Portrait of Alfred Lichtenstein with his flute; Berlin.

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Portrait of Alfred Lichtenstein with his flute; New York.

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Portrait of Alfred Lichtenstein with his flute.

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Portrait of Alfred Lichtenstein with his flute; New York.

Handwritten dedication signed by Lichtenstein

Portrait of Alfred Lichtenstein with his flute; Berlin.

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Portrait of Alfred Lichtenstein with his flute; Berlin.

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Portrait of Alfred Lichtenstein with his flute; New York.

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Portrait of Alfred Lichtenstein with his flute.

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Portrait of Alfred Lichtenstein with his flute; New York.

Handwritten dedication signed by Lichtenstein

Studio portraits of Alfred Lichtenstein with his flute; New York.

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Albert Einstein playing violin with small orchestra; Princeton, New Jersey Portraits; Groups

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Albert Einstein playing violin with small orchestra; Princeton, New Jersey portraits groups

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Violin string shape functions for finite element analysis of rotating Timoshenko beams

Violin strings are relatively short and stiff and are well modeled by Timoshenko beam theory. We use the static part of the homogeneous differential equation of violin strings to obtain new shape functions for the finite element analysis of rotating Timoshenko beams. For deriving the shape functions, the rotating beam is considered as a sequence of violin strings. The violin string shape functions depend on rotation speed and element position along the beam length and account for centrifugal stiffening effects as well as rotary inertia and shear deformation on dynamic characteristics of rotating Timoshenko beams. Numerical results show that the violin string basis functions perform much better than the conventional polynomials at high rotation speeds and are thus useful for turbo machine applications. (C) 2011 Elsevier B.V. All rights reserved.