1 / 38

Harmonic Analysis

Explore setting up, methods, damping, and results review in harmonic analyses. This chapter delves into structure response under steady-state sinusoidal loading and the complexities of harmonic analyses.

stuartj
Download Presentation

Harmonic Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter Ten Harmonic Analysis

  2. Chapter Overview • In this chapter, performing harmonic analyses in Simulation will be covered: • It is assumed that the user has already covered Chapter 4 Linear Static Structural Analysis and Chapter 5 Free Vibration Analysis prior to this chapter. • The following will be covered in this chapter: • Setting Up Harmonic Analyses • Harmonic Solution Methods • Damping • Reviewing Results • The capabilities described in this section are generally applicable to ANSYS Professional licenses and above. • Exceptions will be noted accordingly August 26, 2005 Inventory #002265 10-2

  3. Background on Harmonic Analysis • A harmonic analysis is used to determine the response of the structure under a steady-state sinusoidal (harmonic) loading at a given frequency. • A harmonic, or frequency-response, analysis considers loading at one frequency only. Loads may be out-of-phase with one another, but the excitation is at a known frequency. • One should always run a free vibration analysis (Ch. 5) prior to a harmonic analysis to obtain an understanding of the dynamic characteristics of the model. • To better understand a harmonic analysis, the general equation of motion is provided first: August 26, 2005 Inventory #002265 10-3

  4. Background on Harmonic Analysis • In a harmonic analysis, the loading and response of the structure is assumed to be harmonic (cyclic): • The excitation frequency W is the frequency at which the loading occurs. A force phase shift y may be present if different loads are excited at different phases, and a displacement phase shift f may exist if damping or a force phase shift is present. August 26, 2005 Inventory #002265 10-4

  5. Background on Harmonic Analysis • For example, consider the case on right where two forces are acting on the structure • Both forces are excited at the same frequency W, but “Force 2” lags “Force 1” by 45 degrees. This is a force phase shift y of 45 degrees. • The way in which this is represented is via complex notation. This, however, can be rewritten as:In this way, a real component F1 and an imaginary component F2 are used. • The response {x} is analogous to {F} August 26, 2005 Inventory #002265 10-5 Model shown is from a sample SolidWorks assembly.

  6. Basics of Harmonic Analysis • For a harmonic analysis, the complex response {x1} and {x2} are solved for from the matrix equation:Assumptions: • [M], [C], and [K] are constant: • Linear elastic material behavior is assumed • Small deflection theory is used, and no nonlinearities included • Damping [C] should be included • The loading {F} (and response {x}) is sinusoidal at a given frequency W, although a phase shift may be present • It is important to remember these assumptions related to performing harmonicanalyses in Simulation. August 26, 2005 Inventory #002265 10-6

  7. A. Harmonic Analysis Procedure • The harmonic analysis procedure is very similar to performing a linear static analysis, so not all steps will be covered in detail. The steps in yellow italics are specific to harmonic analyses. • Attach Geometry • Assign Material Properties • Define Contact Regions (if applicable) • Define Mesh Controls (optional) • Set Environment to Harmonic and apply Loads and Supports • Request Harmonic Tool Results • Set Harmonic Analysis Options • Solve the Model • Review Results August 26, 2005 Inventory #002265 10-7

  8. … Geometry • Any type of geometry may be present in a harmonic analysis • Solid bodies, surface bodies, line bodies, and any combination thereof may be used • For line bodies, stresses and strains are not available as output • A Point Mass may be present, although only acceleration loads affect a Point Mass August 26, 2005 Inventory #002265 10-8

  9. … Material Properties • In a harmonic analysis, Young’s Modulus, Poisson’s Ratio, and Mass Density are required input • All other material properties can be specified but are not used in a harmonic analysis • As will be shown later, damping is not specified as a material property but as a global property August 26, 2005 Inventory #002265 10-9

  10. … Contact Regions • Contact regions are available in modal analysis. However, since this is a purely linear analysis, contact behavior will differ for the nonlinear contact types, as shown below: • The contact behavior is similar to free vibration analyses (Ch. 5), where nonlinear contact behavior will reduce to its linear counterparts since harmonic simulations are linear. • It is generally recommended, however, not to use a nonlinear contact type in a harmonic analysis August 26, 2005 Inventory #002265 10-10

  11. … Loads and Supports • Structural loads and supports may also be used in harmonic analyses with the following exceptions: • Loads Not Supported: • Thermal loads • Rotational Velocity • Remote Force Load • Pretension Bolt Load • Compression Only Support (if present, it behaves similar to a Frictionless Support) • Remember that all structural loads will vary sinusoidally at the same excitation frequency August 26, 2005 Inventory #002265 10-11

  12. … Loads and Supports • A list of supported loads are shown below: • Note: ANSYS Professional does not support “Full” solution method, so it does not support a Given Displacement Support in a harmonic analysis. • Not all available loads support phase input. Accelerations, Bearing Load, and Moment Load will have a phase angle of 0°. • If other loads are present, shift the phase angle of other loads, such that the Acceleration, Bearing, and Moment Loads will remain at a phase angle of 0°. August 26, 2005 Inventory #002265 10-12

  13. … Loads and Supports • To specify harmonic loads: • Flag the Environment as “Harmonic” • Enter the magnitude (vector or component method) • Enter an appropriate phase angle • If only real F1 and imaginary F2 components of the load are known, the magnitude and phase y can be calculated as follows: August 26, 2005 Inventory #002265 10-13

  14. … Loads and Supports • The loading (magnitude and phase angle) for two cycles may be visualized by selecting the load, then clicking on the “Worksheet” tab August 26, 2005 Inventory #002265 10-14

  15. B. Solving Harmonic Analyses • Harmonic Setup: • Select the Solution branch and insert a HarmonicTool from the Context toolbar • In the Details view enter the Minimum and Maximum excitation frequency range and Solution Intervals • The frequency range fmax-fmin and number of intervals n determine the freq interval DW • Simulation will solve n frequencies,starting from W+DW. In the example above, with a frequency range of 0 – 10,000 Hz at 10 intervals Simulation will solve for 10 excitation frequencies of 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, and 10000 Hz. August 26, 2005 Inventory #002265 10-15

  16. … Solution Methods • There are two solution methods available in ANSYS Structural and above: • The Mode Superposition method is the default solution option and is available for ANSYS Professional and above • The Full method is available for ANSYS Structural and above • “Solution Method” can be chosen in the Details view of the Harmonic Tool • The Details view of the Solution branch has no effect on the analysis. August 26, 2005 Inventory #002265 10-16

  17. … Mode Superposition Method • The Mode Superposition method solves the harmonic equation in modal coordinates • For linear systems, one can express the displacements x as a linear combination of mode shapes fi : • where yi are modal coordinates (coefficient) for this relation. • For example, one can perform a modal analysis to determine the natural frequencies wi and corresponding mode shapes fi. • As more modes n are included, the approximation for {x} becomes more accurate. August 26, 2005 Inventory #002265 10-17

  18. … Mode Superposition Method • Points to remember: 1. The Mode Superposition method will automatically perform a modal analysis first • Simulation will automatically determine the number of modes n necessary for an accurate solution • The harmonic analysis portion is very quick and efficient, hence, the Mode Superposition method is usually much faster overall than the Full method • Since a free vibration analysis is performed, Simulation knows what the natural frequencies of the structure are and can cluster the harmonic results near them (see next slide) August 26, 2005 Inventory #002265 10-18

  19. In this example, the cluster option captures the peak response better than evenly-spaced intervals (4.51e-3 vs. 4.30e-3) The Cluster Number determines how many results on either side of a natural frequency is solved. … Mode Superposition Method Cluster example: August 26, 2005 Inventory #002265 10-19

  20. … Full Method • The Full method is an alternate way of solving harmonic analyses • In the Full method, this matrix equation is solved for directly in nodal coordinates, analogous to a linear static analysis except that complex numbers are used: August 26, 2005 Inventory #002265 10-20

  21. … Full Method • Points to remember: 1. For each frequency, the Full method must factorize [Kc]. • Because of this, the Full method tends to be more computationally expensive than the Mode Superposition method • Given Displacement Support type is available • The Full method does not calculate modes so no clustering of results is possible. Only evenly-spaced intervals is permitted. August 26, 2005 Inventory #002265 10-21

  22. C. Damping Input • The harmonic equation has a damping matrix [C] • For ANSYS Professional license only a constant damping ratio x is available • For ANSYS Structural licenses and above, either a constant damping ratio x or beta damping value can be input • If both constant damping andbeta damping are input, the effects willbe cumulative • Either damping option can be used witheither solution method (full or modesuperposition) August 26, 2005 Inventory #002265 10-22

  23. … Background on Damping • Damping can be caused by various effects. • Viscous damping is considered here: • The viscous damping force Fdamp is proportional to velocitywhere c is the damping constant • There is a value of c called critical damping ccr where no oscillations will take place • The damping ratiox is the ratio of actual damping c over critical damping ccr. August 26, 2005 Inventory #002265 10-23

  24. … Constant Damping Ratio • The constant damping ratio provides a value of x which is constant over the entire frequency range • The value of x will be used directly in Mode Superposition method • The constant damping ratio x is unitless • In the Full method, the damping ratio x is not directly used and will be converted internally to an appropriate value for [C] August 26, 2005 Inventory #002265 10-24

  25. … Beta Damping • Another way to model damping is to assume that damping value c is proportional to the stiffness k by a constant b: This is related to the damping ratio x: • Beta damping increases with increasing frequency which tends to damp out the effect of higher frequencies • Beta damping is in units of time August 26, 2005 Inventory #002265 10-25

  26. Although a frequency and damping ratio is input in this second case, remember that beta damping will linearly increase with frequency. This means that lower frequencies will have less damping and higher frequencies will experience more damping. … Beta Damping • Beta damping can be input in two ways: • The damping value can be directly input • A damping ratio and frequency can be input and the corresponding beta damping value will be calculated August 26, 2005 Inventory #002265 10-26

  27. … Damping Relationships • Common measures for damping: • The quality factor Qi is 1/(2xi) • The loss factor hi is the inverse of Q or 2xi • The logarithmic decrement di can be approximated for light damping cases as 2pxi • The half-power bandwidth Dwi can be approximated for lightly damped structures as 2wixi August 26, 2005 Inventory #002265 10-27

  28. D. Request Harmonic Tool Results • Results can then be requested from Harmonic Tool branch: • Three types of results are available: • Contour results of components of stresses, strains, or displacements at a specified frequency and phase angle • Frequency response plots of minimum, maximum, or average components of stresses, strains, displacements, or acceleration • Phase response plots of minimum, maximum, or average components of stresses, strains, or displacements at a specified frequency • Results must be requested before solving • If other results are requested after a solution is completed another solution must be re-run August 26, 2005 Inventory #002265 10-28

  29. … Request Harmonic Tool Results • Result notes: • Scope results on entities of interest • For edges and surfaces, specify whetheraverage, minimum, or maximum valuewill be reported • If results are requested between solved-for frequency ranges, linear interpolation will be used to calculate the response • For example, if Simulation solves frequencies from 100 to 1000 Hz at 100 Hz intervals, and the user requests a result for 333 Hz, this will be linearly interpolated from results at 300 and 400 Hz. August 26, 2005 Inventory #002265 10-29

  30. … Request Harmonic Tool Results • Simulation assumes that the response is harmonic (sinusoidal). • Derived quantities such as equivalent/principal stresses or total deformation may not be harmonic if the components are not in-phase, so these results are not available. • No Convergence is available on Harmonic results August 26, 2005 Inventory #002265 10-30

  31. … Solving the Model • The Details view of the Solution branch is not used in a Harmonic analysis. • Only informative status of the type ofanalysis to be solved will be displayed • After Harmonic Analysis options have been set and results have been requested, the solution can be solved as usual with the Solve button August 26, 2005 Inventory #002265 10-31

  32. … Contour Results • Contour results of components of stress, strain, or displacement are available at a given frequency and phase angle August 26, 2005 Inventory #002265 10-32

  33. … Contour Animations • These results can be animated. Animations will use the actual harmonic response (real and imaginary results) August 26, 2005 Inventory #002265 10-33

  34. For scoped results, average, minimum, or maximum values can be requested. Bode plots (shown on right) is the default display method. However, real and imaginary results can also be plotted. The Ctrl-left mouse button allows the user to query results on the graph. Results can also be exported to Excel by right-clicking on the branch Left-click on the graphics window to change the Graph Properties … Frequency Response Plots • XY Plots of components of stress, strain, displacement, or acceleration can be requested August 26, 2005 Inventory #002265 10-34

  35. The average, minimum, or maximum value of the scoped results can be used to track the phase relationship with all of the input forces. In this example, the response is lagging the input forces, as expected, and the user can visually examine this phase difference. Left-click on the graphics window to change the Graph Properties … Phase Response Plots • Comparison of phase of components of stress, strain, or displacement with input forces can be plotted at a given frequency August 26, 2005 Inventory #002265 10-35

  36. … Requesting Results • A harmonic solution usually requires multiple solutions: • A free vibration analysis using the Frequency Finder should always be performed first to determine the natural frequencies and mode shapes • Two harmonic solutions may need to be run: • A harmonic sweep of the frequency range can be performed initially, where displacements, stresses, etc. can be requested. This allows the user to see the results over the entire frequency range of interest. • After the frequencies and phases at which the peak response(s) occur are determined, contour results can be requested to see the overall response of the structure at these frequencies. August 26, 2005 Inventory #002265 10-36

  37. E. Workshop 10 – Harmonic Analysis • Workshop 10 – Harmonic Analysis • Goal: • Explore the harmonic response of the machine frame (Frame.x_t) shown here. The frequency response as well as stress and deformation at a specific frequency will be determined. August 26, 2005 Inventory #002265 10-37

More Related