3.1 Movements analysis when ω₁=ω₂

When ω₁=ω₂, the double eccentricity sleeves have rotations with the same rotation speeds and directions. The polar equation of motion curve of O₂ can be obtained from Eq. (2):

According to Eq. (4), the motion curve of the center point O₂ of swing head is a circle, the center point O is geometric center of external eccentric sleeve, the radius is the sum of eccentricity of internal and external eccentric sleeves, i.e. e₁+e₂, as shown in Fig. 4a.

Figure 4Motion curve of O₂ when ω₁=ω₂. (a) e₁≠e₂, (b) $e_{\mathrm{1}}=e_{\mathrm{2}}=e$.

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From Eq. (3), when ω₁=ω₂, the acceleration equation of O₂ can be obtained as follow:

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the acceleration equation of O₂:

The corresponding acceleration and amplitude curve of O₂ are presented in Figs. 5 and 6, respectively. Figure 5 shows that, when ω₁=ω₂, no matter the eccentricity of internal and external eccentric sleeves is equal or not, the acceleration value always keep constant and the value are equal. Figure 6 reveals the effects of $e_{\mathrm{1}}/e_{\mathrm{2}}(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ on the amplitude of the acceleration. Figure 6 shows that no matter how the eccentricity of internal and external eccentric sleeves changes, the amplitude of acceleration is always equal to 0.

Figure 5Effects of e₁ and e₂ on the acceleration when ω₁=ω₂ ($e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$.

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Figure 6Effects of e₁∕e₂ on the amplitude when ω₁=ω₂ ($e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$.

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3.2 Movements analysis when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$

When ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$, the double eccentricity sleeves have rotations with the same rotation speeds but opposite directions. The polar equation of motion curve of O₂ can be obtained from Eq. (2):

According to Eq. (7), ${\mathit{\rho }}_{max}=e_{\mathrm{1}}+e_{\mathrm{2}}$ and ${\mathit{\rho }}_{min}=\left|e_{\mathrm{1}}-e_{\mathrm{2}}\right|$ can be obtained. The motion curve of point O₂ is an ellipse with a center point located in the geometric center O of external eccentric sleeve, taking 2(e₁+e₂) as the long axis and 2|e₁−e₂| as the short axis, as shown in Fig. 7a.

Figure 7Motion curve of O₂ when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$. (a) e₁≠e₂, (b) $e_{\mathrm{1}}=e_{\mathrm{2}}=e$.

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When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the polar equation of motion curve of O₂:

At this time, the motion curve of O₂ is a straight line with a total length of 4e and a middle point located in the geometric center O of external eccentric sleeve, as shown in Fig. 7b.

From Eq. (3), when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$, the acceleration equation of O₂ can be obtained as follow:

$a_{max}^{\prime }={\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}\left(e_{\mathrm{1}}+e_{\mathrm{2}}\right)$, $a_{min}^{\prime }={\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}\left|e_{\mathrm{1}}-e_{\mathrm{2}}\right|$ and $A^{\prime }=a_{max}^{\prime }-a_{min}^{\prime }=\mathrm{2}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}{\left|e_{\mathrm{1}},e_{\mathrm{2}}\right|}_{min}$ can be obtained from Eq. (9).

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the acceleration equation of O₂:

$a_{max}=\mathrm{2}e{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}$, $a_{min}=\mathrm{0}$ and $A=a_{max}-a_{min}=\mathrm{2}e{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}$ can be obtained from Eq. (10).

Figure 8Effects of e₁∕e₂ on the acceleration when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$ ($e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e$).

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Figure 9Effects of e₁∕e₂ on the amplitude when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$ ($e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$.

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The corresponding acceleration and amplitude curve of O₂ are presented in Figs. 8 and 9, respectively. Figure 8 shows that, when ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$, the acceleration curve changed cyclically with cosine function form over time, and the maximum acceleration value is equal no matter the e₁∕e₂ how to change. When e₁ or e₂ is equal to 0, the straight line will change into circular line, the corresponding acceleration and amplitude curve are the same with ω₁=ω₂. When $e_{\mathrm{1}}/e_{\mathrm{2}}=e_{\mathrm{2}}/e_{\mathrm{1}}$, the acceleration curves will coincide. Figure 9 reveals the effects of $e_{\mathrm{1}}/e_{\mathrm{2}}(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ on the amplitude of the acceleration. Figure 9 shows that the amplitude curve is a piecewise function curve. When $\mathrm{0}<e_{\mathrm{1}}/e_{\mathrm{2}}<\mathrm{1}$, the curve increases rapidly. When $e_{\mathrm{1}}/e_{\mathrm{2}}>\mathrm{1}$, the curve decreases gradually, the amplitude of the acceleration reaches the maximum value when $e_{\mathrm{1}}/e_{\mathrm{2}}=\mathrm{1}$, and when $e_{\mathrm{1}}/e_{\mathrm{2}}=e_{\mathrm{2}}/e_{\mathrm{1}}$, the amplitude value is equal.

3.3 Movements analysis when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$

When ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$ the double eccentricity sleeves have rotations with different rotation speeds and same rotation directions. The polar equation of motion curve of O₂ can be obtained from Eq. (2)

According to Eq. (11), ${\mathit{\rho }}_{max}=e_{\mathrm{1}}+e_{\mathrm{2}}$ and ${\mathit{\rho }}_{min}=\left|e_{\mathrm{1}}-e_{\mathrm{2}}\right|$ can be obtained, the motion curve of point O₂ is a spiral curve with 2(e₁+e₂) as external diameter, 2|e₁−e₂| as internal diameter and a center point located in the geometric center O of external eccentric sleeve, as shown in Fig. 10a.

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the polar equation of motion curve of O₂:

Figure 10Motion curve of O₂ when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/\mathrm{7})$. (a) e₁≠e₂, (b) $e_{\mathrm{1}}=e_{\mathrm{2}}=e$.

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At this time, the motion curve of O₂ is also a spiral with 4e as external diameter and the motion curve goes through the geometric center of external eccentric sleeve, as shown in Fig. 10b.

From Eq. (3), when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$, the acceleration equation of O₂ can be obtained as follow:

$a_{max}^{\prime }=e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}+e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}$, $a_{min}^{\prime }=\left|e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|$ and $A^{\prime }=a_{max}^{\prime }-a_{min}^{\prime }=\mathrm{2}{\left|e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}},e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|}_{min}$ can be obtained from Eq. (13).

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the acceleration equation of O₂:

$a_{max}=e\left({\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right)$, $a_{min}=e\left|{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|$ and $A=a_{max}-a_{min}=\mathrm{2}e{\left|{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}},{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|}_{min}$ can be obtained from Eq. (14)

The corresponding acceleration and amplitude curve of O₂ are presented in Figs. 11 and 12, respectively. Figure 11 shows that, when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/\mathrm{7})$, the acceleration curve changed cyclically with cosine function form over time,and the maximum acceleration value gradually decreases with the increasing of e₁∕e₂; when e₁ or e₂ is equal to 0, the spiral curve will change into circle, the corresponding acceleration curve keeps constant value but the acceleration value is different, and the amplitude is equal to 0. Figure 12 reveals the effects of $e_{\mathrm{1}}/e_{\mathrm{2}}(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ on the amplitude of the acceleration. Figure 12 shows that the amplitude curve is a piecewise function curve. When $\mathrm{0}<e_{\mathrm{1}}/e_{\mathrm{2}}<\mathrm{1.36}$, the curve increases rapidly. When $e_{\mathrm{1}}/e_{\mathrm{2}}>\mathrm{1.36}$, the curve decreases gradually, the amplitude of the acceleration reaches the maximum value when $e_{\mathrm{1}}/e_{\mathrm{2}}=\mathrm{1.36}$, when $e_{\mathrm{1}}/e_{\mathrm{2}}<\mathrm{1}$ and $e_{\mathrm{1}}/e_{\mathrm{2}}>\mathrm{1.72}$, the amplitude of the acceleration will be smaller than that when $e_{\mathrm{1}}/e_{\mathrm{2}}=\mathrm{1}$.

Figure 11Effects of e₁∕e₂ on the acceleration when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$ $(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/\mathrm{7})$.

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Figure 12Effects of e₁∕e₂ on the amplitude when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$ $(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/\mathrm{7})$.

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3.4 Movements analysis when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$

When ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$ the double eccentricity sleeves have rotations with different rotation speeds and same rotation directions. The polar equation of motion curve of O₂ can be obtained from Eq. (2):

According to Eq. (15), ${\mathit{\rho }}_{max}=e_{\mathrm{1}}+e_{\mathrm{2}}$ and ${\mathit{\rho }}_{min}=\left|e_{\mathrm{1}}-e_{\mathrm{2}}\right|$ can be obtained, the motion curve of point O₂ is a rose with 2(e₁+e₂) as external diameter, 2|e₁−e₂| as internal diameter and a center point located in the geometric center O of external eccentric sleeve, as shown in Fig. 13a.

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the polar equation of motion curve of O₂:

At this time, the motion curve of O₂ is also a rose with 4e as external diameter total and the motion curve goes through the geometric center of external eccentric sleeve, as shown in Fig. 13b.

Figure 13Motion curve of O₂ when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/-\mathrm{7})$. (a) e₁≠e₂, (b) $e_{\mathrm{1}}=e_{\mathrm{2}}=e$.

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From Fig. 13, when e₁≠e₂, the petal of rose curve will be more wider and overlap with each other, but the petal number of rose curve keeps unchanged.

From Eq. (3), when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$, the acceleration equation of O₂ can be obtained as follow:

$a_{max}^{\prime }=e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}+e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}$, $a_{min}^{\prime }=\left|e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|$ and $A^{\prime }=a_{max}^{\prime }-a_{min}^{\prime }=\mathrm{2}{\left|e_{\mathrm{1}}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}},e_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|}_{min}$ can be obtained from Eq. (17)

When eccentricity of internal and external eccentric sleeves is equal, i.e. $e_{\mathrm{1}}=e_{\mathrm{2}}=e$, the acceleration equation of O₂:

$a_{max}=e\left({\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}+{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right)$, $a_{min}=e\left|{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|$ and $A=a_{max}-a_{min}=\mathrm{2}e{\left|{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}},{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}\right|}_{min}$ can be obtained from Eq. (18)

Figure 14Effects of e₁∕e₂ on the acceleration when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$ $(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/-\mathrm{7})$.

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Figure 15Effects of e₁∕e₂ on the amplitude when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$ $(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ $({\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/-\mathrm{7})$.

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The corresponding acceleration and amplitude curve of O₂ are presented in Figs. 14 and 15, respectively. Figure 14 shows that, when ω₁≠ω₂ and ${\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$ (${\mathit{\omega }}_{\mathrm{1}}/{\mathit{\omega }}_{\mathrm{2}}=\mathrm{6}/-\mathrm{7}$), the acceleration curve changed cyclically with cosine function form over time,and the maximum acceleration value gradually decreases with the increasing of e₁∕e₂, when e₁ or e₂ is equal to 0, the rose curve will change into circle, the corresponding acceleration curve keeps constant value but the acceleration value is different, and the amplitude is equal to 0. Figure 15 reveals the effects of $e_{\mathrm{1}}/e_{\mathrm{2}}(e_{\mathrm{1}}+e_{\mathrm{2}}=\mathrm{2}e)$ on the amplitude of the acceleration. Figure 15 shows that the amplitude curve is a piecewise function curve. When $\mathrm{0}<e_{\mathrm{1}}/e_{\mathrm{2}}<\mathrm{1.36}$, the curve increases rapidly. When $e_{\mathrm{1}}/e_{\mathrm{2}}>\mathrm{1.36}$, the curve decreases gradually, the amplitude of the acceleration reaches the maximum value when $e_{\mathrm{1}}/e_{\mathrm{2}}=\mathrm{1.36}$, when $e_{\mathrm{1}}/e_{\mathrm{2}}<\mathrm{1}$ and $e_{\mathrm{1}}/e_{\mathrm{2}}>\mathrm{1.72}$, the amplitude of the acceleration will be smaller than that when $e_{\mathrm{1}}/e_{\mathrm{2}}=\mathrm{1}$.

3.5 Comprehensive Analysis

The acceleration and amplitude of the locus point O₂ of swing head reflect the movement of the swing head. The greater the acceleration changes, the more unstable the movement of the swing head and the greater shaking the rotary forging machine will be. In order to reduce the vibration of machine, it needs to reduce the maximum acceleration and the amplitude of acceleration at the same time, under the condition that the sum of eccentricity of internal and external eccentric sleeves and rotation speed of double eccentricity sleeve are constant, the corresponding equations as follow:

The relationship of the eccentricity of the internal and external eccentricity sleeves satisfying Eq. (20) can be obtained as follows:

• When ω₁=ω₂, no matter the eccentricity of internal and external eccentric sleeves is equal or not, the acceleration value keep constant, and the amplitude of acceleration is equal to 0.

• When ${\mathit{\omega }}_{\mathrm{1}}=-{\mathit{\omega }}_{\mathrm{2}}$, the maximum acceleration is equal no matter the eccentricity is equal or not, no matter e₁>e₂ or e₁<e₂, the amplitude of acceleration when e₁≠e₂ is always smaller than that when e₁=e₂.

• When ${\mathit{\omega }}_{\mathrm{1}}\ne {\mathit{\omega }}_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}>\mathrm{0}$, according to Eqs. (14) and (15),

  when ω₁>ω₂ Eq. (21) can be obtained as follow:

  $$\begin{array}{}\text{(20)}& {\displaystyle \frac{e_{\mathrm{1}}}{e_{\mathrm{2}}}}<{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}}}\end{array}$$

  When ω₁<ω₂, Eq. (22) can be obtained as follow:

  $$\begin{array}{}\text{(21)}& {\displaystyle \frac{e_{\mathrm{1}}}{e_{\mathrm{2}}}}>{\displaystyle \frac{\mathrm{2}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}}{{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}}}\end{array}$$

• When ${\mathit{\omega }}_{\mathrm{1}}\ne {\mathit{\omega }}_{\mathrm{2}}{\mathit{\omega }}_{\mathrm{1}}\cdot {\mathit{\omega }}_{\mathrm{2}}<\mathrm{0}$, according to Eqs. (18) and (19), when |ω₁|>|ω₂|, Eq. (23) can be obtained as follow:

  $$\begin{array}{}\text{(22)}& {\displaystyle \frac{e_{\mathrm{1}}}{e_{\mathrm{2}}}}<{\displaystyle \frac{{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}}{\mathrm{2}{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}}}\end{array}$$

  When |ω₁|<|ω₂|, Eq. (24) can be obtained as follow:

  $$\begin{array}{}\text{(23)}& {\displaystyle \frac{e_{\mathrm{1}}}{e_{\mathrm{2}}}}>{\displaystyle \frac{\mathrm{2}{\mathit{\omega }}_{\mathrm{2}}^{\mathrm{2}}-{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}}{{\mathit{\omega }}_{\mathrm{1}}^{\mathrm{2}}}}\end{array}$$

According to the amplitude curves and acceleration curves under different eccentricities, it can be seen that the smaller the e₁∕e₂, the smaller the amplitude and the maximum acceleration, except for the circular trajectory, under the premise of satisfying the optimization formula. However, the maximum value of e₁∕e₂ is less than 5 because of the 1∕3 track curve to be maintained during rotary forging. For the optimization of specific part shape parts, several sets of eccentricities are obtained under the requirements of optimization formulas and trajectory curves, The optimal set of eccentricity is determined by studying the influence of several sets of eccentricity on force and energy parameters and deformation uniformity of rotary forging parts.