Sang Wook Kang1
Department of Mechanical
Systems Engineering,
Hansung University,
389, 2-ga, Samsun-dong,
Sungbuk-gu, Seoul, 136-792, Korea
e-mail: swkang@hansung.ac.kr
Satya N. Atluri
Department of Mechanical and
Aerospace Engineering,
University of California, Irvine,
California 92697
e-mail: satluri@uci.edu
Sang-Hyun Kim
Department of Mechanical
Systems Engineering,
Hansung University,
389, 2-ga, Samsun-dong,
Sungbuk-gu, Seoul, 136-792, Korea
e-mail: shkim@hansung.ac.kr
Application of the
Nondimensional Dynamic
Influence Function Method for
Free Vibration Analysis of
Arbitrarily Shaped Membranes
A new formulation for the NDIF method (the nondimensional dynamic influence function
method) is introduced to efficiently extract eigenvalues and mode shapes of arbitrarily
shaped, homogeneous membranes with the fixed boundary. The NDIF method, which was
developed by the authors for the accurate free vibration analysis of arbitrarily shaped
membranes and plates including acoustic cavities, has the feature that it yields highly
accurate solutions compared with other analytical methods or numerical methods (the
finite element method and the boundary element method). However, the NDIF method
has the weak point that the system matrix of the method is not independent of the
frequency parameter and as a result the method needs the inefficient procedure of searching eigenvalues by plotting the values of the determinant of the system matrix in the frequency parameter range of interest. An improved formulation presented in the paper
does not require the above-mentioned inefficient procedure because a newly developed
system matrix is independent of the frequency parameter. Finally, the validity of the proposed method is shown in several case studies, which indicate that eigenvalues and mode
shapes obtained by the proposed method are very accurate compared to those calculated
by exact, analytica, or numerical methods. [DOI: 10.1115/1.4006414]
Keywords: arbitrarily shaped membrane, eigenvalue, NDIF method, nondimensional
dynamic influence function method, free vibration, algebraic eigenvalue problem
1
Introduction
The authors developed the so-called NDIF method (nondimensional dynamic influence function method) for free vibration analysis of arbitrarily shaped membranes in 1999 [1]. Furthermore,
the authors extended the NDIF method to arbitrarily shaped
acoustic cavities [2], arbitrarily shaped membranes with highly
concave edges [3], and arbitrarily shaped plates with various
boundary conditions [4–6].
As indicated in the above-mentioned papers [1–6], many
researchers have studied the free vibration analysis of membranes
and plates with a variety of shapes [7–20]. However, analytical
approaches dealing with arbitrary shape have been little reported
in the open literature to the authors’ best knowledge. In this paper,
a modified NDIF method is introduced to efficiently extract the
eigenvalues and mode shapes of arbitrarily shaped membranes. In
the future, the method will be extended to arbitrary shaped plates
and acoustic cavities.
Unlike the finite element method (FEM) [21] and the boundary
element method (BEM) [22,23], the NDIF method [1] needs no
integration procedure in its theoretical formulation because the
boundary of the domain of interest is divided with only nodes
(without elements). As a result, the NDIF method needs a small
amount of numerical calculations and yields rapidly converged,
highly accurate results.
1
Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the
JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 22, 2011; final
manuscript received February 15, 2012; published online May 29, 2012. Assoc. Editor: Massimo Ruzzene.
Journal of Vibration and Acoustics
However, the NDIF method has the weak point that its final
system matrix equation does not have a form of the algebraic
eigenvalue problem [24] like Eq. (1):
SM v ¼ Kv
(1)
in which SM, v, and K represent a system matrix, a system vector,
and a frequency parameter, respectively. Since basis functions
used in the NDIF method are a function of the frequency parameter (K), its final system matrix equation has a form of nonalgebraic
eigenvalue problem like Eq. (2) [1]:
SMðKÞ v ¼ 0
(2)
in which the system matrix SMðKÞ is a function of K. As a result,
the NDIF method needs the inefficient procedure of searching
values of the frequency parameter that make the system matrix
singular by sweeping the frequency parameter in the range of
interest. On the other hand, FEM and BEM do not need the aforementioned inefficient procedure because their system matrix equations have forms of the algebraic eigenvalue problem like Eq. (1)
[21–23]. In particular, Nardini described a new procedure which
reduces the problem of free vibrations to an algebraic eigenvalue
problem, solution of which is straightforward [23].
In order to overcome the above-mentioned weak point for the
NDIF method, a modified NDIF method is introduced in the
paper. The validity and accuracy of the proposed method are verified by several case studies in which the results obtained by the
proposed method are compared with those given by other methods
such as the NDIF method, FEM (ANSYS), and the exact method.
C 2012 by ASME
Copyright V
AUGUST 2012, Vol. 134 / 041008-1
On the other hand, the proposed method has a much improved
computational speed than the NDIF method, although the size of
the system matrix in the proposed method is larger than that in the
NDIF method.
2
which is employed as an approximate solution for the eigenfield
of the finite-sized membrane. Note that the approximate solution
also satisfies the Helmholtz equation because the NDIF satisfies
the Helmholtz equation.
Next, the boundary condition given for the membrane is discretized at boundary points P1 , P2 , …, PN as follows:
NDIF Method Reviewed
2.1 Nondimensional Dynamic Influence Function. The
nondimensional dynamic influence function (NDIF) primarily satisfies the governing equation of the eigenfield of interest, and
physically describes the displacement response of a point in an
infinite domain due to a unit displacement excited at another point
[1]. In the case of an infinite membrane (see Fig. 1), the NDIF
between the excitation point Pk and the response point P is
given by the Bessel function of the first kind of order zero as
follows [1]:
NDIF ¼ J0 ðKjr rk jÞ
(3)
Wðri Þ ¼ Ui ;
i ¼ 1; 2; …; N
where Ui denotes a boundary displacement given in boundary
point Pi . Then, applying the discrete boundary condition (Eq. (6))
to the approximate solution (Eq. (5)) gives
Wðri Þ ¼
N
X
Ak J0 ðKjri rk jÞ ¼ Ui ; i ¼ 1; 2; …; N
r2 WðrÞ þ K2 WðrÞ ¼ 0
(4)
where WðrÞ is the transverse displacement of the finite-sized
pffiffiffiffiffiffiffiffi
membrane depicted as the dotted line in Fig. 1, K ¼ x= T=q
denotes the wavenumber in terms of the angular frequency x, the
uniform tension per unit length T, and the mass per unit area q.
Detailed illustrations on the NDIF are given in the previous paper
[1].
2.2 NDIF Method. For free vibration analysis of an arbitrarily shaped membrane, of which the boundary is illustrated by
the dotted line in Fig. 1, N nodes are first distributed at points
P1 , P2 , …, PN along the boundary depicted in an infinite membrane. Assuming that harmonic displacements of amplitudes
A1 , A2 , …, AN are, respectively, generated at points P1 , P2 , …,
PN , the total displacement response at the point P may be obtained
by the sum of responses (the linear combination of NDIFs given
in Eq. (3)) that have resulted from each boundary point, i.e.,
WðrÞ ¼
N
X
Ak J0 ðKjr rk jÞ
(5)
k¼1
(7)
k¼1
Finally, Eq. (7) may be written in a simple matrix form:
SMðKÞ A ¼ U
in which jr rk j denotes the distance between Pk and P. The
NDIF satisfies the Helmholtz equation,
(6)
(8)
where elements of the N N symmetric system matrix SMðKÞ
are given by SMik ¼ J0 ðKjri rk jÞ, the participation vector A represents the participation strength of the NDIFs defined at each
boundary point, and the displacement vector U represents the discrete boundary condition. In the case of the fixed boundary condition (U ¼ 0), Eq. (8) leads to
SMðKÞ A ¼ 0
(9)
which has the same form as Eq. (2). It may be seen that elements
of the system matrix depend on the frequency parameter. As a
result, the aforementioned inefficient procedure is required to
obtain the eigenvalues of the membrane. In the following section,
an improved theoretical formulation for the NDIF method is presented to make the system matrix independent of the frequency
parameter.
3
Improved Theoretical Formulation
In order to overcome the aforementioned weak point of the
NDIF method, a formulation of the NDIF method to the algebraic
eigenvalue problem is newly carried out. First, the Bessel function
included in the approximate solution (Eq. (5)) is expanded in a
Taylor series [25] as follows:
J0 ðKjr rk jÞ M
X
ð1Þj ðK Rk =2Þ2j
½Cðj þ 1Þ2
j¼0
(10)
where M denotes the number of terms of the series and Cðj þ 1Þ
represents the gamma function. For simplicity, Eq. (10) is rewritten as
J0 ðKjr rk jÞ M
X
K2j /j ðRk Þ
(11)
j¼0
where /j ðRk Þ is given by
/j ðRk Þ ¼
ð1Þj ðRk =2Þ2j
½Cðj þ 1Þ2
Rk ¼ jr rk j
(12)
(13)
Substituting Eq. (11) into Eq. (5) yields
Fig. 1 Infinite membrane with harmonic excitation points that
are distributed along the fictitious contour (dotted line) with the
same shape as the finite-sized membrane of interest
041008-2 / Vol. 134, AUGUST 2012
WðrÞ ¼
N
X
k¼1
Ak
M
X
K2j /j ðRk Þ
(14)
j¼0
Transactions of the ASME
N
X
Ak /0 ðRik Þ þ k
N
X
k¼1
Ak /1 ðRik Þ þ k2
k¼1
þ kM
N
X
N
X
Ak /2 ðRik Þ þ k¼1
Ak /M ðRik Þ ¼ 0;
i ¼ 1; 2; …; N
(17)
k¼1
where k ¼ K2 . For simplicity, Eq. (17) is expressed in the matrix
equation
U0 A þ k U1 A þ k2 U2 A þ þ kM UM A ¼ 0
(18)
where elements of matrix Uj are given by
ð1Þj ðRik =2Þ2j
Uj ði; kÞ ¼ /j ðRik Þ ¼
(19)
½Cðj þ 1Þ2
Note that Eq. (18) is called the higher order polynomial eigenvalue problem [26]. Equation (18) may be changed into a linear
matrix equation as follows:
SML B ¼ k SMR B
Fig. 2
(20)
Circular membrane discretized by 16 nodes
which is rearranged as follows:
WðrÞ ¼
M
X
j¼0
K2j
N
X
Ak /j ðRk Þ
(15)
k¼1
Note that the frequency parameter K is explicitly included in the currently developed approximate solution given by Eq. (15), although K
has exponent 2j. Next, applying the fixed boundary condition (Ui ¼ 0
in Eq. (6)) to the approximate solution (Eq. (15)) yields
M
X
K2j
j¼0
N
X
Ak /j ðRik Þ ¼ 0;
i ¼ 1; 2; …; N
where system metrics SML and SMR are given by
3
2
0
I
0
0
6 0
0
I
0 7
7
6
7
6 .
..
..
..
..
7
6 .
SML ¼ 6 .
.
.
.
.
7
7
6 .
..
..
..
7
6 .
5
4 .
.
.
.
I
U0 U1 U2 UM1
2
I 0 0
60 I 0
6
6
.
6
SMR ¼ 6 0 0 . .
6. . .
6. .
..
4. .
0 0 (16)
k¼1
where Rik ¼ jri rk j denotes the distance of boundary points Pi
and Pk .
Next, Eq. (16) is rewritten as the form of a polynomial equation
with respect to k as follows:
and vector B is given by
B ¼ AT kAT
k2 AT
..
.
I
0
0
0
..
.
(21)
3
7
7
7
7
7
7
7
0 5
UM
kM1 AT
(22)
(23)
Table 1 Eigenvalues (Ki ) of the circular membrane obtained by the proposed method, the NDIF method, the exact method, and
FEM (parenthesized values denote errors (%) with respect to the values by the exact method)
Proposed method (N ¼ 16)
FEM (ANSYS)
M ¼ 10
M ¼ 15
M ¼ 20
M ¼ 25
Exact
method [27]
CPU time
0.8 s
1.3 s
2.1 s
3.4 s
–
192.3 s
1.4 s
1.1 s
0.9 s
0.6 s
0.4 s
K1
2.405
(0.00)
2.405
(0.00)
2.405
(0.00)
2.405
(0.00)
2.405
2.405
(0.00)
2.405
(0.02)
2.406
(0.05)
2.417
(0.50)
2.452
(1.95)
2.491
(3.58)
K2
3.842
(0.26)
3.832
(0.00)
3.832
(0.00)
3.832
(0.00)
3.832
3.832
(0.00)
3.833
(0.04)
3.835
(0.07)
3.851
(0.50)
3.911
(2.06)
3.974
(3.71)
K3
None
5.136
(0.00)
5.136
(0.00)
5.136
(0.00)
5.136
5.136
(0.00)
5.139
(0.05)
5.141
(0.10)
5.174
(0.74)
5.293
(3.06)
5.419
(5.51)
K4
None
5.520
(0.00)
5.520
(0.00)
5.520
(0.00)
5.520
5.520
(0.00)
5.528
(0.14)
5.535
(0.27)
5.552
(0.58)
5.647
(2.30)
5.749
(4.15)
K5
None
6.380
(0.00)
6.380
(0.00)
6.380
(0.00)
6.380
6.380
(0.00)
6.386
(0.10)
6.390
(0.16)
6.461
(1.27)
6.708
(5.14)
6.966
(9.18)
K6
None
7.016
(0.00)
7.016
(0.00)
7.016
(0.00)
7.016
7.016
(0.00)
7.029
(0.19)
7.040
(0.35)
7.059
(0.61)
7.192
(2.51)
7.334
(4.53)
No.
Journal of Vibration and Acoustics
NDIF method [1]
(N ¼ 16)
3668 nodes
2378 nodes
1024 nodes 256 nodes 144 nodes
AUGUST 2012, Vol. 134 / 041008-3
Fig. 3 Mode shapes of the circular membrane obtained by the proposed method when
N ¼ 16 and M ¼ 20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e)
fifth mode, and (f) sixth mode
Finally, multiplying Eq. (20) by the inverse matrix of SMR yields
SM1
R SML B ¼ k B
(24)
which may be expressed as the algebraic eigenvalue problem
SM B ¼ k B
(25)
where the system matrix is given by
SM ¼ SM1
R SML
Fig. 4 Rectangular membrane discretized by 24 nodes
041008-4 / Vol. 134, AUGUST 2012
(26)
Note that Eq. (25) has the same form as Eq. (1). It may be seen
that the system matrix is independent of the frequency parameter.
As aforementioned in the introduction of the paper, the final
system matrix (SM) is not a function of the frequency parameter
(K). As a result, the eigenvalues and eigenvectors (mode shapes)
of the membrane of interest can simply be extracted from Eq. (25)
without the inefficient procedure required in the NDIF method.
The ith mode shape of the membrane may be obtained by
Transactions of the ASME
Table 2 Eigenvalues (Ki ) of the rectangular membrane obtained by the proposed method, the NDIF method, the exact method,
and FEM (parenthesized values denote errors (%) with respect to the values by the exact method)
Proposed method ðN ¼ 24Þ
FEM (ANSYS)
Exact
method [27]
NDIF method [1]
ðN ¼ 24Þ
2806 nodes
1813 nodes
1089 nodes
289 nodes
49 nodes
No.
M ¼ 15
M ¼ 20
M ¼ 25
K1
4.363
(0.00)
4.363
(0.00)
4.363
(0.00)
4.363
4.363
(0.00)
4.364
(0.03)
4.364
(0.03)
4.365
(0.05)
4.370
(0.16)
4.413
(1.15)
K2
None
6.292
(0.02)
6.293
(0.00)
6.293
6.293
(0.00)
6.295
(0.03)
6.296
(0.06)
6.301
(0.13)
6.324
(0.49)
6.517
(3.56)
K3
None
7.454
(0.03)
7.456
(0.00)
7.456
7.456
(0.00)
7.461
(0.07)
7.464
(0.11)
7.467
(0.15)
7.500
(0.59)
7.768
(4.18)
K4
None
8.594
(0.01)
8.595
(0.00)
8.595
8.595
(0.00)
8.602
(0.08)
8.606
(0.13)
8.621
(0.30)
8.701
(1.23)
9.129
(6.21)
K5
None
8.732
(0.06)
8.727
(0.00)
8.727
8.727
(0.00)
8.732
(0.06)
8.736
(0.10)
8.741
(0.16)
8.783
(0.64)
9.352
(7.16)
K6
None
10.52
(0.09)
10.51
(0.00)
10.51
10.51
(0.00)
10.518
(0.07)
10.523
(0.13)
10.54
(0.29)
10.62
(1.05)
11.33
(7.80)
Fig. 5 Mode shapes of the rectangular membrane obtained by the proposed method when
N ¼ 24 and M ¼ 20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth
mode, and (f) sixth mode
Journal of Vibration and Acoustics
AUGUST 2012, Vol. 134 / 041008-5
and the NDIF method [1]. Since the proposed method does not
offer any eigenvalue for M ¼ 5, analysis results for M ¼ 5 have
not been presented in Table 1.
On the other hand, it may be seen in Table 1 that the eigenvalues by FEM do not exactly converge to those by the exact method
although FEM uses a large number of nodes. As a result, it may
be concluded that the proposed method is not only highly accurate
like the NDIF method but also very effective in extracting eigenvalues unlike the NDIF method. On the other hand, it may be seen
in Table 1 that the proposed method has improved vastly in computational speed (CPU time) compared with the NDIF method.
In addition, mode shapes produced by the proposed method are
presented in Fig. 3 and they agree well with those given by the
exact method [27], which are omitted in the paper.
Fig. 6
Arbitrarily shaped membrane discretized by 20 nodes
substituting the ith eigenvalue and eigenvector into Eq. (5) and
plotting Eq. (5).
4
Case Studies
The validity of the proposed method is verified through numerical tests of circular, rectangular, and arbitrarily shaped membranes. For each case, the eigenvalues and mode shapes obtained
by the proposed method are compared with those given by the
NDIF method, exact analysis, or FEM (ANSYS).
4.1 Circular Membrane. First, the proposed method is
applied to a circular membrane of unit radius as shown in Fig. 2,
where the boundary of the membrane is discretized with 16 nodes
(N ¼ 16). Eigenvalues calculated by the proposed method when
M ¼ 10, M ¼ 15, M ¼ 20, and M ¼ 25 [M denotes the number of
terms of the series in Eq. (10)] are presented in Table 1 where
eigenvalues computed by the NDIF method, the exact method,
and FEM (ANSYS) are also summarized.
In Table 1, it may be said that only the first two eigenvalues are
obtained for M ¼ 10 but the first six eigenvalues are successfully
obtained when M increases (for M ¼ 15, M ¼ 20, and M ¼ 25)
and fully converge to eigenvalues given by the exact method [27]
4.2 Rectangular Membrane. A rectangular membrane of
which the dimensions are given by a ¼ 1:2 m and b ¼ 0:9 m is
considered in the section. As shown in Fig. 4, the rectangular
membrane is discretized with 24 nodes in order to apply the proposed method. In Table 2, eigenvalues obtained by the proposed
method for M ¼ 15, M ¼ 20, and M ¼ 25 are compared with
eigenvalues obtained by the exact method, the NDIF method, and
FEM. In Table 2, it may be said that only the first eigenvalue is
obtained for M ¼ 15 but the first six eigenvalues are successfully
obtained when M increases (for M ¼ 20 and M ¼ 25) and fully
converge to eigenvalues given by the exact method [27] and the
NDIF method [1].
It may be said that the proposed method yields very accurate
results and is more accurate than the FEM results using 2806
nodes. Note that the proposed method uses a much smaller number of nodes than FEM. On the other hand, it may be said in
Table 2 that the proposed method is a little inaccurate compared
with the NDIF method. This difference between the two methods
may result from the fact that the Bessel function is approximated
in Eq. (10) by using the Taylor series. Figure 5 shows mode
shapes produced by the proposed method. The modes shapes
agree well with those obtained by the exact method [27].
4.3 Arbitrarily Shaped Membrane. In this section, in order
to verify the accuracy of the proposed method for arbitrarily
shaped membranes, an arbitrarily shaped membrane composed of
a semicircle and two straight lines is considered as shown in
Fig. 6, where the membrane is discretized with 20 nodes. Since
the fact that the NDIF method is more accurate than FEM has
already been verified through the author’s previous paper [1], the
eigenvalues by the proposed method and FEM are compared to
those by the NIDF method in Table 3. It may be seen in Table 3
Table 3 Eigenvalues (Ki ) of the arbitrarily shaped membrane obtained by the proposed method, the NDIF method, and FEM
(parenthesized values denote errors (%) with respect to the values by the NDIF method)
Proposed method ðN ¼ 20Þ
No.
M ¼ 15
M ¼ 20
M ¼ 25
K1
2.707
(0.07)
2.708
(0.04)
2.709
(0.00)
K2
4.225
(0.00)
4.225
(0.00)
K3
4.359
(0.02)
K4
FEM (ANSYS)
NDIF method [1]
ðN ¼ 20Þ
2908 nodes
1867 nodes
784 nodes
576 nodes
400 nodes
2.709
2.711
(0.08)
2.712
(0.10)
2.723
(0.52)
2.728
(0.70)
2.735
(0.96)
4.225
(0.00)
4.225
4.234
(0.21)
4.235
(0.24)
4.260
(0.83)
4.270
(1.07)
4.286
(1.44)
4.358
(0.00)
4.358
(0.00)
4.358
4.360
(0.05)
4.362
(0.08)
4.379
(0.48)
4.386
(0.64)
4.399
(0.94)
None
5.559
(0.00)
5.559
(0.00)
5.559
5.577
(0.32)
5.580
(0.37)
5.634
(1.35)
5.656
(1.74)
5.692
(2.39)
K5
None
5.934
(0.00)
5.934
(0.00)
5.934
5.941
(0.11)
5.944
(0.17)
5.985
(0.86)
6.003
(1.16)
6.032
(1.65)
K6
None
6.114
(0.00)
6.114
(0.00)
6.114
6.127
(0.21)
6.131
(0.28)
6.164
(0.82)
6.181
(1.10)
6.208
(1.54)
041008-6 / Vol. 134, AUGUST 2012
Transactions of the ASME
Fig. 7 Mode shapes of the arbitrarily shaped membrane obtained by the proposed method
when N ¼ 20 and M ¼ 20: (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e)
fifth mode, and (f) sixth mode
Table 4 Higher eigenvalues of the arbitrarily shaped membrane obtained by the proposed method, the NDIF method, and
FEM (parenthesized values denote errors (%) with respect to
the values by the NDIF method)
K7
K8
K9
K10
Proposed method
ðN ¼ 20; M ¼ 20Þ
6.984
(0.01)
7.186
(0.00)
7.755
(0.03)
7.833
(0.03)
NDIF method [1]
ðN ¼ 20Þ
6.985
7.186
7.757
7.831
FEM (ANSYS)
(2908 nodes)
7.022
(0.53)
7.198
(0.16)
7.779
(0.28)
7.856
(0.31)
No.
M ¼ 25) and fully converge to the eigenvalues by the NDIF
method. It may be said from this fact that the proposed method
also yields very accurate results for arbitrarily shaped membranes.
Mode shapes of the membrane obtained by the proposed method
are shown in Fig. 7, which were found to agree well with those by
FEM.
To show the feature of the proposed method in a higher frequency range, higher eigenvalues are extracted by the proposed
method, the NDIF method, and FEM. The extracted eigenvalues
are summarized in Table 4, where it may be said that both the
proposed method and the NDIF method also offer accurate eigenvalues in a higher frequency range.
5
that errors of the proposed method with respect to the NDIF
method are much smaller than errors of FEM with respect to the
NDIF method. In Table 3, it may be said that only the first three
eigenvalues are obtained for M ¼ 15 but the first six eigenvalues
are successfully obtained when M increases (for M ¼ 20 and
Journal of Vibration and Acoustics
Conclusion
An analytical method has been presented that can effectively
obtain the accurate eigenvalues and mode shapes of arbitrarily
shaped membrane. Since the final system matrix equation of the
proposed method has the same form as the algebraic eigenvalue
problem, the proposed method is more effective in extracting
AUGUST 2012, Vol. 134 / 041008-7
eigenvalues of the membranes than the NDIF method. Furthermore, the accuracy of the proposed method has been verified
through several case studies. In addition, it is expected that the
proposed method can be extended to analyze homogeneous or
nonhomogeneous membranes and plate with general boundary
conditions using a subdomain approach [3].
Acknowledgment
This research was financially supported by Hansung University.
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