Wave Propagation Approach for Structural Vibration

By Chongjian Wu

This book is intended for researchers, graduate students and engineers in the fields of structure-borne sound, structural dynamics, and noise and vibration control.

Based on vibration differential equations, it presents equations derived from the exponential function in the time domain, providing a unified framework for structural vibration analysis, which makes it more regular and normalized. This wave propagation approach (WPA) divides structures at “discontinuity points,” and the waves show characteristics of propagation, reflection, attenuation, and waveform conversion. In each segment of the system between two “discontinuity points,” the governing equation and constraint are expressed accurately, allowing the dynamic properties of complex systems to be precisely obtained.

Starting with basic structures such as beams and plates, the book then discusses theoretical research on complicated and hybrid dynamical systems, and demonstrates that structural vibration canbe analyzed from the perspective of elastic waves by applying WPA.

• Language: English
• Publisher: Springer
• Release date: Oct 28, 2020
• ISBN: 9789811572371

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© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021

C. WuWave Propagation Approach for Structural VibrationSpringer Tracts in Mechanical Engineeringhttps://doi.org/10.1007/978-981-15-7237-1_1

1. The Basic Theory of Structure–Borne Noise

Chongjian Wu¹  

(1)

Wuhan, PR China

Chongjian Wu

Email: [e-mail]

Depth determines breadth!

—Qian Xuesen

Before we discuss the WPA method, it is necessary to examine the basic theory of structural vibration noise, including basic parameters such as wave number, wavelength, and lateral displacement. The examination of continuous systems such as the bending vibration of a beam and a plate is the focus of this chapter. The vibration modes and natural frequencies of beams and plates are discussed and then the sound pressure, sound power, and sound radiation efficiency of simple structures are analyzed and discussed.

1.1 The Vibration Modes of Beams

1.1.1 Basic Equations

The structural wave is the basic parameter of structural vibration and acoustic radiation, which is directly linked to the target control parameters. The theory of bending vibration of beams and plates is derived from the fourth-order differential equation [1]:

$$\left. {\begin{array}{*{20}c} {\nabla^{2} \left( {\nabla^{2} \tilde{w}} \right) + \rho S\frac{{\partial^{2} \tilde{w}}}{{\partial t^{2} }} = \tilde{p}_{o} } \\ {\tilde{w} = \tilde{w}(x,y,z,t)} \\ {\nabla^{2} = \frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }} + \frac{{\partial^{2} }}{{\partial z^{2} }}} \\ \end{array} } \right\}$$

(1.1)

where

$$\nabla^{2}$$

Laplace operator;

$$\tilde{w}$$

Lateral displacement of the structure;

$$\rho$$

Material density of the beam;

S

Cross-sectional area of the structure;

$$\tilde{p}_{0} (x,y,t)$$

External excitation harmonic force

The displacement $$\tilde{w}$$ is linked not only to the spatial coordinate of the particle $$(x,y,z)$$ but also to the time. It is important to calculate the structural mode and modal frequency when conducting a structural analysis. The excitation features and the generation of the state matrix are based on the structure mode and modal frequency.

With consideration of the Bernoulli–Euler beam as shown in Fig. 1.1, the vibration equation of Eq. (1.1) degenerates to free vibration [2]:

../images/489592_1_En_1_Chapter/489592_1_En_1_Fig1_HTML.png

Fig. 1.1

A schematic diagram of a vibrating beam

$$\frac{{\partial^{4} \tilde{w}(x,t)}}{{\partial x^{4} }} + \frac{\rho S}{EI} \cdot \frac{{\partial^{2} \tilde{w}(x,t)}}{{\partial t^{2} }} = 0$$

(1.2)

where

EI

Bending stiffness of the beam;

E

Young’s modulus of the material;

$$I = bh^{3} /12$$

Cross-sectional moment of inertia of the beam, where b is the width and h is the thickness of the beam.

For harmonic vibration, the displacement response can be divided into two parts: the space function and time function according to the process of separating variables:

$$\tilde{w}(x,t) = w(x) \cdot\Phi (t)$$

(1.3)

where

$$w(x)$$

Structure mode shape function;

$$\Phi (t)$$

Time correlation function

When substituting Eq. (1.3) into Eq. (1.2) and dividing the variable for time t and space x, two ordinary differential equations are attained as

$$\frac{{\partial^{4} w(x)}}{{\partial x^{4} }} - \frac{{\rho S\omega^{2} }}{EI}w(x) = 0$$

(1.4)

$$\frac{{\partial^{2}\Phi (t)}}{{\partial t^{{_{2} }} }} + \omega^{2}\Phi (t) = 0$$

(1.5)

where $$\omega$$ is the circular frequency.

Set

$$k^{4} = \frac{{\rho S\omega^{2} }}{EI}$$

(1.6)

Therefore, Eq. (1.4) can be rewritten as

$$\frac{{\partial^{4} w(x)}}{{\partial x^{4} }} - k_{n}^{4} w(x) = 0$$

(1.7)

where $$k_{n}$$ is the complex wave number of the beam’s bending wave,

$$n = 1,2,3,4$$

.

For beam-type structures, the general boundary conditions are as follows:

(1)

Simply supported boundary condition (S-S beam):

$$\left. {\begin{array}{*{20}l} {\tilde{w}\left( {0,t} \right) = 0,\quad \frac{{\partial^{2} \tilde{w}\left( {0,t} \right)}}{{\partial x^{2} }} = 0} \hfill \\ {\tilde{w}\left( {L_{x} ,t} \right) = 0,\quad \frac{{\partial^{2} \tilde{w}\left( {L_{x} ,t} \right)}}{{\partial x^{2} }} = 0} \hfill \\ \end{array} } \right\}$$

(1.8)

(2)

Clamped boundary condition (C-C beam):

$$\left. {\begin{array}{*{20}l} {\tilde{w}\left( {0,t} \right) = 0,\quad \frac{{\partial \tilde{w}\left( {0,t} \right)}}{\partial x} = 0} \hfill \\ {\tilde{w}\left( {L_{x} ,t} \right) = 0,\quad \frac{{\partial \tilde{w}\left( {L_{x} ,t} \right)}}{\partial x} = 0} \hfill \\ \end{array} } \right\}$$

(1.9)

(3)

Free boundary condition (F-F beam):

$$\left. {\begin{array}{*{20}l} {\frac{{\partial^{2} \tilde{w}\left( {0,t} \right)}}{{\partial x^{2} }} = 0,\quad \frac{{\partial^{3} \tilde{w}\left( {0,t} \right)}}{{\partial x^{3} }} = 0} \hfill \\ {\frac{{\partial^{2} \tilde{w}\left( {L_{x} ,t} \right)}}{{\partial x^{2} }} = 0,\quad \frac{{\partial^{3} \tilde{w}\left( {L_{x} ,t} \right)}}{{\partial x^{3} }} = 0} \hfill \\ \end{array} } \right\}$$

(1.10)

The general solution to the differential equation Eq. (1.7) is

$$w(x) = A\sin (kx) + B\cos (kx) + C\sinh (kx) + D\cosh (kx)$$

(1.11)

where A, B, C, and D are unknown coefficients, respectively,

$$k^{4} = \rho S\omega^{2} /EI$$

.

Using simply supported beams as an example and substituting the displacement of the beam into the boundary conditions, it can be used to solve the unknowns A, B, C, and D.

At the left end $$x = 0$$ , substituting Eq. (1.11) into Eq. (1.8), we get

$$w(0) = B + D = 0$$

(1.12)

$$\frac{{\partial^{2} w(0)}}{{\partial x^{2} }} = k^{2} ( - B + D) = 0$$

(1.13)

Thus,

$$B = D = 0$$

. At the right end $$x = L_{x}$$ , we get

$$w(L_{x} ) = A\sin (kL_{x} ) + C\sinh (kL_{x} ) = 0$$

(1.14)

$$\frac{{\partial^{2} w(L_{x} )}}{{\partial x^{2} }} = k^{2} \left[ { - A\sin (kL_{x} ) + C\sinh (kL_{x} )} \right]{ = 0}$$

(1.15)

From Eqs. (1.14) and (1.15), we get

$$A\sin \left( {kL_{x} } \right) + C\sinh \left( {kL_{x} } \right) = 0$$

(1.16)

Since

$$\sinh (kL_{x} ) \ne 0$$

, provide $$kL_{x} \ne 0$$ , and therefore,

$$\begin{array}{*{20}c} {C = 0,} & {k = \frac{n\pi }{{L_{x} }},} & {A = \sqrt {\frac{2}{{mL_{x} }}} } \\ \end{array}$$

(1.17)

where m is beam mass per unit length.

When substituting Eq. (1.17) into Eq. (1.11), the nth mode shape function for a simply supported beam can be attained as

$$w_{n} (x) = A\sin \left( {\frac{n\pi }{{L_{x} }}x} \right)$$

(1.18)

When substituting Eq. (1.17) into Eq. (1.6), the corresponding natural frequencies can be written as

$$\omega_{n} = \sqrt {\frac{EI}{m}} \cdot \left( {\frac{n\pi }{{L_{x} }}} \right)^{2}$$

(1.19)

The mode shapes are orthogonal with respect to the mass and stiffness distribution [1, 3]:

$$\int\limits_{0}^{{L_{x} }} {mw_{j} (x)w_{k} (x){\text{d}}x = \mu_{j} \delta_{jk} }$$

(1.20)

$$\int\limits_{0}^{{L_{x} }} {EI\frac{{\partial^{2} w_{j} (x)}}{{\partial x^{2} }} \cdot \frac{{\partial^{2} w_{k} (x)}}{{\partial x^{2} }}{\text{d}}x = \mu_{j} \omega_{j}^{2} \delta_{jk} }$$

(1.21)

$$\delta_{jk} = \left\{ {\begin{array}{*{20}c} 1 & {j = k} \\ 0 & {j \ne k} \\ \end{array} } \right.$$

where

$$\delta_{jk}$$

Kronecker delta symbol;

$$\mu_{j}$$

Modal mass of the nth mode.

The generalized mass corresponding to the mode shapes in Eq. (1.18) is $$mL_{x} /2$$ .

As a result of the mode shapes being orthogonal to each other, the response of the beam can be expressed at any arbitrary point as a linear combination of these mode shape functions. This is known as the mode superposition method. The WPA method selects different technical paths, as shown in Sect. 2.​4 of Chap. 2.

$$\tilde{w}(x,t) = \sum\limits_{n = 1}^{\infty } {w_{n} (x) \cdot\Phi _{n} (t)}$$

(1.22)

As a result of the infinite modes in a continuous system, it is necessary to intercept a finite number of modes, such as N. In this way, the analysis of the complex system is simplified, and analytical accuracy is certain.

For other boundary conditions, the structural mode shapes and natural frequencies are listed in Table 1.1.

Table 1.1

The structural mode shapes and natural frequencies

1.1.2 MATLAB Examples

Consider equations in Table 1.1 for this example. Using the MATLAB program to calculate the first 5th mode shapes of the beam structure, we can get the structure modal shape functions corresponding to different boundary conditions listed in the Table, as shown in Figs. 1.2, 1.3, 1.4 and 1.5.

../images/489592_1_En_1_Chapter/489592_1_En_1_Fig2_HTML.png

Fig. 1.2

The first 5th mode shapes of simply supported beams

../images/489592_1_En_1_Chapter/489592_1_En_1_Fig3_HTML.png

Fig. 1.3

The first 5th mode shapes of the clamped supported beams on both sides

../images/489592_1_En_1_Chapter/489592_1_En_1_Fig4_HTML.png

Fig. 1.4

The first 5th mode shapes of clamp-free beams

../images/489592_1_En_1_Chapter/489592_1_En_1_Fig5_HTML.png

Fig. 1.5

The first 5th mode shapes of clamped–simply supported beams

1.2 The Vibration Modes of Plates

1.2.1 Basic Equations

The previous section discussed the vibration of beam-type structures. In this section, we will extend the two-dimensional structure vibration. The governing equation and mode shape for the free vibration of the anisotropic and non-damping plates will be analyzed. Equation (1.1) can also be derived [1] as

$$D\nabla^{4} \tilde{w}(x,y,t) + m_{s} \frac{{\partial \tilde{w}^{2} (x,y,t)}}{{\partial t^{2} }}$$

(1.23)

$$\left. {\begin{array}{*{20}l} {\nabla^{2} = \frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \hfill \\ {D = \frac{{h^{3} E}}{{12(1 - \upsilon^{2} )}}} \hfill \\ \end{array} } \right\}$$

(1.24)

where

$$\tilde{w}(x,y,t)$$

Lateral displacement of the plate;

$$m_{s} = \rho h$$

Area density of the plate, where $$\rho$$ and h are the volume density of the plate and the thickness of the plate, respectively;

$$\upsilon$$

Poisson’s ratio

For harmonic free vibration, the $$\tilde{w}(x,y,t)$$ can be expressed as the superposition of an infinite number of mode shape functions as follows:

$$\tilde{w}(x,y,t) = \sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {w_{mn} (x,y)\Phi _{mn} ( {\text{t)}}e^{j\omega t} } }$$

(1.25)

with the properties

$$\int\limits_{0}^{{L_{x} }} {\int\limits_{0}^{{L_{y} }} {mw_{mn} (x,y)w_{jk} (x,y)} \,} {\text{d}}y{\text{d}}x = \left\{ {\begin{array}{*{20}c} {M_{mn} } & {m = j,n = k} \\ 0 & {\text{other}} \\ \end{array} } \right.$$

(1.26)

where

$$w_{mn}$$

the modal amplitude of the $$(m,n){\text{th}}$$ mode of the plate;

$$M_{mn}$$

the $$(m,n){\text{th}}$$ modal mass

When substituting Eq. (1.25) into Eq. (1.23), this yields the following results:

$$D\left\{ {\frac{{\partial^{4} }}{{\partial x^{4} }} + 2\frac{{\partial^{4} }}{{\partial x^{2} \partial x^{2} }} + \frac{{\partial^{4} }}{{\partial y^{4} }}} \right\}w_{mn} (x,y) - \omega_{mn}^{2} m_{S} w_{mn} (x,y) = 0$$

(1.27)

where $$\omega_{mn}$$ is the $$(m,n){\text{th}}$$ natural frequency.

The structural mode shape functions can be randomly selected as long as they are quasi-orthogonal and both of them satisfy the boundary conditions. This is a method of dimensionality reduction; that is, the binary equation is divided into two univariate equations. The mode shape functions can be written as the product of two independent beam functions [4, 5]:

$$w_{mn} (x,y) = X_{m} (x) \cdot Y_{n} (y)$$

(1.28)

The shape functions $$X_{m} (x)$$ and $$Y_{n} (y)$$ can be randomly selected only if they are quasi-orthogonal and satisfy the boundary conditions. And

$$\begin{array}{*{20}c} {\int\limits_{o}^{{L_{x} }} {X_{j} (x)X_{k} (x){\text{d}}x} = \int\limits_{o}^{{L_{x} }} {\frac{{\partial^{ 2} X_{j} (x)}}{{\partial x^{ 2} }}\frac{{\partial^{ 2} X_{k} (x)}}{{\partial x^{ 2} }}{\text{d}}x = 0} } & {(j \ne k)} \\ \end{array}$$

(1.29)

$$\begin{array}{*{20}c} {\int\limits_{0}^{{L_{x} }} {Y_{j} (y) \cdot Y_{k} (y){\text{d}}y} = \int\limits_{o}^{{L_{x} }} {\frac{{\partial^{ 2} Y_{j} (y)}}{{\partial y^{ 2} }} \cdot \frac{{\partial^{ 2} Y_{k} (y)}}{{\partial y^{ 2} }}{\text{d}}y} = 0} & {(j \ne k)} \\ \end{array}$$

(1.30)

From Eq. (1.25), by using the orthogonal relationship in Eqs. (1.29) and (1.30), the natural frequencies are given by [3]

$$\omega_{mn} = \sqrt {D/m_{S} } \cdot \sqrt {\frac{{I_{1} I_{2} + 2I_{3} I_{4} + I_{5} I_{6} }}{{I_{2} I_{6} }}}$$

(1.31)

where

$$\begin{array}{*{20}c} {I_{1} = \int\limits_{0}^{{L_{x} }} {\frac{{\partial^{ 4} X_{m} (x)}}{{\partial x^{ 4} }}} X_{m} (x){\text{d}}x,} & {I_{2} = \int\limits_{0}^{{L_{y} }} {\left[ {Y_{n} (y)} \right]^{2} {\text{d}}y} } \\ \end{array}$$

(1.32)

$$\begin{array}{*{20}c} {I_{3} = \int_{0}^{{L_{x} }} {\frac{{\partial^{ 2} X_{m} (x)}}{{\partial x^{ 2} }}} X_{m} (x){\text{d}}x,} & {I_{4} = \int_{0}^{{L_{y} }} {\frac{{\partial^{ 2} Y_{n} (y)}}{{\partial y^{ 2} }}Y_{n} (y){\text{d}}y} } \\ \end{array}$$

(1.33)

$$\begin{array}{*{20}c} {I_{5} = \int_{0}^{{L_{y} }} {\frac{{\partial^{ 4} Y_{n} (y)}}{{\partial y^{ 4} }}} Y_{n} (y){\text{d}}y,} & {I_{6} = \int_{0}^{{L_{x} }} {\left[ {X_{n} (x)} \right]^{2} {\text{d}}x} } \\ \end{array}$$

(1.34)

For simply supported boundaries, we can select the shape functions as follows:

$$\left. {\begin{array}{*{20}c} {X_{m} (x) = \sin \left( {k_{m} x} \right)} \\ {Y_{n} (y) = \sin \left( {k_{n} y} \right)} \\ \end{array} } \right\}$$

(1.35)

where

$$k_{m}$$

Wave number in the direction-x,

$$k_{m} = m\pi /L_{x}$$

;

$$k_{n}$$

Wave number in the direction-y,

$$k_{n} = n\pi /L_{y}$$

.

For a clamped plate, the shape functions can be selected as follows:

$$X_{m} (x) = \cosh \left( {\frac{{\lambda_{m} x}}{{L_{x} }}} \right) - \cos \left( {\frac{{\lambda_{m} x}}{{L_{x} }}} \right) - \beta_{m} \left[ {\sinh \left( {\frac{{\lambda_{m} x}}{{L_{x} }}} \right) - \sin \left( {\frac{{\lambda_{m} x}}{{L_{x} }}} \right)} \right]$$

(1.36)

$$Y_{n} \left( y \right) = \cosh \left( {\frac{{\lambda_{n} y}}{{L_{y} }}} \right) - \cos \left( {\frac{{\lambda_{n} y}}{{L_{y} }}} \right) - \beta_{n} \left[ {\sinh \left( {\frac{{\lambda_{n} y}}{{L_{y} }}} \right) - \sin \left( {\frac{{\lambda_{n} y}}{{L_{y} }}} \right)} \right]$$

(1.37)

$$\beta_{m} = \frac{{\cosh \lambda_{m} - \cos \lambda_{m} }}{{\sinh \lambda_{n} - \sin \lambda_{n} }}$$

where $$\lambda_{m}$$ and $$\lambda_{n}$$ are the roots for the equation

$$\cosh \lambda \cos \lambda = 1$$

. Notice that

$$\beta_{Z} \approx (2Z + 1)/2$$

for large values of the integer