Mode shap

El Desarrollo de
Formas de Modos
{
Knowledge is extremely vast and profound, when
it is within the realm of the primordial form.

 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

 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?

 Determine applied forces
 Determine Stiffness Matrices
 Determine Mass Matrix
 Establish Equations of Motion
 Apply Boundary Conditions
Principle System Characteristics

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”

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

 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

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}

 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

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

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

{ 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

 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

 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

Modal Responses and
Required No. of Modes

Symmetric-Plan Buildings
Torsional Excitation

 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

 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

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

Mode shap

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Mode shap