Errata of Vibroacoustic Simulation

This errata page is a collection of all bugs and mistakes that I noticed for the first edition. If you would like to propose some corrections please send a note to author@alexanderpeiffer.de.

A special thank you goes to Valentin who found all the bugs in chapter 7 and the legend bug in figure 8.38.

Chapter 1

Page 9 – Equation (1.37) – In the paragraph before I say signal x(t), thus the equations should be with x(t)

Page 13 – Equation above equation (1.56) – the first integral should read

\Delta E_{\rm cylcle} = \int_0^Tc_v\ddot{u}\,dt=\ldots 

Page 15 – Figure 1.11 – left spring should have stiffness k_{s1}

Page 16 – Line below Equation (1.77) – The second frequency should read \omega_2 = k_{s2}/m_2

Page 31 – Equation (1.147) – The last fraction should read \frac{E^2[fg]}{E[f^2]}

Page 33

Equation (1.162) – It should be \bm{S}_{fg}(\omega)=\bm{S}_{gf}(-\omega)

The last equation must be similar to equation (1.164), thus \bm{G}^*_k(\omega)\bm{F}_k(\omega) should be in boldsymbol

Page 34 – Equation (1.167), \bm{G}^*_m(\omega)\bm{F}_m(\omega) should also be in boldsymbol

Page 38 – Figure 1.24 – To be consistent with the text it should be \ldots f_n(t)\ldots f_N(t) and \ldots g_m(t)\ldots g_M(t)

Chapter 2

Page 44 – Last enumeration – Item 3 should be “The momentum flow out of the volume”

Page 47 – Equation (2.25) – The velocity potential should be \mathbf{v}'= -\nabla\Phi

Page 48 – Equation (2.29) – and consequently … p = \rho_0\frac{\partial\Phi}{\partial t}

Page 53

Table 2.2 – The acoustic velocity formula should read -\frac{1}{j\omega c_0}\nabla\bm{p}

Equation (2.59) – The last expression should be \bm{A}e^{-\frac{\eta}{2}kx}e^{j(-kx+\omega t)}

Page 54

Equation (2.65) – It should read \left( \frac{1}{c_0^2}\frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial r^2} \right) (r\Phi) = 0

The equation reference (2.30) is not correct it should be (2.65)

Page 55

Equation (2.68) – -\frac{\partial\Phi}{\partial r} = v_R

Equation (2.69) – \bm{A} = -\bm{v}_R \frac{R^2}{1-jkR} e^{jkR}

Page 57

Equation (2.85) – It should by 4\pi R^2 instead of 4\pi a^2

Equation (2.86) – The time average must be used \left\langle \Pi \right\rangle_T instead of \Pi

Page 58 – Equations (2.87) and (2.88) – Same correction as for equation (2.86)

Page 62 – Equations (2.116) and (2.117) – The impedance symbols should be boldmath \bm{z}_1,\bm{z}_2

Page 58 – Equation (2.98) – This should read m = -Re (\frac{(4\pi R^2)^2}{j\omega}\frac{\bm{p}}{\bm{Q}}) = \frac{(4\pi R^2)^2}{\omega}Im(\bm{Z}_a)

Page 70 – Above equation (2.155) – It should read kR\le 0.5.

Page 115 – Equation (3.230) – The second term should read \frac{1}{2} Re\left(\frac{\hat{F}^2}{\\bm{Z}_0}\right)

Chapter 3

Page 80 – Equation (3.22) – \sigma_{ij}=\frac{dF_i}{dA_j}

Page 111 – Paragraph below (3.21b) – Insert as first sentence: Considering k_B as the real and positive solution, hence k_B=k_{B2}.

Page 113 – Equations (3.222)-(3.224) – Replace k by k_B

Page 114 – Equation (3.225) – Replace k by k_B

Page 115 – Equation (3.231) – In the nominator it should read \hat{F}^2

Chapter 4

Page 120 Figure 4.1 – The radius is R and not a

Page 135

Figure 4.12 – Missing unit at y-axis label. Use n(\omega)/s

Equation (4.76) – It would be more consistent to use m as index in the sum

\bm{p}(\vec{r}) = \sum_{m=0}^\infty \bm{p}'_m \Phi_m(\bf{r})

Equation (4.77) – The first integral should read as follows

\int_{V} \sum_m^\infty\bm{p}_m'\Phi_m(\bf{r}) (\bm{k}^2 - k_n^2) \Phi_n(\bf{r}) dV

Equation (4.78) – There must be a \bm{p}_m' in the sum over m.

Page 136 Equation (4.84) – Remove conjugate star, not required for real modes

Page 137

Paragraph below Equation (4.86) – it should read R\ll \lambda

Equation (4.87) – Third term must be 1 ⁄ 2 \bm{Z v}^2

Page 141 Last paragraph – first element and not fist element

Page 141 Equation (4.103) – For better consistence with Figure 4.16 {\bm Q}_2 = {\bm Q}_3 = 0 would be better

Chapter 5

Page 151 Section 5.3.1 – First sentence – it should be in-homogenous form.

Chapter 6

Page 161 Equation (6.3) – The square should be removed and it should read \Psi_{\rm mean} = \ldots

Page 166

In the first line the formula should read \bf{k}\Delta\bf{r}=k\Delta r\cos(\vartheta)

Equation 6.21 – The \sin(\vartheta) in the first line should be removed

Page 169

Equations (6.33)-(6.35) – The \frac{1}{M} factor is missing for all sums.

Equations (6.34)-(6.35) – The last \bm{\Phi} should read \bm{\Phi}_i

Page 170

Equations (6.37) – Remove the hat from all pressure symbols p_\text{rms} is correct

Equations (6.38) – Replace \hat{p} by p^2_\text{rms}

Page 175/176 Figure 6.9 and 6.10 – Right hand side. Label of y-axis: z_0 should not be boldsymbol

Page 180

Equation (6.63) – The nominator should read \hat{F}^2

Equation in section 6.4.1.4 – \hat{w}^2(r) should not be boldsymbol. However, for figures 6.17 and 6.18 I used equation (3.225)

Section 6.4.2 – It is worth mentioning that the plate is made of aluminum

Figure 6.14 – The density variation in the caption should be 2% not 1%

Page 192 – Equation below (6.92) – Second sine should include a y and not an x

Chapter 7

Page 207 Equation (7.18) – The last radiation stiffness should be [\bm{D}_{dir}^{(2)}] instead of [\bm{D}_{dir}^{(3)}]

Page 208

Section 7.3.1 heading – Exchange m and n

Globally it would be better to stay with the original heading and exchange m and n in the text and all formulas, because his would correspond better to figure 7.5

Equation (7.22) – The index (n) is missing, it should be [S_{qq}^{(n)}]

Page 210 Equations (7.28) and (7.30) – The index i,j should be used consequently in all formulas as sum index

Page 211 Equation (7.32) – The factor E_m is missing after both large parenthesis

Page 215

Equation (7.40) – \{\bm{q}\} instead of \{\bm{u}\}

Last line of last paragraph – response instead of reponse

Page 218 Figure 7.13 – The SEA-Matrix must read [L'] instead of [A]

Page 220 Equation (7.47) – The modal coordinate \{ {\bm q}'\} is missing

Chapter 8

Page 226 Last paragraph – … is modified by an \cos\vartheta factor instead of \sin\vartheta

Page 227 First sentence – \phi must be replaced by \varphi

Page 231 Figure 8.5 – Label of y-axis: TL/dB instead of TL in dB Page

Page 233 Figure 8.8 – Due to a bug in the code spring curves are wrong. See new figure.

Transmission loss comparison between different spring stiffness's and direct connection
Figure 8.8 Transmission loss of a rectangular 2-beam junction, without and with spring coupling. Source: Alexander Peiffer

Page 234 Figure 8.10 – Label of y-axis: TL/dB instead of TL in dB

Page 238 This item in description – D''_{\rm dir} instead of D_1

Page 242 Equation (8.74) – The equal sign before the \approx must be removed

Page 239 Equation (8.51) – in the first row the integral is missing, The second integral runs over S'

\Pi  = \frac{1}{2} \int_S Re\{ {\bm v}_z^*(x,y){\bm p}(x,y) \} dS 

Page 240 Equation (8.59) – last sine should read \sin(k_y y)

Page 245 Equation (8.73) – The stiffness is complex, thus \bm{D}_{\rm dir}''^{plate}=\ldots

Page 247 Equation (8.94) – The stiffness is complex, thus and there is an \omega missing in the last term

\bm{D}_{\rm tot}''(k_a,\vartheta)=m''\omega^2\left[\frac{ k_a^4\sin^4\vartheta}{k_B^4} -1 \right] 
			- j\omega\frac{2 \rho_0c_0}{\cos\vartheta} 

Page 250

Equation (8.101) – it should read \left[\bm{D}_s\right]=-\omega^2\left[M\right]

Equation (8.103) – remove useless \left\langle\tau\right\rangle_E in the middle

Page 251 Equation (8.105) – use k_s and c_s

Page 253 After Equations (8.108a-d) – it should read B=\frac{Eh^3}{12(1-\nu^2)}

Page 255 Coordinate vector after Equation (8.118) – use bold {\bm\beta}_{x,e}

Page 256

Equation (8.21) – the displacement vector is missing after the first matrix

Equation (8.23) – the amplitude vector is missing after the second matrix

Equation (8.24) – the displacement vector is missing after the first matrix and \bm{D}'_{34}=\mu_{B1}\mu_{B2} + \nu k_x^2

Page 258 Equation (8.131) – is should read k_x = k_v^{(m)} \cos \phi

Page 261

Equation (8.143) – wrong index index in excitation cspd. It should be S^{(m)*}_{e,ff}

Equation (8.147) – wrong index index in center matrix cspd. It should be S^{(m)*}_{e,qq}

Page 263

Figure 8.30 – wavenumber vector labels k_S and k_L must be exchanged

Equation (8.151) – is should read \cos \phi_w^{(n)} = \frac{k_x}{k_w^{(n)}}

Page 267 Figure 8.38 – The legends must be exchanged. The blue line belongs to \tau_{1B,2B}

Chapter 9

Page 287

Equation (9.76) – – Should be \bm{z}/A_c

Paragraph before Equation (9.78) – Should be small \bm{z}

Page 299

Equation (9.107d) – Division by porosity is missing. Use \rho_0c_0^2/\Phi in nominator

Equation (9.107e) – Variables should be in boldsymbol. Use {\bm z}=\sqrt{\bm{K}\bm{\rho}_{eq}}

Equation (9.107f) – Variables should be in boldsymbol. Use {\bm\Gamma}=j\omega\sqrt{\bm{\rho}_{eq}/\bm{K}}

Equation (9.107g) – Variables should be in boldsymbol. Use \bm{c}=j\omega/\bm{\Gamma}

Page 301

Equation (9.109) – The last term of the denominator should read \bm{T}_{22}\frac{\bm{z}_s}{\bm{z}_r}

Equation (9.110) –

{\bm R}(\vartheta)=\frac{{\bm z}(\vartheta)-\frac{{\bm z}_s(\vartheta)}{\cos(\vartheta)}}{{\bm z}(\vartheta)+\frac{{\bm z}_s(\vartheta)}{\cos(\vartheta)}}

Page 302

Paragraph below Equation (6.72) – z = ρ0c0

Next paragraph – z = az0

Page 309 Figure 9.40 – Label of y-axis: First ‘perf’ index should be in roman

Page 313 Figure 9.44 – Figure caption has an index error. It should read m''_1=m''_2=1\text{kg/m}^2

Page 314 Figure 9.45 – Figure caption has an index error. It should read m''_1=m''_2=1\text{kg/m}^2

Page 316 Figure 9.47 – Wrong figure. Should be as follows:

Corrected figure 9.47
General shape of the double wall transmission and the relationship to main parameters.

Chapter 10

Section 10.2.1

I really apologize for all the bugs in this section. This is an excellent example that some mistakes of a derivation are not found, because the final result was know before. In fact all equations of this section are wrong, except the final result!

Page 321

In equation (10.3) the diagonal must be corrected

\omega\begin{bmatrix}
n_1(\eta_{11} + \eta_{12}) & -n_2 \eta_{21}\\
-n_1 \eta_{12} & n_2 (\eta_{22} + \eta_{21})
\end{bmatrix}
\begin{Bmatrix}
E_1/n_1(\omega) \\ E_2/n_2(\omega) 
\end{Bmatrix} = \begin{Bmatrix} \Pi_1 \\ \Pi_2 \end{Bmatrix}

Page 322

This has consequences on equation (10.4)

\begin{Bmatrix} E_1/n_1 \\ E_2/n_2 \end{Bmatrix} 
= -\frac{1}{\omega^2( \eta^2_{12} n_1 - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})) }  
\begin{bmatrix}
\frac{n_2}{n_1}\eta_{22} + \eta_{12} & \eta_{12}\\
			\eta_{12} & \eta_{11} + \eta_{12}
\end{bmatrix}
\begin{Bmatrix} \Pi_1 \\ \Pi_2 \end{Bmatrix}

and on (10.5)

E_1 = - \frac{n_1\eta_{12} +n_2\eta_{22}}{\omega^2(n_1\eta^2_{12}  - n_2(\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1 

and (10.6)

	E_2  = - \frac{n_2\eta_{12} }{\omega^2(n_1\eta^2_{12}  - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

Not to speak about (10.7) and (10.8)

p_{1,\rm rms} = -\frac{\rho_0}{2\pi^2c_0 n_1}
		\frac{n_1\eta_{12} +n_2\eta_{22}}{n_1\eta^2_{12} - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1
p_{2,\rm rms} = -\frac{\rho_0}{2\pi^2c_0}
		\frac{\eta_{12} }{n_1\eta^2_{12} - n_2 (\eta_{11}+\eta_{12})(\eta_{22}+\eta_{21})}\Pi_1

The pressure ratio (10.9) also has totally different look

\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}} = \frac{n_1(\omega)\eta_{12}}{n_1(\omega)\eta_{12}+n_2(\omega)\eta_{22})}

Leaving different result for the coupling loss factor in equation (10.10). Note the involved ratio of modal densities.

\eta_{12} = \frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{n_2(\omega)}{n_1(\omega)\left(1-\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\right)} \approx \eta_{22}\frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{n_2(\omega)}{n_1(\omega)}

To be precise, the volume in equation (10.11) is V_2

\eta_{22}  = \frac{A_s c_0}{4 V_2 \omega}

Page 323

The volume in the \tau\eta relationship must be V_1 and this – in combination with the modal density ratios – gives the final result in equation (10.12).

\left<\tau_{12}\right>  = \frac{4 V_1 \omega}{c_0 S_j} \eta_{12} = \frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{V_1 n_2(\omega)}{V_2 n_1(\omega)} \frac{A_s}{S_j}
                            = \frac{p^2_{2,\rm rms}}{p^2_{1,\rm rms}}\frac{A_s}{S_j}

Page 325 Figure 10.4 – Label at y-axis should be n(\omega)/s

Page 327

Figure 10.7 and 10.8 are not correct due to a bug in the pyva software

Subsystem energy result of two plate example of section 10.4
Figure 10.7 Energy result of two plates
Subsystem velocity result of two plate example of section 10.4
Figure 10.8 Velocity results of two plates

Page 328 Figure 10.9 is not correct due to a bug in the pyva software

Subsystem velocity result of two plate example of section 10.5
Figure 10.9 Velocity results of the two plates example excited by 1N force

Chapter 12

Page 361 Figure 12.1 – The Helmholtz number should be He = \frac{L}{\lambda}

Page 372 – Section 12.3 Last sentence of first paragraph – prototype instead of pototype.

Appendix A

Page 424 Equations (A.68),(A.70)-(A.73) – The force symbol should be in capital letters \bm{F}

Appendix B

Page 426 Equation (B.17) – The upper index should read \Psi_{in,L}

Page 429 Figure B.2 – wavenumber vector labels k_S and k_L must be exchanged

Page 436 Equations (B.79-80) – The upper right coefficient in the matrix requires a second right parenthesis

Index

Page 445 coupling loss factor, non-resonant and resonant instead of resonsant

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