AER 403 Lecture Notes - Lecture 9: Inverse Trigonometric Functions
3 Jan 2018
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Free vibration: free vibration is induced by impulse, the equation of motion is derived as follows: md(cid:884)x/dt(cid:884) = = mg k( +x) at static equilibrium: mg = k md2x/dt2 = -kx. Leading to the equation of motion for free vibration md2x/dt2 + kx = 0 d2x/dt2 + (k/m) x = 0 this is a homogenous 2nd. Ode, with a general solution x = asin nt + bcos nt. With initial condition x(0) and dx/dt (0), we have x(0) = b. X = [( (0)/ n)2+x(0)2]1/2 =arctan[x(0)/( (0)/ n)] x = (0)/ nsin nt (0)/ n = amplitude (cid:523) n = (k/m)1/2) y = 2sin(20t) y = 2 sin(20t + 5) The natural frequency fn (hz) is defined as: N = (cid:523)k/m(cid:524)(cid:883)/(cid:884) = (cid:884) fn mg = k k/m = g/ fn = (cid:523)g/ (cid:524)(cid:883)/(cid:884)/(cid:884) (cid:523)pendulum(cid:524) A spring-mass system consists of a mass of 1 kg and spring stiffness of. 26, | 14 (n/m)/kg = (kg m/s2 /m)/kg.
