Harmonic Vibration Examples
Welcome back to the structural dynamics course. We have been learning about the vibrations of single degree of freedom systems under various scenarios. In the first week we learned about free vibrations, undamped and viscously damped free vibrations. In the previous week we learnt coulomb damped free vibration, we also learned about harmonic vibrations, that is when the system vibrates under a harmonic force. We learned about resonance and the influence of damping on resonant responses. We also learned about transmissibility and vibration isolation.(Refer Slide Time: 00:50)So, before learning something new this week, we would briefly revise what we have learnt in the previous weeks.(Refer Slide Time: 01:07)In the first week we discussed free vibrations. This is the equation of motion of a single degree of freedom system under free vibration, m is the mass of the system, c is damping coefficient, k is the stiffness. And since it is free vibration, there is no external force acting on this system so, the right hand side is 0. So, this is the undamped free vibrations response it is undamped when c becomes 0.So, the displacement response is like this, x naught that is the initial displacement multiplied by cos omega n t plus x dot naught that is the initial velocity divided by omega n multiplied by sin omega n t. And omega n we have seen that it is equal into root of k by m and it is known as the natural frequency of this system. And this is how the response would look like and since there is no damping in the system, this displacement will not decay in time this amplitude will not change at each cycle the amplitude will be same.(Refer Slide Time: 02:27)Then we have damped free vibrations. In damped free vibrations damping is non-zero and this equation can be rewritten as this. If you divide this by mass you will get this equation where zeta is defined as the ratio of the damping coefficient and the critical damping coefficient. This is the response of a damped system under free vibrations. So, we have an exponentially decaying term here it is a function of zeta.}So, depending upon the damping, there is a decay in the vibration response; so, you can see that in the plot. So, the amplitude of the vibration decays with time and here in this equation this x naught is the initial displacement and omega D is natural frequency multiplied by square root of 1 minus zeta square, where zeta is the damping ratio.So, if the damping is high, this value will be lower than the natural frequency. So, here we have the initial velocity and the second term is depending upon the damping and the initial displacement. So, this is how the vibration of a damped system looks like and it decays with time and after some time the vibration stops.(Refer Slide Time: 04:07)Then we discussed one type of forced vibration called harmonic vibration. It so, in forced vibration, a force a time varying force will be acting on the system and a harmonic force is something which can be written like this that is as a function of sin or cosine. So, this is how a harmonic force will look like and p naught is the amplitude of this force and omega is the forcing frequency. So, this period will be equal to 2 pi by this forcing frequency and this is called forcing period.= p0 sin
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