List of Tables
1.3 Analysis Types
There are several analysis types that are available to conduct acoustic and vibration investigations using the ANSYS software. These include:
Modal used to calculate the natural frequencies and mode shapes of systems.
Harmonic used to calculate the acoustic or vibration response of a system due to excitation by a sinusoidally varying driving force, displacement, acoustic pressure, and others, where the excitation is continuous at con-stant frequency. A number of harmonic analyses can be conducted over a frequency range.
Transient used to calculate the time-history response of a system due the application of a time-varying excitation.
Random used to calculate the response of a system due to the application of a prescribed frequency and amplitude spectrum of excitation.
1.3. Analysis Types 7 Each of these analysis types are described further in the following sections.
The focus of this book is covering modal and harmonic analyses for acoustic systems.
1.3.1 Modal
A modal analysis can be conducted to calculate the natural frequencies and mode shapes of an acoustic or structural system, or a combined structural-acoustic system. The results from a modal analysis conducted in ANSYS can be used to calculate a harmonic response, transient, or response spectrum analysis.
The equations of motion for an acoustic or structural system can be written as
−ω2[M] + jω [C] + [K] {p} = {f } , (1.1) where [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, {p} is the vector of nodal pressures for an acoustic system or dis-placements for a structural system, and {f } is the acoustic or structural load applied to the system. For a basic modal analysis, it is assumed that there is no damping and no applied loads, so the damping matrix [C] and the load vector {f } are removed from Equation (1.1), leaving [4, Eq. (17.46)]
−ω2[M] + [K] {p} = {0} . (1.2) For an (undamped) system, the free pressure oscillations are assumed to be harmonic of the form
{p} = {φ}n cos ωnt , (1.3)
where {φ}n is the eigenvector of pressures of the nth natural frequency, ωn is the natural circular frequency (radians/s), t is time. Substitution of Equa-tion (1.3) into (1.2) gives
−ωn2[M] + [K] {φ}n= {0} . (1.4) The trivial solution is {φ}n = 0. The next series of solutions is where the determinant equates to zero and is written as [4, Eq. (17-49)]
[K] − ωn2[M]
= 0 , (1.5)
which is a standard eigenvalue problem and is solved to find the natural fre-quencies (eigenvalues) ωn and mode shapes (eigenvectors) {φ}n. ANSYS will list results of the natural frequencies fn in Hertz, rather than circular fre-quency in radians/s, where
fn=ωn
2π. (1.6)
For many finite element models, the mass and stiffness matrices are sym-metric, and ANSYS has several numerical solvers that can be used to calculate
8 1. Introduction the natural frequencies and mode shapes, which include the supernode, block Lanczos, and Preconditioned Conjugate Gradient PCG Lanczos methods [5].
When the finite element model has unsymmetric matrices, which can occur when the model contains fluid–structure interaction, then an unsymmetric solver must be used. When the system includes damping, it is necessary to use the damped or QR damped solver. The ANSYS online help manual [5, Table 15.1] lists which modal analysis solver is appropriate for the conditions present in the finite element model and is summarized here in Table 1.1.
TABLE 1.1
Modal Analysis Solver Types Available in ANSYS for Determining Natural Frequencies and Mode Shapes
Undamped / Symmetric / Solver APDL
Damped Unsymmetric Name Command
Sections 3.3.1, 3.3.3, 4.4.2, and 4.4.3 describe examples of undamped modal analyses conducted using ANSYS. Section 7.4.2 describes an example of a damped modal analysis of a room that has acoustic absorptive material on the floor.
1.3.2 Harmonic
The harmonic response of a system can be calculated using two methods: full and modal summation (or superposition).
The full method involves forming the mass [M], damping [C], and stiffness [K] matrices and the loading vector {f } of the dynamic equations of motion, combining the matrices, then inverting the combined matrix and multiplying it with the load vector to calculate the nodal displacements {u}, as follows [6]: The modal summation method involves the calculation of the mode shapes
1.3. Analysis Types 9 of a structural or acoustic system, and determining what portion of each mode, called the modal participation factors Pn, contributes to the overall response.
The mathematical derivation of the mode superposition method is presented in the ANSYS theory manual [7] [8].
To illustrate the concept of the modal summation method, consider a sim-ply supported beam that has vibration mode shapes ψn that resemble half sine waves as shown in Figure 1.4. Each mode can be multiplied by a modal participation factor Pn, then summed to calculate the total response of the beam P Pnψn. Any complicated displacement shape can be represented by a weighted sum of a sufficiently large number of modes. A similar acoustic analogy exists where any complicated acoustic response of a system can be represented by a weighted sum of a sufficiently large number of mode shapes of the acoustic system.
Schematic of the concept of modal summation, where fractions (modal par-ticipation factors) Pn of each mode shape ψn contribute to the total response of the system.
Harmonic analysis of acoustic systems using the modal summation method is not available using ANSYS Workbench at Release 14.5 and ACT Acoustics extension version 8, but is expected to be available at Release 15.0. This technique can be employed using ANSYS Mechanical APDL and an example is shown in Section 4.4.3.
1.3.3 Transient Dynamic Analysis
Transient dynamic analysis (sometimes called time-history analysis) is an analysis technique used to determine the response of a system to any time-dependent load. It is used when inputs into the system cannot be considered stationary (unlike a harmonic analysis). The basic equation of motion which is solved by a transient dynamic analysis is [9]
[M]{¨u} + [C]{ ˙u} + [K]{u} = {f (t)} , (1.8)
10 1. Introduction where the terms are the same as defined for the harmonic analysis with the ex-ception of the load vector, {f (t)}, which represents the time-dependent loads applied to the system. As was the case for the harmonic analysis, transient dynamic analysis can be calculated using either a full or mode-superposition method. The advantages and disadvantages of these two methods when un-dertaking transient dynamic analysis are the same as was discussed for the harmonic analysis. However, there are other issues that only affect transient analysis. The mode-superposition method is restricted to fixed time steps throughout the analysis, so automatic time stepping is not allowed. Such a feature is often desirable when there are events of differing time scales. With-out automatic time stepping, the shortest time scale needs to be used over the entire simulation, which can increase solution times. The full method accepts non-zero displacements as a form of load, whereas the mode-superposition method does not.
In Chapter 7 a full transient dynamic analysis is conducted on a model of a damped reverberation room, where the time-varying pressure at a number of locations is predicted in response to a sharp acoustic impulse. The various solver options are discussed in detail in this chapter.
1.3.4 Spectrum Analysis
The spectrum analysis in ANSYS is one in which the results from a modal analysis are used, along with a known spectrum, to calculate the response of a system. It is mainly used in place of transient dynamic analyses to deter-mine the response of a system to either time-dependent or random loading conditions which may be characterized by an input spectrum. This method is linear, so a transient dynamic analysis must be used if the system behavior is non-linear.
There are two broad categories of spectrum analysis: deterministic and probabilistic. The deterministic methods in ANSYS (Response Spectrum and Dynamic Design Analysis Method) use an assumed phase relationship between the various modes, whereas in the probabilistic method (called Power Spectral Density method in ANSYS and also known as random vibration analysis), the way in which the response of the modes are summed is probabilistic.
Although spectrum analyses are commonly used in predicting the response of a system to shock and vibration, these methods are not suitable for acoustic models as of ANSYS Release 14.5, and therefore there are no examples of these methods in this book.