University of Maryland B. Balachandran & E. Magrab

Review of Chapter 3

University of Maryland B. Balachandran & E. Magrab

Lagrange’s Equations

University of Maryland B. Balachandran & E. Magrab

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&

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

Time Response of SDOF Systems

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

Free Vibration

c = 0 , f =0

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

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

= +

University of Maryland B. Balachandran & E. Magrab

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−=

University of Maryland B. Balachandran & E. Magrab

Forced Vibration

c = 0 , f =Fo cos ωt

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

Undamped Single DOF Systems

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

University of Maryland B. Balachandran & E. Magrab

Undamped Single DOF Systems

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

x t tx x t t ω ωω ω δω

− = + + −Ω

=0

= 0

Free Response Forced Response

University of Maryland B. Balachandran & E. Magrab

Damped Single DOFSystems

c = 0 , f =Fo cos ωt

University of Maryland B. Balachandran & E. Magrab

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+ + =

University of Maryland B. Balachandran & E. Magrab

Damped Single DOF Systems

2 22 cosn n nx x x tζω ω δ ω ω+ + =

Equation of Motion

Solution

Homogeneous 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−= +

University of Maryland B. Balachandran & E. Magrab

Damped Single DOF Systems

System Response

cos( )px X tω φ= −Particular Solution

Homogeneous Solution

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

University of Maryland B. Balachandran & E. Magrab

Damped Single DOF Systems

cos( )px X tω φ= −Particular Solution

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+ + =

University of Maryland B. Balachandran & E. Magrab

Damped Single DOF Systems

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

− = −

&

University of Maryland B. Balachandran & E. Magrab

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

x t tx x t t ω ωω ω δω

− = + + −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−=

University of Maryland B. Balachandran & E. Magrab

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

− = −

&

University of Maryland B. Balachandran & E. Magrab

Complete Response of Damped Single DOF Systems

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

University of Maryland B. Balachandran & E. Magrab

Frequency Response of SDOF Systems

University of Maryland B. Balachandran & E. Magrab

Analysis Approaches

System’s Eqn. of Motion

Perform Laplace Transform

Determine Time Response by Inverse Laplace Transform

Determine Frequency Response by

Replacing s=iω

University of Maryland B. Balachandran & E. Magrab

Transfer Function

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

Input OutputSystem

( )x tk

c

m ( )f t

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

Stability of SDOF Systems

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

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

University of Maryland B. Balachandran & E. Magrab

END