System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 4, Problem 4.26P
In Figure P4.26 when
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find the equations of motion using the lagrange equations of motion. The main rod has length l and mass m, the left spring has spring constant k and right spring has spring constant 2k
Q1: 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
128
Using the figure below for a mass-spring system and given that R2 = 2R1, derive the equation of motion for the system in terms of x. Assume that the pulleys are massless, and the answer should be a differential equation that must include x, f, and their (high-order)derivatives.
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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- Q2) Consider a mass (m=5 kg) connected to a massless rod of length (L=2 m) to swing about point O, as shown in Figure (2). The mass is also connected to a spring of spring constant (k=10 N/m). The other end of the spring is connected to a pivot. The unstretched length of the spring is 1 m. Find the equation of motion of the g L-2 m system. m-5 kg 3 m k-10 N/m Figure 2arrow_forwardFigure Q4a Two homogeneous rods ABC and OB of mass 14 kg/m Rajah S4a are welded together to form a T-shaped body as shown 0.3 m in Figure Q4a. At position 1, the spring attached to point C is unstretched and the system is moving with an A C angular velocity w = 3 rad/s. If the system is expected to stop at position 2, whereas point C' is at the same level w = 3 rad/s of point 0, find the stiffness of the spring k. Given AB = ВС 3 ОВ 3 0.5 m. M = 8 Nmarrow_forwardfind the equations of motion using the lagrange equations of motion. The main rod has length l and mass m, the left spring has spring constant k and right end of the rond can be movedarrow_forward
- 3- Draw the free-body diagram and derive the equation of motion using Newton s second law of motion for each of the systems shown in Figure 2 4r Pulley, mass moment of incrtia / Figure 2arrow_forwardDerive the equation of motion of the system shown in the figure below, using the Newton's second law of motion. k1 k2 m Linear mass-spring systemarrow_forwardBy determining the equivalent spring constant of the mass spring system given in the figure, we calculate the equation of motion undamped. write for the situation.arrow_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_forwardfind the equations of motion using the lagrange equations of motion. the bigger circle with the radius R is not moving, the smaller circle with the radius r and mass m1 is connected with the center of the bigger circle with rod of mass m2 and a spiral spring spring constant karrow_forwardA mass weighing 4 pounds is attached to a spring whose spring constant is 36 lb/ft. Find the equation of motion.arrow_forward
- Q2\ A circular disk of mass M and radius R is connected by a spring of modulus k as shown in the figure. If it is free to roll on a horizontal surface without slipping, find its natural frequency. Rarrow_forwardQuestion 4For the dynamic system shown in the figure,derive the equations of motion using Lagrange’s equations.arrow_forwardFind the equivalent spring constant spring constant of the system. Say, k1 = 20 lb/in, k2 = 35 lb/in, k3=18 lb/in,k4 = 50 lb/in, k5 = 45 lb/inarrow_forward
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Ch 2 - 2.2.2 Forced Undamped Oscillation; Author: Benjamin Drew;https://www.youtube.com/watch?v=6Tb7Rx-bCWE;License: Standard youtube license