W. Soedel

Effects of coriolis acceleration on the free and forced in-plane vibrations of rotating rings on elastic foundation

Abstract The vibration of rotating rings is investigated. The equations of motion that cover both transverse and tangential motion are derived from Hamiltons principle. The natural frequencies and modes are then obtained without using the inextensional assumption and an attempt is made to interpret the time dependent natural modes from various viewpoints. The effect of rotation and elastic foundation on system characteristics is examined. Most important, a general solution for forced vibration is formulated and demonstrated by an example. The effect of the Coriolis acceleration component on the forced response is illustrated by comparing the results for a travelling force on a stationary ring with results for a rotating ring subjected to a stationary force.

The receptance method applied to the free vibration of continuous rectangular plates

Abstract Vibrational characteristics of thin rectangular plates continuous over intermediate rigid simple supports and simply supported along two opposite edges with simply supported and/or clamped end conditions have been calculated by using the receptance method. Three different procedures have been illustrated, one of which is an exact solution. Numerical examples of four-equal-span and three-unequal-span plates have been presented and the accuracy of results reported in the literature has been attested.

Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem

Abstract The harmonic and periodic forced vibrations of rotating rings are derived and investigated. The modal expansion technique yields the forced solution, which is characterized by four generalized co-ordinates associated with each n (circumferential wave number). The inextensional assumption is presumed, when flexural vibration is the only important component, to reduce the order of the system. The closed form solutions to the harmonic load cases, once concentrated, once distributed, are demonstrated and interpreted. The approach is then extended to periodic loads, where Fourier sine and cosine series is applied. Examples depict the numerical responses to all the cases being derived. The solutions of a stationary ring subjected to traveling loads are also solved for comparison. Their difference is investigated and interpreted from various viewpoints.

The receptance method applied to ring-stiffened cylindrical shells: Analysis of modal characteristics

Abstract The receptance method is applied to determine the natural frequencies and mode shapes of circular cylindrical shells stiffened by rings. The receptances of cylindrical shell and of a ring to forces in the radial and circumferential directions are derived in terms of the modal characteristics of each. A matrix equation of free vibration, which must be solved by an iterative technique, results by eliminating the angular variable. An iterative solution is practical, since the size of the matrices remains at two times the number of stiffening rings, regardless of the number of modes of the unstiffened cylinder and rings included in the solution. The validity of the method is demonstrated by comparing results for specific cases with the results obtained theoretically and experimentally by others. When various stiffener configurations are being considered for a given cylindrical shell, the modal characteristics of the shell without stiffeners may be calculated once and used repeatedly to calculate the frequencies of the stiffened shell configurations. The form of the results offers potential for simplifications which are presented in a companion paper.

ON THE DYNAMIC RESPONSE OF ROLLING TIRES ACCORDING TO THIN SHELL APPROXIMATIONS

The response of the rolling tire is formulated in terms of the natural modes and frequencies of the non-contacting tire for contact loading due to rolling, braking, accelerating or cornering. The tire is viewed as an equivalent thin shell. Formulation is by way of the three-dimensional dynamic Green function of this shell. As the application example, the response of the tire during pure rolling is evaluated. The critical rolling speed at which the tire develops its first large amplitude standing waves is calculated in a novel fashion, showing the relationship between the standing wave phenomenon and natural tire modes and frequencies. Findings underline the need for measuring or calculating higher order tire modes than those considered in what seems to be the present standard practice.

General formulation of four pole parameters for three-dimensional cavities utilizing modal expansion, with special attention to the annular cylinder

Abstract Four poles for three-dimensional cavities were simulated in terms of eigenfunctions of these cavities, first for general curvilinear co-ordinates, then for the specific example of an annular cylindrical cavity. As an application example, this formulation was applied to an annular cylindrical cavity connected to an anechoic pipe, with a single and a double input. Various responses are illustrated. Many applications come to mind, for example in three-dimensional muffler theory, but application examples were chosen in such a way that they contribute to the understanding of pressure pulsations inside a shell enclosing a refrigeration compressor, with the acoustic input being the suction port of the compressor and the output being the evaporator pipe. The relative locations of input and output ports, pressure distributions in the cavity at resonance condition, and phase cancellations when two input ports are used, are discussed.

Natural frequencies and modes of rings that deviate from perfect axisymmetry

Given the experimental or theoretical natural frequencies and modes of an axisymmetric ring, an analytical method is presented that allows one to obtain the natural frequencies and modes when this ring is non-axisymmetric due to a mass or stiffness non-uniformity of the ring. A ring is the simplest model of an axisymmetric structure and treatment of its non-uniformities paves the way for treatment of more complicated structures such as tires, for example. The receptance method is employed to determine the natural frequencies and mode shapes of a circular ring which is deviating from axisymmetry due to a local mass of stiffness non-uniformity. The receptance of the ring is derived by utilizing the modal expansion method.

Simplified equations and solutions for the vibration of orthotropic cylindrical shells

Abstract Starting with Love type equations of motion for orthotropic circular cylindrical shells, the theory is simplified by assumptions similar to those in the Donnell-Mushtari-Vlasov development for isotropic shells. Closed form solutions for simply supported cases are then obtained. Results of two example cases are compared with finite element results and are shown to agree well. It is argued that this simplified approach allows easy assessment of the influence of design parameter changes.

On the vibration of shells with timoshenko-mindlin type shear deflections and rotatory inertia

Equations of motion of shells are formulated in curvilinear co-ordinates that are consistent with assumptions made in the Timoshenko beam and also the Mindlin plate equations. This consistency is proven by obtaining the Timoshenko and Mindlin equations from the derived shell equations by geometrical reduction. The implication of the Timoshenko type shear is illustrated by constrasting the free vibration behavior of the special case of a cylindrical shell with the behavior of the Timoshenko beam and the Mindlin plate.

On helmholtz resonator effects in the discharge system of a two-cylinder compressor

Discharge pressure oscillations in compressors are a common problem, especially in multicylinder compressors because of the cylinder interaction by way of discharge cavities and collecting plenums. In this paper it is demonstrated that such interactions may be simulated on the computer by viewing the discharge system as a multi-degree-of-freedom Helmholtz resonator. Such an approach is of particular advantage for industrial applications because of the usual irregular shapes of the discharge cavities and plenums. For this purpose a previously developed compressor simulation procedure that computes the dynamic behavior of compressor valves was extended to cover the interaction of two cylinders through the discharge system. Helmholtz resonator type equations were developed and solved simultaneously with the valve dynamic equations. Measured and computed discharge cavity pressures showed good agreement. Reasons for the strong excitation of the asymmetric mode of discharge gas oscillation of the prototype compressor are given and ways of suppression are discussed.