Designing Quiet Structures: A Sound Power Minimization Approach

By Gary H. Koopmann and John B. Fahnline

This book is the first of its kind. It provides the reader with a logical and highly quantitative means of including noise as a parameter in the early design stages of a machine or structure. The unique and unified methodology builds upon the familiar disciplines of acoustics, structural dynamics and optimization. It also exemplifies the art of simplification - the essence of all good engineering design.

Strategies for designing quiet structures require extensive analytical and experimental tools. For computing the sound power from complex structures the authors recommend a new 3-D, lumped parameter formulation. This fully developed, user-friendly program can be applied generally to noise-control-by-design problems. Detailed instructions for running the application are given in the appendix as well as several sample problems to help the user get started.

The authors also describe a new instrument: a specially developed resistance probe used to measure a structure=92s acoustic surface resistance. As an example, the procedure is outlined for measuring the valve cover of an internal combustion engine. Indeed, throughout the book the reader is presented with actual experiments, numerical and physical that they can replicate in their own laboratory.

This is a must-have book for engineers working in industries that include noise control in the design of a product. Its practical and didactic approach also makes it ideally suited to graduate students.

• First text covering the design of quiet structures
• Written by two of the leading experts in the world in the area of noise control
• Strong in its integration of structural dynamics, acoustics, and optimization theory
• Accompanied by a computer program that allows the computation of sound power
• Presents numerous applications of noise-control-by-design methods as well as methods for enclosed and open spaces
• Each chapter is supported by homework problems and demonstration experiments

• Language: English
• Publisher: Academic Press
• Release date: Oct 13, 1997
• ISBN: 9780080504049

Gary H. Koopmann

Professor Gary Koopmann is the Director of the Center for Acoustics and Vibration at Pennsylvania State University, PA, USA. He is also a consultant to industry and government in the area of noise and vibration control.

Book preview

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CHAPTER 1

BASIC EQUATIONS OF ACOUSTICS

To understand the physics of noise control, it is first necessary to understand acoustics, the science of sound. To make the type of problems that we will address in this book tractable, we will only consider linear acoustics and further assume that the ambient quantities of pressure, density, velocity, etc. do not vary as a function of either time or space. As we shall see, these assumptions considerably simplify the associated mathematics. The scope of our required acoustic analysis is further narrowed by the type of problem which is typically encountered in the design of a quiet structure. In many applications, noise, the unwanted sound, is generated by the surface vibrations of a machine which in turn interact with the surrounding acoustic medium. The movement of the surface of the machine compresses the adjacent fluid and causes sound waves to radiate. Only this mechanism for generating noise will be considered, and thus mechanisms such as sound generation by turbulence, combustion, etc. are not considered. Even with these restrictions, a very broad range of design problems can be addressed, as will be seen subsequently.

1.1 Derivation of the Wave Equation

The spatial and time dependence of the sound waves radiating away from a vibrating structure are governed by the physics of the acoustic medium. The change in pressure associated with a sound wave is usually very small in comparison to the ambient pressure, so that the pressure, density, and particle velocity of an acoustic disturbance can be written in terms of an ambient quantity, denoted by a subscript 0, plus a small fluctuating quantity, denoted by a superscript. Thus, the pressure, density, and particle velocity at a point in the fluid are given as p = p0 + p′, ρ = ρ0 + ρ′, and v = v′, respectively, where v0 = 0 because there is no ambient fluid flow. Defining the acoustic variables in this form allows products of the variables to be linearized; for example, p v = (p0 + p′)v′ ≈ p0 v′ because the nonlinear term, p’ v’, is small in comparison to the linear term, p0 v’. The linearization process is revisited later in this section, after the partial differential equations governing sound waves have been derived.

The propagation and radiation of sound waves is governed by three basic equations of fluid dynamics: (1) the equation of continuity or conservation of mass, (2) Euler’s equation of motion of a fluid, and (3) the equation of state. Only a brief derivation of each of these equations is given in the subsequent text because similar derivations can be found in any book on acoustics, including Pierce (1989, pp. 6–20), Beranek (1986, pp. 16–23), Morse (1976, pp. 217–222), Morse and Ingard (1968, 227–305), Temkin (1981, pp. 1–58), and Kinsler et al. (1982, pp. 98–110). These three equations can be combined to give the wave equation, which is the basic partial differential equation governing the spatial and time dependence of an acoustic field.

Conservation of mass

The most physically intuitive of the basic equations of fluid dynamics is the equation of continuity or conservation of mass. To derive this equation, consider a fixed control volume V through which fluid is free to flow, as shown in Figure 1.1.

Figure 1.1 Surface S enclosing a volume of fluid V.

The total mass of fluid within the control volume at a particular moment is given as

so that the rate of change of mass within the control volume is

The rate of change of mass within the control volume can also be written in terms of the velocity of the particles flowing through the surface of the control volume as

where n is the outward normal direction at the point x of the surface S. Equating these quantities gives the integral form of the equation of continuity as

(1.1)

which simply implies that the mass of fluid within the control volume must be conserved. To convert from integral to differential form, take the time derivative within the integral on the left hand side of Equation (1.1) and convert the surface integral on the right hand side to a volume integral using Gauss’ theorem. Rewriting the result as a single volume integral yields

(1.2)

Because Equation (1.2) must be valid for any control volume within the fluid, the integrand must be zero such that

(1.3)

which is the desired result for the differential form of the equation of continuity.

Euler’s equation of motion

The second of the basic equations of fluid dynamics is Euler’s equation of motion, which is simply Newton’s second law for a fluid element in motion. To derive the equation of motion, consider a fixed volume of fluid, Vt, enclosed within an imaginary, deformable surface, St. The volume is referred to as a material volume, because the material within the volume does not change with time, and the subscript t indicates that the shape of the material volume changes with time. The momentum of the fluid within the material volume is affected by surface forces acting on St, and by body forces acting on all the particles within Vt. Mathematically, the rate of change of momentum of the fluid within the material volume can be written a

(1.4)

where fs constitutes the surface forces and fb constitutes the body forces. The surface forces include the pressure and shear forces exerted at the boundary of the material volume and the body force is primarily due to gravity. In the applications considered in this text, the body forces due to gravity and the shear forces due to viscosity are negligible. The surface forces due to the pressure outside the material volume can be written as

(1.5)

such that the surface integral in Equation (1.4) can be rewritten using Gauss’ theorem as

(1.6)

Equation (1.4) then reduces to

(1.7)

To convert Equation (1.7) to a differential form, the total differential of the integral on the left–hand–side needs to be brought inside the integral, which is complicated by the fact that the material volume can change with time. Without going into the details, which can be found in the text by Temkin (1981, pp. 16–20), the term on the left hand side of Equation (1.7) can be transformed