When the external force F(t) is periodic with period = 2/, it can be expanded in a Fourier series

tjbtjaa

tFj

i

j

i sincos2 11

0

2,... 1, 0,,cos2

0 jtdtjtFai

2,... 1, ,sin2

0 jtdtjtFbi

The equation of motion of the system can be expressed as

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

tjbtjaa

kxxcxmj

i

j

i sincos2 11

0

2

0akxxcxm

tjakxxcxm j cos

tjbkxxcxm j sin

The solutions;

k

atxp

2

0

j

j

p tj

jrrj

katx

cos

21

/

2222

j

j

p tj

jrrj

kbtx

sin

21

/

2222

22

1

1

2tan

rj

jrj

The complete steady state solution;

12222

12222

0

sin

21

/

cos

21

/

2

j

j

j

j

j

j

p

tj

jrrj

kb

tj

jrrj

ka

k

atx

22

1

1

2tan

rj

jrj

In the study of vibrations of valves used in hydraulic control systems, the valve and its elastic stem are modeled as a damped spring-mass system, as shown in the next figure. In addition to the spring force and damping force, there is a fluid pressure force on the valve that changes with the amount of opening or closing of the valve. Find the steady-state response of the valve when the pressure in the chamber varies 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 system may be quite irregular and may be only determined only experimentally.

Using numerical integration procedure (trapezoidal rule)

N

i

iFN

a1

0

2...3,2,1 ,

2cos

2

1

jtj

FN

a iN

i

ij

...3,2,1 ,2

sin2

1

jtj

FN

b iN

i

ij

12222

12222

0

sin

21

/

cos

21

/

2

j

j

j

j

j

j

p

tj

jrrj

kb

tj

jrrj

ka

k

atx

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

Method of convolution integral

Method of Laplace Transform

First approximating F(t) by a suitable interpolation model and then using a numerical procedure

Numerically integrating the equation of motion

A nonperiodic exciting force usually has a magnitude 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 very short period of time t

From dynamics:

12 Impulse xmxmtF

tt

tFdtF

~

By designating the magnitude of the impulse;

A unit impulse (f) is defined as

1lim~

0

FdtFdtf

tt

tt

Response to an Impulse

We first consider the response of SDOF system to an impulse excitation

For an underdamped system, the solution of the equation of motion

is

0 kxxcxm

txx

txetx dd

nd

tn

sincos 000

nm

c

2

2

2

21

m

c

m

knd

m

kn

From the impulse-momentum,

and the initial conditions,

01~

impulse xmf

tm

etgtx d

d

tn

sin

mx

10 00 x

0for t 0 xx

Impulse Response Function

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

tFgtm

Fetx d

d

tn

sin

tFgtx

If the impulse F is applied at arbitrary time t =

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

tgFtx

tgFtx

dtgFtxt

0

dteF

mtx d

tt

d

n sin1

0

The Laplace transform method can be used to find the response of a system under any type of excitation, including the harmonic and periodic types. This method can be used for the efficient solution of linear differential equations, particularly those with constant coefficients. It permits the conversion of the differential equations into algebraic ones, which are easier to manipulate. The major advantages of the method are that it can treat discontinuous functions without any particular difficulty and it automatically takes into account the initial conditions.

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

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

Steps; Write the equation of motion of the system.

Transform each term of the equation, using known initial conditions.

dttxetxsx 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,

tdt

dxtx

tFkxxcxm

tdt

xdtx

2

2

00

xsxsdttdt

dxet

dt

dx st

L

0022

2

02

2

xsxsxsdttdt

xdet

dt

xd st

L

dttFetFsF st

0

L

tFtxktxctxm LLLL

0)0(2 xcmsxmsFsxkcsms

Ignoring the homogeneous solution of the differential equation;

kcsmssx

sFsZ 2

0000 xx

222 2

11

nnssmkcsmssF

sxsY

sFsYsx sFsYsxtx 11 LL

Transfer Function

Inverse Laplace transform

Considering the generals solution of the differential equation; 00 00 xxxx

02202222 21

2

2

2x

ssx

ss

s

ssm

sFsx

nnnn

n

nn

dteFm

tex

tex

tx

d

tt

d

d

t

d

d

t

n

nn

sin1

sinsin1

0

012/12

0

1L