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Wave propagation in simple hull-frame structures of ships
Journal of Sound and Vibration (1976) 44(3), 39345
WAVE
PROPAGATION
IN SIMPLE
HULL-FRAME
STRUCTURES
OF SHIPS A. C. NILSSON Division for Marine Technology, Det norske Veritas, Etterstad, Oslo 6, Norway (Received 7 April 1975, and in revisedform 12 July 1975)
The one-dimensional wave propagation parallel to the frames in a hull-structure is investigated. Rotation and shear, and thus also the coupling between flexural, longitudinal and transverse waves, are included in the model. The wave numbers for travelling waves and for near field solutions are derived for two typical frame constructions. The effects due to a water load on the hull plates are also investigated. It is found that for low frequencies the lateral motion of the construction is mainly determined by pure bending. The results indicate that the wave numbers calculated, when shear and rotation are included, exceed by more than 50% the result obtained for pure bending for frequencies above 600 Hz in a typical main frame structure. For high frequencies, the energy flux in the hull plate is dominant. The particle velocities in the hull plate and the frame are also
compared.
1. INTRODUCTION
Noise levels in the living accommodations on board ships are often determined by structureborne sound. The structure-borne sound is chiefly induced by the main and auxiliary engines and by the rotation of the propeller. It has frequently been observed that the structure-borne sound mainly propagates from the sources up through the ship structure in the direction of the frames (see, for example, reference [l]). The energy can be transported in the form of flexural, longitudinal and transverse waves in plates and frames, and also as torsional waves in the frames. However, the flexural waves are of primary interest since these waves are comparatively well coupled to the sound fields in the surrounding media: i.e., air or water. Other wave forms are of importance, as they can be transformed into flexural waves when propagating over a discontinuity, such as a deck or a junction. It has been reported that in a typical ship structure the energy flux due to longitudinal and transverse waves can be as important as the energy carried by the flexural waves (e.g., compare references [2] and [3]). To describe the structure-borne sound propagation in ship structures it is necessary to use approximate models. Sawley [4] has discussed the possibility of using the statistical energy analysis as developed in reference [5]. Chernjawski and Arcidiacano [6] have investigated the transmission of flexural waves in a multipath plane structure. This latter work is based on the results obtained by Cremer and Heckl. These authors have summarized some of the most important aspects of wave transmission through single junctions in reference [7]. Transmission problems concerning waves with normal incidence to a junction have also been discussed by Budrin and Nikiforov [8]. In addition to the models above the finite element method can be a useful tool for finding the resulting vibrational energies in the structure. In particular this is true for low frequency solutions. 393
394
A. C. NILSSON
For any of the models mentioned above it is of vital importance that the wave motion, and thus the wave number, or propagation constant, can be properly defined. Further, to estimate the damping in a structure, not only the loss factor but also the wave number must be known. The classical Bernoulli-Euler theory for beams, and the Kirchhoff theory for plates, can only be used to find the propagation constants as long as the wave length is much larger than the largest dimension of the cross-section of the structure. This condition is generally fulfilled for typical building structures of concrete in the frequency range of interest. In a ship structure, however, this is no longer the case, and consequently shear and rotational effects must be included in the computational model. In order to describe the dominating energy flux in the direction of the frames the hull structures, approximately, can be considered as a system consisting of a number of parallel (unconnected) beams. Each beam is made up of a strip of the hull plate, the web plate and the flange. For low frequencies, an approximate value of the propagation constant in the direction of the frames can be obtained by using classical theory. For higher frequencies, shear and rotation must be taken into account. For certain conditions these effects have been included in the Timoshenko theory, and in the so-called method of internal constraints. However, in order to obtain an explicit expression for the propagation constant, from any of these models, the Timoshenko shear coefficient must be known. This coefficient is mainly a function of the dimensions of the cross-section of the beam. An exact calculation of the coefficient for a given geometry is not readily accomplished; refer to the discussion in section 2 below. An exact theory based on the mathematical theory of elasticity has been discussed by Pochhammer and Chree for the lateral motion of beams (see, for example, reference [9]). This model applies to bars with circular cross-sections, and it has been used by Abramson [lo] to solve exactly the propagation constant for flexural waves. These exact results can thus be compared to approximate solutions. It is found that for low frequencies the propagation constant approaches that predicted by elementary theory for pure bending. The high frequency limit is, however, determined by surface waves of the Rayleigh type. The exact theory is extremely cumbersome if attempted on beams with other than circular crosssections. This is due to the number of boundary conditions that must be taken into account at each surface. For a structure consisting of a frame and a hull plate-where the thickness of each plate is small-the exact theory can be modified and used to calculate the propagation constant for flexural waves travelling in the direction of the axis of symmetry. This approach is used below. Some of the basic theories for wave propagation in beams are summarized in references [9] and [ll]. An alternative to the beam model for the hull structure has been proposed by Fahy and Lindqvist [12]. This model consists of two parallel line stiffeners bounding a strip of plate which does not extend beyond the stiffeners. The model has thus the characteristics of a wave-guide system. However, computational difficulties might possibly occur due to the number of solutions for the motion of the plate perpendicular to the axis of symmetry. The equation being lateral motion of the frames is described in reference [ 121, the Bernoulli-Euler used.
2. APPROXIMATE
SOLUTIONS
FOR BEAMS
For low frequencies, and for beams with small dimensions of the cross-section, the basic Bernoulli-Euler or Timoshenko theories can be used to calculate the propagation constants for the lateral one-dimensional displacement. The first approach only takes into account pure bending, whereas in the second both shear and rotation are included. For free waves,
PROPAGATION IN HULL-FRAME STRUCTURES
395
and for a sinusoidal time variation, general one-dimensional solutions to each of the two equations can be written as ,I = e-‘~‘{A eik,x + Be-i' *x + Ce-‘2* + D ek2”}, (2.1)
where q denotes the displacement perpendicular to the beam (a list of symbols is given in the Appendix). The first two functions inside the bracket correspond to waves travelling along the positive and negative x-axis. The last two functions represent fields which decrease exponentially with the distance from a disturbance or a boundary. These expressions might also be called near field solutions. The amplitudes A, B, Cand Dare determined by the boundary conditions for the entire beam. The resulting propagation constants, according to the Euler-Bernoulli theory, are determined by k, = k, = [/.KLI~/EI]“~=
KB,
(2.2)
where p is the mass per unit length of the beam, E the modulus of elasticity and Z the second moment of area for the cross-section. The suffix B denotes that only bending has been considered. The corresponding results for the Timoshenko model are k, = {k:[0.5 + (1 + v) r] + {k;[0*5 - (1 + v) r]’ + ~;}l’~}l’~,
(2.3)
k, = {{k;[0*5 - (1 + v) z]’ + ~84)~‘-~ k;[0.5 + (1 + v) 7]}1’2.
(2.4)
K~ is defined in equation (2.2), k, is the propagation constant for longitudinal waves in the beam, v is Poisson’s ratio and 7 is the Timoshenko shear coefficient. This coefficient, according to reference [7], page 108, is defined as
7=(1/W!
7:,dS/[(1/S)S5,,dS12.
The expression is valid for lateral one-dimensional motion of the beam: i.e., where 7,. = 0. For low frequencies the effects of shear and rotation are small and k, and k, in equations (2.3) and (2.4) both approach JC~as expected. In the high frequency range, k, is larger than K*, and thus according to equation (2.3) the limit for k, is equal to k, and hence independent of the shape of the cross-section. The corresponding result for k2 is an imaginary expression. A comparison between expression (2.3) and the exact solution obtained by Abramson for a circular cross-section indicates that the exact result is about 1.7 times as large as that predicted through equation (2.3) in the high frequency range. 3. BASIC ASSUMPTIONS Figure 1 shows a hull-frame construction. The structure extends along the x-axis and is considered to consist of parallel sections. Each section is referred to as a beam. The strip of
Figure 1. Configuration
of a hull-frame structure.
396
A. C. NILSSON
the hull plate is for each such beam limited by the centrelines-: = b,l2 and z = -O,i2-on each side of the frame. The thicknesses t,, t2 and t3 of the hull plate, web plate and the Range are assumed to be small compared to the other dimensions. Only wave motions parallel to the x--J)-plane and which correspond to the fundamental mode are considered: no twisting or any other motion in the z-direction is included in the model. Thus the beam model consists of a coupled system made up by the hull plate, web plate and the flange. Following the assumptions above, the following wave motions must be considered: flexural and longitudinal waves in the hull plate and in the flange, longitudinal and transverse waves in the web plate. All these waves are coupled.
4. GOVERNING
4.1. THE HULL-PLATE
EQUATIONS
1
The thickness of the plate is assumed to be small. Thus shear, and rotation in the plate itself, can be neglected. For flexural waves in steel plates, typical for a ship structure, this approximation is valid for frequencies up to 10 kHz. The normal stress components, 6, and oZ, are assumed to vanish on the boundaries and in fact to be zero throughout the plate. The first assumption is a consequence of the plate being thin. The second condition is typical for models describing one-dimensional wave propagation in beams. However, it might be argued that it is more appropriate to assume that the strain E, rather than the stress G, is equal to zero at the boundaries of the hull plate. For the final result concerning the propagation constant the difference between the two boundary conditions is insignificant. Thus with c,, = crZ= 0, the differential equation valid for flexural waves in the plate is a4r7riax4 + (trio,)
aZ0t2
= F,,/D,.
(4.1)
Here q1 is the displacement in the y-direction, pL1the mass per unit length and D, the flexural rigidity. For a strip of hull plate with width br, thickness tr and modulus of elasticity E,, the flexural rigidity can be written as (see Figure 1) D, = El tfbJ12.
(4.2)
Expression (4.2) is valid under the condition that oZ = 0. The external force per unit length, FvI, is caused by the normal stress in the web plate at the boundary (see Figure 2). Plate3
Plate
Figure 2. Resulting forces at the boundaries
-
2 -
flange
web plate
due to normal and shear stresses in the web plate,
PROPAGATION
IN HULL-FRAME
STRUCTURES
397
The corresponding equation for longitudinal waves in the hull plate is
a2 t1/ax2 - (P,/E,)
a2 r,/atz = -F,,ib,
tI Ed.
(4.3)
The displacement along the x-axis is denoted by t1 and the density of the plate by pi. The external force per unit length, Fxl, is caused by shear stress in the web plate and is shown in Figure 2. 4.2.
2
THE WEB PLATE-PLATE
Again the thickness of the plate can be assumed to be small compared to the other dimensions of the plate. This implies that the stress component in the z-direction, or, is equal to zero. According to the theory of elasticity the expressions for normal and shear stresses for a thin plate are : gy = v3,iu
- ~91 [a~,/ay + v ac,m,
ry, = rxy = [~,/2(1 +
41 [x,/ay + atl,bw.
(4.4) (4.5)
The displacements along the x- and y-axes are denoted by t2 and q2, respectively. The displacements are determined by longitudinal and transverse waves in the x-y-plane. Let the velocity potentials @ and Y describe these waves, so that the displacements along the axes are determined by (see reference [7]) c2 = aqax q2 = aqay
+ a!zqay,
(4.6)
- ayylax.
(4.7)
The first part in each expression is thus due to L (longitudinal) waves and the second part to T (transverse) waves. Consequently, the potentials @ and Y must fulfil the following equations for 0 < y < h (see Figure 1) : v2 cp- [p,(l - v2)/E2 ] a2 G/at2 = 0, v2 y - [2p,(l + v)/E~ J a2 ly/afz
= 0.
(4.8) (4.9)
The equations are valid for a thin plate--cr, = 0. 4.3.
THE FLANGE-PLATE
3
The assumptions made for the flange are similar to those made for the hull plate. Thus, with os = cr,,= 0, the governing equations for the displacements due to flexural and longitudinal waves are a4q3/ax4 + (P~/D~) a2q3/at2 = -F,,~D,,
(4.10)
a2 t3ja2
(4.11)
- &/Es)
a2 t/at2 = FxJb3 t, E3.
The flexural rigidity for the plate is determined by (see Figures 1 and 2) D,=E,t,Jb,/l2.
5. BOUNDARY CONDITIONS The resulting forces on the hull plate and the flange due to stresses in the web plate at the boundaries y = 0 and y = h are described by F,, =
t2 hdw,,
(5.1)
Fxs =
f2 (Qy=h,
(5.2)
F,I =
t2 Wy=o,
(5.3)
Fy2
t2 W,=h.
(5.4)
=
398
A. C. NILSSON
Further, the displacement flange. This implies that
is continuous
at the junctions
between
web plate, hull plate and
111 =
q2
for
4’= 0,
(5.5)
13
‘12
for
L’= h,
(5.6)
=
51 = 52
for
4’= 0,
(5.7)
43
for
y = h.
(5.8)
=
(2
6. ASSUMED
WAVEFORMS
Let the propagation constant for the entire wave motion in the x-direction be denoted by k, and let the time dependence be e ‘Of. The general expressions for the flexural and longitudinal waves in the hull plate and flange can consequently be written as, for flexural waves, q1 = A1 ei(ot-kx) 3 (6.1) (6.2) and for longitudinal
waves, (6.3) t3 = B, ei(o+kx).
(6.4)
The suffix refers to the plate in question (see section 4). The amplitudes A and Bare frequency dependent. For the web plate the displacement can be in both y- and x-directions. The potentials describing the L and T waves must satisfy each respective differential equation for 0 < y < h. Because of the boundary conditions the propagation constant in the x-direction must be the same as that for the wave motion in the adjoining plates. The general expression for the L-waves in the web plate is thus of the form
(6.5) This expression
when inserted
in equation
(4.8) yields
I, = [{p,(l - v’)w2/E2} - k2]“2. The corresponding
expression
for the T-waves is y =
where &- is determined
(6.6)
ei(mr-kx)
fj2
e-iAp
+
~~
ef+~]
9
(6.7)
I, = [{p2 2( 1 + v) 02/Ez> - k2]1’2.
(6.8)
[
through
7. RESULTING
SYSTEM OF EQUATIONS
The four boundary conditions and the four wave-equations are sufficient to determine the propagation constant. The forces at the boundaries, included in the wave equations, are obtained by inserting the equations (4.4)-(4.7) and (6.5) and (6.7) in equations (5.1)-(5.4). For convenience the following notations are introduced: the wave number for flexural waves (plates 1 and 3), K, = [P” 02/D,]“4
= [p, 1209/E” t,2]‘/4;
(7.1)
PROPAGATION
IN HULL-FRAME
399
STRUCTURES
the wave number for longitudinal waves (plates 1 and 3), kLn = [p. o’/E,]“‘;
a constant proportional
(7.2)
to the normal stress, L
=
[E,lU
-
v’)lP2~2lEn
b,
(7.3)
&?I;
a constant proportional to the resulting shear stress, M,, = E2 f2/2(l + v) b, t,, E,,.
(7.4)
The suffix it refers to the plate in question-l for the hull plate and 3 for the flange. Now insert equation (6.1) and the final expression for F,,, in the wave equation (4.1). A1 is eliminated by using the boundary condition (5.5). The amplitudes A,, B1 and B3 are eliminated in a similar way by using the remaining wave equations and boundary conditions. The result is a set of four equations. Introduce the symbols X, Y and Z, so that X = AJiB,,
Y = A.,/iB4,
Z = B,lB,.
(7.5)
In terms of these symbols, the resulting system of equations becomes X[L,(Iz
+ vk2) - i1.,(k4 - K:)] f Y[L,(Ii
+ Z[k(k4 - K?) + i&L, XemiALh[L,(iZ, + vk2) + &(k4
k(1 - v)] = -k(k” - K:) + i& kL1(l - v), - v)] = e’“Th[k(k4 - ~2) + i& kL,(l
- v)],
(7.7)
- k2) - i2kll MI ] + Y[k(k,Z, - k2) + i2kl, MI J -
- Z[M,(Af. Xe-‘iLh[k(k,Z,
(74
- K$)] + Ye*‘Lh[L3(ki + vk2) - i&(k4 - K$)]-
_ Ze-*W[k(k4 _ K$)- i& kL,(l X[k(ki,
+ vk2) + i&(k4 - JC?)]+
- k2) + il,(k& - k2)] = Ml($
- k2) - i&.(k~l - k2),
(7.8)
- k2) + i2kl, MS] + Ye’“Lh[k(k& - k2) - i2kJ, MS] +
+ Ze-‘“Th[M,(I$
- k2) - il,(k&
- k2)] = -ei’Th [M&
- k2) + i&(k& k2)].
(7.9)
This system is non-linear in k. There are a number of solutions to the system. As stated before, however, the propagation constant for the fundamental mode is of primary interest. For this mode and for travelling waves the functions 2, and AT are both imaginary. Consequently all the unknown factors are real. From consideration of the definitions of the functions in the system, it is apparent that if a solution k is found then -k is also a solution. This is natural since the velocity of a wave is independent of the direction of propagation along the axis of symmetry. The remaining four amplitudes, AI, A,, B, and B,, can be determined relative to B4 by using the boundary conditions given in the equations (5.5)-(5.8) and the definitions (7.5). The result is AI/B4 = 2=(X - Y) + ik(Z + l), AZ/B4 = %,(Xemi2Lh - Ye‘lLh) + k(Ze-‘“Th + eiATh), B1/B4 = k(X+
Y) - i&(Z - l),
BS/B4 = k(Xe-lALh + YeiALh)- i&.(Ze-iArh - eiATh).
(7.10) (7.11)
(7.12) (7.13)
Expressions giving the velocity level differences for the waves in the various plates are immediately derived by using the equations above. Let vFI and vF3, vL1 and vLj denote the particle velocities for flexural (F) and longitudinal (L) vibrations of the hull plate (1) and
400
A. C. NILSSON
flange (3). The level differences, thus are described by
according
to the equations
(6.1)-(6.4)
and (7. IO)-(7.13),
(7.14)
lOlog lUF1/UF~j2= lOlog /iJ?j3/z = lOlog IA,/A,IZ,
If so required,
lOlog 1UFII~‘lIZ = 101% 1~11~112,
(7.15)
lOlog IuFJ/v‘31z=
(7.16)
similar expressions
lOlog lA3/B312.
can be formed for the velocities
in the web plate.
8. SOLUTIONS An explicit solution to the system of equations for low frequencies and for typical ship structures These solutions are of the form k = ~{k;,(l+
cannot readily be obtained. However, approximate solutions can be derived.
+ v) + [(O-5 + v)‘k,4, + k;,t,(b,
tl + b, t3 + t, h)/hb, tl b, t,]“2}“2.
(8.1)
In equation (8.1) it is assumed that the plates are made of the same material: i.e., k,, is set real and two imaginaryequal to k,,. Thus for low frequencies there are four solutions-two corresponding to the fundamental mode. The absolute value of the imaginary part is, however, not equal to the absolute value of the real part, as in the case for solutions to the elementary equations for flexural waves in a beam (see section 2 above). More complete solutions to the set of equations can be obtained by using an iterative technique-for example a modification of the Newton-Raphson method. The four equations (7.6) to (7.9) are of the form a,, X + a,, Y +
Z = b,,
a,3
where n = 1, 2, 3, 4. The coefficients a,,,, . . . are functions of k. Let the initial value k. be determined by the approximate solution (7.1) in the low frequency range. The first three equations, (7.6) to (7.8), are used to solve for X, Y and Z. Let the second value in the iteration chain be k, = k. + E. The solutions X, Y and Z are inserted in the fourth equation (7.9), and the coefficients a4r, . . . are expanded around k,. With only terms of the first order included, the general result becomes k n+l
a41 =
kn -
Xaa,,/ak
x
+
a42
y
+
a43Z
-
b,
1
+ Y aa,Jak + Z i3a43/ak- ab4/ak ’
k=k,
wheren=0,1,2.... The calculations are continued until a sufficient accuracy say, a result is required for the center frequency in each l/3-octave band, Hz, the initial value for each new band can be set equal to the value obtained band, and so on. The imaginary-or near field-solutions can be derived with ik in the equations (7.6) to (7.9) and by using the appropriate initial equation (7.1). The remaining quantities Al/B,, etc., are found by inserting for k, X, Y and Z in the equations (7.10) to (7.13).
(8.2)
is obtained. If, starting at 31.5 in the preceding by replacing k value for kthe final results
9. RESULTS Normally the hull structure consists of main frames and ordinary frames attached to the hull side. The height of the web plate and the width of the flange are for a main frame about three to four times the corresponding dimensions for an ordinary frame. The spacing between the frames is generally constant. Every fifth frame is in most cases a main frame.
PROPAGATION
IN HULL-FRAME STRUCTURES
401
Two examples corresponding to the different frame types have been investigated. The dimensions of the beam structures and the material parameters are listed in Table 1 (see Figure 1). The same type of steel is assumed to be used for hull, web plate and flange. TABLE
1
Dimensions and material parameters of beams investigated as examples V
(hh3) Beam 1
0.6
0.2
0.6
Beam 2
0.6
0.05
0.2
0.012 0.012
0.012 0.012
0.012 0.012
7.6 x lo3 7.6 x lo3
2 x IO” 2 x 10”
0.3 0.3
For the fundamental mode the equations (7.6)-(7.9) yield four solutions for the propagation constants. Two solutions are real and have the same absolute value and represent waves travelling in opposite directions of the x-axis. The remaining two solutions are imaginary and of the same magnitude, but with opposite signs. These waves are consequently describing the near fields (see section 2). The real and imaginary wave numbers, calculated as described in section 8, are shown in Figures 3(a) and (b) for the beam constructions described above. The wave numbers corresponding to pure bending of the entire beam, and for free flexural waves on the hull plate only, are also drawn in the figures. The resulting velocities due to lateral (F) and longitudinal (L) motion of the hull plate (suffix 1) and the flange (suffix 3) are compared in Figures 4(a) and (b) (see also section 8 above). For decreasing frequencies the wave number for the lateral motion approaches that which was derived for pure bending for the entire construction (see equation (2.2)). Consequently, in the low frequency region the lateral velocities of the hull plate and of the flange are the same (see Figures 4(a) and (b)). Pure bending also implies that the deflection along the line represetting the centre of mass for the cross section is negligible. Hence, the ratio between the longitudinal velocities for the hull plate (vL1) and the flange (vL3) can, for low frequencies, be determined through the relationship vL1IvL3 2: h,/h,, where h, and h, are the respective distances
2.0
from
I
I
I
31.5
63
I25
the plates to the centre of gravity
for the cross-section.
250
31.5
O-
500
KXO
2ooO
4000
6000
16ooo Frcqmcy
63
125
250
500
Kc0
2caJ
4cco
mO0
16ccm
(Hz)
Figure 3. Calculated wave numbers for (a) beam 1 and (b) beam 2. -, Travelling waves; ----, near field solution; -. -, bending only; equation (2.3); . . *. . ., flexural waves in the hull plate; equation (7.1).
A. C. NILSSON
0
31.5
__.. 63
125
250
500
/’ , loo0
2CCO
4ooO
FKWJ
31.5
Bond ct?ntre frequency
63
125
250
500
ICC0
zoo0
‘wx
8003
(Hz)
Figure 4. Velocity level differences in (a) beam 1 and (b) beam 2. The levels are calculated for each l/3-
For increasing frequencies the wave number for the propagating waves increases, whereas the corresponding wave number for the near field solution decreases relative to the value from the simple solution valid for pure bending. The deviations are due to shear and rotational effects. In the mid-frequency range these effects are becoming increasingly pronounced. This causes the wave number for the propagating waves to increase with frequency more rapidly than was the case for the lower frequencies. At the same time the motion of the flange decreases sharply. This implies that the entire motion of the beam becomes more strongly determined by the larger mass: i.e., the hull plate. The lateral velocity of the hull plate has a maximum in the 2000 Hz one-third octave band for beam 1 and in the 2500 Hz band for beam 2. These maxima are functions of the dimensions of the structure. The shear and rotational effects start to dominate at even lower frequencies for higher web plates. In the high frequency range the propagation constant approaches that which is valid for free flexural waves in the uncoupled hull plate and is thus proportional to the square root of the frequency. The same is true for the wave number describing the near field. In the latter case, however, the approach to the asymptotic value is from above. Thus for high frequencies the motion is dominated by a type of wave propagating at the outer boundary of the structure. The analogy with the solutions for the circular bar discussed above is obvious. If the analysis is pursued further, then the propagation constant should again increase more rapidly when-at about 16 kHz--the simple Kirchhoff equation for the plate no longer is valid. Consequently in the very high frequency range-about 20 kHz for a steel plate-the propagation constant should be proportional to the frequency as is the case for ordinary surface waves. The wave motion then would be of the Rayleigh wave type. However, solutions for frequencies above 10 kHz are generally not of interest.
10. COMPARISON
WITH THE TIMOSHENKO
THEORY
The Timoshenko shear coefficient, z, can be derived by inserting the exact solution for the propagation constant for travelling waves in equation (2.3). It is found that z is frequency dependent, although for low frequencies its value is fairly constant. For beam 1 and for frequencies up to 400 Hz, the best agreement between the Timoshenko theory and the exact solution is obtained for T equal to 2.2. Figure 5 shows a comparison between the exact solution and the result obtained by using equation (2.3) with r = 2.2. Also indicated in the figure is the result which is valid for bending only, equation (2.1). Already at about 600 Hz the deviation between the exact and the elementary solutions
403
PROPAGATION IN HULL-FRAME STRUCTURES
O-
-0.5
" 31.5
'I 63
”
“1 125
”
250
1’ 500
Frequency
1” Km0
1 2ooo
”
’
4000
”
6000
(Hz)
Figure 5. Comparison between exact and approximate solutions of the wave numbers for travelling waves on beam 1. --, Exact; ----, approximate, equation (2.3) with T = 2.2; -.-, approximate, equation (2.2).
exceeds 50%. At 2000 Hz the same difference is obtained between the Timoshenko model result and the exact solution. This in spite of the fact that the exact solution has been used to compute the factor r for obtaining good agreement at low frequencies.
11. CONCLUSIONS
For models describing the structure-borne sound propagation in ships it is often essential to include shear and rotational effects, even at comparatively low frequencies. Due to the particular dimensions of the plates in ship structures, accurate and fairly simple models, in which these effects are included, can be used for frequencies up to and even above 10 kHz. The quantities of interest as propagation constants, deflections, moments and forces can be derived by using simple computer programs. Flexural, longitudinal and transverse waves must be considered in order to make it possible to fulfil the necessary boundary conditions in the hull-frame structure. For these coupled waves the resulting (particle) velocities due to flexural waves in the hull plate are considerably higher than the in-plane velocities caused by longitudinal waves. Thus, when measurements indicate that the (particle) velocities of the in-plane waves and the flexural waves are of the same order, then the L and T waves in the hull plate can be assumed, approximately, to be superimposed and uncoupled to the flexural motion of the structure. For low frequencies the energy transport in the ship side is caused by bending of the entire beam or frame structure. In the high frequency range the energy flux in the structure is mainly due to flexural waves in the hull plate. The energy loss per unit length can be obtained in an often-used manner by writing the modulus of elasticity as E( 1 + iv), where v is the loss factor. In this way the resulting wave number for travelling waves is assigned an imaginary part which determines the losses. For low and high frequencies the decay rate for the energy is of the order 2 x k x q dB/m, where k is the wave number. For intermediate frequencies the losses are somewhat higher, or about twice as large. At 1000 Hz for a typical main
A. C.
404
NILSSON
frame structure with a loss factor of 1 % the energy loss per metre becomes 0.07 dB for the forced motion of the entire structure. Even if the loss factor is increased with a damping layer by, say, a factor of ten, the energy loss is still fairly small. The lateral motion of the structure is well coupled to the air-borne sound field. For the entire main frame construction (beam 1) the critical frequency is of the order 16 Hz. The corresponding value for the frame (beam 2) is about 55 Hz. These low critical frequencies thus will ensure a high radiation efficiency over almost the entire frequency range of interest. The model discussed in the preceding sections also can form the basis for calculations of transmission through junctions in ship structures. The calculations can be made according to the principles suggested by Cremer and Heck1 [7]. REFERENCES
5. 6.
7. 8. 9. 10. 11. 12.
J. H. JANSEN1962 Netherlands Ship Research Centre, Deljt, Reports Nos. 44s and 4%. Some acoustical properties of ships with respect to noise control. J. 0. JENSEN1974 Proceedings ofthe 1974 International Conj>rence on Noise Control Engineering. Washington, D.C. Structureborne sound transmission in ships. T. KIHLMAN and J. PLUNT 1974 Proceedings of the 8th International Congress on Acoustics, London. Structureborne sound in ships: study of different wave forms. R. J. SAWLEY1969 Contributed to Stochastic processes in dynamical problems, ASME, New York, 63-73. The evaluation of a shipboard noise and vibration problem using statistical energy analysis. R. H. LYON and G. MAIDANIK 1962 Journal of the Acoustical Society of America 34, 623-639. Power flow between linearly coupled oscillators. M. CHERNJAWSKI and C. ARCIDIACANO1972 Shock and Vibration Bulletin 42,235-243. Simplified method for the evaluation of structure-borne vibration transmission through complex ship structures. L. CREMERand M. HECKL 1967 Kiirperschall. Berlin: Springer-Verlag. S. V. BUDRINand A. S. NIKIFOROV1964 Soviet-Physics Acoustics 9, 333-336. Wave transmission through assorted plate joints. E. VOLTERRAand E. C. ZACHMANOGLOU 1965 Dynamics of Vibrations. Columbus, Ohio : Charles E. Merrill Books Inc. H. N. ABRAMSON1957 Journal of the Acoustical Society of America 29, 42-46. Flexural waves in elastic beams of circular cross section. I. N. SNEDDONand R. HILL (Editors) 1960 Progress in Solid Mechanics, Volume 1. Amsterdam: North-Holland Publishing Co. F. FAHY and E. LINDQVIST1974 Chalmers University of Technology, Gothenburg, Sweden, Report 74-37. Wave propagation in damped, stiffened structures characteristic of ship construction. . APPENDIX A, B, C, D D,, DL, 03
E FL, Fx3, F,.,, Fy3 I L. Mll S x, y, z b,, 63, h, tl, tz, f3 k,, k, k, kL t
: LIST OF SYMBOLS
wave amplitudes flexural rigidity for the plates modulus of elasticity forces per unit length of beam (Figure 2) second moment of area of beam cross section defined in equation (7.3) defined in equation (7.4) area of cross section of beam defined in equation (7.5) beam dimensions (Figure 1) propagation constants for lateral one-dimensional displacement propagation constant in sea water propagation constant for longitudinal waves in a beam time
of beam
PROPAGATION OF,
UL @
!P
e
11
‘;:
IZL,
P
V z c P Pw’ 7 7XY, 7x2 w
IN HULL-FRAME
STRUCTURES
particle velocities for flexural and longitudinal velocity potential for longitudinal waves velocity potential for transverse waves velocity potential in water displacement of beam in the y-direction propagation constant for free bending waves defined in equations (6.6) and (6.8) mass of beam per unit length Poisson’s ratio displacement in x-direction density of beam density of sea water Timoshenko’s constant shear stresses angular frequency
waves
405
WAVE
PROPAGATION
IN SIMPLE
HULL-FRAME
STRUCTURES
OF SHIPS A. C. NILSSON Division for Marine Technology, Det norske Veritas, Etterstad, Oslo 6, Norway (Received 7 April 1975, and in revisedform 12 July 1975)
The one-dimensional wave propagation parallel to the frames in a hull-structure is investigated. Rotation and shear, and thus also the coupling between flexural, longitudinal and transverse waves, are included in the model. The wave numbers for travelling waves and for near field solutions are derived for two typical frame constructions. The effects due to a water load on the hull plates are also investigated. It is found that for low frequencies the lateral motion of the construction is mainly determined by pure bending. The results indicate that the wave numbers calculated, when shear and rotation are included, exceed by more than 50% the result obtained for pure bending for frequencies above 600 Hz in a typical main frame structure. For high frequencies, the energy flux in the hull plate is dominant. The particle velocities in the hull plate and the frame are also
compared.
1. INTRODUCTION
Noise levels in the living accommodations on board ships are often determined by structureborne sound. The structure-borne sound is chiefly induced by the main and auxiliary engines and by the rotation of the propeller. It has frequently been observed that the structure-borne sound mainly propagates from the sources up through the ship structure in the direction of the frames (see, for example, reference [l]). The energy can be transported in the form of flexural, longitudinal and transverse waves in plates and frames, and also as torsional waves in the frames. However, the flexural waves are of primary interest since these waves are comparatively well coupled to the sound fields in the surrounding media: i.e., air or water. Other wave forms are of importance, as they can be transformed into flexural waves when propagating over a discontinuity, such as a deck or a junction. It has been reported that in a typical ship structure the energy flux due to longitudinal and transverse waves can be as important as the energy carried by the flexural waves (e.g., compare references [2] and [3]). To describe the structure-borne sound propagation in ship structures it is necessary to use approximate models. Sawley [4] has discussed the possibility of using the statistical energy analysis as developed in reference [5]. Chernjawski and Arcidiacano [6] have investigated the transmission of flexural waves in a multipath plane structure. This latter work is based on the results obtained by Cremer and Heckl. These authors have summarized some of the most important aspects of wave transmission through single junctions in reference [7]. Transmission problems concerning waves with normal incidence to a junction have also been discussed by Budrin and Nikiforov [8]. In addition to the models above the finite element method can be a useful tool for finding the resulting vibrational energies in the structure. In particular this is true for low frequency solutions. 393
394
A. C. NILSSON
For any of the models mentioned above it is of vital importance that the wave motion, and thus the wave number, or propagation constant, can be properly defined. Further, to estimate the damping in a structure, not only the loss factor but also the wave number must be known. The classical Bernoulli-Euler theory for beams, and the Kirchhoff theory for plates, can only be used to find the propagation constants as long as the wave length is much larger than the largest dimension of the cross-section of the structure. This condition is generally fulfilled for typical building structures of concrete in the frequency range of interest. In a ship structure, however, this is no longer the case, and consequently shear and rotational effects must be included in the computational model. In order to describe the dominating energy flux in the direction of the frames the hull structures, approximately, can be considered as a system consisting of a number of parallel (unconnected) beams. Each beam is made up of a strip of the hull plate, the web plate and the flange. For low frequencies, an approximate value of the propagation constant in the direction of the frames can be obtained by using classical theory. For higher frequencies, shear and rotation must be taken into account. For certain conditions these effects have been included in the Timoshenko theory, and in the so-called method of internal constraints. However, in order to obtain an explicit expression for the propagation constant, from any of these models, the Timoshenko shear coefficient must be known. This coefficient is mainly a function of the dimensions of the cross-section of the beam. An exact calculation of the coefficient for a given geometry is not readily accomplished; refer to the discussion in section 2 below. An exact theory based on the mathematical theory of elasticity has been discussed by Pochhammer and Chree for the lateral motion of beams (see, for example, reference [9]). This model applies to bars with circular cross-sections, and it has been used by Abramson [lo] to solve exactly the propagation constant for flexural waves. These exact results can thus be compared to approximate solutions. It is found that for low frequencies the propagation constant approaches that predicted by elementary theory for pure bending. The high frequency limit is, however, determined by surface waves of the Rayleigh type. The exact theory is extremely cumbersome if attempted on beams with other than circular crosssections. This is due to the number of boundary conditions that must be taken into account at each surface. For a structure consisting of a frame and a hull plate-where the thickness of each plate is small-the exact theory can be modified and used to calculate the propagation constant for flexural waves travelling in the direction of the axis of symmetry. This approach is used below. Some of the basic theories for wave propagation in beams are summarized in references [9] and [ll]. An alternative to the beam model for the hull structure has been proposed by Fahy and Lindqvist [12]. This model consists of two parallel line stiffeners bounding a strip of plate which does not extend beyond the stiffeners. The model has thus the characteristics of a wave-guide system. However, computational difficulties might possibly occur due to the number of solutions for the motion of the plate perpendicular to the axis of symmetry. The equation being lateral motion of the frames is described in reference [ 121, the Bernoulli-Euler used.
2. APPROXIMATE
SOLUTIONS
FOR BEAMS
For low frequencies, and for beams with small dimensions of the cross-section, the basic Bernoulli-Euler or Timoshenko theories can be used to calculate the propagation constants for the lateral one-dimensional displacement. The first approach only takes into account pure bending, whereas in the second both shear and rotation are included. For free waves,
PROPAGATION IN HULL-FRAME STRUCTURES
395
and for a sinusoidal time variation, general one-dimensional solutions to each of the two equations can be written as ,I = e-‘~‘{A eik,x + Be-i' *x + Ce-‘2* + D ek2”}, (2.1)
where q denotes the displacement perpendicular to the beam (a list of symbols is given in the Appendix). The first two functions inside the bracket correspond to waves travelling along the positive and negative x-axis. The last two functions represent fields which decrease exponentially with the distance from a disturbance or a boundary. These expressions might also be called near field solutions. The amplitudes A, B, Cand Dare determined by the boundary conditions for the entire beam. The resulting propagation constants, according to the Euler-Bernoulli theory, are determined by k, = k, = [/.KLI~/EI]“~=
KB,
(2.2)
where p is the mass per unit length of the beam, E the modulus of elasticity and Z the second moment of area for the cross-section. The suffix B denotes that only bending has been considered. The corresponding results for the Timoshenko model are k, = {k:[0.5 + (1 + v) r] + {k;[0*5 - (1 + v) r]’ + ~;}l’~}l’~,
(2.3)
k, = {{k;[0*5 - (1 + v) z]’ + ~84)~‘-~ k;[0.5 + (1 + v) 7]}1’2.
(2.4)
K~ is defined in equation (2.2), k, is the propagation constant for longitudinal waves in the beam, v is Poisson’s ratio and 7 is the Timoshenko shear coefficient. This coefficient, according to reference [7], page 108, is defined as
7=(1/W!
7:,dS/[(1/S)S5,,dS12.
The expression is valid for lateral one-dimensional motion of the beam: i.e., where 7,. = 0. For low frequencies the effects of shear and rotation are small and k, and k, in equations (2.3) and (2.4) both approach JC~as expected. In the high frequency range, k, is larger than K*, and thus according to equation (2.3) the limit for k, is equal to k, and hence independent of the shape of the cross-section. The corresponding result for k2 is an imaginary expression. A comparison between expression (2.3) and the exact solution obtained by Abramson for a circular cross-section indicates that the exact result is about 1.7 times as large as that predicted through equation (2.3) in the high frequency range. 3. BASIC ASSUMPTIONS Figure 1 shows a hull-frame construction. The structure extends along the x-axis and is considered to consist of parallel sections. Each section is referred to as a beam. The strip of
Figure 1. Configuration
of a hull-frame structure.
396
A. C. NILSSON
the hull plate is for each such beam limited by the centrelines-: = b,l2 and z = -O,i2-on each side of the frame. The thicknesses t,, t2 and t3 of the hull plate, web plate and the Range are assumed to be small compared to the other dimensions. Only wave motions parallel to the x--J)-plane and which correspond to the fundamental mode are considered: no twisting or any other motion in the z-direction is included in the model. Thus the beam model consists of a coupled system made up by the hull plate, web plate and the flange. Following the assumptions above, the following wave motions must be considered: flexural and longitudinal waves in the hull plate and in the flange, longitudinal and transverse waves in the web plate. All these waves are coupled.
4. GOVERNING
4.1. THE HULL-PLATE
EQUATIONS
1
The thickness of the plate is assumed to be small. Thus shear, and rotation in the plate itself, can be neglected. For flexural waves in steel plates, typical for a ship structure, this approximation is valid for frequencies up to 10 kHz. The normal stress components, 6, and oZ, are assumed to vanish on the boundaries and in fact to be zero throughout the plate. The first assumption is a consequence of the plate being thin. The second condition is typical for models describing one-dimensional wave propagation in beams. However, it might be argued that it is more appropriate to assume that the strain E, rather than the stress G, is equal to zero at the boundaries of the hull plate. For the final result concerning the propagation constant the difference between the two boundary conditions is insignificant. Thus with c,, = crZ= 0, the differential equation valid for flexural waves in the plate is a4r7riax4 + (trio,)
aZ0t2
= F,,/D,.
(4.1)
Here q1 is the displacement in the y-direction, pL1the mass per unit length and D, the flexural rigidity. For a strip of hull plate with width br, thickness tr and modulus of elasticity E,, the flexural rigidity can be written as (see Figure 1) D, = El tfbJ12.
(4.2)
Expression (4.2) is valid under the condition that oZ = 0. The external force per unit length, FvI, is caused by the normal stress in the web plate at the boundary (see Figure 2). Plate3
Plate
Figure 2. Resulting forces at the boundaries
-
2 -
flange
web plate
due to normal and shear stresses in the web plate,
PROPAGATION
IN HULL-FRAME
STRUCTURES
397
The corresponding equation for longitudinal waves in the hull plate is
a2 t1/ax2 - (P,/E,)
a2 r,/atz = -F,,ib,
tI Ed.
(4.3)
The displacement along the x-axis is denoted by t1 and the density of the plate by pi. The external force per unit length, Fxl, is caused by shear stress in the web plate and is shown in Figure 2. 4.2.
2
THE WEB PLATE-PLATE
Again the thickness of the plate can be assumed to be small compared to the other dimensions of the plate. This implies that the stress component in the z-direction, or, is equal to zero. According to the theory of elasticity the expressions for normal and shear stresses for a thin plate are : gy = v3,iu
- ~91 [a~,/ay + v ac,m,
ry, = rxy = [~,/2(1 +
41 [x,/ay + atl,bw.
(4.4) (4.5)
The displacements along the x- and y-axes are denoted by t2 and q2, respectively. The displacements are determined by longitudinal and transverse waves in the x-y-plane. Let the velocity potentials @ and Y describe these waves, so that the displacements along the axes are determined by (see reference [7]) c2 = aqax q2 = aqay
+ a!zqay,
(4.6)
- ayylax.
(4.7)
The first part in each expression is thus due to L (longitudinal) waves and the second part to T (transverse) waves. Consequently, the potentials @ and Y must fulfil the following equations for 0 < y < h (see Figure 1) : v2 cp- [p,(l - v2)/E2 ] a2 G/at2 = 0, v2 y - [2p,(l + v)/E~ J a2 ly/afz
= 0.
(4.8) (4.9)
The equations are valid for a thin plate--cr, = 0. 4.3.
THE FLANGE-PLATE
3
The assumptions made for the flange are similar to those made for the hull plate. Thus, with os = cr,,= 0, the governing equations for the displacements due to flexural and longitudinal waves are a4q3/ax4 + (P~/D~) a2q3/at2 = -F,,~D,,
(4.10)
a2 t3ja2
(4.11)
- &/Es)
a2 t/at2 = FxJb3 t, E3.
The flexural rigidity for the plate is determined by (see Figures 1 and 2) D,=E,t,Jb,/l2.
5. BOUNDARY CONDITIONS The resulting forces on the hull plate and the flange due to stresses in the web plate at the boundaries y = 0 and y = h are described by F,, =
t2 hdw,,
(5.1)
Fxs =
f2 (Qy=h,
(5.2)
F,I =
t2 Wy=o,
(5.3)
Fy2
t2 W,=h.
(5.4)
=
398
A. C. NILSSON
Further, the displacement flange. This implies that
is continuous
at the junctions
between
web plate, hull plate and
111 =
q2
for
4’= 0,
(5.5)
13
‘12
for
L’= h,
(5.6)
=
51 = 52
for
4’= 0,
(5.7)
43
for
y = h.
(5.8)
=
(2
6. ASSUMED
WAVEFORMS
Let the propagation constant for the entire wave motion in the x-direction be denoted by k, and let the time dependence be e ‘Of. The general expressions for the flexural and longitudinal waves in the hull plate and flange can consequently be written as, for flexural waves, q1 = A1 ei(ot-kx) 3 (6.1) (6.2) and for longitudinal
waves, (6.3) t3 = B, ei(o+kx).
(6.4)
The suffix refers to the plate in question (see section 4). The amplitudes A and Bare frequency dependent. For the web plate the displacement can be in both y- and x-directions. The potentials describing the L and T waves must satisfy each respective differential equation for 0 < y < h. Because of the boundary conditions the propagation constant in the x-direction must be the same as that for the wave motion in the adjoining plates. The general expression for the L-waves in the web plate is thus of the form
(6.5) This expression
when inserted
in equation
(4.8) yields
I, = [{p,(l - v’)w2/E2} - k2]“2. The corresponding
expression
for the T-waves is y =
where &- is determined
(6.6)
ei(mr-kx)
fj2
e-iAp
+
~~
ef+~]
9
(6.7)
I, = [{p2 2( 1 + v) 02/Ez> - k2]1’2.
(6.8)
[
through
7. RESULTING
SYSTEM OF EQUATIONS
The four boundary conditions and the four wave-equations are sufficient to determine the propagation constant. The forces at the boundaries, included in the wave equations, are obtained by inserting the equations (4.4)-(4.7) and (6.5) and (6.7) in equations (5.1)-(5.4). For convenience the following notations are introduced: the wave number for flexural waves (plates 1 and 3), K, = [P” 02/D,]“4
= [p, 1209/E” t,2]‘/4;
(7.1)
PROPAGATION
IN HULL-FRAME
399
STRUCTURES
the wave number for longitudinal waves (plates 1 and 3), kLn = [p. o’/E,]“‘;
a constant proportional
(7.2)
to the normal stress, L
=
[E,lU
-
v’)lP2~2lEn
b,
(7.3)
&?I;
a constant proportional to the resulting shear stress, M,, = E2 f2/2(l + v) b, t,, E,,.
(7.4)
The suffix it refers to the plate in question-l for the hull plate and 3 for the flange. Now insert equation (6.1) and the final expression for F,,, in the wave equation (4.1). A1 is eliminated by using the boundary condition (5.5). The amplitudes A,, B1 and B3 are eliminated in a similar way by using the remaining wave equations and boundary conditions. The result is a set of four equations. Introduce the symbols X, Y and Z, so that X = AJiB,,
Y = A.,/iB4,
Z = B,lB,.
(7.5)
In terms of these symbols, the resulting system of equations becomes X[L,(Iz
+ vk2) - i1.,(k4 - K:)] f Y[L,(Ii
+ Z[k(k4 - K?) + i&L, XemiALh[L,(iZ, + vk2) + &(k4
k(1 - v)] = -k(k” - K:) + i& kL1(l - v), - v)] = e’“Th[k(k4 - ~2) + i& kL,(l
- v)],
(7.7)
- k2) - i2kll MI ] + Y[k(k,Z, - k2) + i2kl, MI J -
- Z[M,(Af. Xe-‘iLh[k(k,Z,
(74
- K$)] + Ye*‘Lh[L3(ki + vk2) - i&(k4 - K$)]-
_ Ze-*W[k(k4 _ K$)- i& kL,(l X[k(ki,
+ vk2) + i&(k4 - JC?)]+
- k2) + il,(k& - k2)] = Ml($
- k2) - i&.(k~l - k2),
(7.8)
- k2) + i2kl, MS] + Ye’“Lh[k(k& - k2) - i2kJ, MS] +
+ Ze-‘“Th[M,(I$
- k2) - il,(k&
- k2)] = -ei’Th [M&
- k2) + i&(k& k2)].
(7.9)
This system is non-linear in k. There are a number of solutions to the system. As stated before, however, the propagation constant for the fundamental mode is of primary interest. For this mode and for travelling waves the functions 2, and AT are both imaginary. Consequently all the unknown factors are real. From consideration of the definitions of the functions in the system, it is apparent that if a solution k is found then -k is also a solution. This is natural since the velocity of a wave is independent of the direction of propagation along the axis of symmetry. The remaining four amplitudes, AI, A,, B, and B,, can be determined relative to B4 by using the boundary conditions given in the equations (5.5)-(5.8) and the definitions (7.5). The result is AI/B4 = 2=(X - Y) + ik(Z + l), AZ/B4 = %,(Xemi2Lh - Ye‘lLh) + k(Ze-‘“Th + eiATh), B1/B4 = k(X+
Y) - i&(Z - l),
BS/B4 = k(Xe-lALh + YeiALh)- i&.(Ze-iArh - eiATh).
(7.10) (7.11)
(7.12) (7.13)
Expressions giving the velocity level differences for the waves in the various plates are immediately derived by using the equations above. Let vFI and vF3, vL1 and vLj denote the particle velocities for flexural (F) and longitudinal (L) vibrations of the hull plate (1) and
400
A. C. NILSSON
flange (3). The level differences, thus are described by
according
to the equations
(6.1)-(6.4)
and (7. IO)-(7.13),
(7.14)
lOlog lUF1/UF~j2= lOlog /iJ?j3/z = lOlog IA,/A,IZ,
If so required,
lOlog 1UFII~‘lIZ = 101% 1~11~112,
(7.15)
lOlog IuFJ/v‘31z=
(7.16)
similar expressions
lOlog lA3/B312.
can be formed for the velocities
in the web plate.
8. SOLUTIONS An explicit solution to the system of equations for low frequencies and for typical ship structures These solutions are of the form k = ~{k;,(l+
cannot readily be obtained. However, approximate solutions can be derived.
+ v) + [(O-5 + v)‘k,4, + k;,t,(b,
tl + b, t3 + t, h)/hb, tl b, t,]“2}“2.
(8.1)
In equation (8.1) it is assumed that the plates are made of the same material: i.e., k,, is set real and two imaginaryequal to k,,. Thus for low frequencies there are four solutions-two corresponding to the fundamental mode. The absolute value of the imaginary part is, however, not equal to the absolute value of the real part, as in the case for solutions to the elementary equations for flexural waves in a beam (see section 2 above). More complete solutions to the set of equations can be obtained by using an iterative technique-for example a modification of the Newton-Raphson method. The four equations (7.6) to (7.9) are of the form a,, X + a,, Y +
Z = b,,
a,3
where n = 1, 2, 3, 4. The coefficients a,,,, . . . are functions of k. Let the initial value k. be determined by the approximate solution (7.1) in the low frequency range. The first three equations, (7.6) to (7.8), are used to solve for X, Y and Z. Let the second value in the iteration chain be k, = k. + E. The solutions X, Y and Z are inserted in the fourth equation (7.9), and the coefficients a4r, . . . are expanded around k,. With only terms of the first order included, the general result becomes k n+l
a41 =
kn -
Xaa,,/ak
x
+
a42
y
+
a43Z
-
b,
1
+ Y aa,Jak + Z i3a43/ak- ab4/ak ’
k=k,
wheren=0,1,2.... The calculations are continued until a sufficient accuracy say, a result is required for the center frequency in each l/3-octave band, Hz, the initial value for each new band can be set equal to the value obtained band, and so on. The imaginary-or near field-solutions can be derived with ik in the equations (7.6) to (7.9) and by using the appropriate initial equation (7.1). The remaining quantities Al/B,, etc., are found by inserting for k, X, Y and Z in the equations (7.10) to (7.13).
(8.2)
is obtained. If, starting at 31.5 in the preceding by replacing k value for kthe final results
9. RESULTS Normally the hull structure consists of main frames and ordinary frames attached to the hull side. The height of the web plate and the width of the flange are for a main frame about three to four times the corresponding dimensions for an ordinary frame. The spacing between the frames is generally constant. Every fifth frame is in most cases a main frame.
PROPAGATION
IN HULL-FRAME STRUCTURES
401
Two examples corresponding to the different frame types have been investigated. The dimensions of the beam structures and the material parameters are listed in Table 1 (see Figure 1). The same type of steel is assumed to be used for hull, web plate and flange. TABLE
1
Dimensions and material parameters of beams investigated as examples V
(hh3) Beam 1
0.6
0.2
0.6
Beam 2
0.6
0.05
0.2
0.012 0.012
0.012 0.012
0.012 0.012
7.6 x lo3 7.6 x lo3
2 x IO” 2 x 10”
0.3 0.3
For the fundamental mode the equations (7.6)-(7.9) yield four solutions for the propagation constants. Two solutions are real and have the same absolute value and represent waves travelling in opposite directions of the x-axis. The remaining two solutions are imaginary and of the same magnitude, but with opposite signs. These waves are consequently describing the near fields (see section 2). The real and imaginary wave numbers, calculated as described in section 8, are shown in Figures 3(a) and (b) for the beam constructions described above. The wave numbers corresponding to pure bending of the entire beam, and for free flexural waves on the hull plate only, are also drawn in the figures. The resulting velocities due to lateral (F) and longitudinal (L) motion of the hull plate (suffix 1) and the flange (suffix 3) are compared in Figures 4(a) and (b) (see also section 8 above). For decreasing frequencies the wave number for the lateral motion approaches that which was derived for pure bending for the entire construction (see equation (2.2)). Consequently, in the low frequency region the lateral velocities of the hull plate and of the flange are the same (see Figures 4(a) and (b)). Pure bending also implies that the deflection along the line represetting the centre of mass for the cross section is negligible. Hence, the ratio between the longitudinal velocities for the hull plate (vL1) and the flange (vL3) can, for low frequencies, be determined through the relationship vL1IvL3 2: h,/h,, where h, and h, are the respective distances
2.0
from
I
I
I
31.5
63
I25
the plates to the centre of gravity
for the cross-section.
250
31.5
O-
500
KXO
2ooO
4000
6000
16ooo Frcqmcy
63
125
250
500
Kc0
2caJ
4cco
mO0
16ccm
(Hz)
Figure 3. Calculated wave numbers for (a) beam 1 and (b) beam 2. -, Travelling waves; ----, near field solution; -. -, bending only; equation (2.3); . . *. . ., flexural waves in the hull plate; equation (7.1).
A. C. NILSSON
0
31.5
__.. 63
125
250
500
/’ , loo0
2CCO
4ooO
FKWJ
31.5
Bond ct?ntre frequency
63
125
250
500
ICC0
zoo0
‘wx
8003
(Hz)
Figure 4. Velocity level differences in (a) beam 1 and (b) beam 2. The levels are calculated for each l/3-
For increasing frequencies the wave number for the propagating waves increases, whereas the corresponding wave number for the near field solution decreases relative to the value from the simple solution valid for pure bending. The deviations are due to shear and rotational effects. In the mid-frequency range these effects are becoming increasingly pronounced. This causes the wave number for the propagating waves to increase with frequency more rapidly than was the case for the lower frequencies. At the same time the motion of the flange decreases sharply. This implies that the entire motion of the beam becomes more strongly determined by the larger mass: i.e., the hull plate. The lateral velocity of the hull plate has a maximum in the 2000 Hz one-third octave band for beam 1 and in the 2500 Hz band for beam 2. These maxima are functions of the dimensions of the structure. The shear and rotational effects start to dominate at even lower frequencies for higher web plates. In the high frequency range the propagation constant approaches that which is valid for free flexural waves in the uncoupled hull plate and is thus proportional to the square root of the frequency. The same is true for the wave number describing the near field. In the latter case, however, the approach to the asymptotic value is from above. Thus for high frequencies the motion is dominated by a type of wave propagating at the outer boundary of the structure. The analogy with the solutions for the circular bar discussed above is obvious. If the analysis is pursued further, then the propagation constant should again increase more rapidly when-at about 16 kHz--the simple Kirchhoff equation for the plate no longer is valid. Consequently in the very high frequency range-about 20 kHz for a steel plate-the propagation constant should be proportional to the frequency as is the case for ordinary surface waves. The wave motion then would be of the Rayleigh wave type. However, solutions for frequencies above 10 kHz are generally not of interest.
10. COMPARISON
WITH THE TIMOSHENKO
THEORY
The Timoshenko shear coefficient, z, can be derived by inserting the exact solution for the propagation constant for travelling waves in equation (2.3). It is found that z is frequency dependent, although for low frequencies its value is fairly constant. For beam 1 and for frequencies up to 400 Hz, the best agreement between the Timoshenko theory and the exact solution is obtained for T equal to 2.2. Figure 5 shows a comparison between the exact solution and the result obtained by using equation (2.3) with r = 2.2. Also indicated in the figure is the result which is valid for bending only, equation (2.1). Already at about 600 Hz the deviation between the exact and the elementary solutions
403
PROPAGATION IN HULL-FRAME STRUCTURES
O-
-0.5
" 31.5
'I 63
”
“1 125
”
250
1’ 500
Frequency
1” Km0
1 2ooo
”
’
4000
”
6000
(Hz)
Figure 5. Comparison between exact and approximate solutions of the wave numbers for travelling waves on beam 1. --, Exact; ----, approximate, equation (2.3) with T = 2.2; -.-, approximate, equation (2.2).
exceeds 50%. At 2000 Hz the same difference is obtained between the Timoshenko model result and the exact solution. This in spite of the fact that the exact solution has been used to compute the factor r for obtaining good agreement at low frequencies.
11. CONCLUSIONS
For models describing the structure-borne sound propagation in ships it is often essential to include shear and rotational effects, even at comparatively low frequencies. Due to the particular dimensions of the plates in ship structures, accurate and fairly simple models, in which these effects are included, can be used for frequencies up to and even above 10 kHz. The quantities of interest as propagation constants, deflections, moments and forces can be derived by using simple computer programs. Flexural, longitudinal and transverse waves must be considered in order to make it possible to fulfil the necessary boundary conditions in the hull-frame structure. For these coupled waves the resulting (particle) velocities due to flexural waves in the hull plate are considerably higher than the in-plane velocities caused by longitudinal waves. Thus, when measurements indicate that the (particle) velocities of the in-plane waves and the flexural waves are of the same order, then the L and T waves in the hull plate can be assumed, approximately, to be superimposed and uncoupled to the flexural motion of the structure. For low frequencies the energy transport in the ship side is caused by bending of the entire beam or frame structure. In the high frequency range the energy flux in the structure is mainly due to flexural waves in the hull plate. The energy loss per unit length can be obtained in an often-used manner by writing the modulus of elasticity as E( 1 + iv), where v is the loss factor. In this way the resulting wave number for travelling waves is assigned an imaginary part which determines the losses. For low and high frequencies the decay rate for the energy is of the order 2 x k x q dB/m, where k is the wave number. For intermediate frequencies the losses are somewhat higher, or about twice as large. At 1000 Hz for a typical main
A. C.
404
NILSSON
frame structure with a loss factor of 1 % the energy loss per metre becomes 0.07 dB for the forced motion of the entire structure. Even if the loss factor is increased with a damping layer by, say, a factor of ten, the energy loss is still fairly small. The lateral motion of the structure is well coupled to the air-borne sound field. For the entire main frame construction (beam 1) the critical frequency is of the order 16 Hz. The corresponding value for the frame (beam 2) is about 55 Hz. These low critical frequencies thus will ensure a high radiation efficiency over almost the entire frequency range of interest. The model discussed in the preceding sections also can form the basis for calculations of transmission through junctions in ship structures. The calculations can be made according to the principles suggested by Cremer and Heck1 [7]. REFERENCES
5. 6.
7. 8. 9. 10. 11. 12.
J. H. JANSEN1962 Netherlands Ship Research Centre, Deljt, Reports Nos. 44s and 4%. Some acoustical properties of ships with respect to noise control. J. 0. JENSEN1974 Proceedings ofthe 1974 International Conj>rence on Noise Control Engineering. Washington, D.C. Structureborne sound transmission in ships. T. KIHLMAN and J. PLUNT 1974 Proceedings of the 8th International Congress on Acoustics, London. Structureborne sound in ships: study of different wave forms. R. J. SAWLEY1969 Contributed to Stochastic processes in dynamical problems, ASME, New York, 63-73. The evaluation of a shipboard noise and vibration problem using statistical energy analysis. R. H. LYON and G. MAIDANIK 1962 Journal of the Acoustical Society of America 34, 623-639. Power flow between linearly coupled oscillators. M. CHERNJAWSKI and C. ARCIDIACANO1972 Shock and Vibration Bulletin 42,235-243. Simplified method for the evaluation of structure-borne vibration transmission through complex ship structures. L. CREMERand M. HECKL 1967 Kiirperschall. Berlin: Springer-Verlag. S. V. BUDRINand A. S. NIKIFOROV1964 Soviet-Physics Acoustics 9, 333-336. Wave transmission through assorted plate joints. E. VOLTERRAand E. C. ZACHMANOGLOU 1965 Dynamics of Vibrations. Columbus, Ohio : Charles E. Merrill Books Inc. H. N. ABRAMSON1957 Journal of the Acoustical Society of America 29, 42-46. Flexural waves in elastic beams of circular cross section. I. N. SNEDDONand R. HILL (Editors) 1960 Progress in Solid Mechanics, Volume 1. Amsterdam: North-Holland Publishing Co. F. FAHY and E. LINDQVIST1974 Chalmers University of Technology, Gothenburg, Sweden, Report 74-37. Wave propagation in damped, stiffened structures characteristic of ship construction. . APPENDIX A, B, C, D D,, DL, 03
E FL, Fx3, F,.,, Fy3 I L. Mll S x, y, z b,, 63, h, tl, tz, f3 k,, k, k, kL t
: LIST OF SYMBOLS
wave amplitudes flexural rigidity for the plates modulus of elasticity forces per unit length of beam (Figure 2) second moment of area of beam cross section defined in equation (7.3) defined in equation (7.4) area of cross section of beam defined in equation (7.5) beam dimensions (Figure 1) propagation constants for lateral one-dimensional displacement propagation constant in sea water propagation constant for longitudinal waves in a beam time
of beam
PROPAGATION OF,
UL @
!P
e
11
‘;:
IZL,
P
V z c P Pw’ 7 7XY, 7x2 w
IN HULL-FRAME
STRUCTURES
particle velocities for flexural and longitudinal velocity potential for longitudinal waves velocity potential for transverse waves velocity potential in water displacement of beam in the y-direction propagation constant for free bending waves defined in equations (6.6) and (6.8) mass of beam per unit length Poisson’s ratio displacement in x-direction density of beam density of sea water Timoshenko’s constant shear stresses angular frequency
waves
405