A new finite element for transverse vibration of rectangular thin plates under a moving mass

Finite Elements in Analysis and Design 66 (2013) 26–35

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A new finite element for transverse vibration of rectangular thin plates under a moving mass _Ismail Esen n Department of Mechanical Engineering, Karab¨ uk University, BalıklarKayasıMevkii, Karab¨ uk 78050, Turkey

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 April 2012 Received in revised form 9 November 2012 Accepted 13 November 2012 Available online 13 December 2012

In this paper a new finite element which can be used in the analysis of transverse vibrations of the plates under a moving point mass is presented. In this technique, which allows for the inclusion of inertial effects of the moving mass, the load is replaced with an equivalent finite element. By means of using the relations between nodal forces and nodal deflections of 16 DOF conforming plate element with C(1) continuity, on the one hand, and shape functions, on the other hand, mass, stiffness, and damping matrices of the new finite element are determined by the transverse inertia force, Coriolis force and centrifuge force, respectively. This method was first applied on a simply supported beam so as to provide a comparison with the previous studies in the literature, and it was proved that the results were within acceptable limits. Second, it was applied on a cantilevered plate so as to determine the dynamic response of the planer entry plate of a high-speed wood-cutting machine. & 2012 Elsevier B.V. All rights reserved.

Keywords: Finite element Plate vibrations Moving mass Moving force

1. Introduction Dynamic response of the structures under moving loads is a critical problem in engineering and has been studied by several researchers. Analysis of the dynamic response of a plate structure has also been an important area of interest for many designers. Fryba [1], as a perfect reference for simple situations with several analytical solution methods, conducted one of the first and most comprehensive studies on the moving load problem. In order to obtain the dynamic response of graded-thickness rectangular plates under moving loads, Takabataka [2] considered the continuous change of bending stiffness and suggested an analytical method which used a characteristic function. Ghafoori and Asghari [3] studied the dynamic analysis of a laminated composite plate overrun by a moving mass depending on a first-order theory. Using finite element method, Shadnam et al. [4] studied this problem for isotropic plates. Wu et al. [5] and Yoshida and Weaver [6] analysed the dynamic response of plates subjected to various types of moving loads. Wu [7] attempted to find the equal beam models of a plate whose transverse directions are supported and longitudinal directions are free. Based on modified generalized finite integral transformations and modified Strubles method, Gbadeyan and Oni [8] obtained some analytical results for the dynamic response of rectangular plates transversed by moving loads. Renard et al. [9] studied the dimensionless

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deflections and tensions of beams and plates under constant moving forces. They provided some asymptotic results of transient deflections and stresses depending on the velocity and arrival time of the force. Esen [10] suggested a solution method that uses new finite element for the dynamic response of a beam under an accelerating moving mass. Esen et al. [11–13] studied the dynamic response of overhead crane beams under moving trolley using analytical and finite element methods. Monographic books and studies for structural and plate dynamics can be found in references [14–23]. Ahmadian et al. [24] studied the dynamic behavior of a laminated composite beam (LCB) supported by a generalized Pasternak-type viscoelastic foundation, subjected to a moving two-degree-of-freedom (DOFs) oscillator with a constant axial velocity, and they used the Galerkin method for the analytical solution. Sharbati and Szyszkowski [25,26] studied the dynamics of beam–moving mass systems and a ‘composite’ beam element was introduced, which explicitly identifies the Coriolis and centripetal effects dependent on the given current relative velocity of the particular mass. Lee, [27] investigated the onset of the separation between the moving mass and beam, and then took into account its effect in calculating the interaction forces and also in calculating the dynamic responses of the beams. He reported that the effects of separation become significant as the velocity of the moving mass and the mass ratio (mass of moving load/mass of structure) increase. Taking into consideration the inertial effects of the mass, moving load problem becomes even more complicated and studies in this area are limited. This paper offers a method of solution which accepts moving mass as a new finite element.

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

The new finite element, representing the motion of a moving mass with all effects, is combined with the classical finite element method. Analytical methods provided in the literature such as Gbadeyan et al. [8] and Takabataka [2] can prove insufficient in the solution of some complex-shaped plates under moving masses. As suggested, the method in this study which can be applied together with classical finite element method can be implemented to plates with different geometries under all kinds of loading, boundary and damping conditions. With the suggested method in this paper, one of recent important subjects that is the vehicle–structure interaction problem, can also be modelled and results can be compared with the studies in the literature. Plates are structural elements which are widely used in machine manufacturing as well as wood processing machines; their dynamic behaviour has to be perfectly determined, so that some design and quality requirements can be met.

2. Mathematical modelling Fig. 1 shows a rectangular plate, a moving lumped mass on it, and a finite element model, and the plate is divided into rectangular plate elements with the dimensions a and b. The plate has lx, ly dimensions in the directions x and y with thickness h. The mass moves on the plate with a constant velocity v through a straight line which is at a distance e from the origin O as being parallel to axis x. The thickness of the plate is low compared to other dimensions; it was considered a Kirchhoff plate. The most important assumption of Kirchhoff plate theory is that the lines vertical to the middle surface of the plate thickness are vertical and straight to the deformed middle surface even after deformation. This theory ignores the deformations caused by share. In this study it is accepted that deflections are small, and within Hook’s law and other assumptions of the Kirchhoff plate theory provided by [14] are valid. Accordingly, in the case that the system is an isotropic plate and there is no damping in the plate and loading system, the differential equation of the motion of the plate is [1,14]: " # @4 w @4 w @4 w @2 w D þ2 2 2 þ 4 þ m 2 ¼ pðx,y,t Þ, ð1Þ 4 @x @x @y @y @t where D¼Eh3/12(1 u2) is the flexural rigidity and E, m and u are Young’s modulus of elasticity, mass per unit area and Poisson’s ratio, respectively. While h is the constant thickness of the plate. 2.1. Finite element model of a moving mass on a plate Fig. 2 shows the four-nodal rectangular sth plate element on which the moving mass m applies at time t as well as the nodal forces and deflections of this element. The mass moves with a constant velocity v and a variable contact point Xm(t) with y axis measure e. In this study it is assumed that the mass is in contact with the plate during all travelling time. While the mass progresses the plate vibrates, the interaction force is caused by

Fig. 1. A rectangular plate on which there is a moving lumped mass that moves with a constant velocity vat distance e from the origin O as being parallel to axis x; and finite element model that is divided into rectangular plate elements at dimensions a and b.

27

Fig. 2. Equivalent node forces and node deflections of sth plate element onto which moving mass m is applied at time t.

gravity and the vibration acceleration of the plate; therefore, the transverse (z) force component between the plate and moving mass induced by the curvature of the deformed plate and gravity is [14]: " # 2 d wðX m ,Y m ,t Þ f ðx,y,t Þ ¼ mgm dðxvtÞdðyeÞ, ð2Þ dt 2 where f(x, y, t) is the net force applied by the moving mass on x, y points at time t. dðxvt Þ and dðyeÞ represent the Dirac delta functions in x and y directions, respectively. When the inertial effect of the moving mass is taken into consideration, the acceleration d2w/dt2 is calculated from the second-order total differential by accepting that x and y coordinates that determine the motion of the load on the plate is a function of time. The acceleration is obtained as follows when differential is taken twice according to the x and y variables. 2

d wðx,y,t Þ € þ 2w00xy x_ y_ þ2w _ 0x x_ ¼ w00x x_ 2 þ w00y y_ 2 þ w dt 2  _ 0x x€ þ w _ 0y y, €  _ 0y y_ þ w þ 2w

x ¼ X m ðtÞ,y ¼ Y m ðtÞ

ð3Þ

In this equation Xm(t) and Ym(t) represent the overall x and y coordinates of the mass on the plate, respectively. As there are no velocity components in y direction in this study, general Ym(t) term is constant and Ym(t)¼e. The derivation procedure of Eq. (3) is given in Appendix A. When Eq. (2) is rearranged according to Eq. (3) the force equation is given by  € þ 2w00xy x_ y_ þ2w _ 0x x_ f ðx,y,t Þ ¼ m½g w00x x_ 2 þ w00y y_ 2 þ w  _ 0x x€ þw0y y€ dðxvt ÞdðyeÞ, _ 0y y_ þ w ð4Þ þ 2w where,  , signs show the derivatives of deflection depending on dimension and time. Eq. (4) shows the general situation of the applied force. If moving mass moves in x direction with a constant velocity and there are no velocity components in y direction, x€ and y€ acceleration terms and y_ velocity term in y direction are equal to zero. In this case, Eq. (4) becomes   € þ 2mvw _ 0x dðxvt ÞdðyeÞ, f ðx,y,t Þ ¼ ½mg mv2 w00x þ mw ð5Þ € where v shows the velocity of the moving mass and mv2 w00x , mw _ 0x show the centripetal force, inertia force and Coriolis and 2mvw force components of the moving mass, respectively. Subscript ‘‘x’’ shows that the deflection function is differentiated according to x. Eq. (5) is similar to the formulas given by Cifuentes [19] and Esen [10] for the beams under a moving mass. The reason for this similarity is the plate problem as two-dimensional situation of the beam. Nevertheless, as the shape functions of beam element and plate elements are different from each other, Eq. (5) will yield different results for the beam element and plate element.

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

28

The rectangular plate element in Fig. 2 is a 16-DOF conforming plate element with C(1) continuity conditions at element boundaries. It includes constant twist @2w/@x@y at corner nodes. Hence each corner nodal displacement is [17]: T

ui ¼ fui1 ui2 ui3 ui4 g ¼ fw yx yy yxy g

h ff g ¼ f s1 h € ¼ u€ s1 fug h

i

( ¼

)T @w @w @2 w  w @y @x @x@y

T

with,

ði ¼ 1, 2, 3, 4Þ,

ð6Þ

_ ¼ u_ s1 fug h fug ¼ us1

iT

f s2

f s3



f s16

u€ s2

u€ s3



u€ s16

u_ s2

u_ s3



u_ s16

us2

us3



us16

, iT iT

iT

, ð12aÞ ,

,

i

where w is the vertical deflection of ith nodal point, and yxi , yyi and yxyi are, respectively the rotation angles of ith nodal point around axes x and y and twist at axes x and y. When the gravitational force and the inertia forces of the plate are taken into consideration, the equivalent nodal forces of the sth plate element are as given in Eq. (7) depending on the relevant shape functions. f sn

  € þ 2mvw _ 0x  ¼ Nn ½mg mv2 w00x þ mw

ðn ¼ 1, 2, 3,:::, 16Þ,

ð7Þ

where Nn (n ¼1, 2, 3, y, 16) are the shape functions of plate element and given by [14]. N1 ¼ p1 q1 ,

N9 ¼ p3 q3 ,

N2 ¼ p2 q1 ,

N10 ¼ p4 q3 ,

N3 ¼ p1 q2 ,

N11 ¼ p3 q4 ,

N4 ¼ p2 q2 ,

N12 ¼ p4 q4 ,

N5 ¼ p1 q3 , N6 ¼ p2 q3 ,

N13 ¼ p3 q1 , N14 ¼ p4 q1 ,

N7 ¼ p1 q4 ,

N15 ¼ p3 q2 ,

N8 ¼ p2 q4 ,

N16 ¼ p4 q2 ,

ð8Þ

where pn and qn (n¼ 1–4) are hermitic polynomial components that represent the shape functions of the plate at axes x and y, respectively. 3

3

p1 ¼ 13xðtÞ2 þ2xðtÞ3 ,

q1 ¼ 1=b ½b 3by2 þ 2y3 ,

p2 ¼ a½xðtÞ2xðtÞ2 þ xðtÞ3 ,

q2 ¼ 1=b ½b y2by2 þ y3 ,

2

3

p3 ¼ 3xðtÞ 2xðtÞ , 3

2

p4 ¼ a½xðtÞ xðtÞ ,

2

2

3

q3 ¼ 1=b ½3by2 2y3 ,

ð9aÞ

2

q4 ¼ 1=b ½y3 by2 ,

xðtÞ ¼ xm ðtÞ=a, y ¼ ym ¼ Z ¼ b=2,

ð9bÞ

Here, as can be seen in Fig. 2, a and b are length and width of sth plate element, respectively, whereas xm(t) is the distance between moving mass and the left end of sth plate element. ym ¼ Z is the constant distance between moving mass and axis x; it is assumed in this paper that this distance is on the line that passes through the middle point of the plate element. At any time t the relation between the deflection of any point at x and y coordinates in plate element and shape functions is

2

N1 2

N1 N2

N1 N3

6 N2 2 N2 N3 6 N2 N1 6 6 N3 N2 N3 2 ½m ¼ m6 N3 N 1 6 : : 4 : N16 N 1 N 16 N2 N16 N 3 2 N1 N02 N 1 N 03 N1 N 01 0 6 N N0 N2 N2 N 2 N 03 6 2 1 6 0 0 N3 N2 N 3 N 03 ½c ¼ 2mv6 6 N3 N1 6 : : : 4 N16 N 01 N 16 N02 N16 N 03 2 N1 N 001 N1 N002 N 1 N 003 00 6 N N 00 N2 N2 N 2 N 003 6 2 1 6 00 2 6 N N 00 N3 N2 N 3 N 003 ½k ¼ mv 6 3 1 6 : : : 4 N16 N 001

N 16 N002

N16 N003

    

N 1 N16

3

7 N 2 N16 7 7 N 3 N16 7 7, 7 : 5 N 16 2



N1 N016

 

N2 N016 N3 N016

 

: N 16 N016



N1 N0016

   

3 7 7 7 7, 7 7 5 3

N2 N0016 7 7 7 N3 N0016 7 7, 7 : 5 00 N 16 N16

where [m], [c] and [k] are mass, damping and stiffness matrices of the new finite element, respectively. As the position of the moving mass changes by depending on time, the values of mass, damping and stiffness matrices, [m], [c] and [k], also change in time. Dimensions of these matrices of the new finite element are the same as dimensions of mass, damping and stiffness matrices of the plate element, because the matrices of the new finite element are calculated from the equivalent nodal deflections and related shape functions of the plate element. As rectangular plate element (see Fig. 2) has 4 deflections at each corner nodal point, the dimensions of the property matrices of the new finite element would be 16  16. The force vector (11) is calculated at each time step according to xm(t) that is the instantaneous position of the moving mass on the element s. When moving mass travels at variable speed (accelerating or decelerating), in this case the _ 0x x€ and w _ 0y y€ in (11) do not vanish and the force vector terms w (11) can be created with the contribution of these terms using the formulation given in Appendix C. For acceleration ( þ), and for deceleration ( ) is used in Eq. (A.5).

3. Equation of the motion of the plate system under moving mass Finite elements expression of the equation of the motion for a system consisting of moving masses and plate is as follows. ½Mfz€ ðtÞg þ ½Cfz_ ðtÞg þ ½KfzðtÞg ¼ fFðtÞg,

wðx,y,t Þ ¼ N1 us1 þN 2 us2 þ N3 us3 þ N4 us4 þ    þN 15 us15 þ N16 us16 , ð10Þ where usi (i¼1–16) is the deflections of nodes of the sth plate element on which moving mass is located. The deflection equation given in Eq. (10) is placed in the equivalent nodal force equation of sth element given in Eq. (7); when the obtained equations are organized in the form of a matrix, the following matrix equation is obtained: € þ ½cfug _ þ ½kfug, ff g ¼ ½mfug

ð11Þ

ð12bÞ

ð13Þ

where ½M, ½C and ½K are mass, damping and stiffness matrices of the entire system at time t, respectively; fz€ ðtÞg, fz_ ðtÞg and fzðtÞg are the acceleration, velocity and deflection vectors of the system, respectively, and fFðtÞg is the external force vector of the system. 3.1. Overall stiffness and mass matrices of the structure In cases where an additional load does not exist, overall stiffness K and mass M matrices of the plate system seen in Fig. 1 can be obtained by combining Ke, Me element matrices of

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

the plate and application of the given boundary conditions. Stiffness coefficients keij of the element Ke matrix can be calculated with the following Equation [14] Z bZ a" 2 @2 Nj @2 Ni @ Ni @2 Nj @2 N @2 Nj e kij ¼ D þu 2 i þu 2 2 2 2 @x @x @x @y @x @y2 0 0 # @2 N @2 N j @2 N i @2 N j þ 2ð1uÞ þ 2i dx dy ð14Þ 2 @x@y @x@y @y @y meij coefficients of the Me kinematically consistent mass matrices are: Z aZ b meij ¼ m Ni ðx,yÞNj ðx,yÞ dx dy ð15Þ 0

0

where m is the mass of unit area of the plate, whereas a and b are dimensions of the plate in the directions x and y. When the moving mass exists, stiffness and mass matrices of the entire system can be obtained by taking into consideration the inertial and centripetal forces caused by moving mass. In this case, instant overall stiffness and mass matrices are: K ij ¼ K ij , M ij ¼ Mij ði,j ¼ 1n : total system DOFÞ,

ð16Þ

except for the coefficients of the sth plate element: K si sj ¼ K si sj þkij , M si sj ¼ M si sj þ mij ði,j ¼ 116Þ,

ð17Þ

When the mass reaches sth element, we add the [m] and [k]matrices of the new finite element given in Eq. (12) to the element mass and stiffness matrices of the sth plate element. In calculating the instant values of [m], [k]and [c] time-dependent matrices, it will be necessary to obtain equation x(t)¼xm(t)/a that represents the position of the mass on sth element and to evaluate the shape functions according to this x(t) value and substitute it in Eq. (9); therefore, instant values of xm(t) and s are determined as follows xm ðtÞ ¼ vtðs1Þlx

    s ¼ integer part of ½ elx =ba þ vt=lx  þ 1,

ð18Þ

3.2. Overall damping matrix of the structure

29

follows: h fFðtÞg ¼ 0

0

0



f s1

f s2

f s3



f s14

f s15

f s16



0

0

0

iT

,

ð22Þ where, f si ¼ mgNi ði ¼ 116Þ

ð23Þ

3.4. Calculations of the property matrices at every time step For the calculation of the instantaneous overall mass and stiffness matrices and force vector (11) of the entire system at every time step of Dt, one may use the following steps: 1. Determine the mass and stiffness matrices of each plate element. 2. For time t, determine the element s on which the moving mass locates with (18). 3. Determine xm(t) which is the time dependent position of the moving mass on the sth element with (18). 4. Calculate the time dependent shape functions with (8), (9a), (9b) by substituting the value xm(t) which is defined in the previous step. 5. Calculate the mass, stiffness and damping matrices of the new finite element with (12b). 6. Calculate the mass and stiffness matricesof the sth element with the help of (16) and (17) by adding the defined mass and stiffness matrices of the new finite element. Calculate the force vector. 7. Calculate the instantaneous overall mass and stiffness matrices of the entire system by combining the mass and stiffness matrices of each plate element. Evaluate the overall force vector with the help of (22). Then impose boundary conditions. 8. For t þ Dt go to step 2.

3.5. Separation between the moving mass and the plate

In order to investigate the effect of damping on the structure, damping matrix C can be obtained by using the Rayleigh damping theory which assumes that the damping matrix is proportionate to a combination of mass and stiffness matrix [15].

When the separation between the moving mass and plate occurs, the interaction force of Eq. (2) must be forced to be zero. That is, during separation t1 rt rt2, the dynamic equation of motion must be replaced by the following two equations [27]:

C ¼ a0 M þ a1 K,

½Mfz€ ðtÞg þ½Cfz_ ðtÞg þ½KfzðtÞg ¼ 0,

with, ( ) a0 a1

2

q€ ¼ g,

3

( ) oi oj 4 oj oi 5 zi ¼2 2 , 1 1 zj oj oi 2 oj oi

ð19Þ

ð20Þ

except for C si sj ¼ C si sj þ cij

ði,j ¼ 116Þ,

with the initial conditions specified as _ 1 ðxÞ, qðt 1 Þ ¼ w1 ðvt 1 Þ, q_ ðt 1 Þ ¼ w _ 1 ðvt 1 Þ, _ ðx,t 1 Þ ¼ w wðx,t 1 Þ ¼ w1 ðxÞ, w

where the terms zi and zj are damping ratios related to the natural frequencies oi and oj. In terms of [15,16], z1 and z2 values are taken in this paper as 0.005 and 0.006, respectively. In this case, the overall damping matrix of the system under moving mass is: C ij ¼ C ij ði,j ¼ 1n : total system DOFÞ,

ð24Þ

ð21Þ

3.3. Overall force vector of the structure Overall force vector of the system is established by equating all coefficients except for the nodal forces of the sth plate element to zero. Thus, the instant force vector of the entire system is as

ð25Þ where vt1 and q represent the motion of moving mass due to gravity. Eq. (25) represent the motion of the moving mass due to the gravity during separation. When two separate solutions of Eq. (24) become equal again at t ¼t2, which implies that the moving mass starts recontacting the beam, then the equation of motion (13) must be solved in sequence by using the solutions of Eq. (24) at t ¼t2 as the new initial conditions for the dynamic response of the plate after t ¼t2. As studied in Fryba [1], the separation and the following impact may be very important in practice for highway and railroad bridges [27].The effects of separation should be studied widely in a special study as reported by [27]. In order not to exceed the limits of the study without aiming at investigating all possible effects of moving mass and the plate system but to make easy the understanding of present

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

30

results, the effects of the separation between the moving mass and plate are not given in the numerical analysis in this study.

4. Results and discussion with numerical solutions In this paper, the Newmark direct integration method [18] is used along with the time step t ¼0.0001, b ¼ 0.25 and g ¼0.5 values to obtain the solution of Eq. (13), where b and g are parameters that manage the sensitiveness and stability of the Newmark procedure. When b takes 0.25 value and g 0.5, this numerical procedure is unconditionally stable [14]. Example 1. Let us take a simple supported isotropic beam-plate transversed by a F¼4.4 N moving load. The dimensional and material specifications of the plate are identical with those chosen in [23], i.e., lx ¼ 10.36 cm; ly ¼0.635 cm, h¼0.635 cm; E¼206.8 GPa, r ¼10,686.9 kg/m3; Tf ¼ 8.149 s, where Tf is the fundamental period. In Table 1, dynamic amplification factors (DAF), which are defined as the ratio of the maximum dynamic deflection to the maximum static Table 1 Dynamic amplification factors (DAF) versus velocity. V (m/s)

Tf/T

1

2

3

4

15.6 31.2 62.4 93.6 124.8 156 250

0.125 0.25 0.5 0.75 1 1.25 2

1.045 1.350 1.273 1.572 1.704 1.716 1.542

1.042 1.082 1.266 – 1.662 – 1.518

1.063 1.151 1.281 1.586 1.704 1.727 1.542

1.025 1.121 1.258 1.572 1.701 1.719 1.548

(1) (2) (3) (4)

Present method using moving finite element. From Ref. [20]. From Ref. [21]. Analytical solution from Ref. [22].

deflection, are compared with several previous numerical, analytical and experimental results available in literature. It is noted that T is the required time for moving load to travel the plate. It is seen that the results obtained by the new finite element (column 3) are very close to the analytical solution [22], and also the results of first order shear deformation theory (FSDT) method [21]. Example 2. In this example, dynamic deflections of a cantilevered plate affected by moving masses are presented by using the new finite element. Dimensions and material properties of the plate are: lx ¼0.15 m, ly ¼0.25 m (clamped), h¼0.01 m E¼210 GPa, r ¼ 7850 kg/m3, u ¼0.3. At each simulation, moving mass moves on the plate in x direction by following the fixed y¼ly/2 middle line at a constant velocity from the clamped end to the right end. First of all, the right hand side of Eq. (13) was taken as zero, and eigenvalues and eigenvectors, meaning vibration modes and frequencies of unloaded plate, were obtained; the first four vibration modes and frequencies of the plate are shown in Fig. 3. In order to obtain dynamic deflections of moving masses of different amounts at different velocities, an independent analysis was made for fixed travelling velocities of different masses between 0 and 100 m/s with 1 m/s increments. As different load, the ratio of the load to the mass of the plate was taken as e ¼m/M 0.25, 0.5 and 1; thus, analysis was obtained for three different mass amounts. The mass of the plate was M¼2.94375 kg, whereas the masses of the load were¼0.7359375, 1.471875 and 2.94375 kg, respectively. Before examining the cases where the mass is moving, it seems like an obligation to examine the natural frequencies of the plate in the rest mass situation. For the rest mass, the first purpose of examining the frequency behaviour of the system is to show the complexity of the behaviour. The change of the first three natural vibration frequencies of the plate depending on the

Fig. 3. The first, second, third and fourth vibration modes and frequencies of the plate.

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35 4.00 ε =1

3.50

ε =0.5

ε =0.25

3.00

DAF

2.50

2.00

1.50

1.00

0.50

0.00

0

10

20

30

40

50

60

70

80

90

100

Velocity of moving mass (m/s) Fig. 5. Vertical non-damped moving mass responses of the middle point of the free right end of the CFFF plate for different travel velocities and mass ratios. Solid line (——) for e ¼ 1. Dashed line (———) for e ¼ 0.5. Dotted line (yy) for e ¼ 0.25.

3.50

3.00

ε=1 ε=0.5

2.50

ε=0.25

2.00

DAF

magnitude and the position of the mass on the plate are given in Fig. 4. Straight lines (——) are for e ¼1, dashed lines (———) are for e ¼ 0.5 and dotted lines (::::::) are for e ¼0.25. When the mass positioned on the modal nodes of plate, the natural frequency of any nth mode becomes equal to the natural frequency in the same mode when no additional mass exists. When the mass lies on any nodal point, the natural vibration frequency decrease, which is expected in plate and mass system, does not occur. As expected, with the exception of nodal points, any mass addition to the plate decreases natural vibration frequency. The more mass is added, the more natural vibration frequency decreases. Places of the minimum values in Fig. 4 depend not only on the amount of the mass but also on the position of the mass on the plate. Therefore, the place of the additional mass on the plate is an important factor in terms of the dynamic behaviour of the plate system. The interesting thing here is that overall maximum values do not occur at 2nd and 3rd modes where the mass is at the right-hand side of the plate. The change interval of the decrease in natural frequency of vibration modes of the plate under constant mass is greater in a higher mode compared to the previous mode. When Fig. 4 is examined, it becomes clear that the range of the frequency change in 2nd mode is bigger than that in 1st mode, whereas the range of the frequency change in 3rd mode is greater when compared to the one in 2nd mode. That is, the increasing mass causes further the range of change in frequency at each successive vibration mode. Vertical deflections of the free end of CFFF plate for different travel velocities, mass ratios and damping and non-damping situations are provided in Figs. 5 and 6. When inertial effects of masses are ignored with the damping situation, which means that the moving force is accepted by merely taking the gravitational effect of the mass, the dynamic deflections are shown in Fig. 8. Depending on the mass and velocity in order to show the maximum deflections of the middle point of the right free end of the plate, the obtained dynamic deflection amount is divided by the deflection amount when the velocity is zero and mass is accepted as static at the right-end point - Dynamic Amplification Factor (DAF). Here DAF is explained with the equation DAF¼ wdin. (lx, ly/2, t)/wstatic (lx, ly/2).These deflections were obtained by a special program in MATLAB by using the Newmark direct integration method. In order to gain these deflections, the effects of

31

1.50

1.00

0.50

0.00

0

10

20

30

40

50

60

70

80

90

100

Velocity of moving mass (m/s) Fig. 6. Vertical damped moving mass responses of the middle point of free end of the CFFF plate for different travel velocities and mass ratios. Solid line (——) for e ¼ 1. Dashed line (———) for e ¼ 0.5. Dotted line (yy) for e ¼0.25. Damping ratios z1 ¼ 0.005 and z2 ¼ 0.006.

Fig. 4. The first three natural frequencies of the plate with a fixed mass in the span for different mass ratios. Solid line (——) for e ¼ 1. Dashed line (———) for e ¼0.5. Dotted line (yy) for e ¼ 0.25.

the first 20 vibration modes of the plate were taken into consideration in numeric analysis. Here the most important factor that determines the sensitiveness of the analysis is the magnitude of time step. The magnitude of time step Dt must be equal to or smaller than the vibration period of the largest vibration mode included in the evaluation. The time interval was chosen as Dt o ¼T20/20, where T20 is the period of 20th vibration mode. If the time interval is chosen larger than the period of the maximum vibration mode included in the calculation, we may not take into account the vibration modes that occur in a time smaller than the time interval; thus, the effect of these modes on total deflection will be ignored, and an inaccurate analytical result will be obtained. If the system is damped as its results are shown in Fig. 6, the effect of the higher modes will be severely damped, so that it will be useless to take into account the higher modes in analysis. Depending on the conditions of damping, one can get acceptable accurate results by considering fewer modes such as

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

32

1.40

4.00 3.50

mg/Mg=1

1.20

2.50

mg/Mg=0.5

2.00

1.00

Undamped

0.50

Damped

0.00 0.00

mg/Mg=0.25

1.00

1.50

0.20

0.40

0.60

0.80

0.80 1.00

ε

Fig. 7. Nonlinear change of the maximum deflection depending on increase of mass at v ¼20 m/s constant travel velocity.

5 or 10. Even considering 2 or 3 modes in analysis maybe sufficient for many heavily damped systems. It can be seen from Figs. 5 and 6 that deflections increase as the mass ratio rises as expected. In Fig. 7, DAF rising graph is given by depending on the increase of mass ratio e. However, the increase in deflection in different mass ratios is not linear with increase of mass as shown in Fig. 7. For example, when we double the mass ratio from 0.25 to 0.5 at 20 m/s travel velocity, the increase rate of deflections (DAF) rises by 2.65. Again, when we increase the mass ratio from 0.5 to 1.0, the increase rate of deflections (DAF) rises by 3.125. The primary effect of an increase in mass is an increase in deflections. As opposed to increase in mass, deflection increase is non-linear; therefore, it is understood that the amount of moving mass on the plate affects the shape of its deformation. In damping (dashed line) and non-damping (straight line) situations, the increase in DAF is not linear, either. However, in non-damping situation, the increase in DAF is bigger when compared to the damping situation. As seen in Figs. 5 and 6, there are many local maximum and minimum values of deflections according to deflection curves depending on the velocity and mass ratio. Nevertheless, for 20 m/s velocity, an overall maximum value is obtained. Dynamic deflections, at the mass ratio e ¼1 and velocity v¼20 m/s, increased by almost 3.66 times compared to the zero velocity. Figures make it clear that the velocity of the mass considerably changes the deflection rates of the plate. Contrary to the effect of mass, the change here occurs in the direction of increase in small increments under 20 m/s. Deflection reaches maximum levels for 20 m/s and begins to decrease in higher velocities. It can be concluded that the vibration pattern of a mass under a moving load is affected by the velocity of the moving mass rather than its magnitude. The reason for this is that the plate vibrates, while the mass moves on the plate. In some cases contrary to the gravity which affects the mass downwards, the movement direction of the point, on which the mass is located, can be in the opposite direction, which is upwards. Moreover, as the position of the mass continuously changes, the frequencies of the vibration modes of the plate also change constantly. Here a complex behaviour occurs in the plate depending on the magnitude, position and velocity of the mass. In addition, a close examination of Fig. 5 reveals that the deflection ratio induced by 100 m/s mass velocity is smaller compared to the deflection ratios under this velocity. This is caused by the increased velocity of the mass and should be evaluated according to the examination of two limit cases, namely v¼0 and v¼N. In the first limit case, we approach the quasi-static loading situation when the maximum deflection of the plate consists merely of the stable mass, and dynamic effects of the mass are negligible. In the second limit case, the plate does not change its shape due to inertial effects and negligible traversing time. The results given in Fig. 6 agree well with the analytical solution results of a composite plate using the Galerkin method [24].

DAF

DAF v=20

3.00

0.60

0.40

0.20

0.00

0

10

20

30

40

50

60

70

80

90

100

Velocity of moving force (m/s) Fig. 8. Vertical damped moving force responses of the middle point of free end of the CFFF plate for different travel velocities and mass ratios when inertial effects of the moving masses are omitted. Damping ratios z1 ¼ 0.005 and z2 ¼ 0.006.

As can be seen from Fig. 6, the viscous damping decreases dynamic deflections. The wavy deflection curve seen in Fig. 5 becomes a relatively much smoother one in Fig. 6 due to the existence of damping. The conclusion drawn from this situation is that increasing internal and external damping decreases deflections and provides several benefits for the control of excessive vibration of the structural system. In the case of a moving force seen in Fig. 8 which is obtained by the ignorance of inertial effects considering only the gravitational effect of the mass, DAF increase is maximum 30 percent. However, in the case of assuming moving mass under the same damping conditions (Fig. 6), DAF increased by 325 percent. Figures look alike at velocities higher than 10 m/s. As can be seen in Fig. 8, in case of assuming moving force, the increase in force does not affect DAF. However, in Figs. 5 and 6, an increase in the mass does alter DAFs. This means that the deflection results of moving mass and assumption of moving force are considerably different from each other. For this reason, the assumption of moving force cannot represent the real dynamic behaviour of the plate which is traversed by a high-speed mass. These results are in agreement with the conclusions of beam problems in literature. Velocities and accelerations of the middle point of the free end of the plate during vibration are given in Fig. 9a and b. In both velocity and acceleration curves, huge differences can be seen between moving mass and moving force assumptions. In smaller mass-travel velocities, i.e., between 0 and 15 m/s, velocity and acceleration change with minor differences for both situations. Under the high velocity, the increases in both velocity and acceleration curves were massive; taking this viewpoint, it can be understood that the ignorance of inertial effect of the mass resulted in a failure to display the real behaviour of the plate under moving mass. The contact between the moving mass and the plate is not actually a point contact. In this paper, the case, when the contact is a point contact, meaning that the mass focuses on a certain point, is examined. The reason for this limitation is to be able to make a comparison with the papers in literature that investigate moving mass and carrier structures like bridges within the vehicle–structure interaction and then to approve of the application and conformity of the suggested method with previous studies in literature.

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

33

force, which is achieved through the ignorance of inertial effects of the mass, indicates inefficiency in the solution of dynamic behaviour of plates under this type of moving mass. This study considers the inertial effects of the mass instead of the methods suggested by some other studies in literature that accept the mass as a moving force in vehicle–structure interaction problems. Using the suggested method in this study will minimize miscalculations. With the new finite element method suggested in this paper, inertial effects of the mass can easily be taken into account, and moving mass–plate and vehicle–structure interaction problems can be easily solved. The mass that moves on deformed surfaces of the plate is modelled as a finite element and adapted to classical finite element method. Thus, inertial effects of the mass are taken into consideration in the finite element model of the entire system. As a result, stiffness, mass and damping matrices of the moving mass and the plate element can easily be combined. Contrary to this convenience, at each step of the Newmark method, stiffness, mass and damping matrices of the plate are updated again, depending on the position of the mass on the plate. The new finite element method offered by this paper requires higher CPU speed and time to obtain the dynamic response of the structural system. If accurate results are desired, this is an indispensable cost.

Acknowledgement The author acknowledges Sabriyaman Inc. (Manufacturer of wood working machineries in Istanbul, Turkey) for all supports to the present research. Fig. 9. (a) velocity and (b) acceleration of the free end of the plate for different velocity of masses and the cases of moving mass and moving force.

Appendix A. Derivation procedures of Eq. (3) Another reason for accepting that the mass is a point mass is to avoid the complexity in the solution of complex motion equations in the cases when the mass contacts the plate in hundreds of points and to attempt to at least partially understand the dynamic behaviour of the plate affected by a moving mass. As a matter of fact, the number of masses is not significant for the model established in this study. It is possible to integrate the suggested method with multiple masses into the system simultaneously or at different times and develop a solution. Moving a large mass can also be examined as a system that consists of sub-masses. In the case of large mass and small plate surface, the assumption that the mass is always in contact with the deformed geometry of the plate becomes invalid. When the obtained results are evaluated, using a lighter wooden part, increasing the mass of the plate, and increasing the damping rate prove critical for decreasing the vibration of planer entry plate of a wood processing machine.

For a vibrating plate, the transverse (z) force components, between a moving mass and the plate, induced by the vibration and curvature of the deflected plate, is given by ! 2 d wðx,y,t Þ f ðx,y,t Þ ¼ mgm dðxvtÞdðyeÞ, ðA:1Þ dt 2 x ¼ X ðtÞ,y ¼ Y ðtÞ m

dwðx,y,t Þ @w dx @w dy @w dt ¼ þ þ , dt @x dt @y dt @t dt 2

ðA:2Þ 2

d wðx,y,t Þ @2 w dx dx @2 w dx dy @w d x þ þ ¼ 2 @x dt 2 @x2 dt dt @x@y dt dt dt 2 2 2 @ w dx @ w dx @ w dy þ þ þ @x@t dt @x@t dt @y@t dt

5. Conclusions It has been obtained in the current study that inertial effects and velocity of the moving mass have considerable influence on the dynamic behaviour of the plate. This influence changes by depending on the mass and velocity of the load. The result of our analysis has shown that the increase in mass causes deflections to rise but this increase is non-linear. The effect of the change in velocity is more significant when compared to the change in mass. When the dynamic behaviour of the plate is changed due to the velocity increase, a more chaotic situation occurs when compared to the effect of mass on plate. Due to the effect of the velocity, dynamic deflections increase. But the increase is not continuously rising: that is, dynamic deflections begin to decrease once the velocity reached a certain level, which is seen when the velocity was more than 20 m/s. It is also shown that the assumption of the moving

m

where w(x,y,t) represents the transverse (z) deflection of the plate at position with coordinates x and y and time t. For the transverse (z) deflection

þ

@2 w @2 w dy dx @2 w dy dy þ 2 þ @x@y dt dt @y dt dt @t 2

þ

@2 w dy @w d y þ , @y@t dt @y dt 2

2

ðA:3Þ

rearranging Eq. (A.3) gives  2  2 d wðx,y,t Þ @2 w dx @2 w dy @2 w ¼ þ þ 2 @x2 dt @y2 dt dt 2 @t 2

þ2

@2 w dx dy @2 w dx @2 w dy þ2 þ2 @x@y dt dt @x@t dt @y@t dt 2

þ

2

@w d x @w d y þ @x dt 2 @y dt 2

_ 0x x_ € þ 2w00xy x_ y_ þ 2w ¼ w00x x_ 2 þ w00y y_ 2 þ w _ 0y y_ þ w _ 0x x€ þ w _ 0y y, € þ2w

ðA:4Þ

_I. Esen / Finite Elements in Analysis and Design 66 (2013) 26–35

34

with, X m ¼ X m0 þ v0 t 8 a2x t=2, v ¼ dX m =dt ¼ v0 8ax t, 2

ax ¼ d X m =dt 2

ðA:5Þ

Substituting Eq. (A.4) into (A.1) gives:  _ 0x x_ € þ 2w00xy x_ y_ þ 2w f ðx,y,t Þ ¼ ½mgm w00x x_ 2 þw00y y_ 2 þ w  _ 0x x€ þ w _ 0y y€ dðxvt ÞdðyeÞ, _ 0y y_ þ w þ 2w

ðA:6Þ

Appendix B. Solution steps for Newmark integration method. Using Newmark integration method, the solution of Eq. (13) is obtained according to the following steps. (16): I. Determine the integration parameters b and g and magnitude of the time interval Dt. In this study b ¼ 0.25 and g ¼ 0.5 is accepted for integration accuracy and stability. II. Calculate integration constants: a0 ¼

1

bDt2

,

a4 ¼ bg 1,

g a1 ¼ bD t,

a5 ¼

Dt 2



a2 ¼ g

b 2

 ,

1

bDt ,

a3 ¼

1 2b 1,

  a6 ¼ Dt 1g ,

ðC:1Þ

When the equities in Eq. (C.1) are organized in the form of a matrix, the following matrix equation is obtained.



 f ¼ ½m u€ þ ½c u_ þ k fug, ðC:2Þ

a7 ¼ gDt: ðB:1Þ

III. Using Sections 2.3 and 2.4, determine the mass, stiffness and damping ½M, ½K and ½C matrices at tn( ¼tn  1 þ Dt) time. IV. Calculate effective stiffness matrix at ( ¼tn  1 þ Dt) time: _ ðB:2Þ ½K  ¼ ½K þa0 ½M þa1 ½C: _ V. Calculate effective force fF ðtÞg at time tn( ¼tn  1 þ Dt): _ fF ðt n Þg ¼ fF ðt n Þg þ ½Mða0 fuðt n1 Þg þ a2 fu_ ðt n1 Þg þa3 fu€ ðt n1 ÞgÞ þ½Cða1 fuðt n1 Þg þ a4 fu_ ðt n1 Þg þ a5 fu€ ðt n1 ÞgÞ:

ðB:3Þ

where fu€ ðt n1 Þg, fu_ ðt n1 Þg, fuðt n1 Þg are, respectively, the initial conditions for the accelerations, velocities and displacements of the structural system at time t¼t0 ¼0. VI. Calculate deflections at tn time: _ _ ðB:4Þ fuðt n Þg ¼ ½K 1 fF ðt n Þg: VII. Calculate accelerations and velocities at tn time: fu€ ðt n Þg ¼ a0 ðfuðt n Þgfuðt n1 ÞgÞa2 fu_ ðt n1 Þga3 fu€ ðt n1 Þg, ðB:5Þ fu_ ðt n Þg ¼ fu_ ðt n1 Þg þ a6 fu€ ðt n1 Þg þ a7 fu€ ðt n Þg:

  þN 1 N01 u_ 1 þ N02 u_ 2 þ    þN 016 u_ 16 ,   f s2 ¼ N 2 mg þ N 2 mv2 N001 u1 þ N002 u2 þ    þ N0016 u16 þN 2 ðN1 u€ 1 þ N2 u€ 2 þ    þ N16 u€ 16 Þ   þN 2 N01 u_ 1 þ N02 u_ 2 þ    þN 016 u_ 16 ,   :f s3 ¼ N3 mg þ N3 mv2 N 001 u1 þN 002 u2 þ    þN 0016 u16 þ N3 ðN1 u€ 1 þ N2 u€ 2 þ    þ N16 u€ 16 Þ   þ N3 N01 u_ 1 þ N02 u_ 2 þ    þ N016 u_ 16 ,:::   f s15 ¼ N15 mg þN 15 mv2 N 001 u1 þ N002 u2 þ    þ N0016 u16 þ N15 ðN1 u€ 1 þ N2 u€ 2 þ    þ N16 u€ 16 Þ   þ N15 N 01 u_ 1 þN 02 u_ 2 þ    þN 016 u_ 16 ,   f s16 ¼ N16 mg þN 16 mv2 N 001 u1 þ N002 u2 þ    þ N0016 u16 þ N16 ðN1 u€ 1 þ N2 u€ 2 þ    þ N16 u€ 16 Þ   þ N16 N 01 u_ 1 þN 02 u_ 2 þ    þN 016 u_ 16 ,

ðB:6Þ

Steps III–VII, t ¼tn ¼tn  1 þ Dt (n¼1, 2, 3,y. and t0 ¼ 0) are repeated for all time steps and deflections fuðt n Þg, velocities fu_ ðt n Þg and accelerations fu€ ðt n Þgof the structural system are calculated.

Appendix C. Derivation procedure of Eq. (11) If the equation which expresses the relation between deflection and shape functions given in Eq. (10) is substituted at the equation for nodal forces of plate element given in Eq. (7), nodal force equations are obtained as follows.   f s1 ¼ N1 mg þN 1 mv2 N001 u1 þ N002 u2 þ    þ N 0016 u16 þ N1 ðN 1 u€ 1 þ N 2 u€ 2 þ    þ N16 u€ 16 Þ

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