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Advanced Structured Materials Farzad Hejazi Tan Kar Chun Conceptual Theories in Structural Dynamics Advanced Structured Materials Volume 135 Series Editors Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of Applied Sciences, Esslingen, Germany Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, Portugal Holm Altenbach , Faculty of Mechanical Engineering, Otto von Guericke University Magdeburg, Magdeburg, Sachsen-Anhalt, Germany Common engineering materials reach in many applications their limits and new developments are required to fulfil increasing demands on engineering materials. The performance ofmaterials can beincreasedby combiningdifferent materials to achieve better properties than a single constituent or by shaping the material or constituents in a specific structure. The interaction between material and structure mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers possible applications in quite diverse fields. Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic etc.) and applications. The topics of Advanced Structured Materials include but are not limited to (cid:129) classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced plastics) (cid:129) metal matrix composites (MMCs) (cid:129) micro porous composites (cid:129) micro channel materials (cid:129) multilayered materials (cid:129) cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere structures) (cid:129) porous materials (cid:129) truss structures (cid:129) nanocomposite materials (cid:129) biomaterials (cid:129) nanoporous metals (cid:129) concrete (cid:129) coated materials (cid:129) smart materials Advanced Structured Materials is indexed in Google Scholar and Scopus. More information about this series at http://www.springer.com/series/8611 Farzad Hejazi Tan Kar Chun (cid:129) Conceptual Theories in Structural Dynamics 123 Farzad Hejazi TanKar Chun Department ofCivil Engineering Department ofCivil Engineering University PutraMalaysia University PutraMalaysia Serdang,Selangor, Malaysia Serdang,Selangor, Malaysia ISSN 1869-8433 ISSN 1869-8441 (electronic) AdvancedStructured Materials ISBN978-981-15-5439-1 ISBN978-981-15-5440-7 (eBook) https://doi.org/10.1007/978-981-15-5440-7 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSingapore PteLtd.2020 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Static and Dynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Degree of Freedom (DOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Components of Structural Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Mass Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Stiffness Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Damping Component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 Idealized Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Free Vibration of Single Degree of Freedom System . . . . . . . . . . . . 29 3.1 Free Vibration of Undamped System . . . . . . . . . . . . . . . . . . . . . 29 3.2 Free Vibration of Damped System. . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Critical Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Overdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.3 Underdamping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 Logarithmic Decremental. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 Force Vibration of Single Degree of Freedom System. . . . . . . . . . . . 61 4.1 Force Vibration of Undamped System . . . . . . . . . . . . . . . . . . . . 62 4.2 Force Vibration of Damped System . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Dynamic Magnification Factor. . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3.1 Frequency Ratio Nearly Equal to Zero ðbffi0Þ . . . . . . . . 83 4.3.2 Frequency Ratio Is Much Greater Than One ðb(cid:3)1Þ. . . . 84 4.3.3 Frequency Ratio Is Equal to One ðb¼1Þ . . . . . . . . . . . . 86 v vi Contents 4.4 Dynamic Response Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5 Single Degree of Freedom Systems Subjected to Various Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1.1 Harmonic Excitation by Generator Machine. . . . . . . . . . . 95 5.1.2 Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.1.3 Harmonic Motion of Foundation (Motion Excitation) . . . . 100 5.1.4 Force Transmissibility (Force Excitation). . . . . . . . . . . . . 106 5.1.5 Vibration Measurement Instrument . . . . . . . . . . . . . . . . . 110 5.2 Impact Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2.1 Duhamel’s Integral for Undamped System. . . . . . . . . . . . 112 5.2.2 Constant Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2.3 Rectangular Load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.2.4 Triangular Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.3 Earthquake Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.3.1 Spectra Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.2 Base Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.4 Numerical Evaluation of Dynamic Response . . . . . . . . . . . . . . . 133 5.4.1 Central Difference Method (CDM) . . . . . . . . . . . . . . . . . 133 5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6 Multi-degree of Freedom System . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.1 Equation of Motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Multi-degree of Freedom for Undamped System Under Free Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3 Normalization of Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 Orthogonality of Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.5 Multi-degree of Freedom System Subjected to Impact Load . . . . 164 6.6 Multi-degree of Freedom System Subjected to Earthquake Excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.7 Iterative Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.8 Rayleigh’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.9 Holzer’s Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 6.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 Appendix. .... .... .... .... ..... .... .... .... .... .... ..... .... 213 References.... .... .... .... ..... .... .... .... .... .... ..... .... 219 List of Figures Fig. 1.1 An at rest pendulum.. ..... .... .... .... .... .... ..... .. 2 Fig. 1.2 A falling pendulum... ..... .... .... .... .... .... ..... .. 2 Fig. 1.3 Change of dead load with respect to time... .... .... ..... .. 3 Fig. 1.4 Degree offreedom of a body .... .... .... .... .... ..... .. 4 Fig. 1.5 Spring–mass system in simple harmonic motion.. .... ..... .. 5 Fig. 2.1 Concept of point mass ..... .... .... .... .... .... ..... .. 8 Fig. 2.2 Reinforce concrete as a heterogeneous material .. .... ..... .. 8 Fig. 2.3 Spring under tension.. ..... .... .... .... .... .... ..... .. 10 Fig. 2.4 Beam under bending.. ..... .... .... .... .... .... ..... .. 11 Fig. 2.5 Deformation of two member in different length .. .... ..... .. 12 Fig. 2.6 Beams deflection. .... ..... .... .... .... .... .... ..... .. 12 Fig. 2.7 Damping of spring-mass system.. .... .... .... .... ..... .. 13 Fig. 2.8 Rotation at joint in frame ... .... .... .... .... .... ..... .. 14 Fig. 2.9 Single concrete member response to movement .. .... ..... .. 15 Fig. 2.10 A functioning viscous damper.... .... .... .... .... ..... .. 16 Fig. 2.11 Structural motion as response to external force... .... ..... .. 17 Fig. 2.12 Symbol of dynamic system components.... .... .... ..... .. 18 Fig. 2.13 Series system ... .... ..... .... .... .... .... .... ..... .. 19 Fig. 2.14 Parallel system .. .... ..... .... .... .... .... .... ..... .. 20 Fig. 2.15 Example 2.1 .... .... ..... .... .... .... .... .... ..... .. 22 Fig. 2.16 Example 2.2 .... .... ..... .... .... .... .... .... ..... .. 22 Fig. 2.17 Example 2.3 .... .... ..... .... .... .... .... .... ..... .. 23 Fig. 2.18 Example 2.4 .... .... ..... .... .... .... .... .... ..... .. 24 Fig. 2.19 Solution for Example 2.4 ... .... .... .... .... .... ..... .. 26 Fig. 2.20 Exercise 2.1 .... .... ..... .... .... .... .... .... ..... .. 26 Fig. 2.21 Exercise 2.2 .... .... ..... .... .... .... .... .... ..... .. 27 Fig. 2.22 Exercise 2.3 .... .... ..... .... .... .... .... .... ..... .. 27 Fig. 2.23 Exercise 2.4 .... .... ..... .... .... .... .... .... ..... .. 28 Fig. 3.1 Analytical models for SDOF system... .... .... .... ..... .. 30 Fig. 3.2 Idealization of structure with a base isolation system .. ..... .. 30 vii viii ListofFigures Fig. 3.3 Analytical models for undamped SDOF system under free vibration... .... .... ..... .... .... .... .... .... ..... .. 31 Fig. 3.4 Displacement response of undamped system under free vibration... .... .... ..... .... .... .... .... .... ..... .. 34 Fig. 3.5 Example 3.1 .... .... ..... .... .... .... .... .... ..... .. 35 Fig. 3.6 Displacement, velocity and acceleration response for the frame in Example 3.1 .... .... .... .... .... ..... .. 37 Fig. 3.7 Example 3.2 .... .... ..... .... .... .... .... .... ..... .. 38 Fig. 3.8 Example 3.3 .... .... ..... .... .... .... .... .... ..... .. 39 Fig. 3.9 Analytical models for damped SDOF system under free vibration... .... .... ..... .... .... .... .... .... ..... .. 41 Fig. 3.10 Displacementresponseofdampedsystem under free vibration (critical damping condition) . .... .... .... .... .... ..... .. 44 Fig. 3.11 Displacementresponseofdampedsystem under free vibration (critical and overdamping condition)... .... .... .... ..... .. 45 Fig. 3.12 Displacement response of damped system under free vibration.... .... ..... .... .... .... .... .... ..... .. 49 Fig. 3.13 Example 3.4 .... .... ..... .... .... .... .... .... ..... .. 50 Fig. 3.14 Dynamic response for Example 3.4 ... .... .... .... ..... .. 52 Fig. 3.15 Undamped system under free vibration. .... .... .... ..... .. 53 Fig. 3.16 Exact and approximated solution of logarithmic decrement .. .. 55 Fig. 3.17 Example 3.5 .... .... ..... .... .... .... .... .... ..... .. 57 Fig. 3.18 Exercise 3.1 .... .... ..... .... .... .... .... .... ..... .. 58 Fig. 3.19 Exercise 3.2 .... .... ..... .... .... .... .... .... ..... .. 58 Fig. 3.20 Exercise 3.3 .... .... ..... .... .... .... .... .... ..... .. 59 Fig. 3.21 Exercise 3.4 .... .... ..... .... .... .... .... .... ..... .. 59 Fig. 4.1 Analytical models for undamped SDOF system under force vibration... .... ..... .... .... .... .... .... ..... .. 62 Fig. 4.2 Example 4.1 .... .... ..... .... .... .... .... .... ..... .. 70 Fig. 4.3 Analytical models for damped SDOF system under force vibration... .... .... ..... .... .... .... .... .... ..... .. 72 Fig. 4.4 Trigonometry relationship for 2nb and 1(cid:4)b2 in sine function.. .... ..... .... .... .... .... .... ..... .. 76 Fig. 4.5 Trigonometry relationship for 2nb and 1(cid:4)b2 in cosine function.... .... .... ..... .... .... .... .... .... ..... .. 81 Fig. 4.6 Damping magnification factor for various frequency ratio and damping ratio..... .... .... .... .... .... ..... .. 83 Fig. 4.7 Corresponding response for B nearly equal to 0.. .... ..... .. 84 Fig. 4.8 Corresponding response for B greater than 1 .... .... ..... .. 86 Fig. 4.9 Correlation between frequency ratio and phase angle.. ..... .. 87 Fig. 4.10 Example 4.2 .... .... ..... .... .... .... .... .... ..... .. 90 Fig. 4.11 Example 4.3 .... .... ..... .... .... .... .... .... ..... .. 91 Fig. 4.12 Exercise 4.1 .... .... ..... .... .... .... .... .... ..... .. 93 Fig. 4.13 Example 4.2 .... .... ..... .... .... .... .... .... ..... .. 93 ListofFigures ix Fig. 5.1 Vibration generated by a rotating machine .. .... .... ..... .. 96 Fig. 5.2 Example 5.1 .... .... ..... .... .... .... .... .... ..... .. 98 Fig. 5.3 Vibration isolation for a force and b motion excitations..... .. 99 Fig. 5.4 Analytical model of ground harmonic motion.... .... ..... .. 100 Fig. 5.5 Trigonometry relationship for cxf and k under motion excitation .. .... .... ..... .... .... .... .... .... ..... .. 101 Fig. 5.6 Transmissibility ratio of structure system ... .... .... ..... .. 103 Fig. 5.7 Example 5.2 .... .... ..... .... .... .... .... .... ..... .. 104 Fig. 5.8 Analytical model offorce transmissibility... .... .... ..... .. 106 Fig. 5.9 Trigonometry relationship for cxf and k under force excitation.. .... ..... .... .... .... .... .... ..... .. 107 Fig. 5.10 Example 5.3 .... .... ..... .... .... .... .... .... ..... .. 108 Fig. 5.11 Analytical model for vibration measurement. .... .... ..... .. 110 Fig. 5.12 General load function as impulsive loading . .... .... ..... .. 112 Fig. 5.13 Mass–spring system acted upon by constant load. .... ..... .. 114 Fig. 5.14 Dynamic response of undamped system due to sudden applied constant force. ..... .... .... .... .... .... ..... .. 115 Fig. 5.15 Rectangular constant force acting on mass–spring system.... .. 116 Fig. 5.16 Triangular dynamic force acted upon undamped oscillator... .. 118 Fig. 5.17 Maximumdynamicfactorforrectangularandtriangularforces (Paz and Leigh 2004). ..... .... .... .... .... .... ..... .. 123 Fig. 5.18 Example 5.4 .... .... ..... .... .... .... .... .... ..... .. 124 Fig. 5.19 Analytical model of ground motion caused by acceleration excitation .. .... .... ..... .... .... .... .... .... ..... .. 126 Fig. 5.20 Response spectra for elastic system based on El Centro earthquake recorded in year 1940 (Paz and Leigh 2004) .... .. 130 Fig. 5.21 Example 5.5 .... .... ..... .... .... .... .... .... ..... .. 132 Fig. 5.22 Central difference method (CDM). .... .... .... .... ..... .. 134 Fig. 5.23 Example 5.6 .... .... ..... .... .... .... .... .... ..... .. 137 Fig. 5.24 Exercise 5.1 .... .... ..... .... .... .... .... .... ..... .. 140 Fig. 5.25 Exercise 5.2 .... .... ..... .... .... .... .... .... ..... .. 140 Fig. 5.26 Exercise 5.3 .... .... ..... .... .... .... .... .... ..... .. 141 Fig. 5.27 Exercise 5.4 .... .... ..... .... .... .... .... .... ..... .. 141 Fig. 6.1 Analytical models for MDOF system .. .... .... .... ..... .. 144 Fig. 6.2 Three-storey building and its MDOF analytical model . ..... .. 144 Fig. 6.3 Undamped 2-DOF system under free vibration... .... ..... .. 146 Fig. 6.4 Normal modes of vibration.. .... .... .... .... .... ..... .. 150 Fig. 6.5 Two-storey building as undamped 2-DOF system under free vibration.... .... ..... .... .... .... .... .... ..... .. 151 Fig. 6.6 Example 6.1 .... .... ..... .... .... .... .... .... ..... .. 156 Fig. 6.7 Idealization of structure for Example 6.1 ... .... .... ..... .. 157 Fig. 6.8 Normal modes for structure in Example 6.1. .... .... ..... .. 164 Fig. 6.9 2-DOF system subjected to impact load .... .... .... ..... .. 164 Fig. 6.10 Example 6.2 .... .... ..... .... .... .... .... .... ..... .. 168

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