Full wedge-type insert mounting method to improve the vibration reproducibility of ultrasonic cutting system

Abstract

The reproducibility of vibration amplitude during insert mounting plays an important role in the controllability of the ultrasonic cutting process. In this study, a novel full wedge-type insert mounting method was developed to suppress the vibration instability induced by insert remounting. An ordinary threaded connection was selected as a counterpart to evaluate the performance of the proposed full wedge-type method through finite element model (FEM) by simulation and experiments. A theoretical model describing the effects of the connection interface on the vibration characteristics of the ultrasonic cutting system was established accordingly. Both the ordinary and full wedge-type threaded connection methods are shown to have a minimum threshold of screw pre-tightening torque, below which the self-relaxation and vibration disturbance of the cutting insert cause transmission failure of vibration energy and the insert itself. The pre-tightening torque threshold of a full wedge-type threaded connection is smaller than that of an ordinary thread connection. The full wedge-type threaded connection shows much less intense vibration amplitude fluctuations than the ordinary threaded connection during repeated insert mounting operations. These performance improvements can be attributed to contact force amplification effects on the wedge-type interface under the same effective pre-tightening torque under the proposed full wedge-type insert mounting conditions.

Appendix

1.1 Equivalent longitudinal vibration model without insert

The system without insert could be equivalent to a longitudinal vibration system of a continuous rod with one end fixed and one end free, as shown in Fig. 1.

Fig. 29

Equivalent structure of ultrasonic horn

In Fig. 29, ignoring the arc transition section between the pitch plane and beam of horn, the length from the fixed pitch plane O to the output end P of the horn was L; the cross section of the horn was square, and the side length was h.

The initial single excitation longitudinal vibration from the transducer generates the external vibration on the plane O as O(t) which could be written as:

$$O(t) = A\mathrm{sin}\;\omega t$$

(4)

Taking O(t) at the fixed nodal plane O as the starting position of external vibration, the horn was equivalent to a quality-spring-damper vibration system with single degree of freedom, as shown in Fig. 30.

Fig. 30

Equivalent structure of ultrasonic horn without turning tool

In Fig. 30, M was the equivalent mass of section OP of the horn; c₁ was the damping coefficient of section OP of the horn in the X direction; k₁ was the stiffness coefficient of the horn in the X direction. According to Fig. 3 and the single-degree-of-freedom vibration response, the steady-state response of longitudinal vibration at point P at the output end of the horn was as follows:

$$\begin{aligned} X_{0} (t) & = \frac{{A{{\left[ {\sqrt {k_{1}^{2} + (c_{1} \omega )^{2} } } \right]} \mathord{\left/ {\vphantom {{\left[ {\sqrt {k_{1}^{2} + (c_{1} \omega )^{2} } } \right]} {k_{1} }}} \right. \kern-\nulldelimiterspace} {k_{1} }}}}{{\sqrt {\left[ {1 - (\frac{\omega }{{\omega_{n} }})^{2} } \right]^{2} + \left[ {2\zeta (\frac{\omega }{{\omega_{n} }})} \right]^{2} } }}\sin (\omega t + \phi ) \hfill \\ & = A_{0} \sin (\omega t + \phi ) \hfill \\ \end{aligned}$$

(5)

where φ = arctan[(− c₁ω)/(k₁ − Mω²)], ω_n = $\sqrt{{k}_{1}/M}$, ζ = c₁/c_n, c_n = 2Mω_n。

When the frequency ω of the external vibration O(t) approximated to natural frequency ω_n of the horn, the horn would resonate, and the steady-state response of point P at the output end appeared the maximum longitudinal vibration amplitude.

1.2 Equivalent longitudinal vibration model with turning tool

After connecting turning tool, the quality of the system changed. It was necessary to analyze the influence of its connection state of the tool with the horn on the ultrasonic longitudinal vibration further. In the case of constant excitation, its structure is shown in Fig. 31.

Fig. 31

Equivalent structure of horn with turning tool

In Fig. 31, m was the mass of insert. In vibration analysis, the insert was regarded as a rigid body, and the influence of its length on vibration was also ignored. The damping coefficient and stiffness coefficient of the section OP of the horn remained unchanged. The focus was the influence of insert on the vibration response of section OP of the horn.

First, insert and section OP of horn was considered as a connected whole part. According to Fig. 31, the equivalent mass of section OP of the horn was increased to M + m. The whole system could be regarded as a quality-spring-damper vibration system with single degree of freedom, and its structure is shown in Fig. 32.

Fig. 32

Equivalent vibration structure of ultrasonic horn with tool

The steady-state response of longitudinal vibration at point P of the output terminal of insert was as follows:

$$\begin{aligned} X_{01} (t) & = \frac{{A{{\left[ {\sqrt {k_{1}^{2} + (c_{1} \omega )^{2} } } \right]} \mathord{\left/ {\vphantom {{\left[ {\sqrt {k_{1}^{2} + (c_{1} \omega )^{2} } } \right]} {k_{1} }}} \right. \kern-\nulldelimiterspace} {k_{1} }}}}{{\sqrt {\left[ {1 - (\frac{\omega }{{\omega_{n1} }})^{2} } \right]^{2} + \left[ {2\zeta_{1} (\frac{\omega }{{\omega_{n1} }})} \right]^{2} } }}\sin (\omega t + \phi_{01} ) \hfill \\ & = A_{1} \sin (\omega t + \phi_{01} ) \hfill \\ \end{aligned}$$

(6)

where φ₀₁ = arctan{(− c₁ω)/[k₁ − (M + m)ω²]}, ζ₁ = c₁/c_n1, c_n1 = 2(M + m)ω_n1, ω_n1 = $\sqrt{{k}_{1}/(M+m)}$.

When the frequency ω of the external vibration O(t) approximated to natural frequency ω_n1 of the horn, it would resonate and the vibration steady-state response of point P at the output end would also appear the maximum longitudinal vibration amplitude.

It could be seen from the all above analysis that when the frequency ω of the external vibration O(t) remained unchanged and after the tool was connected with horn, section OP of the horn increased the equivalent mass and the natural frequency of the longitudinal vibration of whole system decreased and it would not resonate.

When insert and the output end of horn were connected by a “surface-to-surface” contact, the contact surface between tool and horn was regarded as a spring damping system. When analyzing the vibration response of tool in order to ensure the consistency of the vibration analysis, the separation of the contact surface and the self-relaxation of the threaded connection during the vibration transmission process were not considered. The system of horn and tool was regarded as a forced vibration system with damping mass and spring and its equivalent structure is shown in Fig. 33.

Fig. 33

Equivalent vibration structure with two degrees of freedom of spring damping system with tool

In Fig. 33, c1 was the damping coefficient of the connecting surface between insert and horn in the X direction; k₁ was the normal contact stiffness of the connecting surface between insert and horn in the X direction.

The vibration differential equation of the combined system was as follows:

$$\left\{ \begin{array}{l} m\ddot{x}_{2} = k_{2} (x_{1} - x_{2} ) + c_{2} (\dot{x}_{1} - \dot{x}_{2} ) \hfill \\ M\ddot{x}_{1} = k_{1} (O(t) - x_{1} ) + c_{1} (\dot{O}(t) - \dot{x}_{1} ) \hfill \\ \quad \quad \; - k_{2} (x_{1} - x_{2} ) - c_{2} (\dot{x}_{1} - \dot{x}_{2} ) \hfill \\ \end{array} \right.$$

(7)

Then the natural frequencies of the system obtained by the solution were:

$$\left\{ \begin{array}{l} \omega_{1}^{2} = \frac{{\left[ {k_{2} M + (k_{1} + k_{2} )m} \right]}}{2Mm} \hfill \\ \;\;\;\; - \frac{{\sqrt {\left[ {k_{2} M + (k_{1} + k_{2} )m} \right]^{2} - 4k_{1} k_{2} Mm} }}{2Mm} \hfill \\ \omega_{2}^{2} = \frac{{\left[ {k_{2} M + (k_{1} + k_{2} )m} \right]}}{2Mm} \hfill \\ \;\;\;\; + \frac{{\sqrt {\left[ {k_{2} M + (k_{1} + k_{2} )m} \right]^{2} - 4k_{1} k_{2} Mm} }}{2Mm} \hfill \\ \end{array} \right.$$

(8)

When the length of the contact body was greater than the thickness of the contact layer of the contact surface, for the separable elastic contact surface, the normal contact stiffness was less than the material stiffness of the contact body. Therefore, according to Eq. (7), it could be seen that, compared with the natural frequency ω_n of the horn without insert, the natural frequency of the system after connecting tool had dropped to a certain extent.

At the same time, the regular mode matrix for solving the combined system was as follows:

$$U = U^{(1)} + U^{(2)}$$

(9)

where

$$\left\{ \begin{array}{l} U^{(1)} = [\alpha {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\alpha [(k_{1} + k_{2} ) - \omega_{1}^{2} M]}}{{k_{2} }}]^{T} \hfill \\ U^{(2)} = [\beta {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{\beta k_{2} }}{{(k_{2} - \omega_{2}^{2} m)}}]^{T} \hfill \\ \end{array} \right.$$

(10)

In Eq. (7),

$$\left\{ \begin{array}{l} \alpha = \frac{{k_{2} }}{{\sqrt {k_{2}^{2} M + [(k_{1} + k_{2} ) - \omega_{1}^{2} M]^{2} m} }} \hfill \\ \beta = {\kern 1pt} {\kern 1pt} \sqrt {\frac{{(k_{2} - \omega_{2}^{2} M)^{2} }}{{k_{2}^{2} m + M(k_{2} - \omega_{2}^{2} M)^{2} }}} \hfill \\ \end{array} \right.$$

(11)

Based on the mentioned natural frequency and regular mode matrix of the system above, the mode superposition method was used to solve the system and the steady-state response of the system under external vibration O(t) was as follows:

$$\begin{aligned} \left[ \begin{gathered} X_{1} (t) \\ X_{2} (t) \\ \end{gathered} \right] & = U^{(1)} \cdot \alpha H_{1} (\omega )B\sin (\omega t + \phi_{0} - \phi_{1} ) \hfill \\ & + U^{(2)} \cdot \beta H_{2} (\omega )B\sin (\omega t + \phi_{0} - \phi_{2} ) \hfill \\ \end{aligned}$$

(12)

In Eq. (11),

$$B = \sqrt {(c_{1} A\omega )^{2} + (k_{1} A)^{2} }$$

(13)

$$H_{i} (\omega ) = {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{1}{{\sqrt {(\omega_{i}^{2} - \omega^{2} )^{2} + (2\zeta^{\prime}_{i} \omega )^{2} } }};i = 1,2$$

(14)

$$\phi_{0} = \arctan \frac{{c_{1} \omega }}{{k_{1} }}$$

(15)

$$\phi_{{\text{i}}} = \arctan \frac{{2\zeta_{{\text{i}}} \omega }}{{\omega_{{\text{i}}}^{2} - \omega^{2} }};i = 1,2$$

(16)

In Eqs. (13) and (15),

$$\left\{ \begin{array}{l} \zeta^{\prime}_{1} = \frac{{\alpha^{2} }}{{2\omega_{1} }}\{ \frac{{k_{2}^{2} (c_{1} + c_{2} ) - 2k_{2} [(k_{1} + k_{2} ) - \omega_{1}^{2} M]c_{2} }}{{k_{2}^{2} }} \hfill \\ \quad \quad \quad \quad - \frac{{[(k_{1} + k_{2} ) - \omega_{1}^{2} M]^{2} c_{2} }}{{k_{2}^{2} }}\} \hfill \\ \zeta^{\prime}_{2} = \frac{{\beta^{2} }}{{2\omega_{2} }}\{ \frac{{(k_{2} - \omega_{2}^{2} m)^{2} (c_{1} + c_{2} ) - 2k_{2} (k_{2} - \omega_{2}^{2} m)c_{2} }}{{(k_{2} - \omega_{2}^{2} m)^{2} }} \hfill \\ \quad \quad \quad \quad + \frac{{k_{2}^{2} c_{2} }}{{(k_{2} - \omega_{2}^{2} m)^{2} }}\} \hfill \\ \end{array} \right.$$

(17)

From the derived steady-state response with insert, it could be seen that there were two synthetic vibration responses at the same frequency and different phases at the output end of insert and horn. Therefore, in terms of external excited vibration, there were two resonance frequencies ω₁ and ω₂, and the amplitude of the output terminal of insert in the two resonance states were different. In addition, the resonance frequency and amplitude were directly affected by damping in the X direction and normal contact stiffness of the connection state between insert and horn. It could be seen from the analysis above that the resonance frequency of the “horn-insert” system changed compared with it without insert.

1.3 System vibration analysis with insert

Considering tool and section OP as a whole, it could be seen from Fig. 31 and Eq. (5) that the natural frequency of longitudinal vibration of horn with insert was reduced. The “horn-insert” system as a whole did not resonate under the original external excited vibration O(t). Taking the natural frequency ω_n of horn without insert as the benchmark and taking the ratio of the mass of tool to segment OP as the independent variable, trend and difference of the natural frequency of the overall system of “horn-tool” with tool were clarified as shown in Fig. 31.

In Fig. 31, mounting insert was equivalent to increasing the overall mass of section OP of horn, thereby significantly reduced the natural frequency of the system’s longitudinal vibration. The greater the mass ratio of insert to section OP of horn, the greater the drop in the system’s resonance frequency.

Based on whether insert was connected, the resonance amplitude of the output end of horn and the output terminal of insert were written as follows:

$$\left\{ \begin{array}{l} A_{0\max } = A\sqrt {k_{1}^{2} + (c_{1} \omega_{0} )^{2} } /2\zeta \cdot k_{1} \hfill \\ A_{1\max } = A\sqrt {k_{1}^{2} + (c_{1} \omega_{01} )^{2} } /2\zeta_{1} \cdot k_{1} \hfill \\ \end{array} \right.$$

(18)

When the externally excited vibration frequency ω was adjusted to be consistent with the natural frequency of the horn with insert, the ratio of the resonance amplitude of “horn-insert” system to the resonance amplitude of the output end of the horn without insert was as follows:

$$\eta_{0} = \frac{{A_{1\max } }}{{A_{0\max } }} = \frac{{\zeta \sqrt {k_{1}^{2} + (c_{1} \omega_{01} )^{2} } }}{{\zeta_{1} \sqrt {k_{1}^{2} + (c_{1} \omega_{0} )^{2} } }}$$

(19)

According to Fig. 15, the natural frequency of the “horn-insert” overall system decreased, that is, ω₀₁ < ω₀ and relative damping coefficient ζ₁ < ζ. Therefore, when the mass ratio of the insert to section OP was small, the influence of the natural frequency on the amplitude was smaller than the influence of the change of the overall system damping coefficient on the amplitude. The amplitude change at this time was mainly affected by the change of the damping coefficient of the overall system; when the mass of the insert to section OP of the horn was relatively large, the influence of the natural frequency on the amplitude was gradually greater than that of the change of the damping coefficient. The amplitude change at this time was mainly affected by the dual effects of the resonance state after frequency reduction and the change of the damping coefficient.

If insert and section OP section were regarded as a split vibration system with two degree of freedom, it could be seen from Fig. 33 and Eq. (8) that the steady-state vibration responses of section OP of horn with insert and insert respectively contained two vibration superposition states of two natural frequencies, and the two natural frequencies were inconsistent with the natural frequency of the system without insert, so that under the action of the original external excited vibration O(t), the split system of horn and insert disappeared resonance.

According to Eq. (8), after mounting insert, when the external excitation frequency was close to the two natural frequencies ω₁ and ω₂ and both ω₁ and ω₂ were less than ω_n, then the ratio of superimposed resonance amplitudes of the output terminal of insert to the amplitude of output end of horn without insert was as follows:

$$\eta_{1} = \frac{{A_{21\max } }}{{A_{0\max } }} = \frac{{U_{12} \cdot \alpha H_{1} (\omega )B + U_{22} \cdot \beta H_{1} (\omega )B}}{{A\sqrt {k_{1}^{2} + (c_{1} \omega_{0} )^{2} } /2\zeta \cdot k_{1} }}$$

(20)

According to Eq. (19) and the approximation theorem of numerical comparison, the resonance amplitude of output terminal of insert in the “horn-insert” split system increased compared with the output end of the horn without insert because the “horn-insert” split system was equivalent to a threaded connection and a damping system which had a certain amplification effect on the amplitude of high-frequency vibration.

In summary, according to Eq. (12), the output terminal of insert served as the final output end of vibration and its vibration steady-state response was as follows:

$$\begin{gathered} X_{2} (t) = U{}_{12}\left| {H_{1} (\omega )} \right|B{}_{1}\sin (\omega t + \phi_{0} - \phi_{1} ) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + U{}_{22}\left| {H_{2} (\omega )} \right|B{}_{2}\sin (\omega t + \phi_{0} - \phi_{2} ) \hfill \\ \end{gathered}$$

(21)

According to Eqs. (8) to (16), the contact state was unknown. The mass ratio m/M of tool to section OP of the horn, the contact stiffness k₂ and the contact damping c₂ were independent variables and the correlation functions were as follows:

$$\left\{ \begin{array}{l}f_{1} (m/M,k_{2} ,c_{2} ) = U_{12} \cdot \alpha H_{1} (\omega )B \hfill \\ f_{2} (m/M,k_{2} ,c_{2} ) = U_{22} \cdot \beta H_{2} (\omega )B \hfill \\ \theta_{1} (m/M,k_{2} ,c_{2} ) = \phi_{0} - \phi_{1} \hfill \\ \theta_{2} (m/M,k_{2} ,c_{2} ) = \phi_{0} - \phi_{1} \hfill \\ \end{array} \right.$$

(22)

After arranging, the steady-state vibration response of the output terminal of insert was as follows:

$$\begin{aligned} X_{2} (t) & = \sqrt {f_{1}^{2} + f_{2}^{2} + 2f_{1} f_{2} \cos (\theta_{2} - \theta_{1} )} \hfill \\ & \cdot \sin (\omega t + \frac{{\theta_{2} + \theta_{1} }}{2} + \varphi ) \hfill \\ & = F(m,k_{2} ,c_{2} )\sin [\omega t + \alpha (m,k_{2} ,c_{2} )] \hfill \\ \end{aligned}$$

(23)

where

$$\varphi = \arctan (\frac{{f_{1} - f_{2} }}{{f_{1} + f_{2} }}\tan \frac{{\theta_{2} - \theta_{1} }}{2})$$

(24)

According to Eq. (23), the output terminal of insert was the final output end. The amplitude F(m/M,k₂,c₂) of vibration steady-state response of tool and the phase difference α(m/M,k₂,c₂) between it and the external excited vibration O(t) were the function of ratio m/M, contact stiffness k₂ and contact damping c₂. When the mass ratio m/M was determined, the vibration transmission quality of the output terminal of insert was also determined by the contact stiffness and contact damping.

According to Eqs. (5), (11) and (23), after mounting insert, the contact stiffness and contact damping of the contact surface between insert and the output end of horn indirectly had important influence on the vibration transmission state of the tool tip. The contact stiffness and contact damping were also different which affected the resonance frequency of “horn-insert” system and caused a certain deviation between it and the preset output resonance frequency and resonance amplitude, respectively.