Nonlinear Vibration Analysis of Reed Valves

The present paper deals with a nonlinear valve modeling and uses the assumed-modes method to simulate the dynamic behavior of reed valves. To incorporate the geometric nonlinearity due to large deflection of reed valves into the valve dynamic model, the large-deflection strain-displacement relation is considered. As example of use, a parametric analysis of an actual suction valve is presented and the discrepancy between the linear and nonlinear models is numerical compared. The simulation results show that the nonlinear strain term plays an important role on the dynamic behavior of reed valve systems if the thickness of the valve is small. NOMENCLATURE A(x) cross-sectional area of the valve and zdirections respectively Ae effective force area ux,uz traslations of the valve in xand zdirections Av flow area of the suction valve, ndh respectively u strain energy of the valve C,. reflecting coefficient of the valve v instaneous volume of the cylinder E Young's modulus of the valve w external work done by the pressure difference f(t) generalized force x,y,z reference coordinate of the valve I(x) moment of inertia of the valve cross-sectional a valve flow coefficient area 13 pushing force coefficient of the valve I length of the valve Exx longitudinal strain /I distance indicated in Fig. 2 ~(x) admissible function of the cantilever beam p pressure inside the cylinder T] coefficient of the nonlinear term defmed in Eq. Ps pressure in the suction chamber (12) q(t) time-dependent generalized coordinate K isentropic exponent rs radius of the suction orifice p density of the valve R gas constant ro. natural frequency of the valve t time variable s damping ratio T kinetic energy of the valve (-) differentiation with respect to time Ts suction temperature (') differentiation with respect to x u,w translations of neutral axis of the valve in xINTRODUCTION Reed valves in many hermetic reciprocating compressors are usually acted automatically by pressure differences across the valve. They are the most important element for controlling suction and discharge gas flow. When the valve motion is abnormal, compressor efficiency drops dramatically. The highly unsteady characteristic of the reed valves' vibration exacts a proper design in order to achieve higher standards of compressor efficiency, noise, and reliability. Generally, too stiff of the reed valve causes over-compression and too flexible causes unnecessary fluctuation of the reed valve. The reed valve models currently employed usually consider a linear (small amplitude) model [1-3]. Linearity is a conceptual ideal which is never completely realized in the dynamic behavior of a real structure. Although it is Fifteenth International Compressor Engineering Conference at Purdue University, West Lafayette, IN, USAJuly 25-28, 2000 437 known that linearized equations provide no more a frrst approximation of an actual situation, they are sufficient for many practical and engineering purposes. For many phenomena, predictions based on linear models are qualitatively correct and only slightly wrong, quantitatively. This is particularly true for the small amplitudes of motion involved in many vibration problems. Linearized theory is inadequate, however, if the vibration of a reed valve involves amplitude that is not very small, as is assumed in linear theory. Since the displacement of valve is several times to the thickness of valve for typical reciprocating compressors, nonlinear theory should be used to obtain the more accurate results. The present paper considers the large-deflection strain-displacement relation [4] to capture the geometric nonlinearity due to large deflection of reed valves. The assumed-modes method [5] is used to reasonably simplify the govening equation of motion. By means of the Runge-Kutta method [6], the dynamic dispalcement of a typical suction valve is simulated. Numerical results show that the nonlinear strain term has a considerable influence on the dynamic behavior of the reed valve systems. MATHEMATICAL FORMULATION The Lagrangian approach is used to derive the governing equation of motion for a cantilever type valve. The Kinetic energy T, potential energy U, and the work of external load W are written in terms of a time-dependent generalized coordinates with subsequent application of Lagrange's equation. A schematic representation of a hermetic reciprocating compressor is shown in Fig. 1. Figure 2 depicts the geometry of the suction reed valve. According to the geometry of the valve shown in Fig. 2, the suction reed valve can be regarded as a cantilever beam and reasonably modeled by the Bernoulli-Euler beam theory. Under Kirchhoff's hypothesis [4] and referring to the Nomenclature, the displacement of the valve can be written as Ow(x,t) ux=u(x,t)-z , ax uz = w(x,t). (1, 2) Since the valve is assumed to be free to move axially, the effect of the midplane stretching strains can be considered approximately to be zero. Assuming that the transverse displacement is not small compared to the thickness of the valve, the strain components can be computed from the large-deflection strain-displacement relation [4] as