Covering all major aspects of optomechanical engineering - from conceptual design to fabrication and integration of complex optical systems - this handbook is comprehensive. The practical information within is ideal for optical and optomechanical engineers and scientists involved in the design, development and integration of modern optical systems for commercial, space, and military applications. Charts, tables, figures, and photos augment this already impressive handbook.
The text consists of ten chapters, each authored by a world-renowned expert. This unique collaboration makes the Handbook a comprehensive source of cutting edge information and research in the important field of optomechanical engineering. Some of the current research trends that are covered include:. Proceedings of the 34th IMAC, A Conference and Exposition on Dynamics of Multiphysical Systems: From Active Materials to Vibroacoustics, , the tenth volume of ten from the Conference brings together contributions to this important area of research and engineering.
Structural Vibration Author : C. In this book the entire range of methods of control, both by damping and by excitation, is described in a single volume. Clear and concise descriptions are given of the techniques for mathematically modelling real structures so that the equations which describe the motion of such structures can be derived. This approach leads to a comprehensive discussion of the analysis of typical models of vibrating structures excited by a range of periodic and random inputs.
Careful consideration is also given to the sources of excitation, both internal and external, and the effects of isolation and transmissability. A major part of the book is devoted to damping of structures and many sources of damping are considered, as are the ways of changing damping using both active and passive methods.
The numerous worked examples liberally distributed throughout the text, amplify and clarify the theoretical analysis presented. Particular attention is paid to the meaning and interpretation of results, further enhancing the scope and applications of analysis. Over 80 problems are included with answers and worked solutions to most. This book provides engineering students, designers and professional engineers with a detailed insight into the principles involved in the analysis and damping of structural vibration while presenting a sound theoretical basis for further study.
Suitable for students of engineering to first degree level and for designers and practising engineers Numerous worked examples Clear and easy to follow. This book provides a thorough explanation of the principles and methods used to analyse the vibrations of engineering systems, combined with a description of how these techniques and results can be applied to the study of control system dynamics.
Numerous worked examples are included, as well as problems with worked solutions, and particular attention is paid to the mathematical modelling of dynamic systems and the derivation of the equations of motion. All engineers, practising and student, should have a good understanding of the methods of analysis available for predicting the vibration response of a system and how it can be modified to produce acceptable results.
Grllages and Stffened Plates General Case , Cylindrical Shells StiainDeformation Relationships Infinitely Long Cylindrical Shells Other Boundary Conditions Eifect of Axial Constraint Acoustic Cavities Ships and Floating Structures Introduction Added Mass of Coss Sections and Bodies Shells, Internal and External Pipe Flow Solids Liquids The added mass of many slender structures is comparable to the mass of fluid displaced by the structure.
See Chapter Beam-—A structure whose cross-sectional properties and deflection vary along only fa single axis. A slender beam is a beam whose characteristic cross-sectional dimen- sions are much less than the span of the beam and the distance between vibration nodes; therefore, the inertia associated with local rotation is overshadowed by the inertia developed in displacement and the deformation due to shearing of the cross, section is overshadowed by bending deformations.
Boundary Condition—A constraint applied to a structure independent of time. Boundary conditions can be clasified as either geometric or kinetic.
Geometric boundary conditions arise from geometric constraints. For example, the displace- ment of a structure at a joint pinned to a rigid wall is zero. Kinetic boundary conditions arise from force or moments applied to a structure; for example, a pinned joint permits free rotation, so the kinetic boundary condition at a pinned joint is zero moment.
Bulk Modulus of Flasticity—The ratio of the tensile or compressive stress, equal in all directions i. Definitions of symbols are given in Chapter 2. Cable—A massive string, A uniform, massive one-dimensional structure which can bear only tensile loads parallel to its own axis.
The bending rigidity of cables is zero, Cables, unlike chains, may stretch in response to tensile loads. Cable Modulus—The rate of change in the longitudinal stress axial force over cross- sectional area in a cable for a small unit longitudinal strain. If the cable is a solid elastic rod, the cable modulus will be equal to the modulus of elasticity of the rod material. If the cable is woven from fibers, the cable modulus will be less than the modulus of elasticity of the component fibers.
Center of Gravity—The point on which a body can be balanced. See Chapter 5. Centroid—The geometric center of a plane area. The sum over a plane area of all clements of area multiplied by the distance from any axis through the centroid is zero, See Chapter 5. Chain-A uniform, massive one-dimensional structure which can bear only tensile oads parallel to its own axis, The bending rigidity of chains is zero.
Chains, unlike cables, do not stretch in response o tensile loads. Damping—The ability of a structure to absorb vibrational energy.
Density—The mass per unit volume of a materia Elastic—A term applied to a material if deformations of the material increase linearly with increasing load without regard to the sign or magnitude of the load.
For example, the tip of freely vibrating cantilever isa free boundary. Isotropic—A term applied to a material whose properties are unchanged by rotation of the axis of measurement. Only two elastic constants, the modulus of elasticity B and Poisson's ratio v , are required to completely specify the elastic behavior of an isotropic material Linear—A term applied to a structure or material if all deformations inerease in pro- portion to the load without regard to the sign, magnitude, distribution, or direction Of the load.
A membrane may be flat like a drum head or curved like soap bubble. A one-dimensional membrane is a cable. Mode Shape Eigenvector —A function defined over a structure which describes the relative displacement of any point on the structure as the structure vibrates in asingle mode. A mode shape is associated with each natural frequency of a structure. The modulus of elasticity has units of pres- sure.
For most materials, within the limits of linear elasticity, the modulus of elasticity is independent of the sign of the applied stress. Some materials such 9s wood, have a directional modulus of elasticity. See Product of Inertia of a Body and Chapter 5.
Moment of Inertia of a Section—The sum of the products obtained by multiplying cach clement of area within a section by the square of its distance from a given axis.
See Product of Inertia af a Section and Chapter 5. The lowest of these is called the fundamental natural frequency. Each natural frequency is associated with a mode shape of deformation. Natural frequency can be defined either in terms of cycles per second hertz or radians per second. There are 2x radians per cycle. Neutral Axis—The axis of zero stress in the cross section of a structure. Anti-node is a point on a structure where deflection is maximum during vibration in a given mode.
Orthotropic—A term applied to a thin lamina if the material properties of the lamina possess two mutually perpendicular planes of symmetry. Four material constants are required to specify the elastic behavior of an orthotropic lamina, Common examples of orthotropic lamina are sheets of fiber-reinforced plastic or the thin plys of wood that are glued together to form plywood.
Pinned Boundsry—A boundary condition such that the structure is free to rotate but not displace along a given boundary. Poisson's Ratio—The ratio of the lateral shrinkage expansion to the longitudinal expansion shrinkage of a bar of a given material which has been placed under a tuniform longitudinal tensile compressive load.
Poisson's ratio is ordinarily neat 0. Product of Inertia of a Body—The sum of the products obtained by multiplying each clement of mass of a body by the distances from two mutually perpendicular axes. Radius of Gyration of a Section—The square root of the quantity formed by divid- ing the erea moment of inertia of a section by the area of the section.
See Chapter 5 Rotary Inertia—The inertia associated with local rotation of astructure, For example, the rotary inertia of a spinning top maintains its rotation. Seiching—The system of waves in a harbor which is produced as the harbor responds sympathetically to waves in the open sea also see Sloshing..
Shear Beam—A. See Section 8. For most materials the shear modulus is independent of thesign of the applied stress, although some materials, such as wood, may have a directional shear modulus. Shell—A thin clastic structure whose material is confined to the close vicinity of a curved surface, the middle surface of the shell.
A curved plate is «shell. A shell without rigidity in bending is a membrane. Sliding Boundary -A boundary condition such that a structure is fee to displace in a siven direction along a boundary but rotation is prevented. Slshing The «stem of surface wares formed ina lq tank or bana the liquid is excited, a m Speed of Sound—The speed at which very mal pret uctustons propagate ina infinite fluid or solid.
A string is a massless cable. In some cases special symbols have been defined. The symbols listed be- Tow have been consistently applied in all cases. These symbols generally follow those used in the literature. One exception is that here is used to denote all area moments of inertia of seetions and J is used to denote all mass moments of inertia af bodies. For example, if the kilogram is chosen as the unit of force and the unit of accelera- fone centimeter per second per second, then the unit of mass must be such that one kilogram of force will accelerate it at one centimeter per second per second.
Note that it is wrong. These errors can be avoided by using any of the consistent sets of units presented in Table It may be helpful to remember that a smallish apple weighs about one newton. One au of mas weighs The natural frequencies are the result of cyclic exchanges of kinetic and potential energy within the structure. The kinetic energy is associated with velocity of structural mass, while the potential eneray is associated with storage of nergy in the elastic deformations of a resilient structure.
Just asa ball bouncing off 4 hard-wood floor exchanges the potential energy at the apex of its flight for kinetic energy as it plummets, so an elastic structure exchanges the potential energy of elastic deformation for velocity of vibration as it vibrates back and forth.
The rate of energy exchange between the potential and kinetic forms of energy is the natural frequency. Ifa structure is linear, that is, its deformation is propartional to the load it bears regardless of the magnitude, distribution, or direction of load, and has constant mass, then it can be shown that the natural frequency of the structure is independent of the amplitude of vibration.
For example, consider the spring-body system shown in Fig. Mis he mast. IF we assume thatthe spring i massless and the body is perfectly sei then all the potential eneray in the system is asoeated with sprig deformation and al the Kinetic enery inthe system is sociated with velocity of the mas. Sines the system is free of external forees, the total energy of the system is constant. The natal were the natural frequency has wits of eles per nit tine Ne stifles of te spring snd weney of the spring-body syste inereaes withthe si ccheaue: with increasing mos ofthe body.
The natural frequency of the spring-body system is independent of mean deformation due to gravity. There is always some damping in real structures which will make free vibrations decay with time, and there is some amplitude beyond which the structure no longer behaves linearly. For real structures the concept of natural frequency must be tem- pered by some knowledge of the differences between the ideal mathematical model and the actual structure.
Some linear effects which are often neglected are the effect of shearing deformation in slender structures and the effect of surrounding fluid. Nonlinear effects which are often neglected are plasticity due to yielding and the amplitude dependence of damping. For example, if the edge of a plate is riveted at broad intervals to a heavy beam, then the boundary condition on the plate is probably intermediate between a clamped and a pinned edge.
These programs are replacing approximate methods of analysis of complex structures. Finite element programs have not replaced closed form solutions because a frequency can be calculated from a closed form result in a matter of minutes, while computer programs generally require hours to set up and run successively. C marks the centroid of the area, which is located at co- dinates x and ye with respect to the x-y axis. Ixc, Tye: legs and Igone. These axes are called the principal faxes.
An axis of symmetry is always a principal axis, The produet of inertia is zero for any two axes, one of which is a principal axis. If the principal moments of inertia are equal, then the moments of inertia about any rotated axis through the origin of the principal axes are equal to the principal moments of inertia. Properties of Plane Sections. The x-y C marks the i center of gravity of the body and the origin of the coordinate system x'-y'-2'. Continued lS, Fi.
Soi boy. Properties of Homogensous Bodies. These axes ae called he principal axes ofthe body. The SRentation of the principal axco can generally be found by setting the products of space tozero and solving fr the vector coeiients , m,n.
However, any rec ot aymmetry must bea principal axis. Spring Stftness. Spring Stifnes. Spring Stifiness. Continued Table Spring Stifness. Mass, Spring Systams. For this system Table , frame 2 ives: , Tables , , and present natural frequencies and mode shapes of point rmase-spring systems, rigid body-torsion systems, and pendulum systems.
These sys tems possess one natural frequency for each mass in the system and there is a unique mode shape for each natural frequency. The majority of the formulas presented in these tables were developed by writing the linear equations of motion for each sys tem and then solving these equations exactly.
An example of this analysis is pre sented in the following paragraphs. Approximate techniques are discussed in Refs. References and are particularly rich in example problems. References through provide additional background material, 6. If Eq. For vibration at the lowest natural frequency, on x ean and the two masses move in the same direction simultaneously. For vibration at the second natural frequency, Sa and the two masses move in opposite directions.
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