2.0 Outline Free Response of Undamped System Free...

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Ch. 2: Free Vibration of 1-DOF System 2.0 Outline Free Response of Undamped System Free Response of Damped System Natural Frequency, Damping Ratio 2.0 Outline

Transcript of 2.0 Outline Free Response of Undamped System Free...

Ch. 2: Free Vibration of 1-DOF System

2.0 Outline

Free Response of Undamped SystemFree Response of Damped SystemNatural Frequency, Damping Ratio

2.0 Outline

Ch. 2: Free Vibration of 1-DOF System

2.1 Free Response of Undamped System

Free vibration is the vibration of a system in responseto initial excitations, consisting of initial displacements/velocities. To obtain the free response, we must solvesystem of homogeneous ODEs, i.e. ones with zeroapplied forces. The standard form of MBK EOM is

2.1 Free Response of Undamped System

( )0, mx cx kx x x t+ + = =

Ch. 2: Free Vibration of 1-DOF System

If the system is undamped, c = 0. The EOM becomes

2.1 Free Response of Undamped System

( ) ( )

2

0 0

0 0, /subject to the initial conditions 0 , 0

nmx kx x x k mx x x v

ω ω+ = ⇒ + = =

= =

The solutions of homogeneous ODE are in the form( ) , is the amplitude and is constantstx t Ae A s=

Subs. the solution into ODE, we get

( )

( )

2 2 2 2

12

1 2

0 0 ** characteristic equation **

** characteristic roots, eigenvalues **

general solution: by superpositionn n

stn n

ni t i t

Ae s s

s i

x t A e A eω ω

ω ω

ω−

+ = ⇒ + =

= ±

= +

Ch. 2: Free Vibration of 1-DOF System

We can now apply the given i.c. to solve for A1 and A2.However we will use some facts to arrange the solutionsinto a more appealing form.

2.1 Free Response of Undamped System

( )

( ) ( ) ( ) ( )

2 1

1 2

Because is real, .

Let . Therefore .2 2

cos2

n n

i i

i t i tn

x t A AC CA e A e

Cx t e e C t

φ φ

ω φ ω φ ω φ

− − −

=

= =

⎡ ⎤∴ = + = −⎣ ⎦

The given i.c. are then used to solve for the amplitude Cand the phase angle Φ. Note ωn, known as natural frequency, is the system parameter. The system is called harmonic oscillator because of its response to i.c.is the oscillation at harmonic frequency forever.

Ch. 2: Free Vibration of 1-DOF System

2.1 Free Response of Undamped System

( ) ( )

( )

0 0

22 10 0

0 0 00

00

If the initial conditions are 0 and 0

cos and sin and tan

cos sin as the function of i.c. and system parameter

2 / , 1/

nn n

n nn

n

x x x v

v vx C v C C xx

vx t x t t

T f T

φ ω φ φω ω

ω ωω

π ω

= =

⎛ ⎞∴ = = ⇒ = + =⎜ ⎟

⎝ ⎠

∴ = +

= =

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

We normalize the standard MBK EOM by mass m:

2.2 Free Response of Damped System

( )( ) ( )

2

0 0

2 0

/ natural frequency/ 2 viscous damping factor

subject to the initial conditions 0 and 0

n n

n

n

x x x

k mc m

x x x v

ζω ω

ωζ ω

+ + =

= =

= =

= =

The solutions of homogeneous ODE are in the form( ) , is the amplitude and is constantstx t Ae A s=

Subs. the solution into ODE, we get

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

( )

( ) 1 2

2 2 2 2

212

1 2

2 0 2 0 **CHE**

1 ** characteristic roots, eigenvalue **

general solution: by superposition

stn n n n

n n

s t s t

Ae s s s s

s

x t A e A e

ζω ω ζω ω

ζω ω ζ

+ + = ⇒ + + =

= − ± −

= +

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

( ) ( )

( ) ( )

( ) ( )

0 0

12

212

Response of 0, i.c. 0 and 0i) 0 : cos harmonic oscillation with frequency

ii) 0 1: where 1

cos expon

n

n

n

n

n d d n

td

mx cx kx x x x vs i

x t C t

s i

x t Ce tζω

ζ ωω φ

ω

ζ ζω ω ω ω ζ

ω φ−

+ + = = =

= = ±

= −

< < = − ± = −

= −

entially decaying amplitude oscillation with damped frequency and envelope nt

d Ce ζωω −

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

( ) ( )

( ) ( )2 2

0 0

212

1 11 2

Response of 0, i.c. 0 and 0

iii) 1: 1

aperiodic decay with peak more suppressed and decay further slow down as inc

n nn

n n

t tt

mx cx kx x x x v

s

x t e A e A eω ζ ω ζζω

ζ ζω ω ζ

ζ

− − −−

+ + = = =

> = − ± −

= +

rease

( ) ( )12

1 2

iv) 1:

aperiodic decay with highest peak and fastest decay

n

nt

s

x t A A t e ω

ζ ω−

= = −

= +

Ch. 2: Free Vibration of 1-DOF System

2.2 Free Response of Damped System

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

2.3 Natural Frequency, Damping Ratio

The response will partly be dictated by the roots s12, which depend on Root locus diagram givesa complete picture of the manner in which s12 changewith The focus will be the left half s-planewhere the system response is stable.

and .nζ ω

and .nζ ω

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

( )

( )

12

212

constant, vary i) 0 undamped , on the Im-axis far from origin by . The motion is with natural frequency .

ii) 0 1 underdamped , 1 pa

n

n

n

n

n n

s i

harmonic oscillation

s i

ω ζζ ω

ωω

ζ ζω ω ζ

= = ±

< < = − ± −

( ) 12

ir of symmetric points moving on a semicircle of radius . The motion is .iii) 1 critically damped , repeated roots. The motion is .

n

n

oscillatory decays

aperiodic decay

ωζ ω= = −

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

( ) 212iv) 1 overdamped , 1

two negative real roots going to 0 and . The motion is .

constant, vary The symmetric roots will be far from originalong the radius

n n

n

s

aperiodic decay

ζ ζω ω ζ

ζ ω

ω

> = − ± −

−∞

( )1making an angle cos with - axis.n xζ−

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Determination of M-B-K• Mass – directly measure the weight or deduced from

frequency of oscillation• Spring – from measures of the force and deflection

or deduced from frequency of oscillation• Damper – deduced from the decrementing response

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ex. 1 A given system of unknown mass m and spring kwas observed to oscillate harmonically in freevibration with Tn = 2πx10-2 s. When a mass M =0.9 kg was added to the system, the new periodrose to 2.5πx10-2 s. Determine the systemparameters m and k.

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ex. 2 A connecting rod of mass m = 3x10-3 kg and IC =0.432x10-4 kgm2 is suspended on a knife edgeabout the upper inner surface of a wrist-pinbearing, as shown in the figure. When disturbedslightly, the rod was observed to oscillateharmonically with ωn = 6 rad/s. Determine thedistance h between the support and the C.M.

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

3 4 2

22

3 10 kg, 0.432 10 kgm , 6 rad/s

0.2, 0.072 m

Radius of gyration 0.12 m must be greater than

the longest length of the object. 0.072 m

C n

nC

C

m Imgh h

I mh

Im

h

ω

ω

− −= × = × =

= ⇒ =+

= =

∴ =

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ex. 3 A disk of mass m and radius R rolls w/o slipwhile restrained by a dashpot with coefficient ofviscous damping c in parallel with a spring ofstiffness k. Derive the differential equation forthe displacement x(t) of the disk mass center Cand determine the viscous damping factor ζand the frequency ωn of undamped oscillation.

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

FN

mgkx+cv

2

2

0, 2

2 , 36

C C C

x

CC

n

xM I FR IR

F mx F kx cx mx

I Rm x cx kx I mR

c kmkm

θ

ζ ω

⎡ ⎤= − =⎣ ⎦

⎡ ⎤= − − =⎣ ⎦

⎛ ⎞+ + + = =⎜ ⎟⎝ ⎠

= =

∑∑

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ex. 4 Calculate the frequency of damped oscillationof the system for the values m = 1750 kg,c = 3500 Ns/m, k = 7x105 N/m, a = 1.25 m, andb = 2.5 m. Determine the value of critical damping.

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

( )2

2

2

0 weaken spring

10 rad/s 0.12

1 9.95 rad/s2 35,000 Ns/m

nn

d n

cr n

amy cy kyb

a k cb m m

c m

ω ζω

ω ω ζω

+ + =

= = = =

= − =

= =

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ex. 5 A projectile of mass m = 10 kg traveling withv = 50 m/s strikes and becomes embedded in amassless board supported by a spring stiffnessk = 6.4x104 N/m in parallel with a dashpot ofc = 400 Ns/m. Determine the time required forthe board to reach the max displacement andthe value of max displacement.

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

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

( )

2

80 rad/s 0.25 1 underdamped2

cos

cos sin

0 cos 0, / 2

0 50 1 , 0.6455

max displacement when 0 tan , 0.017 s

0.01

n

n n

nn

td

t tn d d d

n

dd

n

k cm m

x t Ce t

x t Ce t Ce t

x C

x C C

x t t t

x

ζω

ζω ζω

ω ζω

ω φ

ζω ω φ ω ω φ

φ φ π

ω ζωωζω

− −

= = = = < ∴

= −

= − − − −

= = =

= = − =

= ⇒ = =

( )7 0.4447 m=

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio

Ch. 2: Free Vibration of 1-DOF System

2.3 Natural Frequency, Damping Ratio