Page 60 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 2 Problems 47

U d
-L- Figure P2-44.

about O is ml^/3), (b) the equation for the undamped natural frequency, and (c) the
expression for critical damping. Use virtual work.
2-45 A thin plate of area A and weight W is attached to the end of a spring and is allowed
to oscillate in a viscous fluid, as shown in Fig. P2-45. If Tj is the natural period of
undamped oscillation (i.e., with the system oscillating in air) and T2 the damped period
with the plate immersed in the fluid, show that
2ttW
M= y fr l^

where the damping force on the plate is = filAv, 2A is the total surface area of
the plate, and u is its velocity.
y////////.

Figure P2-45.
2-46 A gun barrel weighing 1200 lb has a recoil spring of stiffness 20,000 Ib/ft. If the barrel
recoils 4 ft on firing, determine (a) the initial recoil velocity of the barrel, (b) the
critical damping coefficient of a dashpot that is engaged at the end of the recoil stroke,
and (c) the time required for the barrel to return to a position 2 in. from its initial
position.
2-47 A piston of mass 4.53 kg is traveling in a tube with a velocity of 15.24 m /s and engages
a spring and damper, as shown in Fig. P2-47. Determine the maximum displacement of
the piston after engaging the spring-damper. How many seconds does it take?
1/ - 15.24 m/s c=l.75Ns/cm
[ ”1 1^
m = 4.53 kg A = 350 N/cm Figure P2-47.
2-48 A shock absorber is to be designed so that its overshoot is 10% of the initial
displacement when released. Determine fj. If ^ is made equal to what will be the
overshoot?
2-49 Determine the equation of motion for Probs. 2-41 and 2-42 using virtual work.
2-50 Determine the effective stiffness of the springs shown in Fig. P2-50.