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Ch. 1: Introduction of Mechanical Vibrations Modeling

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Ch. 1: Introduction of Mechanical Vibrations Modeling 1.0 Outline That You Should Know Newton’s Second Law Equations of Motion Equilibrium, Stability 1.0 Outline
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Ch. 1: Introduction of Mechanical Vibrations Modeling

1.0 Outline

That You Should KnowNewton’s Second LawEquations of MotionEquilibrium, Stability

1.0 Outline

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Vibration is the repetitive motion of the system relativeto a stationary frame of reference or nominal position.

Principles of Motion Vibration ModelingMath Vibration Analysis

*** design the system to have a particular response ***

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

Spring-Mass Model

Mechanical Energy = Potential + Kinetic

From the energy point of view, vibration is caused bythe exchange of potential and kinetic energy.

When all energy goes into PE, the motion stops.When all energy goes into KE, max velocity happens.

Spring stores potential energy by its deformation (kx2/2).Mass stores kinetic energy by its motion (mv2/2).

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

Springs and Masses connection as the way to modelthe vibrating system.

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Real-life system may not see the springs/massesconnection explicitly! You have to devise the simplemodel smartly, suitable to the requirements.

Ch. 1: Introduction of Mechanical Vibrations Modeling

Various forms of springs: linear, traverse, torsion spring

1.1 That You Should Know

Gravity force can make up the spring!Because work done by the gravity force is a kind of PE.When the altitude change, PE change KE change.

Ch. 1: Introduction of Mechanical Vibrations Modeling

Spring Stiffness (N/m or Nm/rad)

Spring has the characteristic that force is the functionof deformation: F = k(x). If k is constant, the springis linear. Practically it is not constant. k is generallyslope of the F-x curve, and is known as the stiffness.

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Stiffness is the property of the object which dependsmainly on its shape and material properties, e.g. E.

Equivalent massless spring constants

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Equivalent massless spring constants (cont.)

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Equivalent massless spring constants (cont.)

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Equivalent massless spring constants (cont.)

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Equivalent massless spring constants (cont.)

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Spring connection: series or parallel

The calculated stiffness value, keff, is used in themodeling system.

keff

Forces add up

Disp. add up

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ex. 1

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ex. 2

Derive the equivalent spring constant for the systemin the figure.

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

eq

eq 22

2 22 2

2

1 eq

let the system be deviated by torque

at that point

at distance , and =

at distance , and =

1 is in series with 1

l a

l b

l a

l a

Fk

l a F ka a

b al b F k kb a b

k b k bka a

k k k

k

τ

τ τδδ

τ τδδ

τδ

=

=

=

=

= = Δ ⇒ =

= = Δ ⇒ = =

= ∴ =

∴ =2

1 22 2

1 2

1

1l a

k k bk a k b

k=

=++

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

( )

( )

( )

eq 1

1eq

2 1

21 1 2

eq2 2 2 21 2 1 2

Perturb the system by force and observe the displacement

but

need to find relation between and

0 0

O

F xFk F k x ax

k x ak x

xM k b b k x a a

k ax k k bkk a k b k a k b

θ

θθ

θ θ

θ

= = −

−∴ = ⇒

⎡ ⎤= × − − × =⎣ ⎦

= ∴ =+ +

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ex. 3

The system shown in the figure consists of two gears Aand B mounted on uniform circular shafts of equalStiffness GJ/L; the gears are capable of rolling oneach other without slip. Derive an expression for theEquivalent spring constant of the system for theradii ratio Ra / Rb = n.

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

A

A

B

Apply torque at gear A, then gear A rotates .Shaft exerts resistant moment and gear B exerts reaction force .In turns, at gear B, opposite reaction force happens andshaft exerts moment t

MM F

FM

θ

B

eq@A AA

A A B B

A A B B

A B

B A

B

A

o resist rotation .

need to write and as functions of given parameters.

0 and

by spring stiffness, and

by geometry,

Mk M

M M M FR M FR

GJ GJM ML L

RnR

GJGJ LML

θ

θθ

θ θ

θθ

θθ

= ⇒

⎡ ⎤= = + =⎣ ⎦

= =

= =

∴ = +

( )

( )

2A A

B

2eq@A

A

1

1

GJR nR L

M GJk nL

θ

θ

= +

∴ = = +

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

B A

A B trans

2B AB B2

B A

eq@A

torsional spring at gear B is connected to gear A1 gear A sees stiffness at gear B

springs are connected in parallel, equivalent stiffness at gear A

M nM

M MGJk n kL n

k k

θθ

θ θ

⎛ ⎞= =⎜ ⎟⎝ ⎠

= = = ⇒ =

=

( )2 2A B 1GJn k n

L+ = +

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ex. 4

The two gears of the system have mass polar momentof inertia IA and IB. Derive an expression for theequivalent mass polar moment of inertia for the radiiratio RA / RB = n.

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

A A A

B B B

B A

A B

B

Ignore the torsional spring and write the equation of motion at gear A

to eliminate , consider motion at gear B

Employ the geometric constraints:

M I M FR I

F

FR I

RnR

IM

θ θ

θ

θθ

⎡ ⎤= − =⎣ ⎦

=

= =

( )2

2AA A A A B A

A2

eq@A A B

n R I M I n IR

I I n I

θ θ θ= ⇒ = +

∴ = +

Ch. 1: Introduction of Mechanical Vibrations Modeling

Mass, Inertia

To store / release kinetic energy

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

Damper viscous coefficient (Ns/m or Nms/rad)

To dissipate energy

1.1 That You Should Know

df F cx= − = −

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.1 That You Should Know

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.2 Newton’s Second Law

Many ways to derive the equations of motions

Newton-Euler is one, which is suitable for situations…• system in planar motion• force and motion have constant direction• system is simple

1.2 Newton’s Second Law

Ch. 1: Introduction of Mechanical Vibrations Modeling

Relative Velocity

A B A/B rel= + × +v v r vωA

B

A/B

A/B rel

rel

velocity of the particle Avelocity of the particle B

velocity of A relative to B velocity of A as seen from the observer

x yvelocity of A as seen from an obs

==== × + ≠

=

vvv

r vv i + j =

ω

( )( )

A/B rel

rel

A/B

erver fixed to the rotating x-y axes

observed from nonrotating x-y

observed from rotating x-yIf we use nonrotating axes, there will be no term

at anywhere× =

×

r v

vr

ω

ω

rel A/B

A/B rel A/B

.This makes , which means the velocity seen bythe observer is the velocity of A relative to B.If B coincides with A, . This makes ,which means the velocity seen by the observe

=

= =

v v

r 0 v vr is the

velocity of A relative to B, even the observer is rotating.

1.2 Newton’s Second Law

Ch. 1: Introduction of Mechanical Vibrations Modeling

Relative AccelerationDifferentiating the relative velocity equation results inthe relative acceleration equation:

A B A/B A/B rel

A/B rel A/B

rel rel rel

xy XY xy

Recall the vector differentiation relation,

d d d, dt dt dt

angular acceleration observed in fixed frame t

= + × + × +

= + ×

= + ×

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + × ∴ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

→ =

a a r r v

r v rv a v

ω ω

ωω

ω ω ωω ω ω

( )( )

A/B rel rel

rel rel

A B A/B A/B rel rel

hat observed in the rotating frameNote: x y x y x y

and velocity and acceleration seen by the rotating observer

2

= = =

=

∴ = + × + × × + × +

r i + j v i + j a i + jv a

a a r r v aω ω ω ω

ω

V

1.2 Newton’s Second Law

Ch. 1: Introduction of Mechanical Vibrations Modeling

Newton formula for translational motion of the rigid body

1.2 Newton’s Second Law

G

G

the resultant of the external forces forces applied to the rigid body

total mass the acceleration of the center of mass G

F ma

F

ma

=

=

===

FBD

Ch. 1: Introduction of Mechanical Vibrations Modeling

Euler formula for planar rotational motion of the rigid body

1.2 Newton’s Second Law

G G

G

G

moment of the external forces about CM (G) in directioncentroidal mass moment of inertia about axis through CM (G)

rate of change of the angular velocity

z

z

M I

M zI z

ω

ω

=

= −

= −

=

FBD

Ch. 1: Introduction of Mechanical Vibrations Modeling

Pure rotation stationary point in the body

1.2 Newton’s Second Law

O O

G

O

2O G

moment of the external forces about O in directionmass moment of inertia about axis through axis of rotation (O)

rate of change of the angular velocity

z

z

M I

M zI z

I I md

ω

ω

=

= −

= −

=

= +

With this formula, the constraint force is eliminated

FBD

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 5 Simple 1 DOF m-k system

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 6 Simple 2 DOF m-k system

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 7 Simple 1 DOF pendulum

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 8 Compound 2 DOF pendulum

A uniform rigid bar of total mass m and length L2,suspended at point O by a string of length L1, is actedupon by the horizontal force F. Use the angulardisplacement θ1 and θ2 to define the position, velocity,and acceleration of the mass center C in terms ofbody axes and then derive the EOM for the translationof C and the rotation about C.

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

2 222 2 2 1 1 1 2 1 1 1 2 1 2

2 22 2 2 1 1 1 2 1 1 1 2 1 2

22 2 2

2 2 1 2

s c c s c2

c s s c s2

c s2 2 12

eliminate the constraint fo

LF mg T m L L

LF mg T m L L

L L mLF T

θ θ θ θ θ θ θ θ

θ θ θ θ θ θ θ θ

θ θ θ

− − −

− − −

⎡ ⎤+ − = − −⎢ ⎥⎣ ⎦⎡ ⎤− + = + +⎢ ⎥⎣ ⎦

− =

− − − − − − − − − − − − − − − − − −− − − − − − − − − − − − − − −rce T

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 9 Washing machine

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 9 Washing machine

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

st

st

measured from unstrained spring position

measured from static equilibrium positionstatic displacement equilibrium /

0 Weight is balanced at all times by sp

yMy cy ky Mgx

Mg ky xMx cx kx

δδ

+ + = −

=

= −

+ + = st

2

ring force

Imbalance in the washersin

k

Mx cx kx F me t

δ

ω ω+ + = =

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 10 Pitch & Vertical motion simple model of a car

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )

sf b f sf b f

sr b r sr b r b b

sf b f sf b f

sr b r sr b r C

sf b f sf b f f f f f

sr b r sr b r

F k x a x c x a x

k x b x c x b x m x

Fc k x a x c x a x a

k x b x c x b x b I

k x a x c x a x k x m x

k x b x c x b x k

θ θ

θ θ

θ θ

θ θ θ

θ θ

θ θ

− − − − − −

− + − − + − =

⎡ ⎤+ − − + − −⎣ ⎦⎡ ⎤− + − + + − =⎣ ⎦

− − + − − − =

+ − + + − −

2 2

0 0 00 0 0

00 0 000 0 0

r r r r

sf sr sf sr sf srb b b

sf sr sf sr sf srC

sf sf sff f

sr sr srr r

x m x

c c c a c b c cm x xc a c b c a c b c a c bI

c c a cm xc c b cm x

θ θ

=

− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −

+ − + − −⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥ − + + −⎢ ⎥⎢ ⎥ ⎢ ⎥ +⎢ ⎥⎢ ⎥ ⎢ ⎥ −⎢ ⎥⎢ ⎥ ⎢ ⎥ − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

2 2

0 00 0

f

r

sf sr sf sr sf sr b

sf sr sf sr sf sr

sf sf sf f

sr sr sr r

xx

k k k a k b k k x Fk a k b k a k b k a k b Fc

k k a k xk k b k x

θ

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

+ − + − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥− + + −⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − ⎣ ⎦⎣ ⎦⎣ ⎦

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 11 EOM w/ springs in series

A mass m is suspended on a massless beam of uniformflexural rigidity EI. Derive the differential EOM.

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

3

eq 3

3

192equivalent stiffness of the beam at the midspan is

From the arrangement, it is connected in series with the spring 192

192192 0

192

EIL

kkEIk

EI kLkEImx x

EI kL

∴ =+

∴ + =+

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 12 EOM w/ springs in parallel

Devise a lumped model for the n-storey buildingsubjected to a horizontal earthquake excitation.

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1 1i i ix x x+ −> >

1 1 and i i i ix x x x+ −> >

( )1 1i i ik x x+ +−

( )1i i ik x x −−

( )1i i ik x x −−

( )1 1i i ik x x+ + −

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

3

eq 3

Each column acts as clamped-clamped beam12with an end in horizontal deflection .

Two columns for each storey behave like spring in parallel.24

Schematic model is a string of spring/mas

ii

i

EIkH

EIkH

⇒ =

∴ =

( ) ( )

( ) ( )

( ) ( ) ( )

1 1 1 1 1

1 1 1 1 1

t

0 0

s.At each mass

,

or

, and

with the constraint 0

i i i i i i i i i i i

i i i i i i i i i i i i

t

o o

F ma m x k x x k x x x x x

F ma m x k x x k x x x x x x

x t x a t dt

+ + − + −

+ + − + −

⎡ ⎤= = − − − > >⎣ ⎦

⎡ ⎤= = − − − − > >⎣ ⎦

= +

∫ ∫

1.3 Equations of Motion

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

1.4 Equilibrium and Linearization

Linear Time Invariant (LTI) system is convenient to understand and analyzed. It can be described as thesystem of ordinary differential equations with constantcoefficients of the form:

( ) ( ) ( )1 2

0 1 2 11 2

m m

i i mim i m im m

d x d x d x dxa a a a a x f tdt dt dt dt

− −−+ + + + + =…

Very often, the EOM is nonlinear which is difficult tomanipulate. Linearization makes the system becomeLTI using the Taylor’s series expansion around aninterested point. The equilibrium point is commonlychosen.

Ch. 1: Introduction of Mechanical Vibrations Modeling

0.x x= = =…

1.4 Equilibrium

How to find the equilibrium point?

The solution does not change if the system is at theequilibrium. Let that point be and at that point

Substitute into EOM and solve forex x=

.ex

How to linearize the model?

Apply the Taylor’s series expansion to any nonlinear expressions around Assume small motion, whichallows one to ignore the nonlinear terms in the series.In other words, only the constant and linear terms areremained.

.ex

( ) ( ) ( ) ( ) ( ) ( )2

22

12!

e e

e e ex x x x

d df x f x f x x x f x x xdx dx= =

= + − + − +…

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 13 Inverted pendulum

Determine the EOM and linearize it.

1.4 Equilibrium

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

2

2

e

2

22

22 2

Find the equilibrium position

20, 0 or 2

Linearize about 0 for which

2 02 2 2

2

O O

e

e e e e e

e

l lM I k s c mgls ml

kl mgs c mgls s ckl

l l klk s c mgls mgl

klml

α θ θ θ θ

θ θ

θ θ θ θ θ

θ θ θ φ

θ θ θ φ

φ

⎡ ⎤= − × + =⎣ ⎦

=

− + = = =

= = +

⎛ ⎞− × + = + − +⎜ ⎟

⎝ ⎠

∴ + −

2 2

e 2 2

2 2 2

2 2 22 1

0, measured from 0

2 4Linearize about such that , 1 for which

62 02 2 2

6 0, measured from cos2

e e e

mgl

mg m gc skl k l

l l kl m gk s c mglsk

kl m gmlk

φ φ θ

θ θ θ θ θ φ

θ θ θ φ

φ φ φ θ −

⎛ ⎞= =⎜ ⎟

⎝ ⎠

= = − = +

⎛ ⎞− × + = + − +⎜ ⎟

⎝ ⎠⎛ ⎞

∴ + − = =⎜ ⎟⎝ ⎠

2mgkl

⎛ ⎞⎜ ⎟⎝ ⎠

mg2k(l/2)sθ

(l/2)cθ

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

2

22

22 2

02

O Ol lM I k mgl ml

klml mgl

α θ θ θ

θ θ

⎡ ⎤= − × + =⎣ ⎦

⎛ ⎞∴ + − =⎜ ⎟

⎝ ⎠

mg2k(l/2)θ

(l/2)

small angle approx.

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 14

Determine the EOM and linearize it.

1.4 Equilibrium

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

22 1 1 O OM I mgl kl l mlα θ θ θ⎡ ⎤= − − × =⎣ ⎦∑

mg

kl1θ

small angle approx.

Ch. 1: Introduction of Mechanical Vibrations Modeling

Ex. 15

Determine the EOM and linearize it.

1.4 Equilibrium

equilibrium position

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

( )Let the equilibrium position the spring deforms by .

0 0 _____________________ 1

Let be the displacement measured from the equilibrium,corresponding to the spring being deflected b

eq

y eqF mg k c

y

δ

δ α⎡ ⎤= − =⎣ ⎦∑

( ) ( ) ( )

( ) ( )

( ) ( )

22 2 2

2

y angle .

____ 2

From the geometry, is the difference of the spring lengthbetween two postures.

tan 2

From the geometry,

y eqF my mg cy k c my

h h hh h y hy yc c c

h h ys sc c

θ

δ α θ

αα α α

α α θα α θ

⎡ ⎤= − − Δ + − =⎣ ⎦Δ

Δ = + + − = + + −

+= −

( )

( )22 2

22

tantan

1 1 tan

2

hh y

h yc ch hy y

c

αα θ

β α θβ

α

⇒ − =+

⎡ ⎤ += − =⎢ ⎥+⎣ ⎦

+ +

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

( ) ( )

( )

22

2 22

2

2 22 2

2 2

I II

Subs. the findings into 2 and recognize 1 :

22

EOM: 2 2

Linearization

eqh h h ymg cy k hy y my

c c h hy yc

kh h y mg h ymy cy k h y mgc ch hhy y hy y

c c

δα α

α

α αα α

⎛ ⎞ +− − + + − + =⎜ ⎟⎜ ⎟

⎝ ⎠ + +

+ ++ + + − = −

+ + + +

( )( )

( ) ( )

22

2 22

2 22

22

2

0

2

about equilibrium 0

2 22

2 2I 0

2

Iy

y

h y h yh hy yc h hy ykh kh ckh k y O yh hcc hy yc c

kc y

αα

ααα α

α=

=

⎧ ⎫+ ++ + −⎪ ⎪

⎪ ⎪+ +⎪ ⎪⎪ ⎪= − + − − +⎨ ⎬⎪ ⎪+ +⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

≈ ⋅

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium

( )( )

( ) ( )

22

2 22

22

22

2

0

2

2 2

2 22

2 2II 0

2

II

Subs. I and II into EOM,

0

If , i.e. is very small, which r

y

h y h yh hy yc h hy ymgh mg cmg y O yh hcc hy yc c

mg s yh

mgmy cy kc s yh

αα

ααα α

α

α α

α θ α θ

=

⎧ ⎫+ ++ + −⎪ ⎪

⎪ ⎪+ +⎪ ⎪⎪ ⎪= − − − +⎨ ⎬⎪ ⎪+ +⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

≈ ⋅

⎛ ⎞+ + − =⎜ ⎟⎝ ⎠

− ≈ ( )( )( )

2

equires small enough , then 2 becomes

Make use of 1 , it becomes 0Subs. and linearize about 0 to get

0Note that the linearized which makes se

eqmg cy k c my

my cy k cy

my cy kc yyc

α

δ α

α

αα

− − Δ + =

+ + Δ =

Δ =

+ + ⋅ =Δ = nse from the figure

by projecting onto the line of spring.y

Ch. 1: Introduction of Mechanical Vibrations Modeling

1.4 Equilibrium


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