Abstract
How physical systems, e.g., aircraft, respond to a given input, such as a gust wind, under given initial conditions, like cruising altitude and cruising speed, is known as thetime response of the system. Here, we have two major items that come into the picture when determining the time response, namely, the mathematical model of the system and the history of the input. The mathematical model, as studied in Chap. 1, is given as an ODE in the generalized coordinate of the system. Moreover, this equation is usually nonlinear, and hence, rather cumbersome to handle with the purpose of predicting how the system will respond under given initial conditions and a given input. However, if we first find the equilibrium states of the system, e.g., the altitude, the aircraft angle of attack, and cruising speed in our example above, and then linearize the model about this equilibrium state, then we can readily obtain the information sought, as described here.
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.
In a course focusing on vibrations, this section can be skipped. However, its reading is strongly recommended because it helps gain insight into the response of second-order damped systems, studied in Sect. 2.3.2
- 2.
A typical average lifting speed is about 1 storey ( ≈ 3 m) per second.
- 3.
The exponential of a matrix is formally identical to that of a real argument: both have the same expansion.
- 4.
For this simple model, the wheels are assumed to be massless, rigid disks.
- 5.
Aircraft landing gears, like the suspension of terrestrial vehicles are designed with a natural frequency of around 1 Hz.
- 6.
Roads are typically designed with slopes below 6%, although exceptionally roads with slopes of 10% and even 20% exist. The validity of the approximation sinx ≈ x is limited by π ∕ 30 or 6 ∘ , while a slope of 6% entails an angle α = 3. 4 ∘ .
- 7.
Abbreviation of left-hand side.
- 8.
Taken, with some modifications, from [3].
- 9.
Note that an abrupt change in the velocity implies an impulse in the acceleration.
- 10.
See Fig. ??.
- 11.
The integral operation is both additive and homogeneous, and hence, linear.
- 12.
http://www.mathworks.com/moler
- 13.
Reproduced with authorization from http://commons.wikimedia.org/wiki/File:Helicopter_silhouette_AS-355.svg
- 14.
This exercise is drawn from a similar one in [3].
- 15.
The saturation function sat(x) was introduced in Eq. 1.37 and plotted in Fig. 1.26b.
- 16.
This exercise is drawn from a similar one in [3].
- 17.
This exercise is drawn from a similar one in [3].
References
Moler CB, Van Loan C (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev 20(4):801–836
Chicurel-Uziel E (2007) Dirac delta representation by parametric equations. Applications to impulsive vibration systems. J Sound Vibr 305:134–150
Cannon RH (1967) Dynamics of physical systems. McGraw-Hill Book Co., New York
Kahaner D, Moler C, Nash S (1989) Numerical methods and software. Prentice Hall, Inc., Englewood Cliffs, NJ
Moler C (2004) Numerical computing with MATLAB, Electronic edition. The MathWorks, Inc., Nantick
Strang G (1986) Introduction to applied mathematics. Wessley-Cambridge Press, Wessley, pp 274–276
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). Time Response of First- and Second-order Dynamical Systems. In: Dynamic Response of Linear Mechanical Systems. Mechanical Engineering Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1027-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4419-1027-1_2
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-1026-4
Online ISBN: 978-1-4419-1027-1
eBook Packages: EngineeringEngineering (R0)