Chapter Ten Harmonic Analysis Harmonic Analysis Chapter Overview

  • Slides: 38
Download presentation
Chapter Ten Harmonic Analysis

Chapter Ten Harmonic Analysis

Harmonic Analysis Chapter Overview In this chapter, performing harmonic analyses in Simulation will be

Harmonic Analysis 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: A. Setting Up Harmonic Analyses B. Harmonic Solution Methods C. Damping D. Reviewing Results • ANSYS Workbench – Simulation • Training Manual 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

Harmonic Analysis Background on Harmonic Analysis Training Manual – A harmonic, or frequency-response, analysis

Harmonic Analysis Background on Harmonic Analysis Training Manual – 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: ANSYS Workbench – Simulation • A harmonic analysis is used to determine the response of the structure under a steady-state sinusoidal (harmonic) loading at a given frequency. August 26, 2005 Inventory #002265 10 -3

Harmonic Analysis Background on Harmonic Analysis – The excitation frequency W is the frequency

Harmonic Analysis Background on Harmonic Analysis – 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. ANSYS Workbench – Simulation • In a harmonic analysis, the loading and response of the structure is assumed to be harmonic (cyclic): Training Manual August 26, 2005 Inventory #002265 10 -4

Harmonic Analysis Background on Harmonic Analysis For example, consider the case on right where

Harmonic Analysis 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: ANSYS Workbench – Simulation • Training Manual In this way, a real component F 1 and an imaginary component F 2 are used. – The response {x} is analogous to {F} Model shown is from a sample Solid. Works assembly. August 26, 2005 Inventory #002265 10 -5

Harmonic Analysis Basics of Harmonic Analysis Training Manual Assumptions: – [M], [C], and [K]

Harmonic Analysis Basics of Harmonic Analysis Training Manual 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 ANSYS Workbench – Simulation • For a harmonic analysis, the complex response {x 1} and {x 2} are solved for from the matrix equation: • It is important to remember these assumptions related to performing harmonic analyses in Simulation. August 26, 2005 Inventory #002265 10 -6

Harmonic Analysis A. Harmonic Analysis Procedure Training Manual – Attach Geometry – Assign Material

Harmonic Analysis A. Harmonic Analysis Procedure Training Manual – 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 ANSYS Workbench – Simulation • 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. – Review Results August 26, 2005 Inventory #002265 10 -7

Harmonic Analysis … Geometry Training Manual – Solid bodies, surface bodies, line bodies, and

Harmonic Analysis … Geometry Training Manual – 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 ANSYS Workbench – Simulation • Any type of geometry may be present in a harmonic analysis August 26, 2005 Inventory #002265 10 -8

Harmonic Analysis … Material Properties Training Manual – All other material properties can be

Harmonic Analysis … Material Properties Training Manual – 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 ANSYS Workbench – Simulation • In a harmonic analysis, Young’s Modulus, Poisson’s Ratio, and Mass Density are required input August 26, 2005 Inventory #002265 10 -9

Harmonic Analysis … Contact Regions Training Manual • The contact behavior is similar to

Harmonic Analysis … Contact Regions Training Manual • 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 ANSYS Workbench – Simulation • 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: August 26, 2005 Inventory #002265 10 -10

Harmonic Analysis … Loads and Supports Training Manual – Loads Not Supported: • Thermal

Harmonic Analysis … Loads and Supports Training Manual – 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 ANSYS Workbench – Simulation • Structural loads and supports may also be used in harmonic analyses with the following exceptions: August 26, 2005 Inventory #002265 10 -11

Harmonic Analysis … Loads and Supports Training Manual – Note: ANSYS Professional does not

Harmonic Analysis … Loads and Supports Training Manual – 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°. ANSYS Workbench – Simulation • A list of supported loads are shown below: August 26, 2005 Inventory #002265 10 -12

Harmonic Analysis … Loads and Supports To specify harmonic loads: – Flag the Environment

Harmonic Analysis … 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 F 1 and imaginary F 2 components of the load are known, the magnitude and phase y can be calculated as follows: ANSYS Workbench – Simulation • Training Manual August 26, 2005 Inventory #002265 10 -13

Harmonic Analysis … Loads and Supports Training Manual ANSYS Workbench – Simulation • The

Harmonic Analysis … Loads and Supports Training Manual ANSYS Workbench – Simulation • 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

Harmonic Analysis B. Solving Harmonic Analyses Training Manual – Select the Solution branch and

Harmonic Analysis B. Solving Harmonic Analyses Training Manual – Select the Solution branch and insert a Harmonic Tool 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. ANSYS Workbench – Simulation • Harmonic Setup: August 26, 2005 Inventory #002265 10 -15

Harmonic Analysis … Solution Methods – The Mode Superposition method is the default solution

Harmonic Analysis … Solution Methods – 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. ANSYS Workbench – Simulation • There are two solution methods available in ANSYS Structural and above: Training Manual August 26, 2005 Inventory #002265 10 -16

Harmonic Analysis … Mode Superposition Method Training Manual – For linear systems, one can

Harmonic Analysis … Mode Superposition Method Training Manual – 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. ANSYS Workbench – Simulation • The Mode Superposition method solves the harmonic equation in modal coordinates August 26, 2005 Inventory #002265 10 -17

Harmonic Analysis … Mode Superposition Method Points to remember: 1. The Mode Superposition method

Harmonic Analysis … 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 2. Since a free vibration analysis is performed, Simulation knows what the natural frequencies of the structure and can cluster the harmonic results near them (see next slide) ANSYS Workbench – Simulation • Training Manual August 26, 2005 Inventory #002265 10 -18

Harmonic Analysis … Mode Superposition Method Training Manual In this example, the cluster option

Harmonic Analysis … Mode Superposition Method Training Manual In this example, the cluster option captures the peak response better than evenly-spaced intervals (4. 51 e-3 vs. 4. 30 e-3) The Cluster Number determines how many results on either side of a natural frequency is solved. ANSYS Workbench – Simulation Cluster example: August 26, 2005 Inventory #002265 10 -19

Harmonic Analysis … Full Method Training Manual – In the Full method, this matrix

Harmonic Analysis … Full Method Training Manual – 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: ANSYS Workbench – Simulation • The Full method is an alternate way of solving harmonic analyses August 26, 2005 Inventory #002265 10 -20

Harmonic Analysis … Full Method Points to remember: 1. For each frequency, the Full

Harmonic Analysis … 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 2. Given Displacement Support type is available 3. The Full method does not calculate modes so no clustering of results is possible. Only evenly-spaced intervals is permitted. ANSYS Workbench – Simulation • Training Manual August 26, 2005 Inventory #002265 10 -21

Harmonic Analysis C. Damping Input Training Manual – For ANSYS Professional license only a

Harmonic Analysis C. Damping Input Training Manual – 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 and beta damping are input, the effects will be cumulative • Either damping option can be used with either solution method (full or mode superposition) ANSYS Workbench – Simulation • The harmonic equation has a damping matrix [C] August 26, 2005 Inventory #002265 10 -22

Harmonic Analysis … Background on Damping Training Manual • Viscous damping is considered here:

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

Harmonic Analysis … Constant Damping Ratio Training Manual – The value of x will

Harmonic Analysis … Constant Damping Ratio Training Manual – 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] ANSYS Workbench – Simulation • The constant damping ratio provides a value of x which is constant over the entire frequency range August 26, 2005 Inventory #002265 10 -24

Harmonic Analysis … Beta Damping Training Manual This is related to the damping ratio

Harmonic Analysis … Beta Damping Training Manual 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 ANSYS Workbench – Simulation • Another way to model damping is to assume that damping value c is proportional to the stiffness k by a constant b: August 26, 2005 Inventory #002265 10 -25

Harmonic Analysis … Beta Damping Training Manual – The damping value can be directly

Harmonic Analysis … Beta Damping Training Manual – The damping value can be directly input – A damping ratio and frequency can be input and the corresponding beta damping value will be calculated 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. ANSYS Workbench – Simulation • Beta damping can be input in two ways: August 26, 2005 Inventory #002265 10 -26

Harmonic Analysis … Damping Relationships Training Manual – The quality factor Qi is 1/(2

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

Harmonic Analysis D. Request Harmonic Tool Results Training Manual – Three types of results

Harmonic Analysis D. Request Harmonic Tool Results Training Manual – 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 ANSYS Workbench – Simulation • Results can then be requested from Harmonic Tool branch: August 26, 2005 Inventory #002265 10 -28

Harmonic Analysis … Request Harmonic Tool Results Training Manual – Scope results on entities

Harmonic Analysis … Request Harmonic Tool Results Training Manual – Scope results on entities of interest – For edges and surfaces, specify whether average, minimum, or maximum value will 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. ANSYS Workbench – Simulation • Result notes: August 26, 2005 Inventory #002265 10 -29

Harmonic Analysis … Request Harmonic Tool Results Training Manual – Derived quantities such as

Harmonic Analysis … Request Harmonic Tool Results Training Manual – 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 ANSYS Workbench – Simulation • Simulation assumes that the response is harmonic (sinusoidal). August 26, 2005 Inventory #002265 10 -30

Harmonic Analysis … Solving the Model Training Manual – Only informative status of the

Harmonic Analysis … Solving the Model Training Manual – Only informative status of the type of analysis 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 ANSYS Workbench – Simulation • The Details view of the Solution branch is not used in a Harmonic analysis. August 26, 2005 Inventory #002265 10 -31

Harmonic Analysis … Contour Results Training Manual ANSYS Workbench – Simulation • Contour results

Harmonic Analysis … Contour Results Training Manual ANSYS Workbench – Simulation • 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

Harmonic Analysis … Contour Animations ANSYS Workbench – Simulation • These results can be

Harmonic Analysis … Contour Animations ANSYS Workbench – Simulation • These results can be animated. Animations will use the actual harmonic response (real and imaginary results) Training Manual August 26, 2005 Inventory #002265 10 -33

Harmonic Analysis … Frequency Response Plots Training Manual For scoped results, average, minimum, or

Harmonic Analysis … Frequency Response Plots Training Manual 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 ANSYS Workbench – Simulation • XY Plots of components of stress, strain, displacement, or acceleration can be requested August 26, 2005 Inventory #002265 10 -34

Harmonic Analysis … Phase Response Plots Comparison of phase of components of stress, strain,

Harmonic Analysis … Phase Response Plots Comparison of phase of components of stress, strain, or displacement with input forces can be plotted at a given frequency 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 ANSYS Workbench – Simulation • Training Manual August 26, 2005 Inventory #002265 10 -35

Harmonic Analysis … Requesting Results Training Manual – A free vibration analysis using the

Harmonic Analysis … Requesting Results Training Manual – 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. ANSYS Workbench – Simulation • A harmonic solution usually requires multiple solutions: August 26, 2005 Inventory #002265 10 -36

Harmonic Analysis E. Workshop 10 – Harmonic Analysis • Goal: – Explore the harmonic

Harmonic Analysis E. 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. ANSYS Workbench – Simulation • Workshop 10 – Harmonic Analysis Training Manual August 26, 2005 Inventory #002265 10 -37

August 26, 2005 Inventory #002265 10 -38

August 26, 2005 Inventory #002265 10 -38