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Typical harmonic response system. Fo and Ω are known. uo and Φ are unknown (a). Transient and steady-state dynamic response of a structural system (b).
Harmonic response analysis is a linear analysis. Some nonlinearities, such as plasticity will be ignored, even if they are defined. You can, however, have unsymmetric system matrices such as those encountered in a fluid-structure interaction problem (see Chapter 15, “Acoustics” in the ANSYS Coupled-Field Analysis Guide). Harmonic analysis can also be performed on a prestressed structure, such as a violin string (assuming the harmonic stresses are much smaller than the pretension stress). See Section 4.11.1.1: Prestressed Full Harmonic Response Analysis for more information on prestressed harmonic analyses.
4.3. Commands Used in a Harmonic Response Analysis
You use the same set of commands to build a model and perform a harmonic response analysis that you use to do any other type of finite element analysis. Likewise, you choose similar options from the graphical user interface (GUI) to build and solve models no matter what type of analysis you are doing.
Section 4.6: Sample Harmonic Response Analysis (GUI Method) and Section 4.7: Sample Harmonic Response Analysis (Command or Batch Method) show a sample harmonic response analysis done via the GUI and via commands, respectively.
For detailed, alphabetized descriptions of the ANSYS commands, see the ANSYS Commands Reference.
4.4. The Three Solution Methods
Three harmonic response analysis methods are available: full, reduced, and mode superposition. (A fourth, relatively expensive method is to do a transient dynamic analysis with the harmonic loads specified as time-history loading functions; see Chapter 5, “Transient Dynamic Analysis” for details.) The ANSYS Professional program allows only the mode superposition method. Before we study the details of how to implement each of these methods, let's explore the advantages and disadvantages of each method.
4.4.1. The Full Method
The full method is the easiest of the three methods. It uses the full system matrices to calculate the harmonic re-sponse (no matrix reduction). The matrices may be symmetric or unsymmetric. The advantages of the full method are:
• It is easy to use, because you don't have to worry about choosing master degrees of freedom or mode shapes.
• It uses full matrices, so no mass matrix approximation is involved.
• It allows unsymmetric matrices, which are typical of such applications as acoustics and bearing problems.
• It calculates all displacements and stresses in a single pass.
• It accepts all types of loads: nodal forces, imposed (nonzero) displacements, and element loads (pressures and temperatures).
• It allows effective use of solid-model loads.
A disadvantage is that this method usually is more expensive than either of the other methods when you use the frontal solver. However, when you use the JCG solver or the ICCG solver, the full method can be very efficient.
4.4.2. The Reduced Method
The reduced method enables you to condense the problem size by using master degrees of freedom and reduced matrices. After the displacements at the master DOF have been calculated, the solution can be expanded to the original full DOF set. (See Section 3.14: Matrix Reduction, for a more detailed discussion of the reduction procedure.) The advantages of this method are:
• It is faster and less expensive compared to the full method when you are using the frontal solver.
• Prestressing effects can be included.
The disadvantages of the reduced method are:
• The initial solution calculates only the displacements at the master DOF. A second step, known as the expansion pass, is required for a complete displacement, stress, and force solution. (However, the expansion pass might be optional for some applications.)
• Element loads (pressures, temperatures, etc.) cannot be applied.
• All loads must be applied at user-defined master degrees of freedom. (This limits the use of solid-model loads.)
4.4.3. The Mode Superposition Method
The mode superposition method sums factored mode shapes (eigenvectors) from a modal analysis to calculate the structure's response. Its advantages are:
• It is faster and less expensive than either the reduced or the full method for many problems.
• Element loads applied in the preceding modal analysis can be applied in the harmonic response analysis via the LVSCALE command, unless the modal analysis was done using PowerDynamics.
• It allows solutions to be clustered about the structure's natural frequencies. This results in a smoother, more accurate tracing of the response curve.
• Prestressing effects can be included.
• It accepts modal damping (damping ratio as a function of frequency).
Disadvantages of the mode superposition method are:
• Imposed (nonzero) displacements cannot be applied.
• When you are using PowerDynamics for the modal analysis, initial conditions cannot have previously-applied loads.
4.4.4. Restrictions Common to All Three Methods
All three methods are subject to certain common restrictions:
• All loads must be sinusoidally time-varying.
• All loads must have the same frequency.
• No nonlinearities are permitted.
• Transient effects are not calculated.
You can overcome any of these restrictions by performing a transient dynamic analysis, with harmonic loads expressed as time-history loading functions. Chapter 5, “Transient Dynamic Analysis” describes the procedure for a transient dynamic analysis.
4.5. How to Do Harmonic Response Analysis
We will first describe how to do a harmonic response analysis using the full method, and then list the steps that are different for the reduced and mode superposition methods.
4.5.1. Full Harmonic Response Analysis
The procedure for a full harmonic response analysis consists of three main steps:
1. Build the model.
Section 4.5: How to Do Harmonic Response Analysis
2. Apply loads and obtain the solution.
3. Review the results.
4.5.2. Build the Model
See Section 1.2: Building a Model in the ANSYS Basic Analysis Guide. For further details, see the ANSYS Modeling and Meshing Guide.
4.5.2.1. Points to Remember
• Only linear behavior is valid in a harmonic response analysis. Nonlinear elements, if any, will be treated as linear elements. If you include contact elements, for example, their stiffnesses are calculated based on their initial status and are never changed.
• Both Young's modulus (EX) (or stiffness in some form) and density (DENS) (or mass in some form) must be defined. Material properties may be linear, isotropic or orthotropic, and constant or temperature-de-pendent. Nonlinear material properties, if any, are ignored.
4.5.3. Apply Loads and Obtain the Solution
In this step, you define the analysis type and options, apply loads, specify load step options, and initiate the finite element solution. Details of how to do these tasks are explained below.
Note — Peak harmonic response occurs at forcing frequencies that match the natural frequencies of your structure. Before obtaining the harmonic solution, you should first determine the natural frequencies of your structure by obtaining a modal solution (as explained in Chapter 3, “Modal Analysis”).
4.5.3.1. Enter the ANSYS Solution Processor
Command(s): /SOLU GUI: Main Menu> Solution
4.5.3.2. Define the Analysis Type and Options
ANSYS offers these options for a harmonic response analysis: