Abstract
The need of dynamics models in engineering, objectives, scope, and limitations of mechanical analysis in general and of elements of systems (masses, springs, dashpots, energy sources and energy sinks); system; mechanical system; viscous, hysteretic and Coulomb friction.
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Notes
- 1.
http://varan_bhaath.tripod.com/Pages/Saptapadi.htm.
- 2.
Sometimes, dynamical systems are described by difference equations (case of discrete-time systems), integral equations and even by integro-differential equations.
- 3.
These systems “forget” the history of their excitation.
- 4.
Engineering systems are invariablymultibody systems, i.e., systems of rigid and deformable bodies. When a rigid body is constrained to move in one single direction, it can be modeled as a particle, but modern software for mechanical analysis regards all elements as bodies, their constraints being included explicitly in the form of algebraic relations in the mathematical model.
- 5.
a.k.a parallel-axis theorem.
- 6.
For brevity, translational springs and their corresponding stiffness are referred to as “springs” and “stiffness” when no confusion is possible.
- 7.
FBD is the abbreviation of free-body diagram.
- 8.
Henceforth, vectors are represented with lower-case boldfaces, matrices with upper-case boldfaces, scalars with math italics. Thus, while ω is a vector, ω is a scalar.
- 9.
The scalar product of two vectors a and b, of the same dimension, also termed thedot product, is represented by the two alternative expressions a ⋅b and a T b.
- 10.
See a textbook on kinematics of mechanisms.
- 11.
The formal derivation of the Lagrange equations lying beyond the scope of the book, the reader is invited to review this derivation, as pertaining to systems of particles in a mechanics book, e.g., [3].
- 12.
Erroneously, translational springs are sometimes referred to as “linear.”
- 13.
Again, translational dashpots are sometimes erroneously referred to as “linear.”
- 14.
Actually, friction force between pin and plate is present, but it develops no work because the point of application Q of this force is stationary. The sole role of the friction force here is to prevent sliding.
- 15.
The expressions (1.49a–c) are said to be “first-order” because all variations δq and \(\delta \dot{q}\) appear linearly therein.
- 16.
One equilibrium configuration can be obtained from the other by looking at the latter with the aid of a mirror. Mirror-imaging, of course, shouldn’t affect the intrinsic properties of the system.
- 17.
For ease of visualization, δθ and δψ are exaggerated in the figure, but they are both assumed to be “small”.
References
Almen J, László A (1936) The uniform-section disk spring. Trans ASME 58:305–314
Di Benedetto A and Pennestrì E (1993) Introduzione alla cinematica dei meccanismi. Moti infinitesimi. Casa Editrice Ambrosiana, Milan
Meriam JL, Kraige LG (1992) Engineering mechanics. Dynamics, vol 2, 5th edn. Wiley, New York
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Angeles, J. (2011). The Modeling of Single-dof Mechanical Systems. In: Dynamic Response of Linear Mechanical Systems. Mechanical Engineering Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1027-1_1
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DOI: https://doi.org/10.1007/978-1-4419-1027-1_1
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