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Stability of simple discrete systems under nonconservative loading with dynamic follower parameter
Pergamon
0045-7949(95)00421-l
Com~uCrs & Slrucrurrs Vol. 60, No. 4. pp. 653-663, 1996 Copynght 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/96 %lS.OO+ 0.00
STABILITY OF SIMPLE DISCRETE SYSTEMS UNDER NONCONSERVATIVE LOADING WITH DYNAMIC FOLLOWER PARAMETER W. Glabisz Institute
of
Civil Engineering, Technical University of Wroclaw, Wyb. Wyspianskiego 50-370 Wroclaw.
27,
Poland
(Received 6 January 1995) Abstract-The paper presents results of the analysis of the boundaries of parametric resonance areas for simple, linear and nonlinear discrete systems loaded with a static or dynamic nonconservative force with a dynamically variable follower parameter. Copyright 0 1996 Elsevier Science Ltd.
variable in time coefficients, one should mention works by such authors as Lindh and Likins [15], Hsu [16], Hsu and Cheng [17], Friedmann et al. [18], Lau et al. [19], Huang and Hung [20], Lee and Tsay [21], Sleeman and Smith [22], Kondou et al. [23], Hassan [24], Sinha and Wu [25] and Meijaard
1. INTRODU~ION
The analysis of systems with nonconservative forces has been the subject of numerous papers which because of their numbers would be difficult to mention here. The majority of them deal with the problems of the statics, dynamics and stability of linear or nonlinear, continuous or discrete systems usually under static nonconservative loads. As examples one could mention here works by such authors as Ziegler [ 1, 21, Bolotin [3], Leipholz [4-61, Prasad and Herrman [7, 81, Nemat-Nasser and Herrmann [9], Huseyin and Plaut [IO], Galka and Telega [ 1I], Levinson [ 121, Argyris and Symeonidis [13], Argyris et al. [14] and many others. The papers cited above consider a wide range of problems associated with the statics, dynamics and stability of systems with nonconservative forces, and the mathematical description is usually limited to equations or systems of algebraic or differential equations with constant (time-independent) coefficients. When designing some dynamic systems, one often faces the problem of how to analyze their stability when there are time-dependent-usually periodically dependent-coefficients in their mathematical description. The mathematical foundations for the analysis of the stability of linear equations with periodically variable coefficients were laid down by Mathieu in 1862, Hill in 1877 and Floquet who in 1883 presented a general solution which has been used up to this day. It can be easily demonstrated that if the periodicity of a nonconservative load is included in the description of a dynamic problem this will lead to mathematical models with periodically variable coefficients whose precise general solutions, except for some problems concerning linear systems with 1 d.f., are unachievable, As examples of the literature devoted to different approaches to the solving of problems whose description contains
1261. There are few papers w-hich deal with the analysis of systems with dynamically variable nonconservative loads. Diygadlo and Solarz [27] analyzed forced, parametrically self-excited vibration and parametrically forced vibration and parametrically forced vibration of a system with a dynamic follower force, using as an example the transverse vibration of a viscoelastic cantilever bar which is acted upon by a periodically variable follower force and a harmonic transverse load. Kotera 1281 analyzed regions of instability of vibration of columns subjected to a periodic axial follower load by approximation of a characteristic equation with an infinite determinant. Kar and Sujata [29] used the generalized Galerkin method to analyze regions of dynamic stability of an elastically fixed cantilever beam subjected to the action of a pulsating, axial follower force. By studying simple and combined resonance Kar and Sujata [30] determined the regions of dynamic stability of a rotating cantilever beam with end mass under the action of a transverse, pulsating follower force. So far no analysis of a nonconservative load which would characterize the variable in time, with follower parameters that describe the behavior of the load as a function of the structure’s response has appeared in the literature. It turns out that when the dynamic character of follower parameters, which may occur in practice e.g. in some branches of robotics, is taken into account, this results in qualitative and quantitative changes in the regions of stability of the analyzed systems. 653
W. Glabisz
654
49
The aim of this paper is to demonstrate, using simple linear and nonlinear discrete models (which can render, among other things, the behavior of columns), the influence of dynamic follower parameters of static or dynamic nonconservative load on the character of regions of the dynamic stability of these models. Furthermore, the character of these regions is analyzed for different values of damping, and for possible dependence of some parameters of the system on the acting load. The results of the analysis are based on exact formulas derived from Floquet’s theory for a linear system with 1 d.f., and on numerical solutions for a nonlinear system with 2 d.f.
9 a2
t
0
aq
a,
2. LINEAR MODEL WITH ONE DEGREE OF FREEDOM
Let us consider a system with 1 d.f. (Fig. la) in which an undeformable and weightless bar of length 1 is fixed by articulation. A concentrated mass m supported by a spring with spring rate K = k, + k(P), where k, is a constant time-independent component of the spring rate and k(P) is the rate’s component which may be any function of acting load P, has been placed on the top end of the bar. The bar is loaded with a concentrated, nonconservative static or dynamic periodically variable force P(r) whose possible deviation from the perpendicular is described by follower parameter tl(t). It is assumed that follower parameter cc(l) is a periodic function of time t. Deflection x of mass m from the vertical balance point is assumed as the generalized coordinate. By considering the condition of dynamic equilibrium under the assumption that the deflections of the mass are small, the equation of motion of this system will be obtained in the form of this Hill equation: i(t) + wi[l - F(t)]x(t) = 0,
(1)
h
m
t
0 c4
P3
Fig. 2. Forms of the functions
of follower parameter and function F(a, P, p).
load P(t)
cl(r),
and the differentiation along time is designated by dots. For the simplest case, to be considered further on, of the relationship between the spring rate and the
0.3
where
p(t) k(P) F(t) = [l - a(t)] x - 7 0 II
(2) 0.5
0.7
0.6
0.8
0.9
1
0.4
1.5
Fig. I. Diagrams of the analyzed models.
1.75
2
2.25
2.5
2.75
3
Fig. 3. The boundaries of the system’s parametric resonance regions for p, = /iz = -p, = -p., = p. a(t) = 0, for different values of parameter p and 6 = oo/r at T = 2Pi; the dashed line follows from condition N = - I. the solid line from condition N = 1.
Stability of simple discrete systems
0.96
0.98
1.02
1
1.04
1.06
1.08
1.1
Fig. 4. The boundaries
655
0 2.05
2
1.95
2.1
2.2
2.15
of the system’s parametric resonance regions for p, = pLz= -p, = -n,, = p, U(I) = 0, for different values of parameter p and 6 at T = 2Pi; the dashed line represents the system with damping (h = 0.03) the solid line the system without damping.
Fig. 6. The boundaries of the system’s third and fourth parametric resonance region for p, = pL2= -nq = -nj = p and Y, = -CL? = ~1~= -x4 ==0.8 (solid line) and x, = 0 (dashed line).
acting load in the form /c(P) = yP, where constant, function F(t) assumes this form:
When damping proportional to the deflection velocity and characterized by the constant coefficient c is included in the system, eqn (1) will take on the following form:
y is a
n(t) + 2/C(l)
+ ar;;[l - F(t)]x(r)
= 0,
(4)
where h = cj(2m). This equation, by applying a change of variables of the type x(t) = emA’&( can be reduced to the form
where V(f) = 1 -x(t).
4(r) + o’[l - F, (t)]q5(t) = 0,
(5)
where w’ = LO;- /z’, and F, (1) = (w~~/w)‘F(~). Then eqn (1) [(5)] is analyzed for the periodically variable function F(f)(F,(t)) with period T or T/2. Function F(t)(F,(t)) is a product of follower parameter s(f) and load P(r) functions, and it is assumed to be segmentally constant. Under this 0.55
0.5
0.45
0.6
11
(I 0.6 0.4 0.2 6
0 -0.2 -0.4 -0.6 1.12 0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
Fig. 5. The boundaries of the system’s first and second parametric resonance region for p, = flz = -11, = -nL4 = p and LY(= -a? = x2 = -q = 0.8 (solid line) and G(,= 0 (dashed line).
1.14
1.16
1.18
1.2
1.22
1.24
Fig. 7. The boundaries of the system’s first parametric resonance region for static load p, = pz = p1 = p4 = 0.3 and dynamic follower parameter LX,= -a2 = a, = G(; the solid line for the system without damping, the dashed line for damping h = 0.03.
W. Glabisz
656 a
0.6 0.4 0.2 0
-0.4 -0.6 5
2
1.5
1
0.5
Fig. 8. The boundaries of the system’s basic parametric resonance regions for dynamic load p, =pz = -p, = -p4 = 0.6 and dynamic follower parameter OL,= -a? = a,= -cltlq=c(.
assumption-which simplifies substantially the calculations and at the same time allows us to preserve the qualitative character of the phenomenon-it is relatively easy [depending on the number of segments that describe function F(t) in period T(T/2)] to obtain formulas by means of which the regions of parametric resonances can be determined for the problems described by eqns (I) or (5). As it follows from the theory of equations with periodically variable parameters (Floquet’s theory), the particular solutions of eqn (1) have this form (Kaliski [31]) x(f) = e%(t),
(6)
where u(t) is a periodic function with period T equal to the period of function F(t) and constant i is called the characteristic determinant of eqn (1). It follows from relationship (6) that in period T function x(t) changes by the constant value x(t + T) = e”‘x(t) = /Yx(t).
(7)
Of course for I/?1> I, the vibrations grow infinitely in time and parametric resonance will follow. Assuming that the variation of function F(t)(F,(t)) in period T can be described by four parameters pLi (i = 1,2 , ,4), )pLil< 1 and that segments of the variation of this function have the same length-T/4,
5.1
5.2
5.3
5.4
5.5
5.6
Fig. 10. The boundaries of the system’s ninth parametric resonance regions for dynamic load p, =pz = -pl = -p4 = 0.6 and dynamic follower parameter a, = -c(? = lx,= -cLc(4=E.
the
solution
following
of eqn form:
(1) can
be presented
in the
x(t) = x, = A, cos(o, t) + B, sin(w, t)
0 < t < T/4,
x2 = A, cos(o, t) + B2 sin(w, r)
T/4 d t < T/2,
x? = A, cos(w, t) + B, sin(w, t)
T/2 < t < 3/4T,
I xq = A, cos(w, t) + B, sin(w, t)
3/4T Q I cc T, (8)
where o, = CD,,-, (i = 1,2, ,4). Figure 2 shows a representative form of function F(t) for selected functions of follower parameter cc(t) and load P(t) with period T, assuming that y = 0. Of course, functions F(t) and F,(t) can assume other forms depending on the amplitudes and the periods of functions cc(t) and P(t) and parameter 7. Under the assumed division of observation time T into four equal parts, functions z(t) and P(I) can be constant or periodic with period T/2 or T. Equation (8) has to fulfill these conditions of continuity:
0.6 0.6
0.5
0.4
0.4
0.2
0.3
0
0.2
-0.2
0.1
-0.4
6
0 5.1
-0.6 3.6
3.8
4
4.2
4.4
Fig. 9. The boundaries of the system’s seventh and eight parametric resonance regions for dynamic load p, = p2 = -p, = -p4 = 0.6 and dynamic follower parameter a,= -c$=q= -ci‘$=Ix.
5.2
5.3
5.4
5.5
5.6
Fig. 1I. The boundaries of the system’s tenth parametric resonance regions for dynamic load p, =p2 = -p3 = -pd = 0.6 and follower parameter LX,= - az = aj = -Q = a; the solid line represents the system without damping, the dashed line for damping h = 0.08.
Stability
of simple discrete
systems
657
(a)
a
0.6
0.06
0.5
0.04
0.4
0.92
0.3
b
0
0.2
-0.02
0.1 -0.4 OL 0.6
0.45
0.5
0.55 (b)
6.518 6.519 Q 0.6
6.52
0.65
6.521 6.522 6.523 6.524 (b)
0.06 0.04
0.5
0.92
0.4
0
0.3
-0.92
0.2 0.1 n
0.06 0.45
Fig. 12. The the range of load and the the solid line
0.55
0.5
0.6
influence of the magnitude of the first parametric resonance follower parameter as in the for y = 0, the dashed line for y y =0.5 (b).
0.65 0.04
coefficient y on region for the previous figure; = -0.2 (a) and
0.92 0 -0.02 6.517
%(jgJ=.$+,(jg)
j=1,2,3,
(9)
and the equations that follow from relationship
(7)
6.518
6.519
6.52
6.521
6.522
Fig. 14. The boundaries of one of the higher parametric resonance regions for load p, =p2 = -p, = -p4 = 0.3 and follower parameter x, = -a2 = a3 = -a4 = r for ;’ = 0.475
(a). y = 0.493 (b) and y = 0.495 (c).
01
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
/Ii, (0) = &(T). Q
6.518
6.52
6.522
6.524
6.526
6.528
W
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
(IO)
By putting eqn (8) into conditions (9) and (lo), a homogenous as to constants Ai and B, (i = 1,2, ..1 4) system of algebraic equations is obtained whose condition of a nontrivial solution reduces itself to the zeroing of the system’s determinant. The condition produces an equation for fi in the following form: /G*-2Np+l=o,
6.516
6.518
6.52
6.522
(4
a
0.1 0.08 0.06 0.04 0.02 0
::::[T-~ 6.51
6.512
6.514
(11)
where N denotes a complex algebraic expression: N = N(w,, p,, 7). The explicit form of N is not quoted here because of its complexity, the expression was obtained by means of the Mathematics packet of symbolic transformations [32]. The boundaries of the system’s parametric resonance regions are determined for problem without damping [I] by this relationship: INI = 1,
6.516
6.518
6.52
Fig. 13. The boundaries of one of the higher parametric resonance regions for load p, = pz = -p, = -p4 = 0.3 and Follower parameter 3, = -a:, = a3 = -q = a for y = 0.45 (a). :’ = 0.5 fb) and Y = 0.55 (c).
and for eqn (5) with damping, relationship:
(12) by the following
INI > f(edh’ + eh’) > I.
(13)
658
W. Giabisz
Expressions (12) and (13) served as the basis for the analysis carried out in the next point. Making use of the Mathematics packet’s possibilities, similar relationships for segmentally, linearly variable functions F(t) and Fl(t) and for segmentally constant functions with period T divided into 6 equal parts were derived. In the above two cases, the formulas defining the boundaries of regions of instability became greatly complicated whereby the time of the analysis when they were used increased several times. Furthermore, the quality of the obtained solutions remained the same as that of the solutions obtained by means of formulas (12) and (13).
64
, 0
50
Ill’
100
150
200
fh1
ql
3. NUMERICAL ANALYSIS OF SYSTEM WITH ONE DEGREE OF FREEDOM
Here, selected results of the analysis of a system with 1 d.f., as shown in Fig. la, are presented. Figure 2 shows the character of the variation of functions that describe the behavior of follower parameter cc(t) acting on load system P(t), function F(a, y, P) dependent on cc(t) and P(t). and a function which describes the relationship between the rigidity of the system and the acting load-the latter relationship is designated by parameter 7. Each of the functions is periodic (period T or T/2) and described by a set of four parameters a,, p, and p,, i = I, 2, ,4. Symbolically generated general relationships defining the
ql
-0.06 -0.04 -0.02
0
0.M 0.04 0.06
0.1 0.05 0 -0.05 -0.1 1
I
I
-0.1
-0.05
0
0.05
0.1
Fig. 15. Diagrams q, and qz (a) as a function of time t and phase portraits q, (b) and Y2 (c) for P(t) = 0.5 + 0.5 cos(2.25r) and a(f) = 1.
@’ 0.15 0.1 0.05 0
q2
-0.05 -0.1 -0.15 -0.2
-0.1
0
0.1
0.2
Fig. 16. Diagrams q, and y2 (a) as a function of time I and phase portraits q, (b) and 4? (c) for P(t) = 0.5 + 0.5 cos(2.25r) and a(f) = I + 0.5 cos(2.3f).
boundaries of parametric resonance [eqns (12) (I 3)] were tested on a special case of the load described by parameters p,=~~= -p3= -~~=p for cc,=0 and for period T = 2Pi. It was further assumed that k, = 1 = 1 and T = 2Pi, which does not limit in any way the general validity of the presented results. If no mention is made of the dependence of the spring constant on P(t), this means that y = 0 was assumed. Sample results of the analysis for the system without and with damping are presented in Figs 3 and 4. In Figure 3, a dashed line marks the boundaries of resonance areas in which vibrations grow infinitely (N = - I) and a solid line indicates the boundaries in which vibrations are periodic (N = 1). The graphs were plotted for different values of parameter p (different depth of load modulation) and parameter 6 = w,,/r, where o0 and r are the circular frequencies of the free vibration of the system and function F, respectively. The obtained results agree fully with those reported widely in the literature on the subject. Regions of parametric resonance with the dynamic character of the follower coefficient of specified amplitude x, = -‘A* = a3 = -a4 = 0.8 (solid line) included are presented in Fig. 5 against the first and the second area of instability (dashed line) shown in Fig. 4 and in Fig. 6 for the third and the fourth region. One should notice that the frequency of changes of the follower parameter was twice as high as the acting dynamic load P(r). The first region of
Stability of simple discrete systems
659
parametric resonance generated by the dynamically (a) . ql, @ variable follower parameter CC,= - u2 = g1 = -CC+= tl for the constant (static) load P(t) (PI =p2 = p3 =p4 = 0.3) is shown in Fig. 7; the solid line represents the solution without damping and the dashed line-a sample solution with damping h = 0.03. Several further diagrams show the influence of the extent of the amplitude of the dynamic follower parameter on the behavior of the boundaries of parametric resonance regions for dynamic load P(t) (b) ql’ with fixed parameters. The diagrams allow one, 0.2 among other things, to answer the question whether 0.1 the introduction of a variable follower parameter into 0 the system can effect changes in the boundaries of parametric resonance regions. Figure 8 shows the -0.1 relationship between the magnitude of parameter CC -0.2 (x, = --a? = a3 = --x4 = U) and the extension of the four main regions of parametric resonance for dynamic load P(t) with a specified magnitude of parameter p = 0.6 exemplary @, = p: = -pJ = -p4 = p). Selected higher (the seventh and eight) parametric resonance regions are @ shown in Fig. 9 while Fig. 10 shows the ninth such region for the same problem parameters as in Fig. 8. The introduction of a dynamically variable follower parameter results in increased extent of the low -0.75 -0.5 -0.25 0 025 0.5 0.75 parametric resonance regions. For higher regions, the Fig. 18. Diagrams y, and q2 (a) as a function of time ( influence of a variable follower parameter in some of and phase portraits q, and q, (c) for (b) its variation ranges may turn out to be stabilizing f(r)
= 0.5 + 0.5 cos(2.251)
and r(t) = I + 0.5 cos(2.27r).
(Figs 9 and 10) which is evident from the distinct narrowings of the analyzed regions. Damping may even result in the multiple breaking of the continuity of the boundaries of parametric resonance regions and thus bring about their fading in certain ranges of values of follower parameter amplitudes. This situation is illustrated with an example of the boundaries of the tenth region in Fig. I I. If one takes into account the relationship between rigidity and acting load 0 # 0) in the analysis, then the extent of parametric resonance regions may increase or decrease as a function of the magnitude of parameter ;‘. This situation is shown in Fig. 12 using as an example the first of the analyzed regions. Figures 13 and 14 show the boundaries of one of the higher regtons of parametric resonance as a function of the value of the dynamic follower parameter for different values of the coefficient *,I.
0.15 0.1 0.05 0 -0.05 -0.1 -0.15
0.2 4.
0.1
NONLINEAR MODEL WITH TWO DEGREES OF FREEDOM
0 -0.1 -02 -0.4
-0.2
0
0.2
0.4
Fig. 17. Diagrams q, and q2 (a) as a function of time I and phase portraits 4, and (b) q2 (4 for P(r) = 0.5 + 0.5 cos(2.25f) and a(r) = I + 0.5 cos(2.28f).
Let us consider the nonlinear model of a pendulum with 2 d.f. shown in Fig. 1b. The model consists of two elastically connected massless bars of the same length I and two point masses 2m and nr. It is assumed that small imperfections e, and ez may occur in the system that determine its initial nonstress position. Quantities q, and q2 defining the revolutions of the
W. Glabisz
660
system’s bars are adopted as the system’s generalized coordinates. It is assumed that the static or dynamic nonconservative force P(t), whose action direction is determined by the dynamic follower parameter a(t), constitutes the pendulum’s load. It is assumed that the linear viscous damping that occurs in joints A and B of the system is described by parameters c I and c 2, and the nonlinear moments MA and M, that appear in these joints can be described by the following relationships: M,=k,,(q,
-e,)+k,,(q,
-e,)?+k,,(q,
-e,)3>
+k23(q2-e2-91+el)33
(14)
where coefficients k, characterize the rigidities of elastic constraints. Considering possible large deflections of the pendulum formulas for its kinetic energy E, potential energy V and external load work increment Q, on the linear parts of the generalized coordinates displacement increments (variations) 6q, can be expressed as
E=
qLq2
0
10
20
30
40
50
60
'70
0.06 0.04 0.02 0
92
-0.02
T (347+ 4:) + m12cj, coscq, cj2
q2)r
V=fk,,(q,-e,)2+4k,2(q,-e,)3 + ik,,(q,- e,1’ + fk2,(q2-
ql
e2 - q, -t- e, )?
-0.04
Fig. 20. Diagrams q, and q? (a) as a function of time I and phase portraits y, (b) and 9> (c) for P(r) = 0.6007 and r(t) = 0.8.
ql*@
-
/ 0
mgWco@, 1+ cos(q2)l,
1
Qdq,
= P(t)r{sin(q,)cos[cc(r)q,l
-
sW(t)q21cos(ql)Nh
+
P(t)l(sin(q2)cos[a(t)q21
-I
50
100
150
200
(b)
ql' 2 1 0
ql
- sinb (t)q21cM2)Pq2
-1 -2-
-3
-2
-1
q2
0
1
2
3
Cc)
q2
-4
-2
0
2
4
Fig. 19. Diagrams q, and yz (a) as a function of time I and phase portraits q, (b) and q2 (e) for P(f) = 0.5 + 0.5 cos(2.251) and a(r) = 1 + 0.5 cos(2.251).
c,4,@,- c2(42-4,)(h-~q,L
(15)
where g denotes gravitational acceleration. The above formulas can serve as the basis for the derivation of Lagrange equations of motion which then (under assigned initial conditions) can be subjected to numerical integration, or they can constitute a starting point for the application of a direct method based on Hamilton’s law. This second approachpresented, for instance, in the works of Borri et al. [33] and Glabisz [34]-was adopted for the further analysis where the effect of gravitation was neglected, which does not diminish the generality of the obtained results. Formulas from the author’s quoted
Stability
of simple discrete
paper, generalized to cover the case of the occurrence of the dynamically variable follower parameter a(t), were used in the solution. If a variable in time follower parameter is included, this leads to the modification-from the viewpoint of the nonlinear equations that describe the system’s motion--of some coefficients of these equations which become time functions and so the whole problem reduces itself to a complex parametric system of nonlinear differential equations.
systems
661
0.6 0.5 0.4 0.3 0.2 0.1
t
0 0
5
10
15
20
25
30
35
(b)
d’
ANALYSIS OF SYSTEM WITH TWO DEGREES OF FREEDOM
5. NUMERICAL
The numerical analysis of a system with 2 d.f. concentrates chiefly on the search for solutions at the boundaries of dynamic instability, taking into account the dynamic character of the follower parameter. The determination of the boundaries of parametric resonance in this case is associated with a gigantic numerical analysis and these boundaries depend strongly-both qualitatively and quantitatively-on even slight changes in the problem’s parameters. The selected results presented further on illustrate the possible influence of a variable follower parameter on the character of the obtained results. All the figures that follow show consistently the courses of the variation-in time t measured in
(a)
¶l, 92 0.25 0.2 0.15 0.1 0.05
t
0 -0.05 0
20
40
60
80
100 120 140
ql
-0.05
0
0.05
0.1
0.15
0.2
Fig. 21. Diagrams 4, and q2 (a) as a function of time t and phase
portraits
q, (b) and q2 (c) for P(t) = 0.6007 z(f) = 0.8 + 0.1 cos(4r).
and
91 0.1
0.2
0.3
0.4
0.5
0.6
0.06 0.04 0.02 0
$2
-0.02 0
0.1
0.2
0.3
0.4
0.5
Fig. 22. Diagrams q, and yz (a) as a function of time phase portraits q, (b) and q2 (c) for f’(r) = 0.6007 a(t) = 0.8 + 0.1 cos(lt).
I and and
seconds-of coordinates q, and q2 measured in radians (parts (a) of the figures) and phase portraits of the first (q,) and second (q2) generalized coordinate (parts (b) and (c), respectively of the figures). Then m=I=k,,=k,,=l, k,, = -2.5, kzz = -0.75, k,, = kz3 = 0, e, = -e2 = 0.05, c, = C:= 0 and the initial conditions 9,” = -4:” = 0.05 and Q,O= -& = 0 were assumed. Figures 15-19 show solutions for the dynamically variable force P(t) = 0.5 + 0.5 cos(2.25t) and different values of one of the parameters (0) that describe the variable follower parameter a(t) = 1.0 + 0.5 cos(nt); the presented solutions are for parameter o = 0. o = 2.3, o = 2.28, o = 2.27 and o = 2.25, respectively. Stable solutions were obtained for the analyzed higher values of parameter o (o :, 2.3), and the solutions for parameters 0 i o < 2.25 turned out to be unstable. A stable solution was understood as one in which displacements remained restrained [35]. As it was demonstrated in the author’s paper cited before [34], the analyzed system, assuming that a = 0.8, loses its stability at static load Pr.,= 0.600675. Figure 20 shows a solution for P = 0.6007 > Pkr(~ = 0.8). Figures 21 and 22 demonstrate solutions when the dynamic character of follower parameter cc(t) = 0.8 + 0.1 cos(ot) was incorporated for o = 4 and o = I, respectively. Similarly, as in the case of dynamic loading, stable solutions were obtained for o > 4 and unstable solutions--o < 1.
W. Glabisz
662 6. RECAPITULATION
Results of an analysis of the boundaries of parametric resonance regions have been presented for simple discrete systems described by linear or nonlinear differential equations with some periodically variable follower parameters that characterized nonconservative loads acting on either static or dynamic systems. In the case of a linear system with 1 d.f., the presented results were obtained by analyzing exact, closed formulas for determining the boundaries of parametric resonance. The formulas were obtained with the help of the Mathematics packet. The numerical analysis for a nonlinear system with 2 d.f. was carried out using a direct approach based on Hamilton’s law. The results of the analysis of the system with 1 d.f. were treated as the first approximation of selected, potential responses of the system in situations-which have not been analyzed in the literature yet-when dynamically variable follower parameters are taken into account. The presented results allow us to make several important observations: (1) the boundaries of the parametric resonance regions can change substantially under the action of the dynamic, nonconservative load follower parameter (Fig. 5); (2) the change of the boundaries of parametric resonance may be different for different resonance regions; these changes are particularly visible for higher regions (Figs 5 and 6); (3) the influence of viscous damping is similar as for systems without variable follower parameters-it causes, among other things, the narrowing of the areas of parametric resonance (Fig. 7); (4) the taking in of the dynamic character of follower parameters leads-as the parameter’s amplitude increases-to the widening (destabilization) of the first regions of parametric resonance (Fig. 8); (5) the inclusion of dynamic follower parameters in the analysis of higher regions of parametric resonance may lead, in certain ranges of its amplitude, to their destabilization of stabilization (Figs 9 and IO); (6) damping that always occurs in real systems can cause the disruption (depending on the follower parameter’s amplitude) of the continuity of particularly the higher parametric resonance regions of systems with dynamically variable follower parameters (Fig. 11); this phenomenon is not observed for the basic (first) regions of resonance; (7) the inclusion, in a system with a dynamically variable, nonconservative load follower parameter, of simple relationships between rigidity and the acting load level can cause-depending on the character of such a relationship-both the destabilization and the stabilization of parametric resonance regions (Fig. 12), particularly manifest in the case of higher regions (Figs 13 and 14); (8) the response of nonlinear systems is highly sensitive to even small changes in the frequency of the
follower parameter, which may result in the stabilization or destabilization of the system (Figs 16-19); (9) in the case of nonlinear systems, it is possible to stabilize an unstable system by introducing a dynamically variable follower parameter with precisely specified parameters (Figs 20-22). One should treat the presented results and the observations that follow from them only as the initial outcomes of the analysis of nonconservative systems with dynamically variable follower parameters. One can expect further research in this field to discover many new qualitatively significant and as yet not analyzed phonomena both in precritical states and at the boundaries of simple and combined parametric resonances.
REFERENCES
1. H. Ziegler, Die stabilitatskriterien Ing. Arch. 20, 49-56 (1952).
der elastomechanik.
2. H. Ziegler, On the concept of elastic stability. A&. appl. Mech. 4, 351-403
(1956).
3. V. V. Bolotin, Nonconsewutice Problems in the Theor) qf Elastic Stubility. Pergamon Press, Oxford ( 1963). On conservative elastic systems of the first and second kind. hg. Arch. 43, 255-271 (1974). H. H. Leipholz, Variational principles for non-conservative problems. Comput. Meth. oppl. Mrch. Engng 17/l& 609-617 (1979). H. H. Leipholz, On time invariant functionals for certain mechanical systems subject to follower forces. Hadronic J. 3, 962- 1017 (1980). S. Prasad and G. Herrmann. The usefulness of adjoint systems in solving nonconservative stability problems of elastic continua Inf. J. Solids Strucr. 5, 727-737 (1969). S. Prasad and G. Herrmann, Adjoint variational methods in nonconservative stability problems. Inr. J. Solids Struct. 8, 29-40 (1972). S. Nemat-Nasser and G. Herrmann, Some general considerations concerning the destabilizing effect in nonconservative systems. ZAMP 13, 305- 3 13 (1966). K. Huseyin and R. H. Plaut, The elsatic stability of two-parameter nonconservative systems. J. uppl. Mrch. 40, 175 ~180 (1979). A. Galka and J. J. Telega, On variational principles and conservation laws in finite non-potential elasticity. In: Proc. 2nd Int. Con/: Vuriationol Methods in Engineering, Unirersiry of Southampton (Edited by C. A. Brebbia) pp. (3-13))(3-22). Springer, Berlin. M. Levinson, Application of the Galerkin and Ritz methods to non-conservative problems of elastic stability. ZAMP 17, 431 -442 (1966). J. H. Argyris and Sp. Symeonidis, Nonlinear finite analysis of elastic systems under nonconservative loading-natural formulation, part I, quasistatic problems. Comput. Meth. uppl. Mech. Engng 26, 57~ 123 (1981). J. H. Argyris, K. Straub and Sp. Symeonidis, Nonlinear finite analysis of elastic systems under nonconservative loading-natural formulation, part II, dynamic problems. Comput. Merh. oppl. Mech. Engng 28, 241-258 (1981). K. G. Lindh and P. W. Likins, Infinite determinant methods for stability analysis of periodic-coefficient differential equations. AIAA J. 8, 680-686 (1970). C. S. Hsu, Impulsive parametric excitation theory. J. uppl. Mech. 96, 551-558 (1972). C. S. Hsu and W. H. Cheng, Steady-state response of a dynamical system under combined parametric and forcing excitations. J. uppl. Mech. 98, 371-378 (1974).
4. H. H. Leipholz, 5. 6.
7.
8. 9.
10.
II.
12.
13.
14.
15.
16. 17.
Stability
of simple discrete
18. P. Friedmann, C. E. Hammond and T.-H. Woo, Efficient numerical treatment of periodic systems with application to stability problems. Inr. J. namer Me/h. Engng 11, 1117-1136 (1977). 19. S. L. Lau. Y. K. Cheung and S. Y. Woo, A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. J. appl. Mech. 49, 849-853 (1982). 20. J. S. Huang and L. H. Hung, Dynamic stability for a simple supported beam under periodic axial excitation. In/. J. Non-linear M&z. 19, 287-~301 (1984). 21. T.-T. Lee and S.-C. Tsay, Analysis of linear timevarying systems and bilinear systems via shifted Chebyshev polynomials of the second kind. Inr. J. +yr. SC,/. 17, 1757-1766 (1986). 22. B. D. Sleeman and P. D. Smith, Double periodic Floquet theory for a second order system of ordinary differential equations. Q. J. Math. O.vford 37, 347-356 (1986). 23. T. Kondau. H. Tamura and A. Sueoka, On a method of higher approximation and determination of stability criterion for steady oscillations in nonlinear systems. Bull. JSME 29, 525- 532 (1986). 24. H. S. Hassan. Floquet solutions of non-linear ordinary differential equations. Proc. R. Sot. Edinburgh 106A, 267 -275 ( 1987). 25. S. C. Sinha and D.-H. Wu, An efficient computational scheme for the analysis of periodic systems. .I. Sound Vibr. 151, 91 I I7 (1991). 26. J. P. Meijaard, Direct determination of periodic solutions of the dynamical equations of flexible
27.
28.
29.
30.
31. 32.
33.
34.
35.
systems
663
mechanisms and manipulators. Inf. J. numer. Meth. Engng 32, 1691-1710 (1991). Z. Diygadlo and L,. Solarz, On nonautonomous vibrations of self-excited system with tangential force. Proc. Vihr. Probl. 11, 2 (1970). T. Kotera, Regions of instability of vibrations of columns subjected to periodic axial force. Bull. JSME 28, 2056-2063 (I 985). R. C. Kar and T. Sujata, Parametric instability of an elastically restrained cantilever beam. Comput. Struct. 34, 469.-475 (1990). R. C. Kar and T. Sujata, Stability boundaries of a rotating cantilever beam with end mass under a transverse follower excitation. J. Sound Vihr. 154,81-93 (1992). S. Kaliski, (edited with collaboration of L. Solarz), Vibration.r. PWN. Elsevier Amsterdam (1992). Malhemotica. A Syslem for Doing Mathematics hy Cornpurer, Version 2.2.2. Wolfram Research (1994). M. Borri, G. L. Ghiringhelli. M. Lanz, P. Mantegazzd and T. Merlini, Dynamic response of mechanical systems by a weak Hamiltonian formulation. Comput. Strucf. 20, 495-508 (1985). W. Glabisz, Application of Hamilton’s law to the stability analysis of a double pendulum under nonconservative loads. Cornput. Struct. 53, 143-153 (1994). A. N. Kounadis, Nonlinear dynamic buckling and stability of autonomous structural systems. Int. J. mech. Sci. 35, 643-656 (19931.
0045-7949(95)00421-l
Com~uCrs & Slrucrurrs Vol. 60, No. 4. pp. 653-663, 1996 Copynght 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0045-7949/96 %lS.OO+ 0.00
STABILITY OF SIMPLE DISCRETE SYSTEMS UNDER NONCONSERVATIVE LOADING WITH DYNAMIC FOLLOWER PARAMETER W. Glabisz Institute
of
Civil Engineering, Technical University of Wroclaw, Wyb. Wyspianskiego 50-370 Wroclaw.
27,
Poland
(Received 6 January 1995) Abstract-The paper presents results of the analysis of the boundaries of parametric resonance areas for simple, linear and nonlinear discrete systems loaded with a static or dynamic nonconservative force with a dynamically variable follower parameter. Copyright 0 1996 Elsevier Science Ltd.
variable in time coefficients, one should mention works by such authors as Lindh and Likins [15], Hsu [16], Hsu and Cheng [17], Friedmann et al. [18], Lau et al. [19], Huang and Hung [20], Lee and Tsay [21], Sleeman and Smith [22], Kondou et al. [23], Hassan [24], Sinha and Wu [25] and Meijaard
1. INTRODU~ION
The analysis of systems with nonconservative forces has been the subject of numerous papers which because of their numbers would be difficult to mention here. The majority of them deal with the problems of the statics, dynamics and stability of linear or nonlinear, continuous or discrete systems usually under static nonconservative loads. As examples one could mention here works by such authors as Ziegler [ 1, 21, Bolotin [3], Leipholz [4-61, Prasad and Herrman [7, 81, Nemat-Nasser and Herrmann [9], Huseyin and Plaut [IO], Galka and Telega [ 1I], Levinson [ 121, Argyris and Symeonidis [13], Argyris et al. [14] and many others. The papers cited above consider a wide range of problems associated with the statics, dynamics and stability of systems with nonconservative forces, and the mathematical description is usually limited to equations or systems of algebraic or differential equations with constant (time-independent) coefficients. When designing some dynamic systems, one often faces the problem of how to analyze their stability when there are time-dependent-usually periodically dependent-coefficients in their mathematical description. The mathematical foundations for the analysis of the stability of linear equations with periodically variable coefficients were laid down by Mathieu in 1862, Hill in 1877 and Floquet who in 1883 presented a general solution which has been used up to this day. It can be easily demonstrated that if the periodicity of a nonconservative load is included in the description of a dynamic problem this will lead to mathematical models with periodically variable coefficients whose precise general solutions, except for some problems concerning linear systems with 1 d.f., are unachievable, As examples of the literature devoted to different approaches to the solving of problems whose description contains
1261. There are few papers w-hich deal with the analysis of systems with dynamically variable nonconservative loads. Diygadlo and Solarz [27] analyzed forced, parametrically self-excited vibration and parametrically forced vibration and parametrically forced vibration of a system with a dynamic follower force, using as an example the transverse vibration of a viscoelastic cantilever bar which is acted upon by a periodically variable follower force and a harmonic transverse load. Kotera 1281 analyzed regions of instability of vibration of columns subjected to a periodic axial follower load by approximation of a characteristic equation with an infinite determinant. Kar and Sujata [29] used the generalized Galerkin method to analyze regions of dynamic stability of an elastically fixed cantilever beam subjected to the action of a pulsating, axial follower force. By studying simple and combined resonance Kar and Sujata [30] determined the regions of dynamic stability of a rotating cantilever beam with end mass under the action of a transverse, pulsating follower force. So far no analysis of a nonconservative load which would characterize the variable in time, with follower parameters that describe the behavior of the load as a function of the structure’s response has appeared in the literature. It turns out that when the dynamic character of follower parameters, which may occur in practice e.g. in some branches of robotics, is taken into account, this results in qualitative and quantitative changes in the regions of stability of the analyzed systems. 653
W. Glabisz
654
49
The aim of this paper is to demonstrate, using simple linear and nonlinear discrete models (which can render, among other things, the behavior of columns), the influence of dynamic follower parameters of static or dynamic nonconservative load on the character of regions of the dynamic stability of these models. Furthermore, the character of these regions is analyzed for different values of damping, and for possible dependence of some parameters of the system on the acting load. The results of the analysis are based on exact formulas derived from Floquet’s theory for a linear system with 1 d.f., and on numerical solutions for a nonlinear system with 2 d.f.
9 a2
t
0
aq
a,
2. LINEAR MODEL WITH ONE DEGREE OF FREEDOM
Let us consider a system with 1 d.f. (Fig. la) in which an undeformable and weightless bar of length 1 is fixed by articulation. A concentrated mass m supported by a spring with spring rate K = k, + k(P), where k, is a constant time-independent component of the spring rate and k(P) is the rate’s component which may be any function of acting load P, has been placed on the top end of the bar. The bar is loaded with a concentrated, nonconservative static or dynamic periodically variable force P(r) whose possible deviation from the perpendicular is described by follower parameter tl(t). It is assumed that follower parameter cc(l) is a periodic function of time t. Deflection x of mass m from the vertical balance point is assumed as the generalized coordinate. By considering the condition of dynamic equilibrium under the assumption that the deflections of the mass are small, the equation of motion of this system will be obtained in the form of this Hill equation: i(t) + wi[l - F(t)]x(t) = 0,
(1)
h
m
t
0 c4
P3
Fig. 2. Forms of the functions
of follower parameter and function F(a, P, p).
load P(t)
cl(r),
and the differentiation along time is designated by dots. For the simplest case, to be considered further on, of the relationship between the spring rate and the
0.3
where
p(t) k(P) F(t) = [l - a(t)] x - 7 0 II
(2) 0.5
0.7
0.6
0.8
0.9
1
0.4
1.5
Fig. I. Diagrams of the analyzed models.
1.75
2
2.25
2.5
2.75
3
Fig. 3. The boundaries of the system’s parametric resonance regions for p, = /iz = -p, = -p., = p. a(t) = 0, for different values of parameter p and 6 = oo/r at T = 2Pi; the dashed line follows from condition N = - I. the solid line from condition N = 1.
Stability of simple discrete systems
0.96
0.98
1.02
1
1.04
1.06
1.08
1.1
Fig. 4. The boundaries
655
0 2.05
2
1.95
2.1
2.2
2.15
of the system’s parametric resonance regions for p, = pLz= -p, = -n,, = p, U(I) = 0, for different values of parameter p and 6 at T = 2Pi; the dashed line represents the system with damping (h = 0.03) the solid line the system without damping.
Fig. 6. The boundaries of the system’s third and fourth parametric resonance region for p, = pL2= -nq = -nj = p and Y, = -CL? = ~1~= -x4 ==0.8 (solid line) and x, = 0 (dashed line).
acting load in the form /c(P) = yP, where constant, function F(t) assumes this form:
When damping proportional to the deflection velocity and characterized by the constant coefficient c is included in the system, eqn (1) will take on the following form:
y is a
n(t) + 2/C(l)
+ ar;;[l - F(t)]x(r)
= 0,
(4)
where h = cj(2m). This equation, by applying a change of variables of the type x(t) = emA’&( can be reduced to the form
where V(f) = 1 -x(t).
4(r) + o’[l - F, (t)]q5(t) = 0,
(5)
where w’ = LO;- /z’, and F, (1) = (w~~/w)‘F(~). Then eqn (1) [(5)] is analyzed for the periodically variable function F(f)(F,(t)) with period T or T/2. Function F(t)(F,(t)) is a product of follower parameter s(f) and load P(r) functions, and it is assumed to be segmentally constant. Under this 0.55
0.5
0.45
0.6
11
(I 0.6 0.4 0.2 6
0 -0.2 -0.4 -0.6 1.12 0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
Fig. 5. The boundaries of the system’s first and second parametric resonance region for p, = flz = -11, = -nL4 = p and LY(= -a? = x2 = -q = 0.8 (solid line) and G(,= 0 (dashed line).
1.14
1.16
1.18
1.2
1.22
1.24
Fig. 7. The boundaries of the system’s first parametric resonance region for static load p, = pz = p1 = p4 = 0.3 and dynamic follower parameter LX,= -a2 = a, = G(; the solid line for the system without damping, the dashed line for damping h = 0.03.
W. Glabisz
656 a
0.6 0.4 0.2 0
-0.4 -0.6 5
2
1.5
1
0.5
Fig. 8. The boundaries of the system’s basic parametric resonance regions for dynamic load p, =pz = -p, = -p4 = 0.6 and dynamic follower parameter OL,= -a? = a,= -cltlq=c(.
assumption-which simplifies substantially the calculations and at the same time allows us to preserve the qualitative character of the phenomenon-it is relatively easy [depending on the number of segments that describe function F(t) in period T(T/2)] to obtain formulas by means of which the regions of parametric resonances can be determined for the problems described by eqns (I) or (5). As it follows from the theory of equations with periodically variable parameters (Floquet’s theory), the particular solutions of eqn (1) have this form (Kaliski [31]) x(f) = e%(t),
(6)
where u(t) is a periodic function with period T equal to the period of function F(t) and constant i is called the characteristic determinant of eqn (1). It follows from relationship (6) that in period T function x(t) changes by the constant value x(t + T) = e”‘x(t) = /Yx(t).
(7)
Of course for I/?1> I, the vibrations grow infinitely in time and parametric resonance will follow. Assuming that the variation of function F(t)(F,(t)) in period T can be described by four parameters pLi (i = 1,2 , ,4), )pLil< 1 and that segments of the variation of this function have the same length-T/4,
5.1
5.2
5.3
5.4
5.5
5.6
Fig. 10. The boundaries of the system’s ninth parametric resonance regions for dynamic load p, =pz = -pl = -p4 = 0.6 and dynamic follower parameter a, = -c(? = lx,= -cLc(4=E.
the
solution
following
of eqn form:
(1) can
be presented
in the
x(t) = x, = A, cos(o, t) + B, sin(w, t)
0 < t < T/4,
x2 = A, cos(o, t) + B2 sin(w, r)
T/4 d t < T/2,
x? = A, cos(w, t) + B, sin(w, t)
T/2 < t < 3/4T,
I xq = A, cos(w, t) + B, sin(w, t)
3/4T Q I cc T, (8)
where o, = CD,,-, (i = 1,2, ,4). Figure 2 shows a representative form of function F(t) for selected functions of follower parameter cc(t) and load P(t) with period T, assuming that y = 0. Of course, functions F(t) and F,(t) can assume other forms depending on the amplitudes and the periods of functions cc(t) and P(t) and parameter 7. Under the assumed division of observation time T into four equal parts, functions z(t) and P(I) can be constant or periodic with period T/2 or T. Equation (8) has to fulfill these conditions of continuity:
0.6 0.6
0.5
0.4
0.4
0.2
0.3
0
0.2
-0.2
0.1
-0.4
6
0 5.1
-0.6 3.6
3.8
4
4.2
4.4
Fig. 9. The boundaries of the system’s seventh and eight parametric resonance regions for dynamic load p, = p2 = -p, = -p4 = 0.6 and dynamic follower parameter a,= -c$=q= -ci‘$=Ix.
5.2
5.3
5.4
5.5
5.6
Fig. 1I. The boundaries of the system’s tenth parametric resonance regions for dynamic load p, =p2 = -p3 = -pd = 0.6 and follower parameter LX,= - az = aj = -Q = a; the solid line represents the system without damping, the dashed line for damping h = 0.08.
Stability
of simple discrete
systems
657
(a)
a
0.6
0.06
0.5
0.04
0.4
0.92
0.3
b
0
0.2
-0.02
0.1 -0.4 OL 0.6
0.45
0.5
0.55 (b)
6.518 6.519 Q 0.6
6.52
0.65
6.521 6.522 6.523 6.524 (b)
0.06 0.04
0.5
0.92
0.4
0
0.3
-0.92
0.2 0.1 n
0.06 0.45
Fig. 12. The the range of load and the the solid line
0.55
0.5
0.6
influence of the magnitude of the first parametric resonance follower parameter as in the for y = 0, the dashed line for y y =0.5 (b).
0.65 0.04
coefficient y on region for the previous figure; = -0.2 (a) and
0.92 0 -0.02 6.517
%(jgJ=.$+,(jg)
j=1,2,3,
(9)
and the equations that follow from relationship
(7)
6.518
6.519
6.52
6.521
6.522
Fig. 14. The boundaries of one of the higher parametric resonance regions for load p, =p2 = -p, = -p4 = 0.3 and follower parameter x, = -a2 = a3 = -a4 = r for ;’ = 0.475
(a). y = 0.493 (b) and y = 0.495 (c).
01
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
/Ii, (0) = &(T). Q
6.518
6.52
6.522
6.524
6.526
6.528
W
0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04
(IO)
By putting eqn (8) into conditions (9) and (lo), a homogenous as to constants Ai and B, (i = 1,2, ..1 4) system of algebraic equations is obtained whose condition of a nontrivial solution reduces itself to the zeroing of the system’s determinant. The condition produces an equation for fi in the following form: /G*-2Np+l=o,
6.516
6.518
6.52
6.522
(4
a
0.1 0.08 0.06 0.04 0.02 0
::::[T-~ 6.51
6.512
6.514
(11)
where N denotes a complex algebraic expression: N = N(w,, p,, 7). The explicit form of N is not quoted here because of its complexity, the expression was obtained by means of the Mathematics packet of symbolic transformations [32]. The boundaries of the system’s parametric resonance regions are determined for problem without damping [I] by this relationship: INI = 1,
6.516
6.518
6.52
Fig. 13. The boundaries of one of the higher parametric resonance regions for load p, = pz = -p, = -p4 = 0.3 and Follower parameter 3, = -a:, = a3 = -q = a for y = 0.45 (a). :’ = 0.5 fb) and Y = 0.55 (c).
and for eqn (5) with damping, relationship:
(12) by the following
INI > f(edh’ + eh’) > I.
(13)
658
W. Giabisz
Expressions (12) and (13) served as the basis for the analysis carried out in the next point. Making use of the Mathematics packet’s possibilities, similar relationships for segmentally, linearly variable functions F(t) and Fl(t) and for segmentally constant functions with period T divided into 6 equal parts were derived. In the above two cases, the formulas defining the boundaries of regions of instability became greatly complicated whereby the time of the analysis when they were used increased several times. Furthermore, the quality of the obtained solutions remained the same as that of the solutions obtained by means of formulas (12) and (13).
64
, 0
50
Ill’
100
150
200
fh1
ql
3. NUMERICAL ANALYSIS OF SYSTEM WITH ONE DEGREE OF FREEDOM
Here, selected results of the analysis of a system with 1 d.f., as shown in Fig. la, are presented. Figure 2 shows the character of the variation of functions that describe the behavior of follower parameter cc(t) acting on load system P(t), function F(a, y, P) dependent on cc(t) and P(t). and a function which describes the relationship between the rigidity of the system and the acting load-the latter relationship is designated by parameter 7. Each of the functions is periodic (period T or T/2) and described by a set of four parameters a,, p, and p,, i = I, 2, ,4. Symbolically generated general relationships defining the
ql
-0.06 -0.04 -0.02
0
0.M 0.04 0.06
0.1 0.05 0 -0.05 -0.1 1
I
I
-0.1
-0.05
0
0.05
0.1
Fig. 15. Diagrams q, and qz (a) as a function of time t and phase portraits q, (b) and Y2 (c) for P(t) = 0.5 + 0.5 cos(2.25r) and a(f) = 1.
@’ 0.15 0.1 0.05 0
q2
-0.05 -0.1 -0.15 -0.2
-0.1
0
0.1
0.2
Fig. 16. Diagrams q, and y2 (a) as a function of time I and phase portraits q, (b) and 4? (c) for P(t) = 0.5 + 0.5 cos(2.25r) and a(f) = I + 0.5 cos(2.3f).
boundaries of parametric resonance [eqns (12) (I 3)] were tested on a special case of the load described by parameters p,=~~= -p3= -~~=p for cc,=0 and for period T = 2Pi. It was further assumed that k, = 1 = 1 and T = 2Pi, which does not limit in any way the general validity of the presented results. If no mention is made of the dependence of the spring constant on P(t), this means that y = 0 was assumed. Sample results of the analysis for the system without and with damping are presented in Figs 3 and 4. In Figure 3, a dashed line marks the boundaries of resonance areas in which vibrations grow infinitely (N = - I) and a solid line indicates the boundaries in which vibrations are periodic (N = 1). The graphs were plotted for different values of parameter p (different depth of load modulation) and parameter 6 = w,,/r, where o0 and r are the circular frequencies of the free vibration of the system and function F, respectively. The obtained results agree fully with those reported widely in the literature on the subject. Regions of parametric resonance with the dynamic character of the follower coefficient of specified amplitude x, = -‘A* = a3 = -a4 = 0.8 (solid line) included are presented in Fig. 5 against the first and the second area of instability (dashed line) shown in Fig. 4 and in Fig. 6 for the third and the fourth region. One should notice that the frequency of changes of the follower parameter was twice as high as the acting dynamic load P(r). The first region of
Stability of simple discrete systems
659
parametric resonance generated by the dynamically (a) . ql, @ variable follower parameter CC,= - u2 = g1 = -CC+= tl for the constant (static) load P(t) (PI =p2 = p3 =p4 = 0.3) is shown in Fig. 7; the solid line represents the solution without damping and the dashed line-a sample solution with damping h = 0.03. Several further diagrams show the influence of the extent of the amplitude of the dynamic follower parameter on the behavior of the boundaries of parametric resonance regions for dynamic load P(t) (b) ql’ with fixed parameters. The diagrams allow one, 0.2 among other things, to answer the question whether 0.1 the introduction of a variable follower parameter into 0 the system can effect changes in the boundaries of parametric resonance regions. Figure 8 shows the -0.1 relationship between the magnitude of parameter CC -0.2 (x, = --a? = a3 = --x4 = U) and the extension of the four main regions of parametric resonance for dynamic load P(t) with a specified magnitude of parameter p = 0.6 exemplary @, = p: = -pJ = -p4 = p). Selected higher (the seventh and eight) parametric resonance regions are @ shown in Fig. 9 while Fig. 10 shows the ninth such region for the same problem parameters as in Fig. 8. The introduction of a dynamically variable follower parameter results in increased extent of the low -0.75 -0.5 -0.25 0 025 0.5 0.75 parametric resonance regions. For higher regions, the Fig. 18. Diagrams y, and q2 (a) as a function of time ( influence of a variable follower parameter in some of and phase portraits q, and q, (c) for (b) its variation ranges may turn out to be stabilizing f(r)
= 0.5 + 0.5 cos(2.251)
and r(t) = I + 0.5 cos(2.27r).
(Figs 9 and 10) which is evident from the distinct narrowings of the analyzed regions. Damping may even result in the multiple breaking of the continuity of the boundaries of parametric resonance regions and thus bring about their fading in certain ranges of values of follower parameter amplitudes. This situation is illustrated with an example of the boundaries of the tenth region in Fig. I I. If one takes into account the relationship between rigidity and acting load 0 # 0) in the analysis, then the extent of parametric resonance regions may increase or decrease as a function of the magnitude of parameter ;‘. This situation is shown in Fig. 12 using as an example the first of the analyzed regions. Figures 13 and 14 show the boundaries of one of the higher regtons of parametric resonance as a function of the value of the dynamic follower parameter for different values of the coefficient *,I.
0.15 0.1 0.05 0 -0.05 -0.1 -0.15
0.2 4.
0.1
NONLINEAR MODEL WITH TWO DEGREES OF FREEDOM
0 -0.1 -02 -0.4
-0.2
0
0.2
0.4
Fig. 17. Diagrams q, and q2 (a) as a function of time I and phase portraits 4, and (b) q2 (4 for P(r) = 0.5 + 0.5 cos(2.25f) and a(r) = I + 0.5 cos(2.28f).
Let us consider the nonlinear model of a pendulum with 2 d.f. shown in Fig. 1b. The model consists of two elastically connected massless bars of the same length I and two point masses 2m and nr. It is assumed that small imperfections e, and ez may occur in the system that determine its initial nonstress position. Quantities q, and q2 defining the revolutions of the
W. Glabisz
660
system’s bars are adopted as the system’s generalized coordinates. It is assumed that the static or dynamic nonconservative force P(t), whose action direction is determined by the dynamic follower parameter a(t), constitutes the pendulum’s load. It is assumed that the linear viscous damping that occurs in joints A and B of the system is described by parameters c I and c 2, and the nonlinear moments MA and M, that appear in these joints can be described by the following relationships: M,=k,,(q,
-e,)+k,,(q,
-e,)?+k,,(q,
-e,)3>
+k23(q2-e2-91+el)33
(14)
where coefficients k, characterize the rigidities of elastic constraints. Considering possible large deflections of the pendulum formulas for its kinetic energy E, potential energy V and external load work increment Q, on the linear parts of the generalized coordinates displacement increments (variations) 6q, can be expressed as
E=
qLq2
0
10
20
30
40
50
60
'70
0.06 0.04 0.02 0
92
-0.02
T (347+ 4:) + m12cj, coscq, cj2
q2)r
V=fk,,(q,-e,)2+4k,2(q,-e,)3 + ik,,(q,- e,1’ + fk2,(q2-
ql
e2 - q, -t- e, )?
-0.04
Fig. 20. Diagrams q, and q? (a) as a function of time I and phase portraits y, (b) and 9> (c) for P(r) = 0.6007 and r(t) = 0.8.
ql*@
-
/ 0
mgWco@, 1+ cos(q2)l,
1
Qdq,
= P(t)r{sin(q,)cos[cc(r)q,l
-
sW(t)q21cos(ql)Nh
+
P(t)l(sin(q2)cos[a(t)q21
-I
50
100
150
200
(b)
ql' 2 1 0
ql
- sinb (t)q21cM2)Pq2
-1 -2-
-3
-2
-1
q2
0
1
2
3
Cc)
q2
-4
-2
0
2
4
Fig. 19. Diagrams q, and yz (a) as a function of time I and phase portraits q, (b) and q2 (e) for P(f) = 0.5 + 0.5 cos(2.251) and a(r) = 1 + 0.5 cos(2.251).
c,4,@,- c2(42-4,)(h-~q,L
(15)
where g denotes gravitational acceleration. The above formulas can serve as the basis for the derivation of Lagrange equations of motion which then (under assigned initial conditions) can be subjected to numerical integration, or they can constitute a starting point for the application of a direct method based on Hamilton’s law. This second approachpresented, for instance, in the works of Borri et al. [33] and Glabisz [34]-was adopted for the further analysis where the effect of gravitation was neglected, which does not diminish the generality of the obtained results. Formulas from the author’s quoted
Stability
of simple discrete
paper, generalized to cover the case of the occurrence of the dynamically variable follower parameter a(t), were used in the solution. If a variable in time follower parameter is included, this leads to the modification-from the viewpoint of the nonlinear equations that describe the system’s motion--of some coefficients of these equations which become time functions and so the whole problem reduces itself to a complex parametric system of nonlinear differential equations.
systems
661
0.6 0.5 0.4 0.3 0.2 0.1
t
0 0
5
10
15
20
25
30
35
(b)
d’
ANALYSIS OF SYSTEM WITH TWO DEGREES OF FREEDOM
5. NUMERICAL
The numerical analysis of a system with 2 d.f. concentrates chiefly on the search for solutions at the boundaries of dynamic instability, taking into account the dynamic character of the follower parameter. The determination of the boundaries of parametric resonance in this case is associated with a gigantic numerical analysis and these boundaries depend strongly-both qualitatively and quantitatively-on even slight changes in the problem’s parameters. The selected results presented further on illustrate the possible influence of a variable follower parameter on the character of the obtained results. All the figures that follow show consistently the courses of the variation-in time t measured in
(a)
¶l, 92 0.25 0.2 0.15 0.1 0.05
t
0 -0.05 0
20
40
60
80
100 120 140
ql
-0.05
0
0.05
0.1
0.15
0.2
Fig. 21. Diagrams 4, and q2 (a) as a function of time t and phase
portraits
q, (b) and q2 (c) for P(t) = 0.6007 z(f) = 0.8 + 0.1 cos(4r).
and
91 0.1
0.2
0.3
0.4
0.5
0.6
0.06 0.04 0.02 0
$2
-0.02 0
0.1
0.2
0.3
0.4
0.5
Fig. 22. Diagrams q, and yz (a) as a function of time phase portraits q, (b) and q2 (c) for f’(r) = 0.6007 a(t) = 0.8 + 0.1 cos(lt).
I and and
seconds-of coordinates q, and q2 measured in radians (parts (a) of the figures) and phase portraits of the first (q,) and second (q2) generalized coordinate (parts (b) and (c), respectively of the figures). Then m=I=k,,=k,,=l, k,, = -2.5, kzz = -0.75, k,, = kz3 = 0, e, = -e2 = 0.05, c, = C:= 0 and the initial conditions 9,” = -4:” = 0.05 and Q,O= -& = 0 were assumed. Figures 15-19 show solutions for the dynamically variable force P(t) = 0.5 + 0.5 cos(2.25t) and different values of one of the parameters (0) that describe the variable follower parameter a(t) = 1.0 + 0.5 cos(nt); the presented solutions are for parameter o = 0. o = 2.3, o = 2.28, o = 2.27 and o = 2.25, respectively. Stable solutions were obtained for the analyzed higher values of parameter o (o :, 2.3), and the solutions for parameters 0 i o < 2.25 turned out to be unstable. A stable solution was understood as one in which displacements remained restrained [35]. As it was demonstrated in the author’s paper cited before [34], the analyzed system, assuming that a = 0.8, loses its stability at static load Pr.,= 0.600675. Figure 20 shows a solution for P = 0.6007 > Pkr(~ = 0.8). Figures 21 and 22 demonstrate solutions when the dynamic character of follower parameter cc(t) = 0.8 + 0.1 cos(ot) was incorporated for o = 4 and o = I, respectively. Similarly, as in the case of dynamic loading, stable solutions were obtained for o > 4 and unstable solutions--o < 1.
W. Glabisz
662 6. RECAPITULATION
Results of an analysis of the boundaries of parametric resonance regions have been presented for simple discrete systems described by linear or nonlinear differential equations with some periodically variable follower parameters that characterized nonconservative loads acting on either static or dynamic systems. In the case of a linear system with 1 d.f., the presented results were obtained by analyzing exact, closed formulas for determining the boundaries of parametric resonance. The formulas were obtained with the help of the Mathematics packet. The numerical analysis for a nonlinear system with 2 d.f. was carried out using a direct approach based on Hamilton’s law. The results of the analysis of the system with 1 d.f. were treated as the first approximation of selected, potential responses of the system in situations-which have not been analyzed in the literature yet-when dynamically variable follower parameters are taken into account. The presented results allow us to make several important observations: (1) the boundaries of the parametric resonance regions can change substantially under the action of the dynamic, nonconservative load follower parameter (Fig. 5); (2) the change of the boundaries of parametric resonance may be different for different resonance regions; these changes are particularly visible for higher regions (Figs 5 and 6); (3) the influence of viscous damping is similar as for systems without variable follower parameters-it causes, among other things, the narrowing of the areas of parametric resonance (Fig. 7); (4) the taking in of the dynamic character of follower parameters leads-as the parameter’s amplitude increases-to the widening (destabilization) of the first regions of parametric resonance (Fig. 8); (5) the inclusion of dynamic follower parameters in the analysis of higher regions of parametric resonance may lead, in certain ranges of its amplitude, to their destabilization of stabilization (Figs 9 and IO); (6) damping that always occurs in real systems can cause the disruption (depending on the follower parameter’s amplitude) of the continuity of particularly the higher parametric resonance regions of systems with dynamically variable follower parameters (Fig. 11); this phenomenon is not observed for the basic (first) regions of resonance; (7) the inclusion, in a system with a dynamically variable, nonconservative load follower parameter, of simple relationships between rigidity and the acting load level can cause-depending on the character of such a relationship-both the destabilization and the stabilization of parametric resonance regions (Fig. 12), particularly manifest in the case of higher regions (Figs 13 and 14); (8) the response of nonlinear systems is highly sensitive to even small changes in the frequency of the
follower parameter, which may result in the stabilization or destabilization of the system (Figs 16-19); (9) in the case of nonlinear systems, it is possible to stabilize an unstable system by introducing a dynamically variable follower parameter with precisely specified parameters (Figs 20-22). One should treat the presented results and the observations that follow from them only as the initial outcomes of the analysis of nonconservative systems with dynamically variable follower parameters. One can expect further research in this field to discover many new qualitatively significant and as yet not analyzed phonomena both in precritical states and at the boundaries of simple and combined parametric resonances.
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