Machinery's Handbook, 31st Edition
Pendulums
181
For a simple pendulum,
g l
2 = π
(1) where T = period in seconds for one complete cycle; g = acceleration due to gravity = 32.17 ft/sec 2 (approximately); and l is the length of the pendulum in feet as shown on the accompanying diagram. For a physical or compound pendulum, (2) where k o = radius of gyration of the pendulum about the axis of rotation in feet, and r is the distance from the axis of rotation to the center of gravity in feet. The metric SI units that can be used in the two above formulas are T = time in sec onds; g = approximately 9.81 m/s 2 , which is the value for acceleration due to gravity; l = the length of the pendulum in meters; k o = the radius of gyration in meters, and r = the distance from the axis of rotation to the center of gravity in meters. Formulas (1) and (2) are accurate when the angle of oscillation θ shown in the diagram is very small. For θ equal to 22 degrees, these formulas give results that are too small by 1 percent; for θ equal to 32 degrees, by 2 percent. For a conical pendulum, the time in seconds for one revolution is: (3a) or (3b) For a torsional pendulum consisting of a thin rod and a disk as shown in the figure (4) where W = weight of disk in pounds; r = radius of disk in feet; l = length of rod in feet; d = diameter of rod in feet; and G = modulus of elasticity in shear of the rod material in pounds per square inch (psi or lb/in 2 ). The formula using metric SI units is: T d G Mr l 8 4 2 = π where T = time in seconds for one complete oscillation; M = mass in kilograms; r = ra dius in meters; l = length of rod in meters; d = diameter of rod in meters; G = modulus of elasticity in shear of the rod material in pascals (newtons per meter squared). The same formula can be applied using millimeters, provided dimensions are expressed in millimeters throughout, and the modulus of elasticity in megapascals (newtons per millimeter squared). Harmonic.— A harmonic is any component of a periodic quantity which is an integral multiple of the fundamental frequency. For example, a component the frequency of which is twice the fundamental frequency is called the second harmonic. A harmonic, in electricity, is an alternating-current electromotive force wave of higher frequency than the fundamental, and superimposed on the same so as to distort it from a true sine-wave shape. It is caused by the slots, the shape of the pole pieces, and the pulsa tion of the armature reaction. The third and the fifth harmonics, i.e., with a frequency three and five times the fundamental, are generally the predominating ones in three-phase machines. T T gr k o 2 2 = π cos T g l 2 π z = cot T g r 2 π z = T gd G 4 Wr l 2 3 2 = π
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