General
In the 1980s, advancements in ship design technology focused on minimizing excessive vibration However, due to a decline in ship production in Western countries, research and development in ship vibration technology faced a significant pause during the 1990s.
This edition retains many reference sources from the previous version, reflecting the enduring significance of foundational material that remains constant over time.
New material is inserted where appropriate, but engi- neering technology that matured in the late 1980s is still mostly very representative of the state-of-the-art of ship vibration
In the 1990s, the success of European cruise ship development programs showcased significant advancements in vibration control capabilities This innovation is essential for ensuring a comfortable experience for passengers, as vibration avoidance is a critical concern for cruise ships catering to sensitive customers.
The success was achieved via innovation gained by:
1 Placing engine rooms out of the immediate stern region to improve stern lines for low wake gradients.
2 Employing electric drive with the electric motors in articulating podded-propulsors, thereby avoiding the wake of shaft and bearing “shadows” shed into the pro- peller disk from forward.
Despite advancements in ship design, a persistent challenge remains: mitigating undesirable elastic vibrations in hull structures and machinery due to external or internal forces If left unaddressed, these vibrations can lead to passenger discomfort, hinder crew performance, and potentially damage or disrupt the operation of onboard mechanical and electrical systems.
Mechanical vibration refers to the oscillatory motion of both rigid and elastic bodies, making the study of ship vibration a comprehensive topic For naval architects, ship dynamics, excluding maneuvering, primarily focus on various forms of vibration.
A vessel's overall response can be divided into two main components: the rigid body motion in reaction to wave conditions and the elastic response of the hull or structure to various forces Rigid body motions are typically studied under the concept of sea keeping and are generally not classified as vibrations.
Flexural vibration in hull girders can occur through vertical and horizontal bending, torsion, and axial modes, as well as local vibrations in substructures and components A significant concern is the vibration caused by propellers, which is the primary focus of this chapter Additionally, flexural vibrations can be induced by internal forces from rotating machinery and external forces from sea waves, known as springing and whipping This chapter addresses these vibrations in the context of wave motion and structural strength, while highlighting fundamental principles of hull vibration.
Concern about propeller-induced ship vibration has existed since the marine screw propeller was first de- veloped in the mid-19th century; the French textbook
Theorie du Navire (Pollard & Dudebout, 1894) addressed the issue of propeller-induced ship vibration, noting that early propellers had few blades and operated at low RPMs, which caused low-frequency vibrations in ship hulls Early research by Schlick (1884−1911) and Krylov (1936) focused on applying beam theory to develop methods aimed at mitigating these hull vibration issues caused by propellers.
As ships have evolved, propeller-induced vibration has become increasingly complex, particularly in modern oceangoing merchant vessels This complexity is largely due to design advancements, such as the placement of engine rooms and accommodations near the propellers and the increase in ship power While the shift to diesel engines has also contributed to vibration issues, it is not as significant The problem of ship vibration has intensified in recent years due to stricter standards for acceptable vibration levels, with most commercial ship specifications now requiring compliance verification through vibration measurements during builder trials Consequently, extensive studies using both experimental and analytical methods are now standard practice in the design phase of large ships to mitigate vibration problems.
This chapter focuses on the fundamental theory and practical challenges associated with the flexural vibration of ship hulls and their components, emphasizing the impact of propeller-induced vibrations, while also briefly addressing machinery-induced vibrations.
A working knowledge of ship vibration requires the reader to be reasonably well-versed in mathematics and
This chapter on ship vibration theory is designed to cater to a diverse audience, including naval architects and engineers, by providing essential knowledge and practical insights While a deep understanding of the subject may not be necessary for effective application, the content is structured to ensure that readers from various backgrounds can find relevant information to support their work in engineering mechanics.
Section 2, Theory and Concepts, offers a comprehensive overview of the essential principles of ship vibration, serving as a foundational resource for advanced exploration of vibration analysis techniques This section is designed for individuals whose theoretical understanding is still developing within their professional expertise.
Naval architects and shipyard engineers focused on design methods can mitigate theoretical complexities by moving directly to Section 3, Analysis and Design This section is independent yet references formulas from Section 2, providing practical solutions to potential vibration issues that need to be considered during the design phase.
The last section, Criteria, Measurements, and Post-
Trial Corrections offer essential insights for assessing the vibration characteristics of a completed ship and guide necessary adjustments Ship owners or operators, who may not be focused on design processes or vibration theory, can directly access this section for practical solutions.
Basic Definitions
are provided for the uninitiated The definitions are loose and aimed at the context most needed and most often used in the theory of vibration of ships.
Vibration—Vibration is a relatively small amplitude oscillation about a rest position Figure 1 depicts the variation in vibratory displacement with time.
Amplitude—For vibration of a fixed level of severity
(steady-state periodic vibration), amplitude is the maxi- mum repeating absolute value of the vibratory response
(i.e., displacement, velocity, acceleration) Displacement amplitude for steady-state vibration is denoted as A in
Fig 1 For transient vibration, a time-dependent ampli- tude sometimes may be defined.
Cycle—One cycle of vibration is the time between successive repeating points (see Fig 1) The time re- quired for completion of one cycle is its period.
Frequency—Frequency is the number of vibration cycles executed per unit time; it is the inverse of the vi- bration period.
Natural frequency refers to the specific frequencies at which a system vibrates when it is impulsively disturbed from its rest position For a system to exhibit natural vibration, it must have both mass and stiffness Continuous mass and stiffness distributions lead to an infinite number of natural frequencies, although only a few are typically significant in practical applications When a continuous system is impulsively stimulated from rest, it vibrates across all its natural frequencies simultaneously, with the intensity of each vibration depending on the characteristics of the initial stimulus.
Mode—Each different natural frequency of a system defines a mode of system vibration The modes are or- dered numerically upward from the natural frequency with the lowest value.
A mode shape refers to the distribution of relative amplitude or displacement associated with each mode in a structure In the context of a ship hull girder, the typical mode shapes, illustrated in Figure 2, represent the first three vertical plane flexural bending modes Additionally, there are lower modes, not depicted, that correspond to rigid-body heave and pitch, occurring at lower natural frequencies The lower shape in Figure 2 illustrates the one-noded first mode torsional mode shape.
A node refers to a point of zero displacement in a vibratory distribution or mode shape Typically, as the modal order or natural frequency increases, the number of nodes in a mode shape also rises This is illustrated in the ship hull girder vibration, where modes 2V, 3V, and 4V exhibit progressively higher natural frequencies.
Fig 1 Vibration displacement Fig 2 Modes of hull girder vibration.
Vibratory excitation refers to a time-dependent stimulus, such as force or displacement, that induces vibration This excitation can take various forms, including transient, random, or periodic For instance, a steady-state periodic excitation, like that generated by a continuously operating ship propeller, results in a consistent periodic forced vibration.
The exciting frequency in a steady-state periodic excitation refers to the number of cycles completed per unit time, which is the inverse of the excitation period In steady-state conditions, the vibration frequency matches the exciting frequency The system's vibration response at this frequency can be seen as a weighted combination of the mode shapes from all natural modes The extent to which any mode participates is highly influenced by how close its natural frequency is to the imposed exciting frequency.
Resonance occurs in steady-state forced vibration when the exciting frequency matches a system's natural frequency, known as the resonant frequency At this point, the vibration amplitude is primarily influenced by system damping, which is typically minimal in engineering structures like ships As a result, resonant vibratory amplitudes can be significantly larger compared to nonresonant levels This disproportionate amplification leads to a resonant vibration distribution that closely resembles the mode shape of the resonant mode.
Bandwidth—Bandwidth is a range, or band, of fre- quency where a vibration and/or noise is concentrated.
Beat phenomena occur in systems influenced by two or more closely spaced excitation frequencies, resulting in a response characterized by a low beat frequency This frequency fluctuates, with its maximum value corresponding to the bandwidth of the variations in the exciting frequencies.
Octave band—One of the standard frequency bands in which vibration (and noise) signals are analyzed (fil- tered) and presented (see Section 2.2.4).
Decibel (dB)—A quantification of vibration level used primarily in acoustics; dB is defined as 10 times the log10 of a vibration amplitude divided by a reference vibration amplitude (see Section 2.2.4).
Sound pressure level (SPL)—is defined as 20 times the log10 of an absolute value of the sound pressure di- vided by a reference sound pressure (see Section 2.2.4).
Continuous Analysis
Steady-State Response to Periodic Excitation
tion In propeller-induced ship vibration, the steady propeller excitation is, in reality, a random excitation that remains stationary while conditions are unchanged
The periodic nature of the system is characterized by a fundamental frequency that equals the product of the propeller's RPM and the number of blades Consequently, the excitation can be effectively represented as a Fourier series in relation to time When focusing on the steady-state vibratory response to this periodic excitation, the displacement w(x,t) can also be expressed in a Fourier series format.
To solve the equation of motion for steady-state vibration response, the first step is to substitute the Fourier series representations of w(x,t) and f(x,t) into the equation This process eliminates the time dependency, resulting in a series of ordinary differential equations in the variable x Each term is then solved individually to determine the unknown coefficients of the displacement series.
For demonstration purposes, assume a one-term Fourier series (i.e., simple harmonic) representation for the excitation force distribution in time Then optional forms are
The equation R e F ( ) x e i t (4) represents the amplitude distribution of the excitation force along a ship's length, with frequency denoted as F(x) Here, Re indicates the "real part of," and the identity t i t e i t cos + sin is utilized The blade-rate frequency, defined as N⍀ (where ⍀ is the propeller's angular velocity and N is the number of blades), allows for a valid approximation of f(x,t) when the fundamental harmonic of the excitation is dominant This condition is typically met, especially in scenarios where propeller blade cavitation is absent.
For steady-state vibration in response to f(x t), w(x,t) will have the similar form
= (5) where, in view of equation (5),
The unknown complex amplitude, W(x), encompasses both phase and amplitude information and is determined through the solution of the equation of motion By substituting equations (4) and (6) into equation (1) and applying the end conditions from equation (2), while eliminating the time dependency, we can derive the necessary results.
It is convenient to nondimensionalize the variables in equations (7) and (8) before considering solutions for
W(x) Redefine the variables in nondimensional form as
L kL r = is a characteristic rigid-body frequency;
2 is a hydrodynamic damping factor, and denote
This is the nondimensional equation for steady-state vibration amplitude in response to harmonic excitation
Undamped End-Forced Solution-Demonstrations
tions The simplest meaningful solution of equation
(9) is obtained by specializing F(x) to be a concentrated end force and discarding the damping terms This solu- tion, obtained by direct inversion of the reduced equa- tion is
) cosh )(cos sin [(sinh cos cosh 1
Here, the force is concentrated at the stern, x =1 (see
Fig 3) With zero damping, W(x) is pure real and is given by
The solution, equation (10), permits several relevant ob- servations These are developed as described later. 2.1.2.1 RESONANT FREQUENCIES—ADDEDMASS AND BUOY-
The undamped solution reveals that infinite vibration amplitudes occur at specific values where the hyperbolic cosine equals one, identifying these points as the system's resonant frequencies, denoted as n By setting equal to n at these resonant frequencies, we find an infinite number of roots from the equation cosh n cos n = 1 Furthermore, the relationship between the resonant frequencies and the system parameters is expressed in equation (13), where n 4 is calculated as the difference between the squares of the frequency ratio and the radius ratio.
The first root of equation (12) is 0 =0 This implies, from equation (13), that
The rigid-body heave and pitch resonant frequencies are identical for ships with uniform mass and buoyancy distributions At low frequencies, the mass distribution becomes frequency dependent due to surface wave effects in the hydrodynamic component As the vibratory frequency increases, this frequency dependence diminishes In practice, the hydrodynamic added mass of a ship remains largely constant at frequencies associated with hull flexural modes.
The second root of equation (12) is 1 = 4.73, representing the first hull flexural mode, with all subsequent values of n being greater than 1 Assuming frequency independence for n ≥ 1, both ⍀ f and ⍀ r are treated as constants in equation (13) Consequently, the first flexural mode resonant frequency, along with all higher modes, can be directly derived from equation (13).
For ships, the ratio ⍀ r /⍀ f is typically on the order of
The impact of buoyancy on the vertical flexural vibration of a ship's hull is present but typically minimal under normal conditions By neglecting the ratio ⍀ r /⍀ f in equation (14), we can approximate the beam resonant frequencies.
In the analysis of flexural vibration, the effects of buoyancy are considered negligible and thus omitted, leading to the disregard of nonzero rigid-body modes (n = 0) Additionally, the term ⍀ f will be treated as frequency independent, as the hydrodynamic added mass remains constant at high frequencies.
Note that although in the case of wave-excited vibra- tion both rigid-body and flexural vibration occur, the two responses are essentially independent superposi- tions.
2.1.2.2 STERN VIBRATION LEVEL Consider the vibra- tion at the position of the concentrated excitation force by setting x = 1 in equation (10)
( ) cos cosh 1 sin cosh cos sinh )
For exciting frequencies in the range of the beam flex- ural resonant frequencies, the corresponding values of
, as arguments of the hyperbolic functions, can be con- sidered as large That is, for large e
According to Equation (17), for a forcing function with a constant amplitude, the end vibration typically diminishes with frequency, following a trend of −3 or −3/2 Zero vibration at the stern occurs at the antiresonant frequencies, denoted as n_a, which are approximately determined by the condition tan n_a = 1 From Equation (15), these frequencies can be expressed as n_a / f = [(4n + 1) / 4]², where n ranges from 1 onward.
Large vibration occurs only in the immediate vicinity of the resonant frequencies, the flexural values of which, from equation (17), correspond approximately to tan n = ± or n / f = [(2n + 1) /2] 2 ; n = 1, , (19)
As increases, equation (17) implies a limiting state where the vibration is zero except at the resonances
But the resonant frequencies, equation (19), at which the vibration is infinite, occur in the limit of large n, in- finitely far apart.
The trend toward this limiting case is exhibited in
Fig 4, which is a plot of equation (17) in the frequency range of the first few flexural modes.
The relationship between equation (17) and actual ship vibration reveals that, contrary to general assumptions, the spacing of hull girder resonances does not necessarily increase with frequency This discrepancy arises from the omission of shear and rotational inertia in the beam model, as well as the neglect of local vibratory subsystems originating from the hull beam At higher frequencies, these factors significantly influence the vibration characteristics of the ship hull girder.
The vibration is also, in reality, certainly not infinite at the resonant frequencies; this prediction is due to the deletion of damping in the solution to equation (9).
Contrary to the implication of equation (17), propeller-induced vibration does not consistently decrease with frequency In fact, the amplitude of propeller excitation, denoted as F, increases with frequency, approximately following a squared relationship.
2 variation of F in equation (17), W(l) then increases generally as 1/2 , which is more realistic than decreas- ing as −3/2
2.1.2.3 RELATIVE VIBRATION OFBOW ANDSTERN Setting x = 0 in equation (10), the vibration amplitude at the beam end opposite that to which the excitation is ap- plied is
Using equations (16) and (20), the ratio of the end dis- placements is sin cosh cos sinh sin sinh )
W sin cosh cos sinh sin sinh )
Again, replacing the hyperbolic functions by the expo- nential for large
Fig 4 Hull beam response characteristics.
At the antiresonant frequencies, equation (18), W(0)/
W(1) becomes infinite since W(l) = 0 by definition of the antiresonance At the resonant frequencies, equation
(19), W(0)/W(1) = ±1, by equation (12) The minimum absolute value of the displacement ratio occurs at cos
The frequencies at which this minimum value occurs are
The prediction regarding ship vibration contradicts high-frequency observations, as the un-damped end-forced solution suggests that vibration levels at the ship's bow should not fall below a certain threshold.
At high propeller RPM, approximately 70% of the vibration generated by the propeller is concentrated at the stern of the ship This hull girder vibration significantly decreases as it moves forward, often becoming nearly undetectable in the vessel's forebody.
A reconciliation of theory and observation as to this particular point requires a more general solution to equation (9), which includes damping as well as a less restricted characterization of the propeller excitation
However, the direct analytic solution procedure used to produce equation (10) is no longer suitable for providing the desired insight in the more general case.
A More General Solution: Modal Expansion
The modal or eigenfunction expansion technique simplifies the handling of damping and arbitrary excitation characteristics in vibrations This approach illustrates that vibrations can be represented as a superposition of independent natural modes Consequently, the solution to the equation of motion is formulated as an infinite series, contrasting with the alternative closed-form solution presented in equation (10).
The series is expanded in terms of the infinite set of nor- mal modes of the unforced, undamped system.
2.1.3.1 N ATURAL F REQUENCIES AND M ODE S HAPES
Returning to the equation of motion for the Fig 3 uni- form beam, equation (9), the unforced, undamped sys- tem in this case corresponds to equation (9) with zero damping and excitation
W d = = where is defined by equation (11) The solution to the homogeneous differential equation, equation (24), is, for
Applying the two end conditions at x = 0 eliminates two of the four constants in equation (25) as
Application of the remaining end conditions at x = 1 gives the following simultaneous equations for deter- mining C 3 and C 4
3 sinh sin cosh cos cosh cos sinh sin C
To find nonzero values for |C| and W(x), the matrix [B] must be singular, which occurs when its determinant is zero According to equation (28), the determinant of [B] is given by det [B] = –2(1 – cos cosh) The values that satisfy det [B] = 0 are identified as the system's eigenvalues, represented as n An infinite set of eigenvalues is established by solving equation (29), where cos n cosh n = 1 for n = 1, 2, 3, and so on.
Equation (12) defines the system's resonant frequencies, while equation (11) reveals that n^4 = (n/f)^2, where n represents these resonant frequencies In this context, n refers to the natural frequencies of the system, which correspond to unforced and undamped vibrations Consequently, the system's resonant frequencies are synonymous with its natural frequencies.
Nonzero values of constants C3 and C4 from equations (27) exist only at frequencies satisfying equation (29) Although these constants are not zero, their exact values remain indeterminate due to the coefficient determinant being zero at these frequencies This indicates that the equations (27) are linearly dependent at the natural frequencies, meaning there are not two independent equations available to determine the constants Consequently, the only information derivable from equations (27) is the relationship between C3 and C4 at these natural frequencies, and either equation can be used to express this relationship, yielding the same result due to their linear dependency.
Substituting equations (31) and (26) into the homogeneous solution equation (25) yields the beam vibration amplitude at the natural frequencies as a function of x, aside from a constant factor This relative amplitude distribution at the natural frequencies represents the eigenfunction, or mode shape, denoted by n.
From equation (25), the mode shape for the Fig 3 beam is
C x n n n n n n n n n sinh cosh sin cos sinh sin cosh
The constant C 4 is arbitrary and is conventionally set to unity.
Equation (32) is the beam mode shape for ⫽0 This function has the character of the vertical mode shapes depicted in Fig 2 of Section 1.2; increasing n corre- sponds to increasing node number.
For = 0, the solution to the homogeneous system, equation (24) is
The end conditions at x = 0 reduce equation (33) to
W(x) = C 1 + C 2 x (34) which satisfies the end conditions at x = 1 identically
The mode shape derived from equation (34) represents the zeroth order rigid-body heave and pitch mode, with its natural frequency denoted as ⍀ r, previously established in equations (12) and (13) For the analysis of flexural modes, ⍀ r has been assumed to be zero.
2.1.3.2 VIBRATORY DISPLACEMENT Modal expansion expresses the solution of the equations of motion, equa- tion (9), as a weighted summation of the infinite set of mode shapes
By substituting equation (35) into equation (9) and applying the orthogonality property of the mode shapes, the A n terms in equation (35) are effectively uncoupled, leading to their determination as outlined in Section 2.2, equation (72), and the subsequent material.
In equation (36), F n , K n , and n represent the following. Modal exciting force:
Modal damping factor: n = v ( n / f ) + c ( f / n ) (39) Substitution of equation (36) into equation (35) gives the complex vibration amplitude
Substitution of this result into equation (5), and using a trigonometric identity, gives the vibration
2 2 2 displacement at any point x along the beam at any time
The modal phase angle, n , relative to F n , is
The equation (41) illustrates that modal expansion can be interpreted as a combination of the independent responses from an infinite series of equivalent one-degree-of-freedom systems Each of these systems is characterized by stiffness, damping, and excitation values derived from equations (37), (38), and (39), while the equivalent mass corresponds to the modal mass.
M n = K n / n 2 The response of each of the one-degree-of- freedom systems is distributed according to the mode shapes of the respective modes.
2.1.3.3 R ELATIVE VIBRATION OFB OW AND S TERN The rea- sons for the rapid attenuation of hull girder vibration on moving forward from the stern, which were left unex- plained by the simple theory of the last section, can now be reconsidered with the aid of the modal expansion, equation (40).
To simplify the analysis, it is beneficial to normalize the eigenfunction set by assigning specific values to the constant C4 For this purpose, setting n(1) = 1 at the forcing end is a practical choice, leading to the formulation n = 1, , (42).
C 4 in equation (32) is evaluated as n n n
Then from equations (43) and (32), the eigenfunction at x =0 has the values n (0) = (–1) n+1 ; n = 1, , (44)
It will also be notationally convenient to define, W n (x)
A n n (x) where A n is given by equation (36) Equation
By equations (42), (44), and (45), the displacements at the two ends of the beam are
Equation (46) indicates that the absolute values of displacement components at the beam ends are identical, with differences in sums attributed solely to the alternating series form of W(0), reflecting phase changes at the forcing end This characteristic of the displacement series is crucial for understanding the rapid decay of hull girder vibrations moving forward from the stern, as illustrated in Figure 5 This figure features sketches of the Wn components for six modes and their summations across three scenarios The right column of Figure 5 illustrates the displacement of an undamped beam with a concentrated force applied at the extreme end, previously analyzed, where the minimum ratio of end displacements is predicted to be no less than 1/2 The center column shows the case with zero damping and a concentrated force applied at a position slightly less than 1, simulating a typical propeller position Finally, the third column depicts the scenario where the force is applied at the beam end, but with significant nonzero damping.
Fig 5 Hull beam response characteristics.
The exciting frequency is expected to fall between modes 3 and 4, as illustrated in Fig 5 According to equation (23), with n set to 4, this equation provides predictions for the frequencies at which the minimum occurs.
| W(0)/W(1) | = 1/ 2 occurs for the undamped, end-forced case Consider the three cases of Fig 5 individually
Case 1—Undamped, end forced From equation
In general, the modal forces for the three cases of Fig
5 are F n = F n (x 0),by equation (37) For x 0 = 1 in the first case, F n = F for all n since n (1) = 1 by equation (42) For
between the two resonant frequencies, N−1 and N , the beam end displacements can be written from equa- tions (46) and (47) as
The sign change in the denominator of equation (47) at n = N has been clearly identified At x = 1, all modes below share the same sign, contrasting with the opposite sign of the modes above, leading to imperfect cancellation where the lower modes prevail over the upper ones Conversely, at x = 0, interference arises among the groups of modes both below and above due to the alternating signs indicated in equation (48).
The dominant terms W n−1(0) and W n(0) share the same signs, providing support rather than cancellation, which leads to a relatively large value for W(0) Notably, the ratio | W(0)/W(1) | reaches its maximum of 1/2 when x 0 = 1 and n = 0 This maximum occurs due to the effects of repositioning the excitation force forward and allowing for nonzero damping, both of which contribute to a more rapid attenuation of displacement from the forcing point.
Case 2—Undamped, x 0 < 1 Considering the case where x 0 < 1, which corresponds to the center column in Fig 5, the modal force is
In equation (40), F n = F n (x 0) indicates that as the mode number n increases, the modal forces converge This occurs because the last beam nodal point shifts closer to the forcing point, leading to a decrease in n (x 0) values Consequently, higher modes become less responsive to the concentrated force As a result, there is a reduction in the cancellation effect in W(l), as described in equation (48), due to the diminished net displacement from the higher modes compared to the contributions from the lower modes.
Discrete Analysis
Mathematical Models
Vibration analysis primarily utilizes nonuniform and discrete mathematical models instead of uniform and continuous ones These models capture the continuous characteristics of mass, stiffness, damping, and excitation in physical structures at specific locations known as nodal points.
The properties of equivalent nodal points are represented through a collection of discrete or finite elements that connect the nodal points within the structural model It is important to clarify that these nodal points differ from the nodes described in Section 1.2.
In the discrete model analysis, forces and displacements are associated with nodal points, which can exhibit up to six displacement components—three translations and three rotations—alongside six corresponding force components However, due to constraints, the actual displacements permitted at each nodal point are typically fewer than six The permitted displacements at a nodal point are defined as its degrees of freedom.
In dynamic systems, a nodal point displacement linked to mass or mass moment of inertia indicates a dynamic degree of freedom, while a lack of such association signifies a static degree of freedom Continuous systems possess an infinite number of degrees of freedom, whereas discrete models have a finite total, determined by the sum of degrees assigned to each nodal point within the model.
Discrete analysis of ship vibration can be conducted at various levels of detail, with the complexity of the model primarily determined by the computing resources available Typically, the ship hull girder is modeled together with its sprung substructures, such as deckhouses, decks, and double-bottoms, within a single discrete model.
Meaningful estimates of substructure vibration characteristics can often be achieved using a discrete model of the substructure, applying approximate boundary conditions at its connection to the hull girder (Sandstrom & Smith, 1979).
Discrete analysis is conveniently demonstrated by an idealized example of the latter approach noted above
Consider the simple finite element model for a ship deckhouse shown in Fig 6 Here, the house is modeled two-dimensionally as a rigid box of mass m and radius of gyration r¯.
The house front is typically aligned with the forward engine room bulkhead, where the main deck connection functions as a simple pin to allow free rotation The supporting structure behind the house consists of parallel finite elements with axial stiffness and damping, primarily formed by pillars within the engine room cavity This structure experiences base excitation due to the vertical vibratory displacement of the hull girder, denoted as w(, t), which represents the axial coordinate along the hull The base displacements, w(1, t) and w(2, t), correspond to the hull girder displacements at the forward engine room bulkhead and the base of the after foundation, respectively, and are assumed to be predetermined independently.
The Fig 6 model for serious vibration analysis has limitations, particularly because typical deckhouses do not behave as rigid structures at propeller excitation frequencies The flexibility of the underdeck support often dominates the propeller-induced vibrations, while the bending and shear flexibilities of the deckhouse itself are significant Additionally, interactions between the house and hull girder complicate accurate predictions of base displacements Despite these issues, the Fig 6 deckhouse model remains valuable as it effectively illustrates the fundamental characteristics of fore- and aft-deckhouse vibrations, similar to the uniform beam model used for hull girder vertical vibrations discussed previously.
Proceeding as described, the degree-of-freedom assign- ments of the Fig 6 finite element model are shown in Fig
7 Here, x j is used to denote generalized displacement (i.e., rotation or translation) In view of the assumed house rigid- ity, all displacements in the vertical/fore-and-aft plane can be specified in terms of the three assigned in Fig 7 All other possible displacements at the two nodal points of the Fig 7 model are assigned zero values by virtue of their omission
In the analysis presented in Fig 7, three degrees of freedom are identified, with two being dynamic: x1 and x2, which correspond to the house's mass moment of inertia and mass The third degree of freedom, x3, is classified as static.
Fig 6 Ship deckhouse vibration model.
Also, two of the three degrees of freedom are speci- fied from Fig 6 as x 2 w( 1 ,t) and x 3 w( 2 ,t)
Once x 1 is determined, the vertical and fore-and-aft dis- placements at any point (, ) on the house are available, respectively, as w( , ,t) = w( 1 ,t) – x 1(t)( – 1 ) u( , , t) = x 1(t) (50)
Equations of Motion
tion governing the general finite element model are de- rived as follows
It is first required that the model be in dynamic equi- librium in all of its degrees of freedom simultaneously
Application of Newton’s Law in each degree of freedom, in turn, produces
The equation \( |m| \cdot | \ddot{x} | = - | f_s | - | f_d | + | f | \) represents the relationship between the model mass matrix, nodal point acceleration vector, and the stiffness, damping, and excitation force vectors in a system with M total degrees of freedom Here, \( |m| \) denotes the M ⋅ M model mass matrix, \( | \ddot{x} | \) is the M ⋅ 1 nodal point acceleration vector, while \( | f_s | \), \( | f_d | \), and \( | f | \) correspond to the M ⋅ 1 vectors for nodal point stiffness, damping, and excitation forces, respectively.
The model finite elements are designed with predetermined characteristics to ensure compatibility and meet material constitutive requirements at the local level Ensuring these criteria for linear behavior establishes the relationships between the internal forces at the nodal points and their corresponding displacements.
Here, [k] is the model stiffness matrix and [c] is the model damping matrix, both of which are square matri- ces of order M.
Substitution of equation (52) into equation (51) pro- duces the linear equations of motion governing the gen- eral discrete model
This M ×M system of equations can be readily solved for the unknown nodal point displacements once [m], [c], [k], and | f | are specified.
Actually, the equations of motion can be interpreted as a general statement and conveniently used to deter- mine their own coefficients.
For example, if the accelerations and velocities are set to zero, equation (53) reduces to
In the context of nodal point degrees of freedom, the subscripts indicate the specific numerical assignments To define the stiffness matrix \( k_{ij} \), it is essential to ensure that all velocities and accelerations are zero, while all displacements \( x_i \) must be set to zero except for the case where \( i = j \), where \( x_j \) is assigned a value of 1.
Then, for any degree of freedom i, multiplication gives f i = k ij
The kij represents the force in degree of freedom i resulting from a unit displacement in degree of freedom j, while all other degrees of freedom are fully restrained Complete restraint encompasses the prevention of acceleration, velocity, and displacement.
The designation of force in degree of freedom i refers to the specific amount of force needed at that degree of freedom to achieve the intended assignment.
The definitions of m ij and c ij are derived from the general equation (53) by assigning the appropriate degrees of freedom These definitions are similar to those for k ij, but they utilize unit accelerations and velocities in place of unit displacements.
To calculate the components of the excitation force vector, f i, the model is fully constrained in all degrees of freedom Consequently, f i represents the resultant of the applied forces that aim to overcome the restraint in degree of freedom i.
In this model, as illustrated in Fig 7, the displacements across the three degrees of freedom are represented by x1, x2, and x3, where x1 is the variable to be determined while x2 and x3 are predefined By implementing both zero and unit values for accelerations, velocities, and displacements in these three degrees of freedom, we can analyze the system's behavior effectively.
Fig 7 Deckhouse model degrees of freedom. dom, in turn, the mass, damping, stiffness, and excitation force matrices are assembled by the rules stated above as
The excitation force vector includes unknown forces f2 and f3, which correspond to the known displacements x2 and x3 Additionally, f1 is zero since there is no external moment applied at the pin connection.
In this example, either the external force (f i) or the displacement (x i) must be specified for each degree of freedom, as both cannot be known prior to solving the system equations Initially, the equations related to the known forces are solved to find the unknown displacements Subsequently, the unknown forces are calculated using the now fully known displacements through the equations for the unknown forces For the system depicted in Fig 7, this process yields a single equation of motion to determine the unknown displacement x 1.
On solving this equation for x 1, the unknown force com- ponents, f 2 and f 3, are then determined from the two re- maining equations by multiplication as
Solutions
The procedure for continuous analysis remains consistent, utilizing the approximate periodicity of propeller excitation to separate the time variable from the differential equations through Fourier series By defining mN⍀ as the m-th harmonic propeller exciting frequency, we can express the m-th harmonic complex force and displacement amplitude vectors as | F | and | X |, respectively Consequently, the equations of motion can be satisfied on a harmonic basis by solving equation (53) for each harmonic individually.
Define the system dynamic matrix as [D]
Returning to the deckhouse model in Figs 6 and 7 with
| x | = Re | X | e it the system dynamic matrix, from equation (55), is
[D] = – 2 mr¯ 2 + ic 2 + k 2 (61) which is a 1 × 1 matrix on the single unknown complex amplitude, X 1 Likewise, the complex exciting force vec- tor in equation (55) is
| F | F 1 = – 2 m¯ X 2 + (ic + k)(X 2 – X 3) The inversion required by equation (60), using equa- tion (61), is then simply
Equation (62) can be written in the standard form for vibration of systems with one dynamic degree of free- dom by writing its numerator as
F 1 = F 1 R + iF 1 I = mod F 1 e –i and the denominator as
K w w w w + so that x 1 = Re X 1 e it is, from equation (62),
In most cases, achieving an analytic closed-form inversion of the system equations, as done for a simple one dynamic degree-of-freedom system, is not feasible Instead, two alternatives are available, with the most straightforward being the direct numerical inversion of the equations.
Powerful numerical algorithms are readily available for inverting systems of linear simultaneous algebraic equations.
Modal expansion is an alternative solution procedure that involves expressing the series solution of the equations of motion in relation to the natural frequencies and mode shapes of a discrete model This method utilizes eigenvectors to effectively analyze dynamic systems.
2.3.3.1 N ATURAL FREQUENCIES AND M ODE S HAPES By definition, natural frequencies are frequencies of vibra- tion of the free, or unforced, and undamped system
From equation (57), the equations of motion for the free, undamped discrete model are
(Here the system model is considered to have a total of L unknown degrees of freedom, with N dynamic de- grees of freedom (DOF) and L − N static DOF [no mass assigned].)
This equation implies that | X | = 0 unless [D*] is singu- lar But by definition of natural vibration, | X |is not zero
Therefore, the frequencies which make [D*( )] singu- lar are the system natural frequencies; [D*] is singular if its determinant is zero Define
The characteristic polynomial, denoted as P( ), is essential in system dynamics, particularly for systems with N degrees of freedom This polynomial is of order N in 2 and possesses N positive roots The N positive values that satisfy P( ) = 0 correspond to the natural frequencies, denoted as n.
Continuous systems have an infinite number of natural frequencies, whereas discrete models have a finite number of natural frequencies that corresponds to their dynamic degrees of freedom.
All real physical systems exhibit at least piece-wise continuity, which means that discrete systems should be regarded as simplified models of continuous systems This distinction is significant in understanding the nature of these systems.
Proceeding, with the Nmodel natural frequencies in hand, a return to equation (65) gives
At frequency n, the value of |X| is not necessarily zero, as the matrix [D*( )] becomes singular, leading to undefined conditions This singularity indicates a linear dependency among the L equations related to the unknown degrees of freedom (DOF) Consequently, only L - 1 linearly independent equations are available for determining the L unknown components of |X| at frequencies n = 1, , N.
(68) are the relative amplitudes, called mode shapes, or eigenvectors, at each of the Nnatural frequencies. The L× 1 mode shape vector is denoted | n | , n = 1,
N It is determined by assuming any one of its L compo- nents as known Then the L − 1 equations on the remain- ing L − 1 mode shape components at each n are solved in terms of the one presumed known That is, assuming arbitrarily that the L th mode shape component is known, equation (68) is written n L n n
The (L − 1) × (L − 1) system of linear algebraic equa- tions, equation (69), is then solved by standard numeri- cal methods for the (L − 1) component | n | for some or all of the Nmodes of interest.
For the Fig 6 deckhouse example, the above is simple since L and N are 1 The [D*] matrix from equation (61) is
[D*] = – 2 mr¯ 2 + k 2 which is also the characteristic polynomial P( ) P( n )
2 / m r n = k with n = 1 The mode shape | n | is 11, which has an ar- bitrary scale value.
2.3.3.2 MODALEXPANSION At this point in the devel- opment of the solution for the uniform beam of the last section, a brief description of the modal expansion solu- tion procedure, for that simple case, was followed by its statement Here, it is considered worthwhile to develop the solution to illustrate a special difficulty that occurs in the more general case.
As before, the complex displacement amplitude vector is first written as a series of the mode shapes weighted by unknown coefficients, A n n L n
Substitute equation (70) back into the governing equa- tions (57)
Now multiply equation (71) by some | m | T , with T denot- ing transpose and with mnot necessarily equal to n
(73) as the m th mode modal mass By equation (73), the sum- mation of the matrix products involving [m] in equation
(72) is reduced to a single constant, M m Similar reduc- tion of the products involving [k] in equation (72) is ac- complished as follows.
Therefore, in view of equation (73)
K = 2 (74) as the m th mode modal stiffness, such that m m m = K /M
(75) as the m th mode modal exciting force.
Substitute equations (73), (74), and (75) back into equation (72)
To effectively utilize orthogonality in reducing the damping term in equation (76), similar to the mass and stiffness, the required A m in solution (70) can be determined However, orthogonality in the damping matrix is generally absent for N > 1, occurring only in specific cases Notably, orthogonality is present when the damping matrix [c] is proportional to the mass [m] and/or stiffness [k], as demonstrated in the simple distributed model discussed in Section 2.1, which ensured the mode shape orthogonality necessary for equation [36].
[ ] c = n [ ] k + n [ ] m (77) where n and n are constants which are allowed to vary only from mode to mode, then, in equation (76)
C m is called the modal damping coefficient Presuming
C m to exist, the A m are then, from equation (76), m m m m m M i C K
= (80) m is the m th mode modal damping factor Substituting equation (79) into equation (70) completes the deriva- tion
Here, n and n are the modal phase angles
Equations (81) and (82) are fundamental in addressing ship vibration issues, as highlighted in Section 3 They illustrate that modal expansion can be interpreted as the superposition of responses from N equivalent one-degree-of-freedom (DOF) systems, each corresponding to the Nmodes of the discrete model.
The primary distinction between the solutions for continuous and discrete models lies in the length of the series; while discrete models have a finite series, continuous models possess infinite degrees of freedom, resulting in an infinite series.
The restriction imposed upon damping at equation
In continuous analysis for N > 1, it is essential to recognize that the challenges encountered must be addressed; this issue was not explicitly highlighted in the previous section since the beam, characterized by uniform properties, inherently exhibits proportional damping.
The limitation on damping in structural systems is significant, as internal material damping can often be represented by a stiffness-proportional damping matrix This conclusion is supported by the basic theory applied to material damping in continuous beams However, when additional sources of damping are involved, this proportionality is typically disrupted, rendering the modal expansion, as described in equation (81), theoretically invalid.
Despite the challenges, there is a temptation to use the modal formula in scenarios where proportional damping is not justifiable, such as in ship vibration analysis Modal expansion offers three significant advantages over the direct numerical inversion method, especially when dealing with large models.
Propeller Exciting Forces
Propeller Bearing Forces
which depicts a propeller blade rotating with angular velocity ⍀ in the clockwise direction, looking forward
By virtue of the rotation through the circumferentially nonuniform wake, the spanwise blade lift distribution,
L(r, ), fluctuates with time, or with blade position angle
To analyze the forces and moments in the propeller hub due to the simultaneous lift distributions from all N blades, we define the complex function g(r, ;p) = –L(r, )e ip e jbG This function helps in determining the three force and three moment components generated by the time-varying lift across the blades.
In this context, both i and j represent -1, yet they should be considered independent in all algebraic operations for the sake of notation simplicity In equation (86), G signifies the geometric pitch angle of the blade section at radius r, while p is an integer that will be determined later.
The function g(r; p) in equation (86) describes a pseudo-lift distribution on a single blade of an N-bladed propeller To account for the simultaneous action of all N blades, the term is modified by replacing it with + 2(k - 1)/N in equation (86) and summing over k This process establishes a new complex lift function that captures the collective effects of the N blades.
The circumferential wake nonuniformity observed from the blade exhibits a nearly periodic behavior over time, characterized by a fundamental period T = 2 /⍀ Assuming linearity, the lift distribution L(r, t) also demonstrates this periodicity Consequently, L(r, t) can be expressed using a Fourier series representation.
In this context, L q (r) represents the q-th harmonic complex lift amplitude of the blade section at radius r, while L 0 (r) denotes the steady lift distribution linked to steady thrust and torque To determine the L q (r) harmonics, one can choose from various procedures based on the specified harmonics of the wake inflow, as detailed in Section 3 It is assumed that a sufficient number of L q (r) harmonics are accessible from a reliable source.
Fig 8 Propeller blade-element forces.
An alternative representation of equation (88), which is useful for insertion into equation (87), is
(89) where the overbar denotes complex conjugate Discard- ing the steady lift and substituting equation (89) into equation (87) produces
The k summations in equation (90) equal zero when q ± p is not an integer multiple of N (denoted as mN), while they equal N when q ± p equals mN Consequently, equation (90) simplifies to a sum involving only m.
The bearing forces f ip ( ),i = 1, , 6, Fig 8, are now given in terms of G(r, ;p) from equation (91) as
The prefixes Re and Im refer to the real and imaginary parts of the complex quantities involving i and j; the complex lift harmonic is Lq = Lq R + iLq I in this regard.
As an example, consider the vertical bearing force, f 3p Equations (91) and (92) give
This formula presents a contrast to the one proposed by Tsakonas, Breslin, and Miller (1967), differing solely in the sign The variation in sign arises because positive lift is defined here as having a forward axial component, which is a standard interpretation, unlike the convention used by Tsakonas et al.
The following important facts should be observed from equations (91) and (92).
1 Propeller bearing forces are periodic with fun- damental frequency equal to the propeller angular ve- locity times the number of blades The fundamental fre- quency, N⍀, is called blade-rate frequency The bearing forces, as written in equations (91) and (92), are com- posed of terms at blade-rate frequency, plus all of its in- teger multiples, or harmonics, mN⍀.
2 Only certain terms, or harmonics, of the unsteady blade lift, and therefore of the hull wake, contribute to the bearing forces While the forces on a single blade consist of components corresponding to all wake har- monics, a filtering occurs when the blade forces super- impose at the propeller hub Equations (91) and (92) show that the unsteady thrust and torque, f 1p and f 4p , depend only on the lift, or wake, harmonics that are in- teger multiples of blade number The lateral forces and moments, on the other hand, are produced entirely by the wake harmonics corresponding to integer multiples of blade number, plus and minus one.
Propeller-Induced Hull Surface Pressures and Forces
The integration of resultant hull surface forces requires significant effort due to the complexity of the subject Since the foundational experimental work by Lewis in 1973, substantial progress has been made in understanding hull surface excitation and developing predictive methods To illustrate this, we can start with a basic scenario: the pressure exerted on a flat plate by a propeller operating under uniform inflow conditions, as depicted in the water tunnel arrangement shown in Figure 9, with data measured in Figure 10 (Denny, 1967).
In the experiments, two identical three-bladed propellers were utilized, differing only in blade thickness, with one set having blades twice as thick as the other This design enabled the clear distinction of the independent effects of blade thickness and blade lift, based on the experimental data collected The left-side plots in Fig 10 illustrate the amplitude and phase of the pressure induced by blade thickness, while the right-side plots represent the effects of blade lift Additionally, predictions from theories developed in the late 1960s are included in Fig 10 for comparison.
The pressure data presented in Fig 10 corresponds to the blade-rate frequency, where all multiples of this frequency are observed, although higher harmonics diminish rapidly in the uniform wake case The phase in Fig 10 represents the angle of the propeller blade closest to the plate during the positive maximum pressure, defined as counterclockwise when viewed from the front By multiplying the phase angles by three, we can determine the phase relative to a single cycle of the three-cycle-per-revolution blade-rate signal This analysis reveals that the blade thickness pressure is nearly in-phase both upstream and downstream of the propeller, exhibiting an even function in x Conversely, a significant phase shift is noted in the pressure due to blade lift, which behaves as an odd function in x, indicating substantial cancellation in the lift-associated pressure when integrated for the net vertical force on the plate In the case of an infinitely large plate, both thickness pressure and lift pressure independently integrate to yield a net vertical force of zero, illustrating the Breslin condition established in 1959, which was confirmed by integrating theoretical pressures from a noncavitating propeller in uniform inflow over an infinite flat plate.
Figure 11 presents a contour plot of blade-rate pressure amplitude from a uniform wake, flat plate experiment (Breslin & Kowalski, 1964), highlighting only the amplitude while omitting the phase shift distribution that affects integration cancellation Both Figures 10 and 11 indicate that propeller-induced hull surface pressure is significantly localized around the propeller, with pressure values dropping to a small percentage of their maximum within one propeller radius This observation may lead to the misleading conclusion that resultant forces on a ship are also concentrated near the propeller, a misconception that will be addressed in the following section.
It was shown in the propeller bearing force theory that
Fig 9 Flat-plate pressure measurements.
Fig 10 Flat-plate pressure amplitude and phase distributions (A) Compari- sons of theoretical and experimental values, thickness contribution, r / R =
The study presents a comparison of theoretical and experimental values, specifically noting a loading contribution with r/R = 1.10 and J = 0.833 The analysis of flat-plate pressure contours at J = 0.6 reveals that only specific shaft-rate harmonics from the nonuniform wake affect the blade-rate bearing force harmonics In contrast, the propeller-induced hull surface excitation involves contributions from all shaft-rate wake harmonics to each blade-rate excitation harmonic However, certain wake harmonics are more dominant, with their influence primarily determined by the hull form, which will be explored in greater detail later.
The pressure distribution corresponding to the wake operating propeller (without cavitation) has a very similar appearance to the uniform wake case Figure
In Vorus (1974), the study presents both calculated and measured blade-rate pressure amplitudes at various locations within the propeller plane of a DE-1040 model The pressure calculations were based on the assumption that the hull surface was perceived by the propeller as a flat plate extending infinitely.
The upper section of Fig 12 illustrates the measured pressure generated by the wake-operating propeller, alongside the associated calculated results The pressure calculations for both blade rates take into account the uniform wake effects resulting from steady blade lift and blade thickness.
The contributions from the nonuniform part of the wake, represented by wake harmonics 1 through 8, play a significant role alongside the steady blade-lift and blade-thickness components, denoted as the "zeroth" wake harmonic.
The analysis in Fig 12 reveals the blade-rate pressure distribution, highlighting the dominant role of uniform wake effects, which include steady blade lift and blade thickness The pressure attributed to these uniform wake components closely resembles the total pressure depicted in the upper section of the figure In contrast, the contribution from nonuniform wake components, comprising unsteady lift harmonics 1 to 8, remains relatively insignificant, overshadowed by the substantial uniform wake influence.
The integral of pressure exerted by a vertical force on the relatively flat stern exhibits distinct characteristics regarding the contributions from uniform and nonuniform wake components, as illustrated in Fig 13 from Vorus (1974) This figure accurately models the hull surface, ensuring zero pressure at the water surface The second column of Fig 13 presents the total vertical hull surface force calculated for the DE-1040, while the subsequent ten columns detail the force contributions from blade thickness and the first nine harmonics of blade lift Notably, the analysis reveals that the nonuniform wake components, although small, play a significant role in the overall force dynamics.
Fig 12 Blade-rate flat-plate pressures on destroyer stern, station 19.
The calculated vertical hull surface forces on the stern of the destroyer DE 1040 reveal that the integrated surface pressure is significantly influenced by the large uniform wake pressure resulting from steady blade lift and thickness This pressure nearly cancels out over the flat stern surface, adhering to the Breslin condition Consequently, the primary exciting force is attributed to the wake harmonics corresponding to the blade count, as the DE 1040 is equipped with five blades.
Actually, the Breslin condition, as established by
Breslin's 1959 study on uniform inflow can be extended to include nonuniform inflow scenarios It posits that for a general noncavitating propeller, the unsteady vertical force on an infinite plate above the propeller is equal and opposite to the force on the propeller itself, resulting in a net vertical force of zero on the combined plate/propeller system This principle also applies to uniform inflow, where both forces are individually zero The DE-1040 example illustrated in Fig 13 effectively demonstrates the nonuniform inflow case.
The vertical bearing force generated by the propeller is solely derived from the blade-order multiple harmonics of the wake, specifically the plus and minus one harmonics In the case of a propeller operating in a wake created by an infinite flat plate, the vertical force exerted on the plate is equal in magnitude but opposite in direction to the vertical bearing force, and is also composed exclusively of the same blade-order wake harmonics.
The DE-1040 vertical surface force spectrum, depicted in Fig 13, clearly shows pronounced harmonics, particularly in the flat plate-like characteristics of the DE-1040 stern The vertical blade-rate surface force is predominantly influenced by the fourth, fifth, and sixth harmonics, with the fifth harmonic experiencing significant amplification due to the water surface extending beyond the waterplane at the aft.
The DE-1040 experienced a significant cancellation in its net vertical force, with a calculated bearing force amplitude of 0.00205 When this was vectorially added to a surface force of 0.0015 amplitude, the resulting net force amplitude was 0.00055, indicating a considerable reduction in force.
Underwater Radiated Noise
Cavitation Dynamics as a Noise Source
is effectively achieved in modern U.S warships
The analysis of propeller cavitation as a source of vibration and noise begins with essential theoretical foundations This section aims to establish a clear understanding of the problem, referencing key acoustic texts by Beranek to support the discussion.
Propeller sheet cavitation occurs due to the pressure changes from the expanding and collapsing cavity volumes on the propeller blades, exhibiting a periodic dynamic that repeats with each blade revolution Theoretical models suggest that this phenomenon consists of multipole singularities, such as monopoles and dipoles, with higher-order singularities indicating more complex local cavitation patterns However, the sound pressure generated by these higher-order components decreases rapidly with distance, leading to the fundamental monopole component dominating the acoustic field at greater distances In the far field, crucial for acoustic detectability, propeller sheet cavitation can be effectively represented as a series of symmetrically spaced monopoles attached to the rotating blades, with each source strength denoted as q( ), where = −⍀t.
14 The strength of each point source is the periodically pulsating velocity of the cavity volume variation ˙( )
Following the development of the propeller bearing force formula in Section 2.3, equation (94) is first writ- ten in the alternative form
Now replace by + 2 (k − 1)/N in equation (95) and sum over N to obtain the source strength, Q( ), repre- senting all blades collectively.
But as developed on Section 2.3, the k summations in equation (96) are zero if ±p is not an integer multiple of
N, say mN, and the k summations are equal to N for ±p mN This reduces equation (96) to: inN n nN e N
= – t in equation (97), consistent with Section 2.3.
With ˙ q denoting the complete set of single-blade cav- ity volume velocity harmonics, as complex amplitudes,
The fundamental blade-rate frequency, denoted as N, is determined by the number of propeller blades and the angular velocity of the propeller This results in the filtering of the entire cavitation volume velocity spectrum, isolating it to the blade-rate frequency component and its harmonics, which can be observed in the far-field of the propeller.
Far-Field Sound Pressure
The net cavitating propeller generates an oscillating pressure, p(r,t), in its vicinity, where r represents the radius from the source center This sound pressure is described by the general acoustic wave equation, illustrating the relationship between pressure and wave propagation in the field.
= (98) where c is the velocity of sound in water.
At 0° C, sound travels at 1403 m/sec in water compared to 332 m/sec in air, highlighting a significant difference in sound propagation speed This disparity is primarily due to the density differences between air and water, resulting in lower attenuation and greater reach of sound in water Consequently, this characteristic is crucial for subsurface detectability.
For spherical waves with only a radial spatial depen- dence, as produced by the point sources, equation (98) reduces to
A general form of the solution to equation (99), in view of the linearity of the equation and the Fourier se- ries representation of the source disturbance, is
–N in equation (100), with the A n being a set of constants to be determined Equation (100) can also be written in the alternative forms
= (101) n 2c/n and k n n/c = 2 / n in equation (101) are the acoustic wavelength and acoustic wave number, respectively.
Equations (100) and (101) demonstrate that sound waves of varying lengths, n, travel at a consistent speed When the exponential argument in equation (100) equals zero, it indicates that the observer is moving at the same speed as the wave system, positioning the observer at r = ct.
To connect the A n constants in equations (100) and (101) with the cavity volume velocity harmonics in equation (97), it is essential to understand that the governing equation represents an alternative formulation of Newton's Law as it applies to radially expanding particles.
2 r t p= (102) where is the (constant) water density and is the par- ticle radial displacement on spherical surfaces.
Integration of equation (102) in time gives the normal
In the context of an effective spherical cavity influenced by an oscillating source, the radial velocity at the surface, denoted as r = r0(t), must match the radial fluid velocity, vr This surface radial velocity is defined as the instantaneous source strength divided by the spherical surface area, which is calculated as 4r0² Consequently, Equation (104) is established based on this relationship.
For k n r 0=nr 0/c