1 Introduction The special dynamics of the ultrasonic vibration drilling process allows it to obtain good processing results. However, due to the complex design of the ultrasonic vibration drilling system, the operability and reliability of the process are not easy to meet the actual production needs, thus hindering the popularization and application of the vibration drilling technology to some extent. When designing a vibratory drilling system, the drill bit is usually regarded as an iso-section bar, and the length of the transducer and horn is treated at half wavelength or 1/4 wavelength. If the drill bit is short, it can be considered as the equivalent mass when designing the horn; if the drill bit is longer, the length is designed to be an integral multiple of the half wavelength. Obviously, when the bit wears during machining, it must be replaced in time, otherwise the amplitude of the bit will be significantly reduced or even stopped. The theory of local resonance proposed in 1982 breaks through the limitation that slender bit lengths must be designed with an integral multiple of half the wavelength. The theory is that the slender drill bit can resonate independently of the transducer and the horn system used for driving, and it has a speed-magnifying effect. Therefore, when the length of the drill bit changes due to wear, only the ultrasonic generator needs to be adjusted. The frequency can make the drill maintain good machining results. It can be seen that the principle of local resonance has great practical value. Local resonance phenomena have been observed in many vibratory cutting processes and have been applied in actual production. 2 Research on the Mechanism of Local Resonance Phenomenon There are many researchers who have discussed the mechanism of local resonance phenomena from different angles (such as generalized amplifier theory, full resonance point of view, weak coupling vibration of system and tool bar, etc.). The newly published literature analyzes the local resonance mechanism based on the principle of vibration absorption of the power absorber device, and points out that there will always be displacement nodes at the joint between the horn and the tool rod, and the vibration of the tool rod will cause its own law (namely, fixed-free Resonance of mode). This paper intends to analyze the dynamics of the ultrasonic vibration drilling acoustic system by establishing a mathematical model and explore the local resonance phenomenon of the tool bar. The ultrasonic vibration drilling processing acoustic system generally consists of a transducer, a horn, and a tool bar. The transducer is just a driving system that generates high-frequency oscillations, and thus the horn and tool bar can be simplified as equivalent mass-spring-damping models. The structure and mechanical model are shown in FIG. 1 . 
(a) System architecture diagram
(b) Mechanical model without damping
(c) Mechanical model considering damping Figure 1 Structure of vibration drilling acoustics system and mechanical model The variable amplitude rod is m1-k1-c1, and the tool rod is m2-k2-c2, then the horn and the tool rod form a two-freedom Degree system. According to Fig. 1c, the vibrational differential equation of this two-degree-of-freedom system is
(1) 
(2) Or expressed as a matrix 
(3) In the formula, replace u0sinwt with Im(u0eiwt). For brevity, the imaginary symbol Im is omitted without causing ambiguity. Let the steady-state response be [x1] = [X1] eiwt x2 X2 (4) Substitution into differential equations is available 
(5) The phasors of the horn and the tool bar are 
(6) where the frequency equation is f(w2)=[-m1w2+i(c1+c2)w+k1+k2](k2-m2w2+ic2w)-(ic2w+k2)2 (7) Modulus of Phasor for 
(8) The system's steady-state displacement response function is xi(t) = Xi(w) eiwt (i = 1, 2) (9) When excluding damping c1, c2 (see Figure 1b), Equation (6) can be for 
(10) The frequency equation at this time becomes f(w2)=(k1+k2-m1w2)(k2-m2w2)-k22 (11) When the damping is not taken into account, the mode of the phasor of the horn and the tool bar is its amplitude. Let the natural frequency of the horn be 
The natural frequency of the tool bar is From the phasor equation (10), we can see that if we do not count the damping, then when w=w2, |X1|=0 and |X2|≠0. It can be seen that when the natural frequency of the tool bar and the excitation frequency converted by the transducer are equal, the amplitude at the output end of the horn is zero, that is, the connection point of the horn and the tool bar is a displacement node, and this The transducer and tool bar are in resonance. When designing a vibratory drilling system, the horn is generally designed based on the frequency of the transducer. The frequency of the horn is very close to the excitation frequency (ie w≈w1). Therefore, the designed horn and transducer are also close to resonance. That is, when the horn and the tool bar are coupled at the displacement node, there is w=w2≈w1, so the entire vibratory drilling acoustic system is actually in a resonance state. From the phasor equation (10), it can be seen that under the condition of no damping, if and only if w=w2, the joint of the horn and the tool bar is the displacement node. When the natural frequency of the tool bar changes (ie, the drill bit wears out), the output frequency of the ultrasonic generator only needs to be properly adjusted so that w=w2 can always be achieved even if the tool bar is in resonance. From this point of view, in the local resonance, in fact the entire acoustic system is still in a state of resonance. However, there must be damping in the actual system because the system energy is constantly dissipating. If the applied excitation force disappears, the vibration of the system will gradually decrease until it finally stops. From Equation (8), it can be seen that the amplitude of the output end of the horn cannot be zero regardless of the excitation frequency. When w=w1=w2, the entire acoustic system is in a resonant state, but there is also minimal vibration at the junction of the horn and the tool bar. Its amplitude is A2=|c2wu0|/|f(w2)| One point is consistent with the experimental results. 3 The relationship between the wear amount of the tool rod and the frequency adjustment range of the ultrasonic generator 
Figure 2 Longitudinal vibration of a rod with one end fixed and one free end. The range of frequency adjustment required by the sonotrode is now discussed when the tool bar undergoes certain wear during processing. When the rod with one end fixed and one end free is longitudinally vibrated, the force of the rod microelement is shown in FIG. 2 . The axis of the rod is the x-axis, and the longitudinal displacement of each cross-section is u(x,t). Let the rod length be L, the cross-sectional area be A, the rod density be r, and the elastic modulus be E. The longitudinal strain at any x-section of the rod is e(x), and the longitudinal tension is p(x), then e(x) = ∂u ∂ xp(x) = EAe(x) = EA ∂u ∂x The tension at the x+dx cross-section is p+ ∂p dx=EA (∂u + ∂2u dx ) ∂x ∂x ∂x2 According to Newton's second law, the differential equation for the rod element is established as rAdx ∂2u = ( p+ ∂p dx-p ) = ∂2u dx ∂t2 ∂x ∂x2 æ•´ç†2u = 1 ∂2u =wn=const ∂x2 c2 ∂t2 (12) where c2=E/r is inside the slender rod The speed of sound. Equation (12) is a one-dimensional wave equation that can be solved using the discrete variable method. Set the solution of the equation to u(x,t)=Ø(x)q(t) (13) to obtain Ø(x)=A1sin wnx +B1cos wnx cc (14) q(t)=A2sinwnt+B2coswnt ( 15) In the formula, the constants A1, B1 are determined by the boundary conditions; the constants A2, B2 are determined by the initial conditions. Obviously, for a given wn, Ø(x) determines the vibration mode of the rod vibration, ie, the mode shape function, and q(t) corresponds to the motion law of the wn. Therefore, the solution of equation (12) can also be expressed as u(x,t)=(A1sinwnx+B1coswnn) (A2sinwnt+B2coswnt)c c (13') The frequency equation can be determined based on the boundary conditions. For a tool bar with one end fixed and one end free, u(0,t)=0 (14') e(L,t)=∂u (L,t)=0 ∂x (14") will be (14') ), (14") Substitute into (13'), there is B1 = 0 (15') wn A1cos wn L = 0 cc (15") Obviously, by formula (15"), there is wn L = 2n+1 pc 2 (16) Or fn = 2n+1 c (n=0,1,2,...) 4 L (16') Equation (16) or (16') is the frequency equation of a rod with one end fixed and one end free. When the tool bar has local resonance, it vibrates at the frequency expressed by the above formula. It should be noted that the boundary condition (Equation (14')) indicates that the rod end is rigidly clamped, but in general, it is rather difficult to absolutely fix the rod end. When the length of the tool rod changes (ie, when the drill bit wears), assuming that the wear length is ∆L and the wear rate is d, d = (∆L/L) × 100%, the tool rod can be determined by formula (16). The frequency change range is h = fn'-fn × 100% = ∆L × 100% = d fn L-∆L 1-d (17) Conversely, the wear rate d can also be determined according to the frequency range h of the tool rod. , ie, d=h/(1+h) (18) Since the tool bar has local resonance and its frequency is equal to the excitation frequency, the frequency range of the tool bar is the frequency range of the ultrasonic generator, so it can be based on the ultrasonic generator. The adjustable frequency range determines the wear rate of the tool bar without changing the drill bit. For example, when the length of the tool bar is 303.55mm and the diameter is 1mm, and the adjustable frequency range of the ultrasonic generator is ±20%, the wear rate of the tool bar can reach 16.67% and the wear length can reach 50.59mm. In this wear range, the free end of the tool bar can be in a resonant state by appropriately adjusting the frequency of the ultrasonic generator. 4 Designing a Vibratory Drilling System Using the Principle of Local Resonance A typical vibratory drilling system consists of a cutting tool, a chuck and a horn, a horn, a transducer, and an ultrasonic generator. The traditional design method of the vibration drilling device is generally based on the input volume of the ultrasonic generator to establish a corresponding mathematical model according to the no-load condition of the tool, and then determine the equal impedance calculation formula of the tool rod collet and the horn according to the boundary conditions, according to the formula To determine the appropriate size. The chuck is mounted on the horn and can be designed either as a load on the end of the horn or as part of a horn. It has been pointed out in the foregoing that if the drill bit is short, it can be considered as equivalent mass when designing the horn; if the drill bit is longer, it is designed as an integral multiple of a half wavelength. Obviously, in deep hole drilling, the wear of the drill bit necessitates frequent replacement, which affects the production efficiency and processing cost. If the local resonance theory is used to design the acoustic system of the vibratory drilling device, the transducer, the horn and the collet can be designed as the driving part, and the tool bar is separately designed according to the local resonance principle, and the tool bar can also be guaranteed. Matching with the drive section, a large amplitude is obtained at the tool tip. When the drill bit wears out in a certain range, it is not necessary to replace the drill bit, and only the input frequency of the ultrasonic generator can be properly adjusted. According to the principle of local resonance, the author designed an ultrasonic vibration drilling device used on a vertical drilling machine. The device is fixed on the feed box and is used for drilling small diameter deep holes of engineering ceramic workpieces. When designing, the chuck is used as part of the horn, and the impedance matching between the components of the acoustic system is strictly controlled to ensure that the cutting tool rod can resonate with the vibration frequency of the transducer after it is mounted on the horn. Both transducers and horns are designed for a resonant frequency (20 kHz) and their length is half the wavelength. The cutting test shows that the ultrasonic vibration drilling device designed according to the local resonance principle has a good machining effect. 5 Conclusions The horn and tool bar are simplified into an equivalent mathematical model. By analyzing the dynamic law, it is found that when the so-called local resonance occurs in the tool bar, the entire system is actually in a state of resonance. The analysis found that when the impact of damping is not considered, the connection between the horn and the tool bar is a displacement node; when considering the impact of damping, the joint should have a certain vibration. The deep hole vibratory drilling system designed according to the principle of local resonance does not require frequent replacement of tool bars during machining. When the drill bit generates a certain amount of wear, the frequency of the ultrasonic generator can be adjusted properly to ensure that the tool bar maintains a large amplitude, and the adjustable frequency range h of the ultrasonic generator and the wear rate d of the tool bar satisfy the relation d =h/(1+h). Therefore, applying local resonance principle to the design of vibration drilling system has great practical value.
(a) System architecture diagram
(b) Mechanical model without damping
(c) Mechanical model considering damping Figure 1 Structure of vibration drilling acoustics system and mechanical model The variable amplitude rod is m1-k1-c1, and the tool rod is m2-k2-c2, then the horn and the tool rod form a two-freedom Degree system. According to Fig. 1c, the vibrational differential equation of this two-degree-of-freedom system is
(1) (2) Or expressed as a matrix (3) In the formula, replace u0sinwt with Im(u0eiwt). For brevity, the imaginary symbol Im is omitted without causing ambiguity. Let the steady-state response be [x1] = [X1] eiwt x2 X2 (4) Substitution into differential equations is available (5) The phasors of the horn and the tool bar are (6) where the frequency equation is f(w2)=[-m1w2+i(c1+c2)w+k1+k2](k2-m2w2+ic2w)-(ic2w+k2)2 (7) Modulus of Phasor for (8) The system's steady-state displacement response function is xi(t) = Xi(w) eiwt (i = 1, 2) (9) When excluding damping c1, c2 (see Figure 1b), Equation (6) can be for (10) The frequency equation at this time becomes f(w2)=(k1+k2-m1w2)(k2-m2w2)-k22 (11) When the damping is not taken into account, the mode of the phasor of the horn and the tool bar is its amplitude. Let the natural frequency of the horn be The natural frequency of the tool bar is From the phasor equation (10), we can see that if we do not count the damping, then when w=w2, |X1|=0 and |X2|≠0. It can be seen that when the natural frequency of the tool bar and the excitation frequency converted by the transducer are equal, the amplitude at the output end of the horn is zero, that is, the connection point of the horn and the tool bar is a displacement node, and this The transducer and tool bar are in resonance. When designing a vibratory drilling system, the horn is generally designed based on the frequency of the transducer. The frequency of the horn is very close to the excitation frequency (ie w≈w1). Therefore, the designed horn and transducer are also close to resonance. That is, when the horn and the tool bar are coupled at the displacement node, there is w=w2≈w1, so the entire vibratory drilling acoustic system is actually in a resonance state. From the phasor equation (10), it can be seen that under the condition of no damping, if and only if w=w2, the joint of the horn and the tool bar is the displacement node. When the natural frequency of the tool bar changes (ie, the drill bit wears out), the output frequency of the ultrasonic generator only needs to be properly adjusted so that w=w2 can always be achieved even if the tool bar is in resonance. From this point of view, in the local resonance, in fact the entire acoustic system is still in a state of resonance. However, there must be damping in the actual system because the system energy is constantly dissipating. If the applied excitation force disappears, the vibration of the system will gradually decrease until it finally stops. From Equation (8), it can be seen that the amplitude of the output end of the horn cannot be zero regardless of the excitation frequency. When w=w1=w2, the entire acoustic system is in a resonant state, but there is also minimal vibration at the junction of the horn and the tool bar. Its amplitude is A2=|c2wu0|/|f(w2)| One point is consistent with the experimental results. 3 The relationship between the wear amount of the tool rod and the frequency adjustment range of the ultrasonic generator Figure 2 Longitudinal vibration of a rod with one end fixed and one free end. The range of frequency adjustment required by the sonotrode is now discussed when the tool bar undergoes certain wear during processing. When the rod with one end fixed and one end free is longitudinally vibrated, the force of the rod microelement is shown in FIG. 2 . The axis of the rod is the x-axis, and the longitudinal displacement of each cross-section is u(x,t). Let the rod length be L, the cross-sectional area be A, the rod density be r, and the elastic modulus be E. The longitudinal strain at any x-section of the rod is e(x), and the longitudinal tension is p(x), then e(x) = ∂u ∂ xp(x) = EAe(x) = EA ∂u ∂x The tension at the x+dx cross-section is p+ ∂p dx=EA (∂u + ∂2u dx ) ∂x ∂x ∂x2 According to Newton's second law, the differential equation for the rod element is established as rAdx ∂2u = ( p+ ∂p dx-p ) = ∂2u dx ∂t2 ∂x ∂x2 æ•´ç†2u = 1 ∂2u =wn=const ∂x2 c2 ∂t2 (12) where c2=E/r is inside the slender rod The speed of sound. Equation (12) is a one-dimensional wave equation that can be solved using the discrete variable method. Set the solution of the equation to u(x,t)=Ø(x)q(t) (13) to obtain Ø(x)=A1sin wnx +B1cos wnx cc (14) q(t)=A2sinwnt+B2coswnt ( 15) In the formula, the constants A1, B1 are determined by the boundary conditions; the constants A2, B2 are determined by the initial conditions. Obviously, for a given wn, Ø(x) determines the vibration mode of the rod vibration, ie, the mode shape function, and q(t) corresponds to the motion law of the wn. Therefore, the solution of equation (12) can also be expressed as u(x,t)=(A1sinwnx+B1coswnn) (A2sinwnt+B2coswnt)c c (13') The frequency equation can be determined based on the boundary conditions. For a tool bar with one end fixed and one end free, u(0,t)=0 (14') e(L,t)=∂u (L,t)=0 ∂x (14") will be (14') ), (14") Substitute into (13'), there is B1 = 0 (15') wn A1cos wn L = 0 cc (15") Obviously, by formula (15"), there is wn L = 2n+1 pc 2 (16) Or fn = 2n+1 c (n=0,1,2,...) 4 L (16') Equation (16) or (16') is the frequency equation of a rod with one end fixed and one end free. When the tool bar has local resonance, it vibrates at the frequency expressed by the above formula. It should be noted that the boundary condition (Equation (14')) indicates that the rod end is rigidly clamped, but in general, it is rather difficult to absolutely fix the rod end. When the length of the tool rod changes (ie, when the drill bit wears), assuming that the wear length is ∆L and the wear rate is d, d = (∆L/L) × 100%, the tool rod can be determined by formula (16). The frequency change range is h = fn'-fn × 100% = ∆L × 100% = d fn L-∆L 1-d (17) Conversely, the wear rate d can also be determined according to the frequency range h of the tool rod. , ie, d=h/(1+h) (18) Since the tool bar has local resonance and its frequency is equal to the excitation frequency, the frequency range of the tool bar is the frequency range of the ultrasonic generator, so it can be based on the ultrasonic generator. The adjustable frequency range determines the wear rate of the tool bar without changing the drill bit. For example, when the length of the tool bar is 303.55mm and the diameter is 1mm, and the adjustable frequency range of the ultrasonic generator is ±20%, the wear rate of the tool bar can reach 16.67% and the wear length can reach 50.59mm. In this wear range, the free end of the tool bar can be in a resonant state by appropriately adjusting the frequency of the ultrasonic generator. 4 Designing a Vibratory Drilling System Using the Principle of Local Resonance A typical vibratory drilling system consists of a cutting tool, a chuck and a horn, a horn, a transducer, and an ultrasonic generator. The traditional design method of the vibration drilling device is generally based on the input volume of the ultrasonic generator to establish a corresponding mathematical model according to the no-load condition of the tool, and then determine the equal impedance calculation formula of the tool rod collet and the horn according to the boundary conditions, according to the formula To determine the appropriate size. The chuck is mounted on the horn and can be designed either as a load on the end of the horn or as part of a horn. It has been pointed out in the foregoing that if the drill bit is short, it can be considered as equivalent mass when designing the horn; if the drill bit is longer, it is designed as an integral multiple of a half wavelength. Obviously, in deep hole drilling, the wear of the drill bit necessitates frequent replacement, which affects the production efficiency and processing cost. If the local resonance theory is used to design the acoustic system of the vibratory drilling device, the transducer, the horn and the collet can be designed as the driving part, and the tool bar is separately designed according to the local resonance principle, and the tool bar can also be guaranteed. Matching with the drive section, a large amplitude is obtained at the tool tip. When the drill bit wears out in a certain range, it is not necessary to replace the drill bit, and only the input frequency of the ultrasonic generator can be properly adjusted. According to the principle of local resonance, the author designed an ultrasonic vibration drilling device used on a vertical drilling machine. The device is fixed on the feed box and is used for drilling small diameter deep holes of engineering ceramic workpieces. When designing, the chuck is used as part of the horn, and the impedance matching between the components of the acoustic system is strictly controlled to ensure that the cutting tool rod can resonate with the vibration frequency of the transducer after it is mounted on the horn. Both transducers and horns are designed for a resonant frequency (20 kHz) and their length is half the wavelength. The cutting test shows that the ultrasonic vibration drilling device designed according to the local resonance principle has a good machining effect. 5 Conclusions The horn and tool bar are simplified into an equivalent mathematical model. By analyzing the dynamic law, it is found that when the so-called local resonance occurs in the tool bar, the entire system is actually in a state of resonance. The analysis found that when the impact of damping is not considered, the connection between the horn and the tool bar is a displacement node; when considering the impact of damping, the joint should have a certain vibration. The deep hole vibratory drilling system designed according to the principle of local resonance does not require frequent replacement of tool bars during machining. When the drill bit generates a certain amount of wear, the frequency of the ultrasonic generator can be adjusted properly to ensure that the tool bar maintains a large amplitude, and the adjustable frequency range h of the ultrasonic generator and the wear rate d of the tool bar satisfy the relation d =h/(1+h). Therefore, applying local resonance principle to the design of vibration drilling system has great practical value.
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