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### ME3514_Impulse_Function_and_Impulse_Response_Part_1_2

Course: ME 3514, Fall 2011
School: Virginia Tech
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Word Count: 1239

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Transfer ME3514 Topics Function T f F ti Block Diagram Transient Response Analysis using TF with MATLAB p y g Unit Impulse (Delta) Function Impulse Delta f function Momentum Relationship between momentum and impulse Laplace transform of an impulse Examples of impact problems R. Burdisso 1 ME3514 Impulse What i Wh t is an impulse? i l ? f(t) I = Area I f (t ) dt t U ts:[ N .s ] Units:[ t What...

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Transfer ME3514 Topics Function T f F ti Block Diagram Transient Response Analysis using TF with MATLAB p y g Unit Impulse (Delta) Function Impulse Delta f function Momentum Relationship between momentum and impulse Laplace transform of an impulse Examples of impact problems R. Burdisso 1 ME3514 Impulse What i Wh t is an impulse? i l ? f(t) I = Area I f (t ) dt t U ts:[ N .s ] Units:[ t What is the delta function (x)? A mathematical function (t ) dt I (t )dt 1 t 2 R. Burdisso ME3514 Impulse The d lt f Th delta function is used to model a force with short duration ti i dt d l f ith h t d ti f(t) I = Area t I (t ) dt t f (t ) I (t ) R. Burdisso 3 ME3514 Impulse Relationship between I and momentum: What is momentum? v m momentum m v Ns 2 m Units: Ns m s To find relationship between I and momentum, let's solve the following dynamic problem t 0 vi 0 t 0 vf f(t) F m R. Burdisso f (t ) to t 4 ME3514 Impulse EOM: F ma d 2x dv f (t ) m 2 m dt dt Integrating EOM w r t time w.r.t. t f (t )dt m v(t ) o t o I m v(t ) v (0) I mv f mvi Impulse = change in momentum NOTE: force time history f(t) is not important only the impulse of the force R. Burdisso 5 ME3514 Impulse f (t ) I (t ) Example I (t ) dt vi (t 0 ) 0 v f (t 0 ) ? t I (t ) m m m vf at t 0 I I mvf m 0 mvf R. Burdisso 6 ME3514 Impulse (t ) dt t Laplace Transform of the Delta function (t ) 1 Laplace of translated Delta function (t to ) e to s (t t o ) to R. Burdisso t 7 ME3514 Impulse k Example: R E l Response of an undamped system to a delta function f d d t t d lt f ti mx kx I (t ) with xo 0 and Then, Then f (t ) I (t ) m xo 0 b=0 ms X ( s ) kX ( s ) I 2 X (s) I ms 2 k I 1 n X (s) 2 m s 2 n n From table sin n t n 2 s 2 n 8 R. Burdisso ME3514 Impulse Example: R E l Response of an undamped system to a delta function f d d t t d lt f ti x (t ) 1.2 1 0.8 0.6 0.4 I m n sin n t 1(t ) i I m n x(t) [m] 0.2 0 -0.2 -0.4 t 0 xo 0 t 0 0 I x 0 m -1 -0.6 -0.8 -1.2 1.2 0 2 4 6 Time (sec) R. Burdisso 9 ME3514 Impulse k Example: Let's compare the response of a d l function with amplitude I to h f delta f i ih li d that of an initial velocity. mx kx 0 with xo 0 and Then, xo 0 m b=0 m s 2 xo k X ( s ) 0 xo mxo X (s) ms 2 k n X (s) 2 n s 2 n x (t ) n xo sin n t 1(t ) Response to a delta function with amplitude I is the same as the response to an initial velocity R. Burdisso xo I m 10 ME3514 Impulse Example: Response of a system subjected to a short duration force. f(t) k m b=0 f (t ) to t Can I replace f(t) for I(t)? Yes, but only if to 2 n Duration of force is much less than the natural period of the system ! R. Burdisso 11 ME3514 Impulse 10 5 x 50 x f (t ) x with xo 0 and xo 0 Example: R E l Response t several pulses to l l Where the inputs are f(t) F f(t) F f(t) F to t to t to t The inputs have the same impulse, I=2 N.s R. Burdisso 12 ME3514 Impulse 2 Assume: Duration of pulse is much smaller than natural p p period, e.g. , g to T n 0.08 0.06 0.04 0 04 0.02 x( (m) (t) 0 -0.02 -0.04 -0.06 -0.08 for example to T T 2 10 n 2 2.8 sec 10 Pulse 50 1 Pulse 2 Pulse 3 0 5 t (sec) 10 15 The shape of the input force is irrelevant when the duration of the pulse is much smaller than the natural period of the system as long as they have the same impulse. R. Burdisso 13 ME3514 Impulse Pulse 1 Pulse 2 Pulse 3 Assume: Duration of pulse is much smaller than natural p p period, e.g. , g 0.08 0.06 0.04 0.02 x(t) (m) 0 x 10 -3 Response x(t) - Impulse 2 m.s Pulse 1 Pulse 2 Pulse 3 15 -0.02 0 02 -0.04 10 -0.06 -0.08 0 08 0 5 t (sec) 10 x(t) (m) ( 5 15 0 to 0.28 sec -5 -0.2 -0.1 0 t (sec) 0.1 0.2 0.3 R. Burdisso 14 ME3514 Impulse to T 10 0.08 0.06 0.04 0.02 x(t) (m) 0 -0.02 0 02 -0.04 -0.06 0 06 -0.08 Pulse 1 Pulse P l 2 Pulse 3 Impulse Assume: Duration of pulse is much smaller than natural p p period, , Can I model the pulse using a delta function? f (t ) I (t ) 0 5 t (sec) 10 15 YES. The input can be modeled using the delta function because the duration of the pulse is much smaller than the natural period of the system. R. Burdisso 15 ME3514 Impulse to T 10 Assume: Duration of pulse is much smaller than natural p p period, , Can I model the pulse using a delta function? f (t ) I (t ) 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 Pulse 1 Pulse 2 Pulse 3 Impulse x 10 -3 Response x(t) - Impulse 2 m.s Pulse 1 Pulse 2 Pulse 3 Impulse x(t) (m) 15 10 -0.08 0 5 t (sec) 10 15 x(t) (m) 5 0 to 0.28 sec -5 -0.05 0 0.05 0.1 0.15 t (sec) 0.2 0.25 0.3 0.35 R. Burdisso 16 ME3514 Impulse for example to T 0.08 0.06 0.04 0.02 x x(t) (m) 0 -0.02 -0.04 -0.06 -0.08 Pulse 1 Pulse 2 Pulse 3 Impulse Assume: Duration of pulse is NOT much smaller than natural p p period, e.g. , g T 2 n Response x(t) - Impulse 2 m.s 2 2.8 sec 50 10 0 5 t (sec) 10 15 Pulse can not be approximated by an impulse in this case R. Burdisso 17 ME3514 Impulse Impact problems k M b=0 Example 1: A bullet impacts a mass at rest. How do you model this problem? p p y p m, vb The impact is equivalent to a short duration force. Unfortunately, we do not know the f th force time history. ti hi t k m b=0 f(t) f (t ) We want to model the problem using f (t ) I (t ). However, we need to find I. How? Using the relationship between change in momentum and impulse. to t R. Burdisso 18 ME3514 Impulse Impact problems k Assume that the bullet gets embedded into the mass M, then the change in g , g momentum of the bullet is: I change in momentum of m I m vb m xo 0 b=0 M Option 1 Also O ti 1: Al assume: M m then vb xo 0 h The response of the system is and I m vb d mvb I sin n t 1(t ) sin n t 1(t ) x (t ) ( M m) n ( M m) n The velocity of the mass M at the end of the impact, xo 0 , is mvb dx (t ) xo 0 dt t 0 ( M m) mvb xo 0 ( M m) 19 R. Burdisso ME3514 Impulse Impact problems k Assume that the bullet gets embedded into the mass M, then the change in g , g momentum of the bullet is: I change in momentum of m I m vb m xo 0 b=0 M Option 2 O ti 2: The response of the system is m vb m xo 0 I x (t ) sin n t 1(t ) sin nt 1(t ) ( M m) n ( M m) n How do you find xo 0 ? R. Burdisso 20 ME3514 Impulse Impact problems k M b Example 2: Assume a perfectly elastic impact. Find the response? p p y p p m, vb R. Burdisso 21
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