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Wave propagation through right-angled joints with compliance-flexural incident wave
Journal of Sound and Vibrarion (1990) 142( 11, 3 l-46
WAVE
PROPAGATION
WITH
THROUGH
COMPLIANCE-FLEXURAL R. C. N. LEUNG
Institute
RIGHT-ANGLED
AND
of Sound and Vibration Research,
(Received
INCIDENT
JOINTS
WAVE
R. J. PINNINGTON 7’he University, Southampton,
31 October 1988, and in revised form 22 November
England
1989)
The vibration transmission through a right-angled joint with compliance is investigated by using the wave theory approach. The reflection and transmission coefficients are given. The joint is that between two semi-infinite beams, possibly of different materials and different thicknesses, fixed together either by welding, bolts or rivets. A spring-dashpot model is used in each degree of freedom of the joint. Thus, the results can also incorporate the case of a gasket between two plates. The investigation shows that both flexural and longitudinal waves will be induced in the secondary plate by either flexural or longitudinal incident waves alone in the primary plate. The results also show that both transmitted flexural and longitudinal waves can be controlled to some extent by varying either the stiffness of the joint or the thickness of the plates. The given coefficients will have some application to statistical energy analysis.
1. INTRODUCTION
In structures and machinery it is not uncommon to find structural components that meet at right angles-for example, steel beams in structural frames. The beams are normally joined together either by welding, bolts or rivets. Sometimes, there may be gaskets between two structural components. Waves propagating along one component will eventually encounter a joint. Each joint will reflect part of the incident wave, while some of the incident wave will be transmitted through the joint. The attenuation of the incident waves transmitted through a joint is largely governed by the impedance and the orientation of the joint. Cremer et al. [3] have studied the attenuation of an incident wave through a rigid joint. However, it is not always practical to have a rigid joint. Moreover, the possibility of damping joints has created a lot of interest with respect to controlling the vibration of structures; for example, in references [2-81 the damping characteristics of joints have been modelled. In general, a linear spring-dashpot model is adequate in modelling the joint. In this paper the attenuation of a flexural incident wave under the influence of a joint with compliance is predicted. The case of a longitudinal incident wave will be presented in a follow-on paper. The study is confined to two semi-infinite beams of unit width joined together at right angles, with a flexural incident wave. Once transmission efficiencies, 7, have been obtained between the various wave types of the coupled beams it is straightforward to obtain the coupling loss factors, 7),2, required in a statistical energy analysis [ 111; it is given by v,2 = r/2kl, where k is the flexural wave number of the excited beam and 1 is the length of the excited beam. 2. JOINT
It is known local material
MODEL
that riveted, bolted and welded joints damping [4]. Thus, a spring-dashpot 31 0022-460X/90/190031 + 16 $03.00/O
can dissipate more energy than the model could be appropriate to @ 1990 Academic Ress Limited
32
R. C. N. LEUNG
AND
R. J. PINNINGTON
characterize a joint. Yoshimura [9, lo] and Gaul [4] have identified the stiffness and the damping coefficients successfully using linear spring-dashpot models. Tbe same linear spring-dashpot models are adopted in this study. Because there are three degrees of freedom at a right-angle joint-x, y and rotation-three sets of spring-dashpot models are required.
Figure 1. Joint model.
3. TRANSMISSION AND REFLECTION OF A FLEXURAL INCIDENT WAVE THROUGH A RIGHT-ANGLE JOINT WITH COMPLIANCE IN ALL ITS DEGREES OF FREEDOM Cremer et al. [3] have studied two extreme cases-rigid joints and simply supported joints. The same wave approach is to be used here for the cases of a joint with compliance in all three degrees of freedom. Consider a flexural incident wave of amplitude V travelling towards the joint in component 1 (a list of nomenclature is given in the Appendix). The transverse displacements v,(x) along the plate consist of the incident wave, a reflected travelling wave and a near field wave at the joint: i.e., V,(x) = V(eeikBlx+ r, eikalx+ r, ekBIX).
Figure 2. Sign convention.
(1)
WAVE
PROPAGATION
THROUGH
In component 2, its transverse displacements, wave and a near field wave at the joint: i.e.
JOINTS
33
u*(y), consist of a transmitted travelling
uzy(y) = V( ts e-iksl-\’ + 1, emkeIy).
(2)
Because the flexural incident wave has to go through a corner, longitudinal waves will be induced in both components. The longitudinal waves in both components are u,(x) = VrL eikLfix,
Z&J) = VfL e-ikL2v.
(394)
With the sign conventions shown in Figure 2, the relations between the moments, forces and displacements for both components are given as [3] e*=
2;
-;$,
M,=
dy F,,, = -iom,
V, dx,
M2=---.
F,, = -iwm,
B2 d2u2 io dy* ’
u2 dy;
(7)
I
F,, = --m,CLlul,
(56)
52 = m2CL2v2.
(8)
The joint between the two beams is assumed to have the characteristics of three linear spring-dashpot systems as discussed in the previous section. The impedances in the x-direction, the y-direction and the rotation direction are represented as Z, = (K,/iw) + C,,
Z, = (K,/io)
The unknown coefficients in equations conditions at the joint. They are F,, =
Fx2 = Zx(u,
+ C,,
(l)-(4)
- ~21,
Z, = (K,/io)
+ CO.
are found by applying the boundary
?,,I = Fy2 = Zy(v,
- vz),
(10,11)
M, = M2 = Z,( 8, - 0,).
From equations (l)-(12), r,, can be found:
fL
-=
y2
the transmission
4
(12)
and the reflection coefficients, fa, fL, r, and
(l-i)[2+(1+i)(Z,+~,)l
(14)
[2+(1+i)(Z2+~L)l +‘[1+i(Z,+y,)][2+(1+i)(Z2+y2)]rB’ [(I -iW2+ y2)-21
rs=[(1+i)(Z2+y2)+2]-’
[(1+i)(Z,+y,)+21[i(Z2+y2)+11 [i(z,+rl)+11[(1+i)(Z2+Y2)+21
yl(l -i) rL= -[i(Z,+y,)+l]
fB’
(9)
“’
(1%
(16)
Here X = kB2/kB, is the flexural wavelength ratio in the two components and +* = (m2Bz)/(mlBl). The ratios y, = (m2CB2)/m,CL,) and y2 = (m,Ca,)/(m2CL2) relate the longitudinal and flexural wave impedances in the two plates. The dimensionless terms Z, = m2CBz/ZT, Z2 = m,C,,/ZY, and Z, = B,/(CB,Zo) are the ratios of the bending wave impedances, and the impedances of the joint.
34
R. C. N. LEUNG
AND
R. J. PINNINGTON
For a rigid joint, the stiffnesses of the joint, K,, KY and Kg, will be infinite. This results in Z,, 2, and 2, becoming zero. Thus, 2(y1+y,)-i2(I-y1yJ
fg=[JI(-l-yl-2Yz-Y1Y2)+X(-1-2YI-YZ-Y1Y2)1+i[(~+X)(1-Y1Y2)1’
(17)
which is exactly the same as that given by Cremer [3]. Since vibrational power is a much better way to express the severity of vibration in structures than just the amplitude of the vibration displacement, two efficiencies, transmission efficiency (T) and reflection efficiency (p) are defined: 7=
P=
vibrational power transmitted
(18)
vibrational power of the incident wave ’ vibrational power reflected vibrational power of the incident wave ’
For a flexural incident wave, the transmission transmitted and the reflected flexural waves are
(19)
and the reflection efficiencies for the (20921)
P*=IrE?12,
%=X&31*,
and the efficiencies for the longitudinal waves are TL =
4. JOINTS
WITH
PL =
ltLl*PY,,
COMPLIANCE
(CL,/2CB,)
IN THE
(22,231
kL12.
x-DIRECTION
ONLY
The equations (13)-(16) for the transmission and the reflection coefficients are very complicated. It is therefore useful to look at the joint impedance in each orientation separately. This will enable one to understand how the orientation of the joint influences the transmission and reflection of the incident wave. To simplify the problem further, the two components are assumed to be of the same material and of the same cross-section: i.e., X = II,= 1, y, = y2 = y. A joint is assumed to have an impedance of a spring-dashpot type, Z,, in the x-direction only. The joint is rigid in the y-direction, D, = u2 or K,, = 00, and in rotation, 8, = e2 or Ke = 00. The transmission and the reflection coefficients then become
t,=(2y-i2)[1+i(Z,+y)]/D,
t,=y[(iZ-6)-(6+i2)(Z,+y)]/D, rL=[2y-2y2+i(2y+2y2)]/D,
rB=[(2-2y+iy)+(2-2y+i)(Z,+y)]/D, where D= -2(1 -i)-3y-3(2,+y)-2(1 dimensionless impedance
+i)y(Z1+
y). Using equation
ZI =Zk(n+i),
(24,25) (26,27) (9) gives the (28)
with Z, = CBmwKx/(K~+w2C~),
77= WC,/ K,.
(29,30)
The term Z, is the ratio of the impedance of the spring in the x-direction to the impedance of the flexural incident wave. q is the loss factor of the joint. Both Z, and r) are frequency
dependent. By substituting equation (28) into the transmission and the reflection coefficients, and using the definitions for the transmission and reflection efficiencies, equations (20)-(23), the efficiencies can then be found.
WAVE The
PROPAGATION
THROUGH
JOINTS
35
dissipation efficiency, 5, defined as 5=
power dissipated at joint vibrational power of incident wave ’
(31)
can be obtained from the relationship Q+T7L+fpB+pL+[=l.
(32)
For the case where the joint impedance in the x-direction has no damping term, i.e., 2, = C,mw/ K,,the transmission efficiency of the transmitted flexural wave, TV, can be shown to have a minimum when Z, = 1, with y<< 1: i.e.,
CBm=Kx/w.
(33)
The left side of equation (33) is the impedance of the flexural incident wave, and the right side is the magnitude of the impedance of the joint stiffness. Thus, when both impedances match in magnitude there will be no flexural wave transmitted through the joint. A joint with a stiffness in the x-direction acts as a band-stop filter on the flexural incident wave. The transmission and reflection efficiencies are plotted against the non-dimensional term, Z, = CBmo/Kx in Figures 3-6. These plots are for a right-angled joint of two like 10 mm thick steel beams of unit width. The impedance matching effect on the transmitted I.0
O,I
I C,m
-9
IO w/X,
Figure 3. Plots of the transmission efficiency, ~a, with joint impedance in x only. k& = co; --, & = 10” N/m; - - -, K, = 10s N/m; - - - -, K, = IO6N/m.
flexural wave is demonstrated in Figure 3. By using equations (20) and (24), it can be shown that at low frequencies, f+O, ~a +O-5. On the other end of the frequency range, f+ co, r8 + O-5. Together with the results in Figure 3, it is found that the proportion of the incident vibrational power being transmitted as flexural wave into the second component will not exceed a half. From equations (22) and (25), it can be shown that the transmission efficiency of the longitudinal wave Q -2*5/y as f* cc. Since y is proportional to frequency rL will eventually become zero as the frequency increases. Incorporating this result with that in Figure 4, one may draw a conclusion that the peak in rL will be at a frequency well above the frequency range appropriate to the simple flexural wave equations used.
36
R. C. N. LEUNG
AND
C,m -,
R. J. PINNINGTON
w/K,
Figure 4. Plots of the transmission efficiency, 7‘. with joint impedance in x only. K, = CO;--, K, = 10” M/m; - - -, K,x = 10’N/m; - - - -, K, = lo6 N/m.
It is shown in Figures 5 and 6 that when impedance matching occurs, the vibrational power of the flexural incident wave is mainly reflected back as flexural waves. There is also an increase in the amount of power being reflected back as longitudinal waves. On the whole, the secondary transmitted and reflected longitudinal waves remain as a small proportion of the original incident wave. Above the critical frequency at which the impedance matching occurs, there will be negligible longitudinal waves being reflected back at the joint. In Figures 3-6, the curves associated with K, = cc show the limits on the efficiencies as the frequency of the incident waves tends to infinity. Strictly speaking, the term used in the abscissa, C,mo/K,, is zero if K, = 00. Figures 3-6 are plotted with the term I.0
0.6 -
0.4 ------
*_ H-1
0.2 .\
Cfl olK -9
Figure 5. Plots of the reflection efficiency, ~a, with joint impedance in x only. K, = 00; - -, K,v = 10” N/m; - - -, K, = 10’N/m; - - - -, K,v = lob N/m.
WAVE
PROPAGATION
THROUGH
JOINTS
37
2 i g E % c P Y
0.06
0.04
5 0.02
0.00 0.1
I Gm~/K,
-,
Figure 6. Plots of the reflection efficiency, pL, with joint impedance in x only. K,=co; --, K,=lO’ON/m; ---, K, =108N/m; ----, K,=106N/m.
Cam w/K, as the abscissa in order to demonstrate the condition for the minimum transmission efficiency. The reference, the stiffness of a bolted joint for the size of the chosen steel beam will be of the order of 10’ to 10” N/m, depending on the contact pressure of the bolted joint [9].
5. JOINT
WITH
COMPLIANCE
IN y-DIRECTION
ONLY
For a joint with compliance in the y-direction only, Z, = Z, = 0, or ur = u2, 13~= &. With the two components assumed to be of the same material and of the same cross-section, the transmission and reflection coefficients are found to be fa=(2~-i2)[1+i(Zz+y)]/D,
fL= y[(i2-6)-(6+i2)y]/D,
(34335)
rg=[(2+2y+iy)+(-2-2y+i)(Zz+y)]/D,
(36)
r,=i2y(l-i)[l+i(Z,+y)]/D,
(37)
with D=-2(1-i)-3y-3(Z,+y)-2(l+i)y(Z,+y). From equations (34)-(37) and equations (20)-(23), the transmission and the reflection efficiencies can be evaluated. The dimensionless impedance is Z,=Z,(n+i)
(38)
with Z, = CBmwKy/(K~+w2C~),
77= WC,/ K Y'
(39940)
By comparison; equations (24) and (34) are of the same form. Thus, Figure 7 is the same as Figure 3. The only difference is the impedance of the joint. One relates to the joint impedance in the x-direction while the other relates to the impedance in the y-direction. Thus, a joint with impedance either in the x-direction or in the y-direction acts as a band-stop filter limiting the amount of power transmitted as flexural waves through the joint.
38
R. C. N. LEUNG AND R. J. PINNINGTON
0.6
-
0.1
I
C,mo/K
Figure 7. Plots of the transmission efficiency, Q, with joint impedance in y only. -, KY= 00; --, KY= 10” N/m; - - -, KY= 10’N/m; - - - -, K, = IO6 N/m.
The flexural and longitudinal transmission and reflection efficiencies for a flexural incident wave at a right-angled joint with compliance in the y-direction only are plotted against the non-dimensional term, Z,, equation (39), in Figures 7-10. The effect of joint damping is not considered in these plots. The plots are for two 10 mm thick steel beams forming the joint. It is shown in Figure 8 that once the critical frequency, at which the impedance of the flexural wave matches the impedance of the joint, is passed there will be insignificant power transmitted as longitudinal waves through the joint. It also shows that the peak is higher as the stiffness of the joint increases.
C,m O/K
Figure 8. Plots of the transmission efficiency, Q, with joint impedance in y only. -, KY= co; - -, K, = 10” N/m; - - -, K,.= 10”N/m; - - - -, KY = IO6N/m.
WAVE
PROPAGATION
THROUGH
39
JOINTS
It is shown in Figure 9 that the joint stiffness affects the amount of power being reflected as flexural wave. The softer the joint the more likely power is reflected back at the joint as a flexural wave. I.0
‘K\ ‘3
0.6
5
s
‘G
8 G P
-
_.A-/
0.4 --
/
---_/
/
/
A---
---
/
0.2 -
I I
o-o 0.1
IO
C,m. O/ K, Figure 9. Plots of the reflection efficiency, pe, with joint impedance in y only. KY = ~0; --, K,. = 10” N/m; - - -, K, = IO* N/m; - - - -, K,. = lo6 N/m.
-,
In Figure 10 it is indicated that the reflected longitudinal wave is insignificant. It reaches the level of 10% only at very high frequency, 2, > 7, with very high stiffness, ky = 10 GN/m in the present case of two 10mm thick steel beams. At such high frequency, the simple flexural wave equation used will no longer be valid. The cases of KY= cc in Figures 7-10 are for the purpose of showing the limits on the efficiencies as the frequency of the incident wave tends to infinity.
0.10 / /
/’
0 08 -
d
t :
0.06
g
0,04- _-_
5 ._
‘S
/ /
-
/
-..
5 K
\ ooz-
----_ 0.00 0.1
I
IO
Cgmw/KY
-,
in y only. Figure 10. Plots of the reflection efficiency, pL, with joint impedance K,.=co;----, K,=lO’ON/m;-----, K!=lO’N/m;----, K,=106N/m
40
R. C. N. LEUNG
6. JOINT
AND
R. J. PINNINGTON
WITH COMPLIANCE
IN ROTATION
ONLY
In the previous two sections, joints with compliance in either the x- or y-direction were investigated. They are applicable to welded, bolted or riveted joints with no elastic layer separating the two members that form the joint. In such joints it can be assumed that there is no relative angular movement at the joint between the two members. Such an assumption cannot be applied when joints with elastic layers, e.g., gaskets, are used to separate the two members. If gaskets are used, the joints will have compliance in all three orientations, x, y and rotation. The effect of impedance in rotation alone is considered here. For joints with compliance in rotation only, 2, = 2, = 0, or uI = u2, u, = u2. With the two members assumed to be of the same material and of the same cross-section, the transmission and the reflection coefficients are found by applying the boundary conditions at the joint to equations (13)-(16):
fL=y{-6(~+1)+i2(1--y)+iZ~[4+(2+i2)y]}/D,
fB=(2y-i2)(1+iy)/D,
f-,=27(1-y+i+iy)/D,
r,=[(2-2y2+i2y)+Z,(2y+iy2+i2)]/D, with D = [2+(1 +iy)]{(-2yis given as
(41,42)
1 +i)+i(&/2)[2+(1
(43,44)
+iv)]}. The non-dimensional
term Z, (45)
Z3=Zk(77+i), with z, = (B/C,)oK,/(K2,+02C2,),
n=wC,/K,.
(4647)
The term B/Cs is the moment impedance of a flexural wave. Z3 is the ratio between the impedance of the flexural incident wave and the joint rotation impedance. The transmission and the reflection efficiencies can be found by substituting equations (41)-(44) into equations (20)-(23). Then efficiencies are plotted in Figures 11-14 for two steel plates of the same cross-section with no damping.
I.0
0.8 -
0.6 -
\
-\ 0.2 -
-\
\ -\
0.0 ’ IO
1. ‘\__
---__
1 loo
, 1000
--_
_ 10000
Frequency (Hz) -,
Figure 11. Plots of the transmission efficiency, ~a, with joint impedance K, = co; --, K, = lo6 Nm/rad; - - -, Ii, = 10’ Nm/rad; - - - -,
in rotation only. K, = lo4 Nm/rad.
WAVE
PROPAGATION
THROUGH
41
JOINTS
o’53 0.4 i
i g
t
0.9 -
%
5
5
o-2-
ii
P
m
I-
0.1 -
0.0
I
IO
1
1000
loo Frequency
-,
Figure 12. Plots of the transmission efficiency, TV, with joint impedance in rotation only. K, = co; --, K, = lo6 Nm/rad; - - -, K, = lo5 Nm/rad; - - - -, K, = lo4 Nm/rad.
o.o1 IO
100
IO00
Frequency
--,
IC 00
(Hz1
IO 000
(Hz)
Figure 13. Plots of the reflection efficiency, ps, with joint impedance in rotation only. K, = co; --, K, = lo6 Nm/rad; - - -, K, = lo5 Nm/rad; - - - -, K, = lo4 Nm/rad.
It is found that the efficiencies for a joint with only rotation impedance show no peak or trough. Even at the frequency where the moment impedance of the incident wave matches the rotation impedance of the joint, power will still be transmitted as flexural wave through the joint. However, the amount of power transmitted as a flexural wave through the joint is governed by the rotational stiffness of the joint. The softer the joint in rotation, the less power transmitted as flexural wave through the joint. The results show that joints with very soft stiffness in rotation reflect almost zero power back as longitudinal waves. The stiffness in rotation has little influence on the power being transmitted as longitudinal waves through the joint.
42
R. C. N. LEUNG
AND
R. J. PINNINGTON
040
e 5 g .-
0.06
-
oa6
-
5
6
$
0.04-
5 n 0.02 -
0,00
---
.___-I'---,-.
IO
100
1000 Frequency
Figure 14. Plots of the reflectionefficiency, -,
K, = 03; --,
K, = lo6 Nm/rad;
7.
-
- -,
10000
(Hz)
pL, with joint impedance in rotation only. - - - -, K, = lo4 Nm/rad.
K, = lo5 Nm/rad;
DAMPING AT JOINTS
As joints in structures or machinery are an unavoidable source of damping, it is essential to study this effect on the transmission and reflection characteristics of a joint. A bolted joint will have a damping coefficient in the order of about lo6 Ns/m for a unit width and 10 mm thick steel beam [9]. The transmission and reflection efficiencies for two different cases are considered: (1) joints with impedance in the x-direction only, Figures 15-18; (2) joints with impedance in rotation only, Figures 19-22. In the first case, the introduction of damping at the joint smooths out the humps and troughs of the efficiencies. This is similar to the damping in a simple spring-mass-dashpot system. _
0.6 -
E i
5
‘G
0.6 -
%J $ .I? E 2 E k-
04 C-0.2 -
Frequency (Hz)
Figure -,
15. Plots of the transmission efficiency, C, = 0 Ns/m; --, C, = lo4 Ns/m; -
rB, with joint impedance in x only; K..= 10’ N/m. - -, C, = 10’ Ns/m; - - - -, C, = lob Ns/m.
WAVE
PROPAGATION
THROUGH
JOINTS
o.531
43
Frequency (Hz) Figure 16. Plots of the transmission efficiency, TV,with joint impedance in x only; K, = 10”N/m. -, C, = 0 Ns/m; --, C, = IO’Ns/m; - - -, C, = lo5 Ns/m; - - - -, c, = 10’Ns/m.
0.6 -
m i .-
0.6 -
{ 6 ._
0.4 -
g
Frequency
(Hz)
Figure 17. Plots of the reflection efficiency, ps, with joint impedance in x only; K,r = 10s N/m. -, C, = 0 Ns/m; --, C, = lo4 Ns/m; - - -, rZY= lo5 Ns/m; - - - -, C, = 10’Ns/m.
From previous sections, the characteristic equation of the efficiencies for joints with impedance in the x- and y-direction alone will have a minima at z,=2(1-7))/3(1+n2)
(48)
if the terms involving y are ignored. This assumption is valid since y is normally less than one. At n = 1, the minimum occurs at 2, = 0. Therefore, the critical damping of the joint will be c= K/w.
(49)
R. C. N. LEUNG
44
0.06
AND
R. J. PINNINGTON
-
100
1000 Frequency
10 000
(Hz)
Figure 18. Plots of the reflection efficiency, pL, with joint impedance in x only; K, = lo8 N/m. -, C, = 0 Ns/m; - -, C, = lo4 Ns/m; - - -, C, = lo5 Ns/m; - - - -, C, = 10’ Ns/m.
In Figures 19-22 is demonstrated the importance of the joint damping in rotation on a flexural incident wave at the joint with impedance in rotation only. The figures show that the longitudinal waves induced in both members of the joint are affected very little by the introduction of damping in rotation. The damping at the joint affects the llexural waves in the receiving member more. The flexural waves reflected in the first members are affected by the damping at joint only at frequencies above a damping dependent break-away frequency. Below the break-away frequency, the reflection efficiency behaves as if there is no damping. Above the frequency, influence due to the damping on the efficiency is significant. The dissipation efficiency is shown plotted against frequency in Figures 23 and 24.
loo0
Frequency Figure
19. Plots of the transmission -, C, = 0 Nms/rad; --,
10000
(Hz)
efficiency, r,, with joint impedance in rotation only; K, = 10’ Nm/rad. C, = 10 Nms/rad; - - -, C, = lo2 Nms/rad; - - - -, C, = 03.
WAVE
PROPAGATION
THROUGH
JOINTS
45
0.5
c” 0,4 t .-El Q s ._ f ._ E z z I-
I
o’3
0.2 -
Ol-
1 100
L 1000 Frequency
10000
1Hz 1
Figure 20. Plots of the transmission efficiency, TL, with joint impedance in rotation only; K, = IO5Nm/rad. -, C, = 0 Nms/rad; --, C, = 10 Nms/rad; - - -, C, = IO* Nms/rad; - - - -, C, = cc.
1.0.
0.6 -
p” t 5 ‘3
---
__
0.6 \
5 Ei ‘i; 0
_---_
\
L
0.4 -
Yyy_
‘\
% a
-a 0.2 -
0.0
IO
I 1000
1 100 Frequency
IOGQO
(Hz)
Figure 21. Plots of the reflection efficiency, pe, with joint impedance in rotation only; & = 10’Nm/rad. -, C, = 0 Nms/rad; --, C, = 10 Nms/rad; - - -, C, = lo2 Nms/rad; - - - -, C, = 03.
8. CONCLUSIONS General expressions for the transmission and reflection efficiencies for flexural incident waves at a right-angled comer formed by two distinct beams have been presented. The joint between the two beams possesses the characteristics of spring-dashpot systems in three orientations: x, y and rotation. The impedances in the different orientations affect the transmission and reflection efficiencies differently. When the impedance of the tlexural wave matches the impedance
R. C. N. LEUNG
46
AND
R. J. PINNINGTON
.*’
/5/
0.04 -
,’
/
0.02 -
,__--;-/..!--
OcQ I IO
--
_‘-
loo
1000 Frequency
(Hz
/--= 4 la 00
1
Figure 22. Plots of the reflection efficiency, pL, with joint impedance in rotation only; K, = 10’ Nm/rad. -, C,=ONms/rad;--, C,=lONms/rad;---, C,=ldNms/rad;----, C,=co.
.-s z
.a D a
0.2 .
0.1 .
0.0
IO
IO00
IO0
IO 00
Frequency (Hz)
Figure 23. Plots of the dissipation efficiency, l, with joint impedance in x only; & = 10’N/m. -, C, = 10’Ns/m; --, C, = lo5 Ns/m; -- -, C, = 10s Ns/m.
of the joint stiffness, the power of the induced flexural wave in the second member drops to zero if there is no damping at the joint. The critical loss factor at which the transmission efficiency, rB, exhibits no minima is 7) = 1. The orientation of the joint impedance, either x or y, has the same effect on the induced flexural wave in the second member. The orientation of the joint impedance has a profound difference on the amount of power being transmitted as longitudinal waves into the second member. If the joint has impedance in the x-direction only, the power of the transmitted longitudinal wave changes little with different joint &&less. On the other hand, if the joint has impedance in the y-direction only, the power of the transmitted longitudinal wave peaks around the frequency where the flexural wave impedance matches the
WAVE
PROPAGATION
THROUGH
JOINTS
47
0.4 . Ln c g 5
0.3 -
“, 5 z a z b-
02 -
0.1 . / o.oIO
loo
looo Frequency
Figure
10000
(Hz)
24. Plots of the dissipation efficiency, 6, with joint impedance in rotation only; K, = lo5 Nm/rad -, C, = 10 Nms/rad; --, C, = lo2 Nms/rad; - - -, C, = lo3 Nms/rad.
impedance of the joint stiffness. Above the impedance matching frequency, the power of the transmitted longitudinal wave drops to zero. The rotational stiffness of a right-angled joint is a useful means of controlling the transmission of flexural wave into the second member. The softer the joint in rotational stiffness the less flexural waves are transmitted through the joint. A joint soft in rotational stiffness reflects the power of the flexural incident wave back mainly as flexural waves. However, the stiffness in rotation does not affect the amount of power being induced in the second member as longitudinal waves. The most effective way of controlling the transmission of flexural waves into the second member is by varying the rotational stiffness of the joint. However, the joint stiffnesses in the x- and y-directions are also useful in controlling the power of flexural waves into the second member for a limited frequency range. The power of longitudinal waves induced into the second member can only be controlled effectively by the joint impedance in the y-direction. Other orientations of the joint impedance will not stop the longitudinal waves in the second member. Particularly at high frequencies, longitudinal waves are induced more easily in the second member. On the whole, a flexural wave at a right-angled corner induces mainly flexural waves in both members of the joint. The power of the longitudinal waves remains very low, except at high frequencies in the second member. The damping at the joint in the x- and y-directions has effects similar to the damping in a spring-mass-dashpot system. The damping smooths out the peaks and troughs in the transmission and reflection efficiencies. The critical loss factor of the joint is when 77= 1. The rotational damping at the joint has a different effect on the efficiencies. It affects only the efficiencies above a particular frequency. The break-away frequency is a function of the damping coefficient of the joint. ACKNOWLDGMENT This work has been carried out with the support of the Procurement Executive, Ministry of Defence.
48
R. C. N. LEUNG
AND
R. J. PINNINGTON
REFERENCES 1. R. C. N. LEUNG 1986 ISVR Technical Report No. 139. Wave propagation through right-angled joints with compliance. 2. C. F. BEARDS and J. L. WILLIAMS 1977 Journal of Sound and Vibration 53, 333-340. The damping of structural vibration by rotational slip joints. 3. L. CREMER, M. HECKL and E. E. UNGAR 1973 Structure-borne Sound. Berlin: Springer-Verlag. 4. L. GAUL 1983 Transactions of the American Society of Mechanical Engineers Series L 105, 409-496. Wave transmission and energy dissipation at structural and machine joints. 5. B. R. HANKS and D. G. STEPHENS 1967 Shock and Vibration Bulletin 36(4), l-8. Mechanisms and scaling of damping in a practical structural joint. 6. L. JEZEQUEL 1983 Transactions of the American Society of Mechanical Engineers Series L 105, 497-504. Structural damping by slip in joints. 7. G. ROSENHOUSE, H. ERTEL and F. P. MECHEL 1981 Journal of the Acoustical Society of America 70(2), 492-499. Theoretical and experimental investigations of structure-borne sound transmission through a T-joint in a finite system. 8. H. SAITO and H. TANI 1984 Journal of Sound and Vibration 92,299-309. Vibrations of bonded beams with a single lap adhesive joint. 9. M. YOSHIMURA and K. OKUSHIMA 1977 Annals of the CZRP 25, 193-198. Measurement of dynamic rigidity and damping property for simplified joint models and simulation by computer. 10. M. YOSHIMURA and K. OKUSHIMA 1979 Annals of the CZRP 28,241-246. Computer-aided design improvement of machine tool structure incorporating joint dynamics data. 11. R. LYON 1975 Statistical Energy Analysis of Dynamical Systems. Cambridge, Massachusetts, London: MIT Press.
APPENDIX:
k m r t U
P
C F K M P z
J” 8 P
7 W
wavespeed cyclic frequency (Hz) thickness wavenumber mass per unit length reflection coefficient transmission coefficient velocity in x-direction velocity in y-direction flexural rigidity joint damping coefficient force stiffness moment power impedance (force/velocity) loss factor dissipation efficiency rotation speed (rad/ s) reflection efficiency transmission efficiency angular frequency (rad/s)
Subscripts n X
Y e B L
near field x-direction y-direction rotation flexural wave longitudinal wave
NOMENCLATURE
WAVE
PROPAGATION
WITH
THROUGH
COMPLIANCE-FLEXURAL R. C. N. LEUNG
Institute
RIGHT-ANGLED
AND
of Sound and Vibration Research,
(Received
INCIDENT
JOINTS
WAVE
R. J. PINNINGTON 7’he University, Southampton,
31 October 1988, and in revised form 22 November
England
1989)
The vibration transmission through a right-angled joint with compliance is investigated by using the wave theory approach. The reflection and transmission coefficients are given. The joint is that between two semi-infinite beams, possibly of different materials and different thicknesses, fixed together either by welding, bolts or rivets. A spring-dashpot model is used in each degree of freedom of the joint. Thus, the results can also incorporate the case of a gasket between two plates. The investigation shows that both flexural and longitudinal waves will be induced in the secondary plate by either flexural or longitudinal incident waves alone in the primary plate. The results also show that both transmitted flexural and longitudinal waves can be controlled to some extent by varying either the stiffness of the joint or the thickness of the plates. The given coefficients will have some application to statistical energy analysis.
1. INTRODUCTION
In structures and machinery it is not uncommon to find structural components that meet at right angles-for example, steel beams in structural frames. The beams are normally joined together either by welding, bolts or rivets. Sometimes, there may be gaskets between two structural components. Waves propagating along one component will eventually encounter a joint. Each joint will reflect part of the incident wave, while some of the incident wave will be transmitted through the joint. The attenuation of the incident waves transmitted through a joint is largely governed by the impedance and the orientation of the joint. Cremer et al. [3] have studied the attenuation of an incident wave through a rigid joint. However, it is not always practical to have a rigid joint. Moreover, the possibility of damping joints has created a lot of interest with respect to controlling the vibration of structures; for example, in references [2-81 the damping characteristics of joints have been modelled. In general, a linear spring-dashpot model is adequate in modelling the joint. In this paper the attenuation of a flexural incident wave under the influence of a joint with compliance is predicted. The case of a longitudinal incident wave will be presented in a follow-on paper. The study is confined to two semi-infinite beams of unit width joined together at right angles, with a flexural incident wave. Once transmission efficiencies, 7, have been obtained between the various wave types of the coupled beams it is straightforward to obtain the coupling loss factors, 7),2, required in a statistical energy analysis [ 111; it is given by v,2 = r/2kl, where k is the flexural wave number of the excited beam and 1 is the length of the excited beam. 2. JOINT
It is known local material
MODEL
that riveted, bolted and welded joints damping [4]. Thus, a spring-dashpot 31 0022-460X/90/190031 + 16 $03.00/O
can dissipate more energy than the model could be appropriate to @ 1990 Academic Ress Limited
32
R. C. N. LEUNG
AND
R. J. PINNINGTON
characterize a joint. Yoshimura [9, lo] and Gaul [4] have identified the stiffness and the damping coefficients successfully using linear spring-dashpot models. Tbe same linear spring-dashpot models are adopted in this study. Because there are three degrees of freedom at a right-angle joint-x, y and rotation-three sets of spring-dashpot models are required.
Figure 1. Joint model.
3. TRANSMISSION AND REFLECTION OF A FLEXURAL INCIDENT WAVE THROUGH A RIGHT-ANGLE JOINT WITH COMPLIANCE IN ALL ITS DEGREES OF FREEDOM Cremer et al. [3] have studied two extreme cases-rigid joints and simply supported joints. The same wave approach is to be used here for the cases of a joint with compliance in all three degrees of freedom. Consider a flexural incident wave of amplitude V travelling towards the joint in component 1 (a list of nomenclature is given in the Appendix). The transverse displacements v,(x) along the plate consist of the incident wave, a reflected travelling wave and a near field wave at the joint: i.e., V,(x) = V(eeikBlx+ r, eikalx+ r, ekBIX).
Figure 2. Sign convention.
(1)
WAVE
PROPAGATION
THROUGH
In component 2, its transverse displacements, wave and a near field wave at the joint: i.e.
JOINTS
33
u*(y), consist of a transmitted travelling
uzy(y) = V( ts e-iksl-\’ + 1, emkeIy).
(2)
Because the flexural incident wave has to go through a corner, longitudinal waves will be induced in both components. The longitudinal waves in both components are u,(x) = VrL eikLfix,
Z&J) = VfL e-ikL2v.
(394)
With the sign conventions shown in Figure 2, the relations between the moments, forces and displacements for both components are given as [3] e*=
2;
-;$,
M,=
dy F,,, = -iom,
V, dx,
M2=---.
F,, = -iwm,
B2 d2u2 io dy* ’
u2 dy;
(7)
I
F,, = --m,CLlul,
(56)
52 = m2CL2v2.
(8)
The joint between the two beams is assumed to have the characteristics of three linear spring-dashpot systems as discussed in the previous section. The impedances in the x-direction, the y-direction and the rotation direction are represented as Z, = (K,/iw) + C,,
Z, = (K,/io)
The unknown coefficients in equations conditions at the joint. They are F,, =
Fx2 = Zx(u,
+ C,,
(l)-(4)
- ~21,
Z, = (K,/io)
+ CO.
are found by applying the boundary
?,,I = Fy2 = Zy(v,
- vz),
(10,11)
M, = M2 = Z,( 8, - 0,).
From equations (l)-(12), r,, can be found:
fL
-=
y2
the transmission
4
(12)
and the reflection coefficients, fa, fL, r, and
(l-i)[2+(1+i)(Z,+~,)l
(14)
[2+(1+i)(Z2+~L)l +‘[1+i(Z,+y,)][2+(1+i)(Z2+y2)]rB’ [(I -iW2+ y2)-21
rs=[(1+i)(Z2+y2)+2]-’
[(1+i)(Z,+y,)+21[i(Z2+y2)+11 [i(z,+rl)+11[(1+i)(Z2+Y2)+21
yl(l -i) rL= -[i(Z,+y,)+l]
fB’
(9)
“’
(1%
(16)
Here X = kB2/kB, is the flexural wavelength ratio in the two components and +* = (m2Bz)/(mlBl). The ratios y, = (m2CB2)/m,CL,) and y2 = (m,Ca,)/(m2CL2) relate the longitudinal and flexural wave impedances in the two plates. The dimensionless terms Z, = m2CBz/ZT, Z2 = m,C,,/ZY, and Z, = B,/(CB,Zo) are the ratios of the bending wave impedances, and the impedances of the joint.
34
R. C. N. LEUNG
AND
R. J. PINNINGTON
For a rigid joint, the stiffnesses of the joint, K,, KY and Kg, will be infinite. This results in Z,, 2, and 2, becoming zero. Thus, 2(y1+y,)-i2(I-y1yJ
fg=[JI(-l-yl-2Yz-Y1Y2)+X(-1-2YI-YZ-Y1Y2)1+i[(~+X)(1-Y1Y2)1’
(17)
which is exactly the same as that given by Cremer [3]. Since vibrational power is a much better way to express the severity of vibration in structures than just the amplitude of the vibration displacement, two efficiencies, transmission efficiency (T) and reflection efficiency (p) are defined: 7=
P=
vibrational power transmitted
(18)
vibrational power of the incident wave ’ vibrational power reflected vibrational power of the incident wave ’
For a flexural incident wave, the transmission transmitted and the reflected flexural waves are
(19)
and the reflection efficiencies for the (20921)
P*=IrE?12,
%=X&31*,
and the efficiencies for the longitudinal waves are TL =
4. JOINTS
WITH
PL =
ltLl*PY,,
COMPLIANCE
(CL,/2CB,)
IN THE
(22,231
kL12.
x-DIRECTION
ONLY
The equations (13)-(16) for the transmission and the reflection coefficients are very complicated. It is therefore useful to look at the joint impedance in each orientation separately. This will enable one to understand how the orientation of the joint influences the transmission and reflection of the incident wave. To simplify the problem further, the two components are assumed to be of the same material and of the same cross-section: i.e., X = II,= 1, y, = y2 = y. A joint is assumed to have an impedance of a spring-dashpot type, Z,, in the x-direction only. The joint is rigid in the y-direction, D, = u2 or K,, = 00, and in rotation, 8, = e2 or Ke = 00. The transmission and the reflection coefficients then become
t,=(2y-i2)[1+i(Z,+y)]/D,
t,=y[(iZ-6)-(6+i2)(Z,+y)]/D, rL=[2y-2y2+i(2y+2y2)]/D,
rB=[(2-2y+iy)+(2-2y+i)(Z,+y)]/D, where D= -2(1 -i)-3y-3(2,+y)-2(1 dimensionless impedance
+i)y(Z1+
y). Using equation
ZI =Zk(n+i),
(24,25) (26,27) (9) gives the (28)
with Z, = CBmwKx/(K~+w2C~),
77= WC,/ K,.
(29,30)
The term Z, is the ratio of the impedance of the spring in the x-direction to the impedance of the flexural incident wave. q is the loss factor of the joint. Both Z, and r) are frequency
dependent. By substituting equation (28) into the transmission and the reflection coefficients, and using the definitions for the transmission and reflection efficiencies, equations (20)-(23), the efficiencies can then be found.
WAVE The
PROPAGATION
THROUGH
JOINTS
35
dissipation efficiency, 5, defined as 5=
power dissipated at joint vibrational power of incident wave ’
(31)
can be obtained from the relationship Q+T7L+fpB+pL+[=l.
(32)
For the case where the joint impedance in the x-direction has no damping term, i.e., 2, = C,mw/ K,,the transmission efficiency of the transmitted flexural wave, TV, can be shown to have a minimum when Z, = 1, with y<< 1: i.e.,
CBm=Kx/w.
(33)
The left side of equation (33) is the impedance of the flexural incident wave, and the right side is the magnitude of the impedance of the joint stiffness. Thus, when both impedances match in magnitude there will be no flexural wave transmitted through the joint. A joint with a stiffness in the x-direction acts as a band-stop filter on the flexural incident wave. The transmission and reflection efficiencies are plotted against the non-dimensional term, Z, = CBmo/Kx in Figures 3-6. These plots are for a right-angled joint of two like 10 mm thick steel beams of unit width. The impedance matching effect on the transmitted I.0
O,I
I C,m
-9
IO w/X,
Figure 3. Plots of the transmission efficiency, ~a, with joint impedance in x only. k& = co; --, & = 10” N/m; - - -, K, = 10s N/m; - - - -, K, = IO6N/m.
flexural wave is demonstrated in Figure 3. By using equations (20) and (24), it can be shown that at low frequencies, f+O, ~a +O-5. On the other end of the frequency range, f+ co, r8 + O-5. Together with the results in Figure 3, it is found that the proportion of the incident vibrational power being transmitted as flexural wave into the second component will not exceed a half. From equations (22) and (25), it can be shown that the transmission efficiency of the longitudinal wave Q -2*5/y as f* cc. Since y is proportional to frequency rL will eventually become zero as the frequency increases. Incorporating this result with that in Figure 4, one may draw a conclusion that the peak in rL will be at a frequency well above the frequency range appropriate to the simple flexural wave equations used.
36
R. C. N. LEUNG
AND
C,m -,
R. J. PINNINGTON
w/K,
Figure 4. Plots of the transmission efficiency, 7‘. with joint impedance in x only. K, = CO;--, K, = 10” M/m; - - -, K,x = 10’N/m; - - - -, K, = lo6 N/m.
It is shown in Figures 5 and 6 that when impedance matching occurs, the vibrational power of the flexural incident wave is mainly reflected back as flexural waves. There is also an increase in the amount of power being reflected back as longitudinal waves. On the whole, the secondary transmitted and reflected longitudinal waves remain as a small proportion of the original incident wave. Above the critical frequency at which the impedance matching occurs, there will be negligible longitudinal waves being reflected back at the joint. In Figures 3-6, the curves associated with K, = cc show the limits on the efficiencies as the frequency of the incident waves tends to infinity. Strictly speaking, the term used in the abscissa, C,mo/K,, is zero if K, = 00. Figures 3-6 are plotted with the term I.0
0.6 -
0.4 ------
*_ H-1
0.2 .\
Cfl olK -9
Figure 5. Plots of the reflection efficiency, ~a, with joint impedance in x only. K, = 00; - -, K,v = 10” N/m; - - -, K, = 10’N/m; - - - -, K,v = lob N/m.
WAVE
PROPAGATION
THROUGH
JOINTS
37
2 i g E % c P Y
0.06
0.04
5 0.02
0.00 0.1
I Gm~/K,
-,
Figure 6. Plots of the reflection efficiency, pL, with joint impedance in x only. K,=co; --, K,=lO’ON/m; ---, K, =108N/m; ----, K,=106N/m.
Cam w/K, as the abscissa in order to demonstrate the condition for the minimum transmission efficiency. The reference, the stiffness of a bolted joint for the size of the chosen steel beam will be of the order of 10’ to 10” N/m, depending on the contact pressure of the bolted joint [9].
5. JOINT
WITH
COMPLIANCE
IN y-DIRECTION
ONLY
For a joint with compliance in the y-direction only, Z, = Z, = 0, or ur = u2, 13~= &. With the two components assumed to be of the same material and of the same cross-section, the transmission and reflection coefficients are found to be fa=(2~-i2)[1+i(Zz+y)]/D,
fL= y[(i2-6)-(6+i2)y]/D,
(34335)
rg=[(2+2y+iy)+(-2-2y+i)(Zz+y)]/D,
(36)
r,=i2y(l-i)[l+i(Z,+y)]/D,
(37)
with D=-2(1-i)-3y-3(Z,+y)-2(l+i)y(Z,+y). From equations (34)-(37) and equations (20)-(23), the transmission and the reflection efficiencies can be evaluated. The dimensionless impedance is Z,=Z,(n+i)
(38)
with Z, = CBmwKy/(K~+w2C~),
77= WC,/ K Y'
(39940)
By comparison; equations (24) and (34) are of the same form. Thus, Figure 7 is the same as Figure 3. The only difference is the impedance of the joint. One relates to the joint impedance in the x-direction while the other relates to the impedance in the y-direction. Thus, a joint with impedance either in the x-direction or in the y-direction acts as a band-stop filter limiting the amount of power transmitted as flexural waves through the joint.
38
R. C. N. LEUNG AND R. J. PINNINGTON
0.6
-
0.1
I
C,mo/K
Figure 7. Plots of the transmission efficiency, Q, with joint impedance in y only. -, KY= 00; --, KY= 10” N/m; - - -, KY= 10’N/m; - - - -, K, = IO6 N/m.
The flexural and longitudinal transmission and reflection efficiencies for a flexural incident wave at a right-angled joint with compliance in the y-direction only are plotted against the non-dimensional term, Z,, equation (39), in Figures 7-10. The effect of joint damping is not considered in these plots. The plots are for two 10 mm thick steel beams forming the joint. It is shown in Figure 8 that once the critical frequency, at which the impedance of the flexural wave matches the impedance of the joint, is passed there will be insignificant power transmitted as longitudinal waves through the joint. It also shows that the peak is higher as the stiffness of the joint increases.
C,m O/K
Figure 8. Plots of the transmission efficiency, Q, with joint impedance in y only. -, KY= co; - -, K, = 10” N/m; - - -, K,.= 10”N/m; - - - -, KY = IO6N/m.
WAVE
PROPAGATION
THROUGH
39
JOINTS
It is shown in Figure 9 that the joint stiffness affects the amount of power being reflected as flexural wave. The softer the joint the more likely power is reflected back at the joint as a flexural wave. I.0
‘K\ ‘3
0.6
5
s
‘G
8 G P
-
_.A-/
0.4 --
/
---_/
/
/
A---
---
/
0.2 -
I I
o-o 0.1
IO
C,m. O/ K, Figure 9. Plots of the reflection efficiency, pe, with joint impedance in y only. KY = ~0; --, K,. = 10” N/m; - - -, K, = IO* N/m; - - - -, K,. = lo6 N/m.
-,
In Figure 10 it is indicated that the reflected longitudinal wave is insignificant. It reaches the level of 10% only at very high frequency, 2, > 7, with very high stiffness, ky = 10 GN/m in the present case of two 10mm thick steel beams. At such high frequency, the simple flexural wave equation used will no longer be valid. The cases of KY= cc in Figures 7-10 are for the purpose of showing the limits on the efficiencies as the frequency of the incident wave tends to infinity.
0.10 / /
/’
0 08 -
d
t :
0.06
g
0,04- _-_
5 ._
‘S
/ /
-
/
-..
5 K
\ ooz-
----_ 0.00 0.1
I
IO
Cgmw/KY
-,
in y only. Figure 10. Plots of the reflection efficiency, pL, with joint impedance K,.=co;----, K,=lO’ON/m;-----, K!=lO’N/m;----, K,=106N/m
40
R. C. N. LEUNG
6. JOINT
AND
R. J. PINNINGTON
WITH COMPLIANCE
IN ROTATION
ONLY
In the previous two sections, joints with compliance in either the x- or y-direction were investigated. They are applicable to welded, bolted or riveted joints with no elastic layer separating the two members that form the joint. In such joints it can be assumed that there is no relative angular movement at the joint between the two members. Such an assumption cannot be applied when joints with elastic layers, e.g., gaskets, are used to separate the two members. If gaskets are used, the joints will have compliance in all three orientations, x, y and rotation. The effect of impedance in rotation alone is considered here. For joints with compliance in rotation only, 2, = 2, = 0, or uI = u2, u, = u2. With the two members assumed to be of the same material and of the same cross-section, the transmission and the reflection coefficients are found by applying the boundary conditions at the joint to equations (13)-(16):
fL=y{-6(~+1)+i2(1--y)+iZ~[4+(2+i2)y]}/D,
fB=(2y-i2)(1+iy)/D,
f-,=27(1-y+i+iy)/D,
r,=[(2-2y2+i2y)+Z,(2y+iy2+i2)]/D, with D = [2+(1 +iy)]{(-2yis given as
(41,42)
1 +i)+i(&/2)[2+(1
(43,44)
+iv)]}. The non-dimensional
term Z, (45)
Z3=Zk(77+i), with z, = (B/C,)oK,/(K2,+02C2,),
n=wC,/K,.
(4647)
The term B/Cs is the moment impedance of a flexural wave. Z3 is the ratio between the impedance of the flexural incident wave and the joint rotation impedance. The transmission and the reflection efficiencies can be found by substituting equations (41)-(44) into equations (20)-(23). Then efficiencies are plotted in Figures 11-14 for two steel plates of the same cross-section with no damping.
I.0
0.8 -
0.6 -
\
-\ 0.2 -
-\
\ -\
0.0 ’ IO
1. ‘\__
---__
1 loo
, 1000
--_
_ 10000
Frequency (Hz) -,
Figure 11. Plots of the transmission efficiency, ~a, with joint impedance K, = co; --, K, = lo6 Nm/rad; - - -, Ii, = 10’ Nm/rad; - - - -,
in rotation only. K, = lo4 Nm/rad.
WAVE
PROPAGATION
THROUGH
41
JOINTS
o’53 0.4 i
i g
t
0.9 -
%
5
5
o-2-
ii
P
m
I-
0.1 -
0.0
I
IO
1
1000
loo Frequency
-,
Figure 12. Plots of the transmission efficiency, TV, with joint impedance in rotation only. K, = co; --, K, = lo6 Nm/rad; - - -, K, = lo5 Nm/rad; - - - -, K, = lo4 Nm/rad.
o.o1 IO
100
IO00
Frequency
--,
IC 00
(Hz1
IO 000
(Hz)
Figure 13. Plots of the reflection efficiency, ps, with joint impedance in rotation only. K, = co; --, K, = lo6 Nm/rad; - - -, K, = lo5 Nm/rad; - - - -, K, = lo4 Nm/rad.
It is found that the efficiencies for a joint with only rotation impedance show no peak or trough. Even at the frequency where the moment impedance of the incident wave matches the rotation impedance of the joint, power will still be transmitted as flexural wave through the joint. However, the amount of power transmitted as a flexural wave through the joint is governed by the rotational stiffness of the joint. The softer the joint in rotation, the less power transmitted as flexural wave through the joint. The results show that joints with very soft stiffness in rotation reflect almost zero power back as longitudinal waves. The stiffness in rotation has little influence on the power being transmitted as longitudinal waves through the joint.
42
R. C. N. LEUNG
AND
R. J. PINNINGTON
040
e 5 g .-
0.06
-
oa6
-
5
6
$
0.04-
5 n 0.02 -
0,00
---
.___-I'---,-.
IO
100
1000 Frequency
Figure 14. Plots of the reflectionefficiency, -,
K, = 03; --,
K, = lo6 Nm/rad;
7.
-
- -,
10000
(Hz)
pL, with joint impedance in rotation only. - - - -, K, = lo4 Nm/rad.
K, = lo5 Nm/rad;
DAMPING AT JOINTS
As joints in structures or machinery are an unavoidable source of damping, it is essential to study this effect on the transmission and reflection characteristics of a joint. A bolted joint will have a damping coefficient in the order of about lo6 Ns/m for a unit width and 10 mm thick steel beam [9]. The transmission and reflection efficiencies for two different cases are considered: (1) joints with impedance in the x-direction only, Figures 15-18; (2) joints with impedance in rotation only, Figures 19-22. In the first case, the introduction of damping at the joint smooths out the humps and troughs of the efficiencies. This is similar to the damping in a simple spring-mass-dashpot system. _
0.6 -
E i
5
‘G
0.6 -
%J $ .I? E 2 E k-
04 C-0.2 -
Frequency (Hz)
Figure -,
15. Plots of the transmission efficiency, C, = 0 Ns/m; --, C, = lo4 Ns/m; -
rB, with joint impedance in x only; K..= 10’ N/m. - -, C, = 10’ Ns/m; - - - -, C, = lob Ns/m.
WAVE
PROPAGATION
THROUGH
JOINTS
o.531
43
Frequency (Hz) Figure 16. Plots of the transmission efficiency, TV,with joint impedance in x only; K, = 10”N/m. -, C, = 0 Ns/m; --, C, = IO’Ns/m; - - -, C, = lo5 Ns/m; - - - -, c, = 10’Ns/m.
0.6 -
m i .-
0.6 -
{ 6 ._
0.4 -
g
Frequency
(Hz)
Figure 17. Plots of the reflection efficiency, ps, with joint impedance in x only; K,r = 10s N/m. -, C, = 0 Ns/m; --, C, = lo4 Ns/m; - - -, rZY= lo5 Ns/m; - - - -, C, = 10’Ns/m.
From previous sections, the characteristic equation of the efficiencies for joints with impedance in the x- and y-direction alone will have a minima at z,=2(1-7))/3(1+n2)
(48)
if the terms involving y are ignored. This assumption is valid since y is normally less than one. At n = 1, the minimum occurs at 2, = 0. Therefore, the critical damping of the joint will be c= K/w.
(49)
R. C. N. LEUNG
44
0.06
AND
R. J. PINNINGTON
-
100
1000 Frequency
10 000
(Hz)
Figure 18. Plots of the reflection efficiency, pL, with joint impedance in x only; K, = lo8 N/m. -, C, = 0 Ns/m; - -, C, = lo4 Ns/m; - - -, C, = lo5 Ns/m; - - - -, C, = 10’ Ns/m.
In Figures 19-22 is demonstrated the importance of the joint damping in rotation on a flexural incident wave at the joint with impedance in rotation only. The figures show that the longitudinal waves induced in both members of the joint are affected very little by the introduction of damping in rotation. The damping at the joint affects the llexural waves in the receiving member more. The flexural waves reflected in the first members are affected by the damping at joint only at frequencies above a damping dependent break-away frequency. Below the break-away frequency, the reflection efficiency behaves as if there is no damping. Above the frequency, influence due to the damping on the efficiency is significant. The dissipation efficiency is shown plotted against frequency in Figures 23 and 24.
loo0
Frequency Figure
19. Plots of the transmission -, C, = 0 Nms/rad; --,
10000
(Hz)
efficiency, r,, with joint impedance in rotation only; K, = 10’ Nm/rad. C, = 10 Nms/rad; - - -, C, = lo2 Nms/rad; - - - -, C, = 03.
WAVE
PROPAGATION
THROUGH
JOINTS
45
0.5
c” 0,4 t .-El Q s ._ f ._ E z z I-
I
o’3
0.2 -
Ol-
1 100
L 1000 Frequency
10000
1Hz 1
Figure 20. Plots of the transmission efficiency, TL, with joint impedance in rotation only; K, = IO5Nm/rad. -, C, = 0 Nms/rad; --, C, = 10 Nms/rad; - - -, C, = IO* Nms/rad; - - - -, C, = cc.
1.0.
0.6 -
p” t 5 ‘3
---
__
0.6 \
5 Ei ‘i; 0
_---_
\
L
0.4 -
Yyy_
‘\
% a
-a 0.2 -
0.0
IO
I 1000
1 100 Frequency
IOGQO
(Hz)
Figure 21. Plots of the reflection efficiency, pe, with joint impedance in rotation only; & = 10’Nm/rad. -, C, = 0 Nms/rad; --, C, = 10 Nms/rad; - - -, C, = lo2 Nms/rad; - - - -, C, = 03.
8. CONCLUSIONS General expressions for the transmission and reflection efficiencies for flexural incident waves at a right-angled comer formed by two distinct beams have been presented. The joint between the two beams possesses the characteristics of spring-dashpot systems in three orientations: x, y and rotation. The impedances in the different orientations affect the transmission and reflection efficiencies differently. When the impedance of the tlexural wave matches the impedance
R. C. N. LEUNG
46
AND
R. J. PINNINGTON
.*’
/5/
0.04 -
,’
/
0.02 -
,__--;-/..!--
OcQ I IO
--
_‘-
loo
1000 Frequency
(Hz
/--= 4 la 00
1
Figure 22. Plots of the reflection efficiency, pL, with joint impedance in rotation only; K, = 10’ Nm/rad. -, C,=ONms/rad;--, C,=lONms/rad;---, C,=ldNms/rad;----, C,=co.
.-s z
.a D a
0.2 .
0.1 .
0.0
IO
IO00
IO0
IO 00
Frequency (Hz)
Figure 23. Plots of the dissipation efficiency, l, with joint impedance in x only; & = 10’N/m. -, C, = 10’Ns/m; --, C, = lo5 Ns/m; -- -, C, = 10s Ns/m.
of the joint stiffness, the power of the induced flexural wave in the second member drops to zero if there is no damping at the joint. The critical loss factor at which the transmission efficiency, rB, exhibits no minima is 7) = 1. The orientation of the joint impedance, either x or y, has the same effect on the induced flexural wave in the second member. The orientation of the joint impedance has a profound difference on the amount of power being transmitted as longitudinal waves into the second member. If the joint has impedance in the x-direction only, the power of the transmitted longitudinal wave changes little with different joint &&less. On the other hand, if the joint has impedance in the y-direction only, the power of the transmitted longitudinal wave peaks around the frequency where the flexural wave impedance matches the
WAVE
PROPAGATION
THROUGH
JOINTS
47
0.4 . Ln c g 5
0.3 -
“, 5 z a z b-
02 -
0.1 . / o.oIO
loo
looo Frequency
Figure
10000
(Hz)
24. Plots of the dissipation efficiency, 6, with joint impedance in rotation only; K, = lo5 Nm/rad -, C, = 10 Nms/rad; --, C, = lo2 Nms/rad; - - -, C, = lo3 Nms/rad.
impedance of the joint stiffness. Above the impedance matching frequency, the power of the transmitted longitudinal wave drops to zero. The rotational stiffness of a right-angled joint is a useful means of controlling the transmission of flexural wave into the second member. The softer the joint in rotational stiffness the less flexural waves are transmitted through the joint. A joint soft in rotational stiffness reflects the power of the flexural incident wave back mainly as flexural waves. However, the stiffness in rotation does not affect the amount of power being induced in the second member as longitudinal waves. The most effective way of controlling the transmission of flexural waves into the second member is by varying the rotational stiffness of the joint. However, the joint stiffnesses in the x- and y-directions are also useful in controlling the power of flexural waves into the second member for a limited frequency range. The power of longitudinal waves induced into the second member can only be controlled effectively by the joint impedance in the y-direction. Other orientations of the joint impedance will not stop the longitudinal waves in the second member. Particularly at high frequencies, longitudinal waves are induced more easily in the second member. On the whole, a flexural wave at a right-angled corner induces mainly flexural waves in both members of the joint. The power of the longitudinal waves remains very low, except at high frequencies in the second member. The damping at the joint in the x- and y-directions has effects similar to the damping in a spring-mass-dashpot system. The damping smooths out the peaks and troughs in the transmission and reflection efficiencies. The critical loss factor of the joint is when 77= 1. The rotational damping at the joint has a different effect on the efficiencies. It affects only the efficiencies above a particular frequency. The break-away frequency is a function of the damping coefficient of the joint. ACKNOWLDGMENT This work has been carried out with the support of the Procurement Executive, Ministry of Defence.
48
R. C. N. LEUNG
AND
R. J. PINNINGTON
REFERENCES 1. R. C. N. LEUNG 1986 ISVR Technical Report No. 139. Wave propagation through right-angled joints with compliance. 2. C. F. BEARDS and J. L. WILLIAMS 1977 Journal of Sound and Vibration 53, 333-340. The damping of structural vibration by rotational slip joints. 3. L. CREMER, M. HECKL and E. E. UNGAR 1973 Structure-borne Sound. Berlin: Springer-Verlag. 4. L. GAUL 1983 Transactions of the American Society of Mechanical Engineers Series L 105, 409-496. Wave transmission and energy dissipation at structural and machine joints. 5. B. R. HANKS and D. G. STEPHENS 1967 Shock and Vibration Bulletin 36(4), l-8. Mechanisms and scaling of damping in a practical structural joint. 6. L. JEZEQUEL 1983 Transactions of the American Society of Mechanical Engineers Series L 105, 497-504. Structural damping by slip in joints. 7. G. ROSENHOUSE, H. ERTEL and F. P. MECHEL 1981 Journal of the Acoustical Society of America 70(2), 492-499. Theoretical and experimental investigations of structure-borne sound transmission through a T-joint in a finite system. 8. H. SAITO and H. TANI 1984 Journal of Sound and Vibration 92,299-309. Vibrations of bonded beams with a single lap adhesive joint. 9. M. YOSHIMURA and K. OKUSHIMA 1977 Annals of the CZRP 25, 193-198. Measurement of dynamic rigidity and damping property for simplified joint models and simulation by computer. 10. M. YOSHIMURA and K. OKUSHIMA 1979 Annals of the CZRP 28,241-246. Computer-aided design improvement of machine tool structure incorporating joint dynamics data. 11. R. LYON 1975 Statistical Energy Analysis of Dynamical Systems. Cambridge, Massachusetts, London: MIT Press.
APPENDIX:
k m r t U
P
C F K M P z
J” 8 P
7 W
wavespeed cyclic frequency (Hz) thickness wavenumber mass per unit length reflection coefficient transmission coefficient velocity in x-direction velocity in y-direction flexural rigidity joint damping coefficient force stiffness moment power impedance (force/velocity) loss factor dissipation efficiency rotation speed (rad/ s) reflection efficiency transmission efficiency angular frequency (rad/s)
Subscripts n X
Y e B L
near field x-direction y-direction rotation flexural wave longitudinal wave
NOMENCLATURE