Vibrations of a Beam in Variable Contact With a Flat Surface

We study small vibrations of cantilever beams contacting a rigid surface. We study two cases: the first is a beam that sags onto the ground due to gravity, and the second is a beam that sticks to the ground through reversible adhesion. In both cases, the noncontacting length varies dynamically. We first obtain the governing equations and boundary conditions, including a transversality condition involving an end moment, using Hamilton's principle. Rescaling the variable length to a constant value, we obtain partial differential equations with time varying coefficients, which, upon linearization, give the natural frequencies of vibration. The natural frequencies for the first case (gravity without adhesion) match that of a clamped-clamped beam of the same nominal length; frequencies for the second case, however, show no such match. We develop simple, if atypical, single degree of freedom approximations for the first modes of these two systems, which provide insights into the role of the static deflection profile, as well as the end moment condition, in determining the first natural frequencies of these systems. Finally, we consider small transverse sinusoidal forcing of the first case and find that the governing equation contains both parametric and external forcing terms. For forcing at resonance, w find that either the internal or the external forcing may dominate.

Gathering in the home of Dr. Alfred Stern and his wife Lotte nee Wallerstein; Paula Wallerstein nee Molling; Rheydt, Germany. Portraits Groups

Dr. Alfred Stern probably standing far left; Paula Wallerstein nee Molling seated in center

Leo Baeck with unidentified soldiers Military; World War I; Portraits Groups

Center May 2000

Joseph Ritter von Wertheimer, Moritz Lazarus and Abraham Geiger; Officers of the first Jewish Council of Leipzig Portraits; Groups; Men

"Praesidium der ersten israelitischen Synode zu Leipzig. II Vicepraesident Ritter V. Wertheimer aus Wien. Praesident Prof. Dr. M. Lazarus aus Berlin. I Vicepraesident Dr. A. Geiger aus Frankfurt a/M."

Eyeglasses of Moses Mendelssohn.

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Portrait of the six Baum family siblings Portraits Family

Standing are Fritz, Joseph and Lilly. The youngest is Hans. Erich is seated center and Walter is seated on the far right.

Reproduction of photograph of Kaufman and Rosenberg family portrait on the occasion of the wedding of Ludwig Rosenberg and Paula Rosenberg nee Joseph (left); Hamburg.

Includes Ludwig Rosenberg and Paula Rosenberg nee Joseph (left), Thekla (Tekie) Joseph (standing center), Karl Kaufman and Else Kaufman (right)

Officer guards and inmates, including Norbert Pollock, of the Boergermoor Labor Camp; Germany.

Viehhandler (cattle dealer) Norbert Pollock of Ruethen, Westpflahlen (standing center in dark pants, white t-shirt and cap)

Studies of phosphazenes. 28. Reactions of pentachloro- and pentafluoro(triphenylphosphazenyl)cyclotriphosphazenes with sodium methoxide. Mechanistic aspects and their implications for nucleophilic displacement at a tetrahedral phosphorus(V) center

Abstract is not available.

Gans family portraits on Delft platter Portraits Family

Top row: Alexander Gans and his wife

Strauss family portrait; Würzburg.

Seated front center is grandmother Ernestine Strauss nee Uhlfelder (died before 1914, second wife of Leopold (Laemmle) Strauss) surrounded by six grandchildren; Benno Strauss is standing far left

Studio portrait of members of the Provisional Government of Baden (Die badische vorläufig Regierung).

Die badische vorläufig Regierung Stehend von links: Fritz Stockinger, Min d Unterrichts; Leop Rueckert, Min d Verkehrs; Ludw. Marum, Min d Justiz; Herm Rob Dietrich, Min d Auswaertigen; Ad Schwarz, Min f soziale Fuersorge; Bruemmer, Min fur milit Ang; sitzend: Karl Jos Wirth, Min d Finanzen; Gust Jos Ludw Trunk, Min f Ernaeh; sitzend Center: Anton Geiss, Minister President; Ludwig Haas, Interior; Marzloff, Transition & Housing

Class picture of Gisela Kleinermann Portraits Group

The photograph shows Gisela Kleinermann (standing at the right end of the line) with her class at the ‘Juedische Schule’ in Dresden, late summer of 1938, as she and her mother were preparing for her immigration to the USA. Most other children were deported in October 1938 and perished in the Holocaust. The teacher, Jacob Blum (center back) emigrated to Palestine.

Jewish Winterhilfe Berlin; Kueche Pestalozzi-Strasse Jewish Institutions

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Jewish Winterhilfe Berlin; Kueche Choriner-Strasse Jewish Institutions

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