calculating the natural frequency of hollow...

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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

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&Type=01

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].

Calculating the Natural Frequency of Hollow Stepped Cantilever Beam


�� = � ��(�)��

� ��(�(�))�� = � ∑ ��∑ ��(��)�� (1)(1)(1)(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]:

!"# = ($%�&'�)(

)*+,-(., /*++-(0*+,-(

.+ 1 (2)(2)(2)(2)

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:;<)(

)*=>(?)-(@> /*==-(0*=>(?)-(

@= 1 (3)(3)(3)(3)

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)



Table 1 Formula of the deflections of the cantilever beam [20,25,26]

BCD = EFG(HI − F)KLM BDD = EIH

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



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)

Width

(or Diameter)

of Large Part (m)

Width

(or

Diameter)

of Small

1 0.84 0

2 0.72 0.12

3 0.6 0.24

4 0.48 0.36 From

(0.015) to

(0.035)

5 0.84 0.36 0.48 0.01

6 0.24 0.6

7 0.12 0.72

8 0 0.84

9 0.84 0

10 0.72 0.12

11 0.6 0.24

12 0.48 0.36 From (0.02)

to (0.035) 13 0.84 0.36 0.48 0.015

14 0.24 0.6

15 0.12 0.72

16 0 0.84

17 0.84 0

18 0.72 0.12

19 0.6 0.24

20 0.48 0.36 From

(0.025) to

(0.035)

21 0.84 0.36 0.48 0.02

22 0.24 0.6

23 0.12 0.72

24 0 0.84

25 0.84 0

26 0.72 0.12

27 0.6 0.24

28 0.48 0.36 (0.03)

and(0.035)

29 0.84 0.36 0.48 0.025

30 0.24 0.6

31 0.12 0.72

32 0 0.84

33 0.84 0

34 0.72 0.12

35 0.6 0.24

36 0.48 0.36

37 0.36 0.48

38 0.84 0.24 0.6 0.03 0.035



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.



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



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.


DS =0.02 m WS =0.02 m

DS=0.025 m WS=0.025 m



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


Ds=0.025 m WS=0.025 m



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


DS =0.03 m WS =0.03 m

DS =0.035 m. WS =0.035 m.



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


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)

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 %

0.01 0.015 2.690 2.773 0.414 0.445

0.02 2.731 2.939 0.479 0.627

0.025 2.774 3.296 0.552 1.007

0.03 2.832 4.092 0.647 1.822

0.035 2.922 6.323 0.750 4.072

0.015 2.690 2.773 0.414 0.445

0.015

0.02 2.700 2.890 0.427 0.589

0.025 2.774 3.249 0.552 0.972

0.03 2.832 4.023 0.647 1.768

0.035 2.922 6.171 0.750 3.957

0.02 0.025 2.774 3.107 0.552 0.830

0.03 2.832 3.851 0.647 1.599

0.035 2.922 5.906 0.750 3.688



0.025

0.03 2.832 3.485 0.647 1.243

0.035 2.922 5.393 0.750 3.185

0.03

0.035 2.922 4.466 0.750 2.268


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




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




ANSYS ANSYS

CRM CRM

MRM MRM




width (or diameter) (Ws or Ds)is (0.025) m Hollow circular beam Hollow square beam

ANSYS ANSYS



CRM CRM

MRM MRM




width (or diameter) (WS or DS) is (0.03) m. Hollow circular beam Hollow square beam

ANSYS ANSYS

CRM CRM



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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