Vibration Characteristics of Functionally Graded Micro-Beam Carrying an Attached Mass

Document Type: Research Paper

Authors

1 Faculty of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

2 Department of Mechanical Engineering, University of Kentucky, Lexington, KY, 40506, USA

Abstract

In this article, in reference to the modified couple stress theory and Euler-Bernoulli beam theory, the free lateral vibration response of a micro-beam carrying a moveable attached mass is investigated. This is a decent model for biological and biomedical applications beneficial to the early-stage diagnosis of diseases and malfunctions of human body organs and enzymes. The micro-cantilever beam is composed of functionally graded materials (FGMs). The material properties are supposed to show variations through-thickness of the beam in consonance to the power of law. Rayleigh-Ritz method is applied in order to explore the natural frequencies of the first three vibration modes. In order to manifest the accuracy of the proposed method, the results are established and juxtaposed with technical literature. Influences of the material length-scale parameter that captures the size-dependency, ratio of the mass of the beam to the mass of the attached mass and power index of the graded material consequent to the vibrational behavior of the system are contemplated. This technical research denotes the value of the material gradation besides to the inertia of an attached mass in the dynamic behavior of the bio-micro-systems. As a result, the adoption of suitable power index, mass ratio and position of the attached mass lead to the superior design of bio-micro-systems persuading early-stage diagnostics.

Keywords


Vibration Characteristics of Functionally Graded Micro-Beam Carrying an Attached Mass

A. Rahmania, A. Babaeib, S. Faroughia,*

a Faculty of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

b Department of Mechanical Engineering, University of Kentucky, Lexington, KY, 40506, USA

 

 

KEYWORDS

 

ABSTRACT

FG bio-micro-beam

Vibration analysis

Modified couple stress theory

Non-dimensional frequency

Biological and biomedical application of bio-micro-systems

In this article, in reference to the modified couple stress theory and Euler-Bernoulli beam theory, the free lateral vibration response of a micro-beam carrying a moveable attached mass is investigated. This is a decent model for biological and biomedical applications beneficial to the early-stage diagnosis of diseases and malfunctions of human body organs and enzymes. The micro-cantilever beam is composed of functionally graded materials (FGMs). The material properties are supposed to show variations through-thickness of the beam in consonance to the power of law. Rayleigh-Ritz method is applied in order to explore the natural frequencies of the first three vibration modes. In order to manifest the accuracy of the proposed method, the results are established and juxtaposed with technical literature. Influences of the material length-scale parameter that captures the size-dependency, ratio of the mass of the beam to the mass of the attached mass and power index of the graded material consequent to the vibrational behavior of the system are contemplated. This technical research denotes the value of the material gradation besides to the inertia of an attached mass in the dynamic behavior of the bio-micro-systems. As a result, the adoption of suitable power index, mass ratio and position of the attached mass lead to the superior design of bio-micro-systems persuading early-stage diagnostics.

 

1.     Introduction

Functionally graded materials (FGMs) can be interpreted as non-homogeneous graded composites consisted of a mixture of two different materials, usually a metal and a non-metal phase. They dominate the desired continuous variation of mechanical properties as a function of position, in accordance with a certain direction(s). These types of materials are constructed in order to exploit the specific benefits of both constituents. Recently, FGMs are widely applied in micro-structures such as thin films in the form of shape memory alloys, micro-electro-mechanical-systems (MEMS), nano-electro-mechanical-systems (NEMS) and atomic force microscopes (AFMs)[1].

By the compelling improvements contributing to micro-technology, consisted of concepts, modeling, design, and fabrication, a micro-cantilever beam that has the dimensions at the order of micron has been widely applied. The applications chiefly include atomic force microscopes (AFMs), micro and nano-electro-mechanical systems (MEMS, NEMS). Micro-electro-mechanical Systems mainly arranged by miniaturized electro-mechanical and mechanical elements. In the mentioned technology, the elements are being engendered applying the microfabrication techniques. Micro-sensors and micro-actuators are among the most beneficial and pragmatic elements of MEMS [2]. As the structure is scaled down, the size effect phenomena influence the dynamic responses directly. It is experimentally illustrated that the size-dependency of microstructures could not be captured by the classical elasticity theory, consequently non-classical elasticity theories emerge. The modified couple stress theory as one of these non-classical theories, contemplating the rotational characteristics that exert the rotational energy terms into the formulations of energy. By applying a single material length scale parameter, symmetric part of the curvature tensor, it becomes correlated to the couple stress tensor; further this correlation makes the modified couple stress theory as one of the most prevalent non-classical theories for investigation of the micro-structures. Several researchers have inspected the static and dynamic behavior of FGM beams and plates. Babaei and Yang [3]accomplished research related to vibration analysis of rotating rods with applications in MEMS-gyroscopes. They have proposed a novel modified coupled displacement fields optimizing the design of micro-systems in the industry of automotive. Babaei and Ahmadi [4] reported dynamic characteristics of a non-uniform and non-homogenous micro-model of a Timoshenko beam pursuant to the modified couple stress theory (MCST). For the first time, Babaei and Rahmani [5] considered the capricious model of the non-classical length-scale parameter. Moreover, variations impact over dynamic-vibration behavior is revealed as well. Babaei et al. [6] explored the thermal stress effect over the dynamic characteristics of MEMS. They deliberated various slenderness ratios in order to cover a wide range of applications. Babaei [7] proposed an optimized model of MEMS-gyroscope based on his novel modified coupled displacement kinematic field. He presents perceptible and chaotic patterns of the gyroscope that ameliorates the efficiency of navigation systems mostly applied in advanced cars and airplanes.  Chen et al. [8] proposed a new method for free vibration of generally laminated beams based on the state-space-based differential quadrature method. Xiang and Yang [9] inspected free and forced vibration of a laminated FG beam of variable thickness with thermal initial stresses. The effect of diverse boundary conditions was inquired and the beam was presumed to be subjected to one-dimensional steady heat conduction in the thickness direction before undergoing any dynamic deformation. Elastic behavior of FGM ultra-thin films is surveyed by Lü et al. [[10]. They proposed a generalized theory pondered the surface effects in order to capture the size-dependency of the structure’s responses. Amar et al. [11] examined the free vibration analysis of the Euler-Bernoulli beam applying the modified couple stress theory. Pradhan and Chakraverty [12] did research on free vibration analysis of Timoshenko and Euler beams. They used functionally graded materials for modeling their systems and applied the Rayleigh-Ritz method for solution procedure. Aghazadeh et al. [13]  investigated the free vibration and static behavior of small scale FG micro-beams using dissimilar beam theories and the modified couple stress theory. Additionally, they assumed the varying length scale parameter and its effects consequent to the tip deflection and frequency characteristics. Jafari et al. [14] inquired about the bending and vibration analysis of delaminated micro-beams regarding the modified couple stress theory. They utilized the presumed modes method for analysis. Ansari et al. [15] inspected on the free vibration behavior of FG micro-beams applying the strain gradient theory and the Timoshenko beam theory. They attempted to simulate a model including the thick micro-beams theories. The paper which reported the size-dependent behavior of the FG Euler-Bernoulli micro-cantilever beam, using the modified couples stress theory is accomplished by Asghari et al. [16]. Ansari et al. [17] carried out research concerning buckling and vibrations of micro-plates employing the strain gradient theory. Babaei et al. [18] proposed a micro-beam model considering variations of both temperature and material gradation based on modified couple stress and the Euler-Bernoulli beam theories. Shafiei et al. [19] conducted research about size-dependent vibration analysis of non-uniform FG micro-beam. They have assumed Euler-Bernoulli and Timoshenko beam theories for deriving the formulas. Ansari et al. [20] inquired the vibration of postbuckle piezoelectric micro-beams applying non-local theory. Farajpour et al. [21] investigated the nonlinear behavior of nano-tubes conveying nanofluid with both subcritical and supercritical regimes. They presented the model established based on strain gradient theory. Mohammadimehr et al. [22] reported Size-dependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Pre-stress Loading based on DQM. Arabghahestani and Karimian [23] developed the previous research with considerations of uniform liquid argon flow. Fang et al. [24] represented a size-dependent vibration analysis of rotating small-scaled beam. They exployed the modified couple stress theory in order to devisethe formulations along with smooth variations of the mechanical properties. Khaniki and Rajasekaran [25] carried out a research  regarding to vibrational behavior of bi-directional non-uniform beams based on MCST. Jia et al. [26] explored mechanical buckling behavior of small-scaled beams based on MCST under thermal and electrical loads. Their model is supposed to be functionally graded.

In this paper, a functionally graded micro-scale Euler-Bernoulli beam model is proposed for the size-dependent free vibration analysis. This model is presented as a biomedical and biological laboratory set-up to help diagnosis of diseases and detecting body and enzyme malfunctions. It is noteworthy to mention that the said attached mass (proof mass) represents blood or enzyme samples. In order to extend the research available in the literature, the present analysis includes investigation of the influences of inertia generated as a result of the oscillation of the attached mass and varying position of the mass consequent to frequencies of the micro-cantilever beam. The kinetic and strain energy are derived from employing the modified couple stress theory. Due to the usage of the Rayleigh-Ritz method, applying variational approaches is not required. The influences of the volume fraction profiles of the constituents and gradient index (power index) upon the natural frequencies of the micro-cantilever beam are reported.

2.     Functionally Graded Material

A functionally graded micro-beam is illustrated in Figure 1, where length is , width is  and thickness is  . The Cartesian coordinates are defined as  in correspondence to the length, width and thickness directions. The FG micro-beam is formed by Steel (a metal constituent) and Alumina (a non-metal constituent). It is postulated that the effective mechanical properties vary through the thickness direction ( ).

Pursuant to the rule of mixtures, the effective properties ( ) can be computed as follows:

 

(1)

where  and  represent the effective mechanical properties,  and  are the volume fractions of alumina and steel phases. The Volume fractions are constrained by the following equation:

 

(2)

It is postulated that the effective material properties of the FG micro-beam are numerically interpreted by a power-law. The volume fraction of the alumina is defined as Wakashima [46]:

 

(3)

where  is the power-law exponent, which computes the material variation contour in conformity with thickness direction.

Applying Eqs. (1), (2), and (3), the effective material properties of the FG micro-beam take the form below:

 

 

 

 

 

 

Fig 1. Schematic of the FGM beam.

 

(4)

Similarly, Young’s modulus, shear modulus and density of the micro-structure structure can be expressed as follows:

 

(5)

 

(6)

 

(7)

 

3.     Mathematical Formulation

3.1. Modified Couple Stress Theory

Pursuant to the modified couple stress theory developed by Yang et al. [47], the total strain energy of a loaded beam is obtained as follows:

 

(8)

In Eq. (8),  designates the Cauchy stress, tensor  is the Cauchy strain tensor,  represents the deviatoric part of the couple stress tensor and stands for the symmetric curvature tensor. The tensors  and are defined by Eqs. (9), (10).

  

(9)

 

(10)

 in Eq. (9) shows the displacement vector;  in Eq. (10) denotes00000000 the alternating tensor and comma stand for differentiation. Constitutive relations in reference to the Cauchy stress tensor and the deviatoric part of the couple stress tensor is to be written in the following form:

 

(11)

 

(12)

 

(13)

where  and  are classical Lame parameters. Moreover,  is the Poisson’s ratio,  is Young’s modulus and  is shear modulus.

 in Eq. (12) is the material length scale parameter. Employing this non-classical parameter captures the size-dependency of the structure.    

3.2. Euler-Bernoulli beam theory (EBT)

The general displacement components in a Cartesian coordinate, applying EBT can be expressed as below:

  

(14)

 

(15)

 

(16)

In consonance to Eqs. (14)-(16) the elements of , ,  and  are computed as follows:

  

(17)

 

(18)

 

(19)

 

(20)

Substitution of Eqs. (17)-(20) into the Eq. (8), shows the strain energy of the Euler-Bernoulli beam:

 

(21)

The kinetic energy of the Euler- Bernoulli beam carrying an attached mass can be expressed as follows:

  

(22)

4.     Solution Procedure

4.1.  Approximation Method

One of the most applied methods to solve the partial differential governing equations of continuous systems is approximate methods. Rayleigh-Ritz's approach as one of such methods is an extension of Rayleigh’s energy method that can be used only for computing the fundamental frequency; however, Rayleigh-Ritz method can be employed to find all the frequencies. According to this method for a conservative system, the maximum potential and kinetic energies are equal which is accompanied by the fact that natural modes execute harmonic motions. Consequently, with defining  as the frequency of the system, the maximum energy terms are as follows:

  

(23)

  

(24)

In which, the coefficients are:

 

(25)

 

(26)

 

(27)

By separating lateral displacement ( ), Rayleigh’s quotient (square of the frequency) ( ) easily can be defined as follows: (  is the amplitude).

 

(28)

4.2. Rayleigh-Ritz method

Rayleigh-Ritz method is applied in order to appraised frequencies of different modes of vibration. In this method, the actual frequency for each mode is smaller than the appraised one. The method consists of selecting a trial family of admissible functions satisfying the homogenous and geometric boundary conditions and constructing a linear combination for mode shape functions as follows:

 

(29)

where the are known functions and  are the unknown coefficients to be computed. For each mode, the mode shape function is substituted in Rayleigh’s quotient so as to render the quotient stationary, which means that it is required to minimize the estimate. This minimization is done with respect to the unknown coefficients.

 

(30)

Manifesting some mathematical operations leads to the Galerkin’s equations:

 

(31)

where stiffness and mass matrices are defined as follows:

 

(32)

 

(33)

The trial functions satisfying the homogenous boundary conditions are supposed to be polynomial functions.

 

(34)

Exerting the above functions in the quotient and solving the frequency equation calculated from the Galerkin’s equations will result in the frequencies of the system.  

5.     Numerical Results and Discussion

The functionally graded micro-cantilever beam inspected in this paper is a mixture of steel and alumina (aluminium oxide), at which its properties vary through the thickness according to a power-law. The lower surface is pure metal, and the upper surface is pure alumina. The mechanical properties of the two constituents are expressed in Table 1. The beam length is micrometers and its width is micrometers.

Applying the following relation , where  is the second moment of inertia of the beam, and  is the area of the cross-section; the non-dimensional frequency is obtained. In accordance with experimental tests reported by Lame, the material length scale parameter ( ) is taken 15 micrometers. For verification, the results are compared with those of technical references.

In Table 2, by neglecting the material length scale parameter ( ), for the modified couple stress theory and assuming the mass ratio and power index, both equal to zero ( ), comparison with a technical report is done, leading to a good level of accuracy.

Pursuant to Table 2. It can be observed that current results are verified by technical literature. By accounting and neglecting the material length-scale parameter and putting the power index equal to zero, the results are verified. In Tables 3-5 non-dimensional frequency of a micro-cantilever Euler-Bernoulli beam, carrying an attached mass, and using the modified couple stress theory for dissimilar values of power indices are displayed.

The results are reported based on the varying mass ratio of the system (ratio of the mass of the attached mass to the mass of the beam), in which, numerical results for the first mode are portrayed in Table 3, the second mode in Table 4, and the third mode in Table 5. Regularly the change in the values of non-dimensional frequencies is expected by the growth of the power index. Tables 3, 4 and 5 reveal the fact that through increasing the power index, non-dimensional frequency decreases and this is observable for dissimilar values of power index ( ). The second point that is detectable from Tables 3, 4 and 5 is the influence of mass ratio. As the attached mass is heavier, the non-dimensional frequency decreases and this decrement becomes more intense in higher modes of vibration.

Table 1. Mechanical properties

Property

Steel

Alumina

 

7800

3960

(GPa)

210

390

Table 2. Comparison of the non-dimensional frequencies

R

Rao

MCST ( )

MCST ( )

0.01

3.4299

3.4477

3.6305

0.1

2.9687

2.9678

3.1252

1

1.5575

1.5573

1.6399

10

0.5417

0.5414

0.5701

100

0.1731

0.1730

0.1822

 


Table 3. Non-dimensional natural frequencies of a micro-cantilever beam. (First Mode)

                 

0

3.7025

3.4546

3.2715

2.9314

2.6628

2.4457

2.2419

2.1285

0.01

3.6305

3.3928

3.2165

2.8880

2.6276

2.4165

2.2174

2.1063

0.1

3.1252

2.9508

2.8187

2.5666

2.3618

2.1924

2.0276

1.9320

1

1.6399

1.5828

1.5379

1.4491

1.3743

1.3107

1.2419

1.1955

3

1.0139

0.9837

0.9599

0.9127

0.8732

0.8397

0.8021

0.7749

5

0.7971

0.7743

0.7563

0.7207

0.6910

0.6660

0.6374

0.6164

7.5

0.6558

0.6374

0.6230

0.5943

0.5705

0.5504

0.5274

0.5103

10

0.5701

0.5543

0.5419

0.5173

0.4968

0.4796

0.4599

0.4451

30

0.3317

0.3227

0.3157

0.3017

0.2901

0.2804

0.2692

0.2607

50

0.2573

0.2504

0.2450

0.2342

0.2253

0.2178

0.2091

0.2025

100

0.1822

0.1773

0.1735

0.1659

0.1596

0.1543

0.1482

0.1435

 

Table 4. Non-dimensional natural frequencies of a micro-cantilever beam. (Second Mode)

                 

0

23.2030

21.6499

20.5018

18.3709

16.6873

15.3271

14.0494

13.3393

0.01

22.7665

21.2738

20.1670

18.1058

16.4717

15.1478

13.8993

13.2021

0.1

20.3822

19.1507

18.2307

16.5018

15.1178

13.9868

12.9016

12.2811

1

17.1118

16.0291

15.2276

13.7378

12.5600

11.6081

10.7040

10.1897

3

16.5534

15.4703

14.6694

13.1827

12.0087

11.0607

10.1666

9.6647

5

16.4297

15.3455

14.5439

13.0561

11.8810

10.9322

10.0386

9.5389

7.5

16.3662

15.2813

14.4792

12.9905

11.8147

10.8651

9.9716

9.4728

10

16.3341

15.2487

14.4464

12.9571

11.7809

10.8308

9.9372

9.4390

30

16.2689

15.1826

14.3797

12.8892

11.7118

10.7607

9.8669

9.3695

50

16.2557

15.1692

14.3661

12.8754

11.6978

10.7464

9.8525

9.3554

100

16.2458

15.1592

14.3560

12.8651

11.6872

10.7356

9.8417

9.3447

Table 5. Non-dimensional natural frequencies of a micro-cantilever beam. (Third Mode)

                 

0

64.9690

60.6203

57.4058

51.4390

46.7250

42.9163

39.3389

37.3505

0.01

63.7820

59.5952

56.4918

50.7133

46.1332

42.4232

38.9253

36.9724

0.1

58.4624

54.7989

52.0761

46.9892

42.9419

39.6513

36.5167

34.7387

1

53.5948

50.0818

47.4841

42.6610

38.8512

35.7739

32.8721

31.2447

3

52.9576

49.4404

46.8403

42.0139

38.2018

35.1226

32.2264

30.6111

5

52.8224

49.3036

46.7024

41.8741

38.0602

34.9793

32.0831

30.4699

7.5

52.7538

49.2341

46.6323

41.8028

37.9878

34.9059

32.0095

30.3973

10

52.7192

49.1990

46.5969

41.7668

37.9512

34.8687

31.9722

30.3604

30

52.6496

49.1283

46.5255

41.6941

37.8771

34.7934

31.8965

30.2858

50

52.6356

49.1141

46.5111

41.6794

37.8622

34.7782

31.8812

30.2707

100

52.6250

49.1034

46.5003

41.6684

37.8510

34.7668

31.8697

30.2593

 


In Table 6, numerical results for the non-dimensional frequency of the FG micro-beam with respect to the varying position of the attached mass are depicted. Effect of the relative position of the attached mass and the mass ratio for the first, second and third modes of vibrations are presented.   refers to the position of the attached mass with respect to the beam length. The power index is supposed to be zero ( ).

Tables 7 and 8 exhibits the same results for  and .  pursuant to Tables 6, 7, and 8; regardless of the mass ratio, the frequency decreases as the mass reaches the free end of the beam. Furthermore, increment in mass ratio makes the system to vibrate with less frequency.

Figure 2 persuades the variation of non-dimensional frequencies of the first mode of vibration with respect to different mass ratios, in which three power index values are adopted ( ). The variations are such that besides the decrement of frequencies with increment in mass ratios, the gradient of the changes is gentle, means that until reaching the ratio , a sharp decrement can be observed, however, for larger ratios, the gradient is not as intense as before. Eventually, for ratios over 50, the decrement and change could be neglected; better means that the dynamical behavior of the system is independent of inertia of the attached mass. In addition, it can be readily observed that for other values of power index, the above inferences are derived.

The variations of non-dimensional frequencies of the first mode of vibration with respect to power index and for six different mass ratios are portrayed in Fig. 3. Based on this figure, increment in power index leads to decrement in the frequencies, when the mass ratio is smaller than 1 ( ), for the intervals , significant changes take place, while for , the dependency of the frequencies is going to diminish and smooth variations will be experienced. The effects of the mass ratios appear in the way that for ratios larger than one. The variation profile corresponds to a straight line, which means that if a heavy mass in comparison to the beam is attached, the system will vibrate regardless of the magnitude of the power index and the frequencies are such small that it does not matter whether power index is  or . So, the substantial effects of the mass ratios are again observed.

Figure 4 indicates the variations of the non-dimensional frequency with respect to the mass ratio, and assuming , it is accompanied by the consideration of four different positions of the attached mass as well.

 

Fig 2. Variation of the frequency with the mass ratio for different power indices


Table 6. Non-dimensional natural frequencies of a micro-cantilever beam with respect to relative attached mass position. (k=0)

 

Mode No.

Xam/L=0.25

Xam/L=0.5

Xam/L=0.75

Xam/L=1

0.01

 

3.7018

23.1223

64.2982

3.6940

22.9711

64.9686

3.6708

23.1946

64.5392

3.6305

22.7665

63.7820

0.1

 

3.6955

22.4114

59.2055

3.6195

21.2194

64.9649

3.4183

23.1299

61.3967

3.1252

20.3822

58.4624

1

 

3.6326

17.1691

43.6551

3.0446

14.9890

64.9524

2.2386

22.8754

52.2481

1.6399

17.1118

53.5948

3

 

3.4949

12.3594

39.3497

2.3508

12.4169

64.9477

1.4856

22.7620

49.4167

1.0139

16.5534

52.9576

5

 

3.3621

10.3784

38.3757

1.9793

11.6906

64.9464

1.1896

22.7293

48.7006

0.7971

16.4297

52.8224

7.5

 

3.2055

9.0824

37.8793

1.6936

11.2899

64.9457

0.9884

22.7111

48.3187

0.6558

16.3662

52.7538

10

 

3.0610

8.3278

37.6293

1.5038

11.0793

64.9453

0.8636

22.7015

48.1212

0.5701

16.3341

52.7192

30

 

2.2835

6.5903

37.1270

0.9156

10.6364

64.9445

0.5079

22.6810

47.7128

0.3317

16.2689

52.6496

50

 

1.8809

6.2255

37.0262

0.7172

10.5442

64.9444

0.3949

22.6767

47.6288

0.2573

16.2557

52.6356

100

 

1.3941

5.9591

36.9506

0.5115

10.4743

64.9443

0.2800

22.6734

47.5654

0.1822

16.2458

52.6250

 

 

Fig 3. Variation of the frequency with material graduation for different mass ratios

 

 

Fig 4. Variation of the frequency with mass ration for different relative positions  k = 0

Confirming the results are discussed in Fig. 2, it can be concluded that as the attached mass approaches the free end, the decrements of natural frequencies are sharper and more substantial-frequency drops are seen.

Consequently, it is concluded that the inertia influence of the attached mass for vibration diminishes, as the mass gets closer to the clamped end. This inference can be estimated from Fig. 5, too. This Figure reveals the non-dimensional changes with the location of the attached mass. It is clear that as the attached mass approaches the free end, the system vibrates with smaller frequencies and as the power index is assumed larger number, the effects of the attached mass and its relative position are going to be diminished.

 

 

Fig 5. Variation of the frequency with the attached mass location for different power indices

 

 

Table 7. Non-dimensional natural frequencies of a micro-cantilever beam with respect to relative attached mass position. (k=1)

 

Mode No.

Xam/L=0.25

Xam/L=0.5

Xam/L=0.75

Xam/L=1

0.01

 

2.6624

16.6482

46.3983

2.6586

16.5743

46.7247

2.6474

16.6832

46.5153

2.6276

16.4717

46.1332

0.1

 

2.6594

16.3011

43.7769

2.6222

15.6735

46.7228

2.5199

16.6503

44.8799

2.3618

15.1178

42.9419

1

 

2.6289

13.4439

33.1855

2.3181

11.6452

46.7146

1.8083

38.6579

16.4889

1.3743

12.5600

38.8512

3

 

2.5616

10.1257

29.1175

1.8894

9.4802

46.7106

1.2540

16.3947

36.1089

0.8732

12.0087

38.2018

5

 

2.4956

8.5457

28.1108

1.6305

8.7943

46.7094

1.0174

16.3641

35.4038

0.6910

11.8810

38.0602

7.5

 

2.4156

7.4384

27.5886

1.4182

8.3993

46.7087

0.8516

16.3463

35.0168

0.5705

11.8147

37.9878

10

 

2.3389

6.7605

27.3238

1.2713

8.1868

46.7083

0.7471

16.3366

34.8137

0.4968

11.7809

37.9512

30

 

1.8650

5.0582

26.7890

0.7918

7.7287

46.7075

0.4431

16.3156

34.3869

0.2901

11.7118

37.8771

50

 

1.5759

4.6677

26.6815

0.6235

7.6316

46.7073

0.3451

16.3111

34.2981

0.2253

11.6978

37.8622

100

 

1.1941

4.3778

26.6008

0.4464

7.5574

46.7072

0.2451

16.3076

34.2306

0.1596

11.6872

37.8510

 

6.     Conclusions

In the present study, free lateral vibration analysis of the bio-micro-FG beam that carries a movably attached mass is established. Non-classical constitutive terms are applied in order to strain density function. In the absence of the attached mass; the new model which is based on the modified couple stress theory, predicts the frequencies a bit smaller than the classical theory (for example for a classical model with , , meanwhile applying the non-classical effect frequency is 3.6305); meanwhile, the attached mass remarkably decreases the frequency of the system. In detail, when the attached mass reaches 10 times the mass of the system solely, the frequency decreases close to 80%. Furthermore ,through passing the mentioned value ( ), behavior of the entire system is almost independent and vibrations are on the wane regime. Results show that the material length scale parameter, power index and relative position of the attached mass play a major role in the dynamic behavior of FG bio-micro-structures and the presence of an attached mass demonstrates the structural damping effect in the system, exerting direct and explicit influence on vibrational behavior. This effect caused by the attached mass is dependent on the position of the mass along the length of the beam as well. Consequently, it can be realized that a decent and efficient biological laboratory set-up can be devised employed a proper power-law index, and a decent relative position of the attached mass. To put it simple, the material type of bio-sensor and location of the blood/enzyme sample are amongst the crucial factors through the analysis and application of such novel bio-micro-systems. Finally, it is essential to mention that using the proposed model, rapid disease/malfunction detection (diagnosis), is affordable. Addressing key factors of material profile variations and geometrical placement of the sample decent non-classical theorems will lead to eye-catching achievements in clinical diagnostics and biological sciences.

 

 

 

Table 8. Non-dimensional natural frequencies of a micro-cantilever beam with respect to relative attached mass position. (k=10)

 

Mode No.

Xam/L=0.25

Xam/L=0.5

Xam/L=0.75

Xam/L=1

0.01

 

2.1283

13.3146

37.1439

2.1259

13.2678

37.3503

2.1188

13.3367

37.2178

2.1063

13.2021

36.9724

0.1

 

2.1264

13.0950

35.4455

2.1029

12.6840

37.3491

2.0369

13.3155

36.1524

1.9320

12.2811

34.7387

1

 

2.1071

11.1985

27.4987

1.9031

9.7382

37.3431

1.5354

13.1988

31.4744

1.1955

10.1897

31.2447

3

 

2.0646

8.7042

23.7861

1.5985

7.8932

37.3396

1.0965

13.1193

29.2048

0.7749

9.6647

30.6111

5

 

2.0226

7.4017

22.7987

1.4013

7.2619

37.3384

0.8982

13.0913

28.5349

0.6164

9.5389

30.4699

7.5

 

1.9711

6.4471

22.2776

1.2327

6.8867

37.3377

0.7561

13.0745

28.1591

0.5103

9.4728

30.3973

10

 

1.9211

5.8446

22.0115

1.1126

6.6811

37.3374

0.6653

13.0652

27.9595

0.4451

9.4390

30.3604

30

 

1.5905

4.2475

21.4717

0.7049

6.2291

37.3366

0.3972

13.0445

27.5347

0.2607

9.3695

30.2858

50

 

1.3683

3.8567

21.3628

0.5573

6.1317

37.3364

0.3099

13.0400

27.4454

0.2025

9.3554

30.2707

100

 

1.0546

3.5606

21.2811

0.4003

6.0571

37.3363

0.2203

13.0365

27.3773

0.1435

9.3447

30.2593

 

 

References

[1]   Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Composite Structures 2013; 96: 833-49.

[2]   Ghanbari A, Babaei A. The new boundary condition effect on the free vibration analysis of micro-beams based on the modified couple stress theory. International Research Journal of Applied and Basic Sciences 2015; 9(3): 274-9.

[3]   Babaei A, Yang CX. Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field. Microsystem Technologies 2019; 25(3): 1077-85.

[4]   Babaei A, Ahmadi I. Dynamic vibration characteristics of non-homogenous beam-model MEMS. Journal of Multidisciplinary Engineering Science Technology 2017; 4(3): 6807-14.

[5]   Babaei A, Rahmani A. On dynamic-vibration analysis of temperature-dependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mechanics of Advanced Materials and Structures 2018: 1-8.

[6]   Babaei A, Rahmani A, Ahmadi I. Transverse vibration analysis of nonlocal beams with various slenderness ratios, undergoing thermal stress. Archive of Mechanical Engineering 2019.

[7]   Babaei A. Longitudinal vibration responses of axially functionally graded optimized MEMS gyroscope using Rayleigh–Ritz method, determination of discernible patterns and chaotic regimes. SN Applied Sciences 2019; 1(8): 831.

[8]   Chen W, Lv C, Bian Z. Free vibration analysis of generally laminated beams via state-space-based differential quadrature. Composite Structures 2004; 63(3): 417-25.

[9]   Xiang H, Yang J. Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Composites Part B: Engineering 2008; 39(2): 292-303.

[10] Lü C, Lim CW, Chen W. Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures 2009; 46(5): 1176-85.

[11] Amar LHH, Kaci A, Yeghnem R, Tounsi A. A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate. Steel and Composite Structures 2018; 26(1): 89-102.

[12] Pradhan K, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites Part B: Engineering 2013; 51: 175-84.

[13] Aghazadeh R, Cigeroglu E, Dag S. Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of Mechanics-A/Solids 2014; 46: 1-11.

[14] Jafari-Talookolaei R-A, Ebrahimzade N, Rashidi-Juybari S, Teimoori K. Bending and vibration analysis of delaminated Bernoulli–Euler micro-beams using the modified couple stress theory. Scientia Iranica 2018; 25(2): 675-88.

[15] Ansari R, Gholami R, Sahmani S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures 2011; 94(1): 221-8.

[16] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Materials & Design (1980-2015) 2010; 31(5): 2324-9.

[17] Ansari R, Gholami R, Shojaei MF, Mohammadi V, Sahmani S. Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. European Journal of Mechanics-A/Solids 2015; 49: 251-67.

[18] Babaei A, Noorani M-RS, Ghanbari A. Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory. Microsystem Technologies 2017: 1-12.

[19] Shafiei N, Mousavi A, Ghadiri M. Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM. Composite Structures 2016; 149: 157-69.

[20] Ansari R, Oskouie MF, Gholami R, Sadeghi F. Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Composites Part B: Engineering 2016; 89: 316-27.

[21] Farajpour A, Ghayesh MH, Farokhi H. Super and subcritical nonlinear nonlocal analysis of NSGT nanotubes conveying nanofluid. Microsystem Technologies 2019: 1-15.

[22] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M. Size-dependent Effects on the Vibration Behavior of a Ti-moshenko Microbeam subjected to Pre-stress Loading based on DQM. Mechanics of Advanced Composite Structures‎ 2016; 3(2): 99-112.

[23] Arabghahestani M, Karimian S. Molecular dynamics simulation of rotating carbon nanotube in uniform liquid argon flow. Journal of Molecular Liquids 2017; 225: 357-64.

[24] Fang J, Gu J, Wang H. Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences 2018; 136: 188-99.

[25] Khaniki HB, Rajasekaran S. Mechanical analysis of non-uniform bi-directional functionally graded intelligent micro-beams using modified couple stress theory. Materials Research Express 2018; 5(5): 055703.

[26] Jia X, Ke L, Zhong X, Sun Y, Yang J, Kitipornchai S. Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory. Composite Structures 2018; 202: 625-34.    

 

 

[1]   Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Composite Structures 2013; 96: 833-49.

[2]   Ghanbari A, Babaei A. The new boundary condition effect on the free vibration analysis of micro-beams based on the modified couple stress theory. International Research Journal of Applied and Basic Sciences 2015; 9(3): 274-9.

[3]   Babaei A, Yang CX. Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field. Microsystem Technologies 2019; 25(3): 1077-85.

[4]   Babaei A, Ahmadi I. Dynamic vibration characteristics of non-homogenous beam-model MEMS. Journal of Multidisciplinary Engineering Science Technology 2017; 4(3): 6807-14.

[5]   Babaei A, Rahmani A. On dynamic-vibration analysis of temperature-dependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mechanics of Advanced Materials and Structures 2018: 1-8.

[6]   Babaei A, Rahmani A, Ahmadi I. Transverse vibration analysis of nonlocal beams with various slenderness ratios, undergoing thermal stress. Archive of Mechanical Engineering 2019.

[7]   Babaei A. Longitudinal vibration responses of axially functionally graded optimized MEMS gyroscope using Rayleigh–Ritz method, determination of discernible patterns and chaotic regimes. SN Applied Sciences 2019; 1(8): 831.

[8]   Chen W, Lv C, Bian Z. Free vibration analysis of generally laminated beams via state-space-based differential quadrature. Composite Structures 2004; 63(3): 417-25.

[9]   Xiang H, Yang J. Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Composites Part B: Engineering 2008; 39(2): 292-303.

[10] Lü C, Lim CW, Chen W. Size-dependent elastic behavior of FGM ultra-thin films based on generalized refined theory. International Journal of Solids and Structures 2009; 46(5): 1176-85.

[11] Amar LHH, Kaci A, Yeghnem R, Tounsi A. A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate. Steel and Composite Structures 2018; 26(1): 89-102.

[12] Pradhan K, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites Part B: Engineering 2013; 51: 175-84.

[13] Aghazadeh R, Cigeroglu E, Dag S. Static and free vibration analyses of small-scale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of Mechanics-A/Solids 2014; 46: 1-11.

[14] Jafari-Talookolaei R-A, Ebrahimzade N, Rashidi-Juybari S, Teimoori K. Bending and vibration analysis of delaminated Bernoulli–Euler micro-beams using the modified couple stress theory. Scientia Iranica 2018; 25(2): 675-88.

[15] Ansari R, Gholami R, Sahmani S. Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures 2011; 94(1): 221-8.

[16] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the size-dependent behavior of functionally graded micro-beams. Materials & Design (1980-2015) 2010; 31(5): 2324-9.

[17] Ansari R, Gholami R, Shojaei MF, Mohammadi V, Sahmani S. Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. European Journal of Mechanics-A/Solids 2015; 49: 251-67.

[18] Babaei A, Noorani M-RS, Ghanbari A. Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory. Microsystem Technologies 2017: 1-12.

[19] Shafiei N, Mousavi A, Ghadiri M. Vibration behavior of a rotating non-uniform FG microbeam based on the modified couple stress theory and GDQEM. Composite Structures 2016; 149: 157-69.

[20] Ansari R, Oskouie MF, Gholami R, Sadeghi F. Thermo-electro-mechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Composites Part B: Engineering 2016; 89: 316-27.

[21] Farajpour A, Ghayesh MH, Farokhi H. Super and subcritical nonlinear nonlocal analysis of NSGT nanotubes conveying nanofluid. Microsystem Technologies 2019: 1-15.

[22] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M. Size-dependent Effects on the Vibration Behavior of a Ti-moshenko Microbeam subjected to Pre-stress Loading based on DQM. Mechanics of Advanced Composite Structures‎ 2016; 3(2): 99-112.

[23] Arabghahestani M, Karimian S. Molecular dynamics simulation of rotating carbon nanotube in uniform liquid argon flow. Journal of Molecular Liquids 2017; 225: 357-64.

[24] Fang J, Gu J, Wang H. Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences 2018; 136: 188-99.

[25] Khaniki HB, Rajasekaran S. Mechanical analysis of non-uniform bi-directional functionally graded intelligent micro-beams using modified couple stress theory. Materials Research Express 2018; 5(5): 055703.

[26] Jia X, Ke L, Zhong X, Sun Y, Yang J, Kitipornchai S. Thermal-mechanical-electrical buckling behavior of functionally graded micro-beams based on modified couple stress theory. Composite Structures 2018; 202: 625-34.