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2 Double-acting cylinder with mass Hydraulic Systems Analysis 48 whence the equation for the oscillatory system is 1 x = - f3 (1/V1 + 1/V2 ) M d 2x A 2 dt 2 The hydraulic frequency is lowest when V1 = V2 (compared with any position of the piston where V1 ¥- V2 ) and is of special interest. Usually the two volumes are equal with the piston in the midposition. 2a) ~M where l-'; represents the total cylinder volume. 98 in). 32 in 3 ). 3 A Double-acting Cylinder with a Long Exhaust Pipeline An easily overlooked factor in estimating hydraulic frequencies is the mass of oil in pipelines.
Also attached to the mass is a viscous damper of rate f. The mass of the spring and the damper is negligible. The upper end of the spring may be displaced vertically. Derive the operational relation between displacements of the mass x 2 and of the upper end of the spring x 1 • K x2 XI MD 2 +ID+K Determine the amplitude of the oscillations of x 2 when x 1 is moved sinusoidally at frequency 4! rad/s with an amplitude of 2! 288lb/(in/s)). (Approx. 4 em. Note. wn = 5, ( = i-) 6 The pressure of a liquid in a pipeline is varying sinusoidally between 100 and 200 bar at a frequency of 10Hz (20n rad/s).
C. voltage of the same frequency as the input but with a different amplitude and out of phase with the input. 3; if the input is being moved back and forth sinusoidally, the system is being subjected to a harmonic input. The output (in this case the piston motion) will also be simple harmonic at the same frequency but with a different amplitude and out of phase with the input. In the case of this hydraulic system an experiment might show a distorted output waveform but for analysis it is assumed to be a pure sine wave.