Handbook on Stiffness & Damping in Mechanical Design

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1 Handbook on Stiffness & Damping in Mechanical Design

3 Handbook on Stiffness & Damping in Mechanical Design Eugene I. Rivin

4 2010 by ASME, Three Park Avenue, New York, NY 10016, USA ( All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. Information contained in this work has been obtained by the American Society of Mechanical Engineers from sources believed to be reliable. However, neither ASME nor its authors or editors guarantee the accuracy or completeness of any information published in this work. Neither ASME nor its authors and editors shall be responsible for any errors, omissions, or damages arising out of the use of this information. The work is published with the understanding that ASME and its authors and editors are supplying information but are not attempting to render engineering or other professional services. If such engineering or professional services are required, the assistance of an appropriate professional should be sought. ASME shall not be responsible for statements or opinions advanced in papers or... printed in its publications (B7.1.3). Statement from the Bylaws. For authorization to photocopy material for internal or personal use under those circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, tel: , Requests for special permission or bulk reproduction should be addressed to the ASME Publishing Department, or submitted online at: Administration/Permissions.cfm ASME Press books are available at special quantity discounts to use as premiums or for use in corporate training programs. For more information, contact Special Sales at infocentral@asme.org Library of Congress Cataloging-in-Publication Data Rivin, Eugene I. Handbook on stiffness & damping in mechanical design / by Eugene I. Rivin. p. cm. Includes bibliographical references and index. ISBN Engineering design. 2. Dynamics, Rigid. 3. Damping (Mechanics) I. Title. TA174.R dc

5 Contents Foreword xvii Chapter 1 Introductions and Definitions Basic Notions Stiffness Damping Influence of Stiffness and Damping on Strength and Length of Service Influence of Stiffness on Uniformity of Stress Distribution Influence of Stiffness and Damping on Vibration/Dynamics 1.3 Negative Stiffness and Damping Elastic Instability Stick-Slip Mechanical Linkages Mechanisms with Nonlinear Position Functions Electromechanical Systems Stiffness and Damping of Cutting Process a Influence of Machining System Stiffness and Damping on Accuracy and Productivity 1.4 Stiffness and Damping of Some Widely Used Materials

6 vi contents 1.5 General Comments on Stiffness in Mechanical Design 50 References 51 Chapter 2 Modes of Loading and Stiffness of Structural Components Influence of Mode of Loading on Stiffness Influence of Beam Design on its Bending Stiffness and Damping Round versus Rectangular Cross Section Stiffness and Damping of Helically Patterned Tubular Beams Composite/Honeycomb Beams and Plates Torsional Stiffness Influence of Stress Concentrations Stiffness of Frame/Bed Components Local Deformations of Frame Parts General Comments on Stiffness Enhancement of Structural Components Stiffness-Critical Metal Elastic Elements (Springs) Coil Springs Slotted Springs Friction Springs Miniature Tabular Elastic Elements Utilizing Giant Superelasticity Effect 2.8 Static Deformation Characteristics of Quasi-Linear Rubber Elements Stiffness of Bonded Rubber Blocks 102 References 111

7 contents vii Chapter 3 Nonlinear and Variable Stiffness Systems; Preloading Definitions Constant Natural Frequency Nonlinear Characteristic 3.2 Embodiments of Mechanical Elements with Nonlinear Stiffness Material-Related Nonlinearity Geometry-Related Nonlinearity a Coil and Leaf Springs b Belleville (Disc) Springs Nonlinear Spring Elements with Softening Nonlinear Characteristics Statically Nonlinear Rubber Elements Compressed Elements with Controlled Bulging Streamlined Nonlinear Rubber Flexible Elements Thin-Layered Rubber-Metal Laminates Stiffness Management by Preloading (Strength-to-Stiffness Transformation) Embodiments of the Preloading Concept Antagonist Actuators Preloaded Flexible Elements with Variable Stiffness Some Dynamic Effects Caused by Variable Stiffness Assembled Frame-Like and Beam-Like Structures Integrity of Assembled Structures a Integrity of an Assembled Beam b Integrity of an Assembled Preloaded Structure 163

8 viii contents Stiffness of Assembled Structures Dynamics of Assembled Structures 172 References 175 Chapter 4 Contact (Joint) Stiffness and Damping Introduction Contact Deformations Between Non-Conforming Surfaces 4.3 Contact Deformations Between Conforming and Quasi-Comforming Surfaces Contact Stiffness in Structural Analysis Quasi-Conforming Contact Deformations in Cylindrical/Conical Connections Cylindrical Connections a Connections with Clearance Fits b Interference-Fit Cylindrical Connections Elastic Displacements in Conical (Tapered) Connections a Test Data b Computational Evaluation of Contact Deformations in Tapered Connections c Influence of Manufacturing Errors d Finite Element Modeling of 7/24 Taper Connection e Short Taper Connections f Some General Comments on Tapered 221 Connections 4.6 Tangential Contact Compliance Experimental Study of Tangential Compliance of Flat Joints 224

9 contents ix Dynamic Model of Tangential Compliance and Damping Practical Case: Study of a Modular Tooling System Damping of Mechanical Contacts Damping in Flat Joints Damping in Cylindrical and Tapered Connections Energy Dissipation in Power Transmission Components References 251 Chapter 5 Supporting Systems/Foundations Influence of Support Characteristics Rational Location of Supporting/ Mounting Elements Overconstrained (Statically Indeterminate) Systems Influence of Foundation on Structural Deformations General Considerations Machines Installed on Individual Foundations or on Floor Plate Deformations of Long Machine Bases 288 References 292 Chapter 6 Stiffness and Damping of Power Transmission Systems and Drives Basic Notions Compliance of Mechanical Power Transmission and Drive Components Basic Power Transmission Components a Stiffness of Ball Screws 307

10 x contents Compliance of Pneumatic System Components Compliance of Hydraulic System Components Dynamic Parameters of Electric Motors 316 (Actuators) 6.3 Parameter Reduction in Mathematical Models Practical Examples of Structural 329 Compliance Breakdown A Hydraulically Driven Robot Electromechanically Driven Robot 335 of Jointed Structure Electromechanically Driven Parallelogram 339 Robot with Harmonic Drive Electromechanically Driven Spherical 342 Frame Robot Summary More on Stiffness and Damping of Antifriction 348 Bearings and Spindles Stiffness of Spindles Stiffness and Damping of 353 Antifriction Bearings 6.6 Damping in Power Transmission Systems 364 References 369 Chapter 7 Design Techniques for Reducing Structural 371 Deformations (Stiffness Enhancement Techniques ) 7.1 Structural Optimization Techniques Compensation of Structural Deformations Passive Compensation Techniques Active (Servo-Controlled) Systems 389 for Stiffness Enhancement

11 contents xi 7.3 Stiffness Enhancement by Reduction of Stress Concentrations Strength-to-Stiffness Transformation Buckling and Stiffness Reverse Buckling Concept Stiffening of Slender Parts by Axial Tension during Machining Self-Contained Stiffness Enhancement Systems Temporary Stiffness Enhancement Techniques Performance Enhancement of Cantilever Components General Comments Stationary and Rotating Around Longitudinal Axis Cantilever Components Cantilever Components Rotating Around Transverse Axis a Solid Component b Combination Link Damping Enhancement Techniques Introduction Dampers Dynamic Vibration Absorbers 448 References 450 Chapter 8 Use of Managed Stiffness in Design Cutting Edge/Machine Tool Structure Interface Introduction Techniques for Reduction of Cutting Forces Influence of Stiffness and Damping in the Cutting Zone on Cutting Forces and Tool Life 461

12 xii contents Machining Systems with Intentionally Reduced Tool Stiffness 8.1.4a Cutting Tools with Reduced Normal Stiffness 8.1.4b Cutting Tools with Reduced Tangential Stiffness 8.1.4c Trading-off the Stiffness for Damping to Improve the Overall Machining Performance Stiffness of Clamping Devices Introduction General Purpose Clamping Devices Solid State Tool Clamping Devices Modular Tooling Tool/Machine Interfaces. Tapered Connections Managed Stiffness Connections to Reduce Friction-Induced Position Uncertainties /24 Steep Taper Connections Definition of the Problem Tapered Toolholder/Spindle Interfaces for Machine Tools. Practical Sample Cases Other Tapered and Geared Toolholder/ Spindle Interfaces Curvic Coupling Connection KM System HSK System Benefits of Intentional Stiffness Reduction in Design Components Hollow Roller Bearings Stiffness Reduction in Power Transmission Gears 523

13 contents xiii Stiffness Reduction of Chain Transmissions Compliant Bearings for High-Speed Rotors Constant Force (Zero Stiffness) Vibration Isolation Systems 8.7 Anisotropic Elastic Elements as Limited Travel Bearings (Flexures) Elastic Kinematic Connections (Flexures) a Elastic Connections for Rotational Motion 8.7.1b Elastic Connections for Translational Motion c Elastic Motion Transformers Elastic Kinematic Connections Using Thin-Layered Rubber-Metal Laminates 8.7.2a Rubber-Metal Laminates as Anisotropic Elastic Elements 8.7.2b Use of Rubber-Metal Laminates as Limited Travel Bearings c Wedge Mechanism s d Use of Rubber-Metal Laminates as Compensators Modification of Parameters in Dynamic Models Evaluation of Stiffness and Inertia Components in Multi-Degrees-of-Freedom Systems Modification of Structure to Control Vibration Responses References 567 Appendix 1 Single-Degree-of-Freedom Dynamic Systems with Damping 573 References 585

14 xiv contents Appendix 2 Stiffness/Damping/Natural Frequency Criteria 587 A2.1 Introduction 589 A2.2 Self-Excited Vibrations/Dynamic Stability Criterion 590 A2.3 Vibration Isolation of Mechanical Objects 596 A2.3.1 Influence of Isolation on Chatter Resistance 596 A2.3.2 Forced Vibrations 599 A2.3.2a Vibration Level Criteria 599 A2.3.2b Vibration Isolation Criteria for Vibration- Sensitive Objects 601 A2.4 Use of Stiffness-Damping Criteria 603 A2.5 Discussion 605 References 606 Appendix 3 Influence of Axial Force on Beam Vibrations 607 Reference 611 Appendix 4 Characteristics of Elastomeric (Rubberlike) Materials 613 A4.1 Basic Notions 615 A4.2 Static Deformation Characteristics of Rubberlike Materials 618 A4.3 Elastic Stability of Rubber Parts 623 A4.4 Dynamic Characteristics of Rubberlike Materials 627 A4.5 Fatigue Resistance of Elastomeric Elements 633 A4.6 Creep of Rubberlike Materials 638 References 640

15 contents xv Appendix 5 Power Transmission Couplings 643 A5.1 Introduction 645 A5.2 General Classification of Couplings 645 A5.3 Rigid Couplings 647 A5.4 Misalignment-Compensating Couplings 649 A5.5 Torsionally Flexible Couplings and Combination Purpose Couplings A5.5.1 Roles of Torsionally Flexible Coupling in Transmission A5.5.2 Compensation Ability of Combinations Purpose Couplings A5.5.3 Comparison of Combination Coupling Designs References 677 Appendix 6 Systems with Multiple Load-Carrying Components 679 A6.1 Introduction 681 A6.2 Load Distribution Between Rolling Bodies and Stiffness of Antifriction Bearings 681 A6.3 Loading of Spoked Wheels 685 A6.4 Analytical Solution for Bicycle Wheel 688 A6.5 Torsional Systems with Multiple Load-Carrying Connections 689 References 694 Appendix 7 Compliance Breakdown for a Cylindrical (OD) Grinder 695 Reference 706 About the Author 707 Index 709

17 FOREWORD The most important attributes of many a mechanical system/structure are strength, stiffness, and stability. Scores of professional books had been published on various aspects of strength which are important for designers, many books had been published on stability of mechanical and civil engineering structures. Also, the strength and stability issues are heavily represented in all textbooks on machine elements. On the other hand, the stiffness-related issues until recently were practically neglected, with a few exceptions. The author could not find any comprehensive professional publication addressing various important aspects of stiffness in mechanical design, besides his sold-out book on Stiffness and Damping in Mechanical Design (Marcel Dekker, 1999, 512 pp.) The present book is a result of its significant rework and expansion. The author has an extensive experience and expertise in conceptual mechanical design, design components, vibration control, use of elastomers in design, design of production equipment (machine tools and tooling systems, robotics), etc. This expertise touched many practical aspects of stiffness consideration in the design process. The author wants to think that this expertise allows him to write this Handbook on Stiffness and Damping in Mechanical Design. Also, these experiences led to two observations. The first is an observation that there is a need for a comprehensive monograph/handbook addressing various aspects of stiffness in mechanical design. The other observation is that in many important real life cases, wherein the dynamic behavior of the system is important, stiffness and damping attributes cannot and should not be separated. While dynamics as well as forced and self-excited vibrations of mechanical systems are becoming increasingly important, damping and stiffness are usually considered separately. However, frequently, damping and stiffness are closely interrelated, and efforts to improve one parameter while neglecting another are usually ineffective or even counterproductive. Computers are becoming more and more powerful tools assisting the design process. Finite Element Analysis (FEA) and other software packages constituting Computer-Aided Design (CAD) allow quick and realistic

18 xviii foreword visualization and optimization of stresses and deformations inside the component of a structure as well as in the whole structure. However, the results of such analyses are useful only if the adapted analytical models are correct. But they are often not correct, especially for complex systems with critical role of contact stiffness, parallel kinematics machines, etc., if the designer does not possess a broader view of the system. The computer technology, which frees the designer from the tedious drafting and computational chores, not only allows but also forces him to concentrate on general, conceptual issues of design. Some of these issues are so-called conceptual design, reliability, energy efficiency, accuracy, use of advanced materials in the appropriate parts of the system/structure. One of the most important conceptual issues is stiffness of mechanical structures and their components. The book cited above was intended to start correcting the absence of a comprehensive source on stiffness-related issues by addressing various aspects of structural stiffness and structural damping and their roles in design. Several typical cases in which stiffness is closely associated with damping had been addressed. Since stiffness, especially in interaction with damping, is a very large subject, only the basic conceptual issues related to stiffness had been presented, rather than detailed analytical techniques. The same approach is adopted in the present Handbook. A more detailed analytical treatment is given only in a few cases where the results were never published before or had been published in hard-to-obtain sources (e.g., in languages other than English). Many of these concepts are illustrated by practical results either in the text or in Appendices. The first book was based on materials prepared for Stiffness in Design tutorial successfully presented at several Annual Meetings of the American Society for Precision Engineering (ASPE). These materials are, in turn, based to a substantial degree on personal professional experiences and research results of the author. This book is advancing the concepts addressed in the first book on stiffness cited above. The author believes that its contents are much more comprehensive and justify adding the word Handbook in its title. Several conceptual issues are added, such as negative stiffness and damping; close-form analytical expressions for calculating stiffness of typical stiffness critical design elements, both metal springs and elastomeric elements; extensive introduction is given to static and dynamic properties of elastomeric materials; description of behavior of superelastic materials under structural (compressive) loading is added; etc. Interrelation of damp-

19 foreword xix ing with stiffness is addressed much more extensively than before. The important issues of stiffness and damping specific to Micro Electromechanical Systems (MEMS) and nanosystems are not specially addressed since there is not much published material and since they are very specific. The analytical techniques related to consideration of contact stiffness and to constructing 3-D models of effective stiffness of mechanisms, while useful for specific applications, will, undoubtedly, also direct the reader to development of analytical models for other mechanical systems. Many important stiffness- and damping-related issues were studied in-depth in the former Soviet Union. The results are still very relevant, but they were published in Russian and, practically, are not available to the engineering community in the non-russian speaking countries. Some of these results are reflected in the Handbook. A general introduction to the subject is given in Chapter 1. General design areas are described for which the stiffness criterion is critical. Practical embodiments of mechanical systems containing sources of negative stiffness and negative damping are described. It also addresses selection of structural materials for stiffness- and damping-critical applications. Information on influence of the mode of loading and of the component design on stiffness is provided in Chapter 2. Chapter 3 is dedicated to an important subject of nonlinear and variable stiffness systems. Specially addressed is an important issue of preloading. An interesting class of nonlinear elastomeric elements is described, including elements whose preloading does not significantly increase resistance to limited travel motions. Design and performance information on various aspects of normal and tangential contact stiffness, as well as of damping associated with mechanical contacts, is given in Chapter 4. Information on these subjects is very scarce in the English language technical literature. Some important issues related to influence on stiffness of mechanical components by their supporting conditions and devices, as well as by foundations for machines, are addressed in Chapter 5. Chapter 6 concentrates on very specific issues of stiffness (and damping) in power transmission and drive systems which play a significant role in various mechanical systems. Numerous useful design techniques, both passive and active, aimed on enhancing structural stiffness (i.e., reducing structural deformations) are described In Chapter 7.

20 xx FOREWORD Special cases in which performance of stiffness-critical system can be improved by reduction or a proper tuning of the components stiffness are described in Chapter 8. A special attention is given here to tooling sub-systems/structures in machining systems. The Handbook also contains seven Appendices. Appendix 1 gives introductory information on dynamic systems with non-viscous (hysteretic) damping. While the viscous damping model is very convenient for analytical studies of linear dynamic systems, it does not represent real life systems. The hysteretic damping model more properly represents real dynamic systems but, unfortunately, only seldom described in vibration textbooks. Appendix 2 presents the first attempt to consider stiffness and damping in their interrelation, as useful practical criteria. Appendix 3 illustrates a technique complementary to similar techniques addressed in Chapter 7 for increasing effective stiffness of mechanical components (shafts) without changing their material or geometry. Appendix 4 describes basic mechanical characteristics of elastomeric (rubber-like) materials, whose wider use in mechanical design is hampered by difficulties in finding such information in mechanical design literature. Appendix 5 provides principles and criteria for designing and selection of power transmission couplings, important elements of power transmission systems. The proposed analytical approach is expected to help designers of power transmission systems in selecting optimal couplings from a huge variety of coupling designs on the market. Some important issues related to systems with multiple load-carrying components are addressed in Appendix 6. Appendix 7 describes in detail a practical case of constructing a compliance breakdown for a complex precision mechanical system - an OD grinder. The issues related to the Stiffness in Design topic are numerous and very diverse. This Handbook does not pretend to cover all the issues related to stiffness and damping in mechanical design, but it expands significantly compared with the book cited above. It is expected that the book will be useful not only to mechanical designers, especially ones working with precision devices, but also to vibration and dynamic specialists. I hope that instructors in the mechanical engineering field will use numerous examples in the book in the teaching process.

Mechanical Design. The Elements of. by James G. Skakoon. ASME Press New York 2008

Mechanical Design. The Elements of. by James G. Skakoon. ASME Press New York 2008 The Elements of Mechanical Design by James G. Skakoon ASME Press New York 2008 2008 by ASME, Three Park Avenue, New York, NY 10016, USA (www.asme.org) All rights reserved. Printed in the United States