Differential Equations, Input Functions, Complex Exponentials, and Transfer Functions

  • Karl A. SeelerEmail author


Differential system equations describe the dynamic relationship between an input driving the system, and one of the power variables within the energetic system. We simplify, or linearize, the individual energetic element equations, in order to derive a system equation which is an ordinary differential equation with constant coefficients, a form which we can solve for the output or response function. The method of undetermined coefficients superposes or sums the response of a system into the natural or homogeneous response of the system to a disturbance to its energetic equilibrium, and the steady-state or particular response to each input driving the system. Systems with two or more independent energy storage elements yield differential system equations which may describe oscillations or vibrations. Complex numbers, complex exponentials, and Euler’s equations simplify the solution and interpretation of the response of oscillatory systems. The Laplace transformation transforms differential equations into algebraic equations, which can be expressed as multiplicative dynamic operators called transfer functions. The chapter’s appendix introduces Mathcad and MATLAB to plot the solutions or response functions.


Laplace Transformation Step Response Force Function Heaviside Step Function Differential System Equation 
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References and Suggested Reading

  1. Hildebrand FB (1976) Advanced calculus for applications, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  2. Ogata K (2003) System dynamics, 4th edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  3. Ogata K (2009) Modern control engineering, 5th edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  4. Rowell D, Wormley DN (1997) System dynamics: an introduction. Prentice-Hall, Upper Saddle RiverGoogle Scholar
  5. Shearer JL, Murphy AT, Richardson HH (1971) Introduction to system dynamics. Addison-Wesley, ReadingGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentLafayette CollegeEastonUSA

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