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19. Assume that the system described by the equation mu" + y u' + ku = 0 is either critically damped or overdamped. Show that the mass can pass through the equilibrium position at most once, regardless of the initial conditions.
Hint: Determine all possible values of t for which u = 0.
20. Assume that the system described by the equation mu" + y u' + ku = 0 is critically damped and the initial conditions are u(0) = u0, u' (0) = v0.If v0 = 0, show that u ^ 0as t ^ rn, but that u is never zero. If u0 is positive, determine a condition on v0 that will assure that the mass passes through its equilibrium position after it is released.
21. Logarithmic Decrement. (a) For the damped oscillation described by Eq. (26), show that the time between successive maxima is Td = 2n/ p.
(b) Show that the ratio of the displacements at two successive maxima is given by exp (y Td/2m). Observe that this ratio does not depend on which pair of maxima is chosen. The natural logarithm of this ratio is called the logarithmic decrement and is denoted by A.
(c) Show that A = ny/mp. Since m, p, and A are quantities that can be measured easily for a mechanical system, this result provides a convenient and practical method for determining the damping constant of the system, which is more difficult to measure directly. In particular, for the motion of a vibrating mass in a viscous fluid the damping constant depends on the viscosity of the fluid; for simple geometric shapes the form of this dependence is known, and the preceding relation allows the determination of the viscosity experimentally. This is one of the most accurate ways of determining the viscosity of a gas at high pressure.
22. Referring to Problem 21, find the logarithmic decrement of the system in Problem 10.
23. For the system in Problem 17 suppose that A = 3 and Td = 0.3 sec. Referring to Problem 21, determine the value of the damping coefficient y.
24. The position of a certain spring-mass system satisfies the initial value problem
u" + ku = 0, u(0) = 2, u'(0) = v.
If the period and amplitude of the resulting motion are observed to be n and 3, respectively, determine the values of k and v.
? 25. Consider the initial value problem
u" + y ul + u = 0, u(0) = 2, u'(0) = 0.
We wish to explore how long a time interval is required for the solution to become “negligible” and how this interval depends on the damping coefficient y. To be more precise, let us seek the time t such that |u(t)| < 0.01 for all t > t . Note that critical damping for this problem occurs for y = 2.
(a) Let y = 0.25 and determine t, or at least estimate it fairly accurately from a plot of the solution.
(b) Repeat part (a) for several other values of y in the interval 0 < Y < 1.5. Note that t steadily decreases as y increases for y in this range.
(c) Obtain a graph of t versus y by plotting the pairs of values found in parts (a) and (b). Is the graph a smooth curve?
(d) Repeat part (b) for values of y between 1.5 and 2. Show that t continues to decrease until y reaches a certain critical value y0, after which t increases. Find y0 and the corresponding minimum value of t to two decimal places.
3.8 Mechanical and Electrical Vibrations
(e) Another way to proceed is to write the solution of the initial value problem in the form (26). Neglect the cosine factor and consider only the exponential factor and the amplitude R. Then find an expression for t as a function of y. Compare the approximate results obtained in this way with the values determined in parts (a), (b), and (d).
26. Consider the initial value problem
mu" + y u + ku = 0, u(0) = u0, u'(0) = v0.
Assume that y 2 < 4km.
(a) Solve the initial value problem.
(b) Write the solution in the form u(t) = R exp(-y t/2m) cos(ft — S). Determine R in terms of m, y, k, u0, and v0.
(c) Investigate the dependence of R on the damping coefficient y for fixed values of the other parameters.
27. A cubic block of side l and mass density p per unit volume is floating in a fluid of mass density p0 per unit volume, where p0 > p. If the block is slightly depressed and then released, it oscillates in the vertical direction. Assuming that the viscous damping of the fluid and air can be neglected, derive the differential equation of motion and determine the period of the motion.
Hint: Use Archimedes’ principle: An object that is completely or partially submerged in a fluid is acted on by an upward (buoyant) force equal to the weight of the displaced fluid.
? 28. The position of a certain undamped spring-mass system satisfies the initial value problem
u" + 2u = 0, u(0) = 0, u'(0) = 2.
(a) Find the solution of this initial value problem.
(b) Plot u versus t and d versus t on the same axes.
(c) Plot u versus u; that is, plot u(t) and u'(t) parametrically with t as the parameter. This plot is known as a phase plot and the uu'-plane is called the phase plane. Observe that a closed curve in the phase plane corresponds to a periodic solution u(t). What is the direction of motion on the phase plot as t increases?