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. 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 ⇒ =
= = 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
1.2 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
Ex. 5 Simple 1 DOF m-k system
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
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
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
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