Conceptual and mathematical difficulties of the conventional model of damped oscillations

Abstract A mathematical study on the subject of damped oscillations is expounded, focused on the computation of the constants of the equation of movement, their mathematical meaning and their implications on the behavior of systems. Examples are given of the initial value method for computing the constants, both for free and damped oscillations. Algebraic expressions are deduced for the conventional damped model and they are comparatively discussed with those from Crawford's formalism. The meaning of the constants is discussed and answers are offered for questions frequently asked by students and professionals. The concept of secant exponential function is introduced, and the distinction between this function and the envelope of the equation of motion is established. Examples are given of systematic errors committed when constants are arbitrarily imposed, and their meaning and the mathematical reason is discussed. In the appendix, the general equation of damped oscillations is deduced and examples are given for all three possible cases of systems (overdamping, critical damping, and underdamped oscillations). The conventional model of damped oscillations is deduced as a particular case of the general equation.