18.03 Lecture Notes - Lecture 14: Dashpot, Linear Combination
Document Summary
Complex gain: recap, phase lag, driving via the dashpot, complex gain. [1] the story so far: we"re studying solutions of linear constant coefficient equations a_n x^(n) + + a_1 x + a_0 = q(t) (*) A key is the characteristic polynomial p(s) = a_n s^n + + a_1 s + a_0. All homogeneous solutions of (*)_h decay to zero provided that all the roots of p(s) have negative real parts. In this case the solutions to (*)_h are called "transients," By superposition, all solutions to (*) converge together as t gets large, and we say that the equation is "stable. " If we have a system modeled by a stable equation, and we are only interested in what it looks like after the transients have died down, we can eliminate the initial condition: --------------->| system |-------------------------> signal |____________| output signal x_p. So we look for a particular solution x_p . Experiments indicate that sinusoidal in gives sinusoidal out.
