Time Response of First- and Second-order Dynamical Systems

  • Jorge Angeles
Part of the Mechanical Engineering Series book series (MES)


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.


Impulse Response Magnification Factor Bode Plot Unit Impulse Harmonic Response 
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  1. 1.
    Moler CB, Van Loan C (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev 20(4):801–836MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chicurel-Uziel E (2007) Dirac delta representation by parametric equations. Applications to impulsive vibration systems. J Sound Vibr 305:134–150MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cannon RH (1967) Dynamics of physical systems. McGraw-Hill Book Co., New YorkGoogle Scholar
  4. 4.
    Kahaner D, Moler C, Nash S (1989) Numerical methods and software. Prentice Hall, Inc., Englewood Cliffs, NJMATHGoogle Scholar
  5. 5.
    Moler C (2004) Numerical computing with MATLAB, Electronic edition. The MathWorks, Inc., NantickMATHCrossRefGoogle Scholar
  6. 6.
    Strang G (1986) Introduction to applied mathematics. Wessley-Cambridge Press, Wessley, pp 274–276MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringMcGill UniversityMontrealCanada

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