Phys 325 Spring 2011 Lecture 15

Phys 325 Spring 2011 Lecture 15 - Physics 325 Lecture 15...

This preview shows pages 1–3. Sign up to view the full content.

Physics 325 Lecture 15 Impulse Response In the previous lecture, we investigated the response of an oscillator to a periodic driving force. Not all forces are periodic. One can have a force that is impulsive. An impulse is just a force that lasts for a short time and doesn’t repeat. The quintessential case is a force that looks like this: Recall from Physics 111/211 that an impulse causes a change in momentum p f p f t t Therefore, for an oscillator originally at rest, the response to an impulse like this is equivalent to a free oscillator with an initial velocity of 0 f t v m We will solve this problem for such an impulse. Once we’ve accomplished that, we can solve a more general problem of a non-periodic driving force because we can represent it as a sum of impulses and use the principle of superposition. Here we go. Consider an underdamped harmonic oscillator that at t 0 is at rest at 0 x . At t t , it is subjected to a sharp impulse: 0 t 0 t ( ) f t impulse ( ') F t t

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Immediately after the impulse, the oscillator has a velocity of ( ) ( ) ( ) t t t t t t F t F t t v a t dt dt m m     (15.1) For small t , ( lim 0 t   ), we have ( ) ( 0) 0 x t t x t  and the underdamped solution with the initial velocity given by Equation (15.1) is ( ) 1 ( ) ( ) sin ( ) t t d d F t t x t e t t m   (15.2) where d is defined in Equation (12.7). Now suppose that we have a series of impulses of duration t at time t i . Then the response is the superposition of solutions like (15.2): ) 1 ( ) 1 ( ) sin ( ) i N t t i d i i d F t t x t e t t m   (15.3) In the limit where t 0 , the sum in Equation (15.3) becomes an integral ') 0 sin ( ') 1 ( ) ( ') ' t t t d d e t t x t F t dt m   (15.4)
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern