i read in my book that resonance takes place when the frequency of the applied force is equal to the natural frequency of the spring-damper-mass system at a phase angle of 90 between the displacement and the force.
now if i understand correctly a phase angle of 90 means that when the force is applied to the mass, the displacement won't start untill the force f=Fsin(90) , butonce it has reached this maximum value the force value again starts decreasing as it depends on the sine function. so the displacement shouldn't occur at all. if that's the case then how will resonance ever occur?
the book also says for applied frequencies much greater than the natural frequency the phase angle will be 180. now if u start applying force to a system the force will be max when f=Fsin90, but there wont be any displacement since the phase angle between the force applied and displacement is 180, but we continue applying the force and now it takes the value of f=Fsin180 which is 0. then how is it possible that the displacement starts now? when the applied force itself is 0? i'm really confused please help me out. i have similar doubts when the applied frequency is much less than the natural frequency too.
now if i understand correctly a phase angle of 90 means that when the force is applied to the mass, the displacement won't start untill the force f=Fsin(90) , butonce it has reached this maximum value the force value again starts decreasing as it depends on the sine function. so the displacement shouldn't occur at all. if that's the case then how will resonance ever occur?
the book also says for applied frequencies much greater than the natural frequency the phase angle will be 180. now if u start applying force to a system the force will be max when f=Fsin90, but there wont be any displacement since the phase angle between the force applied and displacement is 180, but we continue applying the force and now it takes the value of f=Fsin180 which is 0. then how is it possible that the displacement starts now? when the applied force itself is 0? i'm really confused please help me out. i have similar doubts when the applied frequency is much less than the natural frequency too.
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Hi- so i will try to answer this question by appealing to a common example
take a set of keys on a string, or anything else that substitues as a 'pendulum'
you know that the resonant frequency of the pendulum will be ~sqrt(g/l), and will only be a couple hertz
swing the pendulum very very slowly. do you see how it follows you? when your hand is all the way to the right of its motion, so is the pendulum. this is the case where the driving frequency is much less than the resonant frequency, and the phase difference between the drive and the pendulum is 0. if your hand is a cos, the pendulum's position is also a cos.
now swing the pendulum very quickly. you should see that the mass is exactly opposite your hand at all times- when your hand is at the rightmost of its motion, the mass is at the leftmost of ITS motion. thus, at frequencies much higher than the resonant frequency the response is exactly out of phase (180 degrees) with the drive. if your hand is a cos, the ball's position is a negative cos.
when you are ON resonance, the ball just lags behind your hand- the ball is 90 degrees. if your hand is a cos, the pendulum is a sin.
i think your confusion comes from the fact that this behavior only occurs in steady state. obviously as you start moving your hand the pendulum will not immediately jump to this phase matching behavior, but after a brief period of time you will see this behavior emerge.
take a set of keys on a string, or anything else that substitues as a 'pendulum'
you know that the resonant frequency of the pendulum will be ~sqrt(g/l), and will only be a couple hertz
swing the pendulum very very slowly. do you see how it follows you? when your hand is all the way to the right of its motion, so is the pendulum. this is the case where the driving frequency is much less than the resonant frequency, and the phase difference between the drive and the pendulum is 0. if your hand is a cos, the pendulum's position is also a cos.
now swing the pendulum very quickly. you should see that the mass is exactly opposite your hand at all times- when your hand is at the rightmost of its motion, the mass is at the leftmost of ITS motion. thus, at frequencies much higher than the resonant frequency the response is exactly out of phase (180 degrees) with the drive. if your hand is a cos, the ball's position is a negative cos.
when you are ON resonance, the ball just lags behind your hand- the ball is 90 degrees. if your hand is a cos, the pendulum is a sin.
i think your confusion comes from the fact that this behavior only occurs in steady state. obviously as you start moving your hand the pendulum will not immediately jump to this phase matching behavior, but after a brief period of time you will see this behavior emerge.