Abstract
The principles introduced in Chap. 6 allow the determination of the time response of n-dof systems when n is either small enough or the system possesses symmetries that render its time response analysis handleable in closed form. More general n-dof systems call for a numerical procedure. This is done here upon extension of the techniques introduced in Chap. 3. The aim of this chapter is thus to derive simulation schemes for the total time response of n-dof systems, for an arbitrary integer n. The principles laid down in Chap. 3 will be applied.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
A n ×n positive-definite matrix has 2n square roots, of which one is positive-definite, one negative-definite, and all others sign-indefinite.
- 2.
This section is largely based on material that appeared in Al-Widyan et al. [1], © 2003, with permission from Begell House, Inc.
- 3.
The identity in Eq. 7.11 follows from Fact 4 of Appendix A.
- 4.
Properly speaking, the lower block of z k , as given by Eq. 7.22b, is different from \(\dot{\mathbf{x}}({t}_{k})\) because of the approximation involved when introducing the ZOH.
- 5.
Not to be confused with Z introduced in Exercise 6.8.
- 6.
The interested reader can have a glimpse of the theorem in Åström and Wittenmark [2].
- 7.
Not to be confused with matrix F of Eq. 7.15b!
- 8.
A (statically) unbalanced wheel can be visualized as a disc with its mass concentrated at a point C offset from its center O by a distance e < < r, for a radius r.
- 9.
The material in this section is based largely on Angeles et al. [4].
- 10.
Not to be confused with the state-variable vector introduced in Eq. 7.15a.
- 11.
In proportionally damped systems, the damping matrix is a linear combination of the mass and the stiffness matrices, which then leads to a simple eigenvalue problem.
- 12.
Leonardo da Vinci concluded that the ratio of the forearm length to that of the arm is 71.4%, not too far from the assumption adopted here.
References
Al-Widyan K, Angeles J, Ostrovskaya S (2003) A Robust simulation algorithm for conservative linear mechanical systems. Int J Multiscale Comput Eng 1(2):289–309
Åström K, Wittenmark B (1997) Computer-controlled systems: theory and design. Prentice-Hall Inc, Upper Saddle River
Angeles J, Espinosa I (1981) Suspension-system synthesis for mass transport vehicles with prescribed dynamic behavior. ASME Paper 81-DET-44, In: Proceeding 1981 ASME design engineering technical conference, Hartford, 20–23 September 1981
Angeles J, Etemadi Zanganeh K, Ostrovskaya S (1999) The analysis of arbitrarily-damped linear mechanical systems. Arch Appl Mech 69(8):529–541
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Angeles, J. (2011). Simulation of n-dof Systems. In: Dynamic Response of Linear Mechanical Systems. Mechanical Engineering Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1027-1_7
Download citation
DOI: https://doi.org/10.1007/978-1-4419-1027-1_7
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-1026-4
Online ISBN: 978-1-4419-1027-1
eBook Packages: EngineeringEngineering (R0)