Analytical and Numerical Study for Mass Loading Effect on Natural Frequencies of Aluminum Alloy Propped Cantilever Beam

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Analytical and Numerical Study for Mass Loading Effect on Natural Frequencies of Aluminum Alloy Propped Cantilever Beam

Ayush Kumara, Anuj Yadavb and Hirendra Singhc

a,cDepartment of Mechanical Engineering, Naraina College of Engineering and Technology, Kanpur – 208020, India

bDepartment of Mechanical Engineering, National Institute of Technology, Patna – 800005, India

Abstract A lot of attention has been received to determine the natural frequencies of a continuous system. Engineers are more interested in evaluating the dynamic behavior of a beam since it is one of the most essential continuous structures. The objective of the article is to study the impact of mass loading on the natural frequency of aluminum alloy propped cantilever beam. Analytical and numerical methods are used to model the dynamic behavior of the beam mass system. An Euler-Bernoulli beam of fixed- hinged end conditions carrying a point mass at an arbitrary position has been considered for analytical studies. The non- dimensional frequency equation for beam fixed-hinged end conditions is obtained by satisfying the free vibration equations of motion and by applying the corresponding end and compatibility conditions. This boundary-value problem is solved by a dual frame of reference. Simulation of the problem is done by using ANSYS and the validation of the proposed analytical method is demonstrated utilizing simulated data. It has been observed that the natural frequencies of the beam are influenced by the mass position and its magnitude.

Keywords ANSYS; Propped cantilever beam; MATLAB; Eigenfrequency

  1. INTRODUCTION

    Free vibration of combined structures is a popular issue in today's world. This issue affects a wide range of engineering design disciplines. The most of the research in this field focuses on the discovery and application of methods for determining the combined system's natural frequencies. Rao [1], the basic governing equation for free vibration for simple structure like Euler-Bernoulli beam is available in reference book. Low [2], this study presents a comparative review of the Eigen-frequency analysis for an Euler-Bernoulli beam with a point mass at any position. Mermertas and Erol [3] examined the Eigenfrequency equation of a cracked cantilever beam with attached point mass. Theoretical investigation of the cracked beam was carried out. The governing equation of the cracked beam for free vibrations is developed from the elements of the Euler-Bernoulli beam. Naguleswaran [4], for the free vibrations of a beam carrying two point masses, an empirical approach based on the classical beam eigenvalue method is presented in this article. Öz and Özkaya [5] presented the Euler-Bernoulli beam with masses at various positions is considered in this research. For various boundary conditions, natural frequencies for free transverse

    frequency of a beam in a free-free end conditions was studied in this paper. Prashant et al. [7], experimental modal analysis of cantilever beam was carried out to obtain natural frequencies, modal damping, and mode shapes. Yadav and Singh [8, 9] investigated the mass attachment impact on vibrational frequencies of cantilever and simply- supported magnesium alloy beams. Caker and Sanliturk

    [10] and Ren et al. [11] presented a new approach for removing mass loading effects of transducers from calculated FRFs, based on the ShermanMorrison concept. Any physical problem can be solved with help of exact and approximate methods but most of the problems are solved by an approximate method due to its complexity. In this paper, the modal analysis of the beam-mass system has been examined by approximate analytical and numerical methods.

  2. ANALYTICAL SOLUTION

    An Euler-Bernoulli beam of length (L) 240 mm, fixed- hinged end conditions with attaching a point mass (m) of 23 gm at an arbitrary position have been considered for analytical studies. It is an aluminum alloy beam having a dimension of 240×12×6 mm and beam's material properties like as Young's modulus 71GPa and density is 2770 Kg/m3. A dual frame of reference has been used to solve this beam-mass system (Fig. 1) problem. By satisfying the free vibration equations of motion and applying the related end and compatibility conditions, the non-dimensional frequency equation for beam fixed-hinged end conditions has been obtained.

    Fig. 1. Propped cantilever beam with point mass

    The equation of motion for free vibration of a uniform beam is given as

    4 2

    4 2

    + = 0 (1)

    vibrations are studied. Kotambkar [6], the impact of mass

    4

    2

    loading caused by an accelerometer on the natural

    The technique of separation of variable has been used to find the free-vibration solution.

    (, ) = (). () (2)

    The characteristic function () of the beam is defined as

    ()2

    + () 2

    + () {(

    )1 + ( )1 } = 0 (16)

    () = 1 cosh + 1 sinh + 1 cos + 1 sin

    Let us assume

    Where, 4 = (2 )

    Where is the natural frequency.

    = 2

    (3)

    (4)

    Dimensionless mass location parameter () =

    1 =

    2 =

    3 = +

    The dual frames of references have been taken to solve the problem.

    4 = +

    ( ) = cosh + sinh + cos +

    1 = (1 )

    1 1 1 1 1 1 1 1

    1 sin 1 (5)

    2(2) = 2 cosh 2 + 2 sinh 2 + 2 cos 2 +

    2 sin 2 (6)

    1

    1

    The fixed end coordinate of the beam (1 = 0 =0) and at hinged end coordinate (2 = 0 =) , the fixed end transverse displacement 1 (, ) satisfies boundary conditions (1 = 0 and = 0), 1(, ) can be expressed as

    1(1) = (1 1)1 + (1

    1)1 (7)

    2 = (1 )

    3 = (1 )

    4 = (1 )

    =

    The resulting dimensionless frequency equation in determinant form has been obtained by substituting the equation number (10), (12), (14), and (16).

    The hinged end transverse displacement 2 (, ) satisfies

    boundary conditions (2 = 0 and = 0), 2(, ) can be

    1 2

    4 1

    2 4

    1 3

    2

    expressed as

    | 3 4

    2 + 1. 3 + 2.

    2 4

    1 3

    | = 0 (17)

    2(2) = (2)2 (2)2 (8) For further solution, four additional compatibility equations have been used

    Transverse displacement equation at 1 = or 2 =

    1(1) 1= = 2(2) 2= (9)

    ( )1 + ( )1 + ()2 + ()2 = 0 (10) Slope equation at 1 = or 2 =

    The values of L for different vibrational modes have been found out by solving the equation (17) in MATLAB software. It is observed that the values of L in mode1, 2, and 3 change with respect to the point mass attachment locations. The natural frequencies of the beam-mass system in different modes are obtained by Eq. (4). The changes in the value of L of different modes with respect to the dimensionless mass location parameter () have been represented in Table 1, Fig. 2, Fig. 3, and Fig. 4.

    SI. No.

    Mode I

    Mode II

    Mode III

    1

    0.1

    3.9179

    6.9296

    9.5235

    0.2

    3.8266

    6.2564

    9.0369

    3

    0.3

    3.6220

    6.0833

    9.7016

    4

    0.4

    3.4188

    6.4076

    10.1869

    5

    0.5

    3.2925

    6.9264

    9.3976

    6

    0.6

    3.2599

    6.9929

    9.5153

    7

    0.7

    3.3302

    6.5543

    10.2035

    8

    0.8

    3.5159

    6.2467

    9.5917

    9

    0.9

    3.7846

    6.4669

    9.1929

    10

    1

    3.9266

    7.0685

    10.2102

    SI. No.

    Mode I

    Mode II

    Mode III

    1

    0.1

    3.9179

    6.9296

    9.5235

    2

    0.2

    3.8266

    6.2564

    9.0369

    3

    0.3

    3.6220

    6.0833

    9.7016

    4

    0.4

    3.4188

    6.4076

    10.1869

    5

    0.5

    3.2925

    6.9264

    9.3976

    6

    0.6

    3.2599

    6.9929

    9.5153

    7

    0.7

    3.3302

    6.5543

    10.2035

    8

    0.8

    3.5159

    6.2467

    9.5917

    9

    0.9

    3.7846

    6.4669

    9.1929

    10

    1

    3.9266

    7.0685

    10.2102

    Table 1. Changes in the value of L of different modes with respect to .

    ( )

    = ( )

    (11)

    1 1 1= 2 2 2=

    ( + )1 + ( )1 ()2 ()2 = 0 (12) Moment equation at 1 = or 2 =

    (1) = = (2) = (13)

    1 1 2 2

    ( + )1 + ( + )1 + ()2 ()2 = 0 (14) Shear force at 1 = or 2 =

    (1) = + (2) = + . 1(1) = = 0

    1 1 2 2

    1

    (15)

    ( )1 + ( + )1

    6

    4

    2

    0

    6

    4

    2

    0

    0 0.2 0.4

    0.6

    0.8

    1

    1.2

    0 0.2 0.4

    0.6

    0.8

    1

    1.2

    L

    L

    L

    L

    Fig. 2. Impacts of mass attachment on the natural frequency of the first mode.

    7.5

    7

    6.5

    6

    7.5

    7

    6.5

    6

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    L

    L

    Fig. 3. Impacts of mass attachment on the natural frequency of the second mode.

    10.5

    10

    9.5

    9

    8.5

    10.5

    10

    9.5

    9

    8.5

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    0

    0.2

    0.4

    0.6

    0.8

    1

    1.2

    Fig. 4. Impacts of mass attachment on the natural frequency of the third mode.

  3. NUMERICAL SOLUTION

    Modal analysis module of ANSYS workbench software has been used for numerical modal analysis of a beam- mass system. A prismatic beam model (Fig. 5) of dimension 240×12×6 mm has been developed by the design modular of ANSYS. The appropriate boundary conditions are imposed to make the beam model propped cantilever. A point mass of 23 gm is attached at the desired location on the beam. The major assumption of finite element method is to discretized continuous structure in finite number of elements. Now the model has been discretized (Fig. 6) with Hex20 element, the total elements and nodes number are 10000 and 49581 respectively. Finally, the natural frequencies (Hz) of a propped cantilever beam with an attached point mass at multiple desirable locations are obtained in different bending modes. The results have obtained from an analytical and numerical approach, which has been compared in Table 2, 3 and 4.

    Fig. 5. Beam model

    Fig. 6. Mesh model

    Fig. 7. Mode Shape

    Table 2. Vibrational frequencies of a propped cantilever beam-mass system for Mode I.

    SI. No.

    Mode I

    ANSYS

    MATLAB

    1

    0.1

    371.94

    371.92

    2

    0.2

    354.74

    354.78

    3

    0.3

    317.87

    317.86

    4

    0.4

    283.26

    283.19

    5

    0.5

    262.69

    262.66

    6

    0.6

    257.48

    257.48

    7

    0.7

    268.61

    268.71

    8

    0.8

    299.28

    299.51

    9

    0.9

    346.62

    347.04

    10

    1

    373.19

    373.57

    Table 3. Vibrational frequencies of a propped cantilever beam-mass system for Mode II.

    SI. No.

    Mode II

    ANSYS

    MATLAB

    1

    0.1

    1154.00

    1163.48

    2

    0.2

    939.58

    948.40

    3

    0.3

    890.08

    896.64

    4

    0.4

    987.22

    994.79

    5

    0.5

    1152.40

    1162.41

    6

    0.6

    1175.80

    1184.83

    7

    0.7

    1034.40

    1040.87

    8

    0.8

    939.55

    945.46

    9

    0.9

    1003.30

    1013.29

    10

    1

    1200.00

    1210.59

    Table 4. Vibrational frequencies of a propped cantilever beam-mass system for Mode III.

    SI. No.

    Mode III

    ANSYS

    MATLAB

    1

    0.1

    2112.90

    2197.53

    2

    0.2

    1937.60

    1978.71

    3

    0.3

    2235.90

    2280.49

    4

    0.4

    2459.20

    2514.36

    5

    0.5

    2099.80

    2139.82

    6

    0.6

    2147.40

    2193.75

    7

    0.7

    2464.40

    2522.56

    <>8

    0.8

    2189.20

    2229.12

    9

    0.9

    2001.10

    2047.61

    10

    1

    2326.10

    2525.87

  4. CONCLUSIONS

The effects of a point mass on the frequencies of a propped cantilever beam are investigated. For the first three fundamental frequencies, a single point mass is placed at ten evenly spaced different positions on the beam, and its influence on frequencies is plotted. It is found that the vibrational frequency of this beam is influenced by the point mass location and its magnitude. The addition of point mass reduces the natural frequencies of this beam but, if the mass is placed at a node of the modes then its frequencies do not change. The node positions for Mode II and Mode III are = 0.5 and = 0.333, 0.667 respectively. The simulated results of the numerical approach are similar to analytical results.

REFERENCES

  1. S.S Rao, Mechanical Vibration, Pearson Education, Inc., publishing as Prentice Hall, 2011.

  2. K. H. LowA comparative study of the eigenvalue solutions for mass-loaded beams under classical boundary conditions, International Journal of Mechanical Sciences, Vol. 43, 2001, p. 237- 244.

  3. V. Mermertas, H. ErolEffect of mass attachment on the free vibration of cracked beam, 8th International Congress on Sound and Vibration, Hong Kong, China, 2001, p. 2803-2810.

  4. S. Naguleswaran,Transverse vibrations of an EulerBernoulli uniform beam carrying two particles in-span, International Journal of Mechanical Sciences, Vol. 43, 2001, p. 2737-2752.

  5. H. R. Öz, E. Özkaya,Natural frequencies of beam-mass systems in the transverse motion for different end conditions, Mathematical and Computational Application, Vol. 10, Issue 3, 2005, p. 367-369.

  6. M. S. Kotambkar,Effect of mass attachment on natural frequency of free-free beam, International Journal of Advanced Engineering Research and Studies, 2014, p. 102-105.

  7. S. W. Prashant, V. N. Chougule, A.C. Mitra,Investigation on modal parameters of rectangular cantilever beam using experimental modal analysis, Materials Today Processdings, Vol. 2, Issue 4-5, 2015, p. 2121-2130.

  8. A. Yadav, N. K. Singh,Effects of accelerometer mass on natural frequency of a magnesium alloy cantilever beam, Vibroengineering Procedia, Vol. 29, p. 207-212.

  9. A. Yadav, N. K. Singh,Investigation for Accelerometer Mass Effects on Natural Frequency of Magnesium Alloy Simply Supported Beam, Materials Today Proceedings, Vol. 28, 2020, p. 2561-2565.

  10. O.Cakar, K. Y. Sanliturk,Elimination of transducer mass loading effects from frequency response functions, Mechanical Systems and Signal Processing, Vol. 19, 2005, p. 87-104.

  11. Jun Ren, Jun Wang, Shusheng Bi,Correction of Transducers Mass Effects from the Measured FRFs in Hammer Impact Testing, Shock and Vibration, Vol. 2017, 2017, p. 1-10.

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