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Balachandran Chapter 2 review
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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