Skip to main content
SpringerLink
Log in

Dynamic modeling of the front structure of an excavator

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper presents a new mathematical model of 4 degrees of freedom of links to qualitatively describe the dynamic behavior of the front structure of an excavator. In the model, the effects of couple of forces as new additional effects are involved. The exact forms and solutions for position-varying moments of inertia used in this model are presented. A topologic structure is used for the kinematic analysis of the body frame. The numerical results show that the new additional effects can change the angular kinetic energy of all links to a significant degree when the upper structure swings. The results suggest that the new additional effects should be taken into account for analysis of excavator dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Abbreviations

\(\theta _0 \) :

The angular displacement of the upper structure and is measured from the straight-ahead position by counterclockwise direction

\(\theta _1 \) :

The angular displacement of the center of mass of the boom and is measured from the vertical direction

\(\theta _2 \) :

The angular displacement of the center of mass of the stick and is measured from the vertical direction

\(\theta _3 \) :

The angular displacement of the center of mass of the bucket and is measured from the vertical direction

\(m_1 \) :

The mass of the boom

\(m_2 \) :

The mass of the stick

\(m_3 \) :

The mass of the bucket

\(\rho ( {x,y,z} )\) :

The mass density at the point of (x, y, z)

\(J_{C1X} \) :

The moment of inertia of the rigid body boom about the X axis through the center of mass

\(J_{C1Y} \) :

The moment of inertia of the rigid body boom about the Y axis through the center of mass

\(J_{C1Z} \) :

The moment of inertia of the rigid body boom about the Z axis through the center of mass

\(J_{C2X} \) :

The moment of inertia of the rigid body stick about the X axis through the center of mass

\(J_{C2Y} \) :

The moment of inertia of the rigid body stick about the Y axis through the center of mass

\(J_{C2Z} \) :

The moment of inertia of the rigid body stick about the Z axis through the center of mass

\(J_{C3X} \) :

The moment of inertia of the rigid body bucket about the X axis through the center of mass

\(J_{C3Y} \) :

The moment of inertia of the rigid body bucket about the Y axis through the center of mass

\(J_{C3Z} \) :

The moment of inertia of the rigid body bucket about the Z axis through the center of mass

\(\omega _{C1X} \) :

The angular velocity of the rigid body boom relative to the X axis through the center of mass

\(\omega _{C1Y} \) :

The angular velocity of the rigid body boom relative to the Y axis through the center of mass

\(\omega _{C1Z} \) :

The angular velocity of the rigid body boom relative to the Z axis through the center of mass

\(\omega _{C2X} \) :

The angular velocity of the rigid body stick relative to the X axis through the center of mass

\(\omega _{C2Y} \) :

The angular velocity of the rigid body stick relative to the Y axis through the center of mass

\(\omega _{C2Z} \) :

The angular velocity of the rigid body stick relative to the Z axis through the center of mass

\(\omega _{C3X} \) :

The angular velocity of the rigid body bucket relative to the X axis through the center of mass

\(\omega _{C3Y} \) :

The angular velocity of the rigid body bucket relative to the Y axis through the center of mass

\(\omega _{C3Z} \) :

The angular velocity of the rigid body bucket relative to the Z axis through the center of mass

\(T_\mathrm{boom} \) :

The kinetic energy of the boom

\(T_\mathrm{stick} \) :

The kinetic energy of the stick

\(T_\mathrm{bucket} \) :

The kinetic energy of the bucket

\(V_\mathrm{boom} \) :

The potential energy of the boom

\(V_\mathrm{stick} \) :

The potential energy of the stick

\(V_\mathrm{bucket} \) :

The potential energy of the bucket

\(F_1 \) :

The piston force of the boom cylinder

\(F_2 \) :

The piston force of the stick cylinder

\(F_3 \) :

The piston force of the bucket cylinder

\(F_N \) :

The force of the connecting rod between Q and N

\(F_P \) :

The force of the connecting rod between Q and P

\(\tau _0 \) :

Torque acting on the front structure from the upper structure

\(\tau _1 \) :

Torque acting on the boom from the piston force \(F_{1}\) of the boom cylinder and the piston force \(F_{2}\) of the stick cylinder

\(\tau _2 \) :

Torque acting on the stick from the piston force \(F_{2}\) of the stick cylinder, the piston force \(F_{3}\) of the stick cylinder and the force \(F_{N}\) of the connecting rod between Q and N

\(\tau _3 \) :

Torque acting on the bucket from the force \(F_{P}\) of the connecting rod between Q and P

g:

the acceleration due to gravity toward the surface of the earth

References

  1. Koivo, A.J.: Kinematics of excavators (backhoes) for transferring surface material. J. Aerosp. Eng. 7(1), 17–32 (1994)

    Article  Google Scholar 

  2. Foulon, G., Fourquet, J., Renaud, M.: Coordinating mobility and manipulation using nonholonomic mobile manipulators. Control Eng. Pract. 7(3), 391–399 (1999)

    Article  Google Scholar 

  3. Frimpong, S., Li, Y.: Virtual prototype simulation of hydraulic shovel kinematics for spatial characterization in surface mining operations. Int. J. Surf. Min. Reclam. Environ. 19(4), 238–250 (2005)

    Article  Google Scholar 

  4. Frimpong, S., Hu, Y., Inyang, H.: Dynamic modeling of hydraulic shovel excavators for geomaterials. Int. J. Geomech. 8(1), 20–29 (2008)

    Article  Google Scholar 

  5. Zweiri, Y.H., Seneviratne, L.D., Althoefer, K.: Model-based automation for heavy duty mobile excavator. In: IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 3, pp. 2967–2972. IEEE (2002)

  6. Zhong, G., et al.: Optimal control of the dynamic stability for robotic vehicles in rough terrain. Nonlinear Dyn. 73(1–2), 981–992 (2013)

    Article  MathSciNet  Google Scholar 

  7. Beji, L., Pascal, M.: The kinematics and the full minimal dynamic model of a 6-DOF parallel robot manipulator. Nonlinear Dyn. 18(4), 339–356 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Broersen, P.M.: Automatic spectral analysis with time series models. IEEE Trans. Instrum. Meas. 51(2), 211–216 (2002)

    Article  MathSciNet  Google Scholar 

  9. Šalinić, S., Bošković, G., Nikolić, M.: Dynamic modelling of hydraulic excavator motion using Kane’s equations. Autom. Constr. 44, 56–62 (2014)

    Article  Google Scholar 

  10. Lesser, M.: A geometrical interpretation of Kane’s equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 436(1896), 69–87 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  11. Towarek, Z.: Dynamics of a single-bucket excavator on a deformable soil foundation during the digging of ground. Int. J. Mech. Sci. 45(6), 1053–1076 (2003)

    Article  Google Scholar 

  12. Gottvald, J.: The calculation and measurement of the natural frequencies of the bucket wheel excavator SchRs 1320/4x30. Transport 5(3), 269–277 (2010)

    Article  Google Scholar 

  13. Rusiński, E., Harnatkiewicz, P., Kowalczyk, M., Moczko, P.: Examination of the causes of a bucket wheel fracture in a bucket wheel excavator. Eng. Fail. Anal. 17(6), 1300–1312 (2010)

    Article  Google Scholar 

  14. Savković, M., Gašić, M., Petrović, D., Zdravković, N., Pljakić, R.: Analysis of the drive shaft fracture of the bucket wheel excavator. Eng. Fail. Anal. 20, 105–117 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

We are grateful to the reviewers and the editors for their constructive suggestions. This research is supported by the Research Project of State Key Laboratory of Mechanical System and Vibration (Granted No. MSVZD201401), and sponsored by Qing Lan Project and Science and Technology Innovation Fun of Nanjing Institute of Technology (Granted No. CKJB201408).

Author information

Authors and Affiliations

  1. School of Automobile and Rail Transport, Nanjing Institute of Technology, Nanjing, 211167, People’s Republic of China

    Yuanguo Cao

  2. State Key Laboratory of Mechanical System and Vibration, Institute of Modern Design, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Youbai Xie

Authors
  1. Yuanguo Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youbai Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuanguo Cao.

Appendix

Appendix

The constraint equations for the angles between the force vectors and the lever arm vector are given by,

$$\begin{aligned}&\angle ADB=\frac{\pi }{2}+\theta _1 -\varphi _1 -\angle DBC_1 -\alpha _A \end{aligned}$$
(42)
$$\begin{aligned}&\frac{\sin \varphi _1 }{l_{BC1} }=\frac{\sin \angle ADB}{l_{AB} } \end{aligned}$$
(43)
$$\begin{aligned}&\varphi _E =\pi -\varphi _2 -\angle BEK \end{aligned}$$
(44)
$$\begin{aligned}&\angle EHK=\theta _1 \!-\!\theta _2 -\varphi _2 \!+\!\alpha _1 \!+\!\angle HKC_2 +\angle BKE \end{aligned}$$
(45)
$$\begin{aligned}&\frac{\sin \varphi _2 }{l_{HK} }=\frac{\sin \angle EHK}{l_{EK} } \end{aligned}$$
(46)
$$\begin{aligned}&\angle PRN=\pi +\theta _3 -\theta _2 -\alpha _{PRC3} -\alpha _2 -\alpha _{KRN} \end{aligned}$$
(47)
$$\begin{aligned}&l_{PN} =\sqrt{l_{PR}^2 +l_{NR}^2 -2l_{PR} l_{NR} \cos \angle PRN} \end{aligned}$$
(48)
$$\begin{aligned}&\frac{\sin \angle PNR}{l_{PR} }=\frac{\sin \angle PRN}{l_{PN} } \end{aligned}$$
(49)
$$\begin{aligned}&\angle PNQ=\arccos \left( {\frac{{l_{NQ}}^{2}+l_{PN}^2 -{l_{PQ}}^{2}}{2l_{NQ} l_{PN} }} \right) \end{aligned}$$
(50)
$$\begin{aligned}&\angle NPQ=\arccos \left( {\frac{{l_{PQ}}^{2}+l_{PN}^2 -{l_{NQ}}^{2}}{2l_{PQ} l_{PN} }} \right) \end{aligned}$$
(51)
$$\begin{aligned}&\beta _3 =\alpha _{RNS} -\angle PNR-\angle PNQ \end{aligned}$$
(52)
$$\begin{aligned}&\angle NQS=\pi -\phi _3 -\beta _3 \end{aligned}$$
(53)
$$\begin{aligned}&\frac{\sin \varphi _3 }{l_{NQ} }=\frac{\sin \angle NQS}{l_{NS} } \end{aligned}$$
(54)
$$\begin{aligned}&\varphi _S =\pi -\varphi _3 -\angle KSN \end{aligned}$$
(55)
$$\begin{aligned}&\phi _N =\beta _3 +\alpha _{KNS} \end{aligned}$$
(56)
$$\begin{aligned}&\frac{\sin \angle NPR}{l_{NR} }=\frac{\sin \angle PNR}{l_{PR} } \end{aligned}$$
(57)
$$\begin{aligned}&\varphi _{QPR} =\angle NPQ+\angle NPR \end{aligned}$$
(58)
$$\begin{aligned}&\varphi _Q =\angle NPQ+\angle PNQ-\angle NQS \end{aligned}$$
(59)
$$\begin{aligned}&\angle NQP=\pi -\angle NPQ-\angle PNQ \end{aligned}$$
(60)
$$\begin{aligned}&D\left( \theta \right) \ddot{\theta }=\left[ {{\begin{array}{llll} {D_{11} }&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {D_{22} }&{}\quad {D_{23} }&{}\quad {D_{24} } \\ 0&{}\quad {D_{32} }&{}\quad {D_{33} }&{}\quad {D_{34} } \\ 0&{}\quad {D_{42} }&{}\quad {D_{43} }&{}\quad {D_{44} } \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\ddot{\theta }_0 } \\ {\ddot{\theta }_1 } \\ {\ddot{\theta }_2 } \\ {\ddot{\theta }_3 } \\ \end{array} }} \right] \end{aligned}$$
(61)
$$\begin{aligned} D_{11}= & {} m_1 \left( {l_{BC1} \sin \theta _1 +l_{\hbox {0}B} } \right) ^{2}\nonumber \\&+\,J_{C1Y} +m_2 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. \nonumber \\&\left. +l_{KC2} \sin \theta _2 +l_{\hbox {0}B} \right) ^{2} \nonumber \\&+\,J_{C2Y} +m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. +\,l_{KR} \nonumber \\&\left. \sin \left( {\theta _2 +\alpha _2 } \right) +l_{RC3} \sin \left( {\theta _3 } \right) +l_{\hbox {0}B} \right) ^{2}+J_{C3Y}\nonumber \\ \end{aligned}$$
(62)
$$\begin{aligned} D_{22}= & {} m_1 l_{BC1}^2 +J_{C1Z} +m_2 l_{BK}^2 +m_3 l_{BK}^2 \end{aligned}$$
(63)
$$\begin{aligned} D_{23}= & {} m_2 l_{BK} l_{KC2} \cos \left( {\theta _1 +\alpha _1 -\theta _2 } \right) \nonumber \\&+\,m_3 l_{BK} l_{KR} \cos \left( {\theta _1 +\alpha _1 -\theta _2 -\alpha _2 } \right) \end{aligned}$$
(64)
$$\begin{aligned} D_{24}= & {} m_3 l_{BK} l_{RC3} \cos \left( {\theta _1 +\alpha _1 -\theta _3 } \right) \end{aligned}$$
(65)
$$\begin{aligned} D_{32}= & {} m_2 l_{BK} l_{KC2} \cos \left( {\theta _1 +\alpha _1 -\theta _2 } \right) \nonumber \\&+\,m_3 l_{BK} l_{KR} \cos \left( {\theta _1 +\alpha _1 -\theta _2 -\alpha _2 } \right) \end{aligned}$$
(66)
$$\begin{aligned} D_{33}= & {} m_2 l_{KC2}^2 +J_{C2Z} +m_3 l_{KR}^2 \end{aligned}$$
(67)
$$\begin{aligned} D_{34}= & {} m_3 l_{KR} l_{RC3} \cos \left( {\theta _2 +\alpha _2 -\theta _3 } \right) \end{aligned}$$
(68)
$$\begin{aligned} D_{42}= & {} m_3 l_{BK} l_{RC3} \cos \left( {\theta _1 +\alpha _1 -\theta _3 } \right) \end{aligned}$$
(69)
$$\begin{aligned} D_{43}= & {} m_3 l_{KR} l_{RC3} \cos \left( {\theta _2 +\alpha _2 -\theta _3 } \right) \end{aligned}$$
(70)
$$\begin{aligned} D_{44}= & {} m_3 l_{RC3}^2 +J_{C3Z} \end{aligned}$$
(71)
$$\begin{aligned} C\left( {\theta ,\dot{\theta }} \right) \dot{\theta }= & {} \left[ {{\begin{array}{llll} 0&{} {C_{12} }&{}\quad {C_{13} }&{}\quad {C_{14} } \\ {C_{21} }&{}\quad 0&{}\quad {C_{23} }&{}\quad {C_{24} } \\ {C_{31} }&{}\quad {C_{32} }&{}\quad 0&{}\quad {C_{34} } \\ {C_{41} }&{}\quad {C_{42} }&{}\quad {C_{43} }&{} 0 \\ \end{array} }} \right] \left[ {{\begin{array}{l} {\dot{\theta }_0 } \\ {\dot{\theta }_1 } \\ {\dot{\theta }_2 } \\ {\dot{\theta }_3 } \\ \end{array} }} \right] \end{aligned}$$
(72)
$$\begin{aligned} C_{12}= & {} \left( 2m_1 \left( {l_{BC1} \sin \theta _1 +l_{\hbox {0}B} } \right) l_{BC1} \cos \left( {\theta _1 } \right) +\dot{J}_{C1Y} \right. \nonumber \\&+\,2m_2 \left( {l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KC2} \sin \theta _2 +l_{\hbox {0}B} } \right) \nonumber \\&l_{BK} \cos \left( {\theta _1 +\alpha _1 } \right) +2m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. \nonumber \\&\left. +\,l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) +l_{RC3} \sin \theta _3 +l_{\hbox {0}B} \right) \nonumber \\&\left. \cdot \, l_{BK} \cos \left( {\theta _1 +\alpha _1 } \right) \right) \dot{\theta }_0 \end{aligned}$$
(73)
$$\begin{aligned} C_{13}= & {} \left( 2m_2 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KC2} \sin \theta _2 +l_{\hbox {0}B} \right) \right. \nonumber \\&l_{KC2} \cos \left( {\theta _2 } \right) +\dot{J}_{C2Y} +2m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. \nonumber \\&\left. +\,l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) +l_{RC3} \sin \theta _3 +l_{\hbox {0}B} \right) \nonumber \\&\left. \cdot \, l_{KR} \cos \left( {\theta _2 +\alpha _2 } \right) \right) \dot{\theta }_0 \end{aligned}$$
(74)
$$\begin{aligned} C_{14}= & {} \left( 2m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) \right. \right. \nonumber \\&\left. \left. +\,l_{RC3} \sin \theta _3 +l_{\hbox {0}B} \right) \cdot l_{RC3} \cos \left( {\theta _3 } \right) +\dot{J}_{C3Y} \right) \dot{\theta }_0\nonumber \\ \end{aligned}$$
(75)
$$\begin{aligned} C_{21}= & {} -\left( m_1 \left( {l_{BC1} \sin \theta _1 +l_{\hbox {0}B} } \right) l_{BC1} \cos \theta _1 +\frac{1}{2}\dot{J}_{C1Y}\right. \nonumber \\&+m_2 \left( {l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KC2} \sin \left( {\theta _2 } \right) +l_{\hbox {0}B} } \right) \nonumber \\&l_{BK} \cos \left( {\theta _1 +\alpha _1 } \right) +m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. \nonumber \\&\left. +l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) +l_{RC3} \sin \left( {\theta _3 } \right) +l_{\hbox {0}B} \right) \nonumber \\&\left. \cdot \, l_{BK} \cos \left( {\theta _1 +\alpha _1 } \right) \right) \dot{\theta }_0 \end{aligned}$$
(76)
$$\begin{aligned} C_{23}= & {} \left( m_2 l_{BK} l_{KC2} \sin \left( {\theta _1 +\alpha _1 -\theta _2 } \right) +m_3 l_{BK} l_{KR}\right. \nonumber \\&\left. \sin \left( {\theta _1 +\alpha _1 -\theta _2 -\alpha _2 } \right) \right) \dot{\theta }_2 \end{aligned}$$
(77)
$$\begin{aligned} C_{24}= & {} \left( {m_3 l_{BK} l_{RC3} \sin \left( {\theta _1 +\alpha _1 -\theta _3 } \right) } \right) \dot{\theta }_3 \end{aligned}$$
(78)
$$\begin{aligned} C_{31}= & {} -\left( m_2 \left( {l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KC2} \sin \theta _2 +l_{\hbox {0}B} } \right) \right. \nonumber \\&l_{KC2} \cos \theta _2 +\frac{1}{2}\dot{J}_{C2Y} +m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) \right. \nonumber \\&\left. +l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) +l_{RC3} \sin \left( {\theta _3 } \right) +l_{\hbox {0}B} \right) \nonumber \\&\left. \cdot \, l_{KR} \cos \left( {\theta _2 +\alpha _2 } \right) \right) \dot{\theta }_0 \end{aligned}$$
(79)
$$\begin{aligned} C_{32}= & {} -\left( m_2 l_{BK} l_{KC2} \sin \left( {\theta _1 +\alpha _1 -\theta _2 } \right) \right. \nonumber \\&\left. +m_3 l_{BK} l_{KR} \sin \left( {\theta _1 +\alpha _1 -\theta _2 -\alpha _2 } \right) \right) \dot{\theta }_1 \end{aligned}$$
(80)
$$\begin{aligned} C_{34}= & {} \left( {m_3 l_{KR} l_{RC3} \sin \left( {\theta _2 +\alpha _2 -\theta _3 } \right) } \right) \dot{\theta }_3 \end{aligned}$$
(81)
$$\begin{aligned} C_{41}= & {} -\left( m_3 \left( l_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) +l_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) \right. \right. \nonumber \\&\left. \left. +l_{RC3} \sin \theta _3 +l_{\hbox {0}B} \right) \cdot l_{RC3} \cos \theta _3 +\frac{1}{2}\dot{J}_{C3Y}\right) \dot{\theta }_0\nonumber \\ \end{aligned}$$
(82)
$$\begin{aligned} C_{42}= & {} -\left( {m_3 l_{BK} l_{RC3} \sin \left( {\theta _1 +\alpha _1 -\theta _3 } \right) } \right) \dot{\theta }_1 \end{aligned}$$
(83)
$$\begin{aligned} C_{43}= & {} -\left( {m_3 l_{KR} l_{RC3} \sin \left( {\theta _2 +\alpha _2 -\theta _3 } \right) } \right) \dot{\theta }_2 \end{aligned}$$
(84)
$$\begin{aligned} G\left( \theta \right)= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 0&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{}\quad {-m_1 gl_{BC1} \sin \left( {\theta _1 } \right) }&{}\quad {-m_2 gl_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) }&{}\quad {-m_3 gl_{BK} \sin \left( {\theta _1 +\alpha _1 } \right) } \\ 0&{}\quad 0&{}\quad {-m_2 gl_{KC2} \sin \theta _2 }&{}\quad {-m_3 gl_{KR} \sin \left( {\theta _2 +\alpha _2 } \right) } \\ 0&{}\quad 0&{}\quad 0&{}\quad {-m_3 gl_{RC3} \sin \theta _3 } \\ \end{array} }} \right] \end{aligned}$$
(85)
$$\begin{aligned} \Gamma \left( \theta \right)= & {} \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} 1&{}\quad 0&{}\quad 0&{}\quad 0 \\ 0&{} {-l_{AB} \sin \varphi _1 }&{}\quad {-l_{BE} \sin \varphi _E }&{}\quad 0 \\ 0&{}\quad 0&{}\quad {l_{EK} \sin \varphi _2 }&{}\quad {-l_{KS} \sin \theta _{KS} -l_{KN} \frac{\sin \varphi _Q \sin \left( {\alpha _{KNS} +\beta _3 } \right) }{\sin \left( {\varphi _3 +\beta _3 -\varphi _Q } \right) }} \\ 0&{}\quad 0&{}\quad 0&{}\quad {l_{PR} \frac{\sin \left( {\pi -\varphi _3 -\beta _3 } \right) \sin (\theta _{QPR} )}{\sin \left( {\varphi _3 +\beta _3 -\varphi _Q } \right) }} \\ \end{array} }} \right] \end{aligned}$$
(86)
$$\begin{aligned} F= & {} \left[ {{\begin{array}{l} {\tau _0 } \\ {F_1 } \\ {F_2 } \\ {F_3 } \\ \end{array} }} \right] \end{aligned}$$
(87)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, Y., Xie, Y. Dynamic modeling of the front structure of an excavator. Nonlinear Dyn 91, 233–247 (2018). https://doi.org/10.1007/s11071-017-3865-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3865-7

Keywords

Search

Navigation