Forced Vibration Essay

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1.0 OBJECTIVE To study free vibration and to calculate natural angular frequency Ɯn and Natural frequency fn of simple rod simple rod with spring and load using ‘Equivalent System Method’ 2.0 INTRODUCTION There is a two type of vibration that is free vibration and forced vibration. Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. If there is no external force applied, f(t) = 0, no damping, c = 0 ,the system will experience free vibration and the oscillatory motion will continue forever with a constant amplitude .The mechanical system will then vibrate at one or more of its natural frequency and damp down to zero. Figure 1 : Example of one degree freedom vibration Figure 2: Amplitude versus time From Newton’s second law of motion, F = mass,m x acceleration,a Balancing forces gives, F = ma + kx If the applied force must be zero and this is the requirement for it to be a free natural oscillations. 0 = Ma + kx , rearrange a = -(k/m)x The acceleration is directly proportional to displacement, and it is directed towards the rest position. If the x was plotted against time,a sinusoidal graph would result. The constants proportionally is k/m and this must be the angular frequency squared so wn=km. The frequency of oscillation is f = w/2π =1/2πkm The natural frequency are often denoted as wn and fn.This equation is true for all elastic oscillation \ EXPERIMENTAL-Related formula Equation of motion (EOM), Mo=Jo φ=-Fca Fc=cx=cφa Where, Mo is moment about pivot point o of the beam Fc is spring force,result of the deflection x and spring constant c For small angles, the deflection can be formed from the torsion φ and level arm a Jo=mL33 Where, Jo is mass moment of inertia of the beam about pivot EOM in the form of homogenous differential equation, φ+ 3ca2mL2

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