Dynamics 70 M3 Harmonic

Harmonic Analysis

Module 3

March 14, 2003Inventory

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Module 3

Harmonic Analysis

A. Define harmonic analysis and its purpose.

B. Learn basic terminology and concepts underlying harmonic analysis.

C. Learn how to do a harmonic analysis in ANSYS.

D. Work on a harmonic analysis exercise.


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Harmonic Analysis

A. Definition & Purpose

What is harmonic analysis?

• A technique to determine the steady state response of a structure to sinusoidal (harmonic) loads of known frequency.

• Input:– Harmonic loads (forces, pressures, and imposed displacements) of

known magnitude and frequency.– May be multiple loads all at the same frequency. Forces and

displacements can be in-phase or out-of phase. Surface and body loads can only be specified with a phase angle of zero.

• Output:– Harmonic displacements at each DOF, usually out of phase with the

applied loads.– Other derived quantities, such as stresses and strains.


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Harmonic Analysis

… Definition & Purpose

Harmonic analysis is used in the design of:

• Supports, fixtures, and components of rotating equipment such as compressors, engines, pumps, and turbomachinery.

• Structures subjected to vortex shedding (swirling motion of fluids) such as turbine blades, airplane wings, bridges, and towers.


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Harmonic Analysis

… Definition & Purpose

Why should you do a harmonic analysis?

• To make sure that a given design can withstand sinusoidal loads at different frequencies (e.g, an engine running at different speeds).

• To detect resonant response and avoid it if necessary (by using dampers, for example).


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Harmonic Analysis

B. Terminology & Concepts

Topics covered:

• Equation of motion

• Nature of harmonic loads

• Complex displacements

• Solution methods


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• General equation of motion:

• [F] and {u} are harmonic, with frequency :

• Equation of motion for harmonic analysis:

Harmonic Analysis - Terminology & Concepts

Equation of Motion

FuKuCuM

ti

21tii

max

ti21

tiimax

e)uiu(eeuu

e)FiF(eeFF

)FiF()uiu)(KCiM( 21212


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Nature of Harmonic Loads

• Sinusoidally varying, at known frequencies.

• Phase angle allows multiple, out-of-phase loads to be applied. Defaults to zero.

• All applied loads are assumed to be harmonic, including temperatures and gravity.

Real

Ima

gin

ary


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Complex Displacements

• Calculated displacements will be complex if:– Damping is specified.– Applied load is complex (i.e, imaginary part is non-zero).

• Complex displacements lag by phase angle (with respect to the applied force).

• Results can be viewed in the form of real and imaginary parts or amplitude and phase angle.


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Solution Methods

Three methods of solving the harmonic equation of motion:

• Full method– Default method, easiest of all.

– Uses full structure matrices. Unsymmetric matrices (e.g, acoustics) are allowed.

• Reduced method*– Uses reduced matrices, faster than full method.

– Requires master DOF selection, which results in approximate [M] and [C].

• Mode superposition**– Sums factored mode shapes from a preceding modal analysis.

– Fastest of all methods.

*Not covered in this seminar; see ANSYS Structural Analysis Guide for information.

**Discussed in Module 6.


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… Solution Methods

* generally not recommended


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Harmonic Analysis

C. Procedure

Four main steps:

• Build the model

• Choose analysis type and options

• Apply harmonic loads and solve

• Review results


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Harmonic Analysis Procedure

Build the Model

Model

• Linear elements and materials only. Nonlinearities are ignored.

• Remember density!

• Caution: You may inadvertently include harmonic thermal loads if both ALPX (coefficient of thermal expansion) and T are non-zero. To avoid this, set ALPX to zero.

• See also Modeling Considerations in Module 1.


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Choose Analysis Type & Options

Build the model

Choose analysis type and options

• Enter Solution and choose harmonic analysis.

• Main analysis option is solution method - discussed next.

• Specify damping - discussed next.


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Analysis options

• Solution method - full, reduced, or mode superposition. Defaults to full method.

• DOF printout format - mainly used in batch mode.

• Lumped mass matrix– Recommended if the structure is small in one dimension

compared to the other two dimensions, e.g, slender beams and thin shells.


… Choose Analysis Type & Options


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Analysis options (…continued)

• Choose a solver (default is frontal)

• Option to include pre-stress effects


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Damping

• Choose from alpha damping, beta damping, and damping ratio.

• Damping ratio is most commonly used.


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Apply Harmonic Loads and Solve

Build the model


Apply harmonic loads and solve

• All applied loads vary harmonically at the specified frequency

• “Loads” consist of:– Displacement constraints - zero or non-zero.– Forces– Pressures

• Caution: If gravity and thermal loads are applied, they are also taken as harmonically varying loads!


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… Apply Harmonic Loads and Solve

Specifying harmonic loads requires:

• Amplitude and phase angle

• Frequency

• Stepped vs. ramped specification

Amplitude and phase angle

• The load value (magnitude) represents the amplitude Fmax.

• Phase angle is the phase shift between two or more harmonic loads. Not required if only one load is present. Non-zero only valid for force and displacement harmonic loads.

Real

Ima

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F1max

F2max


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Amplitude and phase angle (continued)

• ANSYS does not allow direct input of amplitude and phase angle. Instead, you specify the real and imaginary components.

• For example, given two harmonic forces F1 and F2 that are out of phase by angle :

F1real = F1max (amplitude of F1)

F1imag = 0

F2real = F2maxcos

F2imag = F2maxsin

• You can use APDL for these calculations, but make sure angular units are set to degrees (default is radians).

Real

Ima

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F1max

F2max


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• Utility Menu > Parameters > Angular Units …


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Frequency of harmonic load:

• Specified in cycles per second (Hertz) by a frequency range and number of substeps within that range.

• For example, a range of 0-50 Hz with 10 substeps gives solutions at frequencies of 5, 10, 15, …, 45, and 50 Hz. Same range with 1 substep gives one solution at 50 Hz.


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Stepped versus ramped loads:

• With multiple substeps, loads can be applied gradually (ramped) or all at once in the first substep (stepped).

• Harmonic loads are usually stepped since the load value represents maximum amplitude.


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• After applying the harmonic loads, the next step is to start the solution.

• Typically one load step, but you may use multiple substeps within a single load step. The frequency range is defined over the entire load step.


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Review Results

Build the model



Review results

• Three steps:– Plot displacement vs. frequency at specific

points in the structure.– Identify critical frequencies and

corresponding phase angles.– Review displacements and stresses over

entire structure at the critical frequencies and phase angles.

Use POST26, the time-history postprocessor

Use POST1, the general postprocessor


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Review Results - POST26

Displacement vs. frequency plots

• First define POST26 variables.– Tables of nodal or element data.– Identified by a number 2.– Variable 1 contains frequencies and is predefined.


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… Review Results - POST26

• Define variables (cont'd.)– Pick nodes that might deform the most, then choose the DOF

direction.– List of defined variables is updated.


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• Define variables.

• Then graph them.A Graphed Response in the Frequency Domain


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Identify critical frequencies and phase angles

• Graph shows frequency at which highest amplitude occurs.

• Since the displacements are out-of-phase with the applied loads (if damping is present), the phase angle at which the peak amplitude occurs needs to be determined.

– To do this, first choose amplitude + phase format ...


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– … then list the variable(s).

• Notice that peak amplitude = 3.7 occurs at 48 Hz, - 85.7°.

• Next step is to review displacements and stresses over the entire model at that frequency and phase angle (using POST1).


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Review Results - POST1

Review results over entire structure

• Enter POST1 and list results summary to identify load step and substep number of critical frequency.


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• Use the HRCPLX command to read in results at desired frequency and phase angle:

– HRCPLX, LOADSTEP, SUBSTEP, PHASE, ...

– Example: HRCPLX,2,4,-88.2754

• Plot deformed shape, stress contours, and other desired results.


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Training ManualHarmonic Analysis Procedure

Build the model



Review results


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Lesson D: Workshop - Harmonic Analysis

• In this workshop, you will examine the harmonic response of a fixed-fixed beam to harmonic forces caused by rotating machinery mounted on the beam.

• See your Dynamics Workshop supplement for detailsHarmonic Analysis Workshop - Fixed-Fixed Beam, Page W-27

Dynamics 70 M3 Harmonic

Documents

harmonic displacements

harmonic analysis module

harmonic analysis exercise

harmonic analysisa

harmonic loads forces

phase loads

applied loads

sinusoidal loads