1. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-1MAR120, Section 14, December 2001 SECTION 14 STRUCTURAL DYNAMICS

2. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-2MAR120, Section 14, December 2001 TABLE OF CONTENTS SectionPage 14.0 Structural Dynamics Overview……………………………………………………………………………………………………………..14-3 Dynamics………………………………………………………………………………………………………….....14-4 Dynamics Differential Equation……………………………………………………………………………………14-5 Dynamics Concepts………………………………………………………………………………………………..14-6 Natural Frequency…………………………………………………………………………………………………..14-8 Natural Frequency Of Free Undamped System…………………………..……………………………………..14-9 Natural Frequency Of Free Damped System………………………………………………………..…………..14-10 Harmonic Oscillations……………………………………………………………………………………………....14-12 Multiple Degree Of Freedom System……………………………………………………………………………..14-14 Multiple DOFs - Free Vibration Analysis…………………………………………………………………..……..14-15 Multiple DOFs - Modal Superposition Method……………………………………………………………….…..14-16 Multiple DOFs – Harmonic Analysis…………………………………………………………….………………..14-17 Natural Frequencies, Preloading And Fem…………………………………………………………………..…..14-18 Frequency Based Dynamics…………………………………………………………………………………..…..14-19 Dynamic Analysis Methods In Msc.Marc…………………………………………………………………..……..14-20 Base Motion………………………………………………………………………………………………………....14-23 Power Transmission Tower Base Motion Example……………………………………………………………..14-24 Damping……………………………………………………………………………………………………………..14-25 Modal Damping……………………………………………………………………………………………………..14-27 Damping In Direct Linear And Nonlinear Dynamics ……………………..……………………………………..14-28 Updated Versus Total Lagrange…………………………………………………………………………………..14-31

3. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-3MAR120, 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

4. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-4MAR120, 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

5. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-5MAR120, 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

6. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-6MAR120, 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.

7. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-7MAR120, 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.

8. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-8MAR120, 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.

9. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-9MAR120, 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

10. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-10MAR120, 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

11. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-11MAR120, Section 14, December 2001 NATURAL FREQUENCY OF FREE DAMPED SYSTEM (CONT.)

12. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-12MAR120, 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

13. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-13MAR120, Section 14, December 2001 HARMONIC OSCILLATIONS (CONT.)

14. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-14MAR120, Section 14, December 2001 MULTIPLE DEGREE OF FREEDOM SYSTEM

15. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-15MAR120, Section 14, December 2001 MULTIPLE DOFS - FREE VIBRATION ANALYSIS

16. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-16MAR120, Section 14, December 2001 MULTIPLE DOFS: MODAL SUPERPOSITION METHOD

17. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-17MAR120, Section 14, December 2001 MULTIPLE DOFS – HARMONIC ANALYSIS

18. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-18MAR120, 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.

19. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-19MAR120, 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)

20. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-20MAR120, 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

21. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-21MAR120, 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

22. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-22MAR120, 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.

23. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-23MAR120, 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.

24. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-24MAR120, Section 14, December 2001 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

25. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-25MAR120, 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

26. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-26MAR120, Section 14, December 2001 Damped natural frequencies are related to undamped frequencies via the following relation: where w d the damped eigenvalue w n the undamped eigenvalue x = c/c o the fraction of critical damping or damping ratio c the damping of that mode shape c o the critical damping 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)

27. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-27MAR120, 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.

28. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-28MAR120, 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

29. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-29MAR120, 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.

30. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-30MAR120, 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

31. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-31MAR120, Section 14, December 2001 UPDATED VERSUS TOTAL LAGRANGE Updated Lagrange Total Lagrange

32. PAT328, Section 3, March 2001MAR120, Lecture 4, March 2001S14-32MAR120, Section 14, December 2001