International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 01, January 2019, pp. 898–914, Article ID: IJMET_10_01_093
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=01
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
Scopus Indexed
CALCULATING THE NATURAL FREQUENCY
OF HOLLOW STEPPED CANTILEVER BEAM
Luay S. Alansari, Hayder Z. Zainy
Department of Mechanical Engineering
The Faculty of Engineering, University of Kufa, Najaf, Iraq
Aya Adnan Yaseen
Al Mansour University College, Baghdad, Iraq
Mohanad Aljanabi
Department of Mechanical Engineering, the Faculty of Engineering, University of Kufa, Najaf,
Iraq
ABSTRACT
Stepped or non-prismatic beams are widely used in many engineering applications
and the calculation of their natural frequency is one of the most important problem.
Several methods were used to calculate the natural frequency of stepped beam. In this
work, the Rayleigh methods (Classical and Modified) and finite element method using
ANSYS software were used for calculating the natural frequency of Hollow stepped
cantilever beam with circular and square cross section area. The comparison between
the results of natural frequency and frequency ratio due to the increasing the length of
the small part for these types of beams and for these three methods were made. The
agreement between ANSYS the classical Rayleigh results was better than the agreement
between ANSYS the Modified Rayleigh results for the two types of cross section area. The
maximum error of MRM was greater than that of CRM and the maximum error of circular
C.S.A. Was greater than that of square C.S.A. At the same dimensions. The natural
frequency of circular C.S.A was smaller than that of square C.S.A. for the same
dimensions
Keyword head: Classical Rayleigh method, Modified Rayleigh method, Finite element
method, ANSYS workbench, Hollow beam, Stepped beam, Frequency.
Cite this Article: Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad
Aljanabi, Calculating the Natural Frequency of Hollow Stepped Cantilever Beam,
International Journal of Mechanical Engineering and Technology, 10(01), 2019, pp.898–
914
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Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi
1. INTRODUCTION
Beams with variable cross-section are widely used in many engineering fields like mechanical
engineering, aeronautical engineering, and civil engineering. There are many examples of
structures that can be modeled as beam-like elements, such as robot arms, crane booms rotor
shafts, columns, and steel composite floor slabs in the single direction loading case.
The free vibrations for stepped or tapered beams (i.e. Non-uniform or Non-Prismatic beam)
has been conducted using various approaches, such as the finite element, transfer-matrix,
Adomian decomposition and other approximate methods [1–13]. Zhou and Cheung [1, 3] and Lu
et al. [12] used the Rayleigh–Ritz method for solving the bending vibrations of tapered beams
and multiple-stepped composite beams respectively.
For stepped beam, the free vibration analysis was studied by Jang and Bert [14, 15],
Naguleswaran [16, 17], Ju et al. [18] and Dong et al. [19] with different boundary conditions.
These papers and additional papers reviewed by Luay AL-Ansari [20] and Xinwei Wang and
Yongliang Wang [21] were dealt with solid stepped beam and/or solid tapered beam.
As theoretical analysis for hollow-sectional beams is more complicated than a solid beam and
these beams are widely used in mechanical engineering and civil engineering. Generally, few
researches dealing with natural frequency of hollow-sectional beams were conducted. Some
researchers (like Murigendrappa et al. [22], Zheng and Fan [23], Naniwadekar et al. [24], and
Peng Liping and Liu Chusheng [25]) focused on vibration of hollow-sectional beams with crack.
Therefore, this paper will focus on calculating the natural frequency hollow-sectional cantilever
beam with internal steps and the circular and square cross section area are used. Three numerical
methods were used and these methods were Classical Rayleigh method, Modified Rayleigh
method and Finite Element Methods using ANSYS-Workbench (17.2).
2. ROBLEM DESCRIPTION
The hollow cantilever beam with internal steps is shown in Fig. (1). The equation of motion of
beam (i.e. Euler-Bernoulli and Timoshenko equations) cannot be solved analytically in this case
because of varying in dimensions (i.e. area and Second Moment of Inertia) along the length of
beam. Several researches were done for deriving new equation of motion described the variation
in dimensions and /or solving it analytically.
For calculating the fundamental natural frequency of the type of beam (or tube) , classical
Rayleigh method (CRM), modified Rayleigh method (MRM) and the finite element method
(ANSYS software) are used in this work in order to avoid the complexity in governing equation
and its solution [20,26,27].
Figure 1. Geometry of hollow beam with internal steps used in this work
2.1. Rayleigh method (RM)
The fundamental natural frequency of the system. The general formula of Rayleigh method was
derived according to equate the potential and kinetic energy of any system and the fundamental
natural frequency of this system can be estimated by the following equation [20, 26, 27].
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Calculating the Natural Frequency of Hollow Stepped Cantilever Beam
( )
=
=∑
( ( ))
∑
(1)
( )
Where:
(ω) is frequency (rad/sec), (y) is Deflection (m) , (M) mass (kg) , (A) is Cross Section Area
(m2) , (ρ) is Density (kg/m3) , (E) is Modulus of Elasticity (N/m2) and (I) is Second Moment of
Inertia (m4).
As mentioned previously, the main problem of the vibration of stepped beam is the varying
of the dimensions along the beam which leads to change in cross section area and second moment
of inertia. Therefore, the methods described in references [20], [26] and [27] are used in order to
calculate the equivalent second moment of inertia and these methods are:
1. Classical Method:
The equivalent second moment of inertia for stepped beam with two internal steps can be
found using the following equation [20, 26, 27]:
!"# =
($% &' )(
(
(
(2)
(
*+ *+ - 0*+ ) ., / + . , 1
,
+
Where (LTotal) is the length of the beam, (LS) is the length of the beam, when the hollow width
or diameter is (WS), calculating from free end and (LL) is the length of the beam, when the hollow
width or diameter is (WL), calculating from free end and in this case equals (X T or
LTotal).Numerical procedure.
Modified Method:
According to the idea described in [20] and [27], the equivalent moment of inertia at any point
in the stepped beam can be calculated by applying the following:
I45 (x) =
(789:;< )(
(
(
(3)
(
*= (?)*= - 0*=> (?)/ =
) >@
1
@
>
=
2.3. Programming Rayleigh methods
The Rayleigh Methods (i.e. Classical Rayleigh Method (CRM) and Modified Rayleigh Method
(MRM)) were programing using MATLAB code [20,25,26]. The general steps are:
1-Input the material properties (i.e. density and modulus of elasticity) and beam dimensions
(see Fig. (1)).
2-Input number of divisions (N) and in this work N=8400 (i.e. the DX=0.1 mm).
3-Calculate the equivalent second moment of inertia according to the method (i.e. CRM or
MRM).
4-Calculate the mass matrix [m] (N+1).
5-Calculate the delta matrix [δ] ((N+1)* (N+1)) using Table (1)
6-Calculate the deflection at each node using the following equation and apply the boundary
conditions.
[y] (N+1) = [δ] ((N+1)* (N+1)) [m] (N+1)
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Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi
Table 1 Formula of the deflections of the cantilever beam [20,25,26]
EFG (HI − F)
BCD =
KLM
EIH
BDD =
HLM
NOP =
QRG (HS − R)
KTU
3. FINITE ELEMENT METHOD (FEM)
In order to build 3D finite element model that shown in Fig. (2), ANSYS – Workbench (17.2)
was used. Cantilever hollow beams with circular and square cross section were used in this work
(see Fig. (2)). generally the number of Tetrahedrons elements was about (40,000) and the size of
element was (2 mm) (see Fig. (3)).
a. Square hollow beam.
b. Circular hollow beam.
Figure 2. Samples of beam geometry built in ANSYS – workbench software
Figure 3. Samples of mesh and result in ANSYS – Workbench software
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4. RESULTS AND DISCUSSION
Generally, the length of beam used in this work is (0.84) m and the outer width (or diameter) is
(0.04) m. Seven values of width (or diameter) for large part (i.e. WL or DL) (WL=0.01, 0.015,
0.02, 0.025, 0.03 and 0.035 m) were used and in the same time the hollow width (or diameter)
for small part (i.e. WS or DS) changed from (WL or DL) to (0.04) m. The dimensions of hollow
stepped beams with square and circular cross section area, used in this work, can be summarized
in Table (2).
Table 2 Cases studied in this Work
NO
Length Beam
(m)
Length of
Large Part (m)
Length of
Small Part (m)
1
0.84
0
2
3
0.72
0.6
0.12
0.24
0.48
0.36
0.36
0.48
6
7
0.24
0.12
0.6
0.72
8
9
10
0
0.84
0.84
0
0.72
0.6
0.12
0.24
0.48
0.36
0.36
0.48
14
15
0.24
0.12
0.6
0.72
16
17
0
0.84
0.84
0
18
19
0.72
0.6
0.12
0.24
0.48
0.36
0.36
0.48
22
23
0.24
0.12
0.6
0.72
24
25
0
0.84
0.84
0
26
27
0.72
0.6
0.12
0.24
0.48
0.36
0.36
0.48
31
0.24
0.12
0.6
0.72
32
33
0
0.84
0.84
0
34
35
0.72
0.6
0.12
0.24
36
37
0.48
0.36
0.36
0.48
0.24
0.6
4
5
0.84
11
12
13
20
21
28
29
30
38
0.84
0.84
0.84
0.84
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Width
(or Diameter)
of Large Part (m)
Width
(or
Diameter)
of Small
0.01
From
(0.015) to
(0.035)
0.015
From (0.02)
to (0.035)
0.02
From
(0.025) to
(0.035)
0.025
(0.03)
and(0.035)
0.03
0.035
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Luay S. Alansari, Hayder Z. Zainy, Aya Adnan Yaseen and Mohanad Aljanabi
39
0.12
0.72
40
0
0.84
Fig. (4) illustrates the comparisons among natural frequencies of Hollow stepped cantilever
beam calculated by classical Rayleigh method (CRM), modified Rayleigh method (MRM) and
the finite element method (ANSYS software) , in additional to the comparison between circular
and square cross section area when the inner width ( or diameter) of larger part is (0.01 m). When
the length of small part is zero, the dimensions of cross section area are Wout (or Dout) = 0.04 m
and Win=WL (or Din=DL) and when the length of small part is (L), the dimensions of cross section
area are Wout (or Dout) = 0.04 m and Win=WS (or Din=DS). These points are found in each curve
and represented the start and end points. When the length of the small part ( i.e. small cross section
area) increases, the natural frequency increases and the natural frequency reaches to its maximum
value when (XS=0.36 m) and then the natural frequency decreases. In the other hand, the natural
frequency increases when the width (or diameter) of the small part increases and width (or
diameter) of the large part is (0.01m). Also, the natural frequencies of square C. S. A. is larger
than that of circular C. S. A. These points can be explain by considering two important parameters
(mass and second moment of Inertia). When the length of small part increases, the mass of beam
decreases and the equivalent second moment of Inertia decreases too and the decreasing rate of
equivalent second moment of Inertia is larger than that of mass ( the equivalent second moment
of Inertia depends on (length of small part)**4) while the mass depends on (length of small part)
only). In the comparisons among the three calculating methods, the ANSYS results are
considered as exact results. For circular C.S.A. , the error of CRM results comparing with ANSYS
results increases when the length of small part increases and the maximum error is found
when(XS=0.36 m). Also, the maximum error increases when the diameter of the small part
increases. In MRM method, the error, also, increases when the length and diameter of small
part increase. Generally the maximum error of MRM is greater than that of CRM and the
maximum error of circular C.S.A. Is greater than that of square C.S.A. At the same dimensions
(see Table (3)). Also, the maximum error increases when the width (or diameter) of small part
increases.
In Figures (5) - (8), the comparisons among natural frequencies of Hollow stepped cantilever
beam calculated by three calculating methods, for circular and square cross section area when
the inner width ( or diameter) of larger part is (0.015, 0.02 , 0.025 and 0.03) m respectively.
Generally, the same behavior can be noted in these figures but with increasing the values of
natural frequencies when the width (or diameter) of small and large parts increase.
For circular and square C.S.A., the variation of frequency ratio (ω/ ωS) [ where ω is the frequency
of beam with any dimensions and ωS is the frequency for beam with dimensions of small part]
are drawn and the comparison among the frequency ratios calculating by ANSYS , CRM and
MRM are illustrated in Figures (9) - (12). From these Figures, the frequency ratio changes along
the dimensionless length of small part with the same way and the maximum frequency ratio of
circular C.S.A equals to that of square C.S.A. for the same calculating method. Also, the
maximum frequency ratio increases when the width (or diameter) of the small part increases and
when the width (or diameter) of the large part increases.
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Hollow circular beam
Hollow square beam
DS =0.015 m
WS =0.015 m
DS =0.02 m
WS =0.02 m
DS=0.025 m
WS=0.025 m
DS =0.03 m
WS =0.03 m
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DS =0.035 m
WS =0.035 m
Figure 4. Comparison among natural frequencies of hollow beams with circular and square cross
section area due to change in length of the small step (xs) for different calculating method and different
values of small width (or diameter) (WS or DS) when the large width (or diameter) (WL or DL) is (0.01)
m.
Hollow circular beam
Hollow square beam
DS =0.02 m
WS =0.02 m
DS=0.025 m
WS=0.025 m
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DS =0.03 m
WS =0.03 m
DS =0.035 m
WS =0.035 m
Figure 5. Comparison among natural frequencies of hollow beams with circular
and square cross section area due to change in length of the small step (Xs) for
different calculating method and different values of small width (or diameter)
(WS or DS) when the large width (or diameter) (WL or DL) is (0.015) m
Hollow circular beam
Hollow square beam
Ds=0.025 m
WS=0.025 m
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DS =0.03 m
WS =0.03 m
Ds =0.035 m
Ws =0.035 m
Figure 6. Comparison among natural frequencies of hollow beams with circular and square
cross section area due to change in length of the small step (XS) for different calculating
method and different values of small width (or diameter) (WS or DS) When the large width
(or diameter) (WL or DL) is (0.02) m
Hollow circular beam
Hollow square beam
DS =0.03 m
WS =0.03 m
DS =0.035 m.
WS =0.035 m.
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Figure 7. Comparison among natural frequencies of hollow beams with circular and
square cross section area due to change in length of the small step (XS) for different
calculating method and different values of small width (or diameter) (WS or DS) when
the large width (or diameter) (WL or DL) is (0.025) m
Hollow circular beam
Hollow square beam
DS =0.035 m
WS =0.035 m
Figure 8. Comparison among natural frequencies of hollow beams with circular
and square cross section area due to change in length of the small step (Xs) for
different calculating method and different values of small width (or diameter)
(Ws or Ds) When the large width (or diameter) (WL or DL) is (0.03) m
Table 3 The maximum error between ANSYS results and classical and modified Rayleigh method
Width
(or
Diameter)
of Large
Part (m)
0.01
0.015
0.02
Width
(or
Diameter
)
of Small
Part (m)
Circle C. S. A.
Classical R.M
Modified R.M
Maximum error %
Square C. S. A.
Classical R.M
Modified R.M
Maximum error %
0.015
0.02
0.025
0.03
0.035
0.015
2.690
2.731
2.774
2.832
2.922
2.690
2.773
2.939
3.296
4.092
6.323
2.773
0.414
0.479
0.552
0.647
0.750
0.414
0.445
0.627
1.007
1.822
4.072
0.445
0.02
2.700
2.890
0.427
0.589
0.025
0.03
0.035
2.774
2.832
2.922
3.249
4.023
6.171
0.552
0.647
0.750
0.972
1.768
3.957
0.025
0.03
0.035
2.774
2.832
2.922
3.107
3.851
5.906
0.552
0.647
0.750
0.830
1.599
3.688
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0.025
0.03
0.03
2.832
3.485
0.647
1.243
0.035
2.922
5.393
0.750
3.185
0.035
2.922
4.466
0.750
2.268
Hollow circular beam
Hollow square beam
ANSYS
ANSYS
CRM
CRM
MRM
MRM
Figure 9. Comparison among frequency ratio of hollow beams with circular and square cross
section area due to change in dimensionless (XS) for different calculating method and different
values of large width (or diameter) (WL or DL) when the small width (or diameter) (Ws or Ds)is
(0.02) m
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Hollow circular beam
Hollow square beam
ANSYS
ANSYS
CRM
CRM
MRM
MRM
Figure 10. Comparison among frequency ratio of hollow beams with circular and
square cross section area due to change in dimensionless (Xs) for different calculating
method and different values of large width (or diameter) (WL or DL) when the small
width (or diameter) (Ws or Ds)is (0.025) m
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Hollow circular beam
Hollow square beam
ANSYS
ANSYS
CRM
CRM
MRM
MRM
Figure 10. Comparison among frequency ratio of hollow beams with circular and
square cross section area due to change in dimensionless (Xs) for different calculating
method and different values of large width (or diameter) (WL or DL) when the small
width (or diameter) (Ws or Ds)is (0.025) m
Hollow circular beam
Hollow square beam
ANSYS
ANSYS
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CRM
CRM
MRM
MRM
Figure 11. Comparison among frequency ratio of hollow beams with circular and
square cross section area due to change in dimensionless (Xs) for different calculating
method and different values of large width (or diameter) (WL or DL) when the small
width (or diameter) (WS or DS) is (0.03) m.
Hollow circular beam
Hollow square beam
ANSYS
ANSYS
CRM
CRM
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MRM
MRM
Figure 12. Comparison among frequency ratio of hollow beams with circular
and square cross section area due to change in dimensionless (XS) for different
calculating method and different values of large width (or diameter) (WL or DL)
when the small width (or diameter) (WS or DS)is (0.035) m
4. CONCLUSION
From the previous results, the following point can be concluded:
• Generally the maximum error of MRM is greater than that of CRM and the maximum error
of circular C.S.A. Is greater than that of square C.S.A. At the same dimensions.
• When the length of the small part increases, the natural frequency increases and the natural
frequency reaches to its maximum value when (XS=0.36 m) and then the natural frequency
decreases.
• The natural frequency increases when the width (or diameter) of the small part increases for
the same width (or diameter) of the large part.
• In hollow stepped cantilever beam, the CRM is better than the MRM for calculating the
natural frequency. While the MRM is better than CRM in stepped cantilever beam (see Ref. [20]).
• The natural frequency of circular C.S.A is smaller than that of square C.S.A. for the same
dimensions.
Finally, the modified Rayleigh method can be used for calculating the natural frequency for
hollow stepped cantilever beam (with number of step larger than two) , non-prismatic beam and
beam with different cross section area .
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