Simple Mathematical Models of the Vibration and Force Transmission of Discharge or Suction Tubes as Function of Discharge and Suction Pressures

On the idealized situation of simply supported discharge tubes it is demonstrated how the discharge and suction pressures influence the resonance frequencies and thus the force transmission from the compressor to the compressor shell. The influence of flow velocity is also briefly discussed. With minor modifications, all results are also applicable to suction tubes in cases of high side compressors. INTRODUCTION It occurs frequently that it is observed that the sound pressure radiated from the shell of a refrigeration compressor changes when operating conditions change. One of the possible mechanisms was investigated by Johnson and Hamilton il]. When the speed of sound of the gas in the shell changes with a condition change, gas resonances change. This effect is typically of importance in the higher frequency range. This paper points out still another mechanism. As the discharge-suction pressure difference changes, discharge or suction tubes change their resonance frequencies and thus their vibration transmission into the shell. The mathematical models are kept as siraple as possible tvithout sacrificing some generali·c:in order to illustrat.e the most important influences to the designer. From a physical viewpoint, the reader is invited to purchase a tube like toy balloon and observe how after blowing the initially limp balloon up it becomes a stiff structure capable of bearrt like vibrations. The balloon beam increases its stiffness with increasing pressure differential, and thus increases its resonance frequency. The effect can also be observed on a Borden tube. 69 Equation of Motion for a Straight Discharge Tube Let us examine the idealized case of a straight discharge tube that is free to expand in axial direction. Following a standard approach of derivation, like Hamilton's principle [2], one obtains the following equation of motion: where a 4 w ~ 2 ~ a 2 EI --4 + pv (Pd ~ Ps)A ~ ax ax aw aw + 2pv axat + (m+p) at2 = 0 w transverse deflection [mm] p mass of gas per unit length [Ns 2 ;mm2 ]