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Vibration Damping of Structural Elements

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Vibration Damping of Structural Elements


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  • provides an introduction into the fundamentals of vibration damping by addressing areas such as the classification of damping, characterization of viscoelastic materials, effects of environmental factors on the viscoelastic material properties, fundamentals of damping material properties, and vibration control techniques such as damping treatments.
  • presents free vibrations, forced vibrations, and transient vibrations of discrete multi-degree-of-freedom systems with non- proportional control techniques such as damping treatments.
  • covers vibration damping of composite materials and structures.
  • examines the vibration of constrained damped beam, damped plate, damped ring, as well as damped cylindrical shell structures.
  • discusses vibration theories, the associated analyses, analytical solutions, experimental data, analytical/experimental results correlation, and vibration characteristics of each damped structural system.
  • formulates a finite element analysis employing off-set finite element technique for structures with constrained viscoelastic damping layer.


  • Copyright 1995
  • Dimensions: 6 x 9
  • Pages: 384
  • Edition: 1st
  • Book
  • ISBN-10: 0-13-079229-2
  • ISBN-13: 978-0-13-079229-7

The book emphasizes on damping improvement of many structural elements such as beams, plates, rods, rings, shells and curved elements by using constrained viscoelastic damping materials.

Sample Content

Table of Contents

Preface, ix.

1. Fundamentals of Vibration Damping, 1.

1.1 Introduction, 1. 1.2 Scope of the Book, 2. 1.3 Classification of Damping, 4. 1.4 Characterization of Viscoelastic Materials, 10. 1.5 Effects of Environmental Factors, 20. 1.6 Fundamentals of Damping Material Properties, 24. 1.7 Fundamentals of Vibration Control Techniques, 29. 1.8 Summary, 37. References, 37.

2. Vibrations of Discrete Systems, 40.

2.1 Introduction, 40. 2.2 Free Vibrations of Single-Degree- of-Freedom Systems with Viscous Damping, 45. 2.3 Forced Vibrations of Single-Degree-of-Freedom Systems with Damping, 48. 2.4 Transient Vibrations of Single-Degree-of-Freedom Systems with Damping, 54. 2.5 Free Vibrations of Two-Degree-of-Freedom Systems with Viscous Damping, 66. 2.6 Forced Vibrations of Two-Degree-of-Freedom Systems with Damping, 77. 2.7 Transient Vibrations of Two-Degree-of-Freedom Systems with Damping, 83. 2.8 Vibrations of Multi-Degree-of-Freedom Systems with Damping, 89. 2.9 Summary, 102. References, 104 .

3. Damping of Fiber-Reinforced Composite Materials, 105.

3.1 Introduction, 105. 3.2 Damping of Aligned Discontinued Fiber Composites, 106. 3.3 Damping of Aligned Off-axis Discontinuous Fiber Composites, 120. 3.4 Damping of Randomly Oriented Short-Fiber Composites, 128. 3.5 Damping of Laminated Composites, 137. 3.6 The Influence of Fiber-Matrix Interface on Damping, 146. 3.7 Summary, 156. References, 156.

4. Vibrations of Constrained Damped Beam Structures, 160.

4.1 Introduction, 160. 4.2 Vibrations of Three-Layered Damped Beam Structures, 161. 4.3 Other Vibration Analyses of Three- Layered Damped Beam Structures, 174. 4.4 Vibration Characteristics of Constrained Three-Layered Damped Beam Structures, 186. 4.5 Vibrations of Multilayered Damped Beam Structures, 192. 4.6 Summary, 193. References, 194.

5. Vibrations of Constrained Damped Plate Structures, 198.

5.1 Introduction, 198. 5.2 Vibrations of Three-Layered Damped Plate Structures, 199. 5.3 Vibration Solutions Using A General Purpose Finite Element Computer Program, 210. 5.4 Vibration Characteristics of Constrained Three-Layered Damped Plate Structures, 226. 5.5 Summary, 236. References, 236.

6. Vibrations of Constrained Damped Ring Structures, 239.

6.1 Introduction, 239. 6.2 Vibrations of Discontinuously Constrained Damped Ring Structures, 239. 6.3 Vibrations of Continuously Constrained Damped Ring Structures, 252. 6.4 Vibration Solutions Using General Purpose Finite Element Computer Program, 272. 6.5 Continuously Versus Discontinuously Constrained Damped Ring Structures, 276. 6.6 Summary, 279. References, 279.

7. Vibrations of Constrained Damped Cylindrical Shell Structures, 281.

7.1 Introduction, 281. 7.2 Effectiveness of Damping Treatments of Constrained Beam-Damped Cylindrical Shell Structures, 282. 7.3 Vibrations of Constrained Beam-Damped Cylindrical Shell Structures, 284. 7.4 Vibration Characteristics of Beam-Damped Cylindrical Shell Structures, 293. 7.5 Vibrations of Constrained Beam- Damped Titanium Cylindrical Shells, 295. 7.6 Vibrations of Constrained Damped Shell Structures with Attached Mass Segments, 303. 7.7 Vibrations of Constrained Damped Shell Structures with Curved Elements, 312. 7.8 Summary, 313. References, 316.

8. Continuous and Discontinuous Constrained Viscoelastic Material on Structural Elements Using Finite Element Method, 318.

8.1 Introduction, 318. 8.2 Finite Element Formulation of Structures with Constrained Viscoelastic Damping Layer, 321. 8.3 Derivation of Elastic Stiffness Matrix of Plate Structures, 327. 8.4 Derivation of Elastic Stiffness Matrix of Beam Structures, 332. 8.5 Derivation of Mass Matrix, 334. 8.6 Evaluation of Loss Factors, 336. 8.7 Constrained Viscoelastic Damping Materials on Structural Elements With Prestress Effects, 349. 8.8 Summary, 362.

References, 362.

Author Index, 367.

Subject Index, 369.


For years, vibration damping as a technology has been well received by various industries, however, very few books exist on this subject. In 1985, Nashif, Jones, and Henderson published an excellent book entitled Vibration Damping which provides practical and detailed information on the research and development work ranging from the fundamentals of vibration damping to design aspects in many practical applications in this subject area. Our intent is that the contents of this book be considered as a continuation of and complementary to the above work rather than competitive in the technical subject of vibration damping.

This book is intended as a reference book for aerospace, mechanical, civil, and acoustical engineers. It should also serve as a valuable reference work for graduate students, professors, and researchers in the area of Aerospace Engineering, Mechanical Engineering, and Engineering Mechanics. This book consists of eight chapters. The first three chapters address the fundamentals of vibration damping, properties of viscoelastic damping materials, vibrations of discrete damped systems, and damping of fiber-reinforced composite materials. Chapters four to seven present vibrations of damped structures for beams, plates, rings, and shells. In chapter eight, a finite element numerical method is presented to solve vibration problems of beam and plate structures with a partially attached damping treatment on the surface of the structures. The effect of initial loading is also included.

Originally, a chapter to address the vibrations of damped cylindrical shells with curved elements was planned. A series of outstanding papers on this specific subject were published by Dr. Michael El-Raheb and his colleague Mr. Paul Wagner. However, the research work was performed for their specific purposes and no numerical solutions on the vibrations of shell structures without incorporating the enclosed fluid medium was available. We regret that these materials are unable to be included in this book. Instead, we have added paragraphs pertaining to this subject topic at the end of Chapter 7 which deals with vibrations of constrained damped cylindrical shell structures.

It should be pointed out that this book deals with vibration damping characteristics of structures or systems employing damping materials. We use the properties of the viscoelastic materials in the vibration analysis, but the detailed analysis regarding their material behaviors will not be our primary concern. The book addresses the vibration damping of structural elements, and is not a materials oriented book. This book emphasizes analyses in the presentation of damped structural systems, their validations and verifications. This is done because the authors feel that analyses are the tools which not only enable us to better understand the complicated physical phenomena, but also can help calculate the physical quantities which are useful in practical applications. One might be critical of the fact that there are not enough tables and figures which may be used directly and readily for design purposes. Our reply is that much of this information for damped beam structures may be found in the book by Nashif, Jones, and Henderson. Additionally, because information pertinent to damped structures other than beams may not be available and because the vibration characteristics of damped structures depend strongly on the realistic (not assumed) properties of the damping materials employed as well as the geometrical parameters of the structures considered, we strongly believe that the presentation of Òdesign dataÓ should be reduced to a minimum unless the geometrical and particularly the damping material parameters of a given damped structural system are specified. Since it is technically difficult to develop and to manufacture viscoelastic materials, the damping material properties can not be assumed for materials not commercially available.

Though vibration damping technology is multi-disciplinary, the researchers and practitioners have, however, formed a rather close-knit community. In preparing this book, we would be remiss if we did not acknowledge the fact that Dr. Lynn Rogers, currently a vibration damping consultant and formerly with the Flight Dynamics Laboratory of the Air Force Wright Aeronautical Laboratories, has for years not only devoted his efforts in promoting vibration technology tirelessly, but also has organized meetings and workshops on a regular basis to provide a forum for the exchange and dissimulation of the latest state-of-the-art technology. His professional and enthusiastic efforts in advancing vibration damping technology certainly deserve our recognition.

We are very happy to take advantage of this opportunity to acknowledge the support and encouragement provided by the managers and individuals of our respective organizations in the Department of Aerospace Engineering, Mechanics and Engineering Science at the University of Florida and at the Carderock Division, and the Naval Surface Warfare Center (formerly David Taylor Research Center). Particularly, we would like to thank Dr. J. M. Bai at the University of Florida, Dr. Bruce Douglas, Director of Research, and Mr. A. J. Roscoe, at the Carderock Division for their assistance, support, and invaluable comments. Finally, the support and cooperation from the staff of Prentice Hall Publication Company, especially our editor Mr. Michael Hays in all phases of the production process are also acknowledged.

C. T. Sun, Gainesville, Florida
Y. P. Lu, Annapolis, Maryland


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