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Analytical solutions of elasto-plastic systems
Journal ofSound and Vibrarion (1990) 142(l), 175-182
ANAL,YTICAL SOLUTIONS OF ELASTO-PLASTIC SYSTEMS 1.
INTRODU(‘TION
In the study of dynamic analysis, different models with various restoring force characteristics are used to describe the resistant members under loading. For general constitutive laws and excitations, the time integration method is often used. This is especially useful in determining the structural responses under earthquake loading. On the other hand, the steady state responses are important in determining the long-term behaviour and material strength under cyclic loading (e.g., fatigue, creep, etc.). The steady state responses of non-linear systems have received much attention in the past few decades. When the geometric non-linearity is considered, the material is elastic and the restoring forces are non-linear functions of displacements. The general solutions have been studied by means of the perturbation method [ 11, the method of multiple scales [ 11, the harmonic balance method [2] and the Krylov-Bogoliubov method of averaging [3]. When the inelastic material non-linearity under periodic excitation is of interest, the system exhibits a phenomenon of hysteresis so that the restoring force is a function of the displacement and the sign of the velocity. The restoring force in general is modelled by a multi-linear relationship for the two directions of loading and unloading. The equivalent viscous damping to model the energy dissipation due to the hysteretic loop is often used. Caughey [4] used the method of averaging to study the response of bi-linear yielding oscillators. Jennings [5] and Iwan [6] used the same approach to study the periodic response of a general yielding oscillator to harmonic excitation. DebChaudhury [7] obtained the response by a linearized iterative method. Badrakhan [8] used the trace method to transform the hysteresis function to an equivalent viscous damping evaluated from the area enclosed by the hysteresis cycle. A generalized Duffing equation with amplitude dependent coefficients resulted. General non-linear equation solvers can be applied to the equation directly. These methods are useful in studying random vibration of hysteretic systems [9]. In this note, a single-degree-of-freedom system exhibiting an elasto-plastic behaviour under periodic excitation is considered. The hysteretic function is approximated by a piecewise linear function. The analytical transient responses of the non-linear governing differential equation are obtained iteratively by solving one set of linear differential equations at a time. Steady state periodic solutions’correspond to the fixed point solutions of the iterative maps. Various kinds of superharmonic responses are studied by setting appropriate constraints to result in a set of non-linear algebraic equations. The superharmanic responses are classified as elasto- or plasto-superharmonic if there are turning points within the elastic or yielded range respectively. Co-existence of elasto- and plastosuperharmonic response is possible. This method has the advantages that the problem is reduced to an equivalent iterative map and no time integration is required. The steady state solutions are determined directly. The non-linear algebraic equations are solved by the Newtonian algorithm. For a given approximate solution, an improved solution is obtained by iterations. The neighbouring solutions with slightly different system parameters are obtained by augmentations. The steady state solutions for different combinations of system parameters can be studied. The steady state solution characteristics over a parametric space are of interest. Instead of evaluating the steady state solutions by repeated application of augmentations and 175 0022-460X/90/190175+08
$03.00/O
@ 1990 Academic
Press Limited
176
LETTERS
TO
THE
EDITOR
iterations, the boundaries separating various characteristic zones are developed by the method of parametric unfolding. All system parameters are allowed to various simultaneously with constraints defining the boundaries so that the characteristic diagram is constructed in an optimal manner. The solution characteristic for different combinations of system parameters can be read directly from the diagram. 2.
PIECEWISE
Consider a single-degree-of-freedom
LINEAR
STIFFNESS
SYSTEM
system,
ii+25oli+f(u,~)=qcos(vt+~,),
(1)
where 4’is the damping ratio, w is the natural frequency corresponding to the instantaneous linear stiffness, q is the amplitude of the periodic excitation having frequency V, 4, is the excitation phase angle and f(u, ri) = F(u) sgn (ti) is the restoring force, where F(u) is piecewise linear with u. Some possible forms of the restoring forces being considered are shown in Figure 1. Figures l(a) and l(b) show the restoring forces for the elastic systems with linear and bi-linear stiffness respectively. The restoring forces are single valued functions of displacement. Figure l(c) represents a system with hysteretic loop and Figure l(d) exhibits a system with non-symmetric constitutive law. The systems are non-conservative and energy is dissipated. In the figures, each straight line can be generalized to a piecewise linear function. The slope of each line segment i is denoted by k; so that the natural frequency corresponding to the line segment is given by oi = dk, and the intersection of the line segment with the u axis is denoted by i&. The static equilibrium position tii, when the restoring force is zero, is determined initially and remains unchanged along the path. f= ki(u - t7i) is the instantaneous restoring force at displacement u along the path kje Because of the piecewise linear stiffness, when the deformation path is along ki, equation (1) can be written as ii+2&oiti+ki(u-n,)=qcos[v(r-ti)+4,].
(cl
(d) f(u.ir)
Figure
f(u.ci)
I. Some possible forms of the restoring forces.
(2)
LETTERS
TO THE
177
EDITOR
The time for the excitation in equation (2) is shifted iteratively and the compensating phase angle is denoted by 4, = 4, + Vti. The system configuration at time t, = 0 is defined by u, , ti, , 4, and f, , with ti, = u, -f,/k,. The analytical solution of equation (2) at time t, with ti < t < ti+l, along the path ki, is given by [lo] (3) where dots denote derivatives with respect to time t, hi(~) = exp (-50,~) sin (&7)/h,,
Af=(1-52)wf,
r=t-ti,
V,(T)=U;+qiCOS(~7-_;+~;),
g,(7)=exp(-5Wi7)COS(hiT)+5W,h;(7), qf= q2/[(wf-
v’)2+45*wfv2],
8, = tan’ [2&~/(~f-
v’)]
and
At time fi+l, equation (3) represents an iterative map for the response u, (4) where Ti = ti+l - ti and U;= U(ti)* There are two points, called the limit points, to be considered in choosing the stiffness path: (i) the yield point-ti( t) does not change sign in the interval ti < t < ti+, and ui+, reaches a yield point, and the stiffness path is sequential; (ii) the zero velocity point-u(t) becomes zero at the first time in the same interval and a new path has to be initiated there. The two conditions constitute the algebraic equations to be solved for the termination time for the present path. in equation form, either
ui+l =Ui+2Ai
or
%+I = ui -24,
or
ti,+, = 0,
(5)
where Ai is the yielding increment to be discussed in the following section. 3.
PERIODIC
SOLUTIONS
periodic solutions of equation (1) correspond to the periodic points of the iterative maps in equations (4). To obtain a periodic steady state solution analytically, additional constraints are added and a set of non-linear algebraic equations is resulted. For a 2n-iteration fixed point solution, i.e., when a periodic solution requires 2n iterations, the constraints are, by considering the presence of symmetry, The
u, = -u,
and
&=+,+r
or
T,+T>+*
* *+T,,=?T/V.
(67)
With these two equations of constraint, the 2n iterative equations and the n + 1 limiting state equations in equations (4) and equations (5) respectively, the 3n +3 unknowns, which are ( ui, ri,), i = 1, n + 1, TV, i = 1, n, and 4, , can be determined. Since the algebraic equations are non-linear, a direct solution procedure is not available. The Newtonian algorithm has been adopted to solve the equations. The steady state solutions of similar nature but with slightly different system parameters can be studied incrementally by using the previous solution as an approximate solution. Note that a solution so obtained may not be physically realistic, since within one iteration
178
LETTERS
TO THE
EDITOR
limit points may be exceeded without being noticed. To accomplish this, the following procedures are adopted. When a solution is obtained, the time trajectory is checked to ensure that the solution is physically possible without either exceeding the yield points or reversing the velocities. Reformulation of the above equations is required to incorporate these changes. A more detailed discussion on the formulation is presented in the numerical examples. 4.
NEWTONIAN
ALGORITHM
The solution for the non-linear algebraic equations (4)-(7) cannot be obtained directly. The Newtonian algorithm has been adopted here. For a given approximate solution, an improved solution is obtained iteratively until the residual error is acceptably small and the solution is converged. Define an unknown variable vector {x} and a residual vector {R}, which have 3( n + 1) elements, as
{X}=[~,,~,,~,,U*,~*,~*,...,U,,~,,~”,U,+,,~,lT, R~i+1~~i~7~~+~i~Ti~~Ui~v~~o~]+hi~Tii)~li~~zj~~0~1~Ui+I~ R,i=Zji(Ti)+fi(Ti)[tli-Vi(O)]fI;i(T;)[lii-tji(O)]-tii+l, R 2nti
=
or
uj+2Ai-ui+,
or
ui-2Ai-ui+, R 3n+2
=
ul+
t&+,=0,
i=l,n+l,
%+I,
R 3n+3=T,+T2+*‘*+T,,-%-/U,
(8)
and the system parameter vector {y} = [q v]‘. 5.
CHARACTERISTIC
DIAGRAM
For neighbouring solutions with slightly different system parameters, {Ay} is non-zero. The previous solution is used as an approximate solution and a new solution is obtained by iterations. The process is called an augmentation. An alternative application of augmentations and iterations can be used for parametric studies. To study the solution characteristics over a parametric space, the method of parametric unfolding can be used. Instead of finding all the solutions over the parametric space of interest, boundaries separating the qualitatively different solutions are constructed parametrically. Additional constraints are included so that one of the system parameters is dependent on the others. A diagram containing regions of different solution characteristics is called a characteristic diagram. The solution characteristics for various combinations of parameters can be read directly from the characteristic diagram. The possibility of an occurrence of a particular response is suggested from the diagram. 6.
NUMERICAL
EXAMPLE
Consider a bi-linear hysteretic system with elastic stiffness k, = 1.0, yielded stiffness forcing frequency Y= O-2. Different superharmonic responses are given by different levels of excitation. The minimum excitation which gives elasto-plastic response is J[ (1 - v’)’ + 0*4v’] = 0.961. A time history for the excitation amplitude q = 0.9 with zero initial displacement, velocity and excitation phase angle is shown in Figure 2. The yielding limit is exceeded during the transient stage, where the yielded portion is shown by the dotted line, and an irrecoverable strain is observed since the vibration is not symmetric about the origin. k, = 0.1, yielding limit A, = 1.0, damping ratio 5 = O-1 and excitation
LETTERS
TO
THE
179
EDITOR
2.0
1.5
+j 3 c x
1.0
5
0.5
0.0
-0.5
I
0
10
I
20
I
I
30
40
/
1
50
60
I
70
I
I
60
90
)O
Time Figure 2. Time history for a bi-linear
system; k, = 1.0, kz = 0.1, 3, = 1.O,5 = 0.I, w = 0.2, q = 0.9.
Figures 3(a)-(d) show the steady state solutions for the excitation amplitudes q = 1.0, 1.1, l-2 and 1a3 respectively. They are obtained incrementally with the elastic solution as an initial approximation. Different kinds of superharmonic solutions are observed. Solid lines and dotted lines denote elastic and yielded portions respectively. Figure 3(a) shows a simple elasto-plastic response similar to a cosine curve for low excitation level. Figure 3(b) shows an elasto-superharmonic response which has turning points in the elastic range. The total displacement before yielding remains the same, that is 24,. Figure 3(c) shows a plasto-superharmonic response which has turning points in the elastic and the yielded portions so that the stiffness restores to k, for a short while. The elastic range ends as soon as the displacement returns to its original reversing point. Figure 3(d) shows
:’
(b)
-2.5 -3,5-
-
; -
-2.5
-
-4.5
-
“‘;,.
..,r/
,
,
Cd)
‘....,
5-
0.5 -0.5
’
:
7-
’
9
:.
2.5
’
“...,
3l-l: :’ ..._...’
-3 -5-
:’ :’
,..’
.. . .
:.
-7-
Timr/(2r/v) Figure 3. Steady state response for a bi-linear (b) q=l.l;
system; k, = 1.O,k, = 0.1,3, = 1.O,J'= 0.1, Y = 0.2. (a) q = I.O; (c) q=1.2; (d) 9=1.3.
180
LETTERS
TO THE
EDITOR
an elasto-plastic trajectory without turning point. The curve is similar to a cosine curve because the dominating stiffness is the yielded stiffness. For symmetric vibration, only half of the cycle is considered. For elasto-plastic solutions, the stiffness path is k, + k, . Equations (8) are explicitly written as U 1+1
~~“~~~i~~~l~i~~iTi)+C~i~iri~~~l~~~7i~+~i~~;~~
ii+1=[U,-Ui(O)]gi(Ti)+[I;lj_i)i(O)]~i(7j)+i)j(7i), ti, = 0,
u~=u,--2A,,
ti, =o,
i=l,2,
u3=-u,
7, + 72= lT/ lJ,
3
+i+, = 4i + VT;, J;+I =h + ki(ui+, - Ui), i = 1,2, and f, = the directly dependent variables, one reduces the number of unknowns to five, which are u, , 4,, ti2, T, and r2, and to be determined from
with
auxiliary
equations
(k, - k2)A, + k2u, .Eliminating
[u,-u,(~)~~,(T,)-~,(~)~~(T,)+U~(T,)-(U,-~A,)=~, [UI [u,+W
-
h(o)li,(T,)
-
&(o)h(Td+
-
~2(O)l&(T2)+[~2-~2(0)lh2(T2)+
i’,(T,)
-
c2 = 0,
02(72)+%
=o, V( 7, + 72) = ?T.
b,+2A,-~2(0)1~2(T2)+[~2-+2(0)1~2(T2)+~2(T2)=’&
For small q, the elastic solution can be used as an initial approximation and the forcing amplitude q is augmented through the range of interest. The above equations are valid for 0.96 < q c 1.08. A turning point is born when q = 1.08. Elasto-superharmonic solutions are resulted by increasing f further. The stiffness path becomes k, + k, + k, + k2. Zero velocity critical states are reached within the elastic portion so that a reformulation is required to incorporate these additional turning points giving superharmonic responses. To this end, the unknowns are ul, u2, u3, ti4, c$,, T, , TV, 73 and TV, and the equations are explicitly given as follows: [UI - u,(O)lg,(r,) - u,(O)h,(r,)+ 1% [U2 [U2-
U,(o)l&(T,)
U2(o)lg2(T2)
-
U2(0)1g2(72)
~,(o)h(T,)+
c2(0)h2(72) -
c2(02(72)+
U,(r,) &(T,) + U2(72)
U2 = 0, = 0,
-
c2(72)
U3 = 0, = 0,
[U3-%(O)lS3(73)-~3(0)h3(73)+~3(73)-(~,-24,)=0, [U3[U,
U3(0)li,(T,)
-
c3(0)h3(73)+
-24,-~4(0)lg,(T4)+[~4-~4(0)1h,(T4)+U4(T4)+U1
br’%-~4(0)1~4(T4)+[~4-~4(0)1~4(T4)+~4(T4)=~,
c3(73)-i4=0, =o, V(T,+T2+73+74)=7K
For further increasing q, the critical point wzmoves into the yielded portion at q = 1.16. Plasto-superharmonic responses are obtained. The stiffness path becomes k, + k2 + k, + k, -* k,. Another set of equations is required to calculate the responses. When q reaches 1.21, the turning points have collided and vanished. Simple elasto-plastic responses are obtained. The stiffness path returns to k, + kl. Figure 4 shows the stationary points described above. Solid lines and dotted lines denote locus of stationary points in the elastic and plastic range respectively.
LE’l-l-ERS
TO
THE
181
EDITOR
9 6?6-
1.00
1.10
1.05
1.15
1.20
1. 10
1.25
Forcing amplitude, 9 Figure 4. Response amplitude
against forcing amplitude
9. Other parameters
as Figures 2 and 3.
To study the effect of the external excitation to the system, the excitation force amplitude and the excitation forcing frequency are chosen as varying parameters. Critical points indicating the birth of inflection points, turning points entering the yielded portion, and collision of turning points are traced incrementally by the method of parametric unfolding. Equations (8) are used to solve for the system parameters on the boundaries. Figure 5 shows the characteristic diagram. The dashed lines correspond to the parameters which give elastic solutions with maximum amplitude. The solid lines correspond to the parameters which have an inflection point in the elastic range. The dotted lines correspond to the parameters which have a stationary yield point. The region below the dashed line gives elastic responses only. The region between the dashed line ane the solid line gives simple elasto-plastic responses without any turning point. In other words, for v > 0.03,
*
~_________-____
0.5
0.0
-------__________
-
0.00
1
0.05
1
010
I
0.15
I
0.20
,
I
0.25
030
0.35
0.40
Forcing frequency, Y Figure 5. Characteristic
diagram
for a bi-linear
system; k, = 1.0, k2 = 0.1, A, = 1.0, 5 =O.l.
182
LETTERS TO THE EDITOR
0.0
01
02
0.3 0.4 0.6 0.6
0.7 0.6 0.9
I.0
Time /(27r/v)
Figure 6. As Figure 3(c), but v = 0.1.
no superharmonic
solution is possible. Solid lines enclose a region which gives superharmanic responses. Elasto-superharmonic and plasto-superharmonic responses are found in the central region and the two side regions respectively. Near the central part, higher order superharmonic responses are possible. Department of Civil and Structural Engineering, University of Hong Kong,
A. Y. T. LEUNG T. C. FUNC
(Received 26 January 1990) REFERENCES
1. A. H. NAYFEH, and D. T. MOOK 1979 Nonlinear Oscillations, New York: Wiley-Interscience. 2. D. W. JARDON and P. SMITH 1983 Nonlinear Ordinary Differential Equations Oxford: Clarendon
Press.
3. J. A. SANDERS and F. VERHULST 1985 Averaging Methods in Nonlinear Dynamical Systems. New York: SpringerVerlag. 1960 Journal of Applied Mechnics 27, 640-643. Sinusoidal excitation of a 4. T. K. CAUGHEY
system with bi-linear hysteresis. 5. P. C. JENNINGS 1968 Journal of the Engineering Mechanics Division, American Society of Mechanical Engineers, 94, 103-l 16. Equivalent viscous damping for yielding structure. 6. W. D. IWAN 1973 International Journal of Nonlinear Mechanics 8, 279-287. A generalization
of the concept of equivalent
linearization.
7. A. DEBCHAUDHURY 1985 Journal of Engineering Mechanics, American Society of Mechanical Engineers 111,977-994. Periodic response of yielding oscillators. 1988 Journal of Sound and Vibration 125,23-42. Dynamic analysis of yielding 8. F. BADRAKHAN
and hysteretic systems by polynomial 9. F. BADRAKHAN
approximation.
1987 International Journal of Nonlinear Mechanics 22,315325.
Rational study of hysteretic systems under stationary random excitation. 10. A. Y. T. LEUNG 1988 Journal of Sound and Vibration 124, 135-139. Direct method for the steady state response of structures.
ANAL,YTICAL SOLUTIONS OF ELASTO-PLASTIC SYSTEMS 1.
INTRODU(‘TION
In the study of dynamic analysis, different models with various restoring force characteristics are used to describe the resistant members under loading. For general constitutive laws and excitations, the time integration method is often used. This is especially useful in determining the structural responses under earthquake loading. On the other hand, the steady state responses are important in determining the long-term behaviour and material strength under cyclic loading (e.g., fatigue, creep, etc.). The steady state responses of non-linear systems have received much attention in the past few decades. When the geometric non-linearity is considered, the material is elastic and the restoring forces are non-linear functions of displacements. The general solutions have been studied by means of the perturbation method [ 11, the method of multiple scales [ 11, the harmonic balance method [2] and the Krylov-Bogoliubov method of averaging [3]. When the inelastic material non-linearity under periodic excitation is of interest, the system exhibits a phenomenon of hysteresis so that the restoring force is a function of the displacement and the sign of the velocity. The restoring force in general is modelled by a multi-linear relationship for the two directions of loading and unloading. The equivalent viscous damping to model the energy dissipation due to the hysteretic loop is often used. Caughey [4] used the method of averaging to study the response of bi-linear yielding oscillators. Jennings [5] and Iwan [6] used the same approach to study the periodic response of a general yielding oscillator to harmonic excitation. DebChaudhury [7] obtained the response by a linearized iterative method. Badrakhan [8] used the trace method to transform the hysteresis function to an equivalent viscous damping evaluated from the area enclosed by the hysteresis cycle. A generalized Duffing equation with amplitude dependent coefficients resulted. General non-linear equation solvers can be applied to the equation directly. These methods are useful in studying random vibration of hysteretic systems [9]. In this note, a single-degree-of-freedom system exhibiting an elasto-plastic behaviour under periodic excitation is considered. The hysteretic function is approximated by a piecewise linear function. The analytical transient responses of the non-linear governing differential equation are obtained iteratively by solving one set of linear differential equations at a time. Steady state periodic solutions’correspond to the fixed point solutions of the iterative maps. Various kinds of superharmonic responses are studied by setting appropriate constraints to result in a set of non-linear algebraic equations. The superharmanic responses are classified as elasto- or plasto-superharmonic if there are turning points within the elastic or yielded range respectively. Co-existence of elasto- and plastosuperharmonic response is possible. This method has the advantages that the problem is reduced to an equivalent iterative map and no time integration is required. The steady state solutions are determined directly. The non-linear algebraic equations are solved by the Newtonian algorithm. For a given approximate solution, an improved solution is obtained by iterations. The neighbouring solutions with slightly different system parameters are obtained by augmentations. The steady state solutions for different combinations of system parameters can be studied. The steady state solution characteristics over a parametric space are of interest. Instead of evaluating the steady state solutions by repeated application of augmentations and 175 0022-460X/90/190175+08
$03.00/O
@ 1990 Academic
Press Limited
176
LETTERS
TO
THE
EDITOR
iterations, the boundaries separating various characteristic zones are developed by the method of parametric unfolding. All system parameters are allowed to various simultaneously with constraints defining the boundaries so that the characteristic diagram is constructed in an optimal manner. The solution characteristic for different combinations of system parameters can be read directly from the diagram. 2.
PIECEWISE
Consider a single-degree-of-freedom
LINEAR
STIFFNESS
SYSTEM
system,
ii+25oli+f(u,~)=qcos(vt+~,),
(1)
where 4’is the damping ratio, w is the natural frequency corresponding to the instantaneous linear stiffness, q is the amplitude of the periodic excitation having frequency V, 4, is the excitation phase angle and f(u, ri) = F(u) sgn (ti) is the restoring force, where F(u) is piecewise linear with u. Some possible forms of the restoring forces being considered are shown in Figure 1. Figures l(a) and l(b) show the restoring forces for the elastic systems with linear and bi-linear stiffness respectively. The restoring forces are single valued functions of displacement. Figure l(c) represents a system with hysteretic loop and Figure l(d) exhibits a system with non-symmetric constitutive law. The systems are non-conservative and energy is dissipated. In the figures, each straight line can be generalized to a piecewise linear function. The slope of each line segment i is denoted by k; so that the natural frequency corresponding to the line segment is given by oi = dk, and the intersection of the line segment with the u axis is denoted by i&. The static equilibrium position tii, when the restoring force is zero, is determined initially and remains unchanged along the path. f= ki(u - t7i) is the instantaneous restoring force at displacement u along the path kje Because of the piecewise linear stiffness, when the deformation path is along ki, equation (1) can be written as ii+2&oiti+ki(u-n,)=qcos[v(r-ti)+4,].
(cl
(d) f(u.ir)
Figure
f(u.ci)
I. Some possible forms of the restoring forces.
(2)
LETTERS
TO THE
177
EDITOR
The time for the excitation in equation (2) is shifted iteratively and the compensating phase angle is denoted by 4, = 4, + Vti. The system configuration at time t, = 0 is defined by u, , ti, , 4, and f, , with ti, = u, -f,/k,. The analytical solution of equation (2) at time t, with ti < t < ti+l, along the path ki, is given by [lo] (3) where dots denote derivatives with respect to time t, hi(~) = exp (-50,~) sin (&7)/h,,
Af=(1-52)wf,
r=t-ti,
V,(T)=U;+qiCOS(~7-_;+~;),
g,(7)=exp(-5Wi7)COS(hiT)+5W,h;(7), qf= q2/[(wf-
v’)2+45*wfv2],
8, = tan’ [2&~/(~f-
v’)]
and
At time fi+l, equation (3) represents an iterative map for the response u, (4) where Ti = ti+l - ti and U;= U(ti)* There are two points, called the limit points, to be considered in choosing the stiffness path: (i) the yield point-ti( t) does not change sign in the interval ti < t < ti+, and ui+, reaches a yield point, and the stiffness path is sequential; (ii) the zero velocity point-u(t) becomes zero at the first time in the same interval and a new path has to be initiated there. The two conditions constitute the algebraic equations to be solved for the termination time for the present path. in equation form, either
ui+l =Ui+2Ai
or
%+I = ui -24,
or
ti,+, = 0,
(5)
where Ai is the yielding increment to be discussed in the following section. 3.
PERIODIC
SOLUTIONS
periodic solutions of equation (1) correspond to the periodic points of the iterative maps in equations (4). To obtain a periodic steady state solution analytically, additional constraints are added and a set of non-linear algebraic equations is resulted. For a 2n-iteration fixed point solution, i.e., when a periodic solution requires 2n iterations, the constraints are, by considering the presence of symmetry, The
u, = -u,
and
&=+,+r
or
T,+T>+*
* *+T,,=?T/V.
(67)
With these two equations of constraint, the 2n iterative equations and the n + 1 limiting state equations in equations (4) and equations (5) respectively, the 3n +3 unknowns, which are ( ui, ri,), i = 1, n + 1, TV, i = 1, n, and 4, , can be determined. Since the algebraic equations are non-linear, a direct solution procedure is not available. The Newtonian algorithm has been adopted to solve the equations. The steady state solutions of similar nature but with slightly different system parameters can be studied incrementally by using the previous solution as an approximate solution. Note that a solution so obtained may not be physically realistic, since within one iteration
178
LETTERS
TO THE
EDITOR
limit points may be exceeded without being noticed. To accomplish this, the following procedures are adopted. When a solution is obtained, the time trajectory is checked to ensure that the solution is physically possible without either exceeding the yield points or reversing the velocities. Reformulation of the above equations is required to incorporate these changes. A more detailed discussion on the formulation is presented in the numerical examples. 4.
NEWTONIAN
ALGORITHM
The solution for the non-linear algebraic equations (4)-(7) cannot be obtained directly. The Newtonian algorithm has been adopted here. For a given approximate solution, an improved solution is obtained iteratively until the residual error is acceptably small and the solution is converged. Define an unknown variable vector {x} and a residual vector {R}, which have 3( n + 1) elements, as
{X}=[~,,~,,~,,U*,~*,~*,...,U,,~,,~”,U,+,,~,lT, R~i+1~~i~7~~+~i~Ti~~Ui~v~~o~]+hi~Tii)~li~~zj~~0~1~Ui+I~ R,i=Zji(Ti)+fi(Ti)[tli-Vi(O)]fI;i(T;)[lii-tji(O)]-tii+l, R 2nti
=
or
uj+2Ai-ui+,
or
ui-2Ai-ui+, R 3n+2
=
ul+
t&+,=0,
i=l,n+l,
%+I,
R 3n+3=T,+T2+*‘*+T,,-%-/U,
(8)
and the system parameter vector {y} = [q v]‘. 5.
CHARACTERISTIC
DIAGRAM
For neighbouring solutions with slightly different system parameters, {Ay} is non-zero. The previous solution is used as an approximate solution and a new solution is obtained by iterations. The process is called an augmentation. An alternative application of augmentations and iterations can be used for parametric studies. To study the solution characteristics over a parametric space, the method of parametric unfolding can be used. Instead of finding all the solutions over the parametric space of interest, boundaries separating the qualitatively different solutions are constructed parametrically. Additional constraints are included so that one of the system parameters is dependent on the others. A diagram containing regions of different solution characteristics is called a characteristic diagram. The solution characteristics for various combinations of parameters can be read directly from the characteristic diagram. The possibility of an occurrence of a particular response is suggested from the diagram. 6.
NUMERICAL
EXAMPLE
Consider a bi-linear hysteretic system with elastic stiffness k, = 1.0, yielded stiffness forcing frequency Y= O-2. Different superharmonic responses are given by different levels of excitation. The minimum excitation which gives elasto-plastic response is J[ (1 - v’)’ + 0*4v’] = 0.961. A time history for the excitation amplitude q = 0.9 with zero initial displacement, velocity and excitation phase angle is shown in Figure 2. The yielding limit is exceeded during the transient stage, where the yielded portion is shown by the dotted line, and an irrecoverable strain is observed since the vibration is not symmetric about the origin. k, = 0.1, yielding limit A, = 1.0, damping ratio 5 = O-1 and excitation
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179
EDITOR
2.0
1.5
+j 3 c x
1.0
5
0.5
0.0
-0.5
I
0
10
I
20
I
I
30
40
/
1
50
60
I
70
I
I
60
90
)O
Time Figure 2. Time history for a bi-linear
system; k, = 1.0, kz = 0.1, 3, = 1.O,5 = 0.I, w = 0.2, q = 0.9.
Figures 3(a)-(d) show the steady state solutions for the excitation amplitudes q = 1.0, 1.1, l-2 and 1a3 respectively. They are obtained incrementally with the elastic solution as an initial approximation. Different kinds of superharmonic solutions are observed. Solid lines and dotted lines denote elastic and yielded portions respectively. Figure 3(a) shows a simple elasto-plastic response similar to a cosine curve for low excitation level. Figure 3(b) shows an elasto-superharmonic response which has turning points in the elastic range. The total displacement before yielding remains the same, that is 24,. Figure 3(c) shows a plasto-superharmonic response which has turning points in the elastic and the yielded portions so that the stiffness restores to k, for a short while. The elastic range ends as soon as the displacement returns to its original reversing point. Figure 3(d) shows
:’
(b)
-2.5 -3,5-
-
; -
-2.5
-
-4.5
-
“‘;,.
..,r/
,
,
Cd)
‘....,
5-
0.5 -0.5
’
:
7-
’
9
:.
2.5
’
“...,
3l-l: :’ ..._...’
-3 -5-
:’ :’
,..’
.. . .
:.
-7-
Timr/(2r/v) Figure 3. Steady state response for a bi-linear (b) q=l.l;
system; k, = 1.O,k, = 0.1,3, = 1.O,J'= 0.1, Y = 0.2. (a) q = I.O; (c) q=1.2; (d) 9=1.3.
180
LETTERS
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an elasto-plastic trajectory without turning point. The curve is similar to a cosine curve because the dominating stiffness is the yielded stiffness. For symmetric vibration, only half of the cycle is considered. For elasto-plastic solutions, the stiffness path is k, + k, . Equations (8) are explicitly written as U 1+1
~~“~~~i~~~l~i~~iTi)+C~i~iri~~~l~~~7i~+~i~~;~~
ii+1=[U,-Ui(O)]gi(Ti)+[I;lj_i)i(O)]~i(7j)+i)j(7i), ti, = 0,
u~=u,--2A,,
ti, =o,
i=l,2,
u3=-u,
7, + 72= lT/ lJ,
3
+i+, = 4i + VT;, J;+I =h + ki(ui+, - Ui), i = 1,2, and f, = the directly dependent variables, one reduces the number of unknowns to five, which are u, , 4,, ti2, T, and r2, and to be determined from
with
auxiliary
equations
(k, - k2)A, + k2u, .Eliminating
[u,-u,(~)~~,(T,)-~,(~)~~(T,)+U~(T,)-(U,-~A,)=~, [UI [u,+W
-
h(o)li,(T,)
-
&(o)h(Td+
-
~2(O)l&(T2)+[~2-~2(0)lh2(T2)+
i’,(T,)
-
c2 = 0,
02(72)+%
=o, V( 7, + 72) = ?T.
b,+2A,-~2(0)1~2(T2)+[~2-+2(0)1~2(T2)+~2(T2)=’&
For small q, the elastic solution can be used as an initial approximation and the forcing amplitude q is augmented through the range of interest. The above equations are valid for 0.96 < q c 1.08. A turning point is born when q = 1.08. Elasto-superharmonic solutions are resulted by increasing f further. The stiffness path becomes k, + k, + k, + k2. Zero velocity critical states are reached within the elastic portion so that a reformulation is required to incorporate these additional turning points giving superharmonic responses. To this end, the unknowns are ul, u2, u3, ti4, c$,, T, , TV, 73 and TV, and the equations are explicitly given as follows: [UI - u,(O)lg,(r,) - u,(O)h,(r,)+ 1% [U2 [U2-
U,(o)l&(T,)
U2(o)lg2(T2)
-
U2(0)1g2(72)
~,(o)h(T,)+
c2(0)h2(72) -
c2(02(72)+
U,(r,) &(T,) + U2(72)
U2 = 0, = 0,
-
c2(72)
U3 = 0, = 0,
[U3-%(O)lS3(73)-~3(0)h3(73)+~3(73)-(~,-24,)=0, [U3[U,
U3(0)li,(T,)
-
c3(0)h3(73)+
-24,-~4(0)lg,(T4)+[~4-~4(0)1h,(T4)+U4(T4)+U1
br’%-~4(0)1~4(T4)+[~4-~4(0)1~4(T4)+~4(T4)=~,
c3(73)-i4=0, =o, V(T,+T2+73+74)=7K
For further increasing q, the critical point wzmoves into the yielded portion at q = 1.16. Plasto-superharmonic responses are obtained. The stiffness path becomes k, + k2 + k, + k, -* k,. Another set of equations is required to calculate the responses. When q reaches 1.21, the turning points have collided and vanished. Simple elasto-plastic responses are obtained. The stiffness path returns to k, + kl. Figure 4 shows the stationary points described above. Solid lines and dotted lines denote locus of stationary points in the elastic and plastic range respectively.
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EDITOR
9 6?6-
1.00
1.10
1.05
1.15
1.20
1. 10
1.25
Forcing amplitude, 9 Figure 4. Response amplitude
against forcing amplitude
9. Other parameters
as Figures 2 and 3.
To study the effect of the external excitation to the system, the excitation force amplitude and the excitation forcing frequency are chosen as varying parameters. Critical points indicating the birth of inflection points, turning points entering the yielded portion, and collision of turning points are traced incrementally by the method of parametric unfolding. Equations (8) are used to solve for the system parameters on the boundaries. Figure 5 shows the characteristic diagram. The dashed lines correspond to the parameters which give elastic solutions with maximum amplitude. The solid lines correspond to the parameters which have an inflection point in the elastic range. The dotted lines correspond to the parameters which have a stationary yield point. The region below the dashed line gives elastic responses only. The region between the dashed line ane the solid line gives simple elasto-plastic responses without any turning point. In other words, for v > 0.03,
*
~_________-____
0.5
0.0
-------__________
-
0.00
1
0.05
1
010
I
0.15
I
0.20
,
I
0.25
030
0.35
0.40
Forcing frequency, Y Figure 5. Characteristic
diagram
for a bi-linear
system; k, = 1.0, k2 = 0.1, A, = 1.0, 5 =O.l.
182
LETTERS TO THE EDITOR
0.0
01
02
0.3 0.4 0.6 0.6
0.7 0.6 0.9
I.0
Time /(27r/v)
Figure 6. As Figure 3(c), but v = 0.1.
no superharmonic
solution is possible. Solid lines enclose a region which gives superharmanic responses. Elasto-superharmonic and plasto-superharmonic responses are found in the central region and the two side regions respectively. Near the central part, higher order superharmonic responses are possible. Department of Civil and Structural Engineering, University of Hong Kong,
A. Y. T. LEUNG T. C. FUNC
(Received 26 January 1990) REFERENCES
1. A. H. NAYFEH, and D. T. MOOK 1979 Nonlinear Oscillations, New York: Wiley-Interscience. 2. D. W. JARDON and P. SMITH 1983 Nonlinear Ordinary Differential Equations Oxford: Clarendon
Press.
3. J. A. SANDERS and F. VERHULST 1985 Averaging Methods in Nonlinear Dynamical Systems. New York: SpringerVerlag. 1960 Journal of Applied Mechnics 27, 640-643. Sinusoidal excitation of a 4. T. K. CAUGHEY
system with bi-linear hysteresis. 5. P. C. JENNINGS 1968 Journal of the Engineering Mechanics Division, American Society of Mechanical Engineers, 94, 103-l 16. Equivalent viscous damping for yielding structure. 6. W. D. IWAN 1973 International Journal of Nonlinear Mechanics 8, 279-287. A generalization
of the concept of equivalent
linearization.
7. A. DEBCHAUDHURY 1985 Journal of Engineering Mechanics, American Society of Mechanical Engineers 111,977-994. Periodic response of yielding oscillators. 1988 Journal of Sound and Vibration 125,23-42. Dynamic analysis of yielding 8. F. BADRAKHAN
and hysteretic systems by polynomial 9. F. BADRAKHAN
approximation.
1987 International Journal of Nonlinear Mechanics 22,315325.
Rational study of hysteretic systems under stationary random excitation. 10. A. Y. T. LEUNG 1988 Journal of Sound and Vibration 124, 135-139. Direct method for the steady state response of structures.