Ae104a_2010_handout_3

Ae104a_2010_handout_3 - 3.3 Definitions and Tools •...

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Unformatted text preview: 3.3 Definitions and Tools • Heaviside (step) function H (x) = 0 1 2 1 x>0 x=0 x>1 (1) The step function can be used to write finite integrals as integral over the whole axis: ∞ b [H (x − a) − H (x − b)] f (x)dx f (x)dx = a −∞ • Dirac delta function 13 δ (x) = 0 ∞ x=0 x=0 (2) such that ∞ ∞ δ (x)dx = 1, −∞ x δf (x)dx = f (0), −∞ δ (x) = lim →0 δ (ζ )dζ = H (x) −∞ 1 H x+ −H x− 2 2 • Ramp function 0 x R(x) = x<0 x≥0 x R(x) = xH (x) = H (ζ )dζ −∞ 14 (3) 3.6.4 Second order harmonic response Figure 1: Amplitude response of a second order system to harmonic forcing. Figure 2: Phase response of a second order system to harmonic forcing. 15 ...
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Ae104a_2010_handout_3 - 3.3 Definitions and Tools •...

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