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Dynamics of size-dependant Timoshenko micro beams subjected to moving loads
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Dynamics of size-dependent Timoshenko micro beams subjected to moving loads Ismail Esen PII: DOI: Reference:
S0020-7403(19)34286-9 https://doi.org/10.1016/j.ijmecsci.2020.105501 MS 105501
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 November 2019 26 December 2019 29 January 2020
Please cite this article as: Ismail Esen , Dynamics of size-dependent Timoshenko micro beams subjected to moving loads, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105501
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Highlights: By adapting the finite element method to modified couple stress theory, the moving load problem was first modelled in the literature for small-scale Timoshenko beams. 2-node and 4 DOF micro-Timoshenko finite element was obtained, in which material scale factor and moving mass effects were represented together. All elements affecting system dynamics such as material scale factor, mass effect, mass velocity, applied beam theory and so on. were integrated into the model and their effects were demonstrated through extensive analysis.
Dynamics of size-dependent Timoshenko micro beams subjected to moving loads. Ismail ESEN Department of Mechanical Engineering, Karabuk University, Karabuk, Turkey [email protected] Abstract In this study, dynamic behaviour of size dependent Timoshenko microbeams under the influence of moving loads is modelled and investigated using FEM method. Firstly, the stiffness and mass property matrices of small-scale size-dependent two-node finite element are obtained by using new shape functions derived from the static state of motion equations created using classical Timoshenko beam theory (TBT) and modified couple stress theory (MCST). Subsequently, the load-beam interaction is modelled and integrated with beam motion equations without neglecting the mass inertia effects of the moving load. The effects of different load velocities, mass of the load and material size parameter on the beam dynamics have been investigated in a wide context. Keywords Small scale, couple stress, moving load, Timoshenko beam, FEM 1 Introduction In recent years, the dynamic behaviour of Nano-sized structures has been of increasing interest. Stresses at one point in small-scale systems are also affected by stresses at other points depending on the size scale. The dimensional dependence of the deformation behavior of metallic materials was experimentally observed in micro-scale beams by [1–3]. Some studies such as [4,5] have been conducted to model the static and dynamic problems based on classical couple stress elasticity theory. The modified couple stress theory (MCST) for isotropic materials has been developed by [6–8] from the strain gradient theory. A Timoshenko beam model based on the microstructure dependence for static bending and free vibration analysis have been developed by [9] using the MCST. Using experimental data based on the MCST, the static bending analysis of Euler-Bernoulli beams has been presented by [10]. [11] have addressed the problem of stability of axially loaded micro dimensional thin beams based on strain gradient elasticity and couple strain theories. [8] have analytically studied the vibrations of FGM beams using the MCST. [12] have studied the small-scale behaviour of micro FGM beams using the MCST, and they investigated the nonlinear behaviour of post buckling of the FGM micro-beams. For FGM Nano-beams [13] have presented the post buckling behaviour using different beam theories with the MCST. [14] has developed a FE model for micro-cantilevered rotating beams and investigated the lead-lag behaviour of the beam, and for rotating micro-beams [15] have investigated the effect of tangential load using the MCST. Based on FE modelling of the small structures, [14] have formulated a two-node with four and six degree of freedom (DOF) beam elements using the MCST. [16] have proposed a nonlinear model for the cantilevered fluid conveying micro-beams and investigated the size-dependent response using the MCST. [17] has investigated the bending and vibrations of Nano-composite FG beams using the MCST. [18] have proposed a 3D model of FG rotating micro-beams using Euler- Bernoulli beam theory (EBT) with the MCST. [19] have investigated the free vibrations of tapered two directional FG rotating micro-beams using Timoshenko beam theory with the MCST. [20] have modelled micro piezo-beams using laminated EBT and MCST and investigated the response
of the cantilevered beams in a thermo-electro coupling environment. [21] have investigated the buckling, bending and free vibrations of the porous FG micro-plates based on the first order shear deformation theory (FSDT) and MCST. [22] have proposed an experimental method for defining the material length scale factor using a laser Doppler measurement system. [23] have presented a mixed FE method with the MCST for the buckling, bending and free vibration analysis of FG micro-beams. [24] have presented the buckling and bending analysis of laminated micro-plates based on the MCST. [25] have presented vibration response analysis of the laminated Timoshenko rotating micro beams using the FEM with remodified couple stress theory for anisotropic elasticity. [26] has presented a general non-local beam theory for bending, buckling and vibration of micro-beams based on the Eringen’s theory. [27] has presented the non-local behaviour of micro-beams under a harmonic moving load using the nonlocal elasticity theory. [28] have presented the dynamic response of Nanocomposite beams subjected to a moving load using EBT and TBT with the classical elasticity theory. [29] have modelled the interaction of a Nano moving mass and single-walled CNT using different beam theories with the Eringen’s nonlocal continuum theory. For the large scale structures [30–35] have presented the dynamic responses of the structures using analytical and FEM methods which accounts the inertial effects of the mass of the moving loads. Every day, new types of micro systems are encountered in practice, and realistic modeling of these systems is important for their accurate operation and development of future versions. Many of these systems are micro-carrier systems with moving mass interaction. In the literature, the moving load problem of small scale structures is modeled using the Eringen’s non-local continuum theory in general. However, the application of this and other non-local stress theories to the moving load problem has not been studied extensively in the literature. Modeling the moving mass problem using the modified couple stress theory and its adaptation to the finite element method are presented first in this study. With the application of MCST to timoshenko beam theory, a method using 4 DOF finite elements with 2 nodes has been developed and all factors such as material scale factor, moving mass, mass velocity etc. which affect dynamic behavior are integrated into this model. 2. Small scale Timoshenko beam model xp xm Z
M
v
X s L, A, ρ, E, I, G, υ, b, h, l Fig. 1. A small-scale Timoshenko beam under a moving load of mass M
Figure 1 shows a uniform rectangular cross-section micro-dimensional Timoshenko beam under the influence of a moving load with the given parameters. The mass of the load M and the load travel from left to right on the beam at a constant speed v, and s indicates the element in which the load is in contact, xp indicates the distance of the load from the left end, and xm indicates the distance on the s element. Here, symbols L, A, ρ, E, I, G, υ, b, h and l are the
beam length, cross-sectional area, mass density, modulus of elasticity, second moment of cross-sectional area, shear modulus, Poisson's ratio, width, height and material size-scale parameter of the beam. 2.1
The couple stress theory
One can use the principle of virtual displacements to obtain the dynamic equations of motion of the modified couple stress theory (MCST): [36] ∫ (
)
(1)
Here T virtual kinetic energy, U virtual strain energy and W is the virtual work done by external forces. According to MCST, the virtual strain energy U is obtained as follows [37] ∫ ∫ (
)
(2)
.
Here denotes the components of the deviatoric portion of the couple stress tensor and are the components of the curvature tensor. . And
(
(3)
), parts of the rotation vector: (
)
Where (
As and
/
(
)
(
)
(4)
), and and are independent of according to the assumptions in Fig. 1, remain from equations (3 and 4) and the others become zero.
2.2 Application to the Timoshenko beam theory In Timoshenko beam theory the non-zero displacements (
) (
( )
)
( (
and
, are [37]
)
(5)
)
where is the arbitrary rotation of the cross-section and is not equal shear strain. The strains and stresses in accordance with Eqs.(3-7) are [37, 38] .
/
due to the
(6)
(
)
.
/
(7)
For bending only, based on Eqs.(2 and 6) the strain energy is ∫ ∫ (
) (8)
∫ (
(
)
(
)
.
Here is the shear correction coefficient and kinetic energy related to and is
∫
∫ ( ̇ ̇
/ )
is the scale factor of the material. The
) (9)
∫ 0 (
)
(
) 1
Since the mass moves on the deflected micro beam, its kinetic energy Tm is [35,39] ( ̇
*
́ )
+
(10)
In the case of the continuous contact, the inter action force in transverse direction is [35]: 0
.
/1 (
)
(11)
Here , and are the inertia, Coriolis and centripetal forces, is the gravity force of the load, δ(.) is the Dirac delta function. When the terms in the second part of Eq. (11) are neglected the moving point force case is obtained. The work done by the force
is (
)
(12)
In terms of Eq.(1) and (8, 9, 10 and 12) with respect to equations of the micro-Timoshenko beam:
.
/
.
/
and
one can obtain the
(13)
(
)
.
/
Without the moving mass, the force boundary conditions are (
)
.
/
(
)
.
,
(14)
/
Here Mx and V are the bending moment and shear force at the ends. The boundary conditions of the micro-beam in Fig. 1 are: (
)
(
)
(15)
2.3. The two-node micro-Timoshenko beam element. For the unloaded case and with the following variables, the governing equations in (13) can be rewritten as follows [40]: .
.
/
/
(
(16)
)
(17)
Here After integrating, Eq. (16) yields:
(18)
The Eq. (17) with the .
/
in Eq. (18) is (
When Eq. (19) is arranged
)
.
/
.
/
(19)
(
(20)
)
here (21) With ̅
Eq. (20) is [40] (
(22)
)
When the higher order term is neglected, the Eq. (22) results in [38,40]
(
(23)
)
After double integration of Eq. (23), it yields:
( Here
and
(24)
)
are integration constants. Substituting Eq. (23) into Eq. (18) results:
( ( After defining and
)
the transverse displacement
(
(
)
)
(
[
(
and rotation angle
can defined as
)
∫( 0
(25)
)
1
)
(26) (
)
)]
Nodal boundary conditons of the micro-beam element are (
)
(
)
(
)
(
)
(27)
When the above baundary conditions are introduced into Eq. (26), the unknown coefficients are derived as )[
( (
) (
] (
)
(28)
)
. Here (
)
(29)
Introducing the coefficients in Eq.(28) into Eq. (26) and arranging according to the nodal displacements yields ( ( C=[ ( (
) ) ], ) )
(30)
* +
From Eq. (30) ( )
(31)
( )
Here ( ) [ ( ) ( )] as given in Eq(26) and ( ) [ ( ) ( )] , and [ ] ( ) [ ] , and , where and ( ) ,(given in Appendix (A.1)), are the derived interpolation functions of the micro-Timoshenko beam. Introducing Eq. (31) into Eq. (9)-(12) results in ( ̇ (
)
(32)
) ̇
(33) (34)
Where ∫ {
.
/ .
/
(
) (
)
(35)
.
/ .
/}
.
/
∫
(36)
∫ (
(37) )
(
[
(38) )
] ,
(39)
-
(40)
Here , and are the stiffness, mass and damping matrices due to the convective acceleration and they are valid only for the element s when x=xp. 2.4 Modelling of the damping The damping matrix for any unloaded element can be derived as
(
)
(
)
(41)
And for beam element s it is:
3.
Dynamics of the micro-beam and moving load system
The governing equation for the micro-beam- moving load system is ̈
̇
(42)
Here, M, C, and K are the assembled matrices of the system, q , q and q are the vectors of acceleration velocity and displacements, respectively; and F is the time dependent applied force vector. Where M, C, and K are obtained from the elemental matrices (35-39 and 41), and F has zero coefficients, except for the force coefficients (40) of for the element s as [
]
(43)
4. Results and discussion with numerical solutions 4.1 Validation and the frequency variation of the micro beam In order to compare the study and the formulation, the calculations were made by considering a micro beam which is given in the literature and whose features are presented below [26,27]. Where E = 1TPa, h = 1e-9 m, ρ = 2300 kg/m3, υ = 0.3. The homogeneous solution of undamped case of Eq. (42) is (
)
(44)
From non-trivial solution of the above equation, =0, where are the eigen values, and for the frequency parameter i of the ith mode of the free vibrations of the micro beam one can write: (45)
√
Firstly, the number of finite elements was determined for the sensitivity analysis of the study based on the frequency parameter and it was determined that 14 finite elements were enough for the required sensitivity as given in Tables 1-3. Using the Timoshenko beam theory (TBT) and Euler-Bernoulli thin beam theory (EBT) (where α= 0, Φ = 0), the analyses were made separately and the results of the analysis of 14 finite elements which were given comparatively, were observed to be very close to the analytical results in the literature which accounts 12 mode in the Galarkin’s approximation. The effect of the material scale parameter on the frequency parameters of 1-4 modes of the microbeam is given in Table (4-6) comparatively. In the tables, the effects of different material scale parameters as l = 1h, 2h and 3h on frequencies based on TBT and EBT beam theories are seen. When the tables are examined, frequency parameters are increasing with respect to beam theory, material scale factor and slenderness ratio. This change in frequencies is not linear, but generally increases as the material scale parameter increases, and the proportions in the increase also depend on the slenderness ratio and the applied beam theory. This non-linear and incremental change will directly lead to the change in beam dynamics and the dynamic responses of the beam under load will be non-linear. In this respect, it is important to make models and designs considering that the linear behaviour known in classical beams does not apply to micro-beams, since the slenderness ratio and applied beam theory will affect the results in addition to the material scale parameter in micro-beams systems. Table 1. Comparisons of the λ1 λ1
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10
9.8702
9.8696
9.8696
9.8696
9.8696
9.8696
9.8696
20
9.8701
9.8698
9.8697
9.8697
9.8696
9.8696
9.8696
50
9.8701
9.8698
9.8697
9.8696
9.8697
9.8696
9.8696
10
9.7304
9.7293
9.7284
9.7281
9.7443
-
9.7443
20
9.8344
9.8338
9.8336
9.8336
9.8381
-
9.8381
50
9.8644
9.8640
9.8639
9.8638
9.8645
-
9.8645
Table 2. Comparisons of the λ2 λ2
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10 20 50
39.5117 39.5103 39.5104
39.4857 39.4888 39.4887
39.4813 39.4829 39.4826
39.4831 39.4807 39.4804
39.4784 39.4784 39.4784
39.4784 39.4784 39.4784
39.4784 39.4784 39.4784
10
37.5263
37.4562
37.4209
37.4069
36.8406
-
36.8406
20
38.9764
38.9403
38.9267
38.9216
38.9645
-
38.9645
50
39.4230
39.3989
39.3917
39.3888
39.3976
-
39.3976
Table 3. Comparisons of the λ3 λ3
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10
89.1879
88.9241
88.8640
88.8610
88.8264
88.8264
88.8264
20
89.1769
88.9416
88.8750
88.8508
88.8264
88.8264
88.8264
50
89.1771
88.9408
88.8738
88.8493
88.8264
88.8264
88.8264
10
80.6980
80.0460
79.7446
79.6096
57.4499
-
57.4499
20
86.7365
86.3625
86.2225
86.1625
85.7483
-
85.7483
50
88.7685
88.5081
88.4288
88.3973
88.4147
-
88.4147
Table 4. Comparisons of the λ1- λ4,for L/h=10, and different scale parameters. L/h=10
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.7443
21.3912
33.7671
41.1633
λ2
36.8406
70.5061
91.8490
99.6557
λ3
57.4499
129.2752
151.7516
158.2101
λ4
125.3644
191.7141
213.1300
218.5310
λ1
9.8696
23.0586
41.4474
57.9812
λ2
39.4784
88.7062
147.0822
186.5037
λ3
88.8264
188.7873
286.6166
336.1991
λ4
148.6366
314.9297
442.9881
493.5297
Table 5. Comparisons of the λ1- λ4,for L/h=20, and different scale parameters. L/h=20
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.8381
22.8495
40.3701
55.2294
λ2
38.9645
85.8730
135.9980
165.9619
λ3
85.7483
177.4611
253.1215
287.0021
λ4
147.1073
287.6220
378.6832
411.8364
λ1
9.8696
23.1270
41.5557
58.1015
λ2
39.4784
89.6498
148.0807
187.1783
λ3
88.8264
192.6495
289.2271
337.3859
λ4
155.5004
324.4917
447.3451
495.0551
Table 6. Comparisons of the λ1- λ4,for L/h=50, and different scale parameters. L/h=50
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.8645
23.3054
43.0225
62.6872
λ2
39.3976
92.2763
166.4244
233.6928
λ3
88.4147
204.3796
356.5508
478.2054
λ4
156.0901
356.2589
598.4990
767.9009
λ1
9.8696
23.1463
41.5858
58.1353
λ2
39.4784
89.9183
148.3589
187.3674
λ3
88.8264
193.7655
289.9524
337.7163
λ4
157.6268
327.2899
448.5478
495.4778
Fig. 2 presents the variation of the first four frequency parameters of micro-beam according to material scale factor and slenderness ratio using TBT. Here, the slenderness ratios are L / h = 10, 20 and 50; and the material scale factor l = 0 -10 is taken as the coefficients of h in 0.1 steps. Fig. 3 shows the variation of the first two frequency parameters obtained by applying the EBT to the microbeam. When EBT is applied, the variation of frequency parameters according to the beam theory and Slenderness ratio is limited. This situation is shown in the graphs given by zooming on the right side in Figure 3. In this case, the magnitude of the material scale factor determines the actual change in frequencies. However, this change is also nonlinear and increases rapidly with the small values of the material scale parameter, the rate of increase continues to decrease after a value and converges to a limit at very large l values. However, as shown in Figure 2, if TBT is applied, in small slenderness ratios, such as L / h = 10, the convergence of the frequency parameter to the upper limit occurs at lower material scale factors. Here, the importance of the beam theory applied according to the slenderness ratio of the microbeam emerges. For example, for L / h = 10, the second frequency parameter, which converges to around 110 in TBT, converges to around 260 in EBT. In large slenderness ratios, such as L / h = 20, 50, it can be seen from the graphs that the frequency parameters are more affected and increased by the material scale factor. And again, interestingly, for the slenderness ratio L / h = 50, as shown in Figure 4c, the fundamental frequency parameter in
TBT results is greater than the results of EBT. In Figures 4 a and b, the situation is reversed at L / h = 10 and 20 and the EBT results for fundamental frequency are greater than TBT results.
a)
c)
b)
d)
Fig.2 Variations of the first four frequency parameters for the material scale parameters of l=0 -10h, Timoshenko beam theory (TBT), and the slenderness ratios of L/h=10,20 and 50; for the frequencies 1-4: a) λ1, b) λ2, c) λ3, d) λ4
a)
b)
c)
d)
Fig.3 Variations of the first two frequency parameters for the material scale parameter of l=0 -10h, Euler Bernoulli Theory (EBT), and the slenderness ratios of L/h=10,20 and 50; a) λ1, b) the zoom of λ1, c) λ2, d) the zoom of λ2
4.2
Forced response of the micro beam due to the moving mass.
The critical velocity of the micro load is described [41] as vcr=ω1 L/π, and the speed parameter is μ=v/ vcr . For the forced response analysis of the same micro beam given above under the effect of the moving micro load, the Eq. (40) has been solved using Newmark’s beta method with 500-time steps, where Δt=tfinal/500, and tfinal=L/v. A micro load of mass M which is equal to the mass of the micro beam, ρAL, has been moved over the beam from left to right with a dimensionless velocity μ between 1/ vcr and 2.5, and with an increment of 1/ vcr. For each analysis the symbol ε states the mass ratio, the ratio of the moving micro load to the mass of the micro beam. In general when only the gravity effect of the load is accounted the results are labelled by moving load (ML) as in the case of the most studies in the literature such as [26,28,42,43]. When the other effects of the mass described in Eqs. (11, 36, 38 and 39) are considered the results are defined by moving mass (MM) such as given in [29]. It is understood from the analyses that the effect of mass is remarkable, even in small micro
dimensions. Therefore, unlike the literature, for the Nano-structures, the results of both cases are presented comparatively in this study and it is shown that the mass effect is as important in the Nano-size applications as it is in the large-size applications [30,34,44] of the moving load. In all subsequent analyses, the L / h ratio is 20 and TBT is applied. As mentioned above, h = 1e-9 m and modulus of elasticity, material density and Poisson’s ratio were not changed. Figure 5 shows the results of moving load analysis depending on the velocity of the load and the material scale parameter as the maximum displacement of the midpoint of the microbeam. Here DAF is defined as wmax (L/2, t) / wst, where wst is the static displacement that occurs when the load is applied at the midpoint of the beam. In the analysis, material scale parameter was taken as l = (0, 0.1, 0.25, 0.5 and 1h) in accordance with the values used in the literature. Figure 6 shows the results obtained by taking the mass effect into account with the above assumptions. This is indicated by the MM symbol in the Figures for analyses which consider the mass effect.
a)
b)
c) Fig.4 Comparisons of the variation of the first frequency parameter λ1 versus material scale parameters of l=0 -10h, and for the applied beam theories of EBT and TBT, and the slenderness ratios: a) L/h=10, b) L/h=20, c) L/h= 50.
Fig.5. DAFs versus speed parameter μ of the micro beam depending on material scale parameters of l=0, 0.1e-9, .25e-9, 0.5e-9 and 1e-9 m, and for ε=1, ML (Moving Load).
Fig.6. DAFs versus speed parameter μ of the micro beam depending on material scale parameters of l=0, 0.1e-9, .25e-9, 0.5e-9 and 1e-9 m, and for the mass ratio ε=1, MM (Moving Mass).
In view of the results given in Figure 6, considering the mass effect increases the DAFs by about 1.4 times in general. In both cases, the increase in material scale factor significantly reduces dynamic displacements. The effect of the increase of this factor is very similar to the effect of increasing the bending stiffness of the beam. In large beams, the increase in bending stiffness of the beam is generally known as the increase of EI. For this, both E and beam cross section width and height should be increased. As can be seen in Tables 4-6 in the previous Tables, the increase of the material scale factor causes the increase of frequency parameters. This means an increase in the natural frequency, and the increase in the natural frequency of
the system caused the vibration pattern as already seen in Figures 5 and 6 to change, and in the analysis results for large material scale factors, the maximums in the DAFs occur at larger velocities, i.e., the velocity at which resonance occurs shifts to higher values. Figures 7, 8, 9 and 10 show the time-dependent vibration response of the microbeam at different mass ratios. Figures 7 and 8 illustrate the effects of l = 0, and Figures 9 and 10 show the effects of the material scale parameter l = 0.1h, i.e., 0.1e-9 m. Where μ = 0.25, the mass ratios are ε= 0.25, 0.5 and 1. As can be seen from the figures (Figs. 9 and 10), the effect of the small material-scale-factor is not as great as the effect of the mass ratio in changing the vibration shape, even if it reduces the vibration amplitude of the microbeam. However, this is true for the relatively low material scale factor, and we will further demonstrate that the effect increases significantly at relatively large values from the analysis results. For a material scale factor of this value, the maximum response in Fig. 7 is about 1.3 and in Fig. 9 is about 1.2. Similarly, the maximum value for l = 0 in Fig. 4 is higher than the maximum for l = 0.1h in Fig.6.
Fig.7. Time response of the micro beam for speed parameter of μ=0.25, and material scale parameter of l=0, and mass ratios of ε=0.25, 0.5 and 1, ML (Moving Load).
Fig.8. Time response of the micro beam for speed parameter of μ=0.25, and material scale parameter of l=0, and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Mass).
Fig.9. Time response of the micro beam for speed parameter of μ=0.25, material scale parameter of l=0.1e-9 and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Load).
Fig.10. Time response of the micro beam for a speed parameter of μ=0.25, material scale parameter of l=0.1e-9 m, and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Mass).
In Figures 11 and 12, the time-dependent responses of the microbeam for velocity parameter μ = 0.25, material scale factor l = 0, 0.1, 0.25, 0.5 and 1h and mass ratio ε= 0.25 are presented. As mentioned above, the effect of the material scale parameter can be better observed in Figures 11 and 12. While small scale factor l = 0.1h, very little shape change of the vibration is observed, whereas the change in shape of vibration in Large scale factor values appears to be obvious. Here we also understand how important the Couple stress theory is in small scale (micro-systems). The need for the development of this theory, as given in the introduction of this study, is triggered from the formation of different behaviours than expected in micro system applications. Especially in micro electro-mechanical systems (mems) and Nanotechnology applications, the effect of material scale factor will continue to be the subject of future scientific studies. As the scale factor of the material grows, it dominates and reduces the mass effect. In Figure 11, the beam was able to vibrate 4.5 times for ML and l = 1h, whereas in Figure 12 the beam was able to vibrate 4 times for the same l. The effect of considering the effect of the mass in terms of vibration characteristics of the micro beam is 12.5%. The effect of the material scale factor of this size is around 378%, as shown in the figures; the number of full vibrations of the beam at l = 0 of is 1.7, whereas for l = 1 the number of vibrations is 4.5. Figures 13 and 14 show the analysis results for the mass ratio ε = 0.5, and Figures 15 and 16 show the results for the mass ratio ε=1. The velocity parameter is the same in all and μ = 0.25, and material scale factors are l = 0, 0.1, 0.25, 0.5 and 1 h. In these analysis results, the dominant effect of the material scale factor is clearly observed. As expected, the increase in mass ratio increases the maximum responses at low material scale factors, but the effect of mass is low at high material scale factors. This is also evident from the figures, and it can be clearly seen that the effect of the material scale factor is more important when looking closely at the vibration patterns in the figures. Figures 13-16 show the effect of the mass ratio size as well as the material scale factor. As can be seen in Figures 14 and 16 MM, the vibrational shape of the micro beam for the small material scale factor and the large mass ratio is almost like half the sine wave. Although the mass ratio is large, in the case of large material scale factors, the vibration shape is in the form of two full sine waves. In any case, the dominance of the material scale factor in the dynamic behaviour of the microbeam is in the foreground.
Fig.11. Comparisons of the time response of the micro beam for a speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9m , and mass ratio of ε=0.25, ML.
Fig.12. Comparisons of the time response of the micro beam for a speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.25, MM.
Fig.13. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.5, ML.
Fig.14. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.5, MM.
Fig.15. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9, and mass ratio of ε=1, ML.
Fig.16. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=1, MM.
5.
Conclusions
In this study, the dynamic behaviour of micro-Timoshenko beams under the effect of moving mass is modelled by using finite element method and modified couple stress theory. In the model, all effects of moving mass are considered and the effects of mass and material scale parameter on the dynamics of the micro beam are examined in a wide framework. The results
of the analyses are presented in detail by taking into consideration the different mass velocities, mass ratios and material scale factors. In general, it has been observed that the magnitude of the material scale parameter increased the natural vibration frequencies of the microbeam. It is known from the general applications in the literature that the mass effect decreases these frequencies. The effect of the material scale factor on the dynamic behaviour of the microbeam is very large and important compared with the effects of speed and mass. As mentioned above, the DAFs of the microbeam are quite low in the large values of the material scale factor which causes the increase of natural frequencies. The effect here is similar to the effect of increasing beam stiffness, and the higher size factor makes the system more rigid and causes the resonance velocity to rise. Although depending on the slenderness ratio, it is observed that the frequency parameters no longer increase but converge to a value as a result of increasing material scale factor to a certain value. In addition to the effect of material scale factor and slenderness ratio, the applied beam theory also leads to changes in the dynamic behaviour of micro beams. Therefore, it is very important to determine the material scale factor correctly according to the material type and to apply the most suitable beam theory according to the slenderness ratio in terms of the accuracy of the results. In the literature, modified couple stress theory has not been studied in detail with these dimensions. In this study, micro beam systems with moving load interaction have been modelled and all the factors affecting dynamic behaviour are integrated in the model and some results have been presented to readers by using the obtained model.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Appendix A ( ) and
( ) in Eq. (31) are ( ) ( )
[ [
], ].
(A.1)
Where ( (
)
)
( )
(
[ ( )
( )]
)( )
(
) ( )] (A.2)
( (
[ ( )
) )
[ ( )
( )
[ ( )
(
(
)
(
]( )( )
[ ( *
) (
)( )
( )(
( )(
)
( )]
(
( )]
) ) ) )+ ( )
(A.3)
Graphical Abstract
Dynamics of size-dependent Timoshenko micro beams subjected to moving loads Ismail Esen PII: DOI: Reference:
S0020-7403(19)34286-9 https://doi.org/10.1016/j.ijmecsci.2020.105501 MS 105501
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
9 November 2019 26 December 2019 29 January 2020
Please cite this article as: Ismail Esen , Dynamics of size-dependent Timoshenko micro beams subjected to moving loads, International Journal of Mechanical Sciences (2020), doi: https://doi.org/10.1016/j.ijmecsci.2020.105501
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1
Highlights: By adapting the finite element method to modified couple stress theory, the moving load problem was first modelled in the literature for small-scale Timoshenko beams. 2-node and 4 DOF micro-Timoshenko finite element was obtained, in which material scale factor and moving mass effects were represented together. All elements affecting system dynamics such as material scale factor, mass effect, mass velocity, applied beam theory and so on. were integrated into the model and their effects were demonstrated through extensive analysis.
Dynamics of size-dependent Timoshenko micro beams subjected to moving loads. Ismail ESEN Department of Mechanical Engineering, Karabuk University, Karabuk, Turkey [email protected] Abstract In this study, dynamic behaviour of size dependent Timoshenko microbeams under the influence of moving loads is modelled and investigated using FEM method. Firstly, the stiffness and mass property matrices of small-scale size-dependent two-node finite element are obtained by using new shape functions derived from the static state of motion equations created using classical Timoshenko beam theory (TBT) and modified couple stress theory (MCST). Subsequently, the load-beam interaction is modelled and integrated with beam motion equations without neglecting the mass inertia effects of the moving load. The effects of different load velocities, mass of the load and material size parameter on the beam dynamics have been investigated in a wide context. Keywords Small scale, couple stress, moving load, Timoshenko beam, FEM 1 Introduction In recent years, the dynamic behaviour of Nano-sized structures has been of increasing interest. Stresses at one point in small-scale systems are also affected by stresses at other points depending on the size scale. The dimensional dependence of the deformation behavior of metallic materials was experimentally observed in micro-scale beams by [1–3]. Some studies such as [4,5] have been conducted to model the static and dynamic problems based on classical couple stress elasticity theory. The modified couple stress theory (MCST) for isotropic materials has been developed by [6–8] from the strain gradient theory. A Timoshenko beam model based on the microstructure dependence for static bending and free vibration analysis have been developed by [9] using the MCST. Using experimental data based on the MCST, the static bending analysis of Euler-Bernoulli beams has been presented by [10]. [11] have addressed the problem of stability of axially loaded micro dimensional thin beams based on strain gradient elasticity and couple strain theories. [8] have analytically studied the vibrations of FGM beams using the MCST. [12] have studied the small-scale behaviour of micro FGM beams using the MCST, and they investigated the nonlinear behaviour of post buckling of the FGM micro-beams. For FGM Nano-beams [13] have presented the post buckling behaviour using different beam theories with the MCST. [14] has developed a FE model for micro-cantilevered rotating beams and investigated the lead-lag behaviour of the beam, and for rotating micro-beams [15] have investigated the effect of tangential load using the MCST. Based on FE modelling of the small structures, [14] have formulated a two-node with four and six degree of freedom (DOF) beam elements using the MCST. [16] have proposed a nonlinear model for the cantilevered fluid conveying micro-beams and investigated the size-dependent response using the MCST. [17] has investigated the bending and vibrations of Nano-composite FG beams using the MCST. [18] have proposed a 3D model of FG rotating micro-beams using Euler- Bernoulli beam theory (EBT) with the MCST. [19] have investigated the free vibrations of tapered two directional FG rotating micro-beams using Timoshenko beam theory with the MCST. [20] have modelled micro piezo-beams using laminated EBT and MCST and investigated the response
of the cantilevered beams in a thermo-electro coupling environment. [21] have investigated the buckling, bending and free vibrations of the porous FG micro-plates based on the first order shear deformation theory (FSDT) and MCST. [22] have proposed an experimental method for defining the material length scale factor using a laser Doppler measurement system. [23] have presented a mixed FE method with the MCST for the buckling, bending and free vibration analysis of FG micro-beams. [24] have presented the buckling and bending analysis of laminated micro-plates based on the MCST. [25] have presented vibration response analysis of the laminated Timoshenko rotating micro beams using the FEM with remodified couple stress theory for anisotropic elasticity. [26] has presented a general non-local beam theory for bending, buckling and vibration of micro-beams based on the Eringen’s theory. [27] has presented the non-local behaviour of micro-beams under a harmonic moving load using the nonlocal elasticity theory. [28] have presented the dynamic response of Nanocomposite beams subjected to a moving load using EBT and TBT with the classical elasticity theory. [29] have modelled the interaction of a Nano moving mass and single-walled CNT using different beam theories with the Eringen’s nonlocal continuum theory. For the large scale structures [30–35] have presented the dynamic responses of the structures using analytical and FEM methods which accounts the inertial effects of the mass of the moving loads. Every day, new types of micro systems are encountered in practice, and realistic modeling of these systems is important for their accurate operation and development of future versions. Many of these systems are micro-carrier systems with moving mass interaction. In the literature, the moving load problem of small scale structures is modeled using the Eringen’s non-local continuum theory in general. However, the application of this and other non-local stress theories to the moving load problem has not been studied extensively in the literature. Modeling the moving mass problem using the modified couple stress theory and its adaptation to the finite element method are presented first in this study. With the application of MCST to timoshenko beam theory, a method using 4 DOF finite elements with 2 nodes has been developed and all factors such as material scale factor, moving mass, mass velocity etc. which affect dynamic behavior are integrated into this model. 2. Small scale Timoshenko beam model xp xm Z
M
v
X s L, A, ρ, E, I, G, υ, b, h, l Fig. 1. A small-scale Timoshenko beam under a moving load of mass M
Figure 1 shows a uniform rectangular cross-section micro-dimensional Timoshenko beam under the influence of a moving load with the given parameters. The mass of the load M and the load travel from left to right on the beam at a constant speed v, and s indicates the element in which the load is in contact, xp indicates the distance of the load from the left end, and xm indicates the distance on the s element. Here, symbols L, A, ρ, E, I, G, υ, b, h and l are the
beam length, cross-sectional area, mass density, modulus of elasticity, second moment of cross-sectional area, shear modulus, Poisson's ratio, width, height and material size-scale parameter of the beam. 2.1
The couple stress theory
One can use the principle of virtual displacements to obtain the dynamic equations of motion of the modified couple stress theory (MCST): [36] ∫ (
)
(1)
Here T virtual kinetic energy, U virtual strain energy and W is the virtual work done by external forces. According to MCST, the virtual strain energy U is obtained as follows [37] ∫ ∫ (
)
(2)
.
Here denotes the components of the deviatoric portion of the couple stress tensor and are the components of the curvature tensor. . And
(
(3)
), parts of the rotation vector: (
)
Where (
As and
/
(
)
(
)
(4)
), and and are independent of according to the assumptions in Fig. 1, remain from equations (3 and 4) and the others become zero.
2.2 Application to the Timoshenko beam theory In Timoshenko beam theory the non-zero displacements (
) (
( )
)
( (
and
, are [37]
)
(5)
)
where is the arbitrary rotation of the cross-section and is not equal shear strain. The strains and stresses in accordance with Eqs.(3-7) are [37, 38] .
/
due to the
(6)
(
)
.
/
(7)
For bending only, based on Eqs.(2 and 6) the strain energy is ∫ ∫ (
) (8)
∫ (
(
)
(
)
.
Here is the shear correction coefficient and kinetic energy related to and is
∫
∫ ( ̇ ̇
/ )
is the scale factor of the material. The
) (9)
∫ 0 (
)
(
) 1
Since the mass moves on the deflected micro beam, its kinetic energy Tm is [35,39] ( ̇
*
́ )
+
(10)
In the case of the continuous contact, the inter action force in transverse direction is [35]: 0
.
/1 (
)
(11)
Here , and are the inertia, Coriolis and centripetal forces, is the gravity force of the load, δ(.) is the Dirac delta function. When the terms in the second part of Eq. (11) are neglected the moving point force case is obtained. The work done by the force
is (
)
(12)
In terms of Eq.(1) and (8, 9, 10 and 12) with respect to equations of the micro-Timoshenko beam:
.
/
.
/
and
one can obtain the
(13)
(
)
.
/
Without the moving mass, the force boundary conditions are (
)
.
/
(
)
.
,
(14)
/
Here Mx and V are the bending moment and shear force at the ends. The boundary conditions of the micro-beam in Fig. 1 are: (
)
(
)
(15)
2.3. The two-node micro-Timoshenko beam element. For the unloaded case and with the following variables, the governing equations in (13) can be rewritten as follows [40]: .
.
/
/
(
(16)
)
(17)
Here After integrating, Eq. (16) yields:
(18)
The Eq. (17) with the .
/
in Eq. (18) is (
When Eq. (19) is arranged
)
.
/
.
/
(19)
(
(20)
)
here (21) With ̅
Eq. (20) is [40] (
(22)
)
When the higher order term is neglected, the Eq. (22) results in [38,40]
(
(23)
)
After double integration of Eq. (23), it yields:
( Here
and
(24)
)
are integration constants. Substituting Eq. (23) into Eq. (18) results:
( ( After defining and
)
the transverse displacement
(
(
)
)
(
[
(
and rotation angle
can defined as
)
∫( 0
(25)
)
1
)
(26) (
)
)]
Nodal boundary conditons of the micro-beam element are (
)
(
)
(
)
(
)
(27)
When the above baundary conditions are introduced into Eq. (26), the unknown coefficients are derived as )[
( (
) (
] (
)
(28)
)
. Here (
)
(29)
Introducing the coefficients in Eq.(28) into Eq. (26) and arranging according to the nodal displacements yields ( ( C=[ ( (
) ) ], ) )
(30)
* +
From Eq. (30) ( )
(31)
( )
Here ( ) [ ( ) ( )] as given in Eq(26) and ( ) [ ( ) ( )] , and [ ] ( ) [ ] , and , where and ( ) ,(given in Appendix (A.1)), are the derived interpolation functions of the micro-Timoshenko beam. Introducing Eq. (31) into Eq. (9)-(12) results in ( ̇ (
)
(32)
) ̇
(33) (34)
Where ∫ {
.
/ .
/
(
) (
)
(35)
.
/ .
/}
.
/
∫
(36)
∫ (
(37) )
(
[
(38) )
] ,
(39)
-
(40)
Here , and are the stiffness, mass and damping matrices due to the convective acceleration and they are valid only for the element s when x=xp. 2.4 Modelling of the damping The damping matrix for any unloaded element can be derived as
(
)
(
)
(41)
And for beam element s it is:
3.
Dynamics of the micro-beam and moving load system
The governing equation for the micro-beam- moving load system is ̈
̇
(42)
Here, M, C, and K are the assembled matrices of the system, q , q and q are the vectors of acceleration velocity and displacements, respectively; and F is the time dependent applied force vector. Where M, C, and K are obtained from the elemental matrices (35-39 and 41), and F has zero coefficients, except for the force coefficients (40) of for the element s as [
]
(43)
4. Results and discussion with numerical solutions 4.1 Validation and the frequency variation of the micro beam In order to compare the study and the formulation, the calculations were made by considering a micro beam which is given in the literature and whose features are presented below [26,27]. Where E = 1TPa, h = 1e-9 m, ρ = 2300 kg/m3, υ = 0.3. The homogeneous solution of undamped case of Eq. (42) is (
)
(44)
From non-trivial solution of the above equation, =0, where are the eigen values, and for the frequency parameter i of the ith mode of the free vibrations of the micro beam one can write: (45)
√
Firstly, the number of finite elements was determined for the sensitivity analysis of the study based on the frequency parameter and it was determined that 14 finite elements were enough for the required sensitivity as given in Tables 1-3. Using the Timoshenko beam theory (TBT) and Euler-Bernoulli thin beam theory (EBT) (where α= 0, Φ = 0), the analyses were made separately and the results of the analysis of 14 finite elements which were given comparatively, were observed to be very close to the analytical results in the literature which accounts 12 mode in the Galarkin’s approximation. The effect of the material scale parameter on the frequency parameters of 1-4 modes of the microbeam is given in Table (4-6) comparatively. In the tables, the effects of different material scale parameters as l = 1h, 2h and 3h on frequencies based on TBT and EBT beam theories are seen. When the tables are examined, frequency parameters are increasing with respect to beam theory, material scale factor and slenderness ratio. This change in frequencies is not linear, but generally increases as the material scale parameter increases, and the proportions in the increase also depend on the slenderness ratio and the applied beam theory. This non-linear and incremental change will directly lead to the change in beam dynamics and the dynamic responses of the beam under load will be non-linear. In this respect, it is important to make models and designs considering that the linear behaviour known in classical beams does not apply to micro-beams, since the slenderness ratio and applied beam theory will affect the results in addition to the material scale parameter in micro-beams systems. Table 1. Comparisons of the λ1 λ1
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10
9.8702
9.8696
9.8696
9.8696
9.8696
9.8696
9.8696
20
9.8701
9.8698
9.8697
9.8697
9.8696
9.8696
9.8696
50
9.8701
9.8698
9.8697
9.8696
9.8697
9.8696
9.8696
10
9.7304
9.7293
9.7284
9.7281
9.7443
-
9.7443
20
9.8344
9.8338
9.8336
9.8336
9.8381
-
9.8381
50
9.8644
9.8640
9.8639
9.8638
9.8645
-
9.8645
Table 2. Comparisons of the λ2 λ2
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10 20 50
39.5117 39.5103 39.5104
39.4857 39.4888 39.4887
39.4813 39.4829 39.4826
39.4831 39.4807 39.4804
39.4784 39.4784 39.4784
39.4784 39.4784 39.4784
39.4784 39.4784 39.4784
10
37.5263
37.4562
37.4209
37.4069
36.8406
-
36.8406
20
38.9764
38.9403
38.9267
38.9216
38.9645
-
38.9645
50
39.4230
39.3989
39.3917
39.3888
39.3976
-
39.3976
Table 3. Comparisons of the λ3 λ3
L/h L/h
TBT
EBT
Number of Elements n 6
8
10
12
14
[27]
[26]
10
89.1879
88.9241
88.8640
88.8610
88.8264
88.8264
88.8264
20
89.1769
88.9416
88.8750
88.8508
88.8264
88.8264
88.8264
50
89.1771
88.9408
88.8738
88.8493
88.8264
88.8264
88.8264
10
80.6980
80.0460
79.7446
79.6096
57.4499
-
57.4499
20
86.7365
86.3625
86.2225
86.1625
85.7483
-
85.7483
50
88.7685
88.5081
88.4288
88.3973
88.4147
-
88.4147
Table 4. Comparisons of the λ1- λ4,for L/h=10, and different scale parameters. L/h=10
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.7443
21.3912
33.7671
41.1633
λ2
36.8406
70.5061
91.8490
99.6557
λ3
57.4499
129.2752
151.7516
158.2101
λ4
125.3644
191.7141
213.1300
218.5310
λ1
9.8696
23.0586
41.4474
57.9812
λ2
39.4784
88.7062
147.0822
186.5037
λ3
88.8264
188.7873
286.6166
336.1991
λ4
148.6366
314.9297
442.9881
493.5297
Table 5. Comparisons of the λ1- λ4,for L/h=20, and different scale parameters. L/h=20
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.8381
22.8495
40.3701
55.2294
λ2
38.9645
85.8730
135.9980
165.9619
λ3
85.7483
177.4611
253.1215
287.0021
λ4
147.1073
287.6220
378.6832
411.8364
λ1
9.8696
23.1270
41.5557
58.1015
λ2
39.4784
89.6498
148.0807
187.1783
λ3
88.8264
192.6495
289.2271
337.3859
λ4
155.5004
324.4917
447.3451
495.0551
Table 6. Comparisons of the λ1- λ4,for L/h=50, and different scale parameters. L/h=50
TBT
EBT
l=0
l=1e-9
l=2e-9
l=3e-9
λ1
9.8645
23.3054
43.0225
62.6872
λ2
39.3976
92.2763
166.4244
233.6928
λ3
88.4147
204.3796
356.5508
478.2054
λ4
156.0901
356.2589
598.4990
767.9009
λ1
9.8696
23.1463
41.5858
58.1353
λ2
39.4784
89.9183
148.3589
187.3674
λ3
88.8264
193.7655
289.9524
337.7163
λ4
157.6268
327.2899
448.5478
495.4778
Fig. 2 presents the variation of the first four frequency parameters of micro-beam according to material scale factor and slenderness ratio using TBT. Here, the slenderness ratios are L / h = 10, 20 and 50; and the material scale factor l = 0 -10 is taken as the coefficients of h in 0.1 steps. Fig. 3 shows the variation of the first two frequency parameters obtained by applying the EBT to the microbeam. When EBT is applied, the variation of frequency parameters according to the beam theory and Slenderness ratio is limited. This situation is shown in the graphs given by zooming on the right side in Figure 3. In this case, the magnitude of the material scale factor determines the actual change in frequencies. However, this change is also nonlinear and increases rapidly with the small values of the material scale parameter, the rate of increase continues to decrease after a value and converges to a limit at very large l values. However, as shown in Figure 2, if TBT is applied, in small slenderness ratios, such as L / h = 10, the convergence of the frequency parameter to the upper limit occurs at lower material scale factors. Here, the importance of the beam theory applied according to the slenderness ratio of the microbeam emerges. For example, for L / h = 10, the second frequency parameter, which converges to around 110 in TBT, converges to around 260 in EBT. In large slenderness ratios, such as L / h = 20, 50, it can be seen from the graphs that the frequency parameters are more affected and increased by the material scale factor. And again, interestingly, for the slenderness ratio L / h = 50, as shown in Figure 4c, the fundamental frequency parameter in
TBT results is greater than the results of EBT. In Figures 4 a and b, the situation is reversed at L / h = 10 and 20 and the EBT results for fundamental frequency are greater than TBT results.
a)
c)
b)
d)
Fig.2 Variations of the first four frequency parameters for the material scale parameters of l=0 -10h, Timoshenko beam theory (TBT), and the slenderness ratios of L/h=10,20 and 50; for the frequencies 1-4: a) λ1, b) λ2, c) λ3, d) λ4
a)
b)
c)
d)
Fig.3 Variations of the first two frequency parameters for the material scale parameter of l=0 -10h, Euler Bernoulli Theory (EBT), and the slenderness ratios of L/h=10,20 and 50; a) λ1, b) the zoom of λ1, c) λ2, d) the zoom of λ2
4.2
Forced response of the micro beam due to the moving mass.
The critical velocity of the micro load is described [41] as vcr=ω1 L/π, and the speed parameter is μ=v/ vcr . For the forced response analysis of the same micro beam given above under the effect of the moving micro load, the Eq. (40) has been solved using Newmark’s beta method with 500-time steps, where Δt=tfinal/500, and tfinal=L/v. A micro load of mass M which is equal to the mass of the micro beam, ρAL, has been moved over the beam from left to right with a dimensionless velocity μ between 1/ vcr and 2.5, and with an increment of 1/ vcr. For each analysis the symbol ε states the mass ratio, the ratio of the moving micro load to the mass of the micro beam. In general when only the gravity effect of the load is accounted the results are labelled by moving load (ML) as in the case of the most studies in the literature such as [26,28,42,43]. When the other effects of the mass described in Eqs. (11, 36, 38 and 39) are considered the results are defined by moving mass (MM) such as given in [29]. It is understood from the analyses that the effect of mass is remarkable, even in small micro
dimensions. Therefore, unlike the literature, for the Nano-structures, the results of both cases are presented comparatively in this study and it is shown that the mass effect is as important in the Nano-size applications as it is in the large-size applications [30,34,44] of the moving load. In all subsequent analyses, the L / h ratio is 20 and TBT is applied. As mentioned above, h = 1e-9 m and modulus of elasticity, material density and Poisson’s ratio were not changed. Figure 5 shows the results of moving load analysis depending on the velocity of the load and the material scale parameter as the maximum displacement of the midpoint of the microbeam. Here DAF is defined as wmax (L/2, t) / wst, where wst is the static displacement that occurs when the load is applied at the midpoint of the beam. In the analysis, material scale parameter was taken as l = (0, 0.1, 0.25, 0.5 and 1h) in accordance with the values used in the literature. Figure 6 shows the results obtained by taking the mass effect into account with the above assumptions. This is indicated by the MM symbol in the Figures for analyses which consider the mass effect.
a)
b)
c) Fig.4 Comparisons of the variation of the first frequency parameter λ1 versus material scale parameters of l=0 -10h, and for the applied beam theories of EBT and TBT, and the slenderness ratios: a) L/h=10, b) L/h=20, c) L/h= 50.
Fig.5. DAFs versus speed parameter μ of the micro beam depending on material scale parameters of l=0, 0.1e-9, .25e-9, 0.5e-9 and 1e-9 m, and for ε=1, ML (Moving Load).
Fig.6. DAFs versus speed parameter μ of the micro beam depending on material scale parameters of l=0, 0.1e-9, .25e-9, 0.5e-9 and 1e-9 m, and for the mass ratio ε=1, MM (Moving Mass).
In view of the results given in Figure 6, considering the mass effect increases the DAFs by about 1.4 times in general. In both cases, the increase in material scale factor significantly reduces dynamic displacements. The effect of the increase of this factor is very similar to the effect of increasing the bending stiffness of the beam. In large beams, the increase in bending stiffness of the beam is generally known as the increase of EI. For this, both E and beam cross section width and height should be increased. As can be seen in Tables 4-6 in the previous Tables, the increase of the material scale factor causes the increase of frequency parameters. This means an increase in the natural frequency, and the increase in the natural frequency of
the system caused the vibration pattern as already seen in Figures 5 and 6 to change, and in the analysis results for large material scale factors, the maximums in the DAFs occur at larger velocities, i.e., the velocity at which resonance occurs shifts to higher values. Figures 7, 8, 9 and 10 show the time-dependent vibration response of the microbeam at different mass ratios. Figures 7 and 8 illustrate the effects of l = 0, and Figures 9 and 10 show the effects of the material scale parameter l = 0.1h, i.e., 0.1e-9 m. Where μ = 0.25, the mass ratios are ε= 0.25, 0.5 and 1. As can be seen from the figures (Figs. 9 and 10), the effect of the small material-scale-factor is not as great as the effect of the mass ratio in changing the vibration shape, even if it reduces the vibration amplitude of the microbeam. However, this is true for the relatively low material scale factor, and we will further demonstrate that the effect increases significantly at relatively large values from the analysis results. For a material scale factor of this value, the maximum response in Fig. 7 is about 1.3 and in Fig. 9 is about 1.2. Similarly, the maximum value for l = 0 in Fig. 4 is higher than the maximum for l = 0.1h in Fig.6.
Fig.7. Time response of the micro beam for speed parameter of μ=0.25, and material scale parameter of l=0, and mass ratios of ε=0.25, 0.5 and 1, ML (Moving Load).
Fig.8. Time response of the micro beam for speed parameter of μ=0.25, and material scale parameter of l=0, and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Mass).
Fig.9. Time response of the micro beam for speed parameter of μ=0.25, material scale parameter of l=0.1e-9 and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Load).
Fig.10. Time response of the micro beam for a speed parameter of μ=0.25, material scale parameter of l=0.1e-9 m, and mass ratios of ε=0.25, 0.5 and 1, MM (Moving Mass).
In Figures 11 and 12, the time-dependent responses of the microbeam for velocity parameter μ = 0.25, material scale factor l = 0, 0.1, 0.25, 0.5 and 1h and mass ratio ε= 0.25 are presented. As mentioned above, the effect of the material scale parameter can be better observed in Figures 11 and 12. While small scale factor l = 0.1h, very little shape change of the vibration is observed, whereas the change in shape of vibration in Large scale factor values appears to be obvious. Here we also understand how important the Couple stress theory is in small scale (micro-systems). The need for the development of this theory, as given in the introduction of this study, is triggered from the formation of different behaviours than expected in micro system applications. Especially in micro electro-mechanical systems (mems) and Nanotechnology applications, the effect of material scale factor will continue to be the subject of future scientific studies. As the scale factor of the material grows, it dominates and reduces the mass effect. In Figure 11, the beam was able to vibrate 4.5 times for ML and l = 1h, whereas in Figure 12 the beam was able to vibrate 4 times for the same l. The effect of considering the effect of the mass in terms of vibration characteristics of the micro beam is 12.5%. The effect of the material scale factor of this size is around 378%, as shown in the figures; the number of full vibrations of the beam at l = 0 of is 1.7, whereas for l = 1 the number of vibrations is 4.5. Figures 13 and 14 show the analysis results for the mass ratio ε = 0.5, and Figures 15 and 16 show the results for the mass ratio ε=1. The velocity parameter is the same in all and μ = 0.25, and material scale factors are l = 0, 0.1, 0.25, 0.5 and 1 h. In these analysis results, the dominant effect of the material scale factor is clearly observed. As expected, the increase in mass ratio increases the maximum responses at low material scale factors, but the effect of mass is low at high material scale factors. This is also evident from the figures, and it can be clearly seen that the effect of the material scale factor is more important when looking closely at the vibration patterns in the figures. Figures 13-16 show the effect of the mass ratio size as well as the material scale factor. As can be seen in Figures 14 and 16 MM, the vibrational shape of the micro beam for the small material scale factor and the large mass ratio is almost like half the sine wave. Although the mass ratio is large, in the case of large material scale factors, the vibration shape is in the form of two full sine waves. In any case, the dominance of the material scale factor in the dynamic behaviour of the microbeam is in the foreground.
Fig.11. Comparisons of the time response of the micro beam for a speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9m , and mass ratio of ε=0.25, ML.
Fig.12. Comparisons of the time response of the micro beam for a speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.25, MM.
Fig.13. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.5, ML.
Fig.14. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=0.5, MM.
Fig.15. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9, and mass ratio of ε=1, ML.
Fig.16. Comparisons of the time response of the micro beam for speed parameter of μ=0.25, material scale parameters of l=0, 0.1e-9, 0.25e-9, 0.5e-9 and 1e-9 m, and mass ratio of ε=1, MM.
5.
Conclusions
In this study, the dynamic behaviour of micro-Timoshenko beams under the effect of moving mass is modelled by using finite element method and modified couple stress theory. In the model, all effects of moving mass are considered and the effects of mass and material scale parameter on the dynamics of the micro beam are examined in a wide framework. The results
of the analyses are presented in detail by taking into consideration the different mass velocities, mass ratios and material scale factors. In general, it has been observed that the magnitude of the material scale parameter increased the natural vibration frequencies of the microbeam. It is known from the general applications in the literature that the mass effect decreases these frequencies. The effect of the material scale factor on the dynamic behaviour of the microbeam is very large and important compared with the effects of speed and mass. As mentioned above, the DAFs of the microbeam are quite low in the large values of the material scale factor which causes the increase of natural frequencies. The effect here is similar to the effect of increasing beam stiffness, and the higher size factor makes the system more rigid and causes the resonance velocity to rise. Although depending on the slenderness ratio, it is observed that the frequency parameters no longer increase but converge to a value as a result of increasing material scale factor to a certain value. In addition to the effect of material scale factor and slenderness ratio, the applied beam theory also leads to changes in the dynamic behaviour of micro beams. Therefore, it is very important to determine the material scale factor correctly according to the material type and to apply the most suitable beam theory according to the slenderness ratio in terms of the accuracy of the results. In the literature, modified couple stress theory has not been studied in detail with these dimensions. In this study, micro beam systems with moving load interaction have been modelled and all the factors affecting dynamic behaviour are integrated in the model and some results have been presented to readers by using the obtained model.
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Appendix A ( ) and
( ) in Eq. (31) are ( ) ( )
[ [
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(A.1)
Where ( (
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( )(
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(A.3)
Graphical Abstract