Class 16: Damped and forced oscillation Class 16: Damped and forced - - PowerPoint PPT Presentation

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Nov 29, 2023 113 likes •203 views

Class 16: Damped and forced oscillation Class 16: Damped and forced oscillation Damped SHM Equation of motion (homogeneous) + + = & & & m x b x k x 0 Causes decay in oscillation Causes decay in oscillation (hence transient)

SLIDE 1

Class 16: Damped and forced oscillation Class 16: Damped and forced oscillation

SLIDE 2

Damped SHM Equation of motion (homogeneous)

k x x b x m = + + & & &

Causes decay in oscillation

Transient (homogeneous) solution:

Causes decay in oscillation (hence transient)

b ) t ( cos Ae ) e (e Ae x(t)

t

t i t

Ω = + =

Ω Ω

φ

β β

2m b = β

Ω ω0 β

SLIDE 3

Damping cases p g

2 2

β

2 2 2 2 2

ω β ω β = <

Under damped Critically damped

2 2

ω β >

Over damped

SLIDE 4

Forced vibration Equation of motion

t cos F k x x b x m ω = + + & & &

General solution = Transient (homogeneous) solution + Steady state (particular) solution solution + Steady state (particular) solution

SLIDE 5

Mechanical oscillation and AC circuit conversion dictionary conversion dictionary

t cos F k x x b x m ω = + + & & & t cos V C Q Q R Q L ω = + + & & & Mechanical oscillation RLC series circuits P i i ( ) Ch Q( ) C Position x(t) Charge Q(t) Velocity v(t) = Current I(t) = (t) Q & (t) x & Force F Voltage V Mass m Inductance L Damping coefficient b Resistance R Spring constant k Reciprocal of Capacitance 1/C Z I ~ V ~ : Law s Ohm' = Z v ~ F ~ =

⇐

SLIDE 6

Impedance model for steady state solution

t cos F k x x b x m ω = + + & & &

~ ~ Z i F (t) x Z F (t) x : Law s Ohm'

p p

ω = ⇒ = &

Z is the total impedance of Zm, Zb, and Zk in series:

Im

ω b Z m i Z

b m

= =

Zm

ω i k Zk

=

Re Zb Zk

Resonance occurs when Zm= Zk

SLIDE 7

t cos F k x x b x m ω = + + & & &

( )

⎞ ⎛ m k m k

( ) ( )

= = − + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + = + + =

2 2 2

F ~ F ~ (t) x ~ m i b m k m i b i k b m i Z ω ω ω ω ω ω ω

( ) ( )

= = − − = =

2 2 2 2 2 2 p 2 2 p

F Z i F ~ (t) x ~ m b i Z i (t) x ω ω ω ω ω

( ) ( )

+ = ∴ − +

2 2 2 2 2 2 p 2 2 2 2 2 2

)

( cos m b F (t) x m b Z i φ ω ω ω ω ω ω ω ω There is a phase between displacement and applied force

( ) ( )⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − +

2 2 1

b

m b ω ω ω φ ω ω ω

( )⎠

⎝

SLIDE 8

Resonance and Q(uality) factor

t cos F k x x b x m ω = + + & & &

( )

m b F

2 2 2 2 2 2

ω ω ω − +

Class 16: Damped and forced oscillation Class 16: Damped and forced oscillation

Damped SHM Equation of motion (homogeneous)

k x x b x m = + + & & &

Causes decay in oscillation

Transient (homogeneous) solution:

Causes decay in oscillation (hence transient)

b ) t ( cos Ae ) e (e Ae x(t)

t

t i t

Ω = + =

Ω Ω

φ

β β

2m b = β

Ω ω0 β

Damping cases p g

β

ω β ω β = <

Under damped Critically damped

ω β >

Over damped

Forced vibration Equation of motion

t cos F k x x b x m ω = + + & & &

General solution = Transient (homogeneous) solution + Steady state (particular) solution solution + Steady state (particular) solution

Mechanical oscillation and AC circuit conversion dictionary conversion dictionary

⇐

Impedance model for steady state solution

t cos F k x x b x m ω = + + & & &

~ ~ Z i F (t) x Z F (t) x : Law s Ohm'

ω = ⇒ = &

Z is the total impedance of Zm, Zb, and Zk in series:

Im

ω b Z m i Z

= =

Zm

ω i k Zk

=

Re Zb Zk

Resonance occurs when Zm= Zk

t cos F k x x b x m ω = + + & & &

( )

⎞ ⎛ m k m k

( ) ( )

= = − + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + = + + =

F ~ F ~ (t) x ~ m i b m k m i b i k b m i Z ω ω ω ω ω ω ω

( ) ( )

= = − − = =

F Z i F ~ (t) x ~ m b i Z i (t) x ω ω ω ω ω

( ) ( )

+ = ∴ − +

)

( cos m b F (t) x m b Z i φ ω ω ω ω ω ω ω ω There is a phase between displacement and applied force

( ) ( )⎟

⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − = − +

b

m b ω ω ω φ ω ω ω

( )⎠

⎝

Resonance and Q(uality) factor

t cos F k x x b x m ω = + + & & &

stored Energy 2 Q Q × = Δ = π ω ω circuit) series RLC for L 1 (or mk Q cycle per dissipated Energy Q = circuit) series RLC for C R (or b Q=