Engineering Vibroacoustic Analysis: Methods and Applications

The book describes analytical methods (based primarily on classical modal synthesis), the Finite Element Method (FEM), Boundary Element Method (BEM), Statistical Energy Analysis (SEA), Energy Finite Element Analysis (EFEA), Hybrid Methods (FEM-SEA and Transfer Path Analysis), and Wave-Based Methods. The book also includes procedures for designing noise and vibration control treatments, optimizing structures for reduced vibration and noise, and estimating the uncertainties in analysis results. Written by several well-known authors, each chapter includes theoretical formulations, along with practical applications to actual structural-acoustic systems. Readers will learn how to use vibroacoustic analysis methods in product design and development; how to perform transient, frequency (deterministic and random), and statistical vibroacoustic analyses; and how to choose appropriate structural and acoustic computational methods for their applications. The book can be used as a general reference for practicing engineers, or as a text for a technical short course or graduate course.

• Language: English
• Publisher: Wiley
• Release date: Feb 18, 2016
• ISBN: 9781118694015

Book preview

1

Overview

Stephen A. Hambric¹, Shung H. Sung² and Donald J. Nefske²

¹ Pennsylvania State University, University Park, PA, USA

² Consultant, Troy, MI, USA

1.1 Introduction

Structural vibrations couple with interior and exterior acoustic fields to produce sound. A vibrating structure generates sound waves in an acoustic field, and conversely, the acoustic pressure affects the structural vibration, along with stresses that may degrade structural integrity. Computational methods for solving vibration and sound problems have been an ongoing development since the early 1960s when digital computers became available. Using computers, complicated analytical formulas that were available to represent structural and acoustic solutions were then able to be solved numerically.

For complicated geometrical systems, the finite element (FE) method was developed, where any shape, source, or boundary condition could be discretized. Structural and acoustic regions may be assembled to capture waveforms and their interactions, while various boundary conditions and forcing functions are generally applied. While the FE method is commonly used to solve interior structural-acoustic problems, the boundary element (BE) method was subsequently developed, which is more suitable for solving exterior structural-acoustic problems, although it is also often used for interior acoustics.

While the FE and BE methods are generally applicable in the low-frequency range, other methods were developed that depend on the frequency range of interest and the level of uncertainty of the structural-acoustic system. These methods include statistical energy analysis (SEA), which was the first such method that was developed for application in the high-frequency range to obtain approximate and statistically relevant solutions. Subsequently, transfer path analysis (TPA), energy FE analysis (EFEA), wave-based structural modeling, among others, have been developed to solve a wide range of structural-acoustic problems [1–3].

This book describes the vibroacoustic methods that are commonly used for predicting the structural and acoustic response in sound–structure interaction applications in transportation vehicles and other mechanical systems. Section 1.2 gives an overview of the traditional FE, BE, and SEA vibroacoustic methods. Section 1.3 gives an overview of the alternative newer methods, hybrid FE/SEA, hybrid TPA, EFEA, and wave-based structural modeling, that have been developed. The modeling, computational, and application considerations for choosing the different methods are then described in Section 1.4, followed by an outline of the book organization in Section 1.5.

1.2 Traditional Vibroacoustic Methods

1.2.1 Finite Element Method

The FE method has been and remains the most popular numerical modeling approach. While the FE method was originally developed to simulate static deformation and stress, it was subsequently extended to model structural vibration by including mass and damping effects. The FE modeling approach was then extended to model sound waves in acoustic enclosures and the structural-acoustic interaction with vibrating structures. It is thereby applicable to solve for the structural and acoustic responses in coupled structural-acoustic systems.

The FE method has also been developed for heavy fluid–structure interaction problems in the nuclear industry and nonlinear structural-acoustic interactions in unbounded acoustic domains such as in underwater acoustics applications. Besides sound pressure response prediction in air, structural-acoustic interaction has been analyzed for acoustic pressure loading effects on structures. This interaction is especially evident in aircraft fuselage designs that sustain intense pressure pulsations during launch or repetitive turbulence pressure loading on the fuselage surface. Similarly, structural-acoustic interaction effects have been assessed for nuclear reactor designs to ensure fatigue design criteria over long lifetimes (30–50 years). More recently, structural-acoustic interactions on medical devices have drawn increased interest.

The FE method was initially implemented in the later 1970s for structural-acoustic analysis of transportation vehicle interiors, such as automobiles, aircrafts, heavy trucks, and so on. With advanced software development for modeling and solving complicated structural systems, the FE method is now commonly used in transportation vehicle interior noise analysis and design. In these applications, the FE solution is generally obtained using the normal-mode synthesis method to predict the coupled structural-acoustic response. This modal approach involves significantly fewer degrees of freedom as compared with the direct solution method and is, therefore, more computationally efficient.

In acoustic noise control, impedance boundary conditions or acoustic interior absorption materials can be modeled using the FE method. Measured material behavior or a model representation of the material is required to represent these in the FE model. Recently, the structural-acoustic wave interaction in multimedia has drawn significant interest in geoacoustics, metamaterials, electroacoustics, and medical acoustics. To solve the highly nonlinear nature of some problems, the FE method is deemed a necessary tool. Finally, developing efficient solution methods to solve large complex problems continues to be a challenging and ongoing research area.

This book mainly covers linear structural-acoustic applications in transportation vehicles and mechanical systems, although as indicated earlier there are a wide range of other structural-acoustic applications using the FE method.

1.2.2 Boundary Element Method

Instead of discretizing a mesh throughout a volume as in the FE method, the BE method decomposes the solution in the integral form only at the boundary of an acoustic region. For exterior acoustics, the traditional FE method is not usually suitable to solve radiation and scattering problems involving an unbounded region. Instead, boundary integral approaches are more direct and straightforward for such problems.

For an unbounded acoustic region in exterior acoustics, the basic solution to the acoustic wave equation that satisfies the Sommerfeld radiation boundary condition of an infinite region is the free-space acoustic Green’s function. By applying the divergence theorem to the acoustic wave equation, the solution can be obtained in the form of the Helmholtz boundary integral equation or the Rayleigh boundary integral equation, which are then discretized using the BE formulation.

The acoustic BE method is also commonly applied to interior acoustics for predicting the sound pressure response resulting from structural vibrations, when the structural-acoustic interaction effect on the response can be neglected. Compared to the use of an FE model of an acoustic region, as in large interior enclosures or unbounded acoustic media, a BE model involves a much smaller mesh as only the boundary is discretized with elements. However, a computational penalty is required to solve the resulting complex fully populated matrices.

1.2.3 Statistical Energy Analysis

While many problems may be solved using simple modal summations computed by the FE method, others may be impractical due to significant number of elements or modes required. For very large structures, like aircraft or ships, it may not be possible to generate very finely meshed FE or BE models unless some form of component reduction method is employed. Even for smaller problems, FE and BE models are also impractical at high frequencies when dense meshes are needed. This is because traditional discretization techniques like FE and BE must subdivide models to the point that all structural and acoustic wavelengths are captured properly over all frequencies of interest. The most commonly cited criterion for this is to ensure at least six to eight subdivisions, or elements, represent each wavelength.

Based on the physics of high-frequency response, approximate methods have been developed which are not based on subdividing structures into small elements, but instead generalize groups of energy or waves that subdivide structural or acoustic regions into subsets. Instead of solving for the sound pressure and vibration response everywhere in a subset, a mean value of the energy response is obtained, from which spatially averaged pressure or vibration responses are calculated. The prevailing method that emerged from these developments is SEA. Instead of modeling vibration or sound directly, SEA tracks the flow of energy between groups of interconnected subsystem modes.

Here, the modal density (number of modes/frequency band) is the important parameter, with subsystems that are modally rich enjoying most of the vibroacoustic energy. Also, more modes in a subsystem result in less variation in the subsystem response. In this case, a mean energy estimate is quite accurate over the full region of the structure or acoustic subsystem, with only minor variations about that mean. As modal density increases with frequency, therefore SEA is most useful at high frequencies. At lower frequencies, however, variability about the mean response is much greater, and SEA is not as useful, particularly when an analyst is most interested in extreme response values at a particular location of interest or from a particular resonant mode.

1.3 New Vibroacoustic Methods

1.3.1 Hybrid FE/SEA Method

In the 1990s, numerical modeling experts began pointing out the so-called mid-frequency gap, where modal density is not very high, so that SEA is not as useful, but where using full FE and/or BE models is still computationally intractable. New investigations suggested various mid-frequency methods, some of which are highlighted in this book. A hybridization of the FEA and SEA methods is one where large-scale vibration is captured with FE models, and coupled to smaller-scale vibrations in connected structures. In this case, the modal response of the large-scale vibration is captured using the FE method, while only the approximated mean value of the smaller-scale vibration is predicted using the SEA method. Framed panels are a typical application to demonstrate the methodology.

1.3.2 Hybrid FE/TPA Method

Transfer path analysis (TPA) is a frequency-based transfer function analysis approach in terms of frequency-response functions (FRFs) computed from an FE model or measured experimentally. The TPA method has been used particularly in the automotive industry for the analysis of different contributions of noise and/or vibration at a particular receiver position. The classical TPA method employs the measured FRFs of the various source-receiver paths by using laboratory-controlled excitation devices. The hybrid TPA (HTPA) method combines the measured FRFs of some substructures with the predicted FRFs of the substructures that are obtained from their FE models. This method is most useful to solve problems in the mid-frequency range where validated FE, BE, or other analytical models of the substructures are not available, so that test-based methods can be used to compensate for analytical limitations and assumptions.

1.3.3 Energy FE Analysis

Energy FE analysis (EFEA) is similar to SEA in that the solution is obtained for the energy distribution in the structural or acoustic system, from which the vibration or sound pressure responses can be calculated. However, while SEA is based on the total energy within subsystems, the EFEA method is based on the derivation of the governing equation of motion in terms of the spatial distribution of energy in structural or acoustic subsystems. The result is a partial differential equation of motion similar to that of heat conduction, from which the energy solution can be obtained from an FE formulation. This provides a more detailed spatial distribution of the energy response and, thereby, more fidelity than SEA in terms of the spatial response in the structural and acoustic systems. This allows the method to be adaptable to a lower frequency range than SEA. Conventional FE models can be used for structural and acoustic analysis, which are readily adaptable to predict the energy distribution in the system. Similar to hybrid FE/SEA, the hybrid EFEA method also combines the FE and EFEA methods for application in the mid-frequency range.

1.3.4 Wave-Based Structural Analysis

Other investigators have recognized that finite elements can represent waveforms in a piecewise fashion, and the so-called spectral element formulation approach has emerged. Spectral elements are ideal for rod and beam structures, where only a few elements may be used to generate exact response throughout a model. Current research is focused on extending the method to two- and three-dimensional problems, where enforcing continuity between elements is more difficult.

1.3.5 Future Developments

There are other emerging vibroacoustic methods that continue to be developed, but none has enjoyed widespread acceptance yet. A conference devoted to noise and vibration emerging methods (NOVEM) tracks these developments, and is a recommended supplementary resource to this book. Details can be found in the NOVEM proceedings on the INCE-USA (Institute for Noise Control Engineering) electronic publications website (see www.inceusa.org).

1.4 Choosing Numerical Methods

There are three main considerations for choosing numerical methods to model structural-acoustic systems: geometrical discretization, solution frequency ranges, and the type of application.

1.4.1 Geometrical Discretization

In present-day applications, the FE method is mainly used for interior structural-acoustics, and the BE method is mainly used for exterior structural-acoustics, as well as for interior acoustics where the coupling effect of the air on the structure can be neglected. Both methods are heavily used today in practical applications, and both require detailed meshing of the structural and acoustic systems. However, with advanced computer pre-processing software, engineers can digitize complicated geometry and generate FE and BE meshes within days (or even hours). By applying boundary conditions and loads, vibration and sound anywhere in a structure or fluid are then computed. Many computer software systems can post-process enormous amounts of data output to assist engineers in examining and interpreting the results. The process is so streamlined that FE and BE are often routinely implemented as requirements in any product design cycle and will be continuously enhanced for the foreseeable future. To be consistent with the same geometry as well as to implement the same preprocessing software, SEA software has also evolved to accept FE-like meshes as inputs, while the EFEA method adapts conventional FE meshes with modified properties.

1.4.2 Solution Frequency Ranges

Table 1.1 categorizes the various analysis methods that are applicable in the low-frequency (LF), mid-frequency (MF), and high-frequency (HF) ranges. In general, one really should not classify analysis methods purely by frequency, but instead by how many waves span a given dimension. This is done by multiplying a wavenumber k (inverse of wavelength) by a characteristic length to get a non-dimensional parameter as ka. Thus, "ka = 2π" means one full wavelength over the dimension a, "ka = 4π means two waves and ka = 6π" means three waves, and so on. By knowing the ka range of a specific problem, one can quickly determine appropriate numerical methods. A high ka value means a large number of waves, so that higher frequency methods like SEA and EFEA are more likely to be applicable.

Table 1.1 Approximate frequency range, computational requirements, and model and response resolution of vibroacoustic analysis methods

Of course, for complicated structural or acoustic systems, it may be difficult to identify the appropriate wavenumber k and characteristic length a, and only rough estimates of the ka ranges can be made. In addition, there may be considerable overlap between the ranges of applicability of the methods as well as the distribution of various wave numbers in multi-coupled subsystems. This means that two or even three methods may be applicable to solve the problem, so that the results overlay in the overlapping frequency range(s). Note that Table 1.1 is only approximate, and other factors like structural damping affect the valid frequency ranges of statistical and hybrid methods.

There is also an increased development of methods that are applicable in the mid-frequency range. In particular, as supercomputing capability continues to grow, the low-frequency computational methods are being extended to the mid-frequency ranges, while the high-frequency computational methods are being enhanced to be better applicable to the mid-frequency ranges. Other newly developed methods also claim their capabilities in the mid-frequency ranges. Despite the computational capabilities or modeling details, the fundamental physics of mid-frequency problem should be fully understood before implementing an appropriate computational method.

Example 1.1

Figure 1.1 is a diagram of the sound pressure response in a vehicle travelling over a rough road. An acoustic mode of the cavity exists at approximately 50 Hz. Determine the LF, MF, and HF ranges of the cavity.

Image described by caption and surrounding text.

Figure 1.1 Low-frequency (LF), mid-frequency (MF), and high-frequency (HF) approximate ranges in sound-pressure-level response in an automotive vehicle passenger compartment

Solution

The acoustic mode of the cavity can be approximated as the first mode of a tube for which . One can then evaluate ka as . Therefore, from Table 1.1, the frequency ranges in Hertz are , , and The FE and BE methods would apply to the acoustic cavity in the low-frequency range, and the SEA or EFEA methods would apply in the high-frequency range. In the mid-frequency range, the hybrid FE/SEA, hybrid FE/EFEA, and hybrid FE/TPA methods are applicable. The LF, MF, and HF frequency ranges are depicted in Figure 1.1. In the low-frequency range, the characteristic modal peak responses are due to a small number of modes (low-modal density). In the high-frequency range, due to the damping of the many modes (high modal density), major modal peak responses are not evident and the response decays due to the damping. In the mid-frequency range, there exists a combination of modal peak responses in the low-frequency range and the damped response in the high-frequency range.

1.4.3 Type of Application

The choice of modeling method also depends on the application and the response resolution that the analyst expects to obtain from the structural and acoustic solutions. The FE and BE methods require detailed modeling of the structural and acoustic systems, and they provide vibration and sound pressure-response solution information in a narrow-band form at all of the grid locations. On the other hand, the SEA and EFEA methods require much less modeling detail of the structural and acoustic systems, and they provide broadband (typically one-third octave) solutions of the spatially averaged energy response, from which the frequency averaged and spatially averaged vibration and sound pressure response of the subsystems are obtained.

Therefore, in applications, the FE and BE methods are most useful for diagnosis of particular modal peak response problems that occur at discrete frequencies in the low-frequency and mid-frequency ranges. On the other hand, the SEA and EFEA methods are most useful for identifying and minimizing the frequency-averaged and spatial-averaged vibration and sound pressure responses. The mid-frequency methods (hybrid FE/SEA, hybrid FE/EFEA) provide a combination of the detailed modal information and averaged response information. The hybrid TPA method provides narrow-band response, but it requires measured FRFs to be obtained.

Example 1.2

Figure 1.2 shows the interior road noise in a vehicle traveling at constant speed (a) on a coarse road and (b) on a smooth road. The interior noise in the vehicle results from a combination of structure-borne noise and airborne noise. The structure-borne noise results from body panel vibrations that are excited by dynamic loads either from road excitation or from powertrain excitation, as in Figure 1.2a. For these types of noise sources, harmonic response in the low-frequency range is of interest where modal density is low and modal-phase interaction is important. The design changes may involve detailed structural architecture modification that would require FE models to optimize the design.

Image described by caption and surrounding text.

Figure 1.2 Road noise sources in vehicle traveling at speed V: (a) Structure-borne noise in vehicle traveling on coarse road and (b) airborne noise in vehicle travelling on smooth road

The airborne noise results from body panel vibrations that are excited by pressure loads acting on the panels from either the airflow excitations around the vehicle or from the radiated pressures from the powertrain and tires, as in Figure 1.2b. For these types of noise sources, broadband response in the high frequency is of interest where modal density is high and modal-phase interaction is no longer relevant. The design changes may only involve add-on treatments, such as damping layers or mass backings on the panels so that SEA or EFEA would be more efficient to analyze the trade-off designs.

Before choosing an appropriate numerical method, one needs to understand the physics of the problem and the expectation of the solution for vibration and noise reduction.

1.5 Chapter Organization

The chapter organization starts by providing the basics of vibration and sound in Chapters 2 and 3, followed by the fundamentals of sound–structure interaction in Chapter 4. A firm understanding of this background is required before studying vibroacoustic methods and applications in the remaining chapters of the book.

Chapter 5 introduces the modal synthesis method of structural-acoustic system to analyze structural-acoustic modal interaction that couples vibrational and acoustic systems. Modes are coupled in multi-structural systems by connections of the substructures, as well as in multiple-acoustic systems by the presence of absorption materials on the boundaries or within the acoustic regions. The subsystem coupling depends on the similarity of the mode shapes at the junction and mating surfaces.

The detailed modeling methods, FE and BE, are described in Chapters 6 and 7. Extensions to FE/BE modeling to assess vibration and sound reductions due to added noise control materials such as elastomers and rubber are provided in Chapter 8. Automated methods for modifying structural designs to optimize noise and vibration reduction are outlined in Chapter 9.

Chapter 10 introduces an important concept, namely nearly all structures vary slightly due to material and/or manufacturing differences and uncertainties. Therefore, a numerical model should be considered as only a single instance of a statistical ensemble of realizations. The methods in Chapter 10 enable evaluating uncertainty bounds in numerical model simulations.

The SEA method is introduced in Chapter 11, and it has much in common with uncertainty analysis, as it assumes averaged coupling between groups of modes. Randomness in this coupling is assumed in the formulation, and SEA solutions are mainly to provide the averaged response over space and time (or frequency band).

The remainder of the book is devoted to advanced and emerging methods, many of which address the mid-frequency analysis range. The hybrid FE/SEA methods in Chapter 12 combines the SEA and FE methods, where FE models represent global modal response, and the SEA models represent a superimposed local statistical response on the global behavior.

In Chapter 13, the HTPA is presented which combines FE-based or test-based frequency-response functions with FE-based or test-based operating powertrain loads.

In Chapter 14, the EFEA method is described where an FE model is used to represent localized designs in more detail than SEA and to capture local distributions and interactions. Conventional FE models are then used to obtain the predicted vibration and acoustic response.

Finally, Chapter 15 presents the basics of wave-based structural modeling, using exact spectral elements to represent the wave forms of simple structure such as beams and rods.

References

[1] N. Lalor and H.-H. Priebsch, The prediction of low- and mid-frequency internal road vehicle noise: a literature survey, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 221, 245–261, 2007.

[2] O. von Estorff, Numerical Methods in Acoustics: Facts, Fears, Future, Plenary Lectures PL004, 19th International Congress on Acoustics, Madrid, 2–7 September 2007, p. 53, 2007.

[3] R. Ohayon and C. Soize, Structural Acoustics and Vibration, Academic Press, San Diego, 1998.

2

Structural Vibrations

Stephen A. Hambric

Pennsylvania State University, University Park, PA, USA

2.1 Introduction

Most beginning vibro-acousticians want answers to the following questions:

How do structures vibrate?

How do vibration patterns over a structure’s surface radiate sound?

Conversely, how do sound waves induce vibrations in structures they impinge on?

Some of the inevitable follow-on questions are:

How can I modify a structure to reduce how much it vibrates or how much sound it makes (noise control engineering)?

Or, for designers of musical instruments and loudspeakers, how can I increase how much sound my structure makes, and craft its frequency dependence to be more pleasing to human listeners?

Also, is my structure vibrating so strongly that it might crack and break?

The answers to all of these questions depend on a structure’s shape and material properties, which define how quickly and strongly different structural waves propagate through it. We will start by studying how structures vibrate in this chapter, and then, in Chapters 3 and 4, how a structural surface’s vibration patterns act on surrounding fluids, radiating acoustic sound fields. Also, in Chapter 4, we will analyze how structures are excited by incident sound fields. I have written mostly about simple structures to define the basic concepts. Of course, more complex structures may be analyzed using some of the methods that you will learn about in other chapters.

Here are some basic definitions and assumptions. First, we will mostly consider homogenous isotropic linear structures. Homogeneity means that material properties, such as elastic modulus and mass density, are the same throughout a structure. Isotropy means that the elastic moduli are identical in all directions. Of course, there are many examples of inhomogeneous, anisotropic structures, and we will briefly discuss some of them later.

Linearity means that a structure’s deformation varies linearly with applied forces. For example, the classic linear forced spring behavior is represented by kx = F, where k is the spring stiffness, F is an applied force, and x is displacement. If the spring stiffness varies with displacement, such that k(x)x = F, we would need to solve a nonlinear problem, iterating on the solution for x and k(x). Another nonlinearity might be with the force, which could increase or decrease with displacement, or even change direction. Once again, while there are many examples of nonlinear structural forced response, we will not consider it further here.

2.2 Waves in Structures

2.2.1 Compressional and Shear Waves in Isotropic, Homogeneous Structures

Structural materials, such as metals, plastics, and rubbers, deform in ways far more complicated than air or water. This is because of one simple fact: structural materials can resist shear deformation, and fluids cannot.¹ This means that both dilatational (or compressive) and shear waves can coexist in structures. By itself, this is not very exciting. However, one more attribute of nearly all practical structures makes the field of structural acoustics so interesting and complicated: most structures have one or two dimensions that are very small with respect to internal wavelengths. We call these structures plates or beams, and they can vibrate in flexure. Why is this so interesting? Because flexural waves are dispersive, meaning that their wave speeds increase with increasing frequency.

Dispersive waves are odd to those not familiar with structural vibrations. Imagine a long plate with two transverse sources at one end which excite flexural waves in the plate. One source drives the plate at a low frequency, while the other vibrates at a high frequency. The sources are turned on at the same time, and somehow the high-frequency wave arrives at the other end of the plate faster than the low-frequency wave! I will show why this is so later, but first we will study the simpler structural wave types.

The simplest structural waves are those that deform an infinite material longitudinally and transversely. Longitudinal waves, sometimes called compressional waves, expand and contract structures in the same way acoustic waves deform fluids. The wave equation and sound speed for a longitudinal wave traveling in the x direction are:

(2.1a)

(2.1b)

where w is the deformation (also in the x direction), B is the elastic bulk modulus, and ρ is the volumetric mass density.

The bulk modulus relates the amount of volumetric contraction (per unit volume) caused by an applied pressure:

(2.2)

Low-volumetric changes mean stiffer structures and faster compressional waves.

So far, we have considered structural waves only in media large in all dimensions with respect to vibrational wavelengths. For audible frequencies and for most practical structures, one or two geometric dimensions are small with respect to a wavelength. As a longitudinal wave expands or contracts a beam or plate in its direction of propagation, the walls of the structure contract and expand transversely due to the Poisson effect, as shown in Figure 2.1. The Poisson’s ratio, which relates in- and out-of-plane strain deformations according to

(2.3)

determines the amount of the off-axis deformation, which for incompressible materials such as rubber approaches the amount of the on-axis deformation (Poisson’s ratio of 0.5).²

Image described by caption.

Figure 2.1 A longitudinal wave passing through a plate or beam (amplitudes highly exaggerated). As the material expands or contracts along the axis of the plate or beam, the Poisson effect contracts and expands the material in the transverse directions

Longitudinal waves are therefore slower in structures such as beams and plates, since the free surfaces of the structural material are exposed to air or fluid. Since the stiffness of most fluids that might surround a beam or plate is smaller than that of the structural material, the free surfaces of the structure act essentially as stress relievers, slowing down the compressional waves. The sound speeds of longitudinal waves in beams and plates are:

(2.4a)

(2.4b)

where cl is defined not by the Bulk modulus, but by the Young’s modulus E, which is related to the volumetric Bulk modulus according to:

(2.5)

For a typical Poissons’ ratio of 0.3, longitudinal wave speeds in plates and beams are 90% and 86% of those in infinite structural media, respectively.

As mentioned earlier, the key difference between waves in structural materials and fluid media is a structure’s ability to resist shear deformation. This shear stiffness allows pure shear waves to propagate through a structure, with the structure deforming in its transverse direction as the wave propagates in the axial direction (Figure 2.2). Shear-wave behavior is governed by the same wave equation as longitudinal waves and acoustic waves in fluid media:

(2.6)

Image described by caption.

Figure 2.2 A shear wave propagating through a plate or beam (amplitudes highly exaggerated). The wave propagates along the plate or beam axis, while deforming the structure transversely

However, shear waves, which travel at the speed

(2.7)

are slower than longitudinal waves, since a structure’s shear modulus is smaller than its Bulk and Young’s moduli. For isotropic materials, the shear modulus G is related to E and Poisson’s ratio according to:

(2.8)

2.2.2 Bending (Flexural) Waves in Beams and Plates

Most sound radiated by vibrating structures is caused by bending, or flexural waves traveling through beams, plates, and shells, like the example shown in Figure 2.3. Bending waves deform a structure transversely, so that they excite acoustic waves in neighboring fluids (we will learn about this phenomenon in future chapters). Although the longitudinal and shear-wave behavior is simple—similar to that of acoustic waves in air or water—bending waves are far more complicated. In particular, the speed of a bending wave depends not only on the elastic moduli and density of the structural material it travels through, but also on the geometric properties of the beam or plate cross section.

Image described by caption.

Figure 2.3 A flexural, or bending wave propagating through a plate or beam (amplitudes highly exaggerated). As with pure shear, the wave propagates along the plate or beam axis, while deforming the structure transversely. Unlike pure shear, however, a bending wave causes the plate or beam cross sections to rotate about the neutral axis

I will not derive the bending wave equations for beams and plates in this chapter, but will show them, along with their corresponding wave speeds. The wave equation and wave speed for flexure in thin³ beams are:

(2.9a)

(2.9b)

Whereas the wave equations for longitudinal and shear waves are second order (both the spatial and time derivatives), the bending wave equation has a fourth-order variation with space. Also, note that the wave speed does not appear explicitly in the flexural wave equation and that the wave speed depends on frequency, as we learned earlier.

Although the thin beam bending wave equation is more complicated than those for pure longitudinal and shear waves, it is still fairly simple. However, at high frequencies where flexural wavelengths become shorter with respect to the beam thickness, other terms become important—such as a cross section’s resistance to shear deformation and its rotary mass inertia. Including these effects complicates the wave equation and sound speed considerably, leading to the thick beam⁴ wave equation and wave speed:

(2.10a)

(2.10b)

Two new components appear in the thick beam-bending wave equation: a fourth-order dependence of motion on both time and space, and a fourth-order dependence on time. Some new combinations of factors also appear: KAG is the shear factor, which is the product of area, shear modulus, and the correction factor K, which is the fraction of the beam cross section which supports shear; and I/A, which represents the rotary inertia of the cross section.

When shear resistance and rotary inertia are negligible (which is the case at low frequencies for waves with long wavelengths with respect to thickness), the wave equation and wave speed reduce to the simpler forms shown earlier for thin, or Bernoulli–Euler beams. Since long wavelengths imply low frequencies, thin beam theory is sometimes called a low-frequency limit of the general, thick beam theory. For very high frequencies, the shear resistance terms become dominant, so that the flexural wave equation reduces to the shear-wave equation, and the bending wave speed approaches the shear-wave speed:

(2.11)

The only difference between a shear wave in a beam and one in an infinite structural material is the shear correction factor K.

Although flexural wave theories for plates are derived in different ways than those for beams, the general thick-plate⁵ wave equation and wave speed are essentially the same as those for beams, but for a wave propagating in two dimensions:

(2.12a)

(2.12b)

D is a combination of elastic modulus and moment of inertia terms called the flexural rigidity, where

(2.13)

As with beams, the low-frequency (thin plate) limits of the thick-plate equations are simpler, but still dispersive:

(2.14a)

(2.14b)

And, just as with beams, at high frequencies, flexural waves in plates approach pure shear waves, where

(2.15)

For homogenous isotropic plates, the shear correction factor is 5/6 (for beams, K depends on the geometry of the cross section).

The wave speeds of thin and thick plates, along with longitudinal and shear-wave speeds in a 10-cm-thick steel plate are shown in Figure 2.4. The low- and high-frequency limits of the general thick-plate wave speed are evident in the plot. So, longitudinal waves are faster than shear waves, which are faster than bending waves. Also, waves in stiffer materials are faster than those in flimsy materials. Finally, structural waves are slower in massive materials than they are in lightweight materials.

Graph of wave speeds in a 10?cm?thick steel plate displaying horizontal line for shear, ascending curve (solid) for flexural, thick plate, dashed curve for flexural, thin plate shear, and longitudinal line atop.

Figure 2.4 Various wave speeds in a 10-cm-thick steel plate. The longitudinal and shear waves are nondispersive, and the bending waves are dispersive (vary with frequency). The thin plate wave speed becomes invalid at high frequencies where rotary inertia and shear resistance become important

Incidentally, if you prefer not to work with the unwieldy Equation 2.12b, you can use this approximate formula for plates instead, which is effectively a curve fit between the low- and high-frequency limits:

(2.16)

2.2.3 Bending Waves in Anisotropic Plates

So far, we have limited our discussions to beams and plates constructed from homogeneous isotropic materials, such as metals. Many modern structures are constructed by combinations of materials that are far from homogenous and isotropic, such as plates made of honeycomb cores and thin outer metal facesheets, or laminated composites, which are cured assemblies of layers of woven fibers and epoxies. Sandwich panels are particularly popular in the transportation and building construction industries, being used for flooring in trains and large transport vehicles, roofing in buildings and rotorcraft, and even in automobiles.

Composite panels are constructed of stacks of fiber mesh embedded in stiff epoxy or resin, as shown in Figure 2.5. The meshes resemble fabric, which are woven with interleaved carbon or glass fiber strands. Each mesh, or layer, is stiff in the directions aligned with the fibers, but soft in other directions. The layers, or laminates, are stacked, with each layer usually oriented differently. The orientations are usually specified in the form of shorthand that looks like this: 0/45/90/…, where each number indicates an angle, in degrees, of each layer about the normal direction to the panel surface. The resin or epoxy that encases the laminate stack provides additional stiffness and mass.

Schematic of composite laminate stacks. Left: Uniaxial with 0° angular orientations. Right: Quasi-isotropic with angular orientations (top-bottom) 0°, 90°, +45°, -45°, -45°, +45, 90°, and 0°.

Figure 2.5 Composite laminate stacks: left—uniaxial, right—quasi-isotropic

Composite panels have anisotropic material properties and have more than one Young’s and Shear modulus, and multiple Poisson’s ratios. An isotropic material has a simple stress–strain relationship:

(2.17)

where σii are extensional stresses (related to longitudinal waves), σij are shear stresses (associated with shear waves), εii are extensional strains and εij are related to the shear strains (shear strain γij = εij + εji). For an isotropic material, the direction of extensional strain does not matter—the elastic moduli are the same. However, an orthotropic material has a far more complicated stiffness matrix E:

(2.18)

In all, there are six elastic (extensional) and shear moduli, related by multiple Poisson’s ratios, which define an orthotropic material’s stiffness and therefore its wave speeds.

Composite panels are often constructed using a series of laminate angular orientations (Figure 2.5) that lead to simpler, quasi-isotropic properties. Quasi-isotropic materials are simpler to define and have an in-plane Young’s modulus that does not vary with orientation (E1 = E2). The in-plane and through-thickness shear moduli and Poissons’ ratios, however, are different. Fortunately, as we will discuss later, the in-plane Young’s modulus and Poissons’ ratio are the only elastic properties that affect the bending rigidity (and therefore the wave speeds) of an important type of panel—the sandwich panel. Composite materials textbooks [1] provide details on how to determine the effective elastic moduli of a composite panel based on the layer material properties, orientations, and thicknesses, as well as the resin or epoxy.

Figure 2.6 shows a cross section of a sandwich panel, used often throughout the aerospace industry due to its high stiffness to mass ratios. The upper and lower face sheets, made of aluminum or composite, provide the flexural stiffness. Offsetting these sheets from the neutral axis increases stiffness by the second power of the offset distance, thanks to the parallel axis theorem for the area moment of inertia. A lightweight core is placed between the sheets to offset them, leading to extremely stiff, lightweight structures. The sheets are bonded to the core with an adhesive, usually stiff enough so that its effects may be ignored.

Schematic of a typical sandwich panel cross section with arrows depicting the upper and lower facesheets (rectangles) and honeycomb core (parallel lines).

Figure 2.6 A typical sandwich panel cross section

While the face sheets provide the flexural rigidity in a sandwich panel (proportional to the in-plane extensional elastic moduli), the cores provide transverse shear stiffness. There are many core types used in sandwich panels. The simplest are stiff foams, with castable materials embedded with air voids. Other cores are corrugated, resembling arrays of S patterns, like cardboard. Perhaps the most popular cores are honeycomb cell patterns, as shown in Figure 2.7. Cores are made from metal, or nonmetallic materials. Nomex, which is a stiffened paper, is quite popular. Most cores are made of stacked corrugated sheets. When assembled, the honeycomb cells are clearly visible. However, the shear stiffnesses of the core are different for loads along the two transverse directions. Core manufacturers therefore provide shear moduli in the ribbon (along the length of a sheet) and warp (along the stack width direction) orientations.

Schematic of honeycomb core (honeycomb cell patterns) with arrows depicting cell size, ribbon direction (top), and warp direction (left).

Figure 2.7 Honeycomb core

Given the face sheet and core dimensions and properties, the effective surface mass density (mass/area), flexural and shear rigidities, and therefore, the effective bending wave speed, may be determined from:

(2.19)

where is the flexural rigidity, is the shear rigidity, Efs is the Young’s Modulus of the facesheet material, hcore is the core thickness, tfs is the facesheet thickness, νfs is the Poisson’s ratio of the face sheets, ρsh is the overall surface mass density of the panel and

(2.20)

where Gribbon and Gwarp are the core shear moduli across the length and width of the honeycomb core. While it is common to use the mean Gcore to compute averaged bending wave speeds, individual wave speeds in the ribbon and warp directions may be also be computed.

The wave speeds in a typical sandwich panel are shown in Figure 2.8. At low frequencies, transverse waves are flexural and depend solely on the flexural stiffness of the face sheets and the overall panel mass. However, the actual effective waves transition to pure shear in the core at higher frequencies. The speed of sound in air is also shown in the example. Notice how the effective wave speed exceeds that of the speed of sound in air at a low frequency (about 300 Hz). When transverse waves become supersonic, they couple extremely well with surrounding acoustic fluids, as we will learn more about in Chapter 4.

Graph of sandwich panel wave speeds displaying line plots for acoustic, air (dashed), shear (dash dotted), and two ascending curves for bending, thin panel and bending, sandwich panel.

Figure 2.8 Typical sandwich panel wave speeds

2.2.4 Bending Waves in Stiffened Panels

Structures built from simple uniform thickness plates are rare. Many structural components take advantage of beam stiffeners (usually called ribs) to increase stiffness with minimal added weight. Figure 2.9 shows a typical aircraft fuselage, with stiffening elements around the circumference (bulkheads) and along the panel length (stringers, or longerons).

Photo displaying a typical aircraft fuselage, with stiffening elements around the circumference (bulkheads) and along the panel length (stringers, or longerons).

Figure 2.9 A stiffened aircraft fuselage

The effects of the ribs on the flexural rigidity and panel mass are simple to estimate. The panel mass is simply the sum of that of the panel and stiffeners, and an equivalent smeared density may be estimated based on the rib spacing. The flexural rigidity may be computed for a given orientation by summing the contributions from the panel and the attached rib stiffeners:

(2.21)

where Dpanel may be computed for isotropic, anisotropic, or sandwich panels, Istiffener is the inertia of the beam stiffener, including the offset from the panel neutral axis, and Δstiffener is the spacing between stiffeners. The equation equally spreads the stiffener added rigidity over the panel.

Note that the added rigidity depends on the individual inertias of the beams, as well as their offset distances from the neutral axis of the overall structure. The inertia increases with the cube of the offset, leading to significant stiffness increases with minimal added mass. The added stiffness increases bending wave speeds at a much higher rate than the added mass reduces them.

2.2.5 Structural Wavenumbers

Many structural acousticians analyze structures in wavenumber space. For those new to the field, wavenumber structural acoustics can be difficult to understand. A wavenumber is simply a measure of how many waves there are along a unit distance and can be defined as a function of wavelength or wave speed:

(2.22)

k has dimensions of radians/length. Wavenumbers may be defined for any wave type—structural and acoustic, and used to describe free waves or standing waves in resonances (we will do this for modes in the next section). It is common to use wavenumber ratios to diagnose how well structures interact with surrounding fluids. We will discuss this more in Chapter 4. It is also common to normalize wavenumber with a characteristic dimension, usually denoted a, computing the quantity ka, which has the dimension of radians. This simplifies describing waves, such that a ka of 2π represents one full wavelength along a characteristic dimension, 4π represents two waves, and so on.

2.3 Modes of Vibration

We have learned generally about how waves propagate in structures. Now, we will consider waves that reflect from the boundaries of finite structures, and how they superpose with waves incident on those boundaries.

Imagine operating a dial that controls frequency, and watching the left and right traveling waves in a finite structure shorten as the dial is turned.⁶ As the wavelengths shorten with increasing frequency, they pass through specific frequencies where the left and right traveling waves either destructively interfere (antiresonance), or constructively interfere (resonance). The constructive interference at resonance causes the appearance of standing waves with high vibration amplitudes, where it does not appear that the waves are traveling at all, but that there is a stationary wave that oscillates in place. Always remember, though, that in reality the standing wave is comprised of left and right traveling waves that move at finite speeds. This concept will be important later when we learn about how well modes radiated sound.

2.3.1 Modes of Beams

The simplest modes in a beam are for longitudinal and torsional (shear) waves. All modes have a resonance frequency, and a characteristic shape. To determine the resonance frequencies for a beam with free ends, simply divide the longitudinal- or shear-wave speed by the length of the structure. For a beam of length a and wave speed c, the fundamental resonance frequency is c/a, which corresponds to one full wave along the length. Actually, a lower resonance frequency occurs for a half wave along the length, so the fundamental resonance frequency is c/2a. Resonances occur in harmonics, or multiples of each other, so you can compute all the longitudinal or torsional resonance frequencies in a beam for a given mode order m from:

(2.22)

The mode shapes are just cosine functions, where the displacement w for a given mode m is:

(2.23)

where mπ/a is the modal wavenumber km, which defines the number of radians spanning the wave motion over distance a. Equation 2.23 defines how a mode shape varies over space (in the x direction) and time (the eiωt term). We assume time-harmonic behavior for all modes, and later for structural vibration and radiated sound. For brevity, I’ll drop the eiωt from future equations and focus on spatial variability. Remember, though, that the time variation is always there, or the modes could not oscillate!

Flexural resonances are more complicated than longitudinal or torsional ones, since the bending wave speeds depend on frequency. We will consider first the simplest flexural resonances—those in a simply supported straight beam of length a. The resonance condition is found from the simply supported boundary conditions, where there is no motion, and no moment resistance by the supports.

The frequencies of resonance for Bernoulli–Euler (thin) beams are found by combining the wave speed equation with the resonance conditions and doing some algebra to sort out the bending wave frequency dependence. For simply supported boundary conditions, the resonance frequencies are:

(2.24)

which correspond to the mode shapes wm(x) = sin(kmx), for 0 < x < a. It is traditional to define analytic beam and plate resonance frequencies in terms of radial frequency ω. Just remember to divide by 2π to get to cyclic frequency in Hertz. There are an infinite number of modes as m increases from 1 to infinity. Note that the resonance frequency is conveniently defined as the product of the square of the modal wavenumber (km = mπ/a) and the square root of the beam parameters EI/ρA. We will see that for other boundary conditions, the resonance frequencies still depend on EI/ρA, but will change as the wavenumber of the mode shape changes.

The mode shapes for m = 1–4 are shown in Figure 2.10 for a simply supported straight beam. In the mode shapes, dashed lines indicate the locations of maximum amplitude (the antinodes). The nodes of the mode shapes are at points of near-zero vibration. A useful way of determining the mode orders of mode shapes measured on structures with nearly simply supported boundary conditions is to count the number of antinodes (try it in the figures). Remember that these mode shapes oscillate over time according to eiωt, and I have just shown snapshots of the shapes at an arbitrary instant in time.

Four mode shapes of a simply supported beam, displaying wave m=1 with one antinode (dashed line), m=2 with two antinodes, m=3 with three antinodes, and m=4 with four antinodes.

Figure 2.10 First four mode shapes of a simply supported beam. The dashed lines indicate the vibration antinodes, or locations of maximum deformation

I often refer to modes of simply supported structures as the analyst’s best friend, since they are easy to incorporate into advanced theories of sound radiation, and into analyses of the flow turbulence acting on structures. It is hard to find something much simpler than a sine wave to integrate! Unfortunately, for the analysts, modes in structures with free (and other) boundary conditions are more complicated than those in ideal simply supported structures, since those boundary conditions impart a near field deformation to the vibration and shapes. For example, for a beam of length a with free or clamped ends, the resonance frequencies may be computed only approximately, where

(2.25)

For m = 2 and 3, Equation 2.25 is inaccurate. Actual values for km are 4.73/a and 7.85/a for m = 2 and 3. Equation 2.25 also applies for cantilevered beams (one end fixed and the other free), but for , and actual values for km of 1.875/a, 4.694/a, and 7.855/a for m = 1, 2, and 3.

The mode shapes of beams with free ends are:

(2.26)

and for clamped ends:

(2.27)

and for cantilevered conditions:

(2.28)

where σ is specified with extreme precision (required for accurate mode shapes) in Table 2.1.

Table 2.1 σm values for free, clamped, and cantilevered beam modes

Figure 2.11 shows sample mode shapes of a straight free beam. Here, dashed lines have been placed at the modal node lines (locations of zero deformation). Whereas counting antinodes can determine mode order for beams with simple supports at their ends, counting nodes determines mode order for free beams.

First four mode shapes of a free beam, displaying wave m=2 with two vibration nodes (dashed lines), m=3 with three vibration nodes, m=4 with four vibration nodes, and m=5 with five vibration nodes.

Figure 2.11 First four mode shapes of a free beam. The dashed lines indicate the vibration nodes, or locations of zero deformation

The only difference between the resonance frequency for beams with free (or other) boundary conditions and those for simply supported boundary conditions is the wavenumber term km. Fortunately, as the mode order m increases and the wavelengths become small with respect to the structural dimensions, the near-field deformations around a structure’s edges influence the mode shapes and their resonance frequencies less. An exercise to confirm this phenomenon is to plot the mode shapes for high m and to compute the resonance frequencies of beams with free and simply supported boundary conditions. Notice how as m increases, the wavenumber for free beam modes approaches the wavenumber term for simply supported modes (the −1 portion of the wavenumber in Equation 2.25 becomes inconsequential, and the wavenumber converges to mπ/a). In Figure 2.12, the mode shapes for a simply supported and free beam are shown for large m and km and are nearly identical away from the boundaries.

Schematic of simply supported labeled SS–SS, m=10, km=31.4 (left) and free beam mode shapes labeled Free–free, m=11, km=33.0 at high mode order (right).

Figure 2.12 Simply supported (left) and free (right) beam mode shapes at high mode order

2.3.2 Modes of Plates

Mode shapes in flat plates look like those in beams, but are two dimensional, as waves can propagate along the length and width. We consider again simply supported boundary conditions (the analyst’s best friend) at the edges of a thin rectangular plate, where the transverse displacement of a given mode shape of order (m,n) is:

(2.29)

and the corresponding resonance frequencies of the modes are

(2.30)

Now, our rigidity/mass term is D/ρh, rather than EI/ρA for beams, and our wavenumber is now a two-dimensional wavevector,

(2.31)

since bending waves travel in the xy plane. The square of the magnitude of the wave vector determines the resonance frequency, as observed in Equation 2.30.

For thick plates, where shear resistance and rotary inertia are important, resonance frequencies cannot be computed with a closed-form equation, since the dependence of wave speed on frequency is so complicated (recall the thick-plate wave speed equation). Fortunately, there is an iterative way of computing resonance frequencies, where the magnitude of the wavenumber of the mode shape |kmn| can be compared to the wavenumber of free bending waves (computed using the thick-plate equation and recalling that kb = ω/cb). At resonance, the modal and free wavenumbers must agree.

Figure 2.13 shows how the resonance frequency of a thick-plate mode may be computed, provided the modal wavenumber kmn is known. First, generate the wavenumber-frequency plot of free bending waves. Next, equate kmn to kb on the ordinate of the plot to find the corresponding resonance frequency. The plot shows the free wavenumber-frequency curves calculated using both thick- and thin-plate theory for a steel 5–mm-thick plate and shows how resonance frequencies computed using thin plate theory become inaccurate at high frequencies.

Graph of resonance frequency of a mode shape in a thick flat plate, provided the modal wave number kmn is known, with ascending solid curve for thick plate and grayed line for thin plate free bending wave numbers.

Figure 2.13 Procedure for finding the resonance frequency of a mode shape in a thick flat plate. The example is for a steel 5-mm-thick plate. Matching the modal wave number kmn to the free bending wave number locates the corresponding resonance frequency

The procedure may be used for any combination of boundary conditions, provided the modal wavenumber can be computed. Also, the procedure applies to other shapes, such as circles and triangles, provided once again that the modal wavenumber can be computed from the mode shapes. The textbooks by Leissa [2] and Blevins [3] are useful references for the resonance frequencies and mode shapes of many plate and beam configurations.

For free or clamped boundary conditions, the mode shapes of rectangular flat plates are not simple standing sine waves, as shown in the examples in Figure 2.14, measured for a 0.2″ thick 12″ × 12″ piece of glass. As with beams, the waves within the plate look like sine functions. However, the free