5.
Vibration under General Forcing Conditions
1. Intr Introd oduc ucti tion on If the excitation is periodic but not harmonic, it can be replaced by a sum of harmonic functions using the harmonic analysis procedure as discussed. By the principle of superposition, the response of the system can then be determined by superposing the responses due to indiidual harmonic forcing functions. !n the other hand, if the system is sub"ected to a suddenly applied nonperiodic force, the response #ill be transient, since steady$state ibrations are not usually produced. %he transient response response of a system can be found using #hat is &no#n as the conolution integral. '. (espon (esponse se )nder )nder a General General *erio *eriodic dic Forc Force e +hen +hen the the exter external nal force force F t - is perio periodi dic c #ith #ith peri period od τ 'π /ω , it can be expanded in a Fourier series0
1#here
, j , 1, ' . . .
'-
and
, j j 1, ' . . . 2 %he e3uation of of motion of the system can be expressed expressed as
4 %he right$hand side of this e3uation is a constant plus a sum of harmonic functions. )sing the principle of superposition, the steady$state solution of 3. 4- is the sum of the steady$state solution of the follo#ing e3uations0
5678oting that the solution of 3. 5- is gien by
1
9and #e can express the solutions of 3s. 6- and 7-, respectiely, as
:-
1#here
11and
1' %hus the complete complete steady$state steady$state solution of 3. 4- is gien gien by
12It can be seen from the solution, 3. 12-, that the amplitude and phase shift corresponding to the j the jth th term depend on j on j.. If j If jω ω n, for any j any j,, the amplitude of the corresponding harmonic #ill be comparatiely large. %his #ill be particularly true true for for smal smalll alu alues es of j and ζ . Further, as " becomes larger, the amplitude becomes smaller and the corresponding terms tend to ;ero. %hus the
transient part of the total solution. xample 8o. 1 > *eriodic Vibration of ?ydraulic Vale In the study of ibrations of ales used in hydraulic control systems, the ale and its elastic stem are modeled as a damped spring$mass systems as sho#n in Fig. 1a-. In additi addition on to the spring spring force force and and dampi damping ng force, force, there there is a @uid @uid pressure force on the ale that changes #ith the amount of opening or closing of the ale. Find the steady$state response of the ale #hen the pressure in
'
the chamber aries as indicated in Fig. 1b-. Assume k '5 8/m, c 1 8$ s/m., and m .'5 &g.
Gien0 ?ydraulic control ale #ith m .'5, & '5 8/m. and c 1 8$s/m and pressure on the ale as gien in Fig. 1b-. (e3uired0 teady$state response of the ale, x -. pt olution0 Find the Fourier series expansion of the force acting on the ale. Add the responses due to indiidual harmonic force components. %he ale can be considered as a mass connected to a spring and a damper on one side and sub"ected to a forcing function F t - on the other side. %he forcing function a#here A is the cross sectional area of the chamber, gien by
band pt - is the pressure acting on the ale at any instant t . ince pt - is periodic #ith period τ ' seconds and A is a constant. F t - is also a periodic function of period τ ' seconds. %he fre3uency of the forcing function is ω 'π/τ - π rad/sec. F t - can be expressed in a Fourier series as0 2
c#here ai and bi are gien by 3s. '- and 2-. ince the function F t - is gien by
dthe Fourier coe=cients ai and bi can be computed #ith the help of 3s '- and 2-0
e-
f-
g-
h-
i-
"-
&Di&e#ise, a4 a6 . . . b4 b5 b6 . . . . By considering only the
l %he steady$state response of the ale to the forcing of 3. m- 0
m8atural fre3uency of the ale0
n4
Forcing fre3uency ω 0
o %hus, fre3uency ratio0
pand damping ratio0
3 %he phase angles φ 1 and φ 3 can be computed as follo#s0
rand
sIn ie# of 3s. b- and n- to s-, the solution can be #ritten as t2. (esponse )nder a *eriodic Force of Irregular Form In some case, the force acting on a system may be 3uite irregular and may be determined only experimentally. xamples of such forces include #ing$and earth3ua&e$induced forces. In such cases, the forces #ill be aailable in graphical form and no analytic expression can be found to describe F t -. ometimes, the alue of F t - may be aailable only at a number of discrete points t 1, t 2, . . ., t N. In all these cases, it is possible to
14-
, j 1,', . . .
15-
5
, j 1,', . . .
16-
!nce the Fourier coe=cients a0, ai, and bi are &no#n, the steady$state response of the system can be found using 3. 12- #ith
4. (esponse )nder 8onperiodic Force %he periodic forces of any general #ae form can be represented by Fourier series as a superposition of harmonic components of arious fre3uencies. %he response of a linear system is then found by superposing the harmonic response to each of the exciting forces. +hen the exciting force F t - is nonperiodic, such as that due to blast from an explosion, a diEerent method of calculating the response is re3uired. Various methods can be used to
x ´
1
and
x ´
denote the elocities of the mass m before and
2
after the application of the impulse, #e hae 6
17By designating the magnitude of the impulse F ∆t by F, #e can #rite, in general,
19A unit impulse f - is de
1:It can be seen that in order for Fdt to hae a
For an underdamped system, the solution of the e3uation of motion 'is gien as follo#s0
'1#here
''7
'2-
'4If the mass is at rest before the unit impulse is applied x
x for t ´
or at t -, #e obtain, from the impulse$momentum relation, '5 %hus the initial conditions are gien by
'6In ie# of 3. '6-, 3. '1- reduces to
'73uation '7- gies the response of a single degree of freedom system to a unit impulse, #hich is also &no#n as the impulse response function, denoted by gt -. %he function gt -, 3. '7-, is sho#n in Fig. 2c-. If the magnitude of the impulse is F instead of unity, the initial elocity
x ´
is F /m and the response of the system becomes
'9If the impulse F is applied at an arbitrary time t τ , as sho#n in Fig. 4a-, it #ill change the elocity at t τ by an amount F /m. Assuming that x until the impulse is applied, the displacement x at any subse3uent time t , caused by a change in the elocity at time t , is gien by 3. '9- #ith t replaced by the time elapsed after the application of the impulse, that is, t > τ .
9
%hus #e obtain ': %his is sho#n in Fig. 4b-.
5.'(esponse to general forcing condition 8o# consider the response of the system under an arbitrary external force Ft-, sho#n in Fig. 5
:
%his force may be assumed to be made up of a series of impulses of arying magnitude. Assuming that at time τ , the force F τ - acts on the system for a short period of time ∆t , the impulse acting at t τ is gien by F τ -∆τ . At any time t, the elapsed time since the impulse is t > τ , so the response of the system at t due to this impulse alone is gien by 3. ':- #ith F F τ -∆τ 0 2 %he total response at time t can be found by summing all the responses due to the elementary impulses acting at all times τ 0 21Detting ∆τ and replacing the summation by integration, #e obtain
2'By substituting 3. '7- into 3. 2'-, #e obtain
22#hich represents the response of an underdamped single degree of freedom system to the arbitrary excitation F t -. 8ote that 3. 22- does not consider the eEect of initial conditions of the system. %he integral in 3. 2'- or 3. 22- is called the conolution or Huhamel integral. In many cases the function F t - has a form that permits an explicit integration of 3. 22-. In case such integration is not possible, it can be ealuated numerically #ithout much di=culty.
1
5.2(esponse to base excitation If a spring$mass damper system is sub"ected to an arbitrary base excitation described by its displacement, elocity, or acceleration, the e3uation of motion can be expressed in terms of the relatie displacement of the mass x > ! as follo#s0 24 %his e3uation is similar to the e3uation 25#ith ariable replacing x and the term >m y replacing the forcing function ´
F . ?ence all of the results deried for the force$excited system are applicable to the base$excited system also for #hen the form F is replaced by > m y . For ´
an underdamped system sub"ected to displacement can be found from 3. 22-.
base
excitation,
the
relatie
xample 8o. ' > Compacting achine )nder Dinear Force Hetermine the response of the compacting machine sho#n in Fig. 6a- #hen a linearly arying force sho#n in Fig. 6b- is applied due to the motion of the cam
11
Gien0 Compacting machine, modeled as single degree of freedom system sub"ected to ramp function. (e3uired0 (esponse of the system olution0 aluate the Huhamel integral #ith F t - δ F"t %he linearly arying force sho#n in Fig. 6b- is &no#n as the ramp function. %he forcing function can be represented as F τ - δ F Jτ , #here dF denotes the rate of increase of force F per unit time. By substituting this into 3. 22-, then
1'
ans#erFor an undamped system,
Figure 6c- sho#s the response. 6. (esponse pectrum %he graph sho#ing the ariation of the maximum response maximum displacement, elocity, acceleration, or any other 3uantity- #ith the natural fre3uency or natural period- of a single degree of freedom system to a speci (esponse pectrum of inusoidal *ulse Find the undamped response spectrum for the sinusoidal pulse force sho#n in Fig. 7a- using the initial conditions x-
12
x - . ´
Gien0 ingle degree of freedom undamped system sub"ected to one$half period of a sinusoidal force. (e3uired0 (esponse spectrum olution0 Find the response and express its maximum alue in terms of its natural time period %he e3uation of motion of an undamped system0
a#here
b %he solution of 3. a- can be obtained by superposing the homogeneous solution x #t - and the particular solution x - as pt cthat is,
d#here A and $ are constants and ω n is the natural fre3uency of the system0
14
e-
x - in 3. d-,
)sing the initial conditions x -
´
$ as
, %hus the solution becomes
f-
,
g-
,
h-
#hich can be re#ritten as
ans#er-
#here
i %he solution gien by 3. h- is alid only during the period of force application K t K t 0. ince there is no force applied for t L t 0, the solution can be expressed as a free ibration solution0 , t L t 0 "#here the constants A% and $% can be found by using the alues of x t t 0- and
x t t - gien by 3. 9-, as initial conditions for the duration t L t . %his 0 0 ´
gies
&-
l#here
m3uations &- and l- can be soled to
n3uations n- can be substituted into 3. "- to obtain
t N t 0
o- ans#er-
3uations h- and o- gie the response of the system in nondimensional form, that is x /δ st is expressed in terms of t /τn. %hus any speci
175 occurs at alue of t 0/τ n-
max
In the example, the input force is simple and hence a closed form solution has obtained for the response spectrum. ?o#eer, if the input force is arbitrary, #e can
266.1(esponse spectrum for base excitation In the design of machinery and structures sub"ected to a ground shoc&, such as that caused by an earth3ua&e, the response spectrum corresponding to the base excitation is useful. If the base of a damped single degree of freedom system is sub"ect to an acceleration
y ´
t -, the e3uation of motion,
in terms of the relatie displacement x > ! , is gien by 3., 24- and the response t - by 3. 26. In the case of a ground shoc&, the elocity response spectrum is generally used. %he displacement and acceleration spectra are then expressed in terms of the elocity spectrum. For a harmonic oscillator an undamped system under free ibration-, #e notice that and 27 %hus the acceleration and displacement spectra 'a and 'd can be obtained in terms of the elocity spectrum '( -0
29 %o consider damping in the system, if #e assume that the maximum relatie displacement occurs after the shoc& pulse has passed, the subse3uent 16
motion must be harmonic. In such a case, #e can use 3. 29-. %he
2:3uation 2:- can be re#ritten as
4#here
41-
4'and
42 %he elocity response spectrum, '( , can be obtained from 3. 4-
44 %hus the pseudo response spectra are gien by
45-
xample 8o. 4 > +ater %an& ub"ected to Base Acceleration %he #ater tan&, sho#n in Fig. 9a-, is sub"ected to a linearly arying ground acceleration as sho#n in Fig. 9b- due to an earth3ua&e. %he mass of the tan& is m, the stiEness of the column is k , and damping is negligible. Find the response spectrum for the relatie displacement, x > ! , of the #ater tan&.
17
Gien0 +ater tan& sub"ected to the base acceleration sho#n in Fig. 9b-. (e3uired0 (esponse spectrum of relatie displacement of the tan&. olution0 odel the #ater tan& as an undamped single degree of freedom system. Find the maximum relatie displacement of the tan& and express it as a function of ω n. %he base acceleration
for K t K 't 0
a-
for t L 't 0
b-
)esponse during K t K 't 00 By substituting 3. a- into 3. 26-, the response for an undamped system0
19
c-
daximum response max 0
e %his e3uation gies the time t m at #hich max occurs0
fBy substituting 3. f- into 3. d-, the maximum response of the tan& 0
g-
ans#er-
)esponse during t L 't 00 ince there is no excitation during this time,
h*roided that #e ta&e the initial displacement and initial elocity as and iusing 3. g-. %he maximum of ;t- gien by 3. 9- can be identi
"#here 0 and z ´
ans#er-
are computed as indicated in 3. i-.
0
7. Daplace %ransformation %he Daplace transform method can be used to
1:
%he Daplace transform of a function x t -, denoted symbolically as x s- D x t-, ´
is de
46#here s is, in general, a complex 3uantity and is called the subsidiary ariable. %he function e *st is called the &ernel of the transformation. ince the integration is #ith respect to t , the transformation gies a function of s. In order to sole a ibration problem using the Daplace transform method, the follo#ing steps are necessary0 a. +rite the e3uation of motion of the system. b. %ransform each term of the e3uation, using &no#n initial conditions. c. ole for the transformed response of the system. d. !btain the desired solution response- by using inerse Daplace transformation. In order to sole the forced ibration e3uation 47by the Daplace transform method, it is necessary to
and %hese can be found as follo#s0
49 %his can be integrated by parts to obtain
4:#here x - x 0 is the initial displacement of the mass m. imilarly, the Daplace transform of the second deriatie of x t - can be obtained0
5#here
x ´
-
x ´
is the initial elocity of the mass m. ince the Daplace
transform of the force Ft- is gien by
51#e can transform both sides of 3. 47- and obtain, using 3s. 46- and 49- to 51'
or 5'#here the right$hand side of 3. 5'- can be regarded as a generali;ed transformed excitation. For the present, #e ta&e x - and x - as ;ero, #hich is e3uialent to ´
ignoring the homogeneous solution of the diEerential e3uation 47-. %hen the ratio of the transformed excitation to the transformed response
Z s- can be ´
expressed as
52 %he function + s- is &no#n as the generali;ed impedance of the system. %he reciprocal of the function + s- is called the admittance or transfer function of the system and is denoted as , s-0
54It can be seen that by letting s iω in Os- and multiplying by k , #e obtain the complex fre3uency response -iω - de
Daplace transform of
x s-, #hich can be de
56In general, the operator D inoles a line integral in the complex domain. Fortunately, #e need to ealuate these integrals separately for each problem such integrations hae been carried out for arious common forms of the function F t - and tabulated. In order to
decompose
x ´
s- coneniently by the method of partial fractions. '1
In the aboe discussion, #e ignored the homogeneous solution by assuming x - and
x - as ;ero. +e no# consider the general solution by ta&ing the ´
initial conditions as x - x 0 and x - x 0. From 3. 5'-, the transformed response
x s- can be obtained0 ´
57+e can obtain the inerse transform of
x s- by considering each term on the ´
right side of 3. 57- separately. +e also ma&e use of the follo#ing relation0
59By considering the
f ´
s- f ´
1
s-,
'
#here
and and by noting that f 1t - D$1 f ´
s- Ft-, #e obtain
1
5:Consider the second term on the right side of 3. 57-, #e
6#here 61Finally, the inerse transform of the coe=cient of
x 0 in the third term on the ´
right side of 3. 57- can be obtained from table of Daplace transform0
6')sing 3s. 47-, 5:-, 6-, and 6'-, the general solution of 3. 47- can be expressed as
''
628H
'2
