Vibration Characteristics of Functionally Graded Micro-Beam 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

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:

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:

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]:

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:

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

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:

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).

 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:

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:

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

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

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

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:

In which, the coefficients are:

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

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:

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.

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

where stiffness and mass matrices are defined as follows:

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

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

Table 2. Comparison of the non-dimensional frequencies

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

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

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

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)

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)

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)

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