Mechatronic system design · 2 Delft WB2414-Mechatronic System Design 2013-2014 3 University of...

1

WB2414-Mechatronic System Design 2013-2014 1DelftUniversity ofTechnology

Mechatronic system design

Mechatronic system design wb2414‐20132014

Course part 4

Profir RHMunnig SchmidtMechatronic System Design

Dynamics of motion systems


Contents

bull Stiffness in Precision Engineering

bull Passive and active stiffness

bull Compliance of (a combination of) dynamic elements

bull Dynamic modelling of damped mass‐spring systems

bull Transmissibility

bull Coupled mass‐spring systems

bull Eigenmodes eigenfrequencies and modeshapes

bull Standard mechanical frequency responses

2


Well known objects

102 Nm Soft pillow

104 Nm Car suspension

Soft couch

105 Nm Table

Bicycle

107 Nm Office building

108 Nm Concrete pillar

109 Nm Steel train wheel on steel rail track

Stiffness of objects


What is stiffness

s

r

d

d

d

F k x

F F

xk

k

F

x

Hookersquos law for force from spring

Hooke‐Newton law for

external force

Where should you place the stiffness if possible

Take the shortest force loop

3


Natural frequency of the resonance of a mass-spring system

bull At resonance the forces are in

balancebull Deformation force (stiffness) plus acceleration force (mass) is zero

m

k

Stationary reference

x02 2

a d 2 2

d d0

d d

x xF F m kx m kx

t t

0

20 0 0

0

ˆ sin ( t)

ˆ ˆsin( ) sin( )

x x

mx t kx t

k

m


The first natural frequency determines the sensitivity to harmonic vibrations

bullThe maximum force needed to follow the acceleration

bullThe maximum error due to this force

bullThe natural frequency

bullWhich results in

20

k

m

20 0 f f

e f 020 e e

ˆ ˆˆ ˆ

ˆ ˆf x x

x x f ff x x

2f

ˆ ˆ ˆF ma mx

2f

e

ˆ ˆ ˆ

m xFx

k k

4


Active stiffness in a CD player bandwidth

bull200 μm radial vibrations at 25 Hz

bullMass lens 10 10‐3 kg

bullMax radial error 02 μm

6r

0 6r

r0

2r

2 50

ˆ 200 10 25 800

0 2 10

4 =25 Nm 1

middot102

xf f Hz

kf k mf

m


Virtual stiffness

bull Measure position

bull Actuate with force proportional and opposite to the deviation (feedback)

bull Gives virtual spring stiffness

m at rcr pF G G G G G

r tr

Fk G

5


Contents










10

Stiffness and compliance

bull Stiffness it the ability of a system to withstand a force by

minimising the resulting motiondeformation

bull Compliance is the opposite

bull Both can be real in phase with a periodic force or complex dynamic frequency dependent 90o out of phase with a periodic

force

bull A spring has a real stiffnesscompliance

s

1xC

F k

6


11

Compliance of (a combination of) dynamic elements

bull k = stiffness of the springbull c = damping coefficient of the damperbull m = mass of the body

m

x0k


c F

x

s

1xC

F k


Stiffness and compliance of a damper

dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7


Stiffness and compliance of a body

22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x


Combined Compliance of body spring and damper

sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8


Overview of the dynamic properties


The separate element responses in a bode plot

Log |xF|

Log ω

Spring line

Mass line (‐2)

Damper line (‐1)Spring line

Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9


No dampingDue to energy conservation response becomes infinite(resonance)

Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase


Extreme damping two first order systems

180

0

Log |xF|

Log ωω0

Phase

11


Contents










Start with second law of NewtonF=ma

2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1

mx kC s sm csF ms cs k sk k

FRF

With only positive imaginary terms (Fourier)

m

x0k


c F

x

Laplace gives

12


The magnitude and the phase

Start with s=jω

2 22

20 0

( )

1 2

st

CC

And phase angle φ is ndash arctan (imaginaryreal of the denominator)


Compliance Bode plot with damping

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13


The damping ratio is related to the pole location in the Laplace plane

s j Poles are those values of s where denominator of Ct is zero

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2

If 0 then 0 no damping

and and 1

c

CC p j p j

ms

k

Re

Im


Critical damping ratio

Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km

C C CkCm cs m m sms s s sk k k k k

14


Above critical damping the mass-spring system reduces to two first order systems

Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m


Effect of the damping ratio on the stepresponse

At ζ = 1 no overshootCritical dampingtime domain related term

15


Energy at natural Frequency

Energy will be ldquotrappedrdquo in the systemWhen excited in this frequency the amplitude will continue to rise

For electrical engineering (frequency domain) the quality factor Q is defined for this property as resonators are also useful But in mechanical engineering the time domain related term ldquodamping ratiordquo is more commonly used

Letrsquos bridge the gap


Q=100 means peak level at 100 times spring-line level

bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x

At Q = 1 no resonance peakfrequency domain related term

16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE

Electrical vs mechanical Time vs frequency domain


Velocity response kinetic energy

2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17


Resonance is energy storage damping is velocity related

bull Driving force is in phase with velocity at resonance

bull Power is Force times speed

bull Max energy transfer at resonance

bull Damper is just the opposite

bull Q tells something about the height of the resonance peak

Maximum energy stored2

Energy lost per cycleQ


Q and energy

18


Contents










Transmissibility transfer of motion through the support of a dynamic system

2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With

The force acting on the body equals

m xm

kxfTable

c


2m f( ) ( )x ms cs k x cs k

19


Bode plot of transmissibility


Contents









20


Coupled mass-spring systems

m1

x1

km2

x2

F


Elastically coupled Multi-body or finite-element non-rigid body dynamics


Which statement is true There is a frequency where

m1

x1

km2

x2

F


1 Only m1 will resonate with k


3 Only m1 will stop moving


21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x

note The domain (s or t) is only mentioned once


Resulting equations in the Laplace domain

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

Low values of s at low frequencies

1 2

21 2

1( ) ( )

x xs s

F F m m s

High values of s at high frequencies

1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22


At mid frequency the driven body shows a strange effect

Combination of poles and zeros

21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w

then the compliance of shows a dip

hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s


m1 = 01 m2 an actuated large mass by a lighter actuator

1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23


m1 = m2 motor mass is optimisedto the driven mass

31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


m1 = 10 m2 parasitic resonances

1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents




bull Dynamic modelling of damped mass spring systems


bull Coupled mass spring systems




Eigenmodes

When excited non‐rigid bodies and complex mass‐spring systems

vibrate in different eigenmodes with two main properties

bull Eigenfrequency the related natural (resonance) frequency

bull Mode‐shape the deformation that corresponds with the eigenmode described in a multiple degree of freedom ldquoshape‐

function rdquo

Modelling is done by discretisation of the system in multiple mass‐

spring systems

bull The shape function reduces to an eigenvector with one value for

each body for the relative motion magnitude and direction (sign)

25


Mode-shape of complex non-rigid body


Eigenfrequencies of multiple eigenmodes in 6-DOF

26


Eigen modes of coupled bodies

1

1

1

2

1

1


Multiplicative expression shows two resonating eigenfrequencies

c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k

F m m s k m m s F m m s k m m s

Starting with

11 2

2

00

m m

k

M

27


Additive expression

bull Can be written as a combination of two eigenmodes with modal

mass and stiffness using eigenvector

21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i


Modes 1 and 2 combined

1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i

Theory on modal decompositionbull General equation of motion (vectormatrix) of finite element system

bull General transfer function

bull In absence of external force

bull Decoupled by eigenvalue problem where is the eigenvector

2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M


Modal mass and stiffness orthogonality of eigenmodes

bull Modal mass

bull Modal Stiffness

20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling

bull Length of eigenvector is not defined (only the direction)

bull Three scaling methods are often applied

1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates

bull When qiequals the motion of eigenmode i then the total displacement vector x(t) will be

where qi is called the modal coordinate

bull This gives the following displacement for DOF xk(t)

1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30


Full set of uncoupled equations

1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )

applied to ( ) ( ) ( )

gives with pre-multiplication with

( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t


This can be transformed into a combined transfer function

bull The response of DOF by

eigenmode i on force equals

bull The total response equals

2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31


Graphical representation

bull The modal coordinate is

represented by the angle of the lever or the displacement at distance 1

bull Modal mass and stiffness are connected to the modal

coordinate

bull The eigenvectors determine the

ratio of ldquocontrollabilityrdquo of actuator and ldquoobservabilityrdquo of sensor

2

i

( ) i k i

k i i

xs

F s


Effective modal values

As perceived at the actuator

32


Mass on leafspring two eigenmodes

F

x

F

x

y

x

a Mode shape 1 b Mode shape 2

Stationary reference Stationary reference


Modes 1 and 2 combined by measuring non co-local with the force

‐2 slope with ‐360o

phase is non‐minimum phase system

33


Non-minimum phase step response


Summary of termsbull Eigenvector

bull Vector with terms that represent the relative motion amplitude of each element in the

discretised structure It is the discretised ldquoshape functionrdquo

bull Scaling is at will as long as the scaling is equal for all properties of that eigenmode

bull Modal mass and stiffnessbull relate to the mass and stiffness matrix by double multiplication with the eigenvector

bull Scaling of the eigenvector also scales the modal mass and stiffness (squared)

bull Modal coordinatesbull Represent the motion of the eigenmode at the location of the modal mass and

stiffness

bull It is used to determine the motion of all elements with the eigenvector and as such it

also depends on the chosen scaling

bull Effective modal mass and stiffness

bull Is the modal mass and stiffness as perceived at the actuator position It is calculated

from the modal mass and stiffness by dividing by the corresponding eigenvector term

squared hence the chosen scaling no longer plays a role

i i i i i iM K

34




Contents









35


Typical combination of two eigenmodes one rigid and one non-rigid

bullActuated at xa

bullSensed at different locations


Different combinations of two eigenmodes

36


4 types of responses result



2


Well known objects

102 Nm Soft pillow

104 Nm Car suspension

Soft couch

105 Nm Table

Bicycle

107 Nm Office building

108 Nm Concrete pillar

109 Nm Steel train wheel on steel rail track

Stiffness of objects


What is stiffness

s

r

d

d

d

F k x

F F

xk

k

F

x

Hookersquos law for force from spring

Hooke‐Newton law for

external force

Where should you place the stiffness if possible

Take the shortest force loop

3





m

k


x02 2

a d 2 2

d d0

d d

x xF F m kx m kx

t t

0

20 0 0

0

ˆ sin ( t)

ˆ ˆsin( ) sin( )

x x

mx t kx t

k

m







20

k

m

20 0 f f

e f 020 e e

ˆ ˆˆ ˆ

ˆ ˆf x x

x x f ff x x

2f

ˆ ˆ ˆF ma mx

2f

e

ˆ ˆ ˆ

m xFx

k k

4






6r

0 6r

r0

2r

2 50

ˆ 200 10 25 800

0 2 10

4 =25 Nm 1

middot102

xf f Hz

kf k mf

m


Virtual stiffness





r tr

Fk G

5


Contents










10






force


s

1xC

F k

6


11



m

x0k


c F

x

s

1xC

F k



dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





3





m

k


x02 2

a d 2 2

d d0

d d

x xF F m kx m kx

t t

0

20 0 0

0

ˆ sin ( t)

ˆ ˆsin( ) sin( )

x x

mx t kx t

k

m







20

k

m

20 0 f f

e f 020 e e

ˆ ˆˆ ˆ

ˆ ˆf x x

x x f ff x x

2f

ˆ ˆ ˆF ma mx

2f

e

ˆ ˆ ˆ

m xFx

k k

4






6r

0 6r

r0

2r

2 50

ˆ 200 10 25 800

0 2 10

4 =25 Nm 1

middot102

xf f Hz

kf k mf

m


Virtual stiffness





r tr

Fk G

5


Contents










10






force


s

1xC

F k

6


11



m

x0k


c F

x

s

1xC

F k



dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





4






6r

0 6r

r0

2r

2 50

ˆ 200 10 25 800

0 2 10

4 =25 Nm 1

middot102

xf f Hz

kf k mf

m


Virtual stiffness





r tr

Fk G

5


Contents










10






force


s

1xC

F k

6


11



m

x0k


c F

x

s

1xC

F k



dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





5


Contents










10






force


s

1xC

F k

6


11



m

x0k


c F

x

s

1xC

F k



dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





6


11



m

x0k


c F

x

s

1xC

F k



dd

d( ) ( )

d( )

( )

( )

1 1( )

( )

xF t c F s scx

tF jc x

xC

k F jc

F t

F t

m

x0k


c F

x

7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





7



22

2

2

m 2m

( )d

( ) ( )d

( )

1 1( )

(

( )

)

xF t m F s ms x

t

F m x

xC

k F

F t

F t

m

m

x0k


c F

x



sts d m t

tt

s d m

d m

1 1 1( ) ( ) ( )

1( ) ( )

1 1 1

xF F x

C C C C

xC

FC

F

C C

F

8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





8





Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m

9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





9



Log |xF|

Log ωω0

Phase

180

0


Low damping

180

0

Log |xF|

Log ωω0

Phase

10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





10


High damping

180

0

Log |xF|

Log ωω0

Phase



180

0

Log |xF|

Log ωω0

Phase

11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





11


Contents











2

2

d d( ) )

d d

x xF t m c kx

t t

2 ( )( ) ( )F s x m ct s sF k

t 2

20 0

( )( ) ( )2 1

sCxC F

s sFt

1sC

k 0

k

m

2

c

km

t 22

11

( ) ( )1


FRF


m

x0k


c F

x

Laplace gives

12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





12



Start with s=jω

2 22

20 0

( )

1 2

st

CC




m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





13




2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

0s

t 1 02

2


and and 1

c

CC p j p j

ms

k

Re

Im



Re

Im

= 1

lt 1

lt 1

gt 1

gt 1

1

2t

s2

20 0

1

1 2 1

Cx kCm cs s sF sk k

1 2 and d dp j p j

s

2

s s2 2t

2

If 2 then 1 and

1

11 2 1 1

c km


14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





14



Log |xF|

Log ω

Spring line

Mass line (‐2)


Damper line

180 Mass line

1s

s

xC

F k

2

1m

m

xC

F m

1d

d

xC

F c

Phase

90

0

180

0

ω0

020

1 1

k

k m m




15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





15








bull k = 104 Nm

bull m = 025 kg

m

x0k


c F

x


16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





16


Relation Q and

1 1

2 2Q

Q

1 1

2 2EE ME

ME EE




2

20 0

dd

2 1

s

xv sxtF F F

s C

s s

m

x0k


c F

x

bull k = 104 Nm

bull m = 025 kg

17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





17











Q and energy

18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





18


Contents











2m f m

tm f m2

d d( )( ) ( )

d d

x x xF t m c k x x

t t

02

20 0

2 1

2 1

m

f

sx

s sx

0

k

m

2

c

km

22

1

1

m

f

csx cs k k

m csx ms cs k sk k

With


m xm

kxfTable

c



19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





19




Contents









20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





20



m1

x1

km2

x2

F





m1

x1

km2

x2

F






21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





21


Equations of Motion

bull First body

bull Second body

m1

x1

km2

x2

F


21

1 1 22

21 1 1 2

22

2 1 22

22 2 1 2

d ( )( ) ( )

d

( ) ( )

d ( )( )

d

( ) ( )

x tm F t k x x

t

m s x s F k x x

x tm k x x

t

m s x s k x x




21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s


1 2

21 2

1( ) ( )

x xs s

F F m m s


1 22 4

1 1 2

1( ) ( )

x x ks s

F m s F m m s

Fourth order system

22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





22




21 2

4 21 2 1 2

24 2

1 2 1 2

2 2

2

1

( )

w


hen

k

m

x m s k

F m m s k m m s

x ks

F m m s k m

s

x

m s



1

2

4

3

3

25 10 kg

25 10

1

kg

Nm0

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

1 2

21 2

1LF ( ) ( )

x x

s sF F m m s

1 22 4

1 1 2

1HF ( ) ( )

x x ks s

F m s F m m s

23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





23



31

32

4

25 10 kg

25 10

10

kg

Nm

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s



1

2

4

3

025 kg

25 10 kg

1 m0 N

m

m

k

21 2

4 21 2 1 2

24 2

1 2 1 2

( )

( )

x m s ks

F m m s k m m s

x ks

F m m s k m m s

24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





24


Contents










Eigenmodes





function rdquo


spring systems



25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





25





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





26



1

1

1

2

1

1



c

c

2122 2

1 2

1 2

1 2

22 2

1 2 c

1 1

with

1

xm s k

F m m s m s k

m mm

m m

x k

F m m s m s k

2

1 2 24 2 4 2

1 2 1 2 1 2 1 2

x m s k x k


Starting with

11 2

2

00

m m

k

M

27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





27


Additive expression



21 1 222 2

1 2 1c

c 2

22 2

1 2 c

1 1 with

1

x m mm s k m

F m m s m s k m m

x k

F m m s m s k

2 22 2

2 2

c 1 1

1 2 1 212 2 2 2

1 2

1 2 1 222 2 2 2

1 2

2 2

1 2 1 22 2

c 1 1 2 2

1 1=

( )

1 1=

( )

m m

m m m mx

F m m s m s k s s

m m

m m m mx

F m m s m s k s s

m m

1

1

1

2

1

1

i i i



1 212 2

1 222

22

2

1 1 2 2

1 22

1 12

2 2

1

1

m

m mx

F s s

m

m mx

F s s

m

28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





28


i





2

2

20

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0

0i i

Mx t Kx t F t

Ms x s Kx s F s

Ms x s Kx s

K M



bull Modal mass


20

0 ( )

0 ( )

i j

i i i

i j

i i i i i

M i j

M

K i j

K

29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





29


Scaling



1 1

2 max 1

3 1

i i i

i

i

i

M


Modal coordinates




1 1 2 2 n n

1 1 2 2 n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )k k k n k

x t q t q t q t

x t q t q t q t

30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





30



1

1 2 n

n

2

( ) ( )

( )

with and ( ) ( )

( )



( ) ( ) ( )

i

x t q t

q t

q t q t

q t

Ms x s Kx s F s

M q t K q t F t






2

i

n n 2

1 1

( )

( )

i k i

k i i

i k i

i ik k i ii

xs

F s

x xs

F F s

xkF

31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





31






coordinate



2

i

( ) i k i

k i i

xs

F s




32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





32



F

x

F

x

y

x







33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





33











stiffness







i i i i i iM K

34




Contents









35



bullActuated at xa




36





34




Contents









35



bullActuated at xa




36





35



bullActuated at xa




36





36





Mechatronic system design · 2 Delft WB2414-Mechatronic System Design 2013-2014 3 University of...

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Transcript of Mechatronic system design · 2 Delft WB2414-Mechatronic System Design 2013-2014 3 University of...