The document summarizes a presentation on acoustic modal analysis of 3D hydrofoils. It discusses degrees of freedom in single and multiple degree of freedom systems, free vibration analysis of SDOF and MDOF systems, dynamic analysis techniques including modal analysis, and determining the natural frequency of hydrofoils through modal analysis. It also covers fluid structure interaction, meshing, stress analysis theories, and presents results of modal and stress analyses of different hydrofoil profiles.

1. Acoustic Modal Analysis of 3D Hydrofoils
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Presented By:
Srijna Singh
Research Scholar (Ph.D.)
Department of Mechanical and Aerospace Engineering
Bennett University, Greater Noida (U.P), India
12 February 2022

2. ▪ Degree of Freedom
▪ Free Vibration of SDOF and MDOF
▪ Dynamic Analysis of System
▪ Modal Analysis
▪ Natural frequency of Hydrofoils.
▪ Failure Theories (Ductile Materials)
2
Outlines

3. Degree of freedom (DOF)
The DOF of any system is the number of independent coordinates required to establish
the position of its particle at any time.
1. Single Degree of Freedom (SDOF) – has a single natural frequency associated
2. Multiple Degree of Freedom (MDOF)- more than one natural frequency
SDOF system MDOF system
https://civildigital.com/structural-dynamics-and-degree-of-freedom/
u is the displacement of the mass
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4. Free vibration of SDOF and MDOF
MDOF
SDOF 1
SDOF 2
SDOF n
Solution 1
Solution 2
Solution n
Final MDOF
solution
n is the number of mode shapes
• Vibration - A particular periodic oscillation about an equilibrium point.
• Free Vibration- No external force, only initial displacement force is applied.
-System starts oscillating and vibrating.
- Each natural frequency is associated with certain shape i.e., Mode shape.
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5. Dynamic analysis of system
1. Analytical approach – closed form of equations
2. Time history analysis – suitable time step is selected, and results are updated for new time.
3. Modal Analysis – The dynamics of structure are physically decomposed by frequency and position.
Modal Analysis
• The process for determining the inherent dynamic characteristics of the system in the
form of:
✓ Natural frequency
✓ Damping factors
✓ Mode shapes
Fu, Zhi-Fang, and Jimin He, “Modal analysis”. Elsevier, 2001.
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6. Modal Analysis
• Modal analysis is based on the fact that vibration response of dynamic system can be expressed
as the linear combination of a set of simple harmonic motions called natural modes of vibration
(sine and cosine waves).
• The natural modes of vibration are inherent to a dynamic system and are determined
completely by its physical properties (mass, stiffness, damping), and their spatial distributions.
• Each mode is described in terms of its modal parameters – natural frequency, modal damping
factor and characteristic displacement pattern (mode shape). Each mode shape corresponds to
a natural frequency.
• The physical model usually comprises of mass, stiffness and damping matrices. These matrices
are incorporated into a set of normal differential equations of motion.
• The superposition principle of linear dynamic system enables the transformation of these
equations into eigen value problem, giving the modal data of the structure.
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7. Governing Equation of Modal Analysis
• The equation of motion for structure
𝑀𝑠 ሷ
𝑥 + 𝐷𝑠 ሶ
𝑥 + 𝐾𝑠 𝑥 = 𝐹 𝑡
Mass
Acceleration
Damping
Velocity
Stiffness
Displacement
Load
• Acceleration, Velocity and Displacement are the unknown values at different nodes of structure
• Since, natural frequency and associated mode shapes are depending on structural properties only,
and independent of any loads the equation (1) is modified as
𝑀𝑠 ሷ
𝑥 + 𝐷𝑠 ሶ
𝑥 + 𝐾𝑠 𝑥 = {0}
https://courses.ansys.com/index.php/courses/modal-analysis/lessons/governing-equations-of-modal-analysis-lesson-2/
(1)
(2)
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𝑀𝑠 ሷ
𝑥 + 𝐾𝑠 𝑥 = {0}
• For free and undamped system Eq (1) can further be modified as
(3)
𝑀𝑠 + 𝑀𝑤 ሷ
𝑥 + 𝐾𝑠 + 𝐾𝑤 𝑥 = {0}
• When structure is kept in still water, Eq (3) has added mass of water (𝑀𝑤) and added stiffness (𝐾𝑤)
due to compressibility effects of the fluid flow
𝑓𝑛𝑤 =
1
2𝜋
𝐾𝑠+𝐾𝑤
𝑀𝑠+𝑀𝑤
• Using Eq (4) the natural frequency of structure in water can be calculated
(4)
(5)
𝐾𝑠 is function of young’s modulus, moment of inertia and characteristic length of structure

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Modal Analysis (Frequency Domain)
• Every node of the structure is assumed to be under harmonic motion (Periodic motion).
• Periodic motion can be described by an amplitude, frequency and phase angle.
• The structure can be represented in terms of simple spring-mass system.
𝑥= 𝐴 𝑠𝑖𝑛(ωt + θ)
Amplitude
Angular Frequency
Phase angle

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(1) f1 (x)
(2)
(3)
f2 (x)
f3 (x)
Fixed end
Free end
Modal Analysis (Cantilever Beam)
First three mode shapes
Zai BA, Park MK, Lim SC, Lee JW, Sindhu RA. Structural Optimization of Cantilever Beam in Conjunction with Dynamic Analysis. In Proceedings of the Computational
Structural Engineering Institute Conference 2008 (pp. 397-401). Computational Structural Engineering Institute of Korea.

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FLUID STRUCTURE INTERACTION (FSI)
Fluid
Fluid
Structure
interface
Fluid-solid interface One way FSI coupling

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• Chord length (c) = 0.1 m
• span (b) = 0.191 m
• cylindrical diameter = 0.015 m
• Density of hydrofoil (𝜌𝑠) = 7800 kg/m3
• Young’s modulus (𝐸) = 210 GPa
• Poisson’s ratio (𝜈) = 0.3
Cantilevered hydrofoil
Discretization of hydrofoil
Modal Analysis (stainless-steel Hydrofoil)
Z. Huang, Y. Xiong, and Y. Xu, “The simulation of deformation and vibration characteristics of a flexible hydrofoil based on static and transient
FSI,” Ocean Eng., vol. 182, no. January, pp. 61–74, 2019.

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Stainless Steel
Fluid domain (water)
Modal Analysis (Hydrofoil)
Z. Huang, Y. Xiong, and Y. Xu, “The simulation of deformation and vibration characteristics of a flexible hydrofoil based on static and transient
FSI,” Ocean Eng., vol. 182, no. January, pp. 61–74, 2019.
Fixed support
Fluid-solid interface

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NATURAL FREQUENCY AND MODE SHAPE OF NACA4418 AND MHKF-180 (WATER)
Frequency (Hz)
Mode NACA4418 MHKF-180
1 202.00 193.77
2 364.87 372.35
3 938.70 910.42
4 1482.50 1417.10
5 1831.50 1836.60
6 2653.90 2469.80
NACA4418
MHKF-180
• The first bending natural frequency for
mode one of NACA4418 and MHKF-180 is
202 Hz and 193.77 Hz, respectively.
Natural frequency for NACA4418 and MHKF-180 at 𝛼= 4 0.

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One-way FSI
NACA4418 and MHKF-180 Hydrofoils

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chord (c) = 0.1 m
span (b) = 0.191 m
cylindrical diameter = 0.015 m
COMPUTATIONAL DOMAIN AND BOUNDARY CONDITIONS (FLUID AND STRUCTURE)
Computational domain and boundary
condition (Fluid)
Cantilevered hydrofoil (Structure)
Z. Huang, Y. Xiong, and Y. Xu, “The simulation of deformation and vibration characteristics of a flexible hydrofoil based on static and transient FSI,” Ocean
Eng., vol. 182, no. January, pp. 61–74, 2019.

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MESHING ( FLUID AND STRUCTURE)
Meshing around the hydrofoil
Discretization of hydrofoil
structure
Tetrahedral mesh is used for fluid part and SOLID186* (hexahedron)is used for the structural discretization
*T. Suzuki, H. Mahfuz, and M. Canino, “Fatigue load and life estimation of composite turbine blades under random ocean current,” Ocean. 2015 - MTS/IEEE
Washingt., 2016.

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Literature work
Z. Huang, Y. Xiong, Y. Xu, The simulation of deformation and vibration characteristics of a flexible hydrofoil based on static and transient FSI, Ocean Eng.
182 (2019) 61–74. https://doi.org/10.1016/j.oceaneng.2019.04.028.

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* Y. Zeng, Z. Yao, J. Gao, Y. Hong, F. Wang, F. Zhang, Numerical
investigation of added mass and hydrodynamic damping on a blunt
trailing edge hydrofoil, J. Fluids Eng. Trans. ASME. 141 (2019).
https://doi.org/10.1115/1.4042759.
* Huang RF, Du TZ, Wang YW, Huang CG. Numerical
investigations of the transient cavitating vortical flow structures
over a flexible NACA66 hydrofoil. Journal of Hydrodynamics.
2020 Oct;32(5):865-78.

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Analysis of Stress
https://nptel.ac.in/content/storage2/courses/112107146/lects%20&%20picts/image/lect1/lecture1.htm
• Investigation of internal resistance developed to balance the externally applied force.
• This external load is due to fluid pressure.
= +
𝜎𝑎𝑣𝑔
𝜎𝑎𝑣𝑔
𝜎𝑎𝑣𝑔
𝜎2 − 𝜎𝑎𝑣𝑔
𝜎1 − 𝜎𝑎𝑣𝑔
𝜎3 − 𝜎𝑎𝑣𝑔
Triaxial state of stress Hydrostatic stress component Deviatoric stress component
• Hydrostatic stress cause change in volume of a body. No shear stress acting on body.
• Deviatoric stress component causes distortion in shape of the body due to presence of shear
stress (responsible for yielding)

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1. Maximum principle stress theory (RANKINE)- this is not consistent with the observation that
yielding is independent of hydrostatic stress.
2. Maximum shear stress theory (TRESCA)- agrees well with experimental data, easier to apply but
more conservative.
3. Maximum distortion energy theory (VON-MISES)- in better agreement with experimental data.
This theory is preferred over Tresca.
Failure Theories (Ductile Materials)
• Failure of ductile materials depends on the deviatoric component.
• There are different failure theories :
Arbitrary orientation of stress element

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Von-Mises Criterion (Maximum
Distortion Energy Theory)
• Structural component is safe as long as the
distortion energy per unit volume is less than
that occurring in a tensile test specimen at
yield.
Or
• If the von-Mises stress of a material under
load is equal to or greater than the yield limit
of same material under tension, material will
yield.
Stress-strain curve for ductile materials
https://upload.wikimedia.org/wikipedia/commons/c/c1/Stress_strain_ductile.svg
𝜎𝑒𝑞 > 𝜎𝑦 (yielding occurs)

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Von-Mises Criterion (Maximum Distortion Energy Theory)

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• Yield strength of Stainless steel is 250 MPa
• Maximum von-Mises stress at different cavitation number (𝜎) and at different angles of attack (𝛼)
Von-Mises Stress on Hydrofoils
𝛼 (o)
Maximum stress (MPa)
NACA4418 MHKF-180
0 34.09 45.48
2 44.59 60.21
4 70.12 65.60
6 74.86 95.63
8 60.12 77.48
10 60.46 89.36
12 78.63 91.27
𝜎
Maximum stress (MPa)
NACA4418 MHKF-180
0.5 42.60 58.11
0.8 64.51 92.51
1.0 60.12 77.48
1.2 93.73 94.47
1.5 92.69 112.88
• All the stresses are less than the yield strength of stainless steel, therefore hydrofoil will not yield.

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THANKS!