Skip to main content
SpringerLink
Log in

A comparison of spherical joint models in the dynamic analysis of rigid mechanical systems: ideal, dry, hydrodynamic and bushing approaches

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

Spherical joints, usually known as ball and socket joints, are utilized in several engineering applications. On the one hand, these types of joints may be found in mechanical systems acting as a pivot element between the wheels and the suspension in cars. On the other hand, ball and socket joints can also be used to model human articulations, as in the case of the hip and shoulder. From more simplistic to more complex scenarios, spherical joints might be modeled using different approaches. Therefore, the objective of this work is to provide a comparative analysis of different spherical joint models and to examine their influence on the dynamic response of mechanical multibody systems. For this purpose, ideal or kinematic formulation, dry, lubricated, and bushing approaches are revised. Additionally, the formulation of the dynamic equations of motion for constrained mechanical multibody systems is succinctly described. Afterward, the kinematic and dynamic characteristics of the considered spherical joint models are comprehensively described. In this regard, normal, tangential, hydrodynamic lubrication and bushing forces experienced by the multibody systems in such cases of spherical joints are examined. The application of the spherical joint models in the dynamic modeling and simulation of multibody systems is investigated. Considering two multibody models as demonstrative examples of application from the outcomes, it is observed that the influence of the spherical joint modeling strategy on the dynamic simulation of mechanical multibody systems strongly depends on the nature of the multibody model analyzed, both in terms of dynamic response and computational efficiency.

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
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25

Similar content being viewed by others

Abbreviations

Symbol :

Description (SI Units)

\(\mathbf{A}_{k}\) :

Transformation matrix of body \(k\)

\(b\) :

Bushing element stiffness proportional to damping parameter

\(c\) :

Radial clearance (m)

\(c_{\mathrm{r}}\) :

Coefficient of restitution

\(\mathbf{D}\) :

Jacobian matrix of the constraint equations

\(\mathbf{e}\) :

Eccentricity vector (m)

\(E_{k}\) :

Young’s modulus of body \(k\) (Pa)

\(e\) :

Magnitude of the eccentricity vector (m)

\(e\) h0, \(e\) h1 :

Tolerances for the eccentricity (m)

\(\dot{e}\) :

Time rate of the eccentricity in the radial direction (m/s)

\(\mathbf{f}_{\mathrm{b}}\) :

Bushing force (N)

\(f_{\mathrm{c}}\) :

Magnitude of the Coulomb friction (N)

\(f_{\mathrm{d}}\) :

Damping force of the shock absorber linked to the A-arm (N)

\(\mathbf{f}_{\mathrm{e}}\) :

External tangential force vector (N)

\(f_{\mathrm{h}}\) :

Hybrid force for lubricated joint (N)

\(f_{\mathrm{k}}\) :

Elastic force of the shock absorber linked to the A-arm (N)

\(f_{\mathrm{l}}\) :

Lubrication force (N)

\(f_{\mathrm{n}}\) :

Normal contact force (N)

\(f_{\mathrm{s}}\) :

Magnitude of the static friction (N)

\(\mathbf{f}_{\mathrm{t}}\) :

Tangential or friction force (N)

\(\mathbf{g}\) :

External generalized force vector (N, N m)

\(K\) :

Generalized contact stiffness (N/m1.5)

\(k\) :

Bushing element stiffness (N/m)

\(\mathbf{M}\) :

Mass matrix (kg, kg m2)

\(\mathbf{n}\) :

Normal unit vector

\(n\) :

Hertzian nonlinear exponent

\(O_{k}\) :

Center of mass of body \(k\)

\(P_{k}\) :

Point representing the center of the ball or socket on body \(k\)

\(p_{k}\) :

Euler parameters, \(k\) = 0, 1, 2, 3

\(Q_{k}\) :

Contact point on body \(k\)

\(\mathbf{r}_{k}\) :

Position vector of the center of mass of body \(k\) described in global coordinates (m)

\(\mathbf{r}_{k}^{P}\) :

Global position vector of point \(P\) located on body \(k\) (m)

\(\mathbf{r}_{k}^{Q}\) :

Position vector of the contact point \(Q_{k}\) expressed in global coordinates (m)

\(R_{k}\) :

Radius of element \(k\) (m)

\(\mathbf{s}_{k}^{P}\) :

Global position vector of point \(P\) located on body \(k\) with respect to local coordinates (m)

\(\mathbf{t}\) :

Tangential direction of the relative velocity associated with the contacting surfaces

\(t\) :

Time variable (s)

\(v_{0}\), \(v_{1}\) :

Tolerances for the velocity (m/s)

\(\dot{\mathbf{v}}\) :

Vector containing the system accelerations (m/s2, rad/s2)

\(\mathbf{v}_{\mathrm{n}}\) :

Relative normal velocity of the contact point (m/s)

\(v_{\mathrm{S}}\) :

Stribeck velocity (m/s)

\(\mathbf{v}_{\mathrm{t}}\) :

Relative tangential velocity of the contact point (m/s)

\(v_{\mathrm{t}}\) :

Magnitude of the relative tangential velocity (m/s)

\(x\) :

Distance between the two ends of the shock absorber (m)

\(x_{\mathrm{B}}\) :

Horizontal position of the bump profile (m)

xyz :

Global coordinate system (m)

\(\dot{x}\) :

Deformation velocity of the shock absorber (m/s)

\(\dot{\mathbf{y}}\) :

Auxiliary vector containing the system accelerations and velocities (m/s2, m/s)

\(\mathbf{y}\) :

Auxiliary vector containing the system velocities and positions (m/s, m)

\(z_{\mathrm{B}}\) :

Vertical position of the bump profile (m)

Symbol :

Description (SI Units)

\(\boldsymbol{\Phi}\) :

Position constraint equations

\(\dot{\boldsymbol{\Phi}} \) :

Velocity constraint equations

\(\ddot{\boldsymbol{\Phi}} \) :

Acceleration constraint equations

\(\alpha\) :

Baumgarte stabilization coefficient

\(\beta\) :

Baumgarte stabilization coefficient

\(\delta\) :

Pseudo-penetration or deformation (m)

\(\dot{\delta} \) :

Pseudo-penetration velocity (m/s)

\(\dot{\delta}^{( - )}\) :

Initial contact velocity (m/s)

\(\varepsilon\) :

Eccentricity ratio

\(\dot{\varepsilon} \) :

Time rate of the eccentricity ratio

\(\boldsymbol{\upgamma}\) :

Right-hand side vector of the acceleration equations

\(\kappa\) :

Parameter representing the negative slope of the sliding state (s/m)

\(\boldsymbol{\lambda}\) :

Lagrange multipliers vector

\(\mu_{\mathrm{k}}\) :

Kinetic coefficient of friction

\(\mu_{\mathrm{s}}\) :

Static coefficient of friction

\(\nu\) :

Lubricant dynamic viscosity (Pa s)

\(\sigma_{k}\) :

Material property of body \(k\) (Pa−1)

\(\upsilon_{k}\) :

Poisson’s ratio of body \(k\)

\(\boldsymbol{\upomega }\) :

Angular velocity vector (rad/s)

\(\xi\eta\zeta\) :

Body fixed coordinate system (m)

Symbol :

Description

\(i\) :

Relative to body \(i\)

\(j\) :

Relative to body \(j\)

\(k\) :

Relative to body \(k\)

n:

Normal direction

t:

Tangential direction

Symbol :

Description

\(P\) :

Generic point \(P\)

\(Q\) :

Generic contact point \(Q\)

\(s\) :

Spherical joint

Symbol :

Description

( )T :

Matrix or vector transpose

(´):

Components of a vector in a body-fixed coordinate system

(\(^{\boldsymbol{\cdot}}\)):

First derivative with respect to time

(⋅⋅):

Second derivative with respect to time

(∼):

Skew-symmetric matrix or vector

References

  1. Nikravesh, P.: Computer-Aided Analysis of Mechanical Systems. Prentice Hall, New York (1988)

    Google Scholar 

  2. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Dynamics of multibody systems with spherical clearance joints. J. Comput. Nonlinear Dyn. 1, 240–247 (2006). https://doi.org/10.1115/1.2198877

    Article  MATH  Google Scholar 

  3. Banerjee, A., Chanda, A., Das, R.: Historical origin and recent development on normal directional impact models for rigid body contact simulation: a critical review. Arch. Comput. Methods Eng. 24, 397–422 (2017). https://doi.org/10.1007/s11831-016-9164-5

    Article  MathSciNet  MATH  Google Scholar 

  4. Pereira, C.M., Ramalho, A.L., Ambrósio, J.A.: A critical overview of internal and external cylinder contact force models. Nonlinear Dyn. 63, 681–697 (2011). https://doi.org/10.1007/s11071-010-9830-3

    Article  Google Scholar 

  5. Rodrigues da Silva, M., Marques, F., Tavares da Silva, M., Flores, P.: A compendium of contact force models inspired by Hunt and Crossley’s cornerstone work. Mech. Mach. Theory 167, 104501 (2022). https://doi.org/10.1016/j.mechmachtheory.2021.104501

    Article  Google Scholar 

  6. Wang, G., Wang, L., Yuan, Y.: Investigation on dynamics performance of multibody system with rough surface. Appl. Math. Model. 104, 358–372 (2022). https://doi.org/10.1016/j.apm.2021.12.012

    Article  MathSciNet  Google Scholar 

  7. Askari, E.: Mathematical models for characterizing non-Hertzian contacts. Appl. Math. Model. 90, 432–447 (2021). https://doi.org/10.1016/j.apm.2020.08.048

    Article  MathSciNet  MATH  Google Scholar 

  8. Gilardi, G., Sharf, I.: Literature survey of contact dynamics modelling. Mech. Mach. Theory 37, 1213–1239 (2002). https://doi.org/10.1016/S0094-114X(02)00045-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Skrinjar, L., Slavič, J., Boltežar, M.: A review of continuous contact-force models in multibody dynamics. Int. J. Mech. Sci. 145, 171–187 (2018). https://doi.org/10.1016/j.ijmecsci.2018.07.010

    Article  Google Scholar 

  10. Machado, M., Moreira, P., Flores, P., Lankarani, H.M.: Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012). https://doi.org/10.1016/j.mechmachtheory.2012.02.010

    Article  Google Scholar 

  11. Alves, J., Peixinho, N., da Silva, M.T., Flores, P., Lankarani, H.M.: A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mech. Mach. Theory 85, 172–188 (2015). https://doi.org/10.1016/j.mechmachtheory.2014.11.020

    Article  Google Scholar 

  12. Corral, E., Moreno, R.G., García, M.J.G., Castejón, C.: Nonlinear phenomena of contact in multibody systems dynamics: a review. Nonlinear Dyn. (2021). https://doi.org/10.1007/s11071-021-06344-z

    Article  Google Scholar 

  13. Ma, J., Dong, S., Chen, G., Peng, P., Qian, L.: A data-driven normal contact force model based on artificial neural network for complex contacting surfaces. Mech. Syst. Signal Process. 156, 107612 (2021). https://doi.org/10.1016/j.ymssp.2021.107612

    Article  Google Scholar 

  14. Ma, J., Chen, G., Ji, L., Qian, L., Dong, S.: A general methodology to establish the contact force model for complex contacting surfaces. Mech. Syst. Signal Process. 140, 106678 (2020). https://doi.org/10.1016/j.ymssp.2020.106678

    Article  Google Scholar 

  15. Zhang, J., Huang, C., Zhao, L., Di, J., He, G., Li, W.: Continuous contact force model with an arbitrary damping term exponent: model and discussion. Mech. Syst. Signal Process. 159, 107808 (2021). https://doi.org/10.1016/j.ymssp.2021.107808

    Article  Google Scholar 

  16. Xiang, W., Yan, S., Wu, J., Gao, R.X.: Complexity evaluation of nonlinear dynamic behavior of mechanisms with clearance joints by using the fractal method. Proc. Inst. Mech. Eng., Part C, J. Mech. Eng. Sci. 228, 3482–3495 (2014). https://doi.org/10.1177/0954406214531251

    Article  Google Scholar 

  17. Li, X., Ding, X., Chirikjian, G.S.: Analysis of angular-error uncertainty in planar multiple-loop structures with joint clearances. Mech. Mach. Theory 91, 69–85 (2015). https://doi.org/10.1016/j.mechmachtheory.2015.04.005

    Article  Google Scholar 

  18. Marques, F., Flores, P., Pimenta Claro, J.C., Lankarani, H.M.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn. 86, 1407–1443 (2016). https://doi.org/10.1007/s11071-016-2999-3

    Article  MathSciNet  Google Scholar 

  19. Marques, F., Flores, P., Claro, J.C.P., Lankarani, H.M.: Modeling and analysis of friction including rolling effects in multibody dynamics: a review. Multibody Syst. Dyn. 45, 223–244 (2019). https://doi.org/10.1007/s11044-018-09640-6

    Article  MathSciNet  Google Scholar 

  20. Pennestrì, E., Rossi, V., Salvini, P., Valentini, P.P.: Review and comparison of dry friction force models. Nonlinear Dyn. 83, 1785–1801 (2016). https://doi.org/10.1007/s11071-015-2485-3

    Article  MATH  Google Scholar 

  21. Olsson, H., Åström, K.J., Canudas de Wit, C., Gäfvert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control 4, 176–195 (1998). https://doi.org/10.1016/S0947-3580(98)70113-X

    Article  MATH  Google Scholar 

  22. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M., Koshy, C.S.: A study on dynamics of mechanical systems including joints with clearance and lubrication. Mech. Mach. Theory 41, 247–261 (2006). https://doi.org/10.1016/j.mechmachtheory.2005.10.002

    Article  MATH  Google Scholar 

  23. Tian, Q., Liu, C., Machado, M., Flores, P.: A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems. Nonlinear Dyn. 64, 25–47 (2011). https://doi.org/10.1007/s11071-010-9843-y

    Article  MATH  Google Scholar 

  24. Flores, P., Lankarani, H.M.: Spatial rigid-multibody systems with lubricated spherical clearance joints: modeling and simulation. Nonlinear Dyn. 60, 99–114 (2010). https://doi.org/10.1007/s11071-009-9583-z

    Article  MATH  Google Scholar 

  25. Li, Y., Wang, C., Huang, W.: Dynamics analysis of planar rigid-flexible coupling deployable solar array system with multiple revolute clearance joints. Mech. Syst. Signal Process. 117, 188–209 (2019). https://doi.org/10.1016/j.ymssp.2018.07.037

    Article  Google Scholar 

  26. Li, Y., Yang, Y., Li, M., Liu, Y., Huang, Y.: Dynamics analysis and wear prediction of rigid-flexible coupling deployable solar array system with clearance joints considering solid lubrication. Mech. Syst. Signal Process. 162, 108059 (2022). https://doi.org/10.1016/j.ymssp.2021.108059

    Article  Google Scholar 

  27. Xiang, W., Yan, S., Wu, J., Niu, W.: Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method. Mech. Syst. Signal Process. 138, 106596 (2020). https://doi.org/10.1016/j.ymssp.2019.106596

    Article  Google Scholar 

  28. Liu, C., Zhang, K., Yang, L.: Normal force-displacement relationship of spherical joints with clearances. J. Comput. Nonlinear Dyn. 1, 160–167 (2005). https://doi.org/10.1115/1.2162872

    Article  Google Scholar 

  29. Wang, G., Liu, H., Deng, P.: Dynamics analysis of spatial multibody system with spherical joint wear. J. Tribol. 137, 021605 (2015). https://doi.org/10.1115/1.4029277

    Article  Google Scholar 

  30. Marques, F., Isaac, F., Dourado, N., Souto, A.P., Flores, P., Lankarani, H.M.: A study on the dynamics of spatial mechanisms with frictional spherical clearance joints. J. Comput. Nonlinear Dyn. 12, 051013 (2017). https://doi.org/10.1115/1.4036480

    Article  Google Scholar 

  31. Chen, X., Gao, W., Deng, Y., Wang, Q.: Chaotic characteristic analysis of spatial parallel mechanism with clearance in spherical joint. Nonlinear Dyn. 94, 2625–2642 (2018). https://doi.org/10.1007/s11071-018-4513-6

    Article  Google Scholar 

  32. Tian, Q., Flores, P., Lankarani, H.M.: A comprehensive survey of the analytical, numerical and experimental methodologies for dynamics of multibody mechanical systems with clearance or imperfect joints. Mech. Mach. Theory 122, 1–57 (2018). https://doi.org/10.1016/j.mechmachtheory.2017.12.002

    Article  Google Scholar 

  33. Flores, P., Ambrósio, P., Pimenta Claro, J.C., Lankarani, H.M.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints - Models and Case Studies. Springer, Berlin (2008)

    MATH  Google Scholar 

  34. Askari, E., Flores, P.: Coupling multi-body dynamics and fluid dynamics to model lubricated spherical joints. Arch. Appl. Mech. 90, 2091–2111 (2020). https://doi.org/10.1007/s00419-020-01711-5

    Article  Google Scholar 

  35. Zhao, B., Zhou, K., Xie, Y.-B.: A new numerical method for planar multibody system with mixed lubricated revolute joint. Int. J. Mech. Sci. 113, 105–119 (2016). https://doi.org/10.1016/j.ijmecsci.2016.04.016

    Article  Google Scholar 

  36. Daniel, G.B., Machado, T.H., Cavalca, K.L.: Investigation on the influence of the cavitation boundaries on the dynamic behavior of planar mechanical systems with hydrodynamic bearings. Mech. Mach. Theory 99, 19–36 (2016). https://doi.org/10.1016/j.mechmachtheory.2015.11.019

    Article  Google Scholar 

  37. Oliveira, M.V.M., Cunha, B.Z., Daniel, G.B.: A model-based technique to identify lubrication condition of hydrodynamic bearings using the rotor vibrational response. Tribol. Int. 160, 107038 (2021). https://doi.org/10.1016/j.triboint.2021.107038

    Article  Google Scholar 

  38. Todres, Z.V.: Organic Mechanochemistry and Its Practical Applications. CRC Press, Boca Raton (2006)

    Book  Google Scholar 

  39. Tian, Q., Zhang, Y., Chen, L., Flores, P.: Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Comput. Struct. 87, 913–929 (2009). https://doi.org/10.1016/j.compstruc.2009.03.006

    Article  Google Scholar 

  40. Askari, E., Flores, P., Dabirrahmani, D., Appleyard, R.: Dynamic modeling and analysis of wear in spatial hard-on-hard couple hip replacements using multibody systems methodologies. Nonlinear Dyn. 82, 1039–1058 (2015). https://doi.org/10.1007/s11071-015-2216-9

    Article  MathSciNet  Google Scholar 

  41. Ledesma, R., Ma, Z.-D., Hulbert, G., Wineman, A.: A nonlinear viscoelastic bushing element in multibody dynamics. Comput. Mech. 17, 287–296 (1996). https://doi.org/10.1007/BF00368551

    Article  MATH  Google Scholar 

  42. Ambrósio, J., Verissimo, P.: Improved bushing models for general multibody systems and vehicle dynamics. Multibody Syst. Dyn. 22, 341–365 (2009). https://doi.org/10.1007/s11044-009-9161-7

    Article  MATH  Google Scholar 

  43. Ambrósio, J., Pombo, J.: A unified formulation for mechanical joints with and without clearances/bushings and/or stops in the framework of multibody systems. Multibody Syst. Dyn. 42, 317–345 (2018). https://doi.org/10.1007/s11044-018-9613-z

    Article  MathSciNet  MATH  Google Scholar 

  44. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, New York (2005)

    Book  MATH  Google Scholar 

  45. Flores, P.: Concepts and Formulations for Spatial Multibody Dynamics. Springer, Berlin (2015)

    Book  MATH  Google Scholar 

  46. Marques, F., Roupa, I., Silva, M.T., Flores, P., Lankarani, H.M.: Examination and comparison of different methods to model closed loop kinematic chains using Lagrangian formulation with cut joint, clearance joint constraint and elastic joint approaches. Mech. Mach. Theory 160, 104294 (2021). https://doi.org/10.1016/j.mechmachtheory.2021.104294

    Article  Google Scholar 

  47. Flores, P., Lankarani, H.M.: Contact Force Models for Multibody Dynamics. Springer, Cham (2016)

    Book  MATH  Google Scholar 

  48. Lidström, P.: On the relative rotation of rigid parts and the visco-elastic torsion bushing element. Math. Mech. Solids 18, 788–802 (2012). https://doi.org/10.1177/1081286512450867

    Article  MathSciNet  MATH  Google Scholar 

  49. Hertz, H.: Über die Berührung fester elastischer Körper. J. Reine Angew. Math. 92, 156–171 (1881)

    MATH  Google Scholar 

  50. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42, 440–445 (1975). https://doi.org/10.1115/1.3423596

    Article  Google Scholar 

  51. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112, 369–376 (1990). https://doi.org/10.1115/1.2912617

    Article  Google Scholar 

  52. Gonthier, Y., McPhee, J., Lange, C., Piedbœuf, J.-C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11, 209–233 (2004). https://doi.org/10.1023/B:MUBO.0000029392.21648.bc

    Article  MATH  Google Scholar 

  53. Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst. Dyn. 25, 357–375 (2011). https://doi.org/10.1007/s11044-010-9237-4

    Article  MATH  Google Scholar 

  54. Threlfall, D.C.: The inclusion of Coulomb friction in mechanisms programs with particular reference to DRAM au programme DRAM. Mech. Mach. Theory 13, 475–483 (1978). https://doi.org/10.1016/0094-114X(78)90020-4

    Article  Google Scholar 

  55. Bengisu, M.T., Akay, A.: Stability of friction-induced vibrations in multi-degree-of-freedom systems. J. Sound Vib. 171, 557–570 (1994). https://doi.org/10.1006/jsvi.1994.1140

    Article  MATH  Google Scholar 

  56. Ambrósio, J.C.: Impact of rigid and flexible multibody systems: deformation description and contact model. In: Schiehlen, W., Valášek, M. (eds.) Virtual Nonlinear Multibody Systems. NATO ASI Series (Series II: Mathematics, Physics and Chemistry), pp. 57–81. Springer, Dordrecht (2003)

    Chapter  MATH  Google Scholar 

  57. Brown, P., McPhee, J.: A continuous velocity-based friction model for dynamics and control with physically meaningful parameters. J. Comput. Nonlinear Dyn. 11, 054502 (2016). https://doi.org/10.1115/1.4033658

    Article  Google Scholar 

  58. Frêne, J., Nicolas, D., Degneurce, B., Berthe, D., Godet, M.: Hydrodynamic Lubrication - Bearings and Thrust Bearings. Elsevier, Amsterdam (1997)

    MATH  Google Scholar 

  59. Pinkus, O., Sternlicht, S.A.: Theory of Hydrodynamic Lubrication. McGraw Hill, New York (1961)

    MATH  Google Scholar 

  60. Flores, P., Ambrósio, J., Claro, J.P.: Dynamic analysis for planar multibody mechanical systems with lubricated joints. Multibody Syst. Dyn. 12, 47–74 (2004). https://doi.org/10.1023/B:MUBO.0000042901.74498.3a

    Article  MATH  Google Scholar 

  61. Frik, S., Leister, G., Schwartz, W.: Simulation of the IAVSD road vehicle benchmark bombardier iltis with FASIM, MEDYNA, NEWEUL and SIMPACK. Veh. Syst. Dyn. 22, 215–253 (1993). https://doi.org/10.1080/00423119308969496

    Article  Google Scholar 

  62. Dormand, J.R., Prince, P.J.: A family of embedded Runge-Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980). https://doi.org/10.1016/0771-050X(80)90013-3

    Article  MathSciNet  MATH  Google Scholar 

  63. Shampine, L.F., Reichelt, M.W.: The MATLAB ODE Suite. SIAM J. Sci. Comput. 18, 1–22 (1997). https://doi.org/10.1137/S1064827594276424

    Article  MathSciNet  MATH  Google Scholar 

  64. Shampine, L.F., Reichelt, M.W., Kierzenka, J.A.: Solving Index-1 DAEs in MATLAB and Simulink. SIAM Rev. 41, 538–552 (1999). https://doi.org/10.1137/S003614459933425X

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been supported by Portuguese Foundation for Science and Technology, under the national support to R&D units grant, with the reference project UIDB/04436/2020 and UIDP/04436/2020, as well as through IDMEC, under LAETA, project UIDB/50022/2020. The first author expresses her gratitude to the Portuguese Foundation for Science and Technology through the PhD grant (2021.04840.BD).

Author information

Authors and Affiliations

  1. CMEMS-UMinho, Departamento de Engenharia Mecânica, Universidade do Minho, Campus de Azurém, Guimarães, 4804-533, Portugal

    Mariana Rodrigues da Silva, Filipe Marques & Paulo Flores

  2. LABBELS–Associate Laboratory, Braga, Guimarães, Portugal

    Mariana Rodrigues da Silva, Filipe Marques & Paulo Flores

  3. IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1, Lisboa, 1049-001, Portugal

    Miguel Tavares da Silva

Authors
  1. Mariana Rodrigues da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Filipe Marques
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Miguel Tavares da Silva
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paulo Flores
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Mariana Rodrigues da Silva: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Resources; Software; Validation; Visualization; Writing – original draft; Writing – review & editing.

Filipe Marques: Conceptualization; Data curation; Formal analysis; Investigation; Methodology; Project administration; Software; Supervision; Validation; Visualization; Writing – review & editing.

Miguel Tavares da Silva: Conceptualization; Data curation; Investigation; Methodology; Supervision; Validation; Writing – review & editing.

Paulo Flores: Conceptualization; Data curation; Investigation; Methodology; Project administration; Software; Supervision; Validation; Writing – review & editing.

Corresponding author

Correspondence to Mariana Rodrigues da Silva.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rodrigues da Silva, M., Marques, F., Tavares da Silva, M. et al. A comparison of spherical joint models in the dynamic analysis of rigid mechanical systems: ideal, dry, hydrodynamic and bushing approaches. Multibody Syst Dyn 56, 221–266 (2022). https://doi.org/10.1007/s11044-022-09843-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-022-09843-y

Keywords

Search

Navigation