Vibration Under General Forcing Conditions

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When the external force F(t) is periodic withperiod τ= 2π/ω, it can be expanded in a

Fourier series

t jbt jaa

t F j

i

j

i sincos2 11

0

2,...1,0,,cos2

0 jtdt jt F ai

2,...1,,sin2

0 jtdt jt F bi



The equation of motion of the system can beexpressed as

Using the superposition principle, the steady statesolution is the sum of the steady state solutions of

t jbt jaakx xc xm j

i

j

i sincos2 11

0

2

0akx xc xm

t jakx xc xm j cos

t jbkx xc xm j sin



The solutions;

k

at x p

2

0

j

j

p t j

jr r j

k at x

cos

21

/

2222

j j

p t j

jr r j

k bt x

sin

21

/ 2222

22

1

1

2tan

r j

jr j



The complete steady state solution;

1

2222

12222

0

sin

21

/

cos21

/

2

j

j

j

j

j

j

p

t j

jr r j

k b

t j jr r j

k a

k

at x

22

1

1

2tan

r j

jr j



In the study of vibrations of valves used inhydraulic control systems, the valve and itselastic stem are modeled as a damped spring-

mass system, as shown in the next figure. Inaddition to the spring force and damping force,there is a fluid pressure force on the valve thatchanges with the amount of opening or closingof the valve. Find the steady-state response of the valve when the pressure in the chambervaries as indicated. Assume k=2500 N/m, c=10 N-s/m, and m=0.25 k.g



In some cases, the force acting on a systemmay be quite irregular and may be only

determined only experimentally.



Using numerical integration procedure (trapezoidal rule)

N

i

iF N

a1

0

2...3,2,1 ,

2cos

2

1

jt j

F N

a i N

i

i j

...3,2,1 ,2

sin2

1

jt j

F N

b i N

i

i j

12222

12222

0

sin

21

/

cos21

/

2

j

j

j

j

j

j

p

t j

jr r j

k b

t j jr r j

k a

k

at x



Some of the methods that can be used to findthe response of a system to an arbitraryexcitation are; Representing the excitation by a Fourier integral

Method of convolution integral

Method of Laplace Transform

First approximating F(t) by a suitableinterpolation model and then using a numericalprocedure

Numerically integrating the equation of motion



A nonperiodic exciting force usually has amagnitude that varies with time; it acts for a

specified period of time and then stop. The simple form is the impulsive force

▪ A force that has a large magnitude F and acts for a veryshort period of timeΔt

▪ From dynamics:

12Impulse xm xmt F



t t

t Fdt F

~

By designating the magnitude of the impulse;

A unit impulse (f) is defined as

1lim~

0

Fdt Fdt f

t t

t t



Response to an Impulse

We first consider the response of SDOF system to

an impulse excitation



For an underdamped system, the solution of theequation of motion

is 0

kx xc xm

t

x xt xet x

d

d

nd

t n

sincos 00

0

nm

c

2

2

2

21

m

c

m

k nd

m

k n



From the impulse-momentum,

and the initial conditions,

01~

impulse xm f

t m

et gt x

d d

t n

sin

m x

10 00 x

0for t0 x x

Impulse Response Function



If the magnitude is F instead of 1, the initial velocity is F/mand the response

t Fgt mFet x d

d

t n

sin

t Fgt x

If the impulse F is applied at

arbitrary time t =τ



Response to a General Forcing Conditions The arbitrary external force may be assumed to be made

up of a series of impulse of varying magnitude

t gF t x

t gF t x

d t gF t x

t

0

d t eF

mt x d

t t

d

n

sin1

0



The Laplace transform method can be used to find theresponse of a system under any type of excitation,including the harmonic and periodic types. This

method can be used for the efficient solution of lineardifferential equations, particularly those withconstant coefficients. It permits the conversion of thedifferential equations into algebraic ones, which areeasier to manipulate. The major advantages of themethod are that it can treat discontinuous functionswithout any particular difficulty and it automaticallytakes into account the initial conditions.



The Laplace Transform of a function x(t) isdefined as,

The integration is with respect to t, thetransformation gives a function of s

Steps;▪ Write the equation of motion of the system.

▪ Transform each term of the equation, using knowninitial conditions.

dt t xet xs x

st

0 L



Steps;

▪ Solve for the transformed response of the system

▪ Obtain the desired solution by using inverse Laplace

transformation.

To solve the forced vibration equation,

t dt

dxt x

t F kx xc xm

t dt

xd t x

2

2



00

xs xsdt t dt

dxet

dt

dx st

L

00

2

2

2

02

2

xsxs xsdt t dt

xd

et dt

xd st

L

dt t F et F sF st

0

L

t F t xk t xct xm L L L L

0)0(2 xcms xmsF s xk csms



Ignoring the homogeneous solution of the differentialequation;

k csmss x

sF s Z

2

0000 x x

222

2

11

nnssmk csmssF

s xsY

sF sY s x sF sY s xt x 11 L L

Transfer Function

Inverse Laplace transform



Considering the generals solution of the differentialequation; 00 00 x x x x

022022222

1

2

2

2 xss xss

s

ssm

sF s x

nnnn

n

nn

d t eF m

t e xt e xt x

d

t t

d

d t

d

d t

n

nn

sin1

sinsin1

0

012 / 12

0

1L

Vibration Under General Forcing Conditions

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