49 Figure 3.3: (a) Experimental setup for input and radiated sound power measurement, (b) Simply supported boundary condition. 63 Figure 3.19: Comparison of sound pressure radiation pattern (left) with vibration displacement pattern (right) on the side wall of the rectangular duct: (a) 54 Hz, (b) 87 Hz.
General introduction
Another drawback is that measuring the sound intensity closer to the surface is not possible due to the presence of the probe. The advantage of this method is that sound pressure, particle velocity and sound intensity or sound power can be obtained at the source plane.
Literature study
- Breakout noise characterization
- Modal parameter estimation
- Near-field Acoustic Holography techniques
- Inverse Numerical Acoustics (INA)
It is important to study modal parameters of the ducts to understand the sound radiation mechanism, which can be obtained by its free vibration analysis. Experimental results can be used as a reference (i) to modify existing structure and to optimize structure design by an iteration process (ii) to validate analytical models for refining future designs [39].
Motivation and research objectives
Research objectives
Investigate the effect of duct connection conditions on the breakout noise and modal parameters of the rectangular ducts. Investigate applicability of the inverse technique to reconstruct sound source at structural-acoustically coupled frequencies.
Structure of the Thesis
Research plan of investigation
Introduction
Theoretical formulation
Pressure field inside the duct and wall vibration velocity
The pressure field inside the duct in terms of uncoupled acoustic mode forms is given as,. The complex amplitude of the nth acoustic mode in structural and acoustic excitation is given as [21].
Calculation of Sound power and Radiation efficiency
Selection of strongly coupled acoustic and structural modes is done based on a transfer factor [18] given as, . 𝑆𝑓 is flexible duct wall area and is given by La x Lc, and vibration velocity of structure is given by 〈ば〉2.
Numerical modelling
- Uncoupled Structural Model
- Uncoupled Acoustical Model
- Acoustic-Structural Coupled Model
- Calculation of Sound power radiation
𝑃 (2.15) 𝑊𝑃 = 𝜌0𝑐0𝑆𝑓 〈 𝒘 〉2 (2.16) where, 𝑊 same root mean square speed as that structure [22]. A numerical model for calculating the sound radiated by flexible duct walls is shown in Fig.
Results and discussion
- Uncoupled structural modes
- Mode shapes representation, grouping and calculating net volume displacement
- Uncoupled Acoustic modes
- Comparison of coupled and uncoupled response
- Calculation of radiation efficiency for strongly coupled structural modes
- Comparison of duct radiation efficiencies with a plate
- Calculation of radiation efficiency for one acoustic mode coupled to multiple
- Total radiation efficiency and Radiated sound power
It is clear that the equivalent plate model can be effectively used to calculate modal radiation efficiency. This calculated radiation efficiency is compared with radiation efficiency of a simply supported plate with dimensions of (1.4 m x 1.5 m).
Summary
Modal grouping based on free vibration research shows that group-1 (Symmetry-Symmetry) modes with (odd-odd) indices are more effective sound emitters and are thus justified based on radiation efficiency results. The proposed equivalent plate model for rectangular channels can be effectively used to predict the free vibration behavior and sound radiation characteristics.
Introduction
Calculation of TTL
- Determination of input sound power
- Determination of radiated sound power
- Radiation efficiency
- Analytical method
Input audio power given to the channel can be determined by calculating an autospectrum of forward progressive plane wave and is given by the following equation, . Sound power radiated from the duct wall surface is determined using, measured intensity over the virtual surface and can be written as [108], .
Numerical model
The sound power radiated by the four flexible duct walls is calculated using the external acoustic BEM method. The structural displacement obtained from the coupled analysis is used as an acoustic boundary condition to estimate the sound power radiated from the duct.
Experimental set-up and measurement procedure
Input and radiated sound power measurement
Radiated sound power from all four flexible surfaces of a duct is measured by two different methods such as P-P (pressure-pressure) and P-U (pressure-velocity). The radiated sound power from all four flexible surfaces of a duct is obtained by using sound intensity over the measured surface.
Directivity measurement
Vibrations measurement
Results and discussion
- Modal Parameters
- Input sound power
- Radiated sound power
- Reactivity
- Pressure-residual intensity index
- Transverse Transmission loss (TTL)
- Vibration displacement and velocity
- Sound radiation efficiency
It is noted that any error in measured sound power in the P-P method is due to background noise (sound pressure), selection of an inadequate spacer length, or phase error between the microphones. This deviation is due to the discrepancy in the radiated sound power measurements, which is due to the phase mismatch error.
Summary
However, the measured radiation efficiency curve is similar to analytical and numerical results and approaches unity at higher frequencies. Transverse transmission loss and radiation efficiency measured by two different experimental intensity methods are in good agreement with analytical and numerical results.
Introduction
Pre-test Analysis
A pre-test analysis is performed based on the NMA solution to find the reference point and response locations for accelerometer placement. Since the modes are closely spaced, 144 measurement points (for all three channel cases) are considered to distinguish the shape of each mode at higher frequencies.
Experimental test setup
Since the weight of these sensors is much smaller than the channel weight, the mass load effect is neglected. In the current study, the m+p VibPilot Data Acquisition System (DAQ) with m+p international analyzer is used for acquisition and post-processing.
Numerical Modal Analysis
If 𝑉𝑖𝐸𝑀𝐴 is modal vector of EMA and 𝑉𝑗𝐹𝐸𝑀 is of FEA, then MAC is given as,. 𝑉𝑖𝐸𝑀𝐴}{𝑉𝑖𝐸𝑀𝐴}𝑇}{{𝑉𝑗𝐹𝐸𝐴}{𝑉𝑗𝐹𝐸𝐴}.’2 is the number of the experimental and numerical results, respectively.
Results and discussion
- Auto-MAC
- Natural frequencies
- Mode shapes
- Modal Assurance Criteria (MAC) plot
- Complexity of modal vector
- Effect of joint on mode shape
- MAC analysis for quarter section of the duct
- MAC analysis for C-section of the duct
- Duct mode shape variation in axial direction
In Duct-3, it is observed that the experimental and numerical shapes agree very well, except for the first mode. In the case of Duct-1, it can be observed that the experimental and numerical shapes agree well at higher frequencies than at lower frequencies.
Summary
For mode-8, it is observed that the modal shift for the channel side plates is similar. 4.15 we can conclude that for mode 3 the shape of the experimental mode matches well with the shape of the numerical mode.
Introduction
Theoretical formulation
- Rayleigh-Ritz approach
- Solution of static beam functions
- Boundary conditions along axial direction
- Boundary conditions along circumferential direction
Static beam functions have four unknowns along the axial direction and seven unknowns in the circumferential direction. The static radius function along the circumferential direction and its first three derivatives are given as follows.
Experimental modal analysis test setup
An impact hammer with a sensitivity of 11.50 mV/N is used for excitation to cause vibrations in the structure. Multivariate Mode Indicator Function (MvMIF) is used for the extraction of modal parameters from the Measured FRFs.
Results and discussion
Modal parameters for duct with joint - Analytical and Experimental
- Comparison of symmetric modes
- Comparison of antisymmetric modes
It can be observed that the deviation in the mode shapes (for antisymmetric modes) due to the presence of the coupling state is nicely captured both in experiment and in . The deviation in the mode shapes is clearly observed on the adjacent surfaces of the node state channel.
Modal parameters for duct with and without joint – Analytical
In the case of antisymmetric modes, it is observed that the channel mode shapes with the junction showed a deviation from the ideal channel mode shapes. It is observed that the vibrational modes for the channel with the junction state deviate from that of the ideal channel only for the antisymmetric modes.
Summary
In order to understand the influence of the duct joint on the modal parameters, a comparative study between the duct with and without joint condition (ideal duct) has been performed. However, the mode shapes (antisymmetric modes only) of the single-node channel deviate from that of an ideal channel.
Introduction
Theoretical formulation
Methodology
In the third step, the vibration speed can be determined by inverse analysis using measured pressures and the ATV. In the inverse analysis, regularization should be applied to overcome the poor condition of ATVs and reduce reconstruction errors.
Results and discussion
- Numerical models for FEM-BEM analysis
- Radiated sound power
- Acoustic Transfer Matrix (ATM)
- Selection of number of measurement points
- Selection of measurement location
- Effect of mesh density
- Reconstruction of vibration velocity for a metal duct
It can be observed that not only the pattern of the reconstructed vibration velocity but also the magnitude is in good agreement with the actual vibration velocity. 6.6-6.12, it is observed that the vibration velocity can be accurately reconstructed at uncoupled and coupled frequencies, taking into account 128 measurement points located at a distance of 0.15 m from the source surface with an element edge length of 0.01 m for the given dimensions of the acrylic channel and boundary conditions.
Summary
It should be noted that the distance in the measuring plane from the source must be greater than the distance between the measuring points. The reactivity in the measuring plane must also be less than 7 dB at the relevant frequency range.
Introduction
Theory of NAH methods
- Fourier NAH
- Statistically Optimized Near-field Acoustic Holography (SONAH)
- Equivalent Source Method (ESM)
- Inverse Boundary Element Method (IBEM)
The sound pressure distribution at new level (𝑧𝑟) in the space domain can be obtained by an Inverse Fourier Transform (IFT) on the left side of Eq. 7.8) yields holographic reconstruction of the 3D sound pressure field 𝑝(𝑥, 𝑦, 𝑧𝑟) in terms of Fourier transform (FT) of measurement data, 𝑝(𝑥, 𝑦, 𝑧ℎ). Then sound pressure at the nodes of the source surface can be expressed in terms of normal velocity and incident pressure for radiation problem.
Experimental setup
Singular Value Decomposition (SVD)
This can also be used to calculate the number of essential reference signals to define the sound field and to diagonalize the cross-spectral reference matrix. The cross-spectral relations (auto-spectrum and cross-spectrum) between measured and reference signals can be given as follows [127],.
Partial field decomposition
Equation (7.50) is rearranged to obtain the singular values of the cross-spectral reference matrix 𝐂𝑟𝑟, which indicates the auto-spectral amplitudes of the virtual references. 7.51), v is virtual reference vector, 𝐂𝑣𝑣 is cross-spectral matrix between virtual references. The sufficient number of reference sets calculated with the SVD technique can be verified by determining the virtual coherence at each frequency of interest [130].
Results and discussion
Singular value decomposition
Decomposed partial fields of the sound (𝐏̂), can be calculated as product of virtual transfer matrix and square root of diagonal matrix consisting of singular values. 7.54), the column vector of 𝐏̂ represents its partial field. The individual SVD values for three references at 1st blade pass frequency (BPF) calculated with reference cross-spectral matrix are shown in Fig.
Reconstructed results by NAH methods
The sound pressure measured in a holographic plane is reconstructed using the various NAH methods described in Sec. The results of the SONAH method showed that there is too much variation in sound pressure levels to measure and reconstruct.
Summary
As can be seen from the reconstructed results of the 1st BPF, ESM is the better method for reconstruction. The reconstructed results obtained by different NAH techniques show that the ESM method is the best for reconstructing fan noise sources with minimal error.
Introduction
Numerical modeling
Simply supported boundary conditions are applied at the edges of the plate and numerical modal analysis is performed using commercial software ANSYS-16. A monopole source with a strength of 0.1 kg/sec2 is used as an acoustic source for excitation.
Structural-acoustic coupled analysis
Uncoupled frequencies
Coupled frequencies
Results and discussion
- Data for the reconstruction
- Radiated sound power calculation
- Regularization
- Reconstructed parameters
- Effect of SNR on reconstruction error
The effect of noise on the reconstruction accuracy is studied in relation to the signal-to-noise ratio. The accuracy of the acoustic parameter reconstruction can be expressed in terms of the reconstruction error.
Summary
It can be observed that for the unbound frequency, the reconstruction error decreases steadily as the SNR values increase. The effect of noise on the reconstruction accuracy is studied for different values of the signal-to-noise ratio (SNR).
Future work
Visualization of vehicle within sound field using near-field acoustic holography based on the Helmholtz equation least squares (HELS) method. Study of the comparison of the methods of equivalent sources and boundary element methods for near-field acoustic holography.
Simplification of quadruple integral
The double integrals involved in Eq. A.5) are expressed in terms of simple integrals by polar coordinate transformation as follows. The integral expressions in Eq. A.5) are expressed in terms of double integrals, evaluated in Eq.
Eigen frequency equation
