## w., "A Digital Simulation of a Reciprocating Hermetic Compressor Including Comparisons With Experiment,

- G. Gatecliff
- Ph.D. Thesis, University of Michigan,
- 1969

- Published 2014

This paper presents a mathematical analysis of the forced vibration of a cantilever leaf valve of non-uniform width. The method of solution is a generalized technique employing the Rayleigh-Ritz procedure to develop an approximate solution to a geometrically accurate formulation of the problem. Sample results are included and the calculations are compared with experimental data measured in a test compressor operating under conditions identical to those used in the analytic model. INTRODUCTION The valves in hermetic refrigeration compressors are thin leaves made of spring steel having a length, width and form determined by the number of ports to be covered and their geometric distribution. Motion is imparted to the valve as a result of pressure and elastic forces. Mathematical analysis of the forced vibration of leaf valves has been undertaken in the past by several investigators (1, 2). In general, these studies involve valves that can be modeled by assuming them to be cantilever beams of uniform width. In cases where the valve under consideration does not differ appreciably from that of a uniform beam, it is not uncommon to define an equivalent uniform beam for the purposes of analysis. Approaches of this type involve exact mathematical solutions to approximate formulations of the problem. Many valves in common usage embody geometries that do not closely approximate that of a uniform beam. This paper presents an exact formulation for the case of a valve of non-uniform width together with an approximate mathematical solution for designs of this type. In addition, experimental data is included illustrating the effectiveness of the model in predicting the time-dependent motion of a valve with geometry of practical significance. 316 MATHEMATICAL MODEL OF THE VALVE For purposes of analysis the valve may be considered as a non-uniform bar in transverse vibration. In addition, it will be assumed that the cross sectional dimensions are small co~pared with the length of the valve and that all loading is in the form of point forces applied at the ports. With these assumptions, both shear and rotary inertia effects may be neglected and the following form of the beam equation employed to describe the valve's behavior: ~ EI(x) ~ + m(x) ~~F. (t) o (x-x.) 2 [ 2 J 2 OX ox ilt J J (1) where the boundary conditions shown below apply to the case of a clamped-free beam:

@inproceedings{Gatecliff2014ForcedVO,
title={Forced Vibration of a Cantilever Valve of Uniform Thickness and Non-Uniform Width},
author={G. W. Gatecliff},
year={2014}
}