pat328, section 3, march 2001mar120, lecture 4, march 2001s14-1mar120, section 14, december 2001...

PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-1MAR120, Section 14, December 2001

SECTION 14

STRUCTURAL DYNAMICS


TABLE OF CONTENTS

Section Page

14.0 Structural DynamicsOverview……………………………………………………………………………………………………………..

14-3Dynamics………………………………………………………………………………………………………….....

14-4Dynamics Differential Equation……………………………………………………………………………………

14-5Dynamics Concepts……………………………………………………………………………………………….. 14-6Natural Frequency…………………………………………………………………………………………………..

14-8Natural Frequency Of Free Undamped System…………………………..……………………………………..

14-9Natural Frequency Of Free Damped System………………………………………………………..…………..

14-10Harmonic Oscillations……………………………………………………………………………………………....

14-12Multiple Degree Of Freedom System……………………………………………………………………………..

14-14Multiple DOFs - Free Vibration Analysis…………………………………………………………………..……..14-15Multiple DOFs - Modal Superposition Method……………………………………………………………….…..

14-16Multiple DOFs – Harmonic Analysis…………………………………………………………….……………….. 14-17Natural Frequencies, Preloading And Fem…………………………………………………………………..…..

14-18Frequency Based Dynamics…………………………………………………………………………………..…..14-19Dynamic Analysis Methods In Msc.Marc…………………………………………………………………..……..

14-20Base Motion………………………………………………………………………………………………………....

14-23Power Transmission Tower Base Motion Example……………………………………………………………..

14-24Damping…………………………………………………………………………………………………………….. 14-25Modal Damping…………………………………………………………………………………………………….. 14-27Damping In Direct Linear And Nonlinear Dynamics ……………………..……………………………………..

14-28Updated Versus Total Lagrange…………………………………………………………………………………..

14-31


OVERVIEW

Dynamic Differential Equation Classic Dynamics versus Nonlinear Dynamics

Inertial Effects Damping Effects Natural Frequency Extraction Free Systems

Harmonic Systems Base Motion Damping Methods Direct Linear Dynamics

Versus Frequency Based Dynamics Damping in Direct Linear Dynamics Controlling Accuracy of Calculations

Nonlinear Dynamics


Linear Dynamics allows effective use of the “natural

modes” of vibration of a structure.

Example of a Modal shape for a flat circular disc with centered circular hole

modeled with shell elements

DYNAMICS

Dynamic analysis differs from static analysis in three fundamental aspects:

Inertial effects are included Dynamic loads vary as a function of time. The time-varying load application induces

a time-varying structural response.

Mass and Density need to be accounted for

Must be in proper (consistent) units


[M]{ü} + [C]{ú} + [K]{u} – {P} = 0

DYNAMICS DIFFERENTIAL EQUATION

Where: [M]{ü} represents the inertial forces

[M] – Mass matrix

{ü} – Acceleration [C]{ú} represents the dissipative forces

[C] – Dissipative matrix{ú} – Velocity

[K]{u} represents the stiffness forces[K] – Stiffness matrix{u} – Displacement

{P} represents the external forces


DYNAMICS CONCEPTS Static

- Events in which time parameters and inertia effects do not play a significant role in the solutions.

Dynamic - A significant time dependent behavior exists in the problem because of inertial forces (d’Alembert forces). Hence, a time integration of the equations of motion is required.

Linear Dynamic - The motion or deformation produced by a dynamic behavior is small enough so that the frequency content of the system remains relatively constant.

Nonlinear Dynamic - The motion or deformation produced by a dynamic behavior

of the structure is large enough that we must account for changes in geometry, material or contact changes in the model.


DYNAMICS CONCEPTS (CONT.)

Direct Integration (over time) - All kinematic variables are integrated through time. It can be used to solve linear or nonlinear problems.

Natural Frequency - The frequencies at which the structure naturally tends to vibrate if it is subjected to a disturbance.

Modal Dynamics - A dynamic solution is obtained by superimposing the natural frequencies and mode shapes of a structure to characterize its dynamic response in the linear regime.

Damping - The dissipative energy produced by a structure’s motion.


NATURAL FREQUENCY

Natural Frequency Solution The natural frequencies of a

structure are the frequencies at which the structure naturally tends to vibrate if it is subjected to a disturbance

When an applied oscillatory load approaches a natural frequency of a structure, the structure will resonate. This is a phenomenon in which the amplitude of the displacement of an oscillating structure will dramatically increase at particular frequencies.


The natural frequency solution, or eigenvalue analysis, is the basis for many types of dynamic analyses.

The structure may include preload before the eigenvalues are calculated. This affects the results.

The natural frequency for a Single Degree Of Freedom (SDOF) system is given by

The frequency procedure extracts eigenvalues of an undamped system:

NATURAL FREQUENCY OF FREE UNDAMPED SYSTEM


The structure may include preload before the eigenvalues are calculated. This affects the results.

The frequency procedure extracts eigenvalues of a damped system:

The natural frequency for the Damped Single Degree Of Freedom (SDOF) system is given by the same equation of the undamped system:

NATURAL FREQUENCY OF FREE DAMPED SYSTEM


NATURAL FREQUENCY OF FREE

DAMPED SYSTEM (CONT.)


When the Damped system is loaded with an exponential function of a single frequency, the resultant oscillations are called harmonic:

HARMONIC OSCILLATIONS


HARMONIC OSCILLATIONS (CONT.)


MULTIPLE DEGREE OF FREEDOM SYSTEM


MULTIPLE DOFS - FREE VIBRATION ANALYSIS


MULTIPLE DOFS:MODAL SUPERPOSITION METHOD


MULTIPLE DOFS – HARMONIC ANALYSIS


Example:Third Modal Shape of a Cantilevered Plate

NATURAL FREQUENCIES, PRELOADING AND FEM

Preloading changes the structural stiffness and as a result, changes the results.

A finite element mesh must be sufficiently fine enough to capture the mode shapes that will be excited in the response.

Meshes suitable for static simulation may not be suitable for calculating dynamic response to loadings that excite high frequencies.

As a general rule of thumb, you should have a minimum of 7 elements spanning a sine wave.


Example:Impact Test using Explicit Dynamics

Reaction Force at Wall

FREQUENCY BASED DYNAMICS

When a linear structural response is dominated by a relatively small number modes, modal superposition can lead to a particularly different method of determining the response.

Modal based solutions require extraction of the natural frequency and mode shapes first (i.e. requires running a Natural Frequency solution first)


DYNAMIC ANALYSIS METHODS IN MSC.MARC

Eigenvalue extractions linear with preloading

Lanczos method Power Sweep

Harmonic response linear with preloading

Real (no Damping) Imaginary (Damping)

Transient analysis linear and nonlinear

Explicit Implicit Contact


DYNAMIC ANALYSIS METHODS IN MSC.MARC (CONT.)

Modal-based Solutions include: Steady State Dynamics (i.e.:

rotating machinery in buildings) Harmonic responses for the steady

state response of a sinusoidal excitation

Modal Linear Transient Dynamics (i.e.: diving board or guitar spring)

Modal superposition for loads known as a function of time

Response Spectrum Analysis (i.e.: seismic events)

Provides an estimate of the peak response when a structure is subjected to a dynamic base excitation


DYNAMIC ANALYSIS METHODS IN MSC.MARC (CONT.)

Frequency based dynamics should have the following characteristics:

The system should be linear. (but for nonlinear preloading)

Linearized material behavior No change in contact conditions No nonlinear geometric effects other than those resulting from preloading.

The response should be dominated by relatively few frequencies. As the frequency of the response increases, such as shock analysis, modal

based dynamics become less effective The dominant loading frequencies should be in the range of the extracted

frequencies to insure that the loads can be described accurately. The initial accelerations generated by any sudden applied loads should be

described by eigenmodes. The system should not be heavily damped.


BASE MOTION

Base motion specifies the motion of restrained nodes.

The base motion is defined by a single rigid body motion, and the displacements and rotations that are constrained to the body follow this rigid body motion.

Example: Launch excitation of mounted electronics packages or hardware.

Base motion is always specified in the global directions.


Frequency Value0.0001 0.00009750.0005 0.00048750.01 0.009750.2 0.1950.3 0.29251 0.975

2.5 2.53 2.5

4.5 2.56.6 2.58 2.25

10 2100 1.1

1000 1.01

This is a typical earthquake spectrum for rocklike material with a soil depth less than 200 ft, as provided by the UBC

POWER TRANSMISSION TOWER BASE MOTION EXAMPLE


[M]{ü} + [C]{ú} + [K]{u} - P = 0 Where

[C]{ú} - Dissipative forces [C] - Damping matrix {ú} - Velocity of the structure

DAMPING

Damping is the energy dissipation due to a

structure’s motion. In an undamped structure, if

the structure is allowed to vibrate freely, the magnitude of the oscillations is constant.

In a damped structure, the magnitude of the oscillations decreases until the oscillation stops.

Damping is assumed to be viscous, or proportional to velocity

Dissipation of energy can be caused by many factors including:

Friction at the joints of a structure Localized material hysteresis


Damped natural frequencies are related to undamped frequencies via the following relation:

wherewd the damped eigenvaluewn the undamped eigenvalue x = c/co the fraction of critical damping or damping ratioc the damping of that mode shapeco the critical damping

21 nd

Damping exhibits three characteristic forms:

DAMPING (CONT.)

Under damped systems (z < 1.0) Critically damped systems (z = 1.0) Over damped systems (z > 1.0)


MODAL DAMPING

Damping in Modal Analysis Direct Damping

Allows definition of damping as a fraction of critical damping.

Typical value is between 1% and 10% of the critical damping.

The same damping values is applied to different modes.


Direct dynamic solutions assemble the mass, damping and stiffness matrices and the equation of dynamic equilibrium is solved at each point in time.

Direct method is favored in wave propagation and shock loading problems, in which many modes are excited and a short time of response is required.

Since these operations are computationally intensive, direct integration is more expensive than the equivalent modal solution.

Direct dynamic solutions can be used to solve linear transient, steady state and nonlinear solutions using Rayleigh damping.

Rayleigh damping is assumed to be made up of a linear combination of mass and stiffness matrices:

[C] = [M] + (+gt)[K]

Many direct integration analyses often define energy dissipative mechanisms as part of the basic model (dashpots, inelastic material behavior, etc.)

For these cases, generic damping is usually not important.

DAMPING IN DIRECT LINEAR AND NONLINEAR DYNAMICS


The damping terms for direct integration are defined in the materials form:

DAMPING IN DIRECT LINEAR AND NONLINEAR DYNAMICS (CONT.)

Mass Proportional Damping

Introduces damping forces caused by absolute velocities in the model

Stiffness Proportional Damping

Introduces damping which is proportional to strain rate.


Nonlinear dynamic procedure uses implicit time integration, such as Central Difference or

Newmark-beta methods.

DAMPING IN DIRECT LINEAR AND NONLINEAR DYNAMICS (CONT.)

Solution includes an automatic impact solution for velocity and acceleration jumps due to contact bodies including rigid structure.

The high frequency response, which is important initially, is damped out rapidly by the dissipative mechanisms in the model


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UpdatedLagrange

TotalLagrange

UPDATED VERSUS TOTAL LAGRANGE

Updated Lagrange

Total Lagrange

pat328, section 3, march 2001mar120, lecture 4, march 2001s14-1mar120, section 14, december 2001...

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