Periodicity in the response of nonlinear plate, under moving mass

Thin-Walled Structures 40 (2002) 283–295 www.elsevier.com/locate/tws

Periodicity in the response of nonlinear plate, under moving mass M.R. Shadnam a, F. Rahimzadeh Rofooei a, M. Mofid b,*, B. Mehri c a

Department of Civil Engineering, Sharif University of Technology, Tehran, Iran Department of Civil Environmental Engineering, University of Kansas, Lawrence, KS 66045-2225, USA c Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b

Received 19 December 2000; received in revised form 1 June 2001; accepted 13 June 2001

Abstract The dynamics of nonlinear thin plates under influence of relatively heavy moving masses is considered. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development of vibration amplitude, which is Duffing’s oscillator with time varying coefficients. Through the application of Banach’s fixed-point theorem, the periodic solutions are predicted. The method presented in this paper is general so that the response of plate to moving force systems can also be considered.  2002 Published by Elsevier Science Ltd. Keywords: Vibration of continuous structures; Moving loads; Non-uniform elastic foundation; Nonlinear elastic foundation; Large deformation; Parimetrically exited system

1. Introduction The vibration of structures under the passage of moving loads is an important consideration in their design. The interaction between the passing load and the structure makes the dynamic response analysis very complex. To understand the phenomenon and to develop rational analysis and design procedures, a number of analytical * Corresponding author. Tel.: +1-785-864-3766; fax: +1-785-864-5631. E-mail address: [email protected] (M. Mofid). 0263-8231/02/$ - see front matter  2002 Published by Elsevier Science Ltd. PII: S 0 2 6 3 - 8 2 3 1 ( 0 1 ) 0 0 0 4 1 - 6

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and experimental investigations were carried out in the past [1–5]. In the analytical studies, the structure has commonly been modeled either as a beam or as a plate. Studies based on the beam model are reported in [1–4] while plate model has been used in [2,5]. A pertinent theory was proposed by Saigal [6] to determine the dynamic response of rectangular linear plates under moving masses. There are also several approximate techniques applied to the problem of moving mass, ranging from perturbation ones to discretization, cell and finite element models, an extended review of them and their related references are available in [7–9]. The validity of such methods depends on existence of a regular solution. Otherwise, the solution could be chaotic, i.e. the solution time history has a sensitive dependence on initial conditions. This sensitivity has implications for numerical computations, e.g. different step sizes, different numerical algorithms, or even the execution of the same algorithm on different machines will introduce small differences in the computed solution, which eventually lead to large deviations [10,11]. Consider the following excerpt from Poincare´ in his essay on Science and Method [12]: It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the later. Prediction becomes impossible. Periodic behavior is the most important regular solution. If the system is acted on by a periodic force, in the classical theory one assumes that the output will also be periodic. Also, nonlinear resonance theory depends on the assumption that periodic input yields periodic output. However, it is this postulate that has been challenged in the new theory of chaotic vibrations [10]. It will be shown that one cannot usually prove the existence of periodic solutions for the governing differential equation of the mechanical system under consideration. A full record of the problems in which inertial effects of the moving load could not be neglected is presented in [9]. A one-term Galerkin method on space is employed and a nonlinear ordinary differential equation is derived from the governing partial differential equation. The behavior of the system is shown to be characterized by a Duffing equation with time varying coefficients. Following the prospects of Mehri et al. [13,14] and Jalali et al. [15], the existence of periodic solutions using Banach’s fixed point theorem is proved. The analysis is carried out in detail for simply supported rectangular plates with stress free edges, however final formulas for analysis of plates, with some other boundary conditions are also presented. The technique is flexible and could easily be extended for analysis of plates, with various boundary conditions, carrying a moving mass.

2. Definition of the problem A rectangular plate with a moving mass and simply boundary conditions is considered. A relatively large mass with considerable inertia is moving on an arbitrary

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trajectory x=xo(t), y=yo(t) on the plate surface, as shown in Fig. 1. The modulus of rigidity D(x,y), and the mass per unit volume of the plate, r(x,y) is assumed to be constant with no damping present in the system. It is further assumed that the mass moving on the plate is a constant quantity. The initial conditions are: w(x,y,0) ⫽ wo(x,y) dw(x,y,0) / dt ⫽ uo(x,y).

3. Formulation of the problem The governing equation of the finite amplitude motion of a rectangular plate shown in Fig. 1 under mass M may be written as: ⵜ4w ⫹

冋冉 冊

ⵜ4F ⫽ E in which,

冋

r·h ∂2w p(x,y,t) ∂2 F ∂ 2 w h ∂2w ∂2F ∂2w ∂2F ⫽ ⫺2 ⫹ ⫹ D ∂t2 D D ∂x2 ∂y2 ∂x·∂y ∂x·∂y ∂y2 ∂x2

册

∂2w 2 ∂2w ∂2w ⫺ 2 2 ∂x·∂y ∂y ∂x

冉

册

(1) (2)

冊

∂2w p(x,y,t) ⫽ q(x,y,t) ⫹ M g⫺ 2 d(x⫺x0(t))d(y⫺y0(t)). ∂t

(3)

In these equations, x and y are Cartesian coordinates, w is the lateral deflection of the plate, and r and h are the lateral distributed force on the plate and r and h are mass density and thickness of the plate, respectively. E is the Young’s modulus of elasticity, D is bending stiffness, q(x,y,t) is general dynamic loading without inertial effects and d is the Dirac Delta function. Also, F is the Airy’s stress function. In derivation of Eqs. (1) and (2), both extensional and rotary inertia effects have been neglected.

Fig. 1.

A moving mass traveling along an arbitrary trajectory on the surface of a rectangular plate.

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For simply supported rectangular plate of sides a and b, the boundary conditions on w are, w ⫽ 0 and v

∂ 2 w ∂2 w ⫹ 2 ⫽ 0 at x ⫽ 0,a, ∂y2 ∂x

(4)

∂2 w ∂2 w w ⫽ 0 and 2 ⫹ v 2 ⫽ 0 at x ⫽ 0,b, ∂y ∂x

where n is the Poisson’s ratio. The boundary conditions for Airy stress function for the case of a plate with stress free edges become, ∂2F ∂2 F and ⫽ 0 at x ⫽ 0,a, ∂y2 ∂x·∂y . ∂2F ∂2 F ⫽ 0 at y ⫽ 0,b and ∂x2 ∂x·∂y

(5)

For simply supported rectangular plate of sides a and b, an approximate single mode assumption is employed in order to derive the decoupled equations for the dynamics of the time dependent amplitudes. mp·x np·y ·sin w(x,y,t) ⫽ h·A(t)·sin a b

(6)

where A(t) is a dimensionless function of time, and m and n are integers. The function (6) obviously satisfies the boundary conditions (4). For other boundary conditions the appropriate mode functions may be found in [16,17]. Employing Eq. (4) into Eq. (2), ⵜ4F ⫽

冉 冊 冋

册

E nmp2h 2 2 2mp·x 2np·y A (t) cos . ⫹ cos 2 ab a b

(7)

The Ritz–Galerkin method is used to obtain the solution of Eq. (7) and Eq. (1). The following solution for the stress function is assumed [18,19],

冉

冊冉

冊

2mp·x 2np·y 1⫺cos . F ⫽ F∗h2A2(t) 1⫺cos a b

(8)

The assumed solution (8) clearly satisfies the stress free edges boundary conditions (5). More general approaches to find the Airy functions could be found in [20,21]. The expressions for the stress function F and the values for w (6) satisfy the boundary conditions and Eq. (2). One cannot, however, expect that they will also exactly satisfy Eq. (7) [18,19]. Employing Eq. (8) with Eq. (7) and multiplying by:

冉

1⫺cos

冊冉

冊

2mp·x 2np·y 1⫺cos a b

and integrating over the surface of the plate results, F∗ ⫽ ⫺

E(mn / ab) . 8[(m / a)2 ⫹ (n / b)2]

(9)

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Employing Eqs. (6) and (8) into Eq. (1), multiplying the resulting equation by the corresponding selected mode shape of the lateral displacement and integrating over the surface of the plate results,

冋冉 冊 冉 冊 册 冉 冊 冋冉 冊 冉 冊 册

m 2 p 4D ⫹ r·a(t)h a 4Rmn(t) Eh2 ⫺ r.a(t)h2ab 2r·a(t)

A⫹

n 2 2 A⫽ b mnp 4 3 A / ab

m a

2

⫹

n b

(10)

2 2

where,

冕 冕 a

Rmn(t) ⫽

b

dx dy(q(x,y,t) ⫹ Mgd(x⫺x0(t))d(y⫺y0(t)))sin

0

mp·x np·y sin a b

(11)

0

and a(t) ⫽ 1 ⫹

mp·x0(t) np·y0(t) 4M sin sin . abr·h a b

(12)

Defining of the following time varying parameters;

冋冉 冊 冉 冊 册 冉 冊 冋冉 冊 冉 冊 册

P(t) ⫽

p 4D m r·a(t)h a

Q(t) ⫽

Eh2 mnp 2p·a(t) ab

F(t) ⫽

4Rmn(t) r·a(t)h2ab

2

⫹ 4

/

n b

2 2

m a

2

⫹

n b

2 2

Eq. (10) can be rewritten as, A¨ ⫹ P(t)A ⫹ Q(t)A3 ⫽ F(t)

(13)

Eq. (13) belongs to the class of “Duffing Equations” with time dependent coefficients. The coefficients are periodic if the trajectory of the mass on beam is periodic. The problem could be solved for other boundary conditions via the same method. The results of some of these problems are listed in the Table 1. The data in first and second rows has been obtained for m=n=1 and the center of the Cartesian coordinate system has been put in the center of the plate. In calculating Rmn(t), cos functions should be used instead of sin ones in (11) in the first row. In calculating Rmn(t), cos2 functions should be used instead of sin ones in (11) in the second row. More complicated models could also be considered with the same procedure. As an example, consider the plate mentioned in the previous section rests on a nonlinear non-uniform elastic foundation obeying the rule F(x,y,t) ⫽ k(x,y,t)·w(x,y,t) ⫹ k1(x,y,t)w(x,y,t)3 as restoring force. Such a general model of foundation could be used to model many mechanical systems, e.g. dynamic analysis of floating runway

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Table 1 The functions of Eq. (13) for a few other cases Case

P(t)

Q(t)

冉冊 冉冊

Plate with p4D 1 simply support, r·a(t)h[ a immovably constrained

2

+

1 22 ] b

冋

Plate with clamped support, immovably constrained

册

16p4D 3 a2 3 +2 2+ 4 4 9r·a(t)h a b b

再 冉 冊 冉冊

Plate with m 2 n 22 4 D 4 simply support, r,a(t)h p [ a + b ] + ab stress-free edges mpx npy on nonlinear k(x,y,t)·sin2( ) sin2 ) dA a b non-uniform A elastic foundation

冕冕

R(t)

冉 冊再 冉 冊冎 冉 冊再 冉

Eh2 p 8r·a(t) ab a4 1 + 1+ 4 2 b

4

b4 + 2na2b2 + a4 1⫺n2

Eh2 p 4 b4 + 2na2b2a4 r·a(t) ab 8(1⫺n2) 2a4 1 9 4 (a + b4) + + 9 8 (1 + a2/b2) a4 a4 + + (1 + 4a2/b2)2 (4 + a2/b2)

冉 冊 冎 冉 冊 冉冊

16Rmn(t) r·a(t)h2ab

16Rmn(t) r,a(t)h2ab

冊冎

Eh2 mnp 4 2r,a(t) ab 4DH + k1 m 2 n 2 2 r,a(t) [ + ] A a b mpx npy 2 ) sin2( ) dA (x,y,t) sin ( a b

冕冕

4Rmn(t) r,a(t)h2ab

under the load of moving aircraft [22] or slab-type supported bridges [5]. More such systems are listed in [9]. Defining k(x,y,t) (and/or k1(x,y,t)) as k·d(x⫺x0)d(y⫺y0) one could take into account the effect of a single elastic column located at (x0,y0) supporting the plate on the plate dynamics. Setting k(x,y,t) (and/or k1(x,y,t)) as k.d(x⫺x0) one could formulate the presence of an elastic wall at x=x0 supporting the plate. Using the time dependence of the k and k1 functions one could go toward considering the time dependent effects like creep, fatigue and damage accumulation.

4. Periodicity in solutions The solution of Eq. (13) can be represented by the following integral equation.

冕 T

A ⫽ ⌫(t,s){F(s)⫺Q(s)A3(s)其 ds, T ⫽ 2p / w

(14)

0

where ⌫(t,s) is the Green’s Function for the operator ⌽ defined as, ⌽A ⫽ A¨ ⫹ P(t)A.

(15)

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289

Green’s Function can be constructed as follows. Let q(t,s) ⫽ ⫺

1 1 (t⫺s)2⫺ (t⫺s); 0ⱕtⱕsⱕT 2T 2

(16)

1 1 q(t,s) ⫽ ⫺ (t⫺s)2⫺ (t⫺s); 0ⱕsⱕtⱕT 2T 2 q(t,s) is a piecewise continuous periodic polynomial of second degree, i.e. q(i)(0,s) ⫽ q(i)(T,s), i ⫽ 0,1

(17)

q˙(t,t⫺)⫺q˙(t,t+) ⫽ 1 Green’s Function can be defined as,

冕 T

⌫(t,s) ⫽ q(t,s) ⫹ q(t,t)R(t,s) dt

(18)

0

where, R(t,s) is the resolvant of ⌽q(t,s), i.e.

冕 T

R(t,s) ⫽ ⫺⌽q(t,s) ⫺ ⫺q(t,t)R(t,s) dt.

(19)

0

In order to estimate ⌫,

冕 T

冕

冕

冕

冕

T

冕

T

T

|R(t,s)| dsⱕ |⌽q(t,s)| ds ⫹ ( |⌽q(t,t)| dt)( |R(t,s)| dt).

0

0

0

(20)

0

Correspondingly,

冕 T

T

T

|R(t,s)| dsⱕ |⌽q(t,s)| ds(1⫺ |⌽q(t,s)| ds).

0

0

(21)

0

From Eqs. (18) and (21), it can be shown that, |⌫|ⱕ

max

t,s苸[0,T] × [0,T]

|q(t,s)|

1

冕 T

.

(22)

1⫺ |⌽q(t,s)| ds 0

Based on Eqs. (15) and (16), ⌽q(t,s) can be calculated. Defining M and m as the maximum and minimum of the function P(t) over the domain of [0,T], results in,

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冕

|⌽q(t,s)| ds ⫽ 1⫺P(t)T2 / 12ⱕ1⫺mT2 / 12 if MT2ⱕ2

冕

|⌽q(t,s)| ds ⫽ ⫺1 ⫹ P(t)T2 / 12ⱕ⫺1 ⫹ MT2 / 12 if MT2ⱖ2.

T

(23)

0

T

0

Substituting Eq. (23) into Eq. (22), |⌫|ⱕT / 2(1⫺1 ⫹ mT2 / 12) ⫽ 6 / mT if MT2ⱕ2

(24)

|⌫|ⱕT / 2(1⫺(⫺1 ⫹ mT / 12)) ⫽ 6T / (24⫺MT ) if MT ⱖ2 and MT ⬍ 24. 2

2

2

2

In the case under consideration 0ⱕq, thus ⌫ was estimated for 0⬍MT2⬍24. Banach’s fixed point theorem can be used to prove the existence of the periodic solutions. Let B ⫽ {A(t):A(t)苸C[0,T],||A(t)|| ⫽ sup |A(t)| ⫽ MA}

(25)

t苸[0,T]

be a Babach space. Define the transformation F:B→B as,

冕 T

FA ⫽ ⌫(t,s){F(s)⫺Q(s)A3(s)} ds.

(26)

0

The solution to be found is the fixed point of the above transformation. By applying the operator F to two sample functions A1 and A2 from B, and subtraction of the results and some algebraic manipulations,

冕 T

|FA1⫺FA2|ⱕ||A1⫺A2|| |⌫(t,s)||Q(s)||A12(s) ⫹ A22(s) ⫹ A1(s)A2(s)| ds.

(27)

0

Finding the supreme of the both sides according to Eq. (25), ||FA1⫺FA2||ⱕb||A1⫺A2||

(28)

in which; 18|Q(t)|M2A mT 18T|Q(t)|M2A ⫽ 24⫺MT2

b⫽

if MT2 ⬍ 2 if MT2ⱖ2 and

(29) 24 ⬎ MT2.

The transformation defined by F is contraction if b⬍1. Under this condition, F has a fixed point denoting T-periodic solution of Eq. (14). The fixed point theorem also guarantees the uniqueness of the solution but it does not mean that under the conditions that the theorem holds, the system will not have any bifurcations. Furthermore, it guarantees the convergence of a successive approximation scheme

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constructed on Eq. (26), i.e. one can start with an arbitrary A0(t), put it in the righthand side of Eq. (26) to obtain A1(t) and go on iteratively and be sure that the series obtained is convergent.

5. Test and example problems In order to verify the results, the following problem from reference [9] is addressed. Consider a square plate, simply supported at corners. It is excited by a mass rotating with a constant angular velocity equal to 10.46 rad/s on a circle of radius 1.27 m centered at the center of the plate. It is desired to determine the transient response of the plate for the first 0.6 second time interval that is time necessary to complete one orbiting cycle. It is assumed that the plate is originally at rest. Other data are listed below. r ⫽ 2133 kg / m2, D ⫽ 16,557 kg / m, a ⫽ b ⫽ 5.08 m, h ⫽ 0.127 m and moving mass ⫽ M ⫽ 4530 kg Fig. 2 shows the displacement of the center point of the linear plate as a function of time. The same problem has been solved by elimination of the nonlinear term from the present method and the results have been shown in Fig. 3. The discontinuous curve should be compared with Fig. 2. The continuous curve in Fig. 3 shows the solution taking into account the nonlinear term. Since MT2⬎24 there is no guarantee for the existence of periodic solution according to the above discussion, though from Fig. 3 it seems to have periodic solution. As the second example steel plate with the following properties is considered. The

Fig. 2. Transient response (displacement) for the first example at the center of the linear plate as a function of time (Ref. [9]).

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Fig. 3. Transient response (displacement) for the first example at the center of the plate as a function of time. The continuous curve is the solution obtained via elimination of nonlinear terms.

dimensions of the plate is 60 m by 30 m with width of 1 m, its density and modulus of elasticity are 2770 kg/m3 and 200 GPa, respectively. The mass of the moving load is 49,860 kg and it is moving on an inscribed ellipse, in the plate with constant angular speed of 18.791 rad/s as is the case familiar with machine tools. Only first vibration mode of the plate is considered. The example is solved via numerical integration of Eq. (10) and the following phase plane is achieved. In this case T and M can be calculated to be 0.334S and 198.98S⫺2, respectively and MT2 is 22.25 and b⬍1. So the above discussion guarantees the existence of periodic solution. Fig. 4 shows the results. It shows that the response has a periodic nature.

Fig. 4.

Phase plane for some inhomogeneous initial conditions (h=1.2 m).

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Fig. 5.

293

Phase plane for homogenous initial condition (h=1.0 m).

The same example is solved for a plate of width 1.0 m. In this case T and M can be calculated to be 0.334S and 238.78S⫺2, respectively and MT2 is 26.7. So the above discussion does not guarantee the existence of T-periodic solution. Fig. 5 shows the phase plane of the solution obtained via numerical integration in [0,T] interval. Fig. 6 shows the same solution curve in [0,8S]. It shows that the response does not have a periodic nature. It should be noted that the above discussion does not mean that the response must be non-periodic.

Fig. 6.

Phase plane for the same inhomogeneous initial condition in a longer time interval (h=1.0 m).

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6. Concluding remarks A simple yet general formulation has been introduced to consider the periodic response of a plate excited by a traveling mass moving along an arbitrary general trajectory on the surface of a rectangular simply supported plate taking into account large deflection (middle plate stretching) effects. The problem was further formulated for a few other cases of boundary conditions and/or foundations. It even includes a relatively complex case of nonlinear plate on nonlinear non-uniform elastic foundation. The method is general so that one could extend it for plates with other conditions. It is shown that the governing partial differential equation could be converted to a nonlinear ordinary differential equation with time varying coefficients. So, this research deals with “Duffing’s Oscillator” with time varying coefficients. Similar equations can arise in the analysis of other mechanical systems. Regular response of dynamical systems has many applications in control of engineering systems. Periodic behavior is the most important regular solution, which is studied in this paper, using Banach’s fixed-point theorem. It is shown that the response of the plate may be nonperiodic even when the moving mass trajectory on the plate surface is periodic. The range of the parameters of the problem for which the existence of periodic solution is guaranteed via application of the Banach’s fixed-point theorem is presented. It should be noted that the above discussion does not mean that the response must be non-periodic outside this range. Outside the specified range the response could be periodic, quasi-periodic or chaotic. A few simple examples conclude the paper. The plate theory used here ignores shear deformation and rotatory inertia effects and is known as the classical plate theory. An extension of the theory so as to include both shear deformation and rotatory inertia effects in the vibration of plates by Mindlin [23] could be used to further improve the present wok. Also considering quasiperiodic and possibly chaotic response of the system could further complete the present work. A detailed consideration of the nature of trajectories and geometric features of the system could be found at [24].

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