pat328, section 3, march 2001mar120, lecture 4, march 2001s14-1mar120, section 14, december 2001...
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PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-1MAR120, Section 14, December 2001
SECTION 14
STRUCTURAL DYNAMICS
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-2MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-3MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-4MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-5MAR120, Section 14, December 2001
[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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-6MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-7MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-8MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-9MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-10MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-11MAR120, Section 14, December 2001
NATURAL FREQUENCY OF FREE
DAMPED SYSTEM (CONT.)
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-12MAR120, Section 14, December 2001
When the Damped system is loaded with an exponential function of a single frequency, the resultant oscillations are called harmonic:
HARMONIC OSCILLATIONS
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-13MAR120, Section 14, December 2001
HARMONIC OSCILLATIONS (CONT.)
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-14MAR120, Section 14, December 2001
MULTIPLE DEGREE OF FREEDOM SYSTEM
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-15MAR120, Section 14, December 2001
MULTIPLE DOFS - FREE VIBRATION ANALYSIS
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-16MAR120, Section 14, December 2001
MULTIPLE DOFS:MODAL SUPERPOSITION METHOD
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-17MAR120, Section 14, December 2001
MULTIPLE DOFS – HARMONIC ANALYSIS
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-18MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-19MAR120, Section 14, December 2001
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)
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-20MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-21MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-22MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-23MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-24MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-25MAR120, Section 14, December 2001
[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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-26MAR120, Section 14, December 2001
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)
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-27MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-28MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-29MAR120, Section 14, December 2001
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.
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-30MAR120, Section 14, December 2001
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
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-31MAR120, Section 14, December 2001
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UpdatedLagrange
TotalLagrange
UPDATED VERSUS TOTAL LAGRANGE
Updated Lagrange
Total Lagrange
PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001 S14-32MAR120, Section 14, December 2001