C in addition to the rotations of masses, all in terms of 0. 2- Find the equation of motion (EOM) in terms of 0. 3- What is the natural frequency of the system? 0 L/2 8 Energy methods A Jo=m L²2 L/2 Joz-m R² R C B C 128arrow_forward
- I In this problem, consider the figure below. A disk with radius r, mass m, and moment of inertia of Io about the point O is on an inclined plane, with the translational coordinates and y and rotation angle 0, as shown. The direction of gravity is shown, and the incline angle is a. A constant input torque is applied at the point O in the direction shown, and it is opposed by a damping torque from a bearing with value co in the direction shown. The point P represents a point of instantaneous contact, rolling without slipping. Your tasks: Direction of Gravity M(t) co 0 P x Mass: m Figure 2: System schematic. Moment of Inertia about 0: lo a: Incline Angle A Draw the FBD for the disk. B Derive the equation of motion with the rotation angle as the dynamic variable. 0 C Take the Laplace transform, and solve for (s), letting the initial conditions be 0(0) = 0 and (0) = 0. Remember that a constant is a step function for a one-sided Laplace transform. D Given your expression for (s), find the…arrow_forwardSolve it correctly and fast please. I will rate and review.arrow_forward4. Consider the following spring-mass systems below with zero initial conditions described as x (0) = 0, and x(t) = 0. Compute the form of the systems. (a) (t) + 4x(t) = 12 cos 2t (b) x(t)+4x(t) = 10 sin 5tarrow_forward
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