On the dynamics of small-sized structures

International Journal of Engineering Science 145 (2019) 103164

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On the dynamics of small-sized structures Hayri Metin Numanog˘ lu, Ömer Civalek∗ Akdeniz University, Civil Engineering Department, Division of Mechanics, Antalya, Turkey

a r t i c l e

i n f o

Article history: Received 31 March 2019 Revised 20 August 2019 Accepted 31 August 2019

Keywords: Nonlocal elasticity Nano/micro truss Nano/micro frame Nonlocal elasticity Size-dependency Nonlocal matrices Dynamic analysis

a b s t r a c t In this study, free vibration analyses of small-scaled trusses and frames are firstly carried out based on nonlocal elasticity of Eringen. Nonlocal matrix motion formulation is derived by using linear algebraic equations. Finite element method based weighted residual is utilized to solve the resulting equations. Various numerical studies are presented for nondimensional natural frequencies of different truss and frame models. A detailed parametric study is performed to investigate the influences of nonlocal parameter, geometric properties, direction angle, mode numbers, and length-to-diameter ratio on the natural frequencies of micro/nano trusses and frames. It is revealed that there is a significant relationship between the size-dependent dynamic response of these structures and the geometrical and structural properties of them. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction In the small–scale material technology and engineering, discovery of carbon nanotube (CNT) and invention of different devices such as atomic force microscope (AFM) and microprocessor have increased importance of mechanical modeling and analysis of all components of nano- and micro- electromechanical systems (NEMS/MEMS). Using atomic scaled tubes, wires, rods, and graphene sheets as fundamental components in such systems has prompted to be investigated mechanical behaviors and characteristics of these structures. It can be stated that the studies are based on simulations and analytical methods. Because of the simulations are expensive and require high expertise, the researchers studied on the mechanical analyses by taking into consideration the different internal and external effects with analytical formulations for nano/micro beams (Akgöz & Civalek, 2011, 2013, 2014a; Khaniki, 2018; Kong, Zhou, Nie, & Wang, 2008; Liu, Liu, & Yang, 2018; Lu, Guo, & Zhao, 2017; Mohamad-Abadi & Daneshmehr 2014; Shafiei, Kazemi, Safi, & Ghadiri, 2016), for nano/micro rods (Akgöz & Civalek, 2014b; Aydog˘ du, 2009, 2012; El-Borgi, Rajendran, Friswell, Trabelssi, & Reddy, 2018; Numanog˘ lu, Akgöz, & Civalek, 2018; Zhu & Li, 2017), for nano/micro plates (Barretta, Faghidian, & de Sciarra, 2019; Ke, Liu, & Wang, 2015; Lu, Guo, & Zhao, 2018; Shahverdi & Barati, 2017; Srividhya, Raghu, Rajagopal, & Reddy, 2018; Taati, 2016; Wang & Li, 2012; Wang, Lin, & Liu, 2013; Zenkour & Sobhy, 2013), and for nano/micro shells (Faleh, Ahmed, & Fenjan, 2018; Farokhi & Ghayesh, 2018; Ghayesh & Farokhi, 2017; Ma et al., 2018; Sahmani, Bahrami, & Aghdam, 2016). It is crucial to achieve the accurate and optimum design of small-sized structures for utilization with the most performance in NEMS/MEMS. Mechanical behaviors of these structures cannot be understood based on classical elasticity because experimental findings are not coherent with classical physics laws. Consequently, their exact design cannot be obtained. To eliminate this serious problem, higher-order continuum theories have been used for nearly 20 years. Nonlocal elasticity ∗

Corresponding author. E-mail address: [email protected] (Ö. Civalek).

https://doi.org/10.1016/j.ijengsci.2019.103164 0020-7225/© 2019 Elsevier Ltd. All rights reserved.

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H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

Fig. 1. The general configuration of a triangular cellular nanostructure.

theory that was developed by Eringen is one of these theories, which is the most popular at the area of computational mechanics of nano/micro structures (Eringen, 1972, 1977a,b, 1983, 1984; Eringen & Edelen, 1972; Hadjo & Eringen, 1979). According to nonlocal elasticity, the stress at a reference point, not only with one region, is associated with stress and strain expressions all point of atomic body. Thus, principles of classical elasticity expand with internal size effect of structure and accurate solutions can be attained under many mechanical constraints such as bending, buckling, torsion, vibration or their coupled effects. On the other hand, we can specify that several higher-order continuum mechanics formulations are also utilized. Modified couple stress theory that was formulated by Yang, Chong, Lam, and Tong (2002) predicates on studies of Mindlin and Tiersten (1962), Koiter (1964) and Toupin (1962). Strain gradient elasticity (Fleck & Hutchinson, 1993, 2001) was modified with three different material scale parameters to analyze nano/micro elastic solids by Lam, Yang, Chong, Wang, and Tong (2003). A general literature review is given about the size-dependent mechanical analyses of continuum models of different atomic structures as follows: Bending response of Euler-Bernoulli beams, which is first application of nonlocal elasticity in the solid mechanics, was investigated by Peddieson, Buchanan, and McNitt (2003). Reddy (2007) performed nonlocal bending, free vibration and buckling of elastic beams by utilizing four different beam formulations. Reddy and Pang (2008) formulated bending, free vibration and buckling of nonlocal Euler–Bernoulli and Timoshenko beam models of carbon nanotubes with different boundary conditions. A new nonlocal nanobeam formulation including shear effect was proposed and implemented to bending, buckling and vibration problems by Thai (2012). Free vibration analysis of micro beam models was given with modified couple stress elasticity by Kong et al. (2008). Buckling problem of Euler-Bernoulli micro beam model was analyzed using modified couple stress (Mohamad-Abadi & Daneshmehr, 2014) and modified strain gradient theories (Akgöz & Civalek, 2011). Ma, Gao, and Reddy (2008) studied bending and vibration for Timoshenko micro beams on the basis of modified couple stress theory. On the other hand, nonlocal free axial vibration of nano rods was examined (Aydogdu, 2009, 2012). Numanog˘ lu et al. (2018) presented nonlocal axial vibration analysis of nano rods whose mass and linear spring attachments at the tip. Axial and torsional vibrations were investigated using strain gradient elasticity (Akgöz & Civalek, 2011; Guven, 2014; Kahrobaiyan, Asghari, & Ahmadian, 2013, 2011). Additionally, it can be stated that static bending (Barretta et al., 2019; Wang & Li, 2012), large deflection (Reddy, 2010; Srividhya et al., 2018), buckling (Zenkour & Sobhy, 2013), and vibration (Daneshmehr, Rajabpoor, & Hadi, 2015; Ke et al., 2015) analyses of nonlocal nano plates. Also, finite element approach was applied to different nonlocal elasticity problems such as axial and torsional free vibrations of rods (Adhikari, Murmu, & McCarthy, 2013, 2014; Demir & Civalek, 2013), vibration (Eltaher, Alshorbagy, & Mahmoud, 2013) and buckling (Civalek & Demir, 2016) of Euler–Bernoulli beams, bending of Timoshenko beams (Hemmatnezhad & Ansari, 2013) and thermomechanical behaviors of Timoshenko beams (Pradhan & Mandal, 2013). There are some inconsistencies for nonlocal analyses of cantilever beam. Different approaches were examined to eliminate this paradoxic case and utilized in the bending and vibration problems of cantilever beams (Barretta, Feo, Luciano, & de Sciarra, 2016; Challamel & Wang, 2008; Challamel et al., 2014; Fernández-Sáez & Zaera, 2017; Fernández-Sáez, Zaera, Loya, & Reddy, 2016; Khodabakhshi & Reddy, 2015; Romano & Barretta, 2017; Tuna & Kirca, 2016a,b). Also, a new and popular size-dependent elasticity formulation, the nonlocal strain gradient theory that contains effects of different atomic parameters was studied (Apuzzo, Barretta, Faghidian, Lucaino, & de Sciarra, 2018; El-Borgi et al., 2018; Faleh et al., 2018; Li & Hu, 2015; Lim, Zhang, & Reddy, 2015; Lu et al., 2017, 2018; Ma et al., 2018; Shahverdi & Barati, 2017; Xu, Zheng, & Wang, 2015; Zhu & Li, 2017). It can be emphasized that nano/micro scaled hierarchical lattice structures and cellular nanostructures are an example for truss system that is known from structural analysis. For instance, hierarchical structures can be utilized as reusable lightweight shock absorber (Frenzel, Findeisen, Kadic, Gumbsch, & Wegener, 2016), lithium-ion battery electrode (Xia, Di Leo, Gu, & Greer, 2016) and biomedical implants (Parthasarathy, Starly, & Raman, 2011). The cellular nanostructures are a significant member of some NEMS organizations that are used in stretching and fracture experiments (Syms, Liu, & Ahmad, 2017). The simple configuration of a triangular cellular nanostructure is demonstrated in Fig. 1. On the other hand, ground electrode taking a part in MEMS structures where aluminum nitride (AlN) piezoelectric resonant plate (Piazza, Stephanou, & Pisano, 2006) occupies to achieve high operation frequency, can be modeled as single-story and single-span frame structure (see Fig. 2a). Electro-thermal actuator bent-beams (Que, Park, & Gianchandani, 2001) which are commonly employed in strain

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

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Fig. 2. (a) Schematic demonstration of a sandwiched AlN piezoelectric resonant plate (b) An actuator bent-beam under electro thermal flow.

sensitivity can be proposed as frame with two bending members. Also, voltage sources can be considered as fixed end connection that anchorages frame to substrate (see Fig. 2b). As aforementioned, a number of studies have been carried out about different mechanical phenomena of size-dependent beam, rod, plate, and shell models of small-scale structures. However, there is no study about small scale effect for truss and frame in the scientific literature. Structures like frame and truss can take on different tasks in the nano/micro electromechanical systems (NEMS/MEMS). In addition to this, many studies containing discovery and properties of hierarchical atomic lattice structures have been existed. Hence, it is a necessity that the determination of size–dependent mechanical behavior characteristics of discrete structures. The present study contains nonlocal free vibration of nano/micro trusses and frames that attracts the attention as a large gap in the literature. With this current study, to understand nonlocal dynamic characteristics of nano/micro structures with multi-members is aimed. In this context, article is organized as follows: The fundamental formulations of nonlocal elasticity theory are explained in Section 2. A size-dependent matrix motion formulation for dynamic analysis of truss and frame elements is examined in Section 3. Nonlocal vibration frequency parameters of discrete structures whose different geometries under different parameters are presented as tables and graphics with a detailed discussion in Section 4. The article is completed by giving general results obtained via numerical analyses in Section 5. 2. Nonlocal elasticity theory According to the nonlocal elasticity, stresses occurred by external forces depend to stresses and strains at the all regions of structure because external forces are in interaction with all atoms of nano/micro structure. The stress and strain equations of nonlocal elasticity are presented as (Eringen, 1983)



tkl =

εi j =

V

1 2

     α x − x , γ σkl dV x



∂ ui ∂ u j + ∂ x j ∂ xi

(1)

 (2)

where tkl is nonlocal stress tensor, |x − x | is distance in the Euclidean form, σ kl is classical (or macroscopic) stress tensor and V(x ) is volume of atomic structure. ɛij indicates nonlocal strain tensor. ui and uj are motion components. On the other hand, γ describes non-dimensional scale parameter and is written as

γ=

e0 a L

(3)

where e0 is atomic material constant, a and L are the internal and external characteristics lengths of atomic body, respectively.

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

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Eq. (1) can be rewritten in the following form (Reddy, 2007)



 ∂2 1−L γ t = σkl ∂ x2 kl 2

2

(4)

Substituting Eq. (3) into Eq. (4), one-dimensional stress is obtained as

txx − (e0 a )2

∂ 2txx = σxx ∂ x2

(5)

3. Nonlocal matrix motion equation In this section, a size-dependent matrix formulation is developed for free vibration analysis of nonlocal truss and frame models. 3.1. Nano/micro trusses Truss structures are discrete systems with axial members. Stress-force relations of axial rods are given as follows

 A

∂ txx ∂ 2u f =ρ 2 − , ∂x A ∂t

txx dA = Nx ,

 A

σxx dA = EA

∂u ∂x

(6)

where A, E, ρ , u, Nx and f explain section-area, modulus of elasticity, mass density, axial motion component, nonlocal axial internal force and axial external distributed force, respectively. After several mathematical operations, nonlocal axial motion equation is attained as

EA

∂ 2u ∂ 2u ∂ 2u ∂2 f − ρ A 2 + f + (e0 a )2 ρ A 2 2 − (e0 a )2 2 = 0 ∂ x2 ∂t ∂ x ∂t ∂x

(7)

where u = u(x, t ) and f = f (x, t ). At the base of vibration analysis of truss structures, there is a finite element formulation. According to this, axial motion of finite rod element with two nodes can be defined by

u = {φ}{ue } where {φ } and



{ue }

(8) are shape function and nodal displacement vectors, respectively and described as

{φ} = φ1

 φ2 = 1 − ξ

 {u e } = u 1

u2

 ξ = 1−

x L

x L



T

(9a) (9b)

where ξ is non-dimensional coordinate. Also,

∂u = Dk u = Bue ∂x

(10)

in which Dk φ = B and Dk is kinematic operator. According to weighted residual formulation, average weighted residual can be written as



I=

xj xi



hRdx =

L



h EA 0

 2 ∂ 2u ∂ 2u ∂ 2u 2 2∂ f − ρ A + f + e a ρ A − e a dx ( ) ( ) 0 0 ∂ x2 ∂t2 ∂ x2 ∂ t 2 ∂ x2

(11)

where h is weighting function and R is residual. Eq. (11) can be separated five different integral expressions and partial integration results of these expressions are attained as

L  L  L  L ∂ 2u ∂ u  ∂h ∂u ∂ 2u I1 = EAh 2 dx = EAh − EA d x, I = ρ Ah d x, I = h f dx, 2 3 ∂ x 0 ∂x ∂x ∂x ∂t2 0 0 0 0 L  L  L ∂ 4u ∂ 3 u  ∂ h ∂ 3u 2 2 2 I4 = − e a ρ A dx, (e0 a ) ρ Ah 2 2 dx = (e0 a ) ρ Ah ( ) 0 2 ∂ x ∂ x∂ t 2 ∂ x ∂t ∂ x∂ t 0 0 0 L  L  L ∂2 f ∂f ∂h ∂ f I5 = dx (e0 a )2 h 2 dx = (e0 a )2 h  − (e0 a )2 ∂x 0 ∂x ∂x ∂x 0 0 

L

(12)

Using of Eq. (12) in Eq. (11) yields weak form of Eq. (7)

 0

L



 ∂h ∂u ∂ 2u ∂ h ∂ 3u 2 2 ∂h ∂ f −EA − ρ Ah 2 + h f − (e0 a ) ρ A + ( e0 a ) dx = 0 ∂x ∂x ∂ x ∂ x∂ t 2 ∂x ∂x ∂t

(13)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

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y u22

p22

y'

u21

u12

x'

2 u11 1

p21 p12 p12

Motion freedoms

α

2 Force freedoms

1

x Fig. 3. Finite axial element and end freedoms.

y

p22

u22

y'

u12

u23

u11

u13 1

u21

x'

p23

2

p11

Motion freedoms

α

p21

p12

2

Force fredooms

p13 1

x Fig. 4. Finite bending element and end freedoms.

By employing Eqs. (9a) and (10), Eq. (13) can be rewritten as



L 0







EA BT B ue dx +

L 0

 L  L  L     ρ A φ T φ u¨ e dx + (e0 a )2 ρ A BT B u¨ e dx − f φ T dx − (e0 a )2 f  BT dx = 0 0

0

(14)

0

The general form of Eq. (14) is presented as

¨ e = fc + fnl K ue + (Mc + Mnl )u

(15)

where K is axial stiffness matrix, Mc is classical mass matrix, Mnl is nonlocal (or size-dependent) mass matrix, fc is classical ¨ e denote axial displacement and acceleration vectors, respectively. force vector, and fnl is nonlocal force vector. ue and u These matrices and vectors are derived as follows



 EA 1 −1 φ1    φ 1 φ 2 dx = K= EA B B dx = EA  , φ2 1 L −1 0 0

  L  L   ρ AL 2 1 φ  Mc = ρ A φ T φ dx = ρ A 1 φ1 φ2 dx = , φ2 6 1 2 0 0

 L  L  T  ( e 0 a )2 ρ A 1 φ1   2 2  φ φ Mnl = e a ρ A B B d x = e a ρ A d x = ( 0 ) ( 0 ) 1 2 φ −1 



L

T





L

0

2

0

1/2 φ1 dx = f L , φ2 1/2 0 0  L  L φ −1 fnl = ( e 0 a )2 f  B T d x = (e0 a )2 f  φ  1 dx = (e0 a )2 f  1 

fc =

L

0

f φ T dx =



L

(16a)

(16b)



−1 , 1

(16c)

L

f

0

(16d)

(16e)

2

in case of free vibration, force components are equal to zero. To achieve the nonlocal free vibration solution of truss structures, expressions in a global coordinate system of above matrices should be obtained. For this, firstly, the following expressions between global and local coordinates for finite axial element are written

d11 = u11 cos α + u12 sin α , d12 = −u11 sin α + u12 cos α

(17a)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

6

6

(a)

(c)

5 3

14

3

7

16 13

1

13

4

α

1

10

3

2

1

15 8

14

2

15

12

10

9 5

6

6 6

10 5

6

9 5

10

13

7

3 1

2

1

4

4

8

8

Lt

7

6

12

α

16 15

3

4

7

4

α

1

4

5

3

7

11

2

8 5

Lt

3

14

3

(b)

11

9

8

Lt

12 11

1

2

8

2

4 1

3

Fig. 5. Nano/micro truss samples and finite element models (a) Truss A (b) Truss B (c) Truss C.

d21 = u21 cos α + u22 sin α , d22 = −u21 sin α + u22 cos α

(17b)

where d11 and d21 are local axial, d12 and d22 local transverse motional freedoms. u11 and u21 are global axial, u12 and u22 global transverse motional freedoms. α is direction angle of element. The finite axial rod and its freedoms are depicted in Fig. 3. According to this, local and global coordinate system are respectively given as x y and xy. The following expression can be described for all freedoms of axial rod

ut =

φ1



φ2

0

φ1

0



0

[φ ]

0

⎧ ⎫ ⎪ ⎨ u11 ⎪ ⎬ u12

(18)

ϕ2 ⎪ u21 ⎪  ⎩u22 ⎭    {u}

where is motion vector, which defines total axial and transverse motion of truss. [φ ] is shape function matrix and {u} explains global displacement vector of end freedoms of truss. Eqs. (17a) and (17b) in matrix form are written as ut

⎧ ⎫ ⎪ ⎨ d11 ⎪ ⎬

⎡

c d12 ⎢−s =⎣ 0 ⎪ ⎪ ⎩d21 ⎭ d22 0

   {d}



s c 0 0



0 0 c −s

⎤⎧

⎫

0 ⎪ u11 ⎪ ⎨ ⎬ 0⎥ u12 ⎦ s ⎪u21 ⎪ ⎩ ⎭ c u22

(19)

    {u }

[T ]

where {d} is local displacements of end freedoms of truss and [T] is transformation matrix (c = cosα and s = sinα ). Similarly, force components in freedoms can be expressed by

⎧ ⎫ ⎪ ⎨ f11 ⎪ ⎬

⎡

c f12 ⎢−s =⎣ 0 ⎪ ⎩ f21 ⎪ ⎭ f22 0

   {f }



s c 0 0

 [T ]

0 0 c −s

⎤⎧

⎫

0 ⎪ p11 ⎪ ⎨ ⎬ 0⎥ p12 ⎦ s ⎪ p21 ⎪ ⎩ ⎭ c p22

    {p }

(20)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

5

8

(a) 2

4

6

2

9

3

(b)

7

5

2

1 2 1

11 1

3

4

5

1

Lf

2

4

6

2

3

Lf

12

α

9

7

3

2

10

1

9

#3

#2

7

#4

#5

2 3

1 8

5

#1

8

8

1

(d)

4

α

1

3

(c)

6

3

Lf

Lf

#1

7

2

6

4

2

3

7

9

#2

#3

#4

#5

Lf

Lf

α 1

2 3

1

Fig. 6. Nano/micro frame samples and finite element models (a) Frame A (b) Frame B (c) Frame C (with different five cases) (d) Frame D (with different five cases).

in which {f} and {p} represent local and global internal force vectors, respectively. Hooke’s law is given as

{f} = [K ]{d}

(21)

[K] is the stiffness matrix obtained in Eq. (16a) for axial rod. Even if ends of axial members perform transverse motion, ends do not take transverse (shear) force. Hence, Eq. (21) is rewritten in the following form

⎧ ⎫ ⎪ ⎨ f11 ⎪ ⎬

⎡

1 AE ⎢ 0 0 = ⎣ L −1 ⎪ ⎩ f21 ⎪ ⎭ 0 0

0 0 0 0

−1 0 1 0

⎤⎧

⎫

0 ⎪ d11 ⎪ ⎨ ⎬ 0⎥ d12 ⎦ 0 ⎪d21 ⎪ ⎩ ⎭ 0 d22

(22)

On the other hand, classical and size-dependent mass matrices are obtained as

⎡ φ1 L L  T  ⎢0 ρ A [φ ] [φ ] dx = ρ A⎣ [M c ] = φ2 0 0

⎤

 φ 1 ⎥ φ1 0 φ 2 0 dx = 0⎦ 0 φ1 0 φ2 0 φ2 ⎡  ⎤ φ1 0

 L  L   φ1⎥ φ1 0 ⎢0 [Mnl ] = (e0 a )2 ρ A [B]T [B] dx = (e0 a )2 ρ A⎣φ  0 ⎦ 0 φ1 2 0 0 0 φ2 ⎡ ⎤ 



0

1

( e 0 a )2 ρ A ⎢ 0 = ⎣−1 L 0

0 1 0 −1

−1 0 1 0

⎡

2 ρ AL ⎢0 ⎣1 6 0

φ2 0

0 2 0 1

0

φ2

1 0 2 0

⎤

0 1⎥ , 0⎦ 2

(23a)

 dx

0 −1⎥ 0⎦ 1

(23b)

To determine expression of stiffness matrix in global coordinates, Eqs. (19) and (20) are substituted into Eq. (21):

[T ]{p} = [K ][T ]{u}

(24)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

8

(a)

(b) γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

1.5

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

2.0

Nondimensional frequency

Nondimensional frequency

2.0

1.5

1.0

1.0

0.5

0.5

0.0

0.0 1

2 Mode number

3

1

2

3

4

5

6 7 8 Mode number

9

10

11

12

13

(c) 2.0 γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

Nondimensional frequency

1.5

1.0

0.5

0.0 1

2

3

4

5

6 7 8 Mode number

9

10

11

12

Fig. 7. Variation of nondimensional frequencies of different nano/micro trusses according to mode number (a) Truss A (b) Truss B (c) Truss C.

T

−1

The transformation matrix have orthogonality property ([T ] = [T ] −1

[T ]

). Hence, the following equation can be written

[T ] {p} = [T ] [K ][T ] {u}

  



[I]

T



(25)



[k]

where [k] is global rigidity matrix of finite axial element and given as T [k] = [T ] [K ][T ]

⎡

c ⎢s =⎣ 0 0

−s c 0 0

0 0 c s

⎤

⎡

0 1 0 ⎥ AE ⎢ 0 × ⎣ −s⎦ L −1 c 0

0 0 0 0

−1 0 1 0

⎤

⎡

0 c 0⎥ ⎢−s × 0⎦ ⎣ 0 0 0

s c 0 0

0 0 c −s

⎤

⎡

c2 0 0⎥ AE ⎢ cs = ⎣ s⎦ L −c2 c −cs

cs s2 −cs −s2

−c2 −cs c2 cs

⎤

−cs −s2 ⎥ cs ⎦ s2 (26)

¨ }): There is a similar case for classical and size-dependent mass matrices because of Newton’s second law ({f} = [M]{d

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

(a)

(b) 1.0

2.0

Mode-1 Mode-3 Mode-5

Mode-1 Mode-2 Mode-3 1.5

Nondimensional frequency

Nondimensional frequency

9

1.0

Mode-2 Mode-4

0.5

0.5

0.0

0.0 0.0

0.1

0.2

0.3

0.4

0.0

0.5

0.1

0.2

e0a/Lt

0.3

0.4

0.5

e0a/Lt

(c) 1.5

Nondimensional frequency

Mode-1 Mode-3 Mode-5

Mode-2 Mode-4

1.0

0.5

0.0 0.0

0.1

0.2

0.3

0.4

0.5

e0a/Lt Fig. 8. Variation of nondimensional frequencies of different nano/micro trusses with respect to nondimensional nonlocal parameter (a) Truss A (b) Truss B (c) Truss C.

T [mc ] = [T ] [Mc ][T ]

⎡

c ⎢s =⎣ 0 0

−s c 0 0

0 0 c s

T [mnl ] = [T ] [Mnl ][T ]

⎡

c ⎢s =⎣ 0 0

−s c 0 0

⎡

0 0 c s

1 ( e0 a ) ρ A ⎢ 0 = ⎣−1 L 0 2

⎤

⎡

0 2 0 ⎥ ρ AL ⎢0 × ⎣1 −s⎦ 6 c 0

⎤

0 2 0 1

⎡

1 0 2 0

0 1 2 0 ⎥ ( e0 a ) ρ A ⎢ 0 × ⎣−1 −s⎦ L c 0 0 1 0 −1

−1 0 1 0

⎤

0 −1⎥ 0⎦ 1

⎤

⎡

0 c 1⎥ ⎢−s × 0⎦ ⎣ 0 2 0

0 1 0 −1

−1 0 1 0

s c 0 0

⎤

0 0 c −s

⎡

0 c −1⎥ ⎢−s × 0⎦ ⎣0 1 0

⎤

0 0⎥ = s⎦ c

s c 0 0

⎡

2 ρ AL ⎢0 ⎣1 6 0

0 0 c −s

0 2 0 1

1 0 2 0

⎤

0 1⎥ , 0⎦ 2

(27a)

⎤

0 0⎥ s⎦ c

(27b)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

10

Fig. 9. Effect of member length on the nondimensional fundamental frequencies of different nano/micro trusses (a) Truss A (b) Truss B (c) Truss C.

here, [mc ] and [mnl ] represent classical mass matrix and size-dependent mass matrix in the global coordinates, respectively. Total mass matrix of element is given as

[m] = [mc ] + [mnl ] Use of

ue



= n

det

Ue

(28)

sin(ωt − α ) in Eq. (15), nonlocal eigenvalue formulation of trusses is presented as

[ki ]n×n − ωi

i=1

!

n 2

[mi ]n×n

=0

(29)

i=1

" " where ωi is ith natural frequency of truss. ni=1 [ki ]n×n and ni=1 [mi ]n×n are reduced stiffness and mass matrices of truss structure under boundary conditions, respectively. 3.2. Nano/micro frames Frame structures consist of elements whose rotational freedom as well as axial and transverse freedoms. In this subsection, nonlocal free vibration formulation of frame structures is given. The general bending finite element is shown in Fig. 4. Because nonlocal finite element matrices of axial rods are derived, we can obtain all matrices of finite bending element by neglecting axial freedoms.

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

(a)

11

(b) 35

Nondimensional frequency

40

30

Nondimensional frequency

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

20

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

25

15

10

0

5 1

2

3 4 Mode number

5

(c)

2 Mode number

3

(d)

70

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

50

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

60

Nondimensional frequency

60 Nondimensional frequency

1

6

40 30 20

50

40

30

20

10

10 0

0 1

2

3 4 Mode number

5

6

1

2

3 4 Mode number

5

6

Fig. 10. Variation of nondimensional frequencies of different nano/micro frames according to mode number (a) Frame A (b) Frame B (α = 45◦ ) (c) Frame C (Case 1) (d) Frame D (Case 1, α = 45◦ ).

Firstly, nonlocal stress-force relations of bending elements are given in the following equations



A

txx ydA = Mx ,

∂ 2txx y ∂ 2w y = −ρ A +f , I ∂t2 I ∂ x2



A

σxx ydA = EI

∂ 2w ∂ x2

(30)

where y, I, Mx , w and f denote transverse coordinate, moment of inertia of cross-section, nonlocal bending moment, transverse motion component and transverse external force component, respectively. After several mathematical operations, nonlocal transverse motion equation is obtained as

EI

2 ∂ 4w ∂ 2w ∂ 4w 2 2∂ f + ρ A − f − e a ρ A + e a =0 ( ) ( ) 0 0 ∂ x4 ∂t2 ∂ x2 ∂ t 2 ∂ x2

(31)

where w = w(x, t ) and f = f (x, t ). General motion of finite bending element is defined as

w = {φ}{we } where {φ } and

{we }

(32) are shape functions and motion freedoms, respectively and given as

⎫T ⎧ ⎫T ⎧ 1 − 3ξ 2 + 2ξ 3 ⎪ ⎪ ⎨φ1 ⎪ ⎬ ⎪ ⎨ ⎬ L −ξ + 2ξ 2 − ξ 3 φ {φ} = φ2 = 2 3 ⎪ ⎩ 3⎪ ⎭ ⎪ ⎩ 3ξ 2 − 2ξ3  ⎪ ⎭ φ4 L ξ −ξ

(33a)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

12

(a)

(b)

40

40 Mode-2 Mode-4

Mode-1 Mode-2 Mode-3

30

Nondimensional frequency

Nondimensional frequency

Mode-1 Mode-3 Mode-5

20

10

30

20

10

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

e0a/Lf

(c)

0.4

0.5

(d) 50

50 Mode-1 Mode-3 Mode-5

Mode-2 Mode-4

Mode-1 Mode-3 Mode-5

40

Nondimensional frequency

40 Nondimensional frequency

0.3

e0a/Lf

30

20

10

Mode-2 Mode-4

30

20

10

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

e0a/Lf

0.2

0.3

0.4

0.5

e0a/Lf

Fig. 11. Variation of nondimensional frequencies of different nano/micro frames with respect to nondimensional nonlocal parameter (a) Frame A (b) Frame B (α = 45◦ ) (c) Frame C (Case 1) (d) Frame D (Case1, α = 45◦ ).

 {w e } = w 1

w2

w3

w4

T

(33b)

where ξ = x/L. In bending elements, kinematic operator can be expressed as

∂w = Dk w = Bwe ∂x

(34)

By using Eq. (31), the average weighted residual is written as

 I=

xj xi

 hRdx =

L 0



 2 ∂ 4w ∂ 2w ∂ 2w 2 2∂ f h EI 4 + ρ A 2 − f − (e0 a ) ρ A 2 2 + (e0 a ) dx ∂x ∂t ∂ x ∂t ∂ x2

Integral expressions in Eq. (35) are described as

L L  L  L ∂ 4w ∂ 3 w  ∂ h ∂ 2 w  ∂ 2h ∂ 2w ∂ 2w I1 = EIh 4 dx = EIh 3  − EI + EI d x, I = ρ Ah 2 dx, 2 2 2 2  ∂ x ∂x ∂x 0 ∂x 0 ∂x ∂x ∂t 0 0 0 L  L  L  L 4 3 ∂ w ∂ w  ∂ h ∂ 3w 2 I3 = f hdx, I4 = − e a ρ A dx, (e0 a )2 ρ Ah 2 2 dx = (e0 a )2 ρ Ah ( ) 0 ∂ x ∂ x∂ t 2 ∂ x ∂t ∂ x∂ t 2 0 0 0 0 

L

(35)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

13

Fig. 12. Effect of member length on the nondimensional fundamental frequencies of different nano/micro frames (a) Frame A (b) Frame B (α = 45◦ ) (c) Frame C (Case 1) (d) Frame D (Case 1, α = 45◦ ).

 I5 =

L 0

L  L  ∂2 f ∂h ∂ f 2 ∂f dx ( e0 a ) h 2 dx = ( e0 a ) h  − (e0 a )2 ∂x 0 ∂x ∂x ∂x 0 2

(36)

By employing Eq. (36) in Eq. (35), weak form of Eq. (31) is achieved:



L 0



−EI

 ∂ 2h ∂ 2w ∂ 2w ∂ h ∂ 3w 2 2 ∂h ∂ f − ρ Ah + h f − e a ρ A + e a dx = 0 ( ) ( ) 0 0 ∂ x ∂ x∂ t 2 ∂x ∂x ∂ x2 ∂ x2 ∂t2

(37)

Substituting Eqs. (33a) and (34) into Eq. (37) yields finite element equation:



L 0

#

$

EI B B we dx + T

 0

L

 L  L  L  e    e ¨ dx + ¨ dx − ρA φTφ w f φ T dx − (e0 a )2 ρ A BT B w (e0 a )2 f  BT dx = 0 0

0

(38)

0

Thus, matrix motion formulation of nonlocal bending elements can be written as

¨ e = fc + fnl K we + (Mc + Mnl )w

(39)

where K, Mc and Mnl are bending stiffness, classical and size-dependent mass matrices in the local coordinates, respectively. ¨ e indicate transverse displacement fc and fnl are classical and size-dependent transverse force vectors, respectively. we and w

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

14

(a)

(b) 20 Case-2 Case-4

Case-1 Case-3 Case-5

10

15

Nondimensional frequency

Nondimensional frequency

Case-1 Case-3 Case-5

10

Case-2 Case-4

5

5

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

e0a/Lf

0.2

0.3

0.4

0.5

e0a/Lf

Fig. 13. Variation of nondimensional fundamental frequencies of nano/micro frames with different cases according to nondimensional nonlocal parameter (a) Frame C (b) Frame D (α = 45◦ ).

and acceleration vectors, respectively. These expressions are calculated as

 K=

L

0

#

$

EI B B dx = T



L

0

⎧  ⎫ ⎪ ⎨φ  1 ⎪ ⎬ φ 2   φ 1 EI  ⎪ ⎩φ  3 ⎪ ⎭ φ 4

⎧ ⎫ ⎪ ⎨φ1 ⎪ ⎬ L L  T  φ  Mc = ρ A φ φ dx = ρ A 2 φ1 ⎪φ3 ⎭ ⎪ 0 0 ⎩ φ4 



 Mnl =

L

0

  ( e 0 a )2 ρ A B T B d x = ⎡

36

( e0 a ) ρ A ⎢ 3L = ⎣−36 30L 2

3L  fc =

L

 fnl =

f φ T dx =

0

L 0



L

0

3L 4L2 −3L −L2



L 0

φ  2

φ  3

12 EI ⎢ 6L  φ 4 dx = 3 ⎣ L −12 6L

⎡

φ2

φ3

156L ρ AL ⎢ 22L2 φ3 d x = ⎣ 54L 420 −13L2

⎧  ⎫ ⎪ ⎨φ  1 ⎪ ⎬ φ  2 ( e0 a ) ρ A φ  2 φ  1 ⎪ ⎩ 3⎪ ⎭ φ4 ⎤

−36 −3L 36 −3L

φ2

3L −L2 ⎥ , −3L⎦ 4L2

⎧ ⎫ ⎧ ⎫ ⎪ ⎪ ⎨ φ1 ⎪ ⎬ ⎨ 1/2 ⎪ ⎬ φ2 −L/12 f dx = f L , ⎪ ⎪ ⎩ φ3 ⎪ ⎭ ⎩ 1/2 ⎪ ⎭ φ4 L/12

(e0 a )2 f  BT dx =

 0

L

⎡

⎧  ⎫ ⎧ ⎫ ⎪ ⎪ ⎨φ  1 ⎪ ⎬ ⎨−1⎪ ⎬ 0 2  φ 2 2  ( e0 a ) f φ  dx = ( e0 a ) f ⎪ ⎪ ⎩ 3⎪ ⎭ ⎩1⎪ ⎭ φ4 0

φ3

6L 4L2 −6L 2L2 22L2 4L3 13L2 −3L3

−12 −6L 12 −6L 54L 13L2 156L −22L2

⎤

6L 2L2 ⎥ , −6L⎦ 2 4L

(40a)

⎤

−13L2 −3L3 ⎥ , −22L2 ⎦ 3 4L

(40b)

φ  4 dx

(40c)

(40d)

(40e)

As in truss systems, contributions to whole structure of all matrices of discrete bending elements should be determined. According to this, in addition to Eqs. (17a) and (17b), relations between rotational freedoms are given as

d13 = u13 , d23 = u23

(41)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

15

Nondimensional frequency

20

15

10

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

5

0 0

10

20

30

40 50 Direction angle (°)

60

70

80

Fig. 14. Influence of direction angle on the nondimensional fundamental frequencies of Frame B.

where d13 and d23 are local rotational freedoms, u13 and u23 are global rotational freedoms. Eqs. (17a), 17b) and (41) in the matrix form can be written as

⎧ ⎫ d11 ⎪ ⎪ ⎪ ⎪d12 ⎪ ⎪ ⎪ ⎨ ⎪ ⎬

⎡

c ⎢−s ⎢0 d13 =⎢ ⎢0 d 21 ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ d 0 ⎪ ⎪ 22 ⎩ ⎭ d23 0

  



{d }

s c 0 0 0 0

0 0 1 0 0 0



0 0 0 c −s 0

⎤⎧

⎫

0 ⎪ u11 ⎪ ⎪u ⎪ ⎪ 0⎥ ⎪ ⎪ ⎨ 12 ⎪ ⎬ ⎥ 0⎥ u13 0⎥ ⎪u21 ⎪ ⎦⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩u22 ⎪ ⎭ 1 u23

0 0 0 s c 0

(42)

    {u }

[T ]

Also, the following equation can be defined for the force components

⎧ ⎫ f11 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ f12 ⎪ ⎬

⎡

c ⎢−s ⎢0 f13 =⎢ ⎢0 f 21 ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ f 0 ⎪ ⎪ 22 ⎩ ⎭ f23 0

  



{f}

s c 0 0 0 0

0 0 1 0 0 0



0 0 0 c −s 0

⎤⎧

⎫

0 ⎪ p11 ⎪ ⎪p ⎪ 0⎥ ⎪ ⎪ 12 ⎪ ⎪ ⎨ ⎬ ⎥ 0⎥ p13 0⎥ ⎪ p21 ⎪ ⎦⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ p22 ⎪ ⎭ 1 p23

0 0 0 s c 0

(43)

    {p }

[T ]

The bending stiffness, classical mass and size-dependent mass matrices separately obtained for trusses and frames are combined as follows under suitable freedoms of the most general finite bending element:

⎡

EA L 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 K=⎢ ⎢ EA ⎢− ⎢ L ⎢ ⎢ 0 ⎣ 0

⎡

0 12EI L3 6EI − 2 L 0 12EI − 3 L 6EI − 2 L

140 ⎢ 0 ρ AL ⎢ ⎢ 0 Mc = 420 ⎢ 70 ⎣ 0 0

0 156 22L 0 54 −13L

0 6EI − 2 L 4EI L 0 6EI L2 2EI L 0 22L 4L2 0 13L −3L2

−

EA L 0

0 EA L 0 0 70 0 0 140 0 0

⎤

0 12EI − 3 L 6EI L2 0 12EI L3 6EI L2

0 ⎥ 6EI ⎥ − 2 ⎥ L ⎥ 2EI ⎥ ⎥ L ⎥, ⎥ 0 ⎥ ⎥ 6EI ⎥ ⎥ L2 ⎦ 4EI L

0 54 13L 0 156 −22L

0 −13L⎥ −3L2 ⎥ ⎥, 0 ⎥ ⎦ −22L 4L2

(44)

⎤

(45)

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

16

(a)

(b)

2.4

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

0.9

Nondimensional frequency

Nondimensional frequency

1.2

0.6

0.3

1.8

1.2

0.6

0.0

0.0 0

10

20

30

40 50 Direction angle (°)

60

70

0

80

(c)

10

20

30

40 50 Direction angle (°)

60

70

80

30

40 50 Direction angle (°)

60

70

80

(d)

2.8

12 γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

2.1

Nondimensional frequency

Nondimensional frequency

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

1.4

0.7

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

9

6

3

0.0

0 0

10

20

30

40 50 Direction angle (°)

60

70

80

0

10

20

(e) γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

Nondimensional frequency

16

12

8

4

0

0

10

20

30

40 50 Direction angle (°)

60

70

80

Fig. 15. Influence of direction angle on the nondimensional fundamental frequencies of various types of Frame D (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4 (e) Case 5.

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

⎡

1 L 0

0 6 5L 1 10 0 6 − 5L 1 10

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 Mnl = (e0 a )2 ρ AL⎢ ⎢ 1 ⎢− ⎢ L ⎢ ⎢ 0 ⎣ 0

⎤

1 L 0

−

0 1 10 2L 15 0 1 − 10 L − 30

17

0 6 − 5L 1 − 10 0 6 5L 1 − 10

0 1 L 0 0

0 ⎥ 1 ⎥ ⎥ 10 ⎥ L ⎥ − ⎥ 30 ⎥ ⎥ 0 ⎥ ⎥ 1 ⎥ − ⎥ 10 ⎦ 2L 15

(46)

Nonlocal eigenvalue formulation of frames is presented as



n

[ki ]n×n − ωi

det

!

n 2

i=1

=0

[mi ]n×n

(47)

i=1

where [ki ] and [mi ] explain global bending stiffness and mass matrices of ith bending element, respectively and are given as follows

⎡

k11

k12 k22

⎢ ⎢

[k] = [T ] [K ][T ] = ⎢ ⎢ T

k13 k23 k33

sym.

⎣

⎡ ⎢ ⎢ [m] = [T ] [Mc ][T ] + [T ] [Mnl ][T ] = ⎢       ⎢ ⎣ [m ] [m ] T

k14 k24 k34 k44

k15 k25 k35 k45 k55

m11

m12 m22

m13 m23 m33

T

c

⎤

k16 k26 ⎥ k36 ⎥ ⎥ k46 ⎥ ⎦ k56 k66

sym.

(48)

m14 m24 m34 m44

⎤

m15 m25 m35 m45 m55

nl

m16 m26 ⎥ m36 ⎥ ⎥ m46 ⎥ ⎦ m56 m66

(49)

The elements of bending stiffness matrix are given as

k11

EA 2 12EI 2 = c + 3 s , k12 = L L













12EI 6EI EA 12EI EA − 3 cs, k13 = − 2 s, k14 = − c2 − 3 s2 , L L L L L

EA 6EI 12EI EA 2 12EI 2 6EI − + 3 cs, k16 = − 2 s, k22 = s + 3 c , k23 = 2 c, L L L L L L

k15 = k24 =

EA EA 12EI 12EI 6EI 4EI 6EI − + 3 cs, k25 = − s2 − 3 c2 , k26 = 2 c, k33 = , k34 = 2 s, L L L L L L L

k35 = −

6EI 2EI EA 2 12EI 2 c, k36 = , k44 = c + 3 s , k45 = L L L2 L

k55 =



m11 =



ρ AL 3

−

 m15 =

 m22 =

 m24 =



12EI EA 6EI − 3 cs, k46 = 2 s, L L L

6EI EA 2 12EI 2 4EI s + 3 c , k56 = − 2 c, k66 = L L L L

(50)

The elements of mass matrix are described as

m13 =



+

(e0 a )2 ρ A L





c2 +





13ρ AL 6 ( e 0 a )2 ρ A 2 + s , m12 = 35 5L

11ρ AL (e0 a )2 ρ A − s, m14 = 210 10 2



4ρ AL ( e0 a ) ρ A + cs, m16 = 105 5L 2



6(e0 a )2 ρ A 2 + c + 3 5L

ρ AL









ρ AL 6

−

(e0 a )2 ρ A L



4ρ AL (e0 a )2 ρ A + cs, m25 = 105 5L









13ρ AL ( e 0 a )2 ρ A 2 + s , m23 = 35 L



6(e0 a )2 ρ A 2 9ρ AL − c + 70 5L



−

c2 +

13ρ AL (e0 a )2 ρ A − s, 420 10 2



 

4ρ AL (e0 a )2 ρ A − cs, 105 5L



6(e0 a )2 ρ A 2 9ρ AL − s , 70 5L



11ρ AL2 (e0 a )2 ρ A + c, 210 10

ρ AL 6

−

( e 0 a )2 ρ A L



s2 ,

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

18

Table 1 First three nondimensional frequencies of nano/micro trusses for different nondimensional nonlocal parameters (α = 45o ). Truss model

Mode

A

1 2 3 1 2 3 1 2 3

B

C

 m26 =

0

0.05

0.1

0.15

0.2

0.54431 1.24442 1.37199 0.13572 0.27022 0.42181 0.09183 0.33408 0.37194

0.54299 1.23928 1.36704 0.13567 0.27016 0.42100 0.09179 0.33400 0.37110

0.53909 1.22414 1.35286 0.13553 0.26998 0.41858 0.09167 0.33379 0.36860

0.53272 1.19985 1.33111 0.13530 0.26967 0.41464 0.09148 0.33343 0.36456

0.52409 1.16798 1.30378 0.13497 0.26924 0.40931 0.09121 0.33293 0.35912



13ρ AL2 ρ AL3 2(e0 a )2 ρ AL (e0 a )2 ρ A − + c, m33 = + , m34 = 420 10 105 15

 m35 =

 m44 =

 m46 =

 m56 =

γ



13ρ AL2 ρ AL3 (e0 a )2 ρ AL (e0 a )2 ρ A − c, m36 = − − , 420 10 140 30

ρ AL 3

+

(e0 a )2 ρ A





c2 +

L





13ρ AL 6(e0 a )2 ρ A 2 + s , m45 = 35 5L

11ρ AL (e0 a )2 ρ A + s, m55 = 210 10 2

−





ρ AL 3



+

6(e0 a )2 ρ A 2 c + 5L

11ρ AL ρ AL3 2(e0 a )2 ρ AL ( e0 a ) ρ A − c, m66 = + 210 10 105 15 2

2





13ρ AL2 (e0 a )2 ρ A − + s, 420 10



−





4ρ AL (e0 a )2 ρ A − cs, 105 5L



13ρ AL (e0 a )2 ρ A 2 + s , 35 L (51)

4. Numerical results and discussions This section contains nonlocal free vibration results of nano/micro scaled trusses and frames. The nondimensional vibration frequency results are given with tables and graphics under different parameters such as mode number, small-scale effect, member length, direction angle, length-to-diameter ratio. Also, the effects of these parameters on the numerical results are discussed in detail. In the scientific literature, any size-dependent mechanical analyses of trusses or frames have not been performed using any higher-order continuum theories. Therefore, all results given in this section are first for scientific literature. Three different truss models and four different frame models, which are displayed in Figs. 5 and 6 respectively, are employed in the numerical analyses. The numerical results are presented by the following expressions

t = ωt Lt

%

ρ E

&

,

f = ωf Lf

2

ρA EI

,

γ=

e0 a L

(52)

where ωt and ωf are nonlocal vibration frequencies of truss and frame, respectively. ϖt and ϖf denote non-dimensional frequency parameters of related systems. Lt and Lf are the specified member length determining geometry of truss and frame. Unless otherwise stated, the specified member length is used as L = Lt = L f = 20 nm and the circular section diameter is chosen as d = 5 nm. In Table 1, first three nondimensional frequencies of different nonlocal truss models are presented under different nondimensional nonlocal parameter values. The nondimensional frequencies are given based on classical elasticity for γ = 0. It is found from the table that the highest frequencies in all modes are obtained for Truss A, which is the truss model of comprising at least number of members, while Truss C, which is the truss model of comprising at the largest number of members, has the lowest frequencies. It is also clearly seen that the nonlocal parameter has a reducing effect on the vibration frequencies in all modes. Moreover, it can be emphasized that nonlocality is more pronounced for Truss A than the others. First three nondimensional frequencies of Trusses A, B, and C are respectively tabulated in Tables 2–4 according to various member lengths and nonlocal parameters. It is clearly seen that the nondimensional classical frequencies for e0 a = 0 are not influenced from the variation of member length. In addition, it is observed that an increase in the truss member length gives rise to an increment in the nondimensional nonlocal frequencies and the nonlocal frequencies approach to the classical ones. On the other hand, the frequencies of Truss A more affected by an increment in the truss member length than those of Trusses B and C, especially in higher modes. Table 5 lists first three nondimensional frequencies for various nano/micro frame models for different nondimensional nonlocal parameters. The highest and the lowest vibration frequencies are obtained for Frames B (α = 45o ) and D (Case 1,

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

Table 2 First three nondimensional frequencies of Truss A for different member lengths and nonlocal parameter (α = 45o ). Mode

Lt (nm)

1

10 20 30 40 50 10 20 30 40 50 10 20 30 40 50

2

3

e0 a (nm) 0

2

5

10

20

0.54431 0.54431 0.54431 0.54431 0.54431 1.24442 1.24442 1.24442 1.24442 1.24442 1.37199 1.37199 1.37199 1.37199 1.37199

0.52409 0.53909 0.54197 0.54299 0.54346 1.16798 1.22414 1.23532 1.23928 1.24113 1.30378 1.35286 1.36327 1.36704 1.36881

0.44225 0.51349 0.53008 0.53620 0.53909 0.92792 1.13064 1.18998 1.21306 1.22414 1.09191 1.27254 1.32253 1.34281 1.35286

0.30515 0.44225 0.49226 0.51349 0.52409 0.64506 0.92792 1.06225 1.13064 1.16798 0.77785 1.09191 1.21498 1.27254 1.30378

0.17188 0.30515 0.39101 0.44225 0.47303 0.37925 0.64506 0.81431 0.92792 1.00675 0.46025 0.77785 0.97326 1.09191 1.16621

Table 3 First three nondimensional frequencies of Truss B for different member lengths and nonlocal parameter (α = 45o ). Mode

1

2

3

Lt (nm)

10 20 30 40 50 10 20 30 40 50 10 20 30 40 50

e0 a (nm) 0

2

5

10

20

0.13572 0.13572 0.13572 0.13572 0.13572 0.27022 0.27022 0.27022 0.27022 0.27022 0.42181 0.42181 0.42181 0.42181 0.42181

0.13497 0.13553 0.13564 0.13567 0.13569 0.26924 0.26998 0.27011 0.27016 0.27018 0.40931 0.41858 0.42037 0.42100 0.42129

0.13121 0.13455 0.13520 0.13543 0.13553 0.26401 0.26868 0.26954 0.26984 0.26998 0.35837 0.40276 0.41301 0.41679 0.41858

0.11985 0.13121 0.13366 0.13455 0.13497 0.24147 0.26401 0.26748 0.26868 0.26924 0.27119 0.35837 0.38963 0.40276 0.40931

0.09245 0.11985 0.12798 0.13121 0.13279 0.15250 0.24147 0.25900 0.26401 0.26627 0.15672 0.27119 0.32558 0.35837 0.37768

Table 4 First three nondimensional frequencies of Truss C for different member lengths and nonlocal parameter (α = 45o ). Mode

Lt (nm)

1

10 20 30 40 50 10 20 30 40 50 10 20 30 40 50

2

3

e0 a (nm) 0

2

5

10

20

0.09183 0.09183 0.09183 0.09183 0.09183 0.33408 0.33408 0.33408 0.33408 0.33408 0.37194 0.37194 0.37194 0.37194 0.37194

0.09121 0.09167 0.09176 0.09179 0.09180 0.33293 0.33379 0.33395 0.33400 0.33403 0.35912 0.36860 0.37045 0.37110 0.37140

0.08812 0.09087 0.09140 0.09158 0.09167 0.30923 0.33230 0.33328 0.33363 0.33379 0.32709 0.35250 0.36289 0.36677 0.36860

0.07893 0.08812 0.09013 0.09087 0.09121 0.22784 0.30923 0.33093 0.33230 0.33293 0.30815 0.32709 0.33941 0.35250 0.35912

0.05814 0.07893 0.08548 0.08812 0.08941 0.14534 0.22784 0.27879 0.30923 0.32772 0.20835 0.30815 0.32191 0.32709 0.32957

19

20

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164 Table 5 First three nondimensional frequencies of nano/micro frames for different nondimensional nonlocal parameters. Frame model

Mode

A

1 2 3 1 2 3 1 2 3 1 2 3

B (α = 45o )

C (Case-1)

D (Case 1) (α = 45o )

γ 0

0.05

0.1

0.15

0.2

3.17908 12.89036 22.02454 16.96097 19.50052 30.12516 1.20664 3.30368 17.10463 0.66288 3.30021 12.95483

3.17344 12.80304 21.86525 16.84658 19.42487 29.37241 1.20340 3.29167 16.74960 0.66210 3.28194 12.91089

3.15669 12.54334 21.40726 16.50528 19.20310 27.46611 1.19382 3.25659 15.74526 0.65975 3.22890 12.76733

3.12934 12.12118 20.70377 15.94442 18.84978 25.11296 1.17829 3.20116 14.30896 0.65589 3.14594 12.13884

3.09222 11.56057 19.82532 15.18386 18.38637 22.85938 1.15742 3.12929 12.76227 0.65059 3.03996 10.92685

Table 6 First three nondimensional frequencies of Frame C with different cases for different nondimensional nonlocal parameters. Case

Mode

1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

2

3

4

5

γ 0

0.05

0.1

0.15

0.2

1.20664 3.30368 17.10463 2.28623 12.12681 24.04446 2.61219 16.66682 19.23151 12.11431 18.89139 25.17324 16.96097 19.50052 30.12516

1.20340 3.29167 16.74960 2.28348 12.01532 23.73471 2.59871 16.36918 19.11753 12.00420 18.81336 24.80708 16.84658 19.42487 29.37241

1.19382 3.25659 15.74526 2.27528 11.69361 22.90602 2.55939 15.49692 18.80953 11.68605 18.58375 23.83827 16.50528 19.20310 27.46611

1.17829 3.20116 14.30896 2.26180 11.19828 21.78337 2.49741 14.18791 18.36518 11.19492 18.21490 22.54826 15.94442 18.84978 25.11296

1.15742 3.12929 12.76227 2.24331 10.58328 20.57626 2.41734 12.71852 17.81873 10.58280 17.72375 21.18650 15.18386 18.38637 22.85938

α = 45o ), respectively. Similar to the small-sized trusses, it is found that the nonlocal parameter is more considerable on the frequencies for nano-scaled frames for higher modes. First three nondimensional frequencies for all cases of Frames C and D (α = 60o ) are respectively presented in Tables 6 and 7. Only difference of Frames C and D is direction angle of vertical element. Case 5 that vibrates in the most rigid conditions has the highest frequency parameters while the lowest frequencies are attained for Case 1 which is the most flexible frame. Also, it is understood that the direction angle with α = 60o more decreases the frequencies of Case 5. According to another deduction from Tables 5 and 6, vibration results of Frame B (α = 45o ) are equal to vibration results of Case 5 of Frame C. Table 8 contains nondimensional fundamental frequencies of Frames B and D for different direction angle values. In Frame B, it is observed that an increase of direction angle firstly leads to an increment in the frequencies but then, the frequencies decrease for higher values of α . This situation is better seen in the graphical results. On the other hand, variation of direction angle for Frame D consistently increases the fundamental frequencies for all cases. In addition to these results, it is observed that the size dependency becomes more evident with high direction angles. Nondimensional fundamental frequencies of different nano/micro frames are given for various length-to-diameter ratios and dimensionless nonlocal parameters in Table 9. It can be stated from the table that in the case of an increase in lengthto-diameter ratio, frequencies of all frames rise. On the other hand, nondimensional fundamental frequencies of Frames C and D (α = 60o ) corresponding to different length-to-diameter ratios are presented in Tables 10 and 11, respectively. It is found from the tables that the nondimensional frequencies of Case 5 are more affected by the increase of the length-todiameter ratio. Additionally, the frequencies of Frame C are always greater than the frequencies of Frame D. Fig. 7 shows the variation of nondimensional frequencies of different nonlocal nano/micro truss models with respect to mode number. It can be easily seen that nondimensional frequencies decline by increasing the nonlocal parameter. It is notable that there is not a regular frequency variation according to increase of mode number. It can be said that the behaviors of frequency variation lines resemble each other for Trusses A and C. However, frequency variation of Truss B displays differences in the first seven modes. On the other hand, it is clearly observed from these graphics that distinct

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

Table 7 First three nondimensional frequencies of Frame D with different cases for different nondimensional nonlocal parameters (α = 60o ). Case

Mode

1

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

2

3

4

5

γ 0

0.05

0.1

0.15

0.2

0.88130 3.54884 15.68763 2.35062 11.42694 17.74034 2.44034 12.81028 16.56727 10.77852 13.64360 24.73122 12.99952 16.79090 26.60581

0.87983 3.53110 15.43921 2.34744 11.32695 17.60029 2.42869 12.73606 16.19185 10.70710 13.53687 24.53395 12.95074 16.62068 26.30389

0.87547 3.47945 14.66154 2.33799 11.03684 17.21875 2.39464 12.51268 15.17977 10.49375 13.24170 23.98605 12.80646 16.12888 25.50723

0.86832 3.39836 13.42963 2.32248 10.58635 16.68447 2.34071 12.11952 13.86254 10.14459 12.82071 23.17845 12.57203 15.37494 24.45022

0.85857 3.29425 12.04496 2.30125 10.02227 16.08417 2.27058 11.44668 12.70175 9.67885 12.34321 22.15819 12.25330 14.45636 23.33303

Table 8 Nondimensional fundamental frequencies of Frame B and Frame D with for different direction angles and nondimensional nonlocal parameters. Frame model

Case

B

–

D

1

α (◦ )

γ

15 30 75 15 30 75 15 30 75 15 30 75 15 30 75 15 30 75

2

3

4

5

0

0.05

0.1

0.15

0.2

9.18633 14.59166 8.55914 0.15689 0.41425 1.06411 0.69527 1.59121 2.39504 1.08835 1.85844 2.55941 1.22585 4.42517 11.83969 1.40636 4.60889 16.62098

9.14603 14.53087 8.52062 0.15685 0.41396 1.06180 0.69481 1.58910 2.39205 1.08566 1.85156 2.54657 1.22501 4.41738 11.73438 1.40531 4.59926 16.55560

9.02824 14.35294 8.40761 0.15672 0.41306 1.05497 0.69343 1.58281 2.38315 1.07768 1.83128 2.50908 1.22252 4.39424 11.42926 1.40217 4.57072 16.36015

8.84165 14.07038 8.22738 0.15650 0.41159 1.04383 0.69114 1.57250 2.36852 1.06474 1.79870 2.44988 1.21839 4.35643 10.95626 1.39698 4.52423 16.00771

8.59884 13.70145 7.99066 0.15620 0.40955 1.02876 0.68798 1.55838 2.34846 1.04734 1.75550 2.37321 1.21268 4.30502 10.36448 1.38982 4.46134 15.23722

Table 9 Nondimensional fundamental frequencies of nano/micro frames for different length-to-diameter ratios and nondimensional nonlocal parameters. Frame model

A

B (α = 45o )

C (Case 1)

D (Case 1) (α = 45o )

L f /d

5 10 20 5 10 20 5 10 20 5 10 20

γ 0

0.05

0.1

0.15

0.2

3.19031 3.20538 3.20916 18.37464 20.05177 20.38944 1.20719 1.20791 1.20809 0.66291 0.66294 0.66295

3.18466 3.19970 3.20348 18.19335 19.74306 20.04976 1.20394 1.20465 1.20483 0.66212 0.66216 0.66216

3.16786 3.18285 3.18661 17.66315 18.88991 19.12280 1.19434 1.19502 1.19519 0.65978 0.65981 0.65982

3.14046 3.15534 3.15907 16.83019 17.67185 17.82478 1.17877 1.17941 1.17958 0.65592 0.65595 0.65596

3.10324 3.11798 3.12168 15.77886 16.29030 16.38066 1.15786 1.15845 1.15860 0.65061 0.65065 0.65065

21

22

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164 Table 10 Nondimensional fundamental frequencies of Frame C with different cases for different nondimensional nonlocal parameters. Case

Lf /d

1

5 10 20 5 10 20 5 10 20 5 10 20 5 10 20

2

3

4

5

γ 0

0.05

0.1

0.15

0.2

1.20719 1.20791 1.20809 2.28893 2.29252 2.29342 2.62834 2.65012 2.65560 12.59373 13.15575 13.27993 18.37464 20.05177 20.38944

1.20394 1.20465 1.20483 2.28617 2.28975 2.29064 2.61459 2.63599 2.64138 12.45998 12.99003 13.10667 18.19335 19.74306 20.04976

1.19434 1.19502 1.19519 2.27794 2.28147 2.28235 2.57451 2.59487 2.60000 12.07912 12.52709 12.62470 17.66315 18.88991 19.12280

1.17877 1.17941 1.17958 2.26440 2.26787 2.26873 2.51140 2.53023 2.53496 11.50607 11.85229 11.92694 16.83019 17.67185 17.82478

1.15786 1.15845 1.15860 2.24584 2.24921 2.25004 2.43000 2.44703 2.45131 10.81342 11.06505 11.11894 15.77886 16.29030 16.38066

Table 11 Nondimensional fundamental frequencies of Frame D with different cases for different nondimensional nonlocal parameters (α = 60o ). Case

1

2

3

4

5

Lf /d

5 10 20 5 10 20 5 10 20 5 10 20 5 10 20

γ 0

0.05

0.1

0.15

0.2

0.88141 0.88155 0.88159 2.35548 2.36190 2.36349 2.47799 2.52987 2.54312 11.65236 12.25667 12.34450 15.55857 17.40111 17.53005

0.87994 0.88008 0.88012 2.35227 2.35865 2.36023 2.46574 2.51674 2.52976 11.53785 12.10812 12.19155 15.46813 17.15827 17.27760

0.87557 0.87571 0.87575 2.34272 2.34896 2.35051 2.42996 2.47848 2.49086 11.20861 11.69244 11.76458 15.18405 16.48490 16.58072

0.86842 0.86855 0.86859 2.32705 2.33307 2.33457 2.37344 2.41827 2.42968 10.70596 11.08452 11.14265 14.67048 15.51617 15.58513

0.85867 0.85879 0.85882 2.30560 2.31134 2.31277 2.30018 2.34061 2.35087 10.08983 10.37208 10.41699 13.91308 14.40414 14.45041

effect of local elasticity in higher modes. The same situation can be emphasized for Fig. 8 where depicted variations of nondimensional frequency of various modes versus increment in the nondimensional nonlocal expression. Also, it can be said that the effect of nonlocality becomes more efficient on the frequencies of Truss A. The influences of member length on the nondimensional fundamental frequencies of different nano/micro-scaled trusses are perused in Fig. 9. It is seen that the decreasing effect of nonlocal parameter e0 a is more considerable in the small member length values. The differences between the classical and nonlocal frequencies diminish due to the increment in the member length. Variation of nondimensional frequencies of different nano/micro frame models against mode number is plotted in Fig. 10. It is clearly seen that both classical and nonlocal frequency curves of all frame types show quite different characteristics from each other. However, it can be given a general deduction: The nonlocal parameter more decreases frequencies in higher modes. Actually, this deduction can be easily figured out from Fig. 11 where illustrates influences of nonlocal expression on the frequencies of various vibration modes. Additionally, it is seen from Fig. 11 that effect of nonlocal parameter is more pronounced for Frame B (α = 45o ). Fig. 12 depicts the effect of member length on the fundamental frequencies of nano/micro frames. In general, frequency variation behaviors are same for Frames A, C, and D (α = 45o ). According to this, classical frequencies are too little influenced by variation of member length while nonlocal frequencies are too influenced. However, member length quite increases classical frequencies as well as nonlocal frequencies for Frame B (α = 45o ). Fig. 13 displays nondimensional fundamental frequencies of Frames C and D (α = 45◦ ) according to nondimensional nonlocal parameter. It can be expressed that difference between frequencies of Cases 2 and 3 is too few. It can be remembered that horizontal elements of Cases 2 and 3 are rigid in directions of only transverse and only axial, respectively. Case 1, whose the related end is free, gives the lowest nondimensional frequency parameters. In Case 4, transverse and axial stiffness are

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

(a)

(b) 3

6 γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

2

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

5 Nondimensional frequency

Nondimensional frequency

23

1

4

3

2

1

0

0 0.1

0.2

0.3

0.4

0.5

Lf /d

0.6

0.7

0.8

0.9

(c)

0.1

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.5

0.6

0.7

0.8

0.9

Lf /d

(d)

1.5

1.0 γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

γ=0 γ = 0.1 γ = 0.2 γ = 0.3 γ = 0.4 γ = 0.5

0.8 Nondimensional frequency

1.2 Nondimensional frequency

0.2

0.9

0.6

0.3

0.6

0.4

0.2

0.0

0.0 0.1

0.2

0.3

0.4

0.5

Lf /d

0.6

0.7

0.8

0.9

0.1

0.2

0.3

0.4

Lf /d

Fig. 16. Effect of length-to-diameter ratio on the nondimensional fundamental frequencies of different nano/micro frames (a) Frame A (b) Frame B (α = 45◦ ) (c) Frame C (Case 1) (d) Frame D (Case 1, α = 45◦ ).

considered together with and its frequency parameters are higher than the frequencies of first three cases. In addition to Case 4, Case 5, that also provides rotational stiffness, has the highest frequency parameters. Fundamental frequencies of Frame B against direction angle values are depicted in Fig. 14. According to this, frequency parameters with low direction angle always rise up to a certain value, but always drop off after the highest value. For instance, this value almost equals to 40° in the classical case (γ = 0) and is reduced by nonlocal parameter. In this case, nondimensional frequencies in a certain range of direction angle are almost same. Fig. 15 presents the fundamental frequencies of Frame D with different cases corresponding to direction angle. It can be stated that direction angle always increases frequencies and size dependency is more effective in larger direction angles. Also, it can be said that direction angle and atomic parameter most change frequencies of Case 5. The effects of length-to-diameter ratio on the nondimensional fundamental frequencies of different nano/micro frames are plotted in Fig. 16. It can be observed that there is an increment in the frequencies due to an increase in the length-todiameter ratio but this increment for Frames C (Case 1) and D (Case 1, α = 45o ) suddenly stops after a certain ratio value. Moreover, effect of dimensionless nonlocal parameter becomes more considerable for higher length-to-diameter ratio. 5. Conclusions Free vibration analyses of nano/micro scaled truss and frame structures are firstly studied with nonlocal elasticity theory and finite element method. A nonlocal matrix motion formulation is obtained utilizing finite element method based

24

H.M. Numanog˘ lu and Ö. Civalek / International Journal of Engineering Science 145 (2019) 103164

weighted residual. And then, nonlocal vibration frequency equation is presented. A detailed numerical investigation is performed for different truss and frame geometries under different parameters. The most general and significant results obtained by considering numerical studies can be summarized as follows:  The nonlocal frequencies are lower than the classical ones due to presence of an internal length scale parameter.  When the sizes of truss and frame structure diminish, the nonlocal parameter becomes significant.  The nonlocal parameter more decreases the frequency values in higher modes. In addition, nonlocal parameter is more significant for rigid supports such as clamped than flexible supports.  The member length does not influence the classical truss frequencies but increases the classical frame frequencies. On the other hand, nonlocality is more significant for higher modes and higher values of e0 a/L ratios.  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