Mode shap
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This document discusses various topics related to structural dynamics and soil-structure interaction, including: 1) Mode shapes and how they define the collective behavior of masses in a system. 2) Methods for analyzing dynamically loaded structures, including the frequency-domain method and using Ritz vectors. 3) Factors that influence soil-structure interaction like soil material damping, frequency-dependent stiffness and damping, and kinematic and inertial interactions between structures and soil. 4) References commonly used in the field like textbooks by Chopra and lecture materials from various universities.
Mode shap
- 1. El Desarrollo de Formas de Modos { Knowledge is extremely vast and profound, when it is within the realm of the primordial form.
- 2. Nuclear power is a global response to restoration of condemned and ancient energies upon our earth, at minute concentrations, and can be restorative of vast legions of conduct within our planet’s system. For the effect, this is the realm of conscience I have been developing in my considerations for the past umpteen years from the precisions of mind and mind-body science. Reviewing the manner with which individual generates well-being, with the harnessing of positive and negative conductivity; while the fashion of Power, for consumption of resources can be reduced, the overall manner with which society generates life and livelihood are two separate matters. ~July 23, 2014~ 106°F A Return on the effects of Nuclear Power
- 3. A mode shape is a unity definition of a directional action composing the joined behavior of masses in a system. The derivation of the mode shape first investigates the overall momentum of the system, the structures easiest response. What is a mode shape?
- 4. Determine applied forces Determine Stiffness Matrices Determine Mass Matrix Establish Equations of Motion Apply Boundary Conditions Principle System Characteristics
- 5. 휑푛is the natural mode corresponding to 휔푛, having elements 휑푗푛 F is called the modal matrix for the eigenvalue problem and is equal to 휑푗푛 W is a spectral matrix Defining “F”
- 6. Factor Γ푛 that multiplies p(t) Referred to as a Modal Participation Factor Is a measure of the degree to which 퓃푡ℎ mode participates Γ푛 is not independent of how the mode is normalized ℇ푞. 1.5.2 : is not independent of how the mode is normalized 퓊 is replaced by Modal Participation Factor
- 7. ℇ푞. 1.5.2 : is not independent of how the mode is normalized 퓊 is replaced by 풟퓃 +o emphasize 퓃푡ℎ mode. 풟퓃 + 2휁퓃휔퓃풟 퓃 + 휔퓃 2 풟퓃 = 퓅(푡) Modal Participation Factor
- 8. Multi-Degree of freedom systems equations of motion in modal coordinates Eq. 12.4.4 where C is a non-diagonal matrix. 1st J modes contribute significantly Size of 12.4.4 Φ is an N x J Matrix M, C & K are J x J Matrices P(t) is a J x 1 Matrix (Vector) Analysis of Non-Classically Damped Systems {Non-Classical ≈Non-Diagonal}
- 9. 2 or more parts with significantly different levels of damping. Structural-soil system underlying soil assumed as rigid in analysis of many structures with very short natural periods Damping matrices could be constructed by any one of procedures in ∮ 11.4 Damping matrix for complete system : utilize two subsystems, a. structure and soil b. dam and water 휔풾 & 휔풿: frequencies of 풾푡ℎ& 풿푡ℎnatural vibration modes Defining a Non-Classical Damping Matrix
- 10. Eq. 12.3.2 Transformation from Coupled Eq’s to a set of uncoupled Eq’s. 푁 휙푟푞푟 푡 = Φ푞 푡 퓊 푡 from 푟=1 Φ are expanded modal coordinates Arrive at Generalized Modal System
- 11. 14.5 Dynamic Analysis Using Ritz Vectors Table 14.4.1 Generation of Force-Dependent Ritz Vectors 1. Determine the 1st Vector 휓1 Solve & Normalize 2. Determine additional Vectors 휓퓃 a. Determine b. Orthogonalize 훼풾퓃 = 휓풾 푇 퓶 퓎퓃 c. Normalize 휓풾 푇 퓶 퓎퓃 Numerical Evaluation of Dynamic Response
- 12. { Numerical Evaluation of Dynamic Response { Peak Responses are estimated by RSA directly from response spectrum for the ground motion. T.13.8.6 Spectral Values & Peak Modal Responses for two response quantities Spectral Values and Peak Modal Responses for two response quantities Example
- 13. Introduce equivalent static forces at any instant of time t, these forces 푓푠 are the external forces that will produce u(t) within.. 푓푠 푡 = 푘푢 푡 Inelastic Systems element forces can be determined by appropriate modifications of these procedures. Such systems are analyzed by incremental time-stepping procedures. 푡푖 푡푖 + Δ푡 is determined by dynamic analysis. Element forces associated with displacements Δ푢푖 are computed from linear force-deformation using secant stiffness valid over the time step Element Forces
- 14. 1.10.3 Frequency-Domain Method Fourier Transform ∞ 푃 휔 = ℱ 푝 푡 = −∞ 푝 푡 푒−푖휔푡풹퓉 풰 휔 = ℋ 휔 푃 휔 Complex frequency-response function ℋ 휔 describes response to harmonic excitation Fig. 1.10.1 : Two RC dome-shaped containment structures house the nuclear reactors of the San Onofre Power Plant in CA. For design purposes, their fundamental natural vibration period was computed to be 0.15s assuming the base as fixed, and 0.50s considering soil flexibility. Frequency Domain
- 15. Modal Responses and Required No. of Modes
- 16. Symmetric-Plan Buildings Torsional Excitation
- 17. Dynamic Impedance Functions express the relation between forces and motions The origin, significance and main factors controlling radiation damping of foundations Soil Material Damping Frequency-independent {k} & {c} approximate interaction effects and facilitate response Interrelationship, structure shaken. Machine foundation problem. ~> Kinematics and Inertial Interactions Structure Soil Interaction
- 18. S 8.5 Occurrence of supporting elastic mass & Elastic mass supporting soil. These principles apply to wide variety. Ch 9 Shallow Foundations Ch 10. Theory pile- caisson-. Buried Structures to dynamic loadings 3-D Wave Propagation, Especially important niche. Theory of dynamic response rigid mass supported by elastic or viscoelastic/ linear-hysteretic body. 1936 work of Reissner, beginning of modern soil dynamics 1970s cover dynamic response of deformable piles and pile groups Theoretical framework . Simple geometries of foundation and subsoil conditions, evaluate dynamic motion Explain important features of load-displacement relationship. Practical analysis of complicated soil-structure systems. Mass
- 19. References Anil K. Chopra, Dynamics of Structures, University of California at Berkeley. Theory and Applications to Earthquake Engineering Lectures of Dr. Enrique Luco; University of California at San Diego, La Jolla Lectures of Dr. Frieder Seible; University of California at San Diego, La Jolla Lectures of Dr. Arup Maji, University of New Mexico Lectures of Dr. Ross and Dr. Gerstle, University of New Mexico. NEHRP, National Institute of Standards and Technology, GCR 12-917-21. US Dept. of Commerce