This paper reviews fundamental investigations of band saw vibration and stability. The literature is spread out in journals and reports in a variety of languages, and consklerable duplication of effort exists. It is this diffusion of references which makes a review necessary. The authors have reviewed all pertinent papers known to them, and apologize to both authors and readers for significant work that is missing. It should be noted that a critical review such as this is necessarily tinged by the authors' interpretation of what is fundamental knowledge in a particular field. The review has been limited to topics of direct, ftmdamental importance to band saw vibration and stability. Topics in which vibration is not the main consideration, such as tooth design, power requirements, fatigue cracking, etc., are not explicitly discussed here. Band saw vibration belongs to a class of vibration problems, called axially moving material vibrations, first investigated by Rudolf Skutch [1897]. This class contains research on magnetic and paper tape vibrations [Miranker 1960], moving belt dynamics [Chubachi 1957, 1958 ; Doyle, Hornung 1969; Rhodes 1970; Lai, Chen 1971], chain drive vibrations [Mahalhlgam 1957], vibration of pipes transporting fhlids [Housner 1952; Niordson 1953; Naguleswaran, Williams 1968b: Thurman, Mote 1969b; Liu, Mote 1974: and many others], moving strings and fibers [Sack 1954; Archibald, Emslie 1958: Swope, Ames 1963: and others], cables and catenaries [Simpson 1972], and of course band saw vibrations. The problems listed above are analogous as a first approximation, because the linearized equation of motion and boundary con-

Principal developments in band saw vibration and stability research

Travelling load response of an axially moving string

The dynamic behavior of an elastic catenary cable due to a moving mass alongits length is investigated. The equations of motions are derived using theHamilton's principle for general supports that include the horizontal andinclined cables with small and large sags and for variable velocity of themoving mass. Those equations of motions are in general nonlinear partialdifferential equations due to the initial curvature of the cable. Theequations are also complex due to the presence of three different types ofaccelerations of the moving mass. Those are the normal, Coriolis andcentrifugal accelerations. Therefore, we used the Galerkin procedure withsine function (Fourier representation) and anti-derivative functions of thecompactly supported wavelets as trial basis and used direct integrationmethods to integrate the discretized equations of motions. Newton–Raphsonmethod is used for iterations. Several examples are studied and the resultsas obtained by Fourier and wavelet representations are compared. Because ofthe localization feature, wavelets are proven to minimize the spuriousoscillations specially those appearing in the cable tension.

Dynamics of an Elastic Cable Carrying a Moving Mass Particle

The evolution of a trapped mode of oscillations in a “string on an elastic foundation-moving inertial inclusion” system

The paper is devoted to the investigation of a possibility of overcoming the critical speed by a moving concentrated load on a string. It is known that exceeding the critical speed is impossible in the framework of the linear statement for this problem. The general geometrically nonlinear statement for the problem is suggested. It allows one to take into account the coupling between transversal and longitudinal waves and the contact friction between the string and the load. It is shown that solutions corresponding to the case in which the load tries to overcome the critical speed are essentially nonlinear. The conception of critical speed for the case of nonlinear waveguide is discussed. It is shown that overcoming the critical speed is impossible in the system without friction. The speed of the load can reach and exceed the value of the critical speed for an unstrained string, but the solution remains subcritical. The case of decelerating load is also considered. Taking into account the contact friction between the string and the load explains that exceeding the critical speed can be possible. The nature of the wave resistance force exerting on the load from the string is discussed.

F.T. Flaherty

Papers

Transient resonance of an ideal string under a load moving with varying speed