Mechanical Vibration

Iva Petríková

Department of Applied MechanicsDepartment of Applied Mechanics

Introduction, basic definitions

• Oscillatory process – alternate increases or decreases of physical

quantities (displacements, velocities, accelerations)

• Oscillatory motion is periodic motion• Oscillatory motion is periodic motion

• Vibration of turbine blades, vibration of machine tools, electrical

oscillation, sound waves, vibration of engines, torsional vibration

of shafts, vibration of automobiles etc.

• Mechanical vibration - mechanisms and machines, buildings,

bridges, vehicles, aircrafts – cause mechanical failure

• Harmonic, periodic general motion

1.3.2018 2

Elastic elements

• Discrete elements (masses) +

linear and torsional springslinear and torsional springs

• Continuos structural elements –

beams and plates

• Number of degrees of freedom

(DOF) – minimum number of

coordinates

Single degree of freedom systems

System with infinite number DOFTwo degree of freedom systems

Oscillatory motions

• Periodic motion with

harmonic components

• Harmonic motion

harmonic components

Periodic motion repeating itself The simplest form of periodic motion isPeriodic motion repeating itself

after a certain time interval.

The simplest form of periodic motion is

harmonic motion – sin, cos

Harmonic motion

• Displacement of harmonic motion is given:

( )sinx X tω ϕ= +

• x, x(t) ... displacement [m]

• X ... amplitude of displacement [m]

• ... phase

• ω ... angular velocity [s-1]

• T ... natural period of oscillation [s]

( )tω ϕ+

2T

πω

=• T ... natural period of oscillation [s]

• f ... frequency [s-1], [Hz] - Hertz

• ϕ ... phase angle

Tω

=

2 fω π=

• velocity:

• ωX ... amplitude of velocity [ms-1]

( )cosdx

x X tdt

ω ω ϕ= = +ɺ

• ωX ... amplitude of velocity [ms ]

• acceleration:

( )2

22

sind x

x X tdt

ω ω ϕ= = − +ɺɺ

• -ω2X ... amplitude of acceleration [ms-2]

Simple degree of freedom systems

mass, spring, damper, harmonic excitation

( )mx bx kx F t+ + =ɺɺ ɺForcing function –

harmonic excitation

0mx bx kx+ + =ɺɺ ɺDamped free vibration

0mx kx+ =ɺɺUndamped free vibration

Simple degree of freedom systems

( )mx bx kx F t+ + =ɺɺ ɺ

0( ) sinF t F tω=

m ... mass

b ... (viscous) damping coefficient

k ... stiffness coefficient

F0 ... amlitudes of force

ω ... frequency of harmonic force

bcr ... critical damping coefficient

ζ ... damping factor

k... natural (circular) frequency

... frequency ratio

k

mΩ =

ωη =Ω

Undamped free vibration0+ =ɺɺmx kx

Solution of the 2nd order differential equation

Assumed solution

Characteristic equation: ( ) Ω − Ω= +i t i tx t Ae Be free vibration

( ) tx t Aeλ=

Characteristic equation: 2 0λ + =m k

1,2

1,2

λ

λ

= ±

= ± Ω

ki

mi

2 0λ + =k

m2λ = − k

m

Two arbitrary constants A a B, determinedfrom initial conditions:

( ) ( )0 00 , 0= =ɺx x x v

( ) ( )i t i tx t i Ae BeΩ − Ω= Ω −ɺ

0x A B= +

( )0v i A B= Ω −1,2

Ω = k

mNatural frequencyof the single DOF systems

0k

x xm

+ =ɺɺ2 0x x+ Ω =ɺɺ

0 01

2

ix vA

i

Ω + = Ω

0 01

2

ix vB

i

Ω − = Ω

Damped free vibration0mx bx kx+ + =ɺɺ ɺ

Solution of linear differential equation of 2nd order:

2 0m b kλ λ+ + = ( ) ( ) ( )2 21 1ζ ζ ζ ζ− + − Ω − − − Ω= +

t tx t Ae Be

21,2

14

2 2

bb mk

m mλ −= ± −

2

2

1,2 2

41

2 2 242

b mk b b k bi i

m m m mm km

m

λ

− − = ± − = ± −

2

21,2 1 1

2 22

b b bi i

m mkmλ ζ− − = ± Ω − = ± Ω −

2 CR

b b

bkmζ= = damping factor

=

1

1

1

ζζζ

><=

overdamped system

underdamped system

critically damped system

0

0.02

0.04

x t( )

x1 t( )2=CRb km critical damping coefficient

21 ζΩ − = Ω Ddamped natural frequency

( )21,2 1λ ζ ζ= − ± − Ω

0 5 10

0.04−

0.02−

0x1 t( )

t

Undamped --- and damped free vibration ---

Damped free vibrationOverdamped system: displacement becomes the sum of

two decaying exponentials with initial value of A+B, no

vibration takes place, the body tends to creep back to the

equilibrium position – APERIODIC MOTION (Fig.1)

1ζ >

Underdamped system: displacement is oscillatory with

diminishing amplitude (Fig.2). Frequency of oscillation is

less than that of the undamped case by the factor .

( ) ( )2 21 1t tx Ae Be

ζ ζ ζ ζ− + − Ω − − − Ω= +

21 ζ− Fig. 1

( )

( )

2 21 11 2

cos sin

[ ]t

i t i tt

D De A t B t

x e C e C eζ

ζ ζζ

− Ω

− Ω − − Ω− Ω

Ω + Ω =

= + =

=

1ζ <

1ζ >

Critical damping:

[ ] −Ω= + tx A Bt eObr. 2Fig. 2

( )

( )2sin 1t

D D

Ce tζ ζ γ− Ω= − Ω +

1ζ =

Damped free vibration – Logarithmic decrement

Natural logarithm of the ratio of any two amplitudes

( )( )

( )( ) ( )

1

2

cos sinln ln ln

cos sin

1 2 2ln

tD D

t TD D

x t e A t B tx

x x t T e A t B t

T

ζ

ζδ

π πζζ ζ

− Ω

− Ω +

Ω + Ω= = = =

+ Ω + Ω

= = Ω = Ω =2

ln1

2

TT

Te ζ ζ ζ

ζδ πζ

− Ω= = Ω = Ω =Ω −

≐ for CRb b<<

( )( )

ln 22

x tn damping factor

x t nT n

δδ π ζ ζπ

= =+

≐

CR

b

b

Forced vibration, harmonic excitation

0

( )

sin

mx bx kx F t

mx bx kx F tω+ + =+ + =ɺɺ ɺ

ɺɺ ɺ

1. Forced undamped vibration1. Forced undamped vibration

- homogenous solution of equation

- particular solution of equation

Solution of dif. equation with right side → steady state oscillation (response)

Assumed solution in the form of harmonic function:

dif. equation:

0+ =ɺɺmx kx

( )1 2

2 21 2

sin cos

sin cos

ω ω

ω ω ω ω

= +

= − = − +ɺɺ

p

p p

x a t a t

x x a t a t

0 sin ω+ =ɺɺmx kx F t

( ) ( )2 2sin cos sinω ω ω ω ω− + − =k m a t k m a t F tdif. equation:

Comparing of coeficients at function sin a cos on the both side of the equation →

amplitude a1

( ) ( )1 2 0sin cos sinω ω ω ω ω− + − =k m a t k m a t F t

01 22

, 0ω

= =−

Fa a

k m

Particular solution:

Forced vibration, harmonic excitation0

2sin ω

ω=

−p

Fx t

k m

0 0 1 1= = = ST

F Fa a

0

10

20

xp t( )

resonance – frequency of exciting force equels

0=ST

Fa

k

2 22 11ω ηω

= = =− −−

STa amkk mk

2

1

1 η=

−ST

a

a

0 0.5 1 1.5 220−

10−

t

Vibration of the single DOF system in resonance.

resonance – frequency of exciting force equels

to natural frquency of the system

1η ω= = Ω → ∞ST

a

aResonance curve – undamped system

Forced vibration

( )0

( )

sin 1

mx bx kx F t

mx bx kx F tω+ + =+ + =ɺɺ ɺ

ɺɺ ɺharmonic force

1. homogenous solution0mx bx kx+ + =ɺɺ ɺ1. homogenous solution

2. → steady-state solution (response)

0mx bx kx+ + =ɺɺ ɺ

( )sin ω ϕ= −x a t

( ) ( )0 0

2 2 22

1F Fa

k= =

0 sin ω+ + =ɺɺ ɺmx bx kx F t

( ) ( ) ( )20sin cos sin sinω ω ϕ ω ω ϕ ω ϕ ω− − + − + − =ma t ba t ka t F t

assumed solution of differential equation (1)

( ) ( )2 2 22221

ak m bk m b

k kω ω ω ω

= = − + − +

2tan

b

k m

ωϕω

=−

Vectors’ diagram

Single degreee of freedom system

Free damped vibration Free undamped vibration Forced damped vibration

Homogenous solution Homogenous solution

( )mx bx kx F t+ + =ɺɺ ɺ0+ + =ɺɺ ɺmx bx kx 0+ =ɺɺmx kx

Homogenous solution Homogenous solution

Initial conditions Initial conditions Amplitude of steady stateoscillation, steady state

response

( )

1 2

cos sin

sin

i t i th

h

h

x C e C e

x A t B t

x C t γ

Ω − Ω= += Ω + Ω= Ω +

( )

( )

2 21 1

2

[ ]

cos sin

sin 1

i t i tth

tD D

t

x e Ae Be

e A t B t

Ce t

ζ ζζ

ζ

ζ ζ γ

− Ω − − Ω− Ω

− Ω

− Ω

= + =

= Ω + Ω =

= − Ω +

( )0 0x t x= ( )0 0x t v=ɺ

( )sinpx a tω ϕ= −

h px x x= +0 sinmx bx kx F tω+ + =ɺɺ ɺ

( )0 0x t x= ( )0 0x t v=ɺ

response

( )( )

( ) ( )

2 21 1

0

2 22

[ ]

sin

i t i ttx t e Ae Be

F t

k m b

ζ ζζ

ω ϕ

ω ω

− Ω − − Ω− Ω= +−

+− +

Forced vibration - magnification factor and phase angle

( ) phx t x x= +

( ) ( )

( ) ( )02

2 22

sinsin 1t F t

x t Ce tk m b

ζ ω ϕζ γ

ω ω− Ω

−= − Ω + +

− +

Solution of equation (1):

a STa

a STa

a STa

a STa

Pha

se A

ngle

φ

( ) ( )0 0

2 2 22 22

1

1

F Fa

k m bk m b

k k

ωω ω ω

= = − + − +

0ST

Fa

k=

Values C a are derived from initial conditions.Amplitudes of steady-state oscillation:

γ

statical deflection of the spring masssystem under the action of steady force F0

η ω=Ω frequency ratio

Ω

ω

Ω

ωΩ

ω

Ω= ωη

Ω= ωη

Ω

ω

Ω

ωΩ

ω

Ω= ωη

Ω= ωη

Pha

se A

ngle

Transient motion, under resonance

( ) ( )2 22

1

1 2ST

a

a η ζη=

− +

Ω

magnification factor

2

2arctan

1

ξηϕη

=− phase angle

Vibration isolation and transmissibilityMachine or engines rigidly attached to a supporting

structure, vibration is transmitted directly to the

support (often undiserable vibration). Disturbing

source must be isolated. Force is trasmitted through

a) ω = const. → η = const. ω >> Ω, η >> 1,

small damping factor ζ

b) ω ≠ const. → isolation, support with

dampingsprings and damper.

damping

ω=η

Pha

se A

ngle

φ

Ω= ωη

Ω

ω=η

Rotating unbalanceRotating unbalance systems (gears, wheels,

shafts disks which are not perfectly uniform,

produce unbalance force which cause

excessive vibrations.

m0 ... unbalance mass

e … eccentricity

m

0

ma

m e

b

0me

m

20

( )

sin

mx bx kx F t

mx bx kx m e tω ω+ + =+ + =

ɺɺ ɺ

ɺɺ ɺ

( ) ( )

2

2 22

01 2

am em

η

η ζη=

− +22

2tan

1

b

k m

ω ξηϕηω

= =−−

ωΩ

Base motion, relative motionForced vibration of mechanical systems can be

caused by the support motion (vehicles,

aircrafts and ships)

harmonic motion 0ω= i ty y eharmonic motion

( ) ( )

20

0

ωω

+ − + − =+ + = −

+ + =

ɺɺ ɺ ɺ

ɺɺ ɺ ɺɺ

ɺɺ ɺ

r r r

i tr r r

mx b x y k x y

mx bx kx my

mx bx kx m y e

m

0

ma

m e

0

solution

( )ω ϕ−= i tp rx a e

20m y

aω=

0

ra

y

( ) ( )

2

2 220 1 2

ra

y

η

η ζη=

− +2

2tan

1

ξηϕη

=−

( ) ( )0

2 22r

m ya

k m b

ω

ω ω=

− +