- Email: [email protected]

RESPONSE

OF NON-LINEAR

SYSTEMS

OF TWO

NON-CONSERVATIVE

DEGREES OF FREEDOM

TRANSIENT M. A. v.

TO

EXCITATIONS

RANGACHARYULUt

Mechanical Engineering Group, Birla Institute of Technology and Science, Pilani 333031, (Received 6 August 1982, and in revised form 15 January

India

1983)

This paper deals with the approximate analysis of non-linear non-conservative systems of two degrees of freedom subjected to transient excitations. By using a transformation of the co-ordinates, the governing differential equations of the system are brought into a form to which the method of averaging of Krylov and Bogoliubov can be applied. The response of a representative spring-mass-damper system to typical pulses like a blast pulse and a half sinusoidal pulse is determined. The validity of the approach is demonstrated by comparison of the approximate solutions with numerical results obtained on a digital computer.

1.

INTRODUCTION

problem of transient forced vibrations in a linear many-degree-of-freedom system can be solved relatively easily. Either the classical approach of solving the system of linear differential equations or some kind of transform techniques can be employed to determine the response. These methods, however, cannot be used when the system exhibits non-linearities as the principle of superposition is not valid for non-linear differential equations. The problem of the response of many-degree-of-freedom systems is of great importance for the design of non-linear elements of machines, buildings, and air and space vehicle structures. Furthermore, loads due to transient excitations and blasts are of the utmost interest for the proper design of such systems. It is important to know the maximum displacements or the relative displacements of a vibrating system. In a previous paper [l] the response of coupled non-linear systems to constant-force-type excitations of infinite duration has been studied. Not many analytical methods have been reported in the literature, however, for dealing with a non-linear system subjected to transient excitations and pulses of finite duration. Evaldson et al. [2] have developed graphical techniques to determine the response of elastically non-linear systems to transient disturbances. Bycroft [3] has determined the transient response of lightly coupled non-linear undamped systems to arbitrary forcing functions using the perturbation method of PoincarC-Lighthill-Kuo. Latterly Bauer [4,5] has used the same technique to determine the response of non-linear systems to transient excitations and pulses of finite duration. In this paper the method of averaging is extended to determine the response of non-linear, non-conservative systems of two degrees of freedom to transient excitations including pulses of finite duration. By using a transformation of co-ordinates, the governing differential equations of the system are brought into a form to which the averaging

The general

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46

M

A. V. RANGACHARYIJLI

method can be applied. The results of the analysis are then applied to a representative system of which the anchoring spring is linear and the coupling spring is hard cubic. The damping is assumed to be viscous. The masses are subjected to typical pulses like a blast pulse, a half sinusoidal pulse and a triangular pulse, and in each case the complete response is determined. The validity of the approach is demonstrated by comparison of the approximate solutions with digital solutions. 2. EQUATIONS

OF MOTION

Consider a non-linear

system

AND

METHOD

of two degrees

OF APPROXIMATION

of freedom

governed

by the equations

~,+a~*x,+a~*xz+Ef(X1,X:,,X2,X2~=F,(t~. ~*+a2,x~+a22x2+Eg~Xl,~l,X2,~2~=~2~~~,

(1)

where Fr(t) and F2(r) are transient excitations, and (‘) A d( )/dt. The transformation xi(t) = ur(t) +gi(t) and x2(f) = u*(t) +g2(f) of the dependent variables reduces equations (1) to iil+allul+al2+Ef~U~+gl,Li1+gl,U2+g2,Li2+g2~+~l+allgl+~12g2=~l~~~r ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

For many of the common

transient

disturbances

~l+Qllgl+~12g2=~l(f),

(2)

particular

solutions

&+a21g,

+a2282

can be found. Let Pi(t) and P2(t) be a set of particular Then equations (2) become

of the equations (3)

=F2(f)

solutions

satisfying

equations

(3).

ii~+u~~u~+ul2U2+&f(U1+P1,~i+f~,U*+P*,~~+P2)=0, ii~+u2~U~+u~2U~+Eg(U~+Pl,~~+~~,.,+~,,~,+~,,=o.

(4)

These equations are a set of quasi-linear, homogeneous and non-autonomous equations. Even though these equations look formidable the method of averaging of KrylovBogoliubov can be applied to arrive at reasonably good approximate solutions. This method has been applied to linear non-autonomous systems by Huston and Doty [6] and they have reported an error analysis giving an indication of the domain of the validity of the approach. Later, Anderson [7] used the averaging method to study a class of non-linear non-autonomous systems to obtain non-resonant solutions. When E f 0 and is small, the solution of equations (4) can be assumed as [8] u,(t) =ui(t)

sin +r(t)+u*(t)

sin (L2(t),

k,(f) = a1(t)w1 cos Li20) a

=

-a21/(a22-w3

=

=

cua1(thJ1

--(a11

$1(f)

=

cos

-w:vu12, Wlf

+

e,(t),

u&)

= crui(f) sin ~r(t)[email protected](t)

$1(t)+az(t)w2 910)

cos

+Pu2(tbJ2 P

=

42(C)

cos

42(f), $2(f),

-a21/(Q22-&= =

W2f

sin &(f),

+ e,(t).

(5) (a11

-&)/&2, (6)

Here ol and w2 are the natural frequencies of the coupled linear system (i.e., for E = 0). Using the method of variation of parameters with expressions (5) as the solutions reduces equations (4) to a set of four first order equations in the amplitudes al, a2 and phases 0i, e2, to which the method of averaging can be applied. It is to be noted that the averaging is done over the region R (0=SIJ1s 21r, 0 s ti2 c 27r) and the functions containing the temporal variable t explicitly are treated as constant while averaging. The validity

NON-LINEAR

SYSTEM TRANSIENT

47

RESPONSE

of this assumption is brought out in the numerical examples discussed at the end. The averaged equations can be written as cil ai1

=f1kJ1,

a2,

t),

62

[email protected],

[email protected],

a2,

t),

a2i2

a2,

=f4b1,

t), a29

(7)

r).

Here -+-

[LF - NG] cos ljl,d4, d&, 1

$-[LF-NG]sin#1d4id42, 1

+

-NGl

PfF

~0s

42

WI

d$2,

2

[MF - NG] sin ti2 d$r d#2,

-G 2

L =u22-01,

2

2

M=ll22--ClJ2,

N =

WI

~12,

F=f(P1+ul,~‘1+til,PZ+~2,Pz+Li2),

=o&iJ:

-to:),

w2

=&J2(&

-09,

G=g(P1+ul,B1+Li1,P2+u2,P2+ri2).

The averaged equations upon integration provide amplitudes and phases and the complete response is given by xi(r) =P1+ul

x2(t) = Pz+aal sin [email protected] sin 1~22.

sin+,+u2sin(L2,

(8)

The initial conditions needed for integrating the averaged equations, ai(O), e,(O) and u2(0), e,(O), can be obtained easily in terms of the initial conditions X,(O), X1(O) and X2(O), X2(O) of the original equations (1) with the help of equations (8). If the forcing function lasts for a finite duration, the final state of the system at the end of the duration of the force should be taken as the initial state to determine the free or the residual response of the system. If to is the duration of the forces, then for 0 c t s to the solution is given by equations (8). For the residual era, i.e., for t > to the solution takes the form x1(7) = ~~(7)sin J1(7)+c2(7) sin &2(F),

x2(7) =acl(i) sin $i(7)+&2(7)

Here 7 is the running variable, $l=o17+~$i(5), equations take the form dcildt = gi(ci, cl gl,

g2,

g3

d&/d7

=

g2k1,

and & =w~?+c$&.

~21,

dc2ldf

cd,

c2

W2ldr

=

g3k1, =

sin I/;;(S). (9)

Now the averaged

c2L

g4tclr

~2).

and g4 retain the same structure as fi, f2, f3 and f4 given earlier except that now

F and G become F =f(xl(S), dxi/dS, x2(S), dxz/dr),

G = g(xi(7), dx,/dY, x2(?), dxJdt).

The averaged equations on integration provide the residual response. The initial conditions for the averaged equations are obtained from the final state of the system (i.e., at t = to) determined from equations (8). Thus the complete solution can be determined. If each degree of freedom is subjected to various pulses, the procedure is the same as discussed above. This yields the response of the system during the application of the

48

M. A.

V. RANGACHARYCILCJ

load and is valid until one of the loads ceases to act, say at t = r,. The following response interval, tl G f d f2, lasts until another forcing function ceases to act. The procedure to determine the response for this interval is the same as in the first interval except for the initial conditions, which are now the end conditions for the previous interval and are given here at t = tl. In the subsequent intervals the same procedure is used until the last of the system now serve as forcing function ceases to act at t = t,. The final conditions the initial conditions for the free oscillations era t > t, of the system. The method of analysis is illustrated with examples in the next section. The response of a non-linear two degree-of-freedom system to a blast pulse, a half sinusoidal pulse, and a triangular pulse is completely determined and the results are presented graphically.

3. APPLICATION

TO SOME TYPICAL

PROBLEMS

Consider a non-linear system of two degrees of freedom as shown in Figure both the masses are excited by transient forces. The equations of motion are

L.f(,) Figure 1. Sketch excitations.

of a non-linear,

LFf(

non-conservative

two degree

The form of the forcing function is chosen typical transient excitations are considered:

of freedom

system

1 wherein

ti

subjected

to transient

as Ffl(t) to facilitate further analysis. Three a blast-type pulse, a half sinusoidal pulse,

and a triangular pulse. In each case the response features are depicted graphically. The approximate results are compared with those obtained by Runge-Kutta fourth order numerical integration of the original non-linear equations. 3.1.

CASE

1: RESPONSE

TO

BLAST

PULSES

Let the transient forces acting on the masses be given by the difference of two exponentially decaying functions, fl(t) = eprr -e-‘I’, r

This represents a blast p& with a slowly rising front followed by an exponential decay. Upon introducing r = de/m& yl = (a/F)ql and y* = (a/F)q2, equations (10) reduce to y;‘+2yr-y2+e y2”-y1+y2+&

[(c0/.5)(2y; [(c~/e)(yG

-y~)+(y1-y2)3]=e-W’-ee-Wlr, -y;)-(yt-y2)[email protected]“‘-e-W”‘,

(11)

NON-LINEAR

SYSTEM

TRANSIENT

RESPONSE

49

where here (‘) ed( )/dr, p = r/q a m, p1 = rJ&&, E = bF2/a3, and co = cl&%. It is assumed that the damping is small as before. Equations (11) are of the form of equations (1) and the response can be obtained as follows. The particular integrals corresponding to the forcing functions are pi(~) = G1 e-“-G? [email protected]“’and P2(7) = G2 ePWT- GF [email protected]’, where Gi = (2+/.~~)/(~~+3~~+1),

~T=(2+&)/(&+3~:+1),

G2=(3+~2)/([email protected]+l),

G; =(3+/&/(~1:+3/~:+1).

Equations (7) for the amplitudes and phases reduce to a; = +?a,,

ai = -Pfa2,

0; =Sfa~+SiTa~+S~[(G1-G2)e-““-(GT-G~)e-c”1’]2, I~~=T~~~+T~~~+T~[(G~--G~)~-“‘--(G~-G~)~-~~’]~.

(12)

The coefficients Pf, ST, TT (i = 1,2, j = 1,2,3) are given in the Appendix. If the masses start from rest, i.e., y&=0 = y2(7=o = dyi/dT],,o = dyJd&o initial conditions associated with equations (12) are found to be

0,(O) = tan-’ Q,(O) = tan-’

G2-G;

= 0, the

-P(Gi-GT)

~~(P(cLG~-cc~GT)-(cLG~-cL~G~))

I’

G2-G;

-cu(Gr-G:) ~~(LY(cLG~-cL~GT)-(cLG~-cL~G~*)) I *

Solution of equations (12) then results in aI

=al(0) epsYT,

a2(7) = a2(0) e-Q’,

~~(~)=~~(O)+[STa~(0)/2~T][1-e-28~’]+[STa~(O)/2~2*][1-e-2”*T] +S~~~(G~-G2)2/2~l(1-ee-2~r)+[G~-G~)2/2~~](1-e~2~~r) -[2(Gi-GJ(GT

-G?)/(,u

&(7) = e2(O)+[TTa:(0)/2PT][l + T3*{[(G1 -G2)2/2b](1 -[2(G1-G2)(GT

+~l)](l-e-(cL+pl)T)},

-e-2P”]+[T2*a~(0)/2P~][1

-e-2pzr]

-e-‘“‘)+[(G?

-e-2r1r)

-G~*)/(P

-G2*)2/2p1(1

+~l)](l-e-(WfF1)T)}.

Then the complete solution takes the form ~~(7) = G1 epCLr-GT e-“17+a1(0) [email protected] ~~(7) = G2 e-“‘-

[or7 + f91(r)]+a2(0) e-84’sin [w27 + &(7)]

G? e-“17+LyaI(0) e-‘:‘sin

+pa2(0) epPz’ Sin

[W2T

+&(T)].

[WIT

+ Ok]

(13)

The response to an exponentially decaying pulses with vertical fronts can be obtained by making GT and Gz go to zero in equations (13).

50

M

3.2.

2:

CASE

RESPONSE

TO

In this case the forcing

A.

t1AL.F

function

V. RANGACHARYULIJ

SINUSOIDAL.

Plll.SES

takes the form sin (x/t&

0 % t d C”

0,

! zt”

(

f‘!(f) =

17

where to is the duration of the force. Introducing the non-dimensional the previous case, one obtains the equations of motion as

where to = oto. The corresponding PI(T) = HI

particular

sin (r/70)7

integrals

parameters

as in

are

and

P*(T)=

and

~2=T~(3T:-~2)/(?T4-37T2T~+T~).

H2sin

(~T/T~)T,

where ~~=T~(2T;-7Tz)/(7T4-3?T*T;+T;)

The averaged

equations

(7) become

a; =-PTa,,

ai =-f3Za2,

0; =STa:+STa:+S2*a~+Sf[(H1-H2) 0; =

The associated

TTa: + Tzaz +T~[(H~-Hz) sin

initial conditions

(w/T~)T]~.

are

a,(o) = (~/W~TOW~-PH~)I(P

-aI,

a2(0)

-P),

The amplitudes

sin(r/~,J~]~,

= (~T/W~TO)(H~-~HI)/(~

e,(o)

= e,(o)

= 0.

and phases are given by

a1(7)=a1(0)e-P7',

U2(T)=

a2(0)e+',

e1(T)=(S~/2~~)U:(0)(1-e-2P")+(S:/2~:)U~(0)(1-e-2"") +

- H2)*[T

($/~)(HI

- (To/27T)

sin

(27T/To)T],

~;(~)=(T~/2~~)a~(O)(l-e~~~~‘)+(T2*/2~$)a~(O)*(l-e~*~~~) + (z-T/2)(H2 The complete y1(7) =HI

solution

for 0 C

-HJ2[r

T S 7.

- (To/277)

sin (27T,/To)].

takes the form

sin (~/T&+a1(0) ePBT7sin [~1~+81(7)]+a2(0) [email protected]’sin [OPT+&],

y2(7)=H2sin(rr/T0)7

02(7)].

+cyal(0)e-B~l sin[W1~+e,(T)]+pU2(0)e-Bt'sin[02~+

(14)

For T 3 response

the forces cease to act and only free vibrations can be determined from

7.

persist.

During

this era the

NON-LINEAR

SYSTEM

TRANSIENT

RESPONSE

51

where 7 is the new running time variable and ci(7) = cl(O) e-‘;+,

c*(7) = ~~(0) e-O*‘, 1 - [email protected]),

+ [S,*Cf (0)/2p2*]( =~,(O)+[STc:(0)/2PTl(l -e- 2BTi)

41(f)

~1(7)=~2(0)+[TT~~(0)/2~~](1-e-2P~~)+[T~c~(0)/2~2*](l-e-2P~i). The constants cl(O), c2(0), 4,(O) and #2(O) are found to be

-~)I[{@ -a)al(70)

cl(O) = [I/@

Sin (Ll(~0))~

+~~I~:~~~~/~o~(P~l-~2~+~P ~20))

-P)lCk

=[l/(a

-Pb2(70)

+ (l/&{(dTo)(ah

c~s~1(~o)~211’27

--CYbl(TO)~l Sin 42(T0H2 -ff2)

+ (a

-p)u2(70)~2

COS

sin(Ll(Toh

(B -ah(To) tan~1(0)=[(~/To)(PH~-H2)+(B-LY)U1(To)01

COS$,(To)]’

(a -P)uz(To) tan

where (12(To)

&(To)

42(o)

= WlTo

= [(?T/To)((YH1-

+ &(To)

are evaluated at

T = 7.

H2)

sin J12(Tob2

+ (a

-/?)u2(To)O2

and $2(To) = [email protected](To). by using equations (14).

1 -t/to,

The non-dimensional

(0,

COS (L2(To)]’

Here

3.3. CASE 3: RESPONSE TO TRIANGULAR PULSES WITH In this case the forcing function can be represented by fl0) =

us,

u2(To),

VERTICAL

O~t~to

and

FRONTS

equations of motion become

+S:(~-T/T~)~,

+S~U;

= 3(l

-T/TO).

I!?;= T:u: + Tfu; + T: (I-

Equations (15) are to.be solved with the following initial conditions: al(o)=13,(o)

Integration

h(To)

1.

t 3 to

and the particular integrals are PI(T) = 2( 1 -T/TO) and P2(7) equations are a; = +ui, a; =Pz*u2, ~9; =STU:

(L2(70)}~1”~,

28-3

&+w:T:

6-a =

OlTo,

tan-’

(-017~)~

u2(0)=-

2a-3m

= U1(0)

e-“‘,

W2To

e2(0) = tan-’

of equations (15) yields the solution for 0 G T (21(T)

,

ff-P

U2(T)

(-0~~7~).

s TO

= U2(0)

e-“‘,

81(T)=e~(o)+[S:U:(0)/2~T](1-e-28”)

+[S~u~(0)/2/3~](1

as

-e-2Bz’>+S$[~-(~2/~o)+(T3/3T~)],

The averaged

T/To)~.

(15)

52

M. A. V. RANGACHARYULU

I

-2L 3

2

I

0

Timt T

Figure 2. Response of initially quiescent system to blast-type excitation. (a) Ye and (b) yz(~) for steep fronted exponentially decaying pulse (p = 1 and CL,= CO in equations (11)); (c) Y,(T) and (d) ~~(7) for finite rise time pulse (p = 0.1 and F 1= 1 .O in equations (11)). - - - , Approximate; -, numerical.

NON-LINEAR

SYSTEM TRANSIENT

RESPONSE

53

4 1

2 c f E “0 0 B a

0

-2

-3 3

Figure 3. Response of initially quiescent system to half sinusoidal pulse. duration rO = 1 (see equations (14)); (c) y 1(~) and (d) yz(~) for pulse duration numerical.

(a) ~~(7) and (b) yz(~) for pulse 70 = 3. - - -, Approximate; -,

54

M. A.

V. RANGACHARYULU

j2(~)=~2(0)+[TTa:(0)/2PT](1-e~‘P:’) +[TZa:(0)/[email protected]~](1 Thus

the complete

response

-em*@

for the interval

)+T;[T+*/~)+

(?/37;)].

0 6 7 s TO is

y1(7)=2(1-~/~~~)+a,(0)e~“~‘sin[0~~+8~(~)]+a~(O)e~“f’sin[~~~+~~(7)l,

Figure 4. Response of initially quiescent system to a steep fronted triangular pulse. (a) Y,(T) and (b) YZ(T) for pulse duration 7C= 2 (see section 3.3); (c) ~~(7) and (d) Y*(T)for pulse duration r. = 4. - - -. Approximate; numerical. -2

NON-LINEAR

SYSTEM TRANSIENT

RESPONSE

55

Time r

Figure 4 (continued)

~~(7) = 3(1-~/7~)+cyu~(O)

e-PTrsin [~~7+f3i(r)]+&z~(O)

The residual response for r >ro can be determined the complete solution.

4. NUMERICAL

RESULTS

AND

[email protected]

[027+&(7)].

as in the previous case to arrive at

DISCUSSION

In order to judge the accuracy of the method of approximation numerical calculations are performed for E = O-1 and 0.2. The variations of the displacements with the nondimensional time r for these cases are presented graphically only for the first few cycles of motion in Figures 2-4. Figures 2(a)-(d) show the response of the systems to pulses of blast type with p = 0.1 and pi = 1.0. The system reacts more strongly to a blast pulse with a finite rise time than to an exponentially decaying pulse with a vertical front, as can be seen by comparing Figures 2(c) and (d) with Figures 2(a) and (b). In both cases the approximate results are in close agreement with the digital computer results. The results for half sinusoidal pulses with pulse durations of 7. = 1 and 7. = 3 are presented in Figures 3(a), (b) and 3(c), (d), respectively, for E = 0.1 and 0.2. In all the cases the approximate results give sufficiently accurate results. Note that the maximum amplitude is found to occur after the pulse ceases to act. Figures 4(a)-(d) show the response to triangular pulses with a vertical front. The numerical computations are carried out for TV= 2 and TV= 4. In the latter case the maximum amplitude occurs within the pulse duration. Again it can be seen that the approximate results agree well with those of the digital computer, validating the analysis. In all the cases increased non-linearity makes the peak displacements decrease. Thus the method of averaging can be used to obtain sufficiently accurate information regarding the response of quasi-linear coupled systems subjected to transient excitations. The method becomes invalid in certain cases. For example, in the case of a half sinusoidal pulse, if the forcing frequency is in the neighbourhood of one of the uncoupled natural frequencies oi or o2 the transformation (3) becomes invalid and Hi and Hz will be very large. In such cases this method does not give good results. In non-resonant cases it gives sufficiently accurate results.

M. A. V. RANGACHARYuLIl

56

5. CONCLUSION

The method of averaging of Krylov and Bogoliubov has been extended to nonconservative two degree of freedom systems subjected to transient excitations. It can be seen from the discussion of the numerical results that the method provides sufficiently accurate results in non-resonant cases. This method is simpler to apply than perturbation techniques [4,5]. The method of averaging based on the ultra-spherical polynomial approximation [8] can be easily applied to the problems discussed in this paper, the method of Krylov and Bogoliubov being a particular case of this method.

ACKNOWLEDGMENTS

The paper is based on the work reported in the doctoral thesis of the author submitted to the Indian Institute of Science, Bangalore, India and the author gratefully acknowledges the help rendered by his supervisor Professor P. Srinivasan.

REFERENCES

P. SRINAVASAN and B. V. DASARATHY 1974 Journal of Approximate analysis of coupled non-linear non-conservative systems subjected to step-function excitation. R. L. EVALSDON, R. S. AVRE and L. S.JACOBSEN 1949 Journal of the Franklin Institute 248,475-494. Response of an elastically non-linear system to transient disturbances. G. N. BYCROFT 1966Journal of Mechanical Engineering Science 8, 252-258. Forced oscillations of non-linear two degree-of-freedom systems. H. F. BAUER 1968 International Journal of Non-linear Mechanics 3, 157-172. The response of a non-linear n-degree-of-freedom system to pulse excitation. H. F. BAUER 1971 International Journal of Non-linear Mechanics 6, 529-543. Vibrational behaviour of non-linear systems to pulse excitations of finite duration. R. L. HUSTON and L. T. DOTY 1966 SIAM Journal of Applied Mathematics 14, 424-428. Note on the Krylov-Bogoliubov method applied to linear differential equations. G. L.ANDERSON 1973Journal of Sound and Vibration 29,463-473. An approximate analysis of non-linear non-conservative systems using orthogonal polynomials. M. A. V. RANGACHARYULU, P. SRINIVASAN and B. V. DASARATHY 1974 Journal of Sound and Vibration 37, 466-473. Transient response of coupled non-linear non-conservative systems.

1. M. A. V. RANGACHARYULU,

Sound and Vibration 37, 359-366.

2. 3. 4. 5. 6. 7. 8.

APPENDIX p:

=(EW1/2W1)[1+(a!-2)0:1,

ST = -_(3~/8 WI)[(~ -(Y)~w:], s:

= -(3e/4W,)[(l T;

-a)w:],

= (3&/4W2)[(1

oi and w2 are the natural E = 0 in equations (4).

-P)3&

frequencies

P2*= h.d2wm-P)w: ST = _(3&/4W,)[(l

-a)(1

TT = (3&/4W2)[(1 -p)(l 7-T = (3/2W2)[(1 of the coupled

linear

- 11, -p,‘w:], -C&J;],

-P)&].

system

obtained

by making