Response to step, ramp and convolution

• Step function, integral of delta function– Forcing function often stepwise continuous– When can you also integrate the response

• Ramp function, integral of step function– Often serves same purpose as highway ramp– Building block

4.2 Step response• Step function

• Useful for notation. Instead of

0 for ( )

1 for t a

u t at a<⎧

− = ⎨ >⎩

1 sin for 0( )

0 f

ntd

d

e t tmg t

t

ζω ωω

−⎧ >⎪= ⎨⎪ or 0

1( ) sin ( )ntd

d

g t e tu tm

ζω ωω

−=write<⎩

( )( ) du t at adt

δ −− =

Solution of step response

• Equation to solve

• Solve from

• Obtain

2

2 ( )d dm c k s u tdt dt

⎛ ⎞+ + =⎜ ⎟

⎝ ⎠

( ) ( )t

s t g dτ τ−∞

= ∫1( ) 1 cos sin ( )nt n

d dd

s t e t t u tk

ζω ζωω ωω

−⎡ ⎤⎛ ⎞= − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

Alternate derivation

• Differential equation

• General solution of homogeneous equation plus particular solution s=1/k

• Find constant from

2

2 ( )d dm c k s u tdt dt

⎛ ⎞+ + =⎜ ⎟

⎝ ⎠

( )1 21 cos sinnt

d ds e C t C tk

ζω ω ω−= + +

( ) ( )0 0 0s s= =

1( ) 1 cos sin ( )nt nd d

d

s t e t t u tk

ζω ζωω ωω

−⎡ ⎤⎛ ⎞= − +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

4.3 Ramp response

• Unit ramp function r(t-a)=(t-a)u(t-a)• Relation to step function

• Differential equation for ramp response• When does 2nd equation hold?

( )( ) ( ) ( )t dr t ar t a u a d u t a

dtτ τ

−∞

−− = − − =∫

2

2 ( )

( ) ( )t

d dm c k u tdt dt

t s dτ τ−∞

⎛ ⎞+ + =⎜ ⎟

⎝ ⎠

= ∫

r

r

Problem 4.5• Find response of spring-

mass-damper to F(t)• Solution:

• Then

• From problem 4.4 (homework)

[ ]0( ) ( ) ( )FF t r t r t TT

= − −

[ ]0( ) ( ) ( )Fx t t t TT

= − −r r

[ ]0( ) ( ) ( )FF t r t r t TT

= − −

21 2 2 2 1( ) cos sin ( )ntd d

n n d

t t e t t u tk

ζωζ ζ ζω ωω ω ω

−⎡ ⎤⎛ ⎞−= − + +⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦r

Reading assignmentSections 4.4, 4.5,4.6

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