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
Keywords
Vibration Characteristics of Functionally Graded MicroBeam Carrying an Attached Mass
A. Rahmani^{a}, A. Babaei^{b}, S. Faroughi^{a}^{,*}
^{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 biomicrobeam Vibration analysis Modified couple stress theory Nondimensional frequency Biological and biomedical application of biomicrosystems 
In this article, in reference to the modified couple stress theory and EulerBernoulli beam theory, the free lateral vibration response of a microbeam carrying a moveable attached mass is investigated. This is a decent model for biological and biomedical applications beneficial to the earlystage diagnosis of diseases and malfunctions of human body organs and enzymes. The microcantilever beam is composed of functionally graded materials (FGMs). The material properties are supposed to show variations throughthickness of the beam in consonance to the power of law. RayleighRitz 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 lengthscale parameter that captures the sizedependency, 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 biomicrosystems. As a result, the adoption of suitable power index, mass ratio and position of the attached mass lead to the superior design of biomicrosystems persuading earlystage diagnostics. 
Functionally graded materials (FGMs) can be interpreted as nonhomogeneous graded composites consisted of a mixture of two different materials, usually a metal and a nonmetal 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 microstructures such as thin films in the form of shape memory alloys, microelectromechanicalsystems (MEMS), nanoelectromechanicalsystems (NEMS) and atomic force microscopes (AFMs)[1].
By the compelling improvements contributing to microtechnology, consisted of concepts, modeling, design, and fabrication, a microcantilever beam that has the dimensions at the order of micron has been widely applied. The applications chiefly include atomic force microscopes (AFMs), micro and nanoelectromechanical systems (MEMS, NEMS). Microelectromechanical Systems mainly arranged by miniaturized electromechanical and mechanical elements. In the mentioned technology, the elements are being engendered applying the microfabrication techniques. Microsensors and microactuators 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 sizedependency of microstructures could not be captured by the classical elasticity theory, consequently nonclassical elasticity theories emerge. The modified couple stress theory as one of these nonclassical 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 nonclassical theories for investigation of the microstructures. 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 MEMSgyroscopes. They have proposed a novel modified coupled displacement fields optimizing the design of microsystems in the industry of automotive. Babaei and Ahmadi [4] reported dynamic characteristics of a nonuniform and nonhomogenous micromodel 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 nonclassical lengthscale parameter. Moreover, variations impact over dynamicvibration 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 MEMSgyroscope 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 statespacebased 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 onedimensional steady heat conduction in the thickness direction before undergoing any dynamic deformation. Elastic behavior of FGM ultrathin films is surveyed by Lü et al. [[10]. They proposed a generalized theory pondered the surface effects in order to capture the sizedependency of the structure’s responses. Amar et al. [11] examined the free vibration analysis of the EulerBernoulli 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 RayleighRitz method for solution procedure. Aghazadeh et al. [13] investigated the free vibration and static behavior of small scale FG microbeams 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 microbeams 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 microbeams applying the strain gradient theory and the Timoshenko beam theory. They attempted to simulate a model including the thick microbeams theories. The paper which reported the sizedependent behavior of the FG EulerBernoulli microcantilever 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 microplates employing the strain gradient theory. Babaei et al. [18] proposed a microbeam model considering variations of both temperature and material gradation based on modified couple stress and the EulerBernoulli beam theories. Shafiei et al. [19] conducted research about sizedependent vibration analysis of nonuniform FG microbeam. They have assumed EulerBernoulli and Timoshenko beam theories for deriving the formulas. Ansari et al. [20] inquired the vibration of postbuckle piezoelectric microbeams applying nonlocal theory. Farajpour et al. [21] investigated the nonlinear behavior of nanotubes conveying nanofluid with both subcritical and supercritical regimes. They presented the model established based on strain gradient theory. Mohammadimehr et al. [22] reported Sizedependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Prestress 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 sizedependent vibration analysis of rotating smallscaled 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 bidirectional nonuniform beams based on MCST. Jia et al. [26] explored mechanical buckling behavior of smallscaled beams based on MCST under thermal and electrical loads. Their model is supposed to be functionally graded.
In this paper, a functionally graded microscale EulerBernoulli beam model is proposed for the sizedependent free vibration analysis. This model is presented as a biomedical and biological laboratory setup 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 microcantilever beam. The kinetic and strain energy are derived from employing the modified couple stress theory. Due to the usage of the RayleighRitz 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 microcantilever beam are reported.
A functionally graded microbeam 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 microbeam is formed by Steel (a metal constituent) and Alumina (a nonmetal 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 microbeam are numerically interpreted by a powerlaw. The volume fraction of the alumina is defined as Wakashima [46]:
(3) 
where is the powerlaw 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 microbeam take the form below:


Fig 1. Schematic of the FGM beam. 


(4) 
Similarly, Young’s modulus, shear modulus and density of the microstructure structure can be expressed as follows:
(5) 


(6) 
(7) 
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 nonclassical parameter captures the sizedependency of the structure.
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 EulerBernoulli beam:
(21) 
The kinetic energy of the Euler Bernoulli beam carrying an attached mass can be expressed as follows:

(22) 
One of the most applied methods to solve the partial differential governing equations of continuous systems is approximate methods. RayleighRitz'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, RayleighRitz 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) 
RayleighRitz 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.
The functionally graded microcantilever 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 powerlaw. 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 crosssection; the nondimensional 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 lengthscale parameter and putting the power index equal to zero, the results are verified. In Tables 35 nondimensional frequency of a microcantilever EulerBernoulli 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 nondimensional frequencies is expected by the growth of the power index. Tables 3, 4 and 5 reveal the fact that through increasing the power index, nondimensional 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 nondimensional 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 nondimensional 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. Nondimensional natural frequencies of a microcantilever 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. Nondimensional natural frequencies of a microcantilever 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. Nondimensional natural frequencies of a microcantilever 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 nondimensional frequency of the FG microbeam 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 nondimensional 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 nondimensional 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 nondimensional 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. Nondimensional natural frequencies of a microcantilever beam with respect to relative attached mass position. (k=0)
Mode No. 
X_{am}/L=0.25 
X_{am}/L=0.5 
X_{am}/L=0.75 
X_{am}/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 substantialfrequency 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 nondimensional 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. Nondimensional natural frequencies of a microcantilever beam with respect to relative attached mass position. (k=1)
Mode No. 
X_{am}/L=0.25 
X_{am}/L=0.5 
X_{am}/L=0.75 
X_{am}/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 
In the present study, free lateral vibration analysis of the biomicroFG beam that carries a movably attached mass is established. Nonclassical 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 nonclassical 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 biomicrostructures 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 setup can be devised employed a proper powerlaw index, and a decent relative position of the attached mass. To put it simple, the material type of biosensor and location of the blood/enzyme sample are amongst the crucial factors through the analysis and application of such novel biomicrosystems. 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 nonclassical theorems will lead to eyecatching achievements in clinical diagnostics and biological sciences.
Table 8. Nondimensional natural frequencies of a microcantilever 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 
[1] Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Composite Structures 2013; 96: 83349.
[2] Ghanbari A, Babaei A. The new boundary condition effect on the free vibration analysis of microbeams based on the modified couple stress theory. International Research Journal of Applied and Basic Sciences 2015; 9(3): 2749.
[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): 107785.
[4] Babaei A, Ahmadi I. Dynamic vibration characteristics of nonhomogenous beammodel MEMS. Journal of Multidisciplinary Engineering Science Technology 2017; 4(3): 680714.
[5] Babaei A, Rahmani A. On dynamicvibration analysis of temperaturedependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mechanics of Advanced Materials and Structures 2018: 18.
[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 statespacebased differential quadrature. Composite Structures 2004; 63(3): 41725.
[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): 292303.
[10] Lü C, Lim CW, Chen W. Sizedependent elastic behavior of FGM ultrathin films based on generalized refined theory. International Journal of Solids and Structures 2009; 46(5): 117685.
[11] Amar LHH, Kaci A, Yeghnem R, Tounsi A. A new fourunknown refined theory based on modified couple stress theory for sizedependent bending and vibration analysis of functionally graded microplate. Steel and Composite Structures 2018; 26(1): 89102.
[12] Pradhan K, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites Part B: Engineering 2013; 51: 17584.
[13] Aghazadeh R, Cigeroglu E, Dag S. Static and free vibration analyses of smallscale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of MechanicsA/Solids 2014; 46: 111.
[14] JafariTalookolaei RA, Ebrahimzade N, RashidiJuybari S, Teimoori K. Bending and vibration analysis of delaminated Bernoulli–Euler microbeams using the modified couple stress theory. Scientia Iranica 2018; 25(2): 67588.
[15] Ansari R, Gholami R, Sahmani S. Free vibration analysis of sizedependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures 2011; 94(1): 2218.
[16] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the sizedependent behavior of functionally graded microbeams. Materials & Design (19802015) 2010; 31(5): 23249.
[17] Ansari R, Gholami R, Shojaei MF, Mohammadi V, Sahmani S. Bending, buckling and free vibration analysis of sizedependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. European Journal of MechanicsA/Solids 2015; 49: 25167.
[18] Babaei A, Noorani MRS, Ghanbari A. Temperaturedependent free vibration analysis of functionally graded microbeams based on the modified couple stress theory. Microsystem Technologies 2017: 112.
[19] Shafiei N, Mousavi A, Ghadiri M. Vibration behavior of a rotating nonuniform FG microbeam based on the modified couple stress theory and GDQEM. Composite Structures 2016; 149: 15769.
[20] Ansari R, Oskouie MF, Gholami R, Sadeghi F. Thermoelectromechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Composites Part B: Engineering 2016; 89: 31627.
[21] Farajpour A, Ghayesh MH, Farokhi H. Super and subcritical nonlinear nonlocal analysis of NSGT nanotubes conveying nanofluid. Microsystem Technologies 2019: 115.
[22] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M. Sizedependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Prestress Loading based on DQM. Mechanics of Advanced Composite Structures 2016; 3(2): 99112.
[23] Arabghahestani M, Karimian S. Molecular dynamics simulation of rotating carbon nanotube in uniform liquid argon flow. Journal of Molecular Liquids 2017; 225: 35764.
[24] Fang J, Gu J, Wang H. Sizedependent threedimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences 2018; 136: 18899.
[25] Khaniki HB, Rajasekaran S. Mechanical analysis of nonuniform bidirectional functionally graded intelligent microbeams 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. Thermalmechanicalelectrical buckling behavior of functionally graded microbeams based on modified couple stress theory. Composite Structures 2018; 202: 62534.
[1] Jha D, Kant T, Singh R. A critical review of recent research on functionally graded plates. Composite Structures 2013; 96: 83349.
[2] Ghanbari A, Babaei A. The new boundary condition effect on the free vibration analysis of microbeams based on the modified couple stress theory. International Research Journal of Applied and Basic Sciences 2015; 9(3): 2749.
[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): 107785.
[4] Babaei A, Ahmadi I. Dynamic vibration characteristics of nonhomogenous beammodel MEMS. Journal of Multidisciplinary Engineering Science Technology 2017; 4(3): 680714.
[5] Babaei A, Rahmani A. On dynamicvibration analysis of temperaturedependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mechanics of Advanced Materials and Structures 2018: 18.
[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 statespacebased differential quadrature. Composite Structures 2004; 63(3): 41725.
[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): 292303.
[10] Lü C, Lim CW, Chen W. Sizedependent elastic behavior of FGM ultrathin films based on generalized refined theory. International Journal of Solids and Structures 2009; 46(5): 117685.
[11] Amar LHH, Kaci A, Yeghnem R, Tounsi A. A new fourunknown refined theory based on modified couple stress theory for sizedependent bending and vibration analysis of functionally graded microplate. Steel and Composite Structures 2018; 26(1): 89102.
[12] Pradhan K, Chakraverty S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh–Ritz method. Composites Part B: Engineering 2013; 51: 17584.
[13] Aghazadeh R, Cigeroglu E, Dag S. Static and free vibration analyses of smallscale functionally graded beams possessing a variable length scale parameter using different beam theories. European Journal of MechanicsA/Solids 2014; 46: 111.
[14] JafariTalookolaei RA, Ebrahimzade N, RashidiJuybari S, Teimoori K. Bending and vibration analysis of delaminated Bernoulli–Euler microbeams using the modified couple stress theory. Scientia Iranica 2018; 25(2): 67588.
[15] Ansari R, Gholami R, Sahmani S. Free vibration analysis of sizedependent functionally graded microbeams based on the strain gradient Timoshenko beam theory. Composite Structures 2011; 94(1): 2218.
[16] Asghari M, Ahmadian M, Kahrobaiyan M, Rahaeifard M. On the sizedependent behavior of functionally graded microbeams. Materials & Design (19802015) 2010; 31(5): 23249.
[17] Ansari R, Gholami R, Shojaei MF, Mohammadi V, Sahmani S. Bending, buckling and free vibration analysis of sizedependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory. European Journal of MechanicsA/Solids 2015; 49: 25167.
[18] Babaei A, Noorani MRS, Ghanbari A. Temperaturedependent free vibration analysis of functionally graded microbeams based on the modified couple stress theory. Microsystem Technologies 2017: 112.
[19] Shafiei N, Mousavi A, Ghadiri M. Vibration behavior of a rotating nonuniform FG microbeam based on the modified couple stress theory and GDQEM. Composite Structures 2016; 149: 15769.
[20] Ansari R, Oskouie MF, Gholami R, Sadeghi F. Thermoelectromechanical vibration of postbuckled piezoelectric Timoshenko nanobeams based on the nonlocal elasticity theory. Composites Part B: Engineering 2016; 89: 31627.
[21] Farajpour A, Ghayesh MH, Farokhi H. Super and subcritical nonlinear nonlocal analysis of NSGT nanotubes conveying nanofluid. Microsystem Technologies 2019: 115.
[22] Mohammadimehr M, Mohammadi Hooyeh H, Afshari H, Salarkia M. Sizedependent Effects on the Vibration Behavior of a Timoshenko Microbeam subjected to Prestress Loading based on DQM. Mechanics of Advanced Composite Structures 2016; 3(2): 99112.
[23] Arabghahestani M, Karimian S. Molecular dynamics simulation of rotating carbon nanotube in uniform liquid argon flow. Journal of Molecular Liquids 2017; 225: 35764.
[24] Fang J, Gu J, Wang H. Sizedependent threedimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences 2018; 136: 18899.
[25] Khaniki HB, Rajasekaran S. Mechanical analysis of nonuniform bidirectional functionally graded intelligent microbeams 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. Thermalmechanicalelectrical buckling behavior of functionally graded microbeams based on modified couple stress theory. Composite Structures 2018; 202: 62534.