Estimation of dynamic behaviors of hydraulic forging press machine in slow-motion manufacturing process

Abstract

In this paper, stick–slip phenomenon of hydraulic forging press machine (HFPM), a nonlinear manufacturing system subjected to weak rigidity and negative friction velocity gradient, was investigated in slow-motion process. Dynamic characteristics of non-smooth motion were investigated experimentally under various operating conditions in slow-motion process. Taking into account the quadratic and cubic nonlinearities, the governing equation of the HFPM was derived. The method of multiple scales was employed to obtain approximate solution for velocity oscillation during slow motion. The steady-state responses for each case (i.e., non-resonance, primary, super-harmonic and sub-harmonic oscillation) were examined to investigate the correlation between velocity oscillation and nonlinear vibration in slow motion. The effects of viscous damping coefficient, hydraulic stiffness, friction parameters, excitation amplitude and frequency on the amplitude–frequency response characteristics were studied, and the interaction between system dynamics and tribological effects were presented. The stability of operational conditions and the critical velocity of the forging process were analyzed through the bifurcation techniques. Experimental results conducted on the HFPM verify the effectiveness of the proposed method.

Appendix: Derivations regarding stability of response behaviors for primary resonance

Firstly, the steady-state solution for the primary resonance case is defined by $(a_0 ,\omega _0 )$; thus, the solution yields

$$\begin{aligned}&-\eta -\varphi \cos \gamma _1 -\xi \sin \gamma _1 =0, \end{aligned}$$

(A.1)

$$\begin{aligned}&a\omega _0 \sigma -\phi -\xi \cos \gamma _1 +\varphi \sin \gamma _1 =0. \end{aligned}$$

(A.2)

In order to examine the stability of the steady-state response behaviors for the case of primary resonance, the small perturbations, denoted by $\delta a$ and $\delta \gamma$, are incorporated $a = a_0 +\delta a$, $\gamma _1 =\gamma _0 +\delta \gamma$, in Eqs. (37) and (38). Therefore, the linearized equations are expressed as

$$\begin{aligned} \frac{\hbox {d}\left( {\delta a} \right) }{\hbox {d}t}= & {} -\left( \frac{\phi _1 }{\omega _0 }+3\lambda \omega _0 a_0 \rho _1 \omega _1 \cos \gamma _0 \right. \nonumber \\&\quad \left. +\,\frac{3}{\omega _0 }\alpha _1 a_0 \rho _1 \sin \gamma _0 \right) \delta a\nonumber \\&\quad +\,\left( {\frac{\varphi _0 }{\omega _0 }\sin \gamma _0 -\frac{\xi _0 }{\omega _0 }\cos \gamma _0 } \right) \delta \gamma , \end{aligned}$$

(A.3)

$$\begin{aligned}&\frac{\hbox {d}\left( {\delta \gamma } \right) }{\hbox {d}t}=\left( -\frac{3\alpha _1 a_0 }{4\omega _0 }{-}\left( {\frac{3\alpha _1 \rho _1 }{2\omega _0 }{-}\frac{3\alpha _1 \rho _1^3 }{a_0^2 \omega _0 }{-}\frac{6\alpha _1 \rho _1 \rho _2^2 }{a_0^2 \omega _0 }} \right) \right. \nonumber \\&\quad \cos \gamma _0 \left. +\,\left( {-\frac{\mu \omega _1 \rho _1 }{a_0^2 \omega _0 }+\frac{3}{2}\lambda \omega _0 \rho _1 \omega _1 -\frac{3\lambda \omega _1^3 \rho _1^3 }{a_0^2 \omega _0 }} \right. \right. \nonumber \\&\quad \left. \left. -\,\frac{6\lambda \omega _1 \rho _1 \omega _2^2 \rho _2^2 }{a_0^2 \omega _0 } \right) \sin \gamma _0 \right) \delta a\nonumber \\&\quad +\,\left( {\left( {\frac{3\alpha _1 a_0 \rho _1 }{2\omega _0 }+\frac{3\alpha _1 \rho _1^3 }{a_0 \omega _0 }+\frac{6\alpha _1 \rho _1 \rho _2^2 }{a_0 \omega _0 }} \right) } \right. \sin \gamma _0 \nonumber \\&\quad +\,\left( {\frac{\mu \omega _1 \rho _1 }{a_0 \omega _0 }+\frac{3}{2}\lambda \omega _0 a_0 \rho _1 \omega _1 } \right. \nonumber \\&\quad \left. {\left. {+\frac{3\lambda \omega _1^3 \rho _1^3 }{a_0 \omega _0 }+\frac{6\lambda \omega _1 \rho _1 \omega _2^2 \rho _2^2 }{a_0 \omega _0 }} \right) \cos \gamma _0 } \right) \delta \gamma , \end{aligned}$$

(A.4)

with the variables $\sin \gamma _0$ and $\cos \gamma _0$, obtained from Eqs. A.1 and A.2,

$$\begin{aligned} \sin \gamma _0= & {} \frac{\left( {\phi _0 -a_0 \omega _0 \sigma } \right) \varphi _0 -\eta _0 \xi _0 }{\xi _0^2 +\varphi _0^2 }, \end{aligned}$$

(A.5)

$$\begin{aligned} \cos \gamma _0= & {} \frac{-\left( {\phi _0 -a_0 \omega _0 \sigma } \right) \xi _0 -\eta _0 \varphi _0 }{\xi _0^2 +\varphi _0^2 }, \end{aligned}$$

(A.6)

where

$$\begin{aligned} \phi _0= & {} \frac{3}{8}\alpha _1 a_0^3 +3\alpha _1 a_0 \rho _1^2 +3\alpha _1 a_0 \rho _2^2 , \\ \phi _1= & {} \frac{1}{2}\mu \omega _0 +\frac{9}{8}\lambda a_0^2 \omega _0^2 +3\lambda \omega _0 \left( {\rho _1^2 \omega _1^2 +\rho _2^2 \omega _2^2 } \right) , \\ \varphi _0= & {} \mu \omega _1 \rho _1 {+}\frac{3}{2}\lambda \omega _0^2 a_0^2 \rho _1 \omega _1 {+}3\lambda \omega _1^3 \rho _1^3 {+}6\lambda \omega _1 \rho _1 \omega _2^2 \rho _2^2 , \\ \xi _0= & {} \frac{3}{2}\alpha _1 a_0^2 \rho _1 +3\alpha _1 \rho _1^3 +6\alpha _1 \rho _1 \rho _2^2 , \\ \eta _0= & {} \frac{1}{2}\mu a_0 \omega _0 +\frac{3}{8}\lambda a_0^3 \omega _0^3 +3\lambda a_0 \omega _0 \left( {\rho _1^2 \omega _1^2 +\rho _2^2 \omega _2^2 } \right) . \end{aligned}$$

And the corresponding eigenvalue equation is defined by

$$\begin{aligned} \varsigma ^{2}+p^{*}\varsigma +q^{*}=0, \end{aligned}$$

(A.7)

where

$$\begin{aligned} p^{*}= & {} -(A_1 +A_2 ), q^{*}=A_1 A_4 -A_2 A_3 , \end{aligned}$$

(A.8)

$$\begin{aligned} A_1= & {} -\left( \frac{\phi _0 }{\omega _0 }+3\lambda \omega _0 a_0 \rho _1 \omega _1 \cos \gamma _0 \right. \nonumber \\&\left. +\frac{3}{\omega _0 }\alpha _1 a_0 \rho _1 \sin \gamma _0 \right) , \end{aligned}$$

(A.9)

$$\begin{aligned} A_2= & {} \frac{\varphi _0 }{\omega _0 }\sin \gamma _0 -\frac{\xi _0 }{\omega _0 }\cos \gamma _0 , \end{aligned}$$

(A.10)

$$\begin{aligned} A_3= & {} -\frac{3\alpha _1 a_0 }{4\omega _0 }-\left( {\frac{3\alpha _1 \rho _1 }{2\omega _0 }-\frac{3\alpha _1 \rho _1^3 }{a_0^2 \omega _0 }-\frac{6\alpha _1 \rho _1 \rho _2^2 }{a_0^2 \omega _0 }} \right) \nonumber \\&\quad \times \cos \gamma _0 +\,\left( {-\frac{\mu \omega _1 \rho _1 }{a_0^2 \omega _0 }+\frac{3}{2}\lambda \omega _0 \rho _1 \omega _1 } \right. \nonumber \\&\quad \left. -\frac{3\lambda \omega _1^3 \rho _1^3 }{a_0^2 \omega _0 }- {\frac{6\lambda \omega _1 \rho _1 \omega _2^2 \rho _2^2 }{a_0^2 \omega _0 }} \right) \sin \gamma _0 , \end{aligned}$$

(A.11)

$$\begin{aligned} A_4= & {} \left( {\left( {\frac{3\alpha _1 a_0 \rho _1 }{2\omega _0 }+\frac{3\alpha _1 \rho _1^3 }{a_0 \omega _0 }+\frac{6\alpha _1 \rho _1 \rho _2^2 }{a_0 \omega _0 }} \right) } \right. \sin \gamma _0\nonumber \\&+\left( \frac{\mu \omega _1 \rho _1 }{a_0 \omega _0 }+\frac{3}{2}\lambda \omega _0 a_0 \rho _1 \omega _1 \right. \nonumber \\&+\frac{3\lambda \omega _1^3 \rho _1^3 }{a_0 \omega _0 }+ \left. {\frac{6\lambda \omega _1 \rho _1 \omega _2^2 \rho _2^2 }{a_0 \omega _0 }} \right) \cos \gamma _0 . \end{aligned}$$

(A.12)

The stability of steady-state solutions of primary resonance can be estimated based upon the sign of root of Eq. (A-7). Therefore, it can be deduced from Eq. (A-7) that the stability of the periodic solution for case I can be expressed by

$$\begin{aligned} \left\{ {{\begin{array}{l} {q^{*}<0\rightarrow \hbox {Perfectly unstable}} \\ {q^{*}>0\left\{ {{\begin{array}{l} {p^{*}>0\rightarrow \hbox {Asymptotically stable}\left\{ {{\begin{array}{l} {p^{{*}^{2}}-4q^{*}\ge 0\rightarrow \hbox {Stable node}} \\ {p^{{*}^{2}}-4q^{*}<0\rightarrow \hbox {Stable focus}} \\ \end{array} }} \right. } \\ {p^{*}<0\rightarrow \hbox {Perfectly unstable}\left\{ {{\begin{array}{l} {p^{{*}^{2}}-4q^{*}\ge 0\rightarrow \hbox {Unstable node}} \\ {p^{{*}^{2}}-4q^{*}<0\rightarrow \hbox {Unstable focus}} \\ \end{array} }} \right. } \\ \end{array} }} \right. } \\ \end{array} }} \right. . \end{aligned}$$

(A.13)

In order to investigate the stability boundaries of corresponding response curves of primary resonance, differentiating Eqs. A.1 and A.2 to detuning parameter $\sigma$ results in

$$\begin{aligned}&\left( {\phi _0 +3\lambda \omega _0^2 a_0 \rho _1 \omega _1 \cos \gamma _0 +3\alpha _1 a_0 \rho _1 \sin \gamma _0 } \right) \frac{\hbox {d}a}{\hbox {d}\sigma }\nonumber \\&\quad +\,\left( {\xi _0 \cos \gamma _0 -\varphi _0 \sin \gamma _0 } \right) \frac{\hbox {d}\gamma }{\hbox {d}\sigma }=0, \end{aligned}$$

(A.14)

$$\begin{aligned}&\left( \omega _0 \sigma -\left( {\frac{9\alpha _1 a_0^2 }{8}+3\alpha _1 \rho _1^2 +3\alpha _1 \rho _2^2 } \right) \right. \nonumber \\&\quad \left. -\,3\alpha _1 a_0 \rho _1 \cos \gamma _0 +3\lambda \omega _0^2 a_0 \rho _1 \omega _1 \sin \gamma _0 \right) \frac{\hbox {d}a}{\hbox {d}\sigma }\nonumber \\&\quad +\,\left( {\xi _0 \sin \gamma _0 +\varphi _0 \cos \gamma _0 } \right) \frac{\hbox {d}\gamma }{\hbox {d}\sigma }=0. \end{aligned}$$

(A.15)

By eliminating ${\hbox {d}\gamma }/{\hbox {d}\sigma }$, and substituting Eqs. A.5 and A.6 to A.14 and A.15, one derives

$$\begin{aligned}&\left[ \left( {\phi _0 +3\lambda \omega _0^2 a_0 \rho _1 \omega _1 \cos \gamma _0 +3\alpha _1 a_0 \rho _1 \sin \gamma _0 } \right) \eta _0 \right. \nonumber \\&\quad \left. +\,H_0 \left( {a_0 \omega _0 \sigma -\phi _0 } \right) \right] \left( {\frac{\hbox {d}a}{\hbox {d}\sigma }} \right) \nonumber \\&\quad =-a_0 \omega _0 \left( {a_0 \omega _0 \sigma -\phi _0 } \right) , \end{aligned}$$

(A.16)

where

$$\begin{aligned} \mathrm{H}_0= & {} \omega _0 \sigma -\left( {\frac{9\alpha _1 a_0^2 }{8}+3\alpha _1 \rho _1^2 +3\alpha _1 \rho _2^2 } \right) \\&-3\alpha _1 a_0 \rho _1 \cos \gamma _0 +3\lambda \omega _0^2 a_0 \rho _1 \omega _1 \sin \gamma _0 . \end{aligned}$$

Let

$$\begin{aligned} \varTheta= & {} \left( {\phi _0 +3\lambda \omega _0^2 a_0 \rho _1 \omega _1 \cos \gamma _0 +3\alpha _1 a_0 \rho _1 \sin \gamma _0 } \right) \eta _0 \nonumber \\&\quad +\,H_0 \left( {a_0 \omega _0 \sigma -\phi _0 } \right) . \end{aligned}$$

(A.17)

Based on Eqs. A.13 and A.16, one can conclude that the boundary of stability of response curves is

$$ \begin{aligned} \left\{ {{\begin{array}{l} {a_0 \omega _0 \sigma -\phi _0<0\left\{ {{\begin{array}{l} {\frac{\hbox {d}a}{\hbox {d}\sigma }<0\rightarrow \varTheta<0\left\{ {{\begin{array}{l} {q^{*}<0\left| {q^{*}>0,p^{*}<0} \right. \rightarrow \hbox {Untable}} \\ {q^{*}>0\, \& \,p^{*}>0\rightarrow \hbox {Stable}} \\ \end{array} }} \right. } \\ {\frac{\hbox {d}a}{\hbox {d}\sigma }>0\rightarrow \varTheta>0\left\{ {{\begin{array}{l} {q^{*}<0\left| {q^{*}>0,p^{*}<0} \right. \rightarrow \hbox {Untable}} \\ {q^{*}>0\, \& \,p^{*}>0\rightarrow \hbox {Stable}} \\ \end{array} }} \right. } \\ \end{array} }} \right. } \\ {a_0 \omega _0 \sigma -\phi _0>0\left\{ {{\begin{array}{l} {\frac{\hbox {d}a}{\hbox {d}\sigma }<0\rightarrow \varTheta>0\left\{ {{\begin{array}{l} {q^{*}<0\left| {q^{*}>0,p^{*}<0} \right. \rightarrow \hbox {Untable}} \\ {q^{*}>0\, \& \,p^{*}>0\rightarrow \hbox {Stable}} \\ \end{array} }} \right. } \\ {\frac{\hbox {d}a}{\hbox {d}\sigma }>0\rightarrow \varTheta<0\left\{ {{\begin{array}{l} {q^{*}<0\left| {q^{*}>0,p^{*}<0} \right. \rightarrow \hbox {Untable}} \\ {q^{*}>0\, \& \,p^{*}>0\rightarrow \hbox {Stable}} \\ \end{array} }} \right. } \\ \end{array} }} \right. } \\ \end{array} }} \right. .\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \end{aligned}$$

(A.18)