Ordinary differential Equation In vibration theory there is a problem to define position and velocity of a rod which loaded with distributed bending force q(x). If to assume that this force acts by sine law the vibration equation will be written as follows: Where v(t,x) – function of vertical position of the rod points, E – elasticity modulus, J – second area moment, m – mass per unit length, ω – circular frequency of acting force.Note that J and E assumed to constant.If to substitute now v(t,x)=y(x)·sin(ω·t) one can obtain an ordinary differential equation which describes a form of oscillation:Solve this equation assuming:q=10 kgf/mm;Titanium alloy E=100 GPa;Second area moment J=1440 mm4. Which is equal to rectangle with sides 10 mm and 12 mm, and force acts along longer side;Circular freq. ω=0.1 1/s;Length l=100 mm;Rigid constraint on left bound y(0)=0, y’(0)=0;Bending force on the left bound ;Bending moment on the left bound ;
