# A 3D Free Vibration Analysis of the Horn-Gear System through Chebyshev–Ritz Method in Ultrasonic Gear Honing

Authors: Yi Li; Ming Lv; Shiying Wang; Huibin Qin
Journal: Shock and Vibration
 (2021)

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## Abstract

Applying the ultrasonic machining in gear honing can improve honing speed, reduce cutting force, and avoid blocking. There are two problems leading to the decrease of calculation accuracy in the traditional nonresonant theory of the ultrasonic gear honing. One is that one-dimensional longitudinal vibration theory and two-dimensional theory cannot reflect the vibration characteristics of ultrasonic horn and gear comprehensively. And, the other one is that the difference of the analysis dimension between the two theories leads to mismatch of the coupling condition dimension between ultrasonic horn and gear. A free vibration analysis through Chebyshev–Ritz method based on three-dimensional elasticity theory was presented to analyze the eigenfrequencies of the horn-gear system in ultrasonic gear honing. In the method, the model of the horn-gear system was divided into four parts: a solid circular plate, an annular plate, a solid cylinder, and a cone with hole. The eigenvalue equations were derived by using displacement coupling condition between each part under completely free boundary condition. It was found that the eigenfrequencies were highly convergent through convergence study. The hammering method for a modal experiment was used to test the horn-gear systems’ eigenfrequencies. And, the finite element method was also applied to solve the eigenfrequencies. Through a comparative analysis of the frequencies obtained by these three methods, it showed that the results achieved by the Chebyshev–Ritz method were close to those obtained from the experiment and finite element method. Thus, it was feasible to use the Chebyshev–Ritz method to solve the eigenfrequencies of the horn-gear system in ultrasonic gear honing.


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## Body

## 1. Introduction

Gear is an important mechanical component to transmit power in most mechanical equipment [1, 2]. The trends of gear development are miniaturization and high speed [3]. The hardened surface gear machining is widely used for improving gears’ load capacity to satisfy the gear’s developing trends [4]. Hardened surface gear honing is applied for machining hardened tooth surface gear because it can improve surface quality and machining accuracy and reduce costs. However, the honing speed between honing wheel and gear is too slow to meet the required honing speed of hardened surface gear honing. The abrasive grit falls off from the honing wheel leading to block because of the cutting force increasing caused by the slow honing speed [5, 6].The ultrasonic machining can improve honing speed, reduce cutting force, and avoid blocking to enhance the processing efficiency. Therefore, it is necessary to introduce the ultrasonic machining into hardened surface gear honing to avoid the disadvantages mentioned above [7–9]. Ultrasonic gear honing equipment consists of ultrasonic power supply, transducer, vibrating rod, ultrasonic horn, and gear. The horn-gear system is made of the ultrasonic horn and the gear. To ensure the ultrasonic gear honing equipment works normally, the horn-gear system should vibrate with high frequency within the frequency modulation range of ultrasonic power supply. So, it is important to analyze the vibration characteristic of the horn-gear system accurately. There are already two theories analyzing the vibration characteristic of the horn-gear system. At first, the whole-resonant theory was used in the ultrasonic gear honing to analyze the vibration characteristics of the horn-gear system. Because of the large size of gear, the gear cannot be neglected in the analysis process. The whole-resonant theory requires that ultrasonic horn and gear should have the same resonant frequency within the frequency modulation range of ultrasonic power supply. The whole-resonant theory has limitation on the size of gear and ultrasonic horn. However, the size of gear is decided by the gear’s application requirements in manufacturing, not by the resonant frequency of the ultrasonic gear honing equipment.The nonresonant theory can solve the deficiency of the whole-resonant theory. The gear of the horn-gear system can be simplified as a uniform annular plate whose outside diameter is the gear’s pitch diameter [10, 11]. The dynamic equation is derived based on force coupling of the plate and the ultrasonic horn in nonresonant theory. We can adjust the size of the ultrasonic horn to ensure the resonant frequency of the horn-gear system within the frequency modulation range of ultrasonic power supply. In past analyzing process of the nonresonant theory, one-dimensional longitudinal vibration beam theory was adopted for the ultrasonic horn, and thin-plate theory was adopted for the gear [12, 13]. The thin-plate theory based on three hypotheses is only applicable to the plate with the thickness-diameter ratio less than 0.2 [14–17]. The one-dimensional longitudinal vibration beam theory just gives the slender rods’ frequencies of longitudinal vibration mode. If we want to obtain more frequencies of other kinds of vibration mode, this theory will be no longer applicable [18]. The other theories for analyzing the vibration characteristics of beams still cannot conduct the three-dimensional researches [19–24]. The traditional nonresonant theory mentioned above cannot reflect the vibration of the horn-gear system comprehensively and cause the mismatching of boundary condition at the coupling location of ultrasonic horn and gear. The finite element method is also applied to analyze the horn-gear system, but this method cannot obtain a solution in theory.Many scholars have conducted the three-dimensional research to analyze the vibration characteristics of rod and plate. Leissa and Kang conducted three-dimensional research to analyze the eigenfrequencies of thick, linearly, tapered, annular plates, and rods with arbitrary size using Algebraic–Ritz method. The admissible function of the Algebraic–Ritz method consists of algebraic polynomial multiplying boundary functions [25–27]. The poor numerical stability of the algebraic polynomial leads to the truncation order cannot be large enough to ensure enough eigenfrequencies converging to the accurate. Because the Chebyshev polynomial is as simple as the algebraic polynomial and has outstanding numerical stability, Zhou replaces the algebraic polynomial to the Chebyshev polynomial in admissible functions. This method, the so-called Chebyshev–Ritz method, can obtain more accurate eigenfrequencies of single plate [28–30]. The Ritz method above is still not adopted for analyzing the eigenfrequencies of the horn-gear system yet.It is essential to apply Chebyshev–Ritz method into analyzing the eigenfrequencies of the horn-gear system to solve the problems owned by the traditional nonresonant theory. A free vibration analysis of the horn-gear system using Chebyshev–Ritz method based on three-dimensional elasticity theory was presented. The model of the horn-gear system was divided into four parts. The eigenvalue equations were derived, and the convergent study was conducted. The comparison study of the natural frequencies obtained by the traditional nonresonant theory, and the method mentioned in this paper was made. The experimental platform of hammering method for testing the horn-gear system’s frequencies in a completely free condition was established. The finite element method was also used to obtain the horn-gear system’s eigenfrequencies. The comparative analysis of the frequencies obtained by the three methods was made.


## 2. Method Application

The homogeneous, isotropic, and simplified model of horn-gear system is shown in Figure1. A cylindrical coordinate (r, θ, z) is defined. The model of the horn-gear system is divided into four parts, namely, Ω1, Ω2, Ω3, and Ω4 shown in Figure 2. The first part Ω1 is the coupling part of the gear and the ultrasonic horn with radius R1 and thickness h. The second part Ω2 is the gear except the coupling part, and it is simplified as an annular plate with pitch radius R2 and thickness h. The third part Ω3 is a solid cylinder as a part of the ultrasonic horn; its left end is at z = 0, and its right end is at z = L with radius R1, and L is the length of the solid cylinder. The rest of the horn is the fourth part Ω4 which is regarded as a cone with hole. The cone’s left end is at z = 0 with radius R3 and right end at z = L with radius R1. The cone’s height between r=R1 and r=R3 can be expressed as L4r. The zero point of radial (r) and axial (z) coordinates is measured from central axis and θ is the circumferential angle. The corresponding displacement components are u, v, and w in the r, θ, and z direction, respectively. And, L4r can be expressed as follows:(1)L4r=−LR1−R3r−LR3R1−R3,R1≤r≤R3.Figure 1
The simplified model of the horn-gear system having circular cross-section with its cylindrical coordinate systemr,θ,z.Figure 2
(a) The first part Ω1. (b) The second part Ω2. (c) The third part Ω3. (d) The fourth part Ω4.
(a)(b)(c)(d)According to the theory of three-dimensional elasticity theory [31], the strain energy V of the horn-gear system can be expressed as follows:(2)V=∑q=14Vq=∑q=14G2∫RllqRulq∫02π∫LenllqLenulq2ν1−2νεrrq+εθθq+εzzq2+2εrrq2+εθθq2+εzzq2+εrθq2+εrzq2+εθzq2rdzdθdr,where G is the shear modulus and v is Poisson’s ratio and the other parameters are expressed as follows:(3)Rul1=Rul3=R1,Rul2=R2,Rul4=R3,Rll1=Rll3=0,Rll2=R1,Rll4=R1,Lenul1=Lenul2=h+L,Lenul3=L,Lenul4=L4r,Lenll1=Lenll2=L,Lenll3=Lenll4=0.The strain componentsεijq (i, j = r, θ, z; q = 1, 2, 3, 4) are written as follows [31]:(4)εrrq=∂uq∂r,εθθq=uqr+∂vqr∂θ,εzq=∂wq∂z,εrθq=∂uqr∂θ+∂vq∂r−vqr,εrzq=∂uq∂z+∂wq∂r,εθzq=∂vq∂z+∂wqr∂θ.The kinetic energyT of the horn-gear system can be expressed as follows [31]:(5)T=∑q=14Tq=∑q=14ρ2∫RllqzRulqz∫02π∫LenllqLenulq∂uq∂t2+∂vq∂t2+∂wq∂t2rdzdθdr,where ρ is the density of each part and t is the time.For the convenience of calculation, the dimensionless variables can be defined as follows:(6)r¯1=r¯3=2rR1−1,r¯2=2rR2−R1−R2+R1R2−R1,r¯4=2rR3−R1−R3+R1R3−R1,θ¯1=θ¯2=θ¯3=θ¯4=θ,z¯1=z¯2=2z−L−h/2h,z¯3=2zL−1,z¯4=2zL4r−1.Thus, equation (1) can be simplified as(7)L4r¯4=−L2r¯4+L2.According to the free vibration analysis method, the vibration displacement at any parts of the horn-gear system can be assumed as follows:(8)uqr,θ,z,t=Uqr¯q,θ¯,z¯qeiωt,vqr,θ,z,t=Vr¯q,θ¯,z¯qeiωt,wqr,θ,z,t=Wr¯q,θ¯,z¯qeiωt,where ω is the angular frequency of the horn-gear system and i=−1.Since each part of the horn-gear system is a rotator and has symmetry characteristics, the amplitude displacement of each part of the system can be expressed as(9)Uqr¯q,θ¯,z¯q=U¯qr¯q,z¯qcossθ¯,Vqr¯q,θ¯,z¯q=V¯qr¯q,z¯qsinsθ¯,Wqr¯q,θ¯,z¯q=W¯qr¯q,z¯qcossθ¯,where s is the circumferential wave number, which is an integer (namely, s=0,1,2,…,∞) to ensure the period of vibration in the θ¯ direction.Substituting equations (4), (6)–(9) into equations (2) and (5), the maximum potential energy Vmaxq and kinetic energy Tmaxq of the horn-gear system can be calculated, respectively (q = 1, 2, 3, and 4)(10)Vmaxq=GΛVq4∫−1+1∫−1+12ν1−2νΓ1ε¯r¯r¯q+ε¯θ¯θ¯q+ε¯z¯z¯q2+2Γ1ε¯r¯r¯q2+ε¯θ¯θ¯q2+ε¯z¯z¯q2+Γ1ε¯r¯z¯q2+Γ2ε¯r¯θ¯q2+ε¯θ¯z¯q2r¯q+δqdz¯qdr¯q,whenq=1,2,3,Tmaxq=ρΛVqΛTq2ω216∫−11∫−11Γ1U¯q2+Γ2V¯q2+Γ1W¯q2r¯q+δqdz¯qdr¯q,whenq=1,2,3,Tmax4=ρR3−R1ω216∫−11∫−11Γ1U¯42+Γ2V¯42+Γ1W¯42r¯4R3−R1+R3+R1fr¯4dz¯qdr¯q,Vmax4=GR3−R116∫−1+1∫−1+12ν1−2νΓ12R3−R1∂U¯4∂Ur¯4−2R3−R1z¯+1L4r¯4∂L4r¯4∂r¯4∂U¯4∂z¯4+2r¯R3−R1+R3+R1U¯4+2sr¯4R3−R1+R3+R1V¯4+2L4r¯4∂W¯4∂z¯42+2Γ12R3−R1∂U¯4∂Ur¯4−2R3−R1z¯+1L4r¯4∂L4r¯4∂r¯4∂U¯4∂z¯42+2r¯4R3−R1+R3+R1U¯4+2sr¯4R3−R1+R3+R1V¯42+2L4r¯4∂W¯4∂z¯42+Γ12L4r¯4∂U¯4∂z¯4+2R3−R1∂W¯4∂r¯4−2R3−R1z¯+1L4r¯4∂L4r¯4∂r¯4∂W¯4∂z¯42+Γ2−2sr¯R3−R1+R3+R1U¯4+2r¯4R3−R1+R3+R1V¯4+2R3−R1∂V¯4∂r¯4−2R3−R1z¯+1L4r¯4∂L4r¯4∂r¯4∂V¯4∂z¯42+2L4r¯4∂V¯4∂z¯4−2sr¯4R3−R1+R3+R1W¯42r¯4R3−R1+R3+R1L4r¯4dz¯4dr¯4,whereΓ1=∫02πcos2sθ¯qdθ¯q=2π,s=0,π,s>0,Γ2=∫02πsin2sθ¯qdθ¯q=0,s=0,π,s>0,ΛV1=ΛV2=h,ΛV3=L,δ1=δ3=1,δ2=R2+R1R2−R1,ΛT1=ΛT3=R1,ΛT2=R2−R1,ε¯r¯r¯q=∂U¯q∂r¯q,ε¯θ¯θ¯q=U¯qr¯q+δq+sV¯qr¯q+δq,ε¯z¯z¯q=ΛTq∂W¯qΛVq∂z¯q,ε¯r¯θ¯q=−sU¯qr¯q+δq+∂V¯q∂r¯q−V¯qr¯q+δq,ε¯r¯z¯q=ΛTq∂U¯qΛVq∂z¯q+∂W¯q∂r¯q,ε¯θ¯z¯q=ΛTq∂VqΛVq∂z¯q−sW¯qr¯q+δq.Each part’s vibration displacement function of the horn-gear system can be expressed as the two Chebyshev polynomials multiplied by corresponding boundary conditions:(11)U¯qr¯q,z¯q=ηuqr¯qηuqz¯q∑i=1Iq∑j=1JqAijqPir¯qPjz¯q,V¯qr¯q,z¯q=ηvqr¯qηvqz¯q∑k=1Kq∑l=1LqBklqPkr¯qPlz¯q,W¯qr¯q,z¯q=ηwqr¯qηwqz¯q∑m=1Mq∑n=1NqCmnqPmr¯qPnz¯q,where Iq, Jq, Kq, Lq, Mq, and Nq are the truncation orders of the Chebyshev polynomial, Aijq,Bklq,Cmnq are to be determined parameter, Ppχp=1,2,3,…,χ=r¯q,z¯q is the pth order Chebyshev polynomial in the one-dimensional which can be described as follows:(12)Ppχ=cosp−1arccosχ.ηuqr¯q,ηuqz¯q,ηvqr¯q,ηvqz¯q,ηwqr¯q, and ηwqz¯q are the boundary condition functions for the four parts of the horn-gear system. In the improved method, each part of the horn-gear system is recognized as completely free, so the displacement parameters uq,vq, and wq should satisfy the geometric boundary conditions of each part, then the boundary characteristic functions ηuqr¯q,ηuqz¯q,ηvqr¯q,ηvqz¯q,ηwqr¯q,ηwqz¯q=1. Each part’s vibration displacement function can be obtained by substituting the boundary conditions into equation (11).The energy equation of the horn-gear system can be defined as follows:(13)Πq=Vmaxq−Tmaxq.The minimum of the coefficient can be calculated from the deriving the energy equation:(14)∂Πq∂Aijq=0,∂Πq∂Bklq=0,∂Πq∂Cmnq=0.Then, the following eigenvalue equation can be obtained:(15)Kq−Ω2MqXq=0,where(16)Ω=ωρG,Kq=KuuqKuvqKuwqKvvqKvwqsymKwwq,Mq=Muuq00Mvvq0symMwwq,Xq=AqBqCqT,fors>1,Kq=KuuqKuwqsymKwwq,Mq=Muuq0symMwwq,Xq=AqCqT,s=0for the axisymmetric vibration,Kq=Kvvq;Mq=Mvvq;Xq=BqT,s=0for the torsional vibration.In equations (15) and (16), Kαβq and Mαβqα,β=u,v,w are the stiffness submatrices and the mass submatrices of each part of the horn-gear system.The vectorsAq,Bq, and Cq can be described as follows:(17)Aq=A11q…A1JqA21q…A1Jq…AI1q…AIJqT,Bq=B11q…B1LqB21q…B2Lq…BK1q…BKLqT,Cq=C11q…C1NqC21q…C2Nq…CM1q…CMNqT,when q = 1, 2, 3, are respectively, the submatrices Kαβq and Mαβqα,β=u,v,w when can be described as follows:(18)Kuuq=ΛVq1−ν1−2νDui¯uiq111+Dui¯uiq00−1+ν1−2νDui¯uiq010+Dui¯uiq100+s22Dui¯uiq00−1Huj¯ujq00+12ΛTqΛVq2Dui¯uiq001Huj¯ujq11,Kuvq=ΛVqν1−2νDui¯vkq100+1−ν1−2νDui¯vkq00−1+12Dui¯vkq00−1−Dui¯vkq010sHuj¯vlq00,Kuwq=ΛVqν1−2νΛTqΛVqDui¯wmq101+Dui¯wmq000Huj¯wnq01+12ΛTqΛVqDui¯wmq011Huj¯wnq10,Kvvq=ΛVq1−νs21−2νDvk¯vkq00−1Hvl¯vlq00+12Dvk¯vkq111−Dvk¯vkq010−Dvk¯vkq100+Dvk¯vkq00−1Hvl¯vlq00+12ΛTqΛVq2Dvk¯vkq001Hvl¯vlq11,Kvwq=ΛVqνs1−2νΛTqΛVqDvk¯wmq000Hvl¯wnq01−s2ΛTqΛVqDvk¯wmq000Hvl¯wnq10,Kwwq=ΛVq1−ν1−2νΛTqΛVq2Dwm¯wmq001Hwn¯wnq11+12Dwm¯wmq111+s2Dwm¯wmq00−1Hwn¯wnq00,Muuq=ΛTqΛTq2Dui¯uiq001Huj¯ujq008,Mvvq=ΛTqΛTq2Dvk¯vkq001Hvl¯vlq008,Mwwq=ΛTqΛTq2Dwm¯wmq001Hwn¯wnq008,where(19)Dασβτqefa=∫−11dePσr¯qdr¯qedfPτr¯qdr¯qfr¯q+1adr¯q,Hασβτqef=∫−11dePσz¯qdz¯qedfPτz¯qdz¯qfdz¯q,α,β=u,v,w,σ=i¯,j¯,k¯,l¯,m¯,n¯,τ=i,j,k,l,m,n,Whenq = 4, the submatrices Kαβq and Mαβqα,β=u,v,w can be described as follows:(20)Kuuq=1R3−R11−ν1−2νDui¯uiq01111+ν1−2νDui¯uiq01001+Dui¯uiq01010+1−ν1−2νR3−R1Dui¯uiq01−100Huj¯ujq000−1R3−R11−ν1−2νDui¯uiq10101+ν1−2νDui¯uiq10000Huj¯ujq110−1R3−R11−ν1−2νDui¯uiq10110+ν1−2νDui¯uiq10000Huj¯ujq101+1R3−R11−ν1−2νDui¯uiq2−1100Huj¯ujq211+12R3−R1Dui¯uiq0−1100Huj¯ujq011+s22R3−R1Dui¯uiq01−100Huj¯ujq000,Kuvq=ν1−2νDui¯vkq01010+1−ν1−2νR3−R1Dui¯vkq01−100sHuj¯vlq000−ν1−2νDui¯vkq10000sHuj¯vlq110+12R3−R1Dui¯vkq01−100−12Dui¯vkq01001sHuj¯vlq000,Kuwq=ν1−2νDui¯wmq00110+ν1−2νR3−R1Dui¯wmq00000Huj¯wnq001−121−2νDui¯wmq1−1100Huj¯wnq111+12Dui¯wmq00101Huj¯wnq010,Kvvq=1−ν1−2νR3−R1Dvk¯vkq01−100s2Hvl¯vlq001+12R3−R1Dvk¯vkq01−100−Dvk¯vkq01010−Dvk¯vkq01001+1R3−R1Dvk¯vkq01111Hvl¯vlq000+12Dvk¯vkq10000−1R3−R1Dvk¯vkq10101Hvl¯vlq110+12Dvk¯vkq10000−1R3−R1Dvk¯vkq10110Hvl¯vlq101+121R3−R1Dvk¯vkq2−1100Hvl¯vlq211+12R3−R1Dvk¯vkq−11000Hvl¯vlq011,Kvwq=ν1−2νR3−R1Dvk¯wmq00000sHvl¯wnq001+12R3−R1Dvk¯wmq00000sHvl¯wnq010,Kwwq=1−ν1−2νR3−R1Dwm¯wmq0−1100Hwn¯wnq011+121R3−R1Dwm¯wmq01111Hwn¯wnq000−121R3−R1Dwm¯wmq10101Hwn¯wnq110−121R3−R1Dwm¯wmq10110Hwn¯wnq101+121R3−R1Dwm¯wmq2−1100Hwn¯wnq211+12R3−R1Dwm¯wmq01−100s2Hwn¯wnq000,Muuq=R3−R1Dui¯uiq01100Huj¯ujq0008,Mvvq=R3−R1Dvk¯vkq01100Hvl¯vlq0008,Mwwq=R3−R1Dwm¯wmq01100Hwn¯wnq0008,where(21)Dασβτqabcef=∫−11∂L4r¯4∂r¯4aL4r¯4br¯4R3−R1+R3+R1cdePσr¯4dr¯4edfPτr¯4dr¯4fdr¯4,Hασβτgef=∫−11z¯4+1gdePσz¯4dz¯4edfPτz¯4dz¯4fdz¯4α,β=u,v,w,σ=i¯,j¯,k¯,l¯,m¯,n¯,τ=i,j,k,l,m,n.According to the above formulas, equation (15) can be further simplified to(22)K−Ω2MX=0,where(23)K=K10000K20000K30000K4,M=M10000M20000M30000M4,X=X1X2X3X4.It is obvious that the four parts of the horn-gear system are coupled with each other, and it results that the matrix {X} are also not independent of each other, and the eigenfrequencies of the horn-gear system could not be obtained directly from equation (22). However, the four parts of the horn-gear system need to meet the conditions of equal displacement conditions at each coupling, and it can be defined as follows:(24)U¯1r¯1,z¯1=U¯2r¯2,z¯2,V¯1r¯1,z¯1=V¯2r¯2,z¯2,W¯1r¯1,z¯1=W¯2r¯2,z¯2,atr¯1=1andr¯2=−1,U¯1r¯1,z¯1=U¯3r¯3,z¯3,V¯1r¯1,z¯1=V¯3r¯3,z¯3,W¯1r¯1,z¯1=W¯3r¯3,z¯3,atz¯1=−1andz¯3=1,U¯3r¯3,z¯3=U¯4r¯4,z¯4,V¯3r¯3,z¯3=V¯4r¯4,z¯4,W¯3r¯3,z¯3=W¯4r¯4,z¯4,atr¯3=1andr¯4=−1.Substituting equation (11) into (24),(25)∑i=1I1∑j=1J1Aij1Pi1Pjz¯1=∑i=1I2∑j=1J2Aij2Pi−1Pjz¯2,∑k=1K1∑l=1L1Bkl1Pk1Plz¯1=∑k=1K2∑l=1L2Bkl2Pk−1Plz¯2,∑m=1M1∑n=1N1Cmn1Pm1Pnz¯1=∑m=1M2∑n=1N2Cmn2Pm−1Pnz¯2,∑i=1I1∑j=1J1Aij1Pir¯1Pj−1=∑i=1I3∑j=1J3Aij3Pir¯3Pj1,∑k=1K1∑l=1L1Bkl1Pkr¯1Pl−1=∑k=1K3∑l=1L3Bkl3Pkr¯3Pl1,∑m=1M1∑n=1N1Cmn1Pmr¯1Pn−1=∑m=1M3∑n=1N3Cmn3Pmr¯3Pn1,∑i=1I3∑j=1J3Aij3Pi1Pjz¯3=∑i=1I4∑j=1J4Aij4Pi−1Pjz¯4,∑k=1K3∑l=1L3Bkl3Pk1Plz¯3=∑k=1K4∑l=1L4Bkl4Pk−1Plz¯4,∑m=1M3∑n=1N3Cmn3Pm1Pnz¯3=∑m=1M4∑n=1N4Cmn4Pm−1Pnz¯4.According to the properties of Chebyshev polynomial, it can be obtained that(26)Px1=1,Px−1=−1x−1,Gxy=∫−11Pxz¯qPyz¯q1−z¯q2dz¯q=0,x≠y,1,x=y,Hxy=∫−11Pxr¯qPyr¯q1−r¯q2dr¯q=0,x≠y,1,x=y,and then the two sides of equation (25) are multiplied by the Chebyshev polynomial, it can be changed to(27)∑i=1I2−1i−1Aij2=∑i=1I1∑j¯=1J1Aij¯1Gjj¯j=1,2,…,J2,∑k=1K2−1k−1Bkl2=∑k=1K1∑l¯=1L1Bkl¯1Gll¯l=1,2,…,L2,∑m=1M2−1m−1Cmn2=∑m=1M1∑n¯=1N1Cmn¯1Gnn¯n=1,2,…,N2,∑j=1J3Aij3=∑i¯=1I1∑j=1J1−1j−1Ai¯j1Hii¯i=1,2,…,I3,∑l=1L3Bkl3=∑k¯=1K1∑l=1L1−1l−1Bk¯l1Hkk¯k=1,2,…,K3,∑n=1N3Cmn3=∑m¯=1M1∑n=1N1−1n−1Cm¯n1Hmm¯m=1,2,…,M3,∑i4=1I4−1i4−1Ai4j44=Gj41∑i1=1I1∑i3=1I3Hi3i1∑j1=1J1−1j1−1Ai1j11−∑i3=1I3∑j3=2J3Ai3j33Gj4j3−Gj41j4=1,2,…,J4,∑k4=1K4−1k4−1Bk4l44=Gl41∑k1=1K1∑k3=1K3Hk3k1∑l1=1L1−1l1−1Bk1l11−∑k3=1K3∑l3=2L3Bk3l33Gl4l3−Gl41l4=1,2,…,L4,∑m4=1M4−1m4−1Cm4n44=Gn41∑m1=1M1∑m3=1M3Hm3m1∑n1=1N1−1n1−1Cm1n11−∑m3=1M3∑n3=2N3Cm3n33Gn4n3−Gn41n4=1,2,…,N4.From equation (27), the number of independent unknown parameters is F=∑q=14Iq×Jq+Kq×Lq+Mq×Nq−J2+L2+N2+I3+K3+M3+J4+L4+N4. The independent unknown parameters F which extracted from X can form a new matrix X¯. It can be described as(28)X=SX¯,where the coefficients s of the matrix [S] can be calculated by equation (27). Finally, equation (22) can be simplified to(29)K¯−Ω2M¯X¯=0,where(30)K¯=STKS,M¯=STMS.


## 3. Convergence and Comparison Analysis

### 3.1. Convergence Analysis

It is necessary to check the convergence of the eigenfrequencies using Chebyshev polynomial as admissible function. Firstly, a convergence study is performed for the completely free horn-gear system withh = 0.012, R1 = 0.014, R2 = 0.040, R3 = 0.028, L = 0.174, and s = 0t. In the following study, Poisson’s ration is taken as ν=0.3. For convenience, the displacement amplitude functions U¯, V¯, and W¯ in each coordinate direction have equivalent Chebyshev polynomial terms which means Iq=Kq=Mq and Jq=Lq=Nq. In the following article, all the eigenvalues have four significant figures. In order to express and compare easily, the eigenvalue parameter Ω=ωρ/G is adopted.Table1 shows the convergence of the first four eigenfrequencies. The first four eigenfrequencies can basically meet the requirements of ultrasonic gear honing. The values of Iq and Jq start at six and ten, respectively, increasing Iq from 6 to 16 and Jq from 10 to 16. The results showed that when Iq×Jq=6×10, the first, second, and fourth eigenfrequencies were close to the final values when Iq×Jq=16×16. This phenomenon indicated the outstanding numerical stability of this method. The eigenfrequencies who were the easiest to converge to the final value are the first and second eigenfrequencies. Iq = 15 and Jq = 14 were the smallest terms to obtain accurate values in the r and z directions. If we want to obtain higher order eigenfrequencies, the numbers of Chebyshev polynomial terms Iq and Jq should be larger. The first four eigenfrequencies all converging to the final values were at Iq×Jq=16×14. This result proved that the Chebyshev–Ritz method can ensure the first four eigenfrequencies converge to the final values at least. It can be adopted for analyzing the vibration characteristic of the horn-gear system in Figure 1.Table 1
Convergence of the first four eigenvalues inΩ=ωρ/G of a completely free horn-gear system with h = 0.012, R1 = 0.014, R2 = 0.040, R3 = 0.028, L = 0.174, s = 0t, ν=0.3, and s = 0t.
Iq×JqΩ1Ω2Ω3Ω46×106.70125.8533.0544.486×126.67425.8432.8444.116×146.66425.8332.7444.096×166.65725.8232.6844.0710×106.62325.7330.6844.0610×126.60625.7230.4944.0410×146.59225.7030.3444.0210×166.58225.6830.2244.0014×106.59425.6729.9144.0414×126.56725.6529.5344.0114×146.55025.6429.3243.9914×166.53825.6329.1743.9815×106.59125.6629.8344.0415×126.56125.6329.4144.0115×146.52225.9028.8543.9815×166.52225.5928.8543.9716×106.55425.6629.7744.0416×126.52225.6029.3144.0016×146.52225.5928.8543.9716×166.52225.5928.8543.97


### 3.2. Experiment Analysis

In order to verify the accuracy of the eigenfrequencies achieved by Chebyshev–Ritz method, we designed one ultrasonic horn and three gears for the modal experiment. The ultrasonic horn can be assembled with the three gears separately to form three horn-gear systems. The ultrasonic horn and the three gears are made of 45# steel. Poisson’s ratio of the horn-gear system wasν=0.31, Young’s modulus was E=210GPa, and density ρ=7850kg/m3. The one ultrasonic horn and three gears are shown in Figure 3. The dimension parameters of the horn-gear systems are listed in Table 2.Figure 3
(a) The three gears and (b) the one horn.
(a)(b)Table 2
Dimension parameters of the three horn-gear systems.
Parameters (mm)Horn-gear systemhR1R2R3LHorn-gear system 1 (HGS1)12144028174Horn-gear system 2 (HGS2)20144028174Horn-gear system 3 (HGS3)28144028174In Table2, the parameter h/2R2 changed from 0.15 to 0.35 representing the plates became thicker and thicker. The plates in HGS 2 and HGS 3 were not the thin plates any more. The parameter L/R1+R3=4.14 meant the ultrasonic horn was not thin rod. If we needed the natural frequencies of other kinds of vibration mode of the ultrasonic horn, the one-dimensional longitudinal vibration beam theory was no longer applicable.We established an experimental platform to acquire the natural frequencies of the horn-gear system through hammering method. To simulate the completely free condition, we hung the horn-gear system with two ropes located at both ends of the horn-gear system. A microphone connected to computer was hung near the horn-gear system to receive the acoustic signal generated by knocking the horn-gear system. The audio signal acquisition system based on sound card was installed on the computer. The experimental platform was established as shown in Figure4.Figure 4
The experimental platform.The frequencies of the three horn-gear system obtained by Chebyshev–Ritz method, the experiment, and the finite element method are listed in Table3–5. The frequencies obtained by the Chebyshev–Ritz method are expressed by FC. The frequencies obtained by the experiment are expressed by FE and frequencies obtained by the finite element method are expressed FF. The finite element analysis type was a modal analysis with Block Lanczos. The selected unit type is 20-node solid 95, and the meshing method is intelligent meshing. The index FCR represents the relative error between the natural frequency FC obtained from Chebyshev–Ritz method and the natural frequency FF obtained by the finite element method. The index FER represents the natural frequency FE obtained by the hammering experiment and the natural frequency FF obtained by the finite element method. Poisson’s ratio of the ultrasonic horn and gear is ν = 0.31, Young’s modulus is E = 210 GPa, and density ρ = 7850 kg/m3.Table 3
Frequencies between 1 kHz and 20 kHz of HGS1 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC3324.013019FE3380.513254FF3458.013462FCR3.883.29FER2.241.55s = 0aFC8340.41455518959FE8248.91456218435FF8324.61522118255FCR0.194.383.86FER0.914.330.99s = 1FC3086.01175314896FE3101.81171415175FF3054.71196315448FCR1.021.763.57FER1.542.081.77s = 2FC1109719173FE1094219024FF1093218391FCR1.514.25FER0.093.44Table 4
Frequencies between 1 kHz and 20 kHz of HGS2 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC2955.31284114469FE2931.11303414476FF3034.01340614725FCR2.594.211.74FER3.392.771.69s = 0aFC8045.515394FE8074.715270FF8179.215384FCR1.630.07FER1.280.74s = 1FC2508.7117681617719320FE2566.1117141615419186FF2689.4115111549318792FCR6.722.234.412.81FER4.581.764.272.10s = 2FC14957FE14835FF14735FCR1.51FER0.68Table 5
Frequencies between 1 kHz and 20 kHz of HGS3 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC2777.81276014489FE2739.51315814546FF2818.61338815153FCR1.454.694.58FER2.811.724.01s = 0aFC7567.414998FE7688.515156FF7726.615378FCR2.062.47FER0.491.44s = 1FC2088.2116921685919330FE2092.6116091703319745FF2398.4114431726019968FCR12.932.182.323.20FER12.751.451.321.12s = 2FC17047FE17247FF17266FCR1.27FER0.11In Table3, 6 FERs are not greater than 2%, and no FER is greater than 5% in the 10 data sets. In Figure 5(a), the percentage of FER no more than 2% which means (6⁄10) × 100% is 60%. At the same time, in Table 3, 4 FCRs are not greater than 2%, and no FCR is greater than 5%. In Figure 5(a), the percentage of FCR no more than 2% which means (4⁄10) × 100% is 40%. The percentage of FCR and FER in Figures 5(b) and 5(c) are the same as that of Figure 5(a). In Table 4, 4 FCRs and 5 FERs are no more than 2%, and 1 FCR and 0 FER are greater than 5%. In Table 5, there are 2 FCRs and 7 FERs no more than 2%, and 2 FCRs and 1 FER greater than 5%. As can be seen from the results in Table 3, there is no FCR greater than 5% which means the simplified model of the horn-gear system is reasonable, the coupling conditions of each part are accurate, and the Chebyshev–Ritz method is effective for multimodule coupling. This phenomenon indicates that the Chebyshev–Ritz method based on the three-dimensional elastic theory not only has higher precision for analyzing a horn-gear system consists of the thinner plate but also has sufficient accuracy to analyze the horn-gear system composed of thicker plates.Figure 5
Bar graphs of the percentage of FER and FCR, in total, for each horn-gear system according to Tables3–5. (a) Percentage of FER and FCR as a total of HGS1. (b) Percentage of FER and FCR as a total of HGS2. (c) Percentage of FER and FCR as a total of HGS3.
(a)(b)(c)WhenL/R1+R3 is less than ten, the accuracy of the horn’s eigenfrequencies become unacceptable if we use the classical one-dimensional beam theory. Thus, it is necessary to use three-dimensional beam theory to analyze the eigenfrequencies of the ultrasonic horn. It can be seen from Figure 5, the percentage of FCRs greater than 5% of each horn-gear system was low. This phenomenon indicated that the accuracy rate of frequencies obtained by Chebyshev–Ritz method were acceptable. The Chebyshev–Ritz method can be adopted for analyzing the horn-gear system consist of thick rod and thick plate. Because the percentage of FERs greater than 5% was low, the experimental results can be considered reasonable and reliable. In Figure 5, the FERs were mainly concentrated in the range of 0% to 2% and rarely beyond the range of 5%. It indicated that the system error of the experimental platform has little effect on the measurement accuracy of frequencies. And, among the total of 30 sets of data, 33% of the FCRs is no more than 2% and less than 7% of the FCRs is more than 5%. This phenomenon proved that the frequencies obtained by Chebyshev–Ritz method, and the finite element method are not much different. The accuracy of frequencies obtained by the two methods was basically at the same level.


## 3.1. Convergence Analysis

It is necessary to check the convergence of the eigenfrequencies using Chebyshev polynomial as admissible function. Firstly, a convergence study is performed for the completely free horn-gear system withh = 0.012, R1 = 0.014, R2 = 0.040, R3 = 0.028, L = 0.174, and s = 0t. In the following study, Poisson’s ration is taken as ν=0.3. For convenience, the displacement amplitude functions U¯, V¯, and W¯ in each coordinate direction have equivalent Chebyshev polynomial terms which means Iq=Kq=Mq and Jq=Lq=Nq. In the following article, all the eigenvalues have four significant figures. In order to express and compare easily, the eigenvalue parameter Ω=ωρ/G is adopted.Table1 shows the convergence of the first four eigenfrequencies. The first four eigenfrequencies can basically meet the requirements of ultrasonic gear honing. The values of Iq and Jq start at six and ten, respectively, increasing Iq from 6 to 16 and Jq from 10 to 16. The results showed that when Iq×Jq=6×10, the first, second, and fourth eigenfrequencies were close to the final values when Iq×Jq=16×16. This phenomenon indicated the outstanding numerical stability of this method. The eigenfrequencies who were the easiest to converge to the final value are the first and second eigenfrequencies. Iq = 15 and Jq = 14 were the smallest terms to obtain accurate values in the r and z directions. If we want to obtain higher order eigenfrequencies, the numbers of Chebyshev polynomial terms Iq and Jq should be larger. The first four eigenfrequencies all converging to the final values were at Iq×Jq=16×14. This result proved that the Chebyshev–Ritz method can ensure the first four eigenfrequencies converge to the final values at least. It can be adopted for analyzing the vibration characteristic of the horn-gear system in Figure 1.Table 1
Convergence of the first four eigenvalues inΩ=ωρ/G of a completely free horn-gear system with h = 0.012, R1 = 0.014, R2 = 0.040, R3 = 0.028, L = 0.174, s = 0t, ν=0.3, and s = 0t.
Iq×JqΩ1Ω2Ω3Ω46×106.70125.8533.0544.486×126.67425.8432.8444.116×146.66425.8332.7444.096×166.65725.8232.6844.0710×106.62325.7330.6844.0610×126.60625.7230.4944.0410×146.59225.7030.3444.0210×166.58225.6830.2244.0014×106.59425.6729.9144.0414×126.56725.6529.5344.0114×146.55025.6429.3243.9914×166.53825.6329.1743.9815×106.59125.6629.8344.0415×126.56125.6329.4144.0115×146.52225.9028.8543.9815×166.52225.5928.8543.9716×106.55425.6629.7744.0416×126.52225.6029.3144.0016×146.52225.5928.8543.9716×166.52225.5928.8543.97


## 3.2. Experiment Analysis

In order to verify the accuracy of the eigenfrequencies achieved by Chebyshev–Ritz method, we designed one ultrasonic horn and three gears for the modal experiment. The ultrasonic horn can be assembled with the three gears separately to form three horn-gear systems. The ultrasonic horn and the three gears are made of 45# steel. Poisson’s ratio of the horn-gear system wasν=0.31, Young’s modulus was E=210GPa, and density ρ=7850kg/m3. The one ultrasonic horn and three gears are shown in Figure 3. The dimension parameters of the horn-gear systems are listed in Table 2.Figure 3
(a) The three gears and (b) the one horn.
(a)(b)Table 2
Dimension parameters of the three horn-gear systems.
Parameters (mm)Horn-gear systemhR1R2R3LHorn-gear system 1 (HGS1)12144028174Horn-gear system 2 (HGS2)20144028174Horn-gear system 3 (HGS3)28144028174In Table2, the parameter h/2R2 changed from 0.15 to 0.35 representing the plates became thicker and thicker. The plates in HGS 2 and HGS 3 were not the thin plates any more. The parameter L/R1+R3=4.14 meant the ultrasonic horn was not thin rod. If we needed the natural frequencies of other kinds of vibration mode of the ultrasonic horn, the one-dimensional longitudinal vibration beam theory was no longer applicable.We established an experimental platform to acquire the natural frequencies of the horn-gear system through hammering method. To simulate the completely free condition, we hung the horn-gear system with two ropes located at both ends of the horn-gear system. A microphone connected to computer was hung near the horn-gear system to receive the acoustic signal generated by knocking the horn-gear system. The audio signal acquisition system based on sound card was installed on the computer. The experimental platform was established as shown in Figure4.Figure 4
The experimental platform.The frequencies of the three horn-gear system obtained by Chebyshev–Ritz method, the experiment, and the finite element method are listed in Table3–5. The frequencies obtained by the Chebyshev–Ritz method are expressed by FC. The frequencies obtained by the experiment are expressed by FE and frequencies obtained by the finite element method are expressed FF. The finite element analysis type was a modal analysis with Block Lanczos. The selected unit type is 20-node solid 95, and the meshing method is intelligent meshing. The index FCR represents the relative error between the natural frequency FC obtained from Chebyshev–Ritz method and the natural frequency FF obtained by the finite element method. The index FER represents the natural frequency FE obtained by the hammering experiment and the natural frequency FF obtained by the finite element method. Poisson’s ratio of the ultrasonic horn and gear is ν = 0.31, Young’s modulus is E = 210 GPa, and density ρ = 7850 kg/m3.Table 3
Frequencies between 1 kHz and 20 kHz of HGS1 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC3324.013019FE3380.513254FF3458.013462FCR3.883.29FER2.241.55s = 0aFC8340.41455518959FE8248.91456218435FF8324.61522118255FCR0.194.383.86FER0.914.330.99s = 1FC3086.01175314896FE3101.81171415175FF3054.71196315448FCR1.021.763.57FER1.542.081.77s = 2FC1109719173FE1094219024FF1093218391FCR1.514.25FER0.093.44Table 4
Frequencies between 1 kHz and 20 kHz of HGS2 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC2955.31284114469FE2931.11303414476FF3034.01340614725FCR2.594.211.74FER3.392.771.69s = 0aFC8045.515394FE8074.715270FF8179.215384FCR1.630.07FER1.280.74s = 1FC2508.7117681617719320FE2566.1117141615419186FF2689.4115111549318792FCR6.722.234.412.81FER4.581.764.272.10s = 2FC14957FE14835FF14735FCR1.51FER0.68Table 5
Frequencies between 1 kHz and 20 kHz of HGS3 obtained by the Chebyshev–Ritz method and the experiment compared with those obtained by the finite element method.
Frequency (Hz)f1f2f3f4s = 0tFC2777.81276014489FE2739.51315814546FF2818.61338815153FCR1.454.694.58FER2.811.724.01s = 0aFC7567.414998FE7688.515156FF7726.615378FCR2.062.47FER0.491.44s = 1FC2088.2116921685919330FE2092.6116091703319745FF2398.4114431726019968FCR12.932.182.323.20FER12.751.451.321.12s = 2FC17047FE17247FF17266FCR1.27FER0.11In Table3, 6 FERs are not greater than 2%, and no FER is greater than 5% in the 10 data sets. In Figure 5(a), the percentage of FER no more than 2% which means (6⁄10) × 100% is 60%. At the same time, in Table 3, 4 FCRs are not greater than 2%, and no FCR is greater than 5%. In Figure 5(a), the percentage of FCR no more than 2% which means (4⁄10) × 100% is 40%. The percentage of FCR and FER in Figures 5(b) and 5(c) are the same as that of Figure 5(a). In Table 4, 4 FCRs and 5 FERs are no more than 2%, and 1 FCR and 0 FER are greater than 5%. In Table 5, there are 2 FCRs and 7 FERs no more than 2%, and 2 FCRs and 1 FER greater than 5%. As can be seen from the results in Table 3, there is no FCR greater than 5% which means the simplified model of the horn-gear system is reasonable, the coupling conditions of each part are accurate, and the Chebyshev–Ritz method is effective for multimodule coupling. This phenomenon indicates that the Chebyshev–Ritz method based on the three-dimensional elastic theory not only has higher precision for analyzing a horn-gear system consists of the thinner plate but also has sufficient accuracy to analyze the horn-gear system composed of thicker plates.Figure 5
Bar graphs of the percentage of FER and FCR, in total, for each horn-gear system according to Tables3–5. (a) Percentage of FER and FCR as a total of HGS1. (b) Percentage of FER and FCR as a total of HGS2. (c) Percentage of FER and FCR as a total of HGS3.
(a)(b)(c)WhenL/R1+R3 is less than ten, the accuracy of the horn’s eigenfrequencies become unacceptable if we use the classical one-dimensional beam theory. Thus, it is necessary to use three-dimensional beam theory to analyze the eigenfrequencies of the ultrasonic horn. It can be seen from Figure 5, the percentage of FCRs greater than 5% of each horn-gear system was low. This phenomenon indicated that the accuracy rate of frequencies obtained by Chebyshev–Ritz method were acceptable. The Chebyshev–Ritz method can be adopted for analyzing the horn-gear system consist of thick rod and thick plate. Because the percentage of FERs greater than 5% was low, the experimental results can be considered reasonable and reliable. In Figure 5, the FERs were mainly concentrated in the range of 0% to 2% and rarely beyond the range of 5%. It indicated that the system error of the experimental platform has little effect on the measurement accuracy of frequencies. And, among the total of 30 sets of data, 33% of the FCRs is no more than 2% and less than 7% of the FCRs is more than 5%. This phenomenon proved that the frequencies obtained by Chebyshev–Ritz method, and the finite element method are not much different. The accuracy of frequencies obtained by the two methods was basically at the same level.


## 4. Conclusion

In this paper, the three-dimensional coupling model of the horn-gear system is established, and the vibration characteristics are analyzed in three dimensions. The natural frequencies of the horn-gear system are achieved accurately through applying the Chebyshev–Ritz method. The frequencies are compared with the results obtained by the experiment of hammering method and the finite element method, respectively. It is proved that the Chebyshev–Ritz method based on the three-dimensional elasticity theory is feasible to analyze the vibration characteristics of the horn-gear system. The application of the Chebyshev–Ritz method further extends the traditional nonresonant theory of mixed using 1D and 2D to 3D analysis, which fully reflects the vibration characteristics of the horn-gear system and solves the mismatching of the coupling boundary conditions and the size limitation. This is proved by the index FCR that the percentage of FCR, in total, greater than 5% is low. Since Chebyshev–Ritz method is easy to parameterize, we only need to input the different parameters in the program to get different frequencies in different vibration modes by adjusting the value ofs. The comparison of frequencies obtained by the experiment shows that the frequencies obtained by the Chebyshev–Ritz method are reasonable and reliable. It realizes the three-dimensional analysis of the horn-gear system used in ultrasonic gear honing.



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