Abstract

In this paper, we study experimentally the impact of a vibrating wire on the free abrasive machining (FAM) process in removing material from the surface of brittle materials, such as silicon. An experimental setup was designed to study the FAM process on silicon substrate surface by using a slurry-fed wire with a periodic excitation. An analytical solution of a wire moving axially, subject to an oscillating boundary condition with damping from abrasive slurry, was derived based on the partial differential equation of motion. Experiments were conducted on the apparatus using a wire with an oscillating boundary. It was found that the amplitudes of vibration were larger at the side of the oscillatory boundary, which caused more FAM interaction near the edge of the oscillatory boundary with larger material removal that was measured and validated. Furthermore, experiments were conducted to elucidate the effectiveness of brittle material removal using FAM with abrasive grits: (i) under dry condition, (ii) with water, and (iii) with abrasive slurry. Experimental results showed that the vibration of wire resulted in plastic deformation on the surface of silicon wafer. The abrasive grits in slurry driven by a vibrating wire generated material removal through observable grooves and fractures on the surface of silicon due to FAM in just a few minutes. The grooves from FAM process is an outcome of brittle machining through fracture formation and concatenation, generated by the indentation of abrasive grits on the silicon surface.

Introduction

When operating slurry wiresaws, an axially or longitudinally moving wire carrying abrasive slurry interacts with the surface of workpiece in the slicing region to remove material. This is commonly known as the free abrasive machining (FAM) process [1]. FAM removes material very slowly and is typically used in machining brittle materials, such as silicon, III-V and II-VI compounds, and ceramic materials [2,3,18,20]. The wire in slurry wiresaw can be considered as being immersed in viscous fluid. It is of interests to study the impact of vibration of wire on the FAM process in material removal. In this paper, we investigate the impact of vibration on FAM through modeling and experimental study using a vibrating wire on a silicon surface.

In a slurry wiresaw system, the wire can be modeled as an axially moving continuum with damping. Huang and Mote [4] presented the model of a damped axially moving wire system for the computer memory disks. Chen [5] presented a literature review on the research of transverse vibrations of an axially moving string and control. Kao and co-authors [68,1820] applied the gyroscopic model in the modeling of wiresaw by introducing a damping force to represent the hydrodynamic effect. Wei and Kao [6] obtained numerical results of damped vibration under external harmonic excitation to study the vibration of wire in a slurry wiresaw system. The hydrodynamic effect was taken into consideration in vibration analysis by Zhu and Kao [7]. Chung and Kao [8,20] derived the analytical solution of a moving wire with damping. Their analytical solution of damped moving wire in 2011 using the Green’s function is the first known closed-form solution of such vibration problem with damping.

Indentation or scratching by abrasive indenters on the surface of brittle materials is a common model to study the stress and material removal process in abrasive machining [9,10,20]. Indentation and scratching of brittle materials result in localized zone for cracks to form [11]. Such crack formation, including radial/lateral cracks and other subsurface cracks, is quite typical in abrasive machining of brittle materials. However, it is difficult to observe directly the abrasive interaction on machined surface in real-time and in-situ during abrasive machining process. Although observation on machined surface of abrasive machining is a good way to evaluate the macro-scale outcome of the process [1214], the overlapping interaction of many abrasives poses a challenge.

In this paper, an axially moving wire is modeled with a harmonic oscillating boundary condition. The analytical solution of this model is obtained and discussed. An experimental setup is designed to perform FAM by a slurry-fed wire with a periodic excitation at one boundary. The results are presented and analyzed to correlate with the theoretical modeling. In addition, experimental study of the role of abrasive grits on material removal is also presented.

Model of an Axially Moving Wire With an Oscillating Boundary

Some industrial wiresaw systems utilize oscillating movements of wire guides to enhance slicing performance [15,16]. Figure 1(a) illustrates the oscillating (rocking) motion of wire to move wire web in slicing.2 To study the vibration of wire, the rocking motion is modeled as an oscillating boundary condition at one end with the other boundary fixed, as illustrated in Fig. 1(b). The oscillating boundary is at X = 0 with the fixed boundary at X = L.

Fig. 1
(a) Schematic of a wire from a wire web with rocking motions and (b) the model of a damped moving wire with an oscillating boundary condition at the left wire guide in a wiresaw system is shown. The left wire guide moves along the transverse direction in a harmonic oscillation.
Fig. 1
(a) Schematic of a wire from a wire web with rocking motions and (b) the model of a damped moving wire with an oscillating boundary condition at the left wire guide in a wiresaw system is shown. The left wire guide moves along the transverse direction in a harmonic oscillation.
In the following modeling equations, the oscillating boundary condition is represented by a harmonic function, U(0, T) = CeT, where Ω is the frequency of oscillation of the moving boundary and C is the half range of oscillation, as shown in Fig. 1(b). The right boundary is fixed with U(L, T) = 0. We further assume that the elasto-hydrodynamic interaction with the slurry results in a distributed damping force along the wire when it moves and immersed in slurry. The equation of motion is Refs. [8,20]
ρ(U,TT+2VU,XT+V2U,XX)PU,XX+ηd(VU,X+U,T)=Fwithboundaryconditions{U(0,T)=CeiΩTU(L,T)=0
(1)
To simplify the derivation, the non-dimensionalized parameters in Eq. (1) are defined as follows.
x=XL,u=UL,t=TPρL2,f=FLP,v=VρP,η=ηdLPρ,c=CL,ω=ΩρL2P
(2)
All non-dimensionalized parameters are in lower-case. From Eqs. (1) and (2), the non-dimensionalized model is
u,tt+2vu,xt(1v2)u,xx+ηvu,x+ηu,t=fwithboundaryconditions{u(0,t)=ceiωtu(1,t)=0
(3)
where u,tt = ∂2u/∂t2, u,xt = ∂2u/(∂xt), and u,xx = ∂2u/∂x2 denote the second-order partial derivatives of u with respect to t and x. Likewise, the notations of u,t = ∂u/∂t and u,x = ∂u/∂x denote the first-order partial derivatives of u with respect to t and x, respectively. The solution of the differential continuum problem in Eq. (3) is assumed to be
u(x,t)=Ampeκpxeλmt
(4)
where Amp is an amplitude, κp is the parameter pertaining to the independent variable x in the spatial domain, and λm is the parameter pertaining to the independent variable t in the time domain.
The details of derivation are presented in the  Appendix. The analytical solution is the real part of Eq. (A9) when the excitation of oscillating boundary condition is u(0,t)=ccosωt. This solution can be written as follows:
u(x,t)=ce(ηv/2(1v2))xe2a2cos2b+e2a{e2aaxcos[ωt+(ωv1v2b)x]+e2a+axcos[ωt+(ωv1v2+b)x]eaxcos[ωt2b+(ωv1v2+b)x]eaxcos[ωt+2b+(ωv1v2b)x]}
(5)
where a=(4ω2v2η2)2+16ω2η2(4ω2v2η2)/22(1v2) and b=(4ω2v2η2)2+16ω2η2+(4ω2v2η2)/22(1v2).
According to the solution in Eq. (5), the frequency of vibration is the same as the oscillating frequency of the left boundary at the steady-state. The amplitude of the response is proportional to the half range of oscillation at the oscillating boundary, c. As an example, the analytical equation of vibration response with the non-dimensionalized parameters of c = 0.01, ω=2.5π, v = 0.1, and η=50 is
u(x,t)=3.48×1014[e26.39+15.72xcos(7.85t+15.98x)+e26.3910.67xcos(7.85t14.39x)e15.72xcos(7.85t30.37+15.98x)e10.67xcos(7.85t+30.3714.39x)]

The Amplitudes of Vibration Based on Analytical Solution

In a typical slurry wiresaw, the range of the damping factor is ηd=0.73.5Ns/m2 [8]. The corresponding non-dimensionalized damping factor is η=6.1730.87 based on Eq. (2). Figure 2 shows four snap shots of vibration responses corresponding to one full cycle of oscillation at the boundary x = 0, within the range of practical damping factors η according to the analytical solution in Eq. (5). The amplitudes of vibration response are reduced exponentially when the damping is increased, as shown in Figs. 2(a)2(c). It is also noted that the amplitudes of vibration are larger near the side of oscillatory boundary in all cases.

Fig. 2
The waveform of vibration with the following parameters: v = 0.1, u(0, t) = 0.01cos(2.5π t), with a damping factor of (a) η=1 (ηd=0.113Ns/m2), (b) η=10 (ηd=1.13Ns/m2), and (c) η=30 (ηd=3.40Ns/m2), respectively. For typical physical parameters of L = 0.54 m, ρ=1.874×10−4kg/m and P = 20 N, the period at the oscillating boundary is T = 1.32 × 10−3 s. WFi/4 is the vibration response at i-fourth (i/4) of the period.
Fig. 2
The waveform of vibration with the following parameters: v = 0.1, u(0, t) = 0.01cos(2.5π t), with a damping factor of (a) η=1 (ηd=0.113Ns/m2), (b) η=10 (ηd=1.13Ns/m2), and (c) η=30 (ηd=3.40Ns/m2), respectively. For typical physical parameters of L = 0.54 m, ρ=1.874×10−4kg/m and P = 20 N, the period at the oscillating boundary is T = 1.32 × 10−3 s. WFi/4 is the vibration response at i-fourth (i/4) of the period.

Frequency of Oscillating Boundary and its Impact.

Increasing the wire speed or decreasing the frequency of oscillation is found to be equivalent to the reduction of damping. By comparing Figs. 3(a) and 3(b), the amplitude of vibration is enlarged by increasing the wire speed from v = 0 to v = 0.3 under the same damping of η=15, equivalent to a reduction in damping. In Figs. 3(a) and 3(c), the oscillating frequency is reduced from ω=2.5π to ω=0.25π with the same damping of η=15. This decrease of frequency at the oscillating boundary increases the amplitude of vibration from Figs. 3(a) to 3(c), also equivalent to a reduction in damping.

Fig. 3
The waveform of vibration under damping factor η=15 with the left boundary conditions u(0,t)=0.01cos(ωt). For typical physical parameter of L = 0.54, ρ=1.874×10−4kg/m, and P = 20 N, the period (T) at the oscillating boundary is T = 1.32 × 10−3 s for cases (a) and (b), and T = 1.32 × 10−2 s for case (c). (a) v = 0 and ω=2.5π, (b) v = 0.3 and ω=2.5π, and (c) v = 0 and ω=0.25π.
Fig. 3
The waveform of vibration under damping factor η=15 with the left boundary conditions u(0,t)=0.01cos(ωt). For typical physical parameter of L = 0.54, ρ=1.874×10−4kg/m, and P = 20 N, the period (T) at the oscillating boundary is T = 1.32 × 10−3 s for cases (a) and (b), and T = 1.32 × 10−2 s for case (c). (a) v = 0 and ω=2.5π, (b) v = 0.3 and ω=2.5π, and (c) v = 0 and ω=0.25π.

Experimental Study of Free Abrasive Machining Using a Vibrating Wire

In order to study the impacts of vibration of wire on FAM process, experiments were conducted using a setup of a slurry-fed wire with a periodic excitation on one side of the wire to emulate oscillating boundary condition. The experiments investigate the role of a vibrating wire on silicon substrate surface under the conditions of (i) dry substrate surface, (ii) surface fed with water, and (iii) surface fed with slurry. Results are compared and discussed.

Experimental Setup.

An apparatus was designed to perform experiments with a vibrating steel wire on wafer surface, as illustrated in Fig. 4. A wire of a diameter of 200 μm was maintained at a constant tension of 7.5 N by an overhanging weight. The wire was excited for harmonic vibration by a DC motor, where a tip was attached to its shaft and oscillated the wire at 600–2000 rpm. The excitation was placed at a distance of l = 12 − 16 cm from the left support for all experiments to produce wire vibration.

Fig. 4
The apparatus for experiments: (a) schematic of setup with a vibrating wire on silicon wafer surface and (b) zoom-in view of the inerface area
Fig. 4
The apparatus for experiments: (a) schematic of setup with a vibrating wire on silicon wafer surface and (b) zoom-in view of the inerface area

The polished surface of commercial single-crystalline n-type <100> silicon prime wafers, which are free of defect for device fabrication, are used as the specimen for experiments. The wafer is placed directly under the wire, constrained by walls to form a channel for the slurry flow and a barrier to encourage the vertical movement of the wire so as to interact with the wafer surface, as illustrated in Fig. 4(b). A small container with a hole at its bottom provides a constant drip of DI water or slurry, with a flowrate between 40 and 50 mL/min.

The wafer surfaces after experiments are observed under a scanning electron microscope (HITACHI S-4800 and JEOL 7600F) to study the surface morphology. A stylus profilometer (Dektak 150 stylus profilometer) is used to obtain the surface profile.

Experimental Results.

Experimental results are presented in the following including: (i) vibrating wire on dry wafer surface, (ii) vibrating wire on wafer surface with water, and (iii) vibrating wire on wafer surface with slurry.

(i) Vibrating wire on dry wafer surface.

The experiment of vibrating wire on dry wafer surface, without water or slurry, lasted for 10 h. Only the area close to the edge of the wafer has machining feature on the surface, as shown in Fig. 5. The vibration generates only little plastic deformation on the surface of silicon wafer. More deformation exists at the edge close to the oscillating boundary, as shown in Fig. 6(a) at Y ≅ 0 mm.

Fig. 5
(a) Surface of silicon wafer at the edge close to the oscillating boundary on dry surface and (b) a zoom-in view of the rectangular area in (a)
Fig. 5
(a) Surface of silicon wafer at the edge close to the oscillating boundary on dry surface and (b) a zoom-in view of the rectangular area in (a)
Fig. 6
The surface profile of wafer from the experiments under dry condition: (a) the wafer surface along the wire direction Y (refer to Fig. 5(a)) with the origin at the edge closed to the excitation and (b) the surface profile of wafer along the X-direction close to the edge at Y ≅ 0 mm in (a)
Fig. 6
The surface profile of wafer from the experiments under dry condition: (a) the wafer surface along the wire direction Y (refer to Fig. 5(a)) with the origin at the edge closed to the excitation and (b) the surface profile of wafer along the X-direction close to the edge at Y ≅ 0 mm in (a)

(ii) Vibrating wire on surface with water.

The results of wafer surface after 10 h of vibration with DI water is shown in Fig. 7. The wire could generate some plastic deformation at the edge of silicon wafer specimen. However, due to the damping effect of water, the depth of groove, as shown in Fig. 8, is smaller compared with that of the dry wafer surface shown in Fig. 6.

Fig. 7
(a) Surface of silicon wafer at the edge close to the oscillating boundary with wire vibrating in water and (b) a zoom-in view of the rectangular area in (a)
Fig. 7
(a) Surface of silicon wafer at the edge close to the oscillating boundary with wire vibrating in water and (b) a zoom-in view of the rectangular area in (a)
Fig. 8
The surface profile of wafer from the experiments of wire vibration in water: (a) the wafer surface along the wire direction Y (refer to Fig. 7(a)) with the origin at the edge close to the excitation and (b) the surface profile of the wafer along the X-direction close to the edge at Y ≅ 0 mm in (a)
Fig. 8
The surface profile of wafer from the experiments of wire vibration in water: (a) the wafer surface along the wire direction Y (refer to Fig. 7(a)) with the origin at the edge close to the excitation and (b) the surface profile of the wafer along the X-direction close to the edge at Y ≅ 0 mm in (a)

(iii) Vibrating wire on surface with slurry.

The slurry applied in this study is a mixture of F400 silicon carbide abrasive grits and DI water, with a mixing ratio of 8 g of abrasive with 3 L of water. We conducted the experiments of wire with slurry at different time durations to observe the change on the wafer surface. Figure 9 illustrates the progression of surface topology under FAM with slurry as a function of time.

Fig. 9
Surface of wafer under vibrating wire in slurry are shown for a time duration of (a) 5 min, (b) 15 min, and (c) 30 min in experiments. All SEM photos are taken at the middle of the wafer surface.
Fig. 9
Surface of wafer under vibrating wire in slurry are shown for a time duration of (a) 5 min, (b) 15 min, and (c) 30 min in experiments. All SEM photos are taken at the middle of the wafer surface.

In the experiment with 5 min time duration, sporadic and discrete indentations appear on the surface, as shown in Fig. 9(a). Few indentation has enough energy to generate the chip-off fractures, while most defects observed are just shallow indenting marks or tiny scratches on the surface.

For the experiment with 15-min time duration, fractures and cracks from indentation begin to appear and overlap with one another, with fractures observed in Fig. 9(b). With 30 min for abrasives to indent on the surface, more fractures and cracks are formed and concatenated with one another, leading to chips and materials removal from the substrate surface. Thus, the surface topology of indentation and cracks becomes much finer and more random, as shown in Fig. 9(c). It is noted that the indentation with fractures have higher density across the surface with slurry and FAM, as compared with those in the cases of dry surface and surface with water previously.

Figure 10 plots the surface profiles of wafers under a vibrating wire in slurry with time durations of 5, 15, and 30 min. In comparison with the surface profile of wire on dry surface and in water, as shown in Figs. 6 and 8, respectively, the results with slurry in Fig. 10 show much deeper grooves, even with a much shorter time duration. The vibrating wire with slurry can generate a groove with a depth of 15.31 μm within only 5 min and reach a depth of 38.84 μm in 30 min, far exceeding the depths of 5.17 μm and 2.47 μm of plastic deformation under the dry condition and in water for 10 h. Thus, the abrasive grits play a significant role in the material removal in FAM.

Fig. 10
The surface profiles under different durations of experiment (a) 5 min, (b) 15 min, and (c) 30 min of vibrating wire in slurry with FAM
Fig. 10
The surface profiles under different durations of experiment (a) 5 min, (b) 15 min, and (c) 30 min of vibrating wire in slurry with FAM

Furthermore, the material removal is more pronounced near the edges of silicon specimen, as shown in Fig. 10. This is due to the larger amplitudes of vibration as also discussed and illustrated in Figs. 2 and 3.

Figure 11 plots the depth of machining as a function of the time duration. Each data point in Fig. 11 is a new and independent experiment with different time durations of 5, 10, 15, 20, and 30 min, starting from the time at t = 0. The depths of machining increase with the time duration, shown by the data points with linear regression in the figure.

Fig. 11
The depths of machining as a function of the time durations of 5, 10, 15, 20, and 30 min, and linear regression
Fig. 11
The depths of machining as a function of the time durations of 5, 10, 15, 20, and 30 min, and linear regression

Figures 9 and 10 are cross-referenced to show the correlation between the following: (1) the progression of the surface topology in Fig. 9 under FAM, by using SEM images of wafer surface at different time of machining and (2) the resulting surface profiles of FAM, by using a stylus profilometer shown in Fig. 10.

Discussions

The experimental results of vibrating wire in all three cases of dry, water, and slurry in Figs. 6,8, and 10 show consistently that the deformation or depth of machining toward Y = 0 is the largest and reduces as Y increases. This is consistent with the theoretical results of vibration analysis presented in Figs. 2 and 3 in which the amplitudes of vibration are higher toward the oscillating boundary, resulting in more impact on the surface, or more indentation in the case of slurry with abrasive grits, toward the edge of the surface. It demonstrates that the results of the theoretical analysis and experimental observation match with each other well. The damping in the modeling plays a role to damp the vibration away from the oscillating boundary, resulting in this edge effect. The theoretical modeling suggests that the axial moving speed, oscillating frequency at boundary and damping factor influence the vibration of wire. However, the vibration amplitude is always larger at the side closer to excitation, as shown in Fig. 3.

This also explains the effectiveness of rocking motion of moving boundaries (moving wire guides) in industrial practice of wiresawing. When the wire traverses on the cutting interface between the wire and crystal, the shifting edge resulting from the rocking motion will enhance the material removal around the edge of moving contact. As such edge continues to traverse along the cutting span, the rate of material removal is increased and the efficiency of slicing is enhanced.

Plastic deformation can be observed due to the impact of vibrating wire in Figs. 6 and 8 without abrasive grits; however, there is no apparent removal of materials from the surface, other than plastic deformation.3 While plastic deformation alone does not cause material removal, the presence of abrasive grits in FAM does cause material removal, as clearly shown in Figs. 9 and 10. Even though the experimental setup does not flush out all the spent slurry, the abrasive grits in the contact interface excited by the vibrating wire are capable of continuously removing material from the surface. This elucidates the role of material removal by the abrasive grits in FAM, especially in the machining of brittle materials. Although FAM does not have as high a material removal rate as the traditional machine tools on ductile materials, FAM is still the choice of machining for many brittle materials, such as the various semiconductor and optoelectronic materials.

Figure 11 shows that the depth of machining, and therefore the material removal, increases with the time duration. A linear regression best-fit shows the trend of such increase.

The limitation of this experimental study may be on the lack of accurate control of normal load onto the wafer surface during the FAM process. In a typical FAM process, normal force is an important machining parameter. This limitation, however, is mitigated by the vibration of the wire and the force of impact onto the surface. In an industrial wiresaw process, the normal load is controlled through the tension in the wire and the bow angle of wire on the silicon ingot to sustain the interaction between the cutting tool (wire) and the workpiece (silicon crystal).

Conclusions

In this paper, we conducted experimental study on the impacts of a vibrating wire on FAM processes, as correlated to the theoretical analysis of vibration. The theoretical analysis predicts that larger vibration amplitudes along the edge near the oscillating boundary would lead to more material removal near the boundary. This was confirmed by the experimental results of FAM. Moreover, the experimental results also can explain the industrial practice of rocking movements of wire guides to increase the efficiency of material removal. This is because the edge, which has higher material removal rate due to the boundary condition, continues to shift through rocking, resulting in more material removal across the span of shifting contact and boundary.

Furthermore, the experimental results also suggest that (1) plastic deformation is generated without apparent material removal when abrasive grits are not present; (2) the vibrating wire with abrasive slurry can result in material removal on the surface of silicon wafer through FAM; and (3) FAM process removes brittle materials through indentation and concatenation of fractures, caused by the indentation of abrasive grits carried in the slurry and by the vibrating wire. It is also found that the material removal is proportional to the time of machining.

Footnotes

2

The readers may refer to Refs. [18,20] for a schematic and desciption of wire web in a wiresaw system.

3

Plastic deformation of silicon occurs at a shear stress of 12–14 GPa. When the shear stress exceeds 30 GPa, cracks begin to extend from the crack nucleus [17]. Since single-crystalline silicon is anisotropic, the shear stress for plastic deformation may vary for different orientations of silicon substrate.

Acknowledgment

This research utilized the resources of the Center for Functional Nanomaterials (CFN), which is a US DOE Office of Science Facility, at Brookhaven National Laboratory (BNL) under Contract No. DE-SC0012704. The authors appreciate the training and help of Dr. Fernando Camino, Ms. Gwen Wright and Dr. Ming Lu of BNL.

Conflict of Interest

There are no conflicts of interest.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

Nomenclature

     
  • L =

    the span (or distance) between two wire guides, in meter

  •  
  • P =

    the tension in wire, in Newtons

  •  
  • T =

    the time, in seconds

  •  
  • V =

    the axial speed of the moving wire, in meters per second

  •  
  • X =

    the coordinate defined along the axial direction, with a range of 0 ≤ XL, in meters

  •  
  • F(X, T) =

    the external excitation per unit length along the wire, in Newtons per meter

  •  
  • U(X, T) =

    the transverse (or lateral) displacement perpendicular to the X direction of the wire, in meters; U is also the amplitude of vibration

  •  
  • ρ =

    the spatial density of wire, in kilograms per meter

All physical parameters are in upper-case, except ρ. In a typical wiresaw system, the density of wire is ρ=1.11.2g/cm3, the length of wire L is between 0.5 to 1 m and the tension in wire is P = 20 ∼ 35 N. The axial speed is V = 10 ∼ 30 m/s [1820].

Appendix: Derivation of the analytical solution

Substitute Eq. (4) into Eq. (3) to obtain
Amp{λm2+2vλmκp(1v2)κp2+ηvκp+ηλm}eκpxeλmt=f
(A1)
Considering the case of free vibration without external excitation, we obtain
(1v2)κp2(2vλm+ηv)κp(λm2+ηλm)=0
(A2)
The roots of κp can be found from the quadratic equation in Eq. (A2) as follows:
κ1,2=2vλm+vη±4λm2+4ηλm+v2η22(1v2)=αm±βm
(A3)
where αm=v(2λm+η)/2(1v2) and βm=4λm2+4ηλm+v2η2/2(1v2). Note that the solution of κ and λ can not be readily separated due to the coupled partial differential equation of motion.
The general solution of the partial differential equation in Eq. (3) is
u(x,t)=m=eλmteαmx(A1meβmx+A2meβmx)
(A4)
where A1m and A2m are constants corresponding to the parameter λm.
The general solution with homogeneous boundary condition was solved by Chung and Kao in Ref. [1]. Here, we deal with the vibration response with an oscillating boundary. We obtain the following equations based on the oscillating boundary condition at the left end
u(0,t)=(A1m+A2m)eλmt=ceiωt
(A5)
and the fixed boundary condition at the right end
u(1,t)=(A1meβm+A2meβm)e(αm+λmt)=0
(A6)
After rearranging the equations, we have
{A1m+A2m=cA1meβm+A2meβm=0λm=iω
(A7)
The coefficients and frequency in Eq. (A7) are obtained as follows:
{A1m=ceβmeβmeβmA2m=ceβmeβmeβmλm=iω
(A8)
Thus, the solution of vibration due to the oscillating boundary is
u(x,t)=c[(eβ(1x)eβ(1x))eαxeβeβ]eiωt
(A9)
where α=v(2iω+η)/2(1v2) and β=4ω2+4iωη+v2η2/2(1v2).

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