# Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Development

_{s}as shear correction, the potential energy becomes (when Equations (3) and (4) are considered) [69]:

_{r}and c

_{d}denote the damping coefficient related to rotation and displacements, respectively. The external-force one becomes:

## 3. Discretized Equations of Motion and Solution Procedure

_{1}, inserting Equation (14) into Equations (10)–(12) gives:

## 4. Asymmetric Size-Dependent Vibrations

_{R}= h

_{L}= 2 μm; l

_{R}= 0.2 μm; l

_{L}= 0.8 μm; L/h = 60; b

_{L}/h = 1; b

_{R}/h = 3]; a modal damping ratio of ζ=0.009 is used throughout; ζ is defined as the ratio of damping coefficient (c) to critical damping coefficient (c

_{cr}). The AFG microsystem is made of Aluminium at the left end and ceramics (SiC) at the right end; [ρ

_{L}= 2700 kg/m

^{3}; E

_{L}= 69 GPa; ν

_{L}= 0.33], and [ρ

_{R}= 3100 kg/m

^{3}; E

_{R}= 427 GPa; ν

_{R}= 0.17]; K

_{s}=5/6. These values have been used unless otherwise stated.

_{1}= 42.2806. The asymmetry in material distribution and nonuniform geometry give rise to asymmetric modes of vibration. The peak amplitude in the q

_{1}motion is almost 5 times that of the q

_{2}motion; the contribution of the q

_{2}motion is quite strong. The motion (in all the symmetric/asymmetric modes) is hardening with two saddle-type bifurcations at Ω/ω

_{1}= 1.2254 and 1.0447. The number of stable branches is two and that of unstable is one; as the frequency is increased, the response amplitudes in the frequency diagrams increase until hitting point A, where practically a jump occurs to a lower motion amplitude, then the asymmetric/symmetric vibration amplitudes decrease with further frequency increment. A decrease in the frequency from the upper-bound in the figure, the AFG system shows a reverse scenario characterized by an amplitude jump at point B.

_{1}= 42.2806. For all the symmetric and asymmetric components of axial/rotational/transverse motions, there are two bifurcations at f

_{1}= 137.3 and 22.9, representing the jumps. For sufficiently small forces, the vibration motion is stable with increasing asymmetric/symmetric for larger forces; this scenario is violated when hitting point A, where mathematically speaking, there is a jump for both asymmetric/symmetric modes; bifurcation point B plays the same role but for decreasing forces when there is a reverse sweep.

## 5. Concluding Remarks

- Asymmetry and stretching-type nonlinearity in the AFG microsystem cause necessity for a high DOF analysis.
- The size effect is more significant for larger gradient indices.
- For larger gradient indices, the symmetric transverse and rotational-mode peak-amplitudes are larger.
- All the asymmetric and symmetric modes in the coupled transverse/axial/rotational motion display hardening motions.
- The modified CST, in all the asymmetric and symmetric modes, reduces the peak-amplitude and shifts the frequency diagrams to the right.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Axially functionally graded (AFG) microscale nonuniform Timoshenko beam: (

**a**) Side view; (

**b**) top view.

**Figure 2.**Frequency diagrams for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}(b

_{R}= 3b

_{L}, (l

_{s})

_{R}= 0.1, f

_{1}= 40.0, n = 2.0 (l

_{s})

_{L}= 0.4).

**Figure 3.**Frequency diagrams for different degrees of freedom (DOFs); b

_{R}= 3b

_{L}, (l

_{s})

_{R}= 0.1, n = 2.0, f

_{1}= 40.0, (l

_{s})

_{L}= 0.4.

**Figure 4.**Force diagrams for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}(b

_{R}= 3b

_{L}, (l

_{s})

_{R}= 0.1, n = 2.0, Ω/ω

_{1}= 1.1000, (l

_{s})

_{L}= 0.4).

**Figure 5.**Frequency diagrams for classical ((l

_{s})

_{L}= 0.0; (l

_{s})

_{R}= 0.0) and modified couple stress theory (CST) for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}; ((l

_{s})

_{L}= 0.4; (l

_{s})

_{R}= 0.1): f

_{1}= 40.0, b

_{R}= 3b

_{L}, n = 1.0.

**Figure 6.**Frequency diagrams for classical ((l

_{s})

_{R}= 0.0; (l

_{s})

_{L}= 0.0) and modified CST ((l

_{s})

_{R}= 0.1; (l

_{s})

_{L}= 0.4) for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}; f

_{1}= 40.0, b

_{R}= 3b

_{L}, n = 8.0.

**Figure 7.**Frequency diagrams for different tapered status and for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}; n = 4.0, (l

_{s})

_{R}= 0.1, (l

_{s})

_{L}= 0.4, f

_{1}= 40.0.

**Figure 8.**Frequency diagrams for different n and for (

**a**) q

_{1}, (

**b**) q

_{2}, (

**c**) p

_{1}and (

**d**) r

_{1}; b

_{R}= 3b

_{L}, f

_{1}= 40.0, (l

_{s})

_{R}= 0.1, (l

_{s})

_{L}= 0.4.

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**MDPI and ACS Style**

Ghayesh, M.H.; Farajpour, A.; Farokhi, H.
Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams. *Vibration* **2019**, *2*, 201-221.
https://doi.org/10.3390/vibration2020013

**AMA Style**

Ghayesh MH, Farajpour A, Farokhi H.
Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams. *Vibration*. 2019; 2(2):201-221.
https://doi.org/10.3390/vibration2020013

**Chicago/Turabian Style**

Ghayesh, Mergen H., Ali Farajpour, and Hamed Farokhi.
2019. "Asymmetric Oscillations of AFG Microscale Nonuniform Deformable Timoshenko Beams" *Vibration* 2, no. 2: 201-221.
https://doi.org/10.3390/vibration2020013