dynamic response of an elastically connected double

Khayyam J. Math. 5 (2019), no. 1, 58–77 DOI: 10.22034/kjm.2018.73854

DYNAMIC RESPONSE OF AN ELASTICALLY CONNECTED DOUBLE NON-MINDLIN PLATES WITH SIMPLY-SUPPORTED END CONDITION DUE TO MOVING LOAD J.A. GBADEYAN1 , O.M. OGUNMILORO∗2 AND S.E. FADUGBA3 Communicated by P.I. Naumkin Abstract. In this paper, the dynamic response of two identical parallel nonmindlin (i.e., not taking into account the effect of shear deformation and rotatory inertia) plates which are elastically connected and subjected to a constant moving load is considered. The fourth order coupled partial differential governing equations is formulated and solved, using an approximate analytical method by assuming; firstly, a series solution later on treating the resulting coupled second order ordinary differential equations with an asymptotic method of Struble. The differential transform method, being a semi-analytical technique, is applied to the reduced coupled second order ordinary differential equations, to get a non-oscillatory series solution. An after treatment technique, comprising of the Laplace transform and Pade approximation techniques, is finally used via MAPLE ODE solver to make the series solution oscillatory. The dynamic deflections of the upper and lower plates are presented in analytical closed forms. The effect of the moving speed of the load and the elasticity of the elastic layer on the dynamic responses of the double plate systems is graphically shown and studied in details. The graphs of the plate’s deflections for different speed parameters were plotted. It is however observed that the transverse deflections of each of the plates increase with an increase in different values of velocities for the moving load for a fixed time t.

1. Introduction and preliminaries Plates are solid structural elements having the geometry of two dimensions, whose thickness is very small compared to its planar dimensions. Also, the effects of load on the plates generate stresses normal to the thickness of the plate. Date: Received: 31 May 2018; Revised: 8 September 2018; Accepted: 29 September 2018. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 39B82; Secondary 44B20, 46C05. Key words and phrases. Moving load, non-mindlin, simply-supported, Struble’s method. 58

DYNAMIC RESPONSE OF AN ELASTICALLY

59

In practical terms, the vibration analysis of plates have been explored and still being studied by many researchers in the field of applied mathematics and engineering, where, different constitutive governing equations with different end conditions were formulated. In several literature previously written, single beam or double beam systems, [12], [24], [17], [8], [21], [4] under moving loads acted upon by moving forces have been studied. But much work have not been carried out on double or multiplate systems, due to the problems encountered while trying to solve the coupled fourth order partial differential equations governing the systems. Fryba, [9] in his excellent monograph discussed the vibration of solids and structures under moving loads. Furthermore, Gbadeyan and Oni [12] considered the dynamic behavior of beams and rectangular plates under moving loads where it was shown that the response amplitude of the moving force is greater than that of the moving mass. Abu-Hilal [2], investigated the response of a double Euler–Bernoulli beams, due to a moving loads. Moreover, literatures of [22], [14], [5], [19], [16], [20], [15], [24], [8], and [10] are work done on the theory and analysis of vibrations that are still producing positive development in the field of engineering. However, double-plate systems are very much applicable in engineering which includes, decking systems for railways viaducts and bridges, tunnels, anti slides and avalanche guards, industrial flooring systems, commercial flooring with high loading capacities, construction of trolleys and girders, and so many other applications. Also, its advantages include, provision of convenient transport, heavy loading capacity, great stability and long fatigue life, and so on. Having gone through the work of Gbadeyan and Hammed [11], where the influence of a moving mass of the dynamic behavior of a visco-elastically connected prismatic double Raighley beam system having arbitrary end support, in addition with the work of [12], [13] [18], [3], [17], [23], this paper considered, the problem of the effect of the influence of the mass of the moving load of constant magnitude and velocity of the dynamic behavior of the finite double elastically connected non-mindlin’s plate with simply supported end conditions. The considered system is governed by a pair of fourth order partial differential equations, which is reduced to a pair of second order ordinary differential equations by using a pair of assumed series solutions. The reduced second order differential equation is further simplified, employing the approximate analytical method of Struble, which is commonly used to solve a weakly non-linear oscillatory system. The differential transform method (DTM) [1], is applied to obtain the solutions of the reduced coupled ordinary differential equations. Hence, a technique referred to as an after treatment (AT) [6], [7], comprising of the Laplace transform and Pade approximation is applied to the differential transform series solution, enlarging the convergence domain of the series solution by truncating it, thereby making the solution oscillatory. The simply supported end condition is considered as an example. The rest of the paper is organized into different sections which includes. Section 2 involves the mathematical model formulation with the corresponding boundary and initial conditions and defining the parameters involved in the systems. Section 3 involves the discussion of the method of solution . Section 4 presents the simplification of the coupled second order differential equations. Section 5 involves the solution of the two elastically connected double non-mindlin

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

plate using DTM. Section 6, the simply supported condition is considered as an example. In section 7, AT technique is employed to obtain the solutions of upper plate W1 (x, y; t) and lower plate W2 (x, y; t) generated with computational software MAPLE ODE solver and numerical computations, is performed, and graphs were presented analytically and conclusively. 2. Mathematical model formulation A double plate system modeled as a two finite parallel upper and lower nonmindlin plates inter-connected by an elastic core is considered. Figure 1, shows that the upper plate is subjected to a load P1 (x, y; t) having a mass Ml Figure 1. Diagram of the interconnected plate system

  2  ∂ 4 W1 (x, y; t) ∂ 4 W1 (x, y; t) ∂ 4 W1 (x, y; t) ∂ W1 (x, y; t) D +2 + +µ ∂x4 ∂x2 ∂y 2 ∂y 4 ∂t2 + K1 [W˙ 2 (x, y; t) − W˙ 1 (x, y; t)] = P1 (x, y; t)   2   4 ∂ 4 W1 (x, y; t) ∂ 4 W1 (x, y; t) ∂ W1 (x, y; t) ∂ W1 (x, y; t) D +2 + +µ ∂x4 ∂x2 ∂y 2 ∂y 4 ∂t2 + K1 [W˙ 2 (x, y; t) − W˙ 1 (x, y; t)] = 0. 

(2.1)

(2.2)

The simply supported boundary conditions subjected to the pairs of fourth order coupled partial differential equations in (2.1) and (2.2) are: W1 (lx , y; t) = W2 (x, ly ; t) = 0 W1 (0, y; t) = W2 (0, y; t) = 0 ∂ 2 W1 (0, y; t) ∂ 2 W1 (x, y; t) = =0 ∂x2 ∂x2 ∂ 2 W1 (0, y; t) ∂ 2 W1 (x, y; t) = = 0. ∂y 2 ∂x2

(2.3)

DYNAMIC RESPONSE OF AN ELASTICALLY

While the corresponding initial conditions are: ( W1 (x, y; t)|t=0 = 0 = W˙ 1 (x, y; t)|t=0 W2 (x, y; t)|t=0 = 0 = W˙ 2 (x, y; t)|t=0 .

61

(2.4)

Several parameters involved in the governing equations presented in (2.1)–(2.2) are defined as follows: The concentrated moving load P1 (x, y; t) is   2 ∂ W1 (x, y; t) P1 (x, y; t) = Ml g − Ml ∂t2  (2.5) 2 ∂ 2 W2 (x, y; t) 2 ∂ W1 (x, y; t) +2V1 + V1 δ(x − vt)(y − s) ∂t∂y ∂t2 The Dirac delta function, being an even function expressed as a Fourier cosine series, is defined as: ∞ ∞ nπvt nπx 1 2 X mπs mπy 1 2X cos cos and δ(y−s) = + cos cos , δ(x−vt) = + lx lx n=1 lx lx ly ly m=1 ly ly

where ∞ ∞ 4 XX mπs nπvt mπy nπvt δ(x − vt)δ(y − s) = cos cos cos cos lx ly lx ly lx ly n=1 m=1 ∞ X

2 + lx ly

∞ nπvt mπs nπx 2 X mπy 1 cos cos cos + cos + , lx lx lx ly ly ly lx ly m=1 m=1 (2.6)

where  δ(x) =

0, ∞,

x 6= 0, x = 0. 3

Eh D is the flexural rigidity of the plate given as D = 12(1−θ) , and Ml is the mass of the load P1 (x, y; t) moving with a constant velocity V , also E is Young’s modulus, θ is Poisson’s ratio ( θ < 1), x and y are the position coordinate in x and y direction, K is the stiffness constant, µ is the constant mass per unit length of the plates,  is the fixed length of the plates, and t is the time. W1 (x, y; t) is the traversed displacement of the upper plate and W2 (x, y; t) is also the traversed displacement of the lower plate. Also, from (2.5), g is the acceleration due to gravity, V12 W100 (x, y; t) represents the centrifugal acceleration, 2V W10 (x, y; t) is ¨1 (x, y; t) represents the local acceleration. the Coriolis acceleration, while W

3. Method of solution To solve the transverse dynamic responses W1 (x, y; t) and W2 (x, y; t), (2.1)– (2.2) are transformed into a set of two ordinary differential equations. The method

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of solution of employed in this paper, involves assuming the series of the form:

W1 (x, y; t) =

∞ X ∞ X

φm,n (m, n; t)Vm (x)Vn (y),

n=1 m=1

W2 (x, y; t) =

∞ X ∞ X

(3.1) βm,n (m, n; t)Vm (x)Vn (y),

n=1 m=1

where

∞ ∞ ∂ 4 W1 (x, y; t) X X φm,n (m, n; t)Vmiv (x)Vn (y), = ∂x4 m=1 n=1 ∞ ∞ ∂ 4 W1 (x, y; t) X X = φm,n (m, n; t)Vm (x)Vniv (y), ∂y 4 m=1 n=1 ∞ ∞ ∂ 4 W1 (x, y; t) X X φm,n (m, n; t)Vm00 (x)Vn00 (y), = 2 2 ∂x ∂y m=1 n=1 ∞ ∞ ∂ 2 W2 (x, y; t) X X ¨ βm,n (m, n; t)Vm (x)Vn (y), = ∂t2 m=1 n=1 ∞ ∞ ∂ 2 W1 (x, y; t) X X ¨ φm,n (m, n; t)Vm (x)Vn (y), = ∂t2 m=1 n=1 ∞ ∞ ∂ 2 W2 (x, y; t) X X ¨ = βm,n (m, n; t)Vm00 (x)Vn00 (y), ∂x2 m=1 n=1 ∞ ∞ ∂ 4 W2 (x, y; t) X X = βm,n (m, n; t)Vm00 (x)Vn00 (y), 2 2 ∂x ∂y m=1 n=1 ∞ ∞ ∂ 4 W1 (x, y; t) X X = βm,n (m, n; t)Vm (x)Vniv (y), 4 ∂y m=1 n=1 ∞

∞

∞

∞

∂ 4 W1 (x, y; t) X X ˙ = φk,l (k, l; t)Vk0 (x)Vl (y), ∂t∂x k=1 l=1 ∂ 2 W1 (x, y; t) X X = φk,l (k, l; t)Vk00 (x)Vl00 (y). ∂x2 k=1 l=1

(3.2)

DYNAMIC RESPONSE OF AN ELASTICALLY

63

Substituting (3.2) into (2.1) yields " ∞ ∞ ∞ X ∞ XX X iv φm,n (m, n; t)Vk (x)Vn (y) + 2 φm,n (m, n; t)Vm00 (x)Vn00 (y) m=1 n=1 ∞ X ∞ X

+

m=1 n=1 ∞ X ∞ X

φm,n (m, n; t)Vm (x)Vniv (y) + µ

m=1 n=1 " ∞ ∞ XX

+ K1

# φm,n (m, n; t)Vm00 (x)Vn00 (y)

m=1 n=1 ∞ X ∞ X

φm,n (m, n; t)Vm (x)Vn (y) −

m=1 n=1

βm,n (m, n; t)Vm (x)Vn (y)

m=1 n=1

" = Ml g − Ml

#

∞ X ∞ X

φ¨k,l (k, l; t)Vk0 (x)Vl (y) + 2V1

φ˙ k,l (k, l; t)Vk0 (x)Vl (y)

k=1 l=1

k=1 l=1

+V12

∞ X ∞ X

∞ X ∞ X

# φk,l (k, l; t)Vk00 (x)Vl00 (y)

δ(x − vt)(y − s).

k=1 l=1

(3.3) We assume that the load function P1 (x, y; t) can also be expressed as P1 (x, y; t) =

∞ X ∞ X

φm,n (m, n; t)Vm (x)Vn (y),

(3.4)

n=1 m=1

where, φm,n (m, n; t), βm,n (m, n; t), and ψm,n (m, n; t) are unknown functions of time t and Vm (x), Vn (y) are the known deflection curves of a vibrating plate. For the free vibration of a rectangular plate, we have, for the homogeneous part of the governing equation in terms of the assumed solution,   2 D Vmiv (x)Vn (y) + 2Vm00 (x)Vn00 (y) + Vm (x)Vniv (y) − µωm,n Vm (x)Vn (y) = 0, (3.5) 2 where the frequency ωm,n for a simply supported plate is  4 4  mπ m2 n2 π 2 n4 π 4 2 ωm,n = +2 2 2 + 4 . L4x Lx Ly Ly

(3.6)

Also, the Kth mode of vibration of a uniform plate in x-direction is given as Vk (x) = sin

λk λk λk λk (x) + Ak cos (x) + Bk sinh (x) + Ck cosh (x). lx lx lx lx

While the corresponding Lth mode of vibration of a uniform plate in x and also y direction are Vl (x) = sin Vl (y) = sin

λy λy λy λy (y) + Ak cos (y) + Bk sinh (y) + Ck cosh (y) ly ly ly ly

λy λy λy λy (y) + Ak cos (y) + Bk sinh (y) + Ck cosh (y), ly ly ly ly

(3.7)

where the constants Ak , Bk , Ck , and λk in (3.7) are determined by the simply supported boundary conditions.

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

After rearrangement, simplification and modification, (2.1) reduces to K1 2 φ¨m,n (m, n; t) + ωm,n (m, n; t)φm,n (m, n; t) + φm,n (m, n; t) µ ∞ X ∞ X K1 + βm,n (m, n; t) + [ql∗ (a, b; t) + V q2∗ (a, b; t) + V q3∗ (a, b; t)] µ k=1 l=1

(3.8)

Mg Vm vtVm S . µ

=

Following the same procedure for the lower plate W2 (x, y; t), (2.2) becomes K1 K1 2 (m, n; t)βm,n (m, n; t) + β¨m,n (m, n; t) + ωm,n φm,n (m, n; t) − βm,n (m, n; t) = 0, µ µ (3.9) where ∞ X ∞ X nπvt mπs ∗ ql (a, b; t) =4 φ¨k,l (m, n; t) cos cos ∆q (R, S)∆t (U, V ) lx ly k=1 l=1 + 2

∞ X

nπvt φ¨k,l (m, n; t) cos ∆q (R, S)∇1 (U, V ) lx k=1

(3.10)

∞ X

mπs φ¨k,l (m, n; t) cos ∆t (U, V )∇1 (J, K) l x m=1 mπs + cos ∆t (U, V )∇1 (J, K) + φ¨k,l ∆1 (J, K)∇1 (U, V ), lx + 2

q2∗ (a, b; t)

=8

∞ ∞ X X

φk,l (m, n; t) cos

k=1 l=1 ∞ X

+ 4

φk,l (m, n; t) cos

k=1

+ 4

∞ X

φk,l (m, n; t) cos

m=1

+ cos q3∗ (a, b; t)

=4

nπvt ∆2c (J, K)∇t (U, V ) lx

(3.11)

mπs ∆t (U, V )∇2 (J, K) ly

mπs ∆t (U, V )∇1 (J, K) + φk,l ∆1 (J, K)∇1 (U, V ), lx

∞ X ∞ X

φk,l (m, n; t) cos

k=1 l=1 ∞ X

+ 2

+ 2

nπvt mπs cos ∆2c (J, K)∆t (U, V ) lx ly

φk,l (m, n; t) cos

k=1 ∞ X m=1

φk,l (m, n; t) cos

nπvt mπs cos ∆3c (J, K)∆k (U, V ) lx ly

nπvt ∆3c (J, K)∇1 (U, V ) lx mπs ∆t (U, V )∇3 (J, K) + φk,l ∆1 (J)∇1 (P ). ly (3.12)

DYNAMIC RESPONSE OF AN ELASTICALLY

65

The following are defined as Z

lx

nπx Vk (x)Vj (x)dx, lx 0 Z ly mπy cos Vl (y)Vp (y)dy, ∆t (U, V ) = lx 0 Z lx Vk (x)Vj (x)dx, ∆1 (J, K) = 0 Z ly Vl (x)Vp (x)dy, ∇1 (U, V ) = 0 Z lx nπx 0 cos ∆2c (J, K) = V (x)Vj (x)dx, lx k 0 Z lx Vk0 (x)Vj (x)dx, ∆2 (J, K) = 0 Z lx nπx 00 cos ∆3c (J, K) = V (x)Vj (x)dx, lx k 0 Z lx Vk00 (x)Vj (x)dx, ∆3 (J, K) = cos

∆q (R, S) =

0

Z

lx

nπx Vj (x)dx, lx 0 Z ly mπy cos ∇k (P ) = Vp (y)dy, lx 0 Z lx Vj (x)dx, ∆1 (J) = 0 Z ly ∇1 (P ) = Vp (y)dy. ∆k (J) =

cos

(3.13)

0

Equations (3.8)–(3.13) are the coupled transformed second order ordinary differential equations governing the behavior of a double non-mindlin plate system interconnected by an elastic core traversed by a moving load. To be able to simplify further, by (3.8) and (3.9), we assume, for the time being, that the plate is disconnected. That is, we set the connecting elastic core term to zero, to become; K1 [φm,n (m, n; t) − βm,n (m, n; t)] = 0. (3.14) µ So that the resulting equations for the upper and lower plates are such that φ¨m,n (m, n; t) + ω 2 (m, n; t)φm,n (m, n; t) m,n ∞ X ∞ X

+ 1

k=1 l=1

[ql∗ (a, b; t) + V q2∗ (a, b; t) + V q3∗ (a, b; t)] =

Mg Vm vtVm S µ (3.15)

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

and

2 β¨m,n (m, n; t) + ωm,n (m, n; t)βm,n (m, n; t) = 0,

(3.16)

respectively.

4. Simplification of the coupled second order differential equations We have been able to reduce the coupled fourth order partial differential equation to a second order differential equations, using an assumed series solution method. However, it is still observed that while the reduced lower plate equation can be solved analytically, the upper plate has no exact analytical solution, and to this effect, an approximate analytical method, which is the modified asymptotic technique developed by Struble was employed. We set

1 < 1, 1 + 1

(4.1)

1 = λ + 0(λ2 )

(4.2)

1 = [1 + λRa t]−1 = 1 − λRa t + (λ)2 . 1 + 1 Ra t

(4.3)

λ=

and it can also be shown that

and

DYNAMIC RESPONSE OF AN ELASTICALLY

67

Substituting (4.3) into the homogeneous part of (3.15), we obtain φ¨m,n (m, n; t) + 1 Ra t(1 − λRa t)φ¨m,n (m, n; t) 2 + 1 Rc t)(1 − λRa t)φm,n (m, n; t) + (ωm,n ∞ ∞ X X nπvt mπs + λ(1 − λRa t) 4 cos cos ∆q (R, S)∇1 (U, V ) l l x y k=1k6=m l=1l6=n

+2

∞ X

cos

nπvt ∆q (R, S)∇1 (U, V ) lx

cos

nπvt ∆t (U, V )∇1 (J, K) lx

n=1

+2 +

∞ X

m=1 ∞ X

∆1 (J, K)∇1 (U, V )φ¨k,l (k, l; t)

k=1

+ 8V + 4V

∞ X ∞ X m=1 n=1 ∞ X

+ 4V

nπvt mπs cos ∆2c (J, K)∇t (U, V ) lx ly

cos

nπvt ∆2c (J, K)∇t (U, V ) lx

cos

mπs ∆t (U, V )∇2 (J, K) ly

k=1 ∞ X

cos

k=1

+ 2V ∆2 (J, K)∇1 (U, V )φ˙ k,l (k, l; t) ∞ X ∞ X nπvt mπs 2 + 4V cos cos ∆3c (J, K)∇k (P ) l l x y n=1 m=1 + 2V

2

+ 2V 2

∞ X n=1 ∞ X m=1

cos

nπvt ∆3c (J, K)∇1 (U, V ) lx

cos

mπs nπvt cos ∆3c (U, V )∇2 (J, K) ly ly

+ ∇1 (P )∆1 (J)φ˙ k,l (k, l; t) = 0. (4.4) Setting λ = 0 in (4.4), we obtain a case corresponding to that when the mass ratio effect of the plates is regarded negligible. In this case we have φ¨m,n (m, n; t) + 1 Ra tφ˙ m,n (m, n; t) (4.5) 2 + ωm,n (m, n; t)βm,n (m, n; t) + [1 Rc t]φ˙ m,n (m, n; t) = 0, whose solution is of the form φ¨m,n (m, n; t) = D0 [ωm,n t − αm,n ],

(4.6)

where D0 , ω(m,n) , and α(m,n) in (4.6) are all constants. Since λ = 1, for any arbitrary mass ratio 1 , Struble’s method requires that the asymptotic solution

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

of the (4.4) is of the form φm,n (t) = θ(m, n; t) cos[ωm,n t − αm,n ] +

N X

λr φr (m, n; t) + 0(λN ),

(4.7)

n=1

where N is any positive number less than infinity and θ(m, n; t) and α(m, n; t) are slowly varying time functions, which implies that, ˙ θ(m, n; t) → 0(λ), ¨ θ(m, n; t) → 0(λ2 ), α(m, ˙ n; t) → 0(λ),

(4.8)

α ¨ (m, n; t) → 0(λ2 ). Substituting, without loss of generality, N = 1, it becomes φm,n (t) = θ(m, n; t) cos[ωm,n t − αm,n ] + λφ1 (m, n; t) + 0(λ2 ).

(4.9)

Hence φm,n (t) = θ(m, n; t) cos[ωm,n t − αm,n ] + λφ1 (m, n; t) + 0(λ2 )

(4.10)

such that ˙ φ˙ m,n (t) =θ(m, n; t) cos[ωm,n t − αm,n ] − θ(m, n; t) sin[ωm,n t − αm,n ] + λφ˙ 1 (m, n; t) + 0(λ2 )φ˙ 1 (m, n; t)

(4.11)

and 2 ¨ φ¨m,n (t) =[θ(m, n; t) − ωm,n (m, n; t) + 2ωm,n θ(m, n; t)α(m, ˙ n; t) 2 ˙ − ωm,n (t) − α˙ (m, n; t)] cos[ωm,n t − αm,n ] − 2ωm,n θ(m, n; t)α(m, ˙ n; t) m,n

˙ − 2θ(m, n; t)¨ α(m, n; t) − θ(m, n; t)¨ α(m, n; t)] sin[ωm,n t − αm,n ] + 0(λ2 )φ˙ 1 (m, n; t). (4.12) The variational equations are obtained, by equating the coefficients of sin [ωm,n t− αm,n (m, n; t)] and cos [ωm,n t − αm,n (m, n; t)] terms on both sides of the (4.12) to zero. Neglecting those terms that do not contribute to the variational equations, we obtain ˙ −2ωm,n θ(m, n; t) sin[ωm,n t − αm,n ] + 2ωm,n θ(m, n; t)α(m, ˙ n; t) cos[ωm,n t − αm,n ] − 2λV ωm,n θ(m, n; t)∆2 (J, K)∇1 (U, V ) sin[ωm,n t − αm,n ] 2 − λωm,n ∆1 (J, K)∇1 (U, V )θ(m, n; t) cos[ωm,n t − αm,n ]

− λV 2 ∆2c (J, K)∇t (U, V )θ(m, n; t) cos[ωm,n t − αm,n (m, n; t)] = 0, (4.13) where the variational equations are − 2ωm,n θ˙ − 2λV ωm,n θ(m, n; t)∆2 (J, K)∇1 (U, V ) = 0 (4.14)

DYNAMIC RESPONSE OF AN ELASTICALLY

and

2ωm,n θα(m, ˙ n; t) − λα2 ωm,n ∆2 (J, K)∇1 (U, V )θ(m, n; t)

+ λV 2 ∆2c (J, K)∇t (U, V )θ(m, n; t) = 0 solving (4.14) and (4.15), we obtain ˙ θ(m, n; t) = −λθ(m, n; t)∆2 (J, K)∇1 (U, V ),

69

(4.15)

(4.16)

That is, dθ(m, n; t) = λ∆2 (J, K)∇1 (U, V )θ, θ

(4.17)

which implies that, 0

θ(m, n; t) = −λV ∆2 (J,K)∇1 (U,V ) = A0 −r t ,

(4.18)

r0 = λV ∆2 (J, K)∇1 (U, V )

(4.19)

where 0

and A is constant. Equation (4.18) implies that, dα(m, n; t) dt   λωm,n θ(m, n; t) λ = ∆1 (J, K)∇1 (U, V ) − 2 2α(m, n)∆3 (J, K)∇1 (U, V ) t + φm,n , 2 V (4.20)

where φ is constant. The first approximation to the homogeneous system is from (4.6) φm,n (m, n; t) = D0 cos[ωm,n t − αm,n (m, n; t)]. (4.21) That is, 0 φm,n (m, n; t) = A0 −r t cos[δm,n t − ψm,n (m, n; t)], (4.22) where   λ V2 δm,n = 1 − ∆1 (J, K)∇1 (U, V ) − 2 ∆3 (J, K)∇1 (U, V ) . (4.23) 2 ωm,n (4.23) is the desired modified frequency corresponding to the frequency of free system involving moving mass effect. Hence, according to Struble’s technique, (3.15) is reduced to φ¨m,n (m, n; t) + δ 2 φm,n (m, n; t) = PRT Vm (vt)Vn (s), (4.24) m,n

where PRT = g1 lx ly (1 − 1 Ra (t)). 5. Solution of two elastically connected double non-mindlin plates At this juncture, the connecting elastic core is restored and the reduced pair of coupled second order differential equation for the non-mindlin plates   K K 1 1 2 φ¨m,n (m, n; t) + ωm,n φm,n (m, n; t) φm,n (m, n; t) − βm,n (m, n; t) µ µ (5.1) = PRT Vm (vt)Vn (s)

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and   K1 K1 2 ¨ φm,n (m, n; t) + δm,n φm,n (m, n; t) φm,n (m, n; t) − βm,n (m, n; t) = 0 (5.2) µ µ 2 2 are considered, respectively. and ωm,n in terms of the modified frequencies δm,n A semi analytical method known as DTM is now employed. To this end, we will state briefly, the basic theory of method as follows. The transform of the mth derivative of a function W (t) is given as   1 dm W (t) W (m) = (5.3) m! dt2

and the corresponding inverse transformation is defined as W (t) =

∞ X

(t − t0 ) W (m).

(5.4)

m=1

Hence, the above equations will yield   ∞ X t − t0 dm W (t) W (t) = . m! dt2 m=1

Table 1. Basic properties of DTM for equations of motion Original Function Transformed Function w(t) = c1 u(t) ± c2 v(t) W (k) = c1 U (k) ± c2 V (k) du(t) w(t) = W (k) = (k + 1)U (k + 1) dt r d u(t) w(t) = W (k) = (k + 1)(k + 2) · · · (k + n)U (k + n) dtr k X U (n)V (k − n) w(t) = u(t)v(t) W (k) = r=0

w(t) = u(t)v(t)y(t)

k X k−n X

U (n)V (r)V (k − n − r)  1 W (k) = δ(k − m) = 0 k = m and k 6= m   1 k kπ W (k) = a sin k! 2  1 k kπ W (k) = a cos k! 2  1  k W (k) = a − (−(a)k ) 2k!  1  k W (k) = a + (−(a)k ) 2k!

W (k) =

n=0 r=0

w(t) = tm

w(t) = sin at w(t) = cos at w(t) = sinh at w(t) = cosh at

(5.5)

DYNAMIC RESPONSE OF AN ELASTICALLY

71

It is well known that in application the series in (5.4) is finite and usually written as

W (t) =

p X

(t − t0 )m W (m),

(5.6)

m=1

such that the series

W (t) =

p+1 X

(t − t0 )

(5.7)

m=1

is considered unimportantly small. Furthermore, it can be readily shown that the relationships in Table 1 between the original function W (t) and the transformed function W (m), for t0 = 0, hold. Applying the DTM on (5.1) and (5.2), gives

φm,n (k + 2) =

1 Prt Vn (s) (k + 1)(k + 2) "  k    k   kπ Am λm V kπ 1 λm V sin cos × k! L 2 k! L 2 #  k Bm λm V + 2k! L "  # k  k "  k # λm V C m λm V λm V 2 φ(k) − − + − − − δm,n L 2k! L L −

K1 K1 φm,n (m, n; t) + βm,n (m, n; t) µ µ (5.8)

βm,n (k + 2) =

1 (k + 1)(k + 2)   K1 K1 2 ωm,n − βm,n (m, n; t) − βm,n (m, n; t) + φm,n (m, n; t) µ µ (5.9)

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

Subjecting (5.8) and (5.9) to the transformed form of the initial conditions earlier stated in (2.4), our assumed solutions earlier stated in (3.2) will become ∞ X ∞ X
2 PRT Vn (s) 1 2 2 W 1 (x, y; t) = (A + C ) α − δ m m m m,n t 2 α2 − δm,n 2! p=1 q=1 m


1 2 2 αm − δm,n (α(1 + Bm )) t3 3!    1 K1 2 2 + (Am + Cm ) αm − δm,n t4 4! µ    1 K1 2 2 + (Am + Cm ) αm − δm,n (αm 1 + Bm Vn s)t5 5! µ   λm x λm x λm x λm x × sin + Am cos + Bm sinh + Cm cosh lx lx lx lx   λn y λn x λn y λn y × sin + An cos + Bn sinh + Cn cosh , ly ln ly ly (5.10)    ∞ X ∞ X PRT Vn (s) 1 K1 K1 2 2 W 1 (x, y; t) = (Am + Cm ) αm − δm,n t4 2 2 α − δ 4! µ µ m,n p=1 q=1 m +


 1 K1  2 2 α(1 + Bm ) αm − δm,n (αm 1 + Bm Vn s)t5 5! µ   λm x λm x λm x λm x + Am cos + Bm sinh + Cm cosh × sin lx lx lx lx   λn y λn x λn y λn y × sin + An cos + Bn sinh + Cn cosh . ly ln ly ly (5.11) Therefore, (5.10) and (5.11) represent the transverse displacement of the nonmindlin’s plates interconnected by an elastic layer and traversed by a moving mass and having arbitrary end supports. +

6. Simply-supported boundary condition as illustrative example The considered system is made up of two plates interconnected by an elastic core. The eigen functions or normal modes are: Vm (0) = 0 = Vm (lx ), Vm00 (0) = 0 = Vm00 (lx ),

(6.1)

as well as, Vn (0) = 0 = Vn (lx ), Vn00 (0) = 0 = Vn00 (lx ).

(6.2)

Using the boundary conditions earlier stated in (2.4), we obtain Am = Bm = Cm = An = Bn = Cn , Ak = Bk = Ck = Al = Bl = Cl ,

(6.3)

DYNAMIC RESPONSE OF AN ELASTICALLY

73

and sin λm = 0 = sin λn , sin λk = 0 = sin λl ,

(6.4)

so that, λm = mπ and λn = nπ, where (m, n = 1, 2, . . . ), λk = kπ, λl = lπ. Following the same procedure from the generalized equation, the resulting variational equation on applying Struble’s technique gives − 2ωm,n A(m, n; t) sin[ωm,n t − αm,n (m, n; t)] + 2ωm,n A(m, n; t)α˙ m,n   lx ly 2 nπs 2 cos[ωm,n t − αm,n (m, n; t)] − λ lx ly sin + ωm,n ly 4   v 2 m2 π 2 lx ly 2 nπs A(m, n; t) cos[ωm,n t − αm,n (m, n; t)] − λ lx ly sin + lx2 ly 4 A(m, n; t) cos[ωm,n t − αm,n (m, n; t)]

(6.5)

On further simplification, ωm,n A(m, n; t) = 0

(6.6)

and nπs 1 v 2 m2 π 2 + + 2 2 sin ly 4 lx ωm,n

  1 2 nπs sin + = 0. ly 4 (6.7) Therefore the modified frequency corresponding to the frequency of system, due to the presence of moving mass, will be     1 v 2 m2 π 2 1 λlx ly 2 nπs 2 nπs φm,n = ωm,n 1 − + + 2 2 + sin = 0, (6.8) sin 2 ly 4 lx ωm,n ly 4 2 −2ωm,n A(m, n; t)α˙ m,n −λlx ly ωm,n



2

and the reduced pair of second order coupled differential equations is   K1 K1 2 φm,n (m, n; t) − βm,n (m, n; t) φm,n (m, n; t) + ψm,n φm,n (m, n; t) + µ µ (6.9) 1 lx ly mπvt nπs = sin 1 + 1 Ra t lx ly and   K1 K1 2 ¨ βm,n (m, n; t) + ωm,n φm,n (m, n; t) + φm,n (m, n; t) − βm,n (m, n; t) = 0. µ µ (6.10) Applying DTM on (6.6) and (6.9) above gives a resulting recurrence series relation as "   k   1 nπs 1 nπs kπ φm,n (k + 2) = 1 lx ly g sin sin (k + 1)(k + 2) ly k! ly 2  K1 K1 2 − ψm,n φm,n (k) − φm,n (k) − βm,n (k) µ µ (6.11)

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

and

βm,n (k + 2) =

1 (k + 1)(k + 2)   K1 K1 2 −ωm,n βm,n (k) − βm,n (k) + φm,n (k) . µ µ

(6.12)

7. Concluding remarks The numerical results for the dynamic response of W1 (x, y; t) and W2 (x, y; t) were generated and presented in plotted curves with the aid of mathematical software (MAPLE V). The following values were used in the computation: lx = 0.457, ly = 0.914, D = 5751, µ = 0.075, Ml = 2, x = 0.2285, y = 0.457, K = 10, , t = 0.5, 1 = 0.5 h = 0.12. g = 10, s = 0.4, π = 22 7 Figure 2 shows the deflection profile of a finite simply supported non-mindlins plate, due to moving loads, where the amplitude of the plates increases as the speed increases at different times t by varying K (layer stiffness) (i.e., K = 40, K = 400, K = 4000).

Figure 2.

DYNAMIC RESPONSE OF AN ELASTICALLY

Figure 3. deflection profile of a finite simply supported plate. The amplitude increases with time t at v = 5

Figure 4. deflection profile of a finite simply supported plate. The amplitude increases with time t at v = 7

75

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J.A. GBADEYAN, O.M. OGUNMILORO AND S.E. FADUGBA

Figure 5. Deflection profile of a finite simply supported plate. The amplitude increases with time t at v = 6

References 1. I.H. Abdel-Halim Hassan, Application of DTM to solving systems of differential equations, Appl. Math. Model.12 (2012). 2. M. Abu-Hilal, Dynamic response of a double Euler–Bernoulli beams due to a moving constant load, J. Sound Vib. 297 (2006), 477–491. 3. S.W. Alisjahbana. Dynamic response of clamped orthotropic plates to dynamic moving loads Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, Canada, 2004. 4. R. Bogacz and W. Czyczula, Response of beams on visco-elastic foundation to moving distributed load, J. Theoret. Appl. Mech., 46 (2008), 763-775. 5. V.V. Bolotin, Problem of Bridge Vibration under the Action of a Moving Load (in Russian). Izvestiya AN USSSR, 4, (1961) 109–115. 6. C.-K. Chen and S.-H. Ho , Application of differential transformation to eigenvalue problem, Appl. Math. Comp. 79 (1996):173–188. 7. A.E. Ebaid, E. Ali, On a new after treatment technique for DTM and its applications to non-linear Oscillatory System, Int. J. Nonlinear sci. 18 (2009), 488–497. 8. E. Esmailzadeh and M. Gorashi, Vibration analysis of Beams traversed by uniform partially distributed Moving Masses, J. Sound Vib. 184, (1995), 9–17. 9. L. Fr´ yba, Vibrations of Solids and Structures under Moving Loads, Academia, Prague, 1972. 10. J.A. Gbadeyan and M.S. Dada, Dynamic response of plates on Pasternak foundation to distributed moving loads, J. Nigerian Association Math. Phys. 5 (2001), 184–200. 11. J.A. Gbadeyan and F.A. Hammed , Influence of a moving mass on the dynamic behaviour of a visco-elastically connected prismatic double-rayleigh beam system having arbitrary end support Chinese J. Math. 2017, Article ID6055035. 12. J. A. Gbadeyan and S.T. Oni Dynamic behaviour of beams and rectangular plates under moving loads, J. Sound Vib. 182 (1995), 677–695.

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77

13. Y. Huang and X.-F. Li, A New Approach for free vibration of axially functionally graded beams with non-uniform cross-section, J. Sound Vib. 329 (2010), 2291–2303. 14. C. Inglis, A Mathematical Treatise on Vibrations of Railway bridge, Cambridge University Press, 1934. 15. V. Kolousek, Civil Engineering Structures subjected to Dynamic loads(In Slovak), SVT4 Bratislava, 1967. 16. M. Olsson, On the Fundamental Moving Load Problem, J. Sound Vib. 145( 1991), 299–307. 17. S.T. Oni and T.O. Awodola, Vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation, J. Nigerian association Math. Phys. 7 (2003), 191–206. 18. S.T Oni and T.O. Awodola, The dynamic response to moving masses of rectangular plates with general boundary conditions and also resting on a winkler foundation, J. Sound Vib. 297 (2012), 477–491. 19. S. Sadiku and H. H. E. Leipholz, On the Dynamics of elastic systems with moving concentrated masses, Ingenieur Archive, 57, (1981) 223–242. 20. J.J. Tuma, Schaum’s Outline of Theory and Problems of Structural Analysis With an Introduction to Transport, Flexibility and Stiffness Matrices and Their Applic (Schaum Outline Series), McGraw-Hill, Inc, Book company, New York, 1960. 21. M.A. Usman, Dynamic Response of a Bernoulli Beam on Winkler Foundation under the Action of Partially Distributed Load, Nigerian J. Math. Appl. 16 (2003), 128–150. 22. W. Weaver Jr., S.P. Timoshenko and D.H. Young, Vibration Problems in Engineering, Fourth Edition, New York, Wiley,1974. 23. J.J. Wu, Dynamic analysis of a Rectangular Plate under a moving line Load using scale Beams and scaling laws, Comput. Structures, 83 (2005) 1646–1658. 24. D. Y. Zheng, Y. K. Cheung, F. T. K. Au and Y. S. Cheng, Vibration of multi-span nonuniform beams under moving loads by using modified beam vibration functions, J. Sound Vib. 212 (1998) 455–467. 1

Department of Mathematics, University of Ilorin, Ilorin, Kwara-State, Nigeria. E-mail address: [email protected] 2

Department of Mathematics, Ekiti - State University(EKSU), Ado-Ekiti, Ekiti - State, Nigeria. E-mail address: [email protected] 3

Department of Mathematics, Ekiti - State University(EKSU), Ado-Ekiti, Ekiti - State, Nigeria. E-mail address: [email protected]