Engineering Vibrations Chapter 2

University of Maryland B. Balachandran & E. Magrab

Review of Chapter 3


Lagrange’s Equations


General Form:

1,...,knc

k k k

d L L DQ k ndt q q q ∂ ∂ ∂

− = − = ∂ ∂ ∂

General Lagrange’s Equations

L = T – U = Lagrangian

T = Kinetic Energy, U = Potential Energy

Where

= Rayleigh Dissipation Function12k kD c =

r.r

= Generalized force. 1,2knc

k

Q kq∂

= =∂

rF&


NO. Newton’s Equations Lagrange’ s Equations

1 Vector-based Scalar-based

2 Requires free-body analysis Analysis of entire system

3 More equations than DOF Equations = DOF

4 Requires internal forces & reactions

Formulation is independent of internal forces & reactions

5 Requires deriving expressions of velocities & accelerations

Requires deriving expressions of ONLY the velocities

Lagrange’s Equations


Time Response of SDOF Systems


Behavior of Single DOF Systems

Derive the Equations of Motion

Determine Natural Frequencies

Determine System Stability

Predict System Response

Optimize System Response

Control System Response


Free Vibration

c = 0 , f =0


Free Vibration of a Single DOF System

Equation of Motion0mx kx+ =

2 0nx xω+ =

where 2 kn mω =

Assume tx Ceλ=where C & λ are to be determined


Single DOF Systems

But, as cos sinie iθ θ θ= +

1 2 1 2cos sinn ni t i tn nx C e C e A t A tω ω ω ω−= + = +

If 0( 0)x t x= = & 0( 0)x t x= =

0 1( 0)x t x A= = = & 0 2( 0) nx t x Aω= = =

00 cos sinn n

n

xx x t tω ωω

= +


Single DOF Systems

00 cos sinn n

n

xx x t tω ωω

= +

2 20 0

0 02 20 0

( / )( cos / sin )

( / )n

n n n

n

x xx x t x t

x x

ωω ω ω

ω

+= +

+

φ

x0

0 / nx ω

2 20 0( / )nx x ω+

But, as

[ ]10 0tan / ( / )nx xϕ ω−=


Forced Vibration

c = 0 , f =Fo cos ωt


Undamped Single DOF Systems

0 cosm x k x F tω+ =

Equation of Motion

1 2cos sinh n nx A t A tω ω= +Solution

Homogeneous Solution

cospx X tω=Particular Solution

1 2cos sin cosn nx A t A t X tω ω ω= + +General Solution



0 cosm x k x F tω+ =

cospx X tω=Particular Solution

Substitute in

20cos cos cosm X t kX t F tω ω ω ω− + =

0 02 2 2

/1 ( / ) 1 ( / )

s

n

F F kXk m m k

δω ω ω ω

= = =− − −

1 2 2cos sin cos1 ( / )

sn n

n

x A t A t tδω ω ωω ω

= + +−

or



As 00 2 2cos sin cos

1 1s s

n nn

xx x t t tδ δω ω ωω

= − + + −Ω −Ω

00 2

cos coscos sin1

nn n s

n

x t tx x t t ω ωω ω δω

− = + + −Ω

If ω= ωn0

0 2

cos coscos sin1

nn n s

n


− = + + −Ω

=0

= 0

Free Response Forced Response


Damped Single DOFSystems

c = 0 , f =Fo cos ωt


Damped Single DOF Systems

0 cosm x cx k x F tω+ + =

Equation of Motion

Rearrangement of the homogeneous part:

0c kx x xm m

+ + =

Characteristic Eqn.:2 0c k

m mλ λ+ + =



2 22 cosn n nx x x tζω ω δ ω ω+ + =

Equation of Motion

Solution


cos( )px X tω φ= −Particular Solution

General Solution sin( ) cos( )nth d hx X e t X tζω ω φ ω φ−= + + −

( )sinnth h d hx X e tζω ω φ−= +



System Response



sin( ) cos( )nth d hx X e t X tζω ω φ ω φ−= + + −General Solution

( )sinnth h d hx X e tζω ω φ−= +

Transient Solution Steady-State Solution0 =

Causal

Mathematical




Substitute in

2 2 2[( )cos( ) 2 sin( )] cosn n nX t t tω ω ω φ ζω ω ω φ δ ω ω− − − − =

But cos( ) cos cos sin sint t tω φ ω φ ω φ− = +

& sin( ) sin cos cos sint t tω φ ω φ ω φ− = −

2 22 cosn n nx x x tζω ω δ ω ω+ + =



ωt( )2 cosn X tω ω φ−φ

2 sin( )n X tζω ω ω φ−( )2 cosn tδω ω

( )2 cosX tω ω φ− −

Damping

Spring

Inertia

Excitation

2 2 2 2 2/ ( ) (2 )n n nX δω ω ω ζω ω= − + 12 2

2tan n

n

ζω ωφω ω

− = −

&


SummaryNo Case Basic Governing Equations1 Undamped- Free

Vibration2 Undamped-Forced

Vibrationa Resonance

b Beat Phenomenon

3 Damped-Forced Vibration

* Steady-state Response

2 20 0( / ) sin( )n nx x x tω ω ϕ= + +

00 2

cos coscos sin1

nn n s

n


− = + + −Ω

0 / sin .sin2

F mx t tε ωεω

=

00

1cos sin [ sin ]2n n n s n n

n

xx x t t t tω ω ω ω δ ω ωω=

= + +

sin( ) cos( )nth d hx X e t X tζω ω φ ω φ−= + + −

2 2 2/ 1/ (1 ) (2 )sX δ ζ= −Ω + Ω 12

2tan1ζφ − Ω = −Ω

&

& [ ]10 0tan / ( / )nx xϕ ω−=


Complete Response of Damped Single DOF Systems

where 2 2 2 2 2/ ( ) (2 )n n nX δω ω ω ζω ω= − +

12 2

2tan n

n

ζω ωφω ω

− = −

&


Complete Response of Damped Single DOF Systems

Also, Xo and ϕ0 are determined from the initial conditions as follows:0 0&x x


Frequency Response of SDOF Systems


Analysis Approaches

System’s Eqn. of Motion

Perform Laplace Transform

Determine Time Response by Inverse Laplace Transform

Determine Frequency Response by

Replacing s=iω


Transfer Function

, ,m k c( )f t ( )x t

Input OutputSystem

( )x tk

c

m ( )f t


Transfer Function

( )x tk

c

m ( )f t

m x cx k x f+ + = Equation of Motion

( )2 ( ) ( )m s cs k X s F s+ + =Laplace Transformation

( )2

( ) 1( )

X sF s m s cs k

=+ +Transfer Function


Frequency Response Function

( )x tk

c

m ( )f t

( )2

( ) 1( )

X sF s m s cs k

=+ +

Transfer Function

( )2

( ) 1( )

XF k m i cωω ω ω

=− +

Frequency Response Function

Replace s = iω


Stability of SDOF Systems


O

θ

m

L

kKinetic Energy

2 212

T mL θ=

Potential Energy

( ) ( )

( )

2

2 2

1 1212

U k L mgL cos

kL mgL

θ θ

θ

= − −

= −

Equation of Motion

( )2 2 0mL kL mgLθ θ+ − =

System Stability


Equation of Motion 0k gm L

θ θ + − =

System Stability

Characteristic Equation 2 0k gm L

λ + − =

2 0g kL m

λ − + =

or

2

11 0kgmL

λ

+ = −

( )2

2 211 0m

pω

λ ω+ =

−

2 2m p

k g,m l

ω ω= =where


System StabilityRoots of Characteristic Equation for different when 2 20pω =2

mω

( )2

2 211 0m

pω

λ ω+ =

−

MATLAB>> n=1;>> d=[1 0 -20];>> rlocus(n,d)

-6 -4 -2 0 2 4 6

-6

-4

-2

0

2

4

6

0.160.340.50.64

0.76

0.86

0.94

0.985

0.160.340.50.64

0.76

0.86

0.94

0.985

1

2

3

4

5

6

1

2

3

4

5

6

Real Axis (seconds-1

)

Imag

inar

y A

xis

(sec

onds

-1) System: sys

Gain: 20

Pole: 0

Damping: -1

Overshoot (%): 0

Frequency (rad/s): 0 xx


System StabilityRoots of Characteristic Equation for different when 2 20pω =

2mω

( )2

2 211 0

1m

pω

λ λ ω+ =

+ −

MATLAB>> n=1;>> d=[1 1 -20];>> rlocus(n,d)

-6 -4 -2 0 2 4 6

-5

-4

-3

-2

-1

0

1

2

3

4

5

0.955

0.81

0.20.40.560.7

0.81

0.9

0.955

0.988

0.20.40.560.7

0.9

0.988

123456

Real Axis (seconds-1

)

Imag

inar

y A

xis

(sec

onds

-1) System: sys

Gain: 20

Pole: 0.0242

Damping: -1

Overshoot (%): 0

Frequency (rad/s): 0.0242 xx


END

Engineering Vibrations Chapter 2

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Transcript of Engineering Vibrations Chapter 2