Random Response of a Single Degree of Freedom Oscillator

Abstract

A single input single output (SISO) time invariant linear system is entirely characterized by its impulse response h(t), which represents the response of the system, initially at rest, to a unit Dirac delta function δ(t). Equivalently, the system is completely characterized by its transfer function, H(s), which is the Laplace transform of h(t):

EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaabm % aabaGaam4CaaGaayjkaiaawMcaaiabg2da9maapedabaGaamiAamaa % bmaabaGaamiDaaGaayjkaiaawMcaaiaadwgadaahaaWcbeqaaiabgk % HiTiaadohacaWG0baaaaqaaiaaicdaaeaacqGHEisPa0Gaey4kIipa % kiaadsgacaWG0baaaa!47E1! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ H\left( s \right) = \int_0^\infty {h\left( t \right){e^{ - st}}} dt $$

(5.1)

Usually, H(s) can be determined directly from the differential equation of the system.