1. Mode Superposition Module 7

2. Training Manual January 30, 2001 Inventory #001447 7-2 Module 7 Mode Superposition A. Define mode superposition. B. Learn how to use the mode superposition method. C. Work on a mode superposition exercise.

3. Training Manual January 30, 2001 Inventory #001447 7-3 Mode Superposition A. Definition & Purpose A solution technique for transient or harmonic analyses. It sums factored mode shapes from a modal analysis to calculate the dynamic response. A fast, efficient method that can be used for linear dynamics problems. The alternative is to use the direct integration method which can be time consuming. The two methods are compared next.

4. Training Manual January 30, 2001 Inventory #001447 7-4 General equation of motion: Mode Superposition … Definition & Purpose Mode superposition assumes that U(t) can be represented as a linear combination of mode shapes of the structure. where [  is the matrix of mode shapes         

5. Training Manual January 30, 2001 Inventory #001447 7-5 The general equation of motion can be premultiplied by [  T and written as: Mode Superposition … Definition & Purpose Orthogonality of natural modes means: If proportional damping is specified, then:

6. Training Manual January 30, 2001 Inventory #001447 7-6 Mode Superposition … Definition & Purpose Defining ‘m’ as the number of modes, this reduces the problem to a system of ‘m’ single DOF uncoupled equations : These equations are then solved for y i (t) using an ‘undamped’ solver (e.g. the Frontal solver). If non-proportional damping is specified, then the system of ‘m’ single DOF equations are coupled by the damping matrix. This system of equations must be solved using the QR Damped solver. The final solution (regardless of the specified damping) is:

7. Training Manual January 30, 2001 Inventory #001447 7-7 Mode Superposition … Definition & Purpose Mode Superposition +Fast solution regardless of whether equations of motion are uncoupled (proportional damping only ) or coupled (non-proportional damping). +Effective when only a few modes are needed to describe response. ±Requires eigenvectors from a modal solution. –Linear only, no nonlinearities. –Deciding how many modes to use may be difficult. Too few modes may give good displacements but poor stresses Direct Integration –Fully coupled equation of motion. Solution can be time consuming. +Effective for most problems. ±No eigenvectors required. However, most dynamic analyses begin with a modal solution. +Nonlinearities allowed in transient analysis. +Easier to determine  t, the integration time step, than number of modes.

8. Training Manual January 30, 2001 Inventory #001447 7-8 Mode Superposition B. Procedure Five main steps: Build the model Obtain the modal solution Switch to harmonic or transient analysis Apply loads and solve Review results

9. Training Manual January 30, 2001 Inventory #001447 7-9 Mode Superposition Build the Model Model Same considerations as a modal analysis. Linear elements and materials only. Nonlinearities are ignored. Remember density! Also, if material-dependent damping is present, it must be defined in this step. See also Modeling Considerations in Module 1.

10. Training Manual January 30, 2001 Inventory #001447 7-10 Mode Superposition Obtain the Modal Solution 3Build the model Obtain the modal solution Same procedure as a normal modal analysis. A few differences, discussed next.

11. Training Manual January 30, 2001 Inventory #001447 7-11 Mode Superposition … Obtain the Modal Solution Mode extraction: –Only valid methods are Block Lanczos, subspace, reduced, powerdynamics, or QR damped. –Extract all modes that may contribute to the dynamic response. –Mode expansion is needed to view mode shapes but not required for the mode superposition solution. –If QR damped mode extraction method is used, the damping must be specified during preprocessing or in the modal analysis. Damping specified during the mode superposition transient or harmonic analysis will be ignored.

12. Training Manual January 30, 2001 Inventory #001447 7-12 Mode Superposition … Obtain the Modal Solution Loads and BC’s: –All displacement constraints must be applied in this step. Zero valued only; non-zero displacements are not allowed. –If element loads (pressures, temperatures, and accelerations) are to be applied in the harmonic or transient analysis, they must be specified in this step. The solver ignores the loads for the modal solution, but writes a load vector to the.mode file.

13. Training Manual January 30, 2001 Inventory #001447 7-13 Mode Superposition Switch to Harmonic or Transient Analysis 3Build the model 3Obtain the modal solution Switch to harmonic or transient analysis Exit and re-enter Solution New analysis: Harmonic or Transient Analysis options: Discussed next Damping: Discussed next

14. Training Manual January 30, 2001 Inventory #001447 7-14 Mode Superposition … Switch to Harmonic or Transient Analysis options - same as for a full harmonic or transient except: Solution method: Mode superposition Maximum mode number: Highest mode number to be used for solution. Defaults to highest mode extracted. Minimum mode number: Lowest mode number. Defaults to 1. Also for harmonic analysis: –Solution clustering option for a smooth response curve. –Option to print mode contributions at each frequency.

15. Training Manual January 30, 2001 Inventory #001447 7-15 Mode Superposition … Switch to Harmonic or Transient Damping Specified here if QR Damped mode extraction method not used. Damping in some form should be specified in most cases. All four forms are available for mode superposition: –Alpha (mass) damping –Beta (stiffness) damping both global and material dependent –Constant damping ratio –Frequency dependent damping ratio (modal damping)

16. Training Manual January 30, 2001 Inventory #001447 7-16 Mode Superposition Apply Loads and Solve 3Build the model 3Obtain the modal solution 3Switch to harmonic or transient analysis Apply loads and solve Only forces and accelerations, no displacements. Load vector from modal analysis (discussed next). Conditions for initial static solution in a transient analysis (discussed next). Integration time step is constant throughout transient. Start solution calculations (SOLVE).

17. Training Manual January 30, 2001 Inventory #001447 7-17 Mode Superposition … Apply Loads and Solve Load vector Gives a way to apply element loads (pressures, accelerations, and temperatures) in a mode superposition analysis. Calculated during the modal solution based on loads specified in the modal analysis. Can be applied with a scale factor (which defaults to 1.0).

18. Training Manual January 30, 2001 Inventory #001447 7-18 Mode Superposition … Apply Loads and Solve Initial static solution in a transient analysis The initial solution (at time = 0) in a mode superposition transient analysis is always a static solution (using the frontal solver). Can take a long time and much disk space for large models. To avoid it (and get {U} t=0 = {0}), do not apply any loads at time = 0.

19. Training Manual January 30, 2001 Inventory #001447 7-19 Mode Superposition … Apply Loads and Solve Solve Same procedure as for a full transient or harmonic analysis. Only displacement results are calculated during solution (no stresses or reaction forces). The displacement solution is written to: jobname.rdsp for a transient analysis jobname.rfrq for a harmonic analysis Next step is to review results.

20. Training Manual January 30, 2001 Inventory #001447 7-20 Mode Superposition Review Results 3Build the model 3Obtain the modal solution 3Switch to harmonic or transient analysis 3Apply loads and solve Review results. Three steps: Review the displacement solution Expand the solution Review the expanded solution

21. Training Manual January 30, 2001 Inventory #001447 7-21 Mode Superposition … Review Results Review displacement solution Enter POST26, the time-history postprocessor. First identify the results file - jobname.rdsp or jobname.rfrq. TimeHist Postpro > Settings > File or FILE command Define displacement variables at specific points in the model and obtain displacement-versus-time (or frequency) plots. Using graphs and listings, identify the critical time-points (or frequencies and phase angles).

22. Training Manual January 30, 2001 Inventory #001447 7-22 Mode Superposition … Review Results Expand the solution A process in which derived data (stresses, reaction forces, etc.) are calculated from the primary data (displacement solution). Three steps: 1.Enter Solution and activate the expansion pass. Solution > Expansion Pass... or EXPASS,ON

23. Training Manual January 30, 2001 Inventory #001447 7-23 2.Specify the solution or range of solutions to be expanded. For harmonic analysis, remember to specify the phase angle(s) or request expansion of both real and imaginary parts (which can then be combined in POST1 using the HRCPLX command). –Solution > Expansion Pass > 3.Start expansion pass solution –Solution > -Solve-Current LS or SOLVE –Results are written to the.rst file (jobname.rst) and can then be reviewed using POST1, the general postprocessor. Mode Superposition … Review Results

24. Training Manual January 30, 2001 Inventory #001447 7-24 Review the expanded solution Use POST1, the general postprocessor. Procedure is the same as for a full transient or harmonic analysis. –Read the desired results set from the results file, then plot deformed shape, stress contours, etc. –For a harmonic analysis, if you chose to expand both real and imaginary parts, use the HRCPLX command to combine them at the desired phase angle. (No need to do this if you chose to expand the displacement solution at a specified phase angle.) Mode Superposition … Review Results

25. Training Manual January 30, 2001 Inventory #001447 7-25 Mode Superposition Review Results 3Build the model 3Obtain the modal solution 3Switch to harmonic or transient analysis 3Apply loads and solve 3Review results

26. Training Manual January 30, 2001 Inventory #001447 7-26 C. Workshop - Mode Superposition In this workshop, you will run the Galloping Gertie example again, but this time follow and understand each step as you go. See your Dynamics Workshop supplement for details ( Introductory Workshop - Galloping Gertie, Page W-4. ). Introductory Workshop - Galloping Gertie, Page W-4.