Response of Grilage-Like Structures by the Mobility Approach

Response of Grilage-Like Structures by the Mobility Approach

Fábio Fiates

Arcanjo Lenzi

Laboratório de Vibrações e Acústica (LVA)

Departamento de Engenharia Mecânica

Universidade Federal de Santa Catarina (UFSC)

Caixa Postal 476, Campus Universitário

88040-090 Trindade, Florianópolis, SC. Brazil

[e-mail], [e-mail]

This work presents an application of the Mobility Approach to the analysis of the power flow through grillage-like structures. Such structures are usually found in offshore platforms, supporting large and heavy machines. Different wave kinds (longitudinal, flexural and torsional) were initially considered in the power flow analysis between two beams joined in L. Beams excited by an in-plane point force showed strong coupling between longitudinal-flexural waves, while that for out-of-plane point force excitation, flexural-torsional waves coupling represents the most important mechanism of energy transmission. The response determination of grillages by the mobility approach requires the structure to be separated into simple beam-like structural components. Equations for rotations and displacements at the joints of all beams are written for as mobility functions, and moments and forces acting at the joints. A system of equations relating all such internal forces and moments is obtained. This approach was applied to simple grillages. Response results showed good agreement when compared to those provided by Finite Elements.

Keywords: Power Flow, waves, coupled beams, mobility, mobility approach

Introduction

Vibratory energy propagation problems in offshore structures are frequently found in operating platforms and are caused by the large number of heavy and large machines and equipment installed directly on their main structure. The structure vibrations generate unpleasant and very often, excessively high noise levels at the accommodation area, which are responsible for stress and deafness problems usually showed by workers exposed to such noises.

The structures supporting large and heavy machines are made of large cross section beams placed in orthogonal directions, in a grillage shape. Studies on the vibration response of beams networks, as used in buildings structures, were carried out by Sablik et al. (1985), using Statistical Energy Analysis (SEA). Sablik pointed out the importance of considering torsional waves coupled with flexural waves in the transmission of power across joined beams. Works by Goyder & White (1980), and by Davies & Wahab (1981) contributed significantly to the understanding of the power flow mechanisms.

Two methods currently in use, the Finite Elements Method (FEM) and SEA, are usually applied to different ranges of frequency. While FEM yields to accurate analyses at frequencies corresponding to the first resonances of the structure, SEA is applied to higher frequencies and in bands having at least five to six modes, in each subsystem. Both techniques usually fail to provide accurate results at frequencies in between the ranges mentioned above.

The mobility approach, presented by Cuschieri (1987, 1990), allows a deterministic calculation of the response, as well as of the power flowing across the joints, based on the use of mobility response functions of beams when considered separately.

This work represents an extension of Cushieri's work to more complex structures made up of several beams assembled in the shape of grillages. The relative importance of the several wave types is initially analyzed. The response of two grillages is determined by the mobility approach. Results are compared to those obtained by FEM, showing good agreement.

Nomenclature

A = beam cross section, m2

b = base of square or I beam, m

c_f = flexural wave speed, m/s

c_L = longitudinal wave speed, m/s

c_t = torsional wave speed, m/s

d = height of I beam, m

E = elasticity modulus, N/m2

f = frequency, Hz

F = shear force, N

G = shear modulus, N/m2

h = height of square beam, m

I = inertia moment, m4

j = imaginary part, dimensionless

k_L = longitudinal wave number, m-1

k_t = torsional wave number, m-1

L = beam length, m

m' = mass per unit length, kg/m

M = flexural moment, N.m

Q = torsional stiffness, N.m2

r = Timoshenko's shear coefficient, dimensionless

t = time, s

t_a = thickness of I beam, m

T = torsional moment, N.m

u = axial displacement, m

u = axial displacement, m

U = axial force, N

v = transversal flexural displacement, m

w = flexural angular displacement, rrad

W = power flow transmitted in bending, W

W_F = power flow transmitted by shear force, W

W_L = power flow transmitted by longitudinal force, W

W_M = power flow transmitted by flexural moment, W

W_t = power flow transmitted by torsional moment, W

Y = Mobility, (m/s)/N

Ymn - m = n = pont mobility

m ¹ n = transfer mobility

Greek Symbols

l_a, l_b = Timoshenko theory bending wave number, m-1

L = mass inertia moment per unit length, kg.m

f = angular flexural displacement for Timoshenko theory, rad

h = loss factor, dimensionless

n = Poisson coefficient, dimensionless

w = frequency, rad/s

r = density, kg/m3

q = angular torsional displacement, rad

Subscripts

x = in x direction

y = in y direction

z = in z direction

1, 2, ,7 = refers to each point (beam) at a junction

w_m = angular velocity (w) caused by a moment (m)

w_f = angular velocity (w) caused by a force (f)

v_m = transverse velocity (v) caused by a moment (m)

v_f = transverse velocity (v) caused by a force (f)

Superscripts

FF = flexural - flexural

TF = torsional - flexural

TT = torsional – torsional

Signals

__ (dash over a variable) = complex variable

. (dot over a variable) = temporal rate

* = complex conjugate

Basic Wave Equations

Longitudinal Waves

Longitudinal waves are characterized by all points of a given cross section having same displacement and direction of motion, coinciding with that of the wave propagation. This is true when cross sections dimensions are much smaller than the length of the beam.

Internal damping is here assumed to be the only mechanism of energy dissipation. It is considered of hysteric type, frequency independent and is analytically represented by the complex modulus of elasticity , where h is the loss factor. The bar symbol over the variable represents a complex (damped) variable.

The longitudinal displacement, (x,t), of a general beam cross section, in the frequency domain, can be expressed by the following differential equation (Cremer, Heckl & Ungar, 1973)

where is the complex longitudinal wave number; , is the complex longitudinal wave speed, and f, is the frequency.

The solution for the longitudinal displacement can be expressed in the form

where A1 and A2 are constants to be determined from two boundary conditions.

The power flowing through beams by the action of the axial force, is proportional to the axial velocity , as given by

the symbol * meaning complex conjugate.

Torsional Waves

In structural components performing torsional vibrations, all points of a given cross section have the same rotation about the axis where the moment is applied.

The torsional wave differential equation for the complex rotation, (x,t), in the frequency domain, is given by (Cremer, Heckl & Ungar, 1973)

where is the complex torsional wave number; is the complex torsional wave speed and f, the frequency.

Square cross sections beams have torsional stiffness, , and mass inertia moment per unit length, L, given by (Sablik, 1982; Timoshenko & Goodier, 1980)

For I cross section beams,

The solution of the differential equation can be expressed in the form

where A1 and A2 are determined from boundary conditions.

Torsional waves can also transmit power through the action of the torsional moment, (Tx), according to the expression

Flexural Waves

Analysis of flexural waves in beams whose dimension of the cross section are significant compared to their length, must consider shear deformation and rotatory inertia (Rao, 1986, Timoshenko & Goodier, 1980). Flexural moment and shear force are determined from the expressions

Equilibrium equations for forces and moments yield to the following expressions

The solutions for the coupled equations can be expressed as (Flügge, 1962)

where

It is noticed that power can be transmitted by the simultaneous action of the moment and shear force, in the form

Mobility Method

This method consists in modeling a whole structure as a number of coupled substructures, which are analyzed separately considering the forces and moments acting at the joints, and their corresponding mobility response functions (represented by Y_ij). These represent the behavior of the velocity response at one particular point of the structure to an excitation either at the same point (point mobility i = j) or at any other point (transfer mobility i ¹ j). Mobility is defined as the ratio between the Fourier Transform of the (translational or rotational) velocity response and the Fourier Transform of the corresponding (moment or force) excitation (Baars, 1996), as shown in Fig. 1.

Consider a two subsystems case, joined at one single point (as showed in Fig. 2), and that energy can be transmitted through this point by one force only.

The displacement of the junction points, u2 and u3, can be expressed as functions of components point mobilities Y22 and Y33 and the transfer mobility Y12, in the form

where F1 is an externally, applied excitation (force) and F2 and F3 are the internal forces at the joint. At the joint, the following physical conditions must be met: u2 = u3 and F2 = - F3, from which the following expression results

This simple example shows the application procedure by the mobility approach. It can be seen that the unknown force acting at the joint can be determined from the point and transfer mobilities of the components, and from the external force. Once F2 is determined, the velocity u2 and the power transmitted at the joint can also be calculated, as well as the response at any point of both components.

Beams Joined in L

Consider two beams joined in L, excited by a transversally (out-of-plane) harmonic point force, applied at the free end of beam 1, as shown in Fig. 3.

Power can now be transmitted through the joint (points 1 and 2) by shear force, and flexural and torsional moments. These three unknown variables will lead to a system of equations, resultant from three physical conditions to be met by the angular and transverse displacements, as follows

Equilibrium conditions for forces and moments acting at the joint, result in the following

The coordinate system used for writing the wave equations of both beams is also indicated in Fig. 3. The angular and transverse velocities at the joint of each beam are expressed as function of the mobilities response functions, where the subscripts 1 refers to point 1 in beam 1 and 2 to point 2 in beam 2, as follows

where is the mobility related to the ratio between a torsional (first superscript T) angular velocity (w) at point 1 and a torsional (second superscript T) moment (m) also at point 1, and so on.

Using the identities derived from the equilibrium conditions one can write the three remaining equations in a matrix form

Solving this system for the three unknowns moments and force acting at the joint, the response at any point of any beam can de determined as well as the power flowing through the joint.

Power Flow between Coupled Beams

Energy can be transmitted between coupled beams by the various types of propagating waves. As already mentioned, flexural, torsional and longitudinal waves must be considered in the power flow analysis of beam-like structures (Fiates, 1996).

In this section the results of power flow calculations between two beams joined in L, having dimensions as typically found in structures, are presented. All combinations of wave types coupling were considered (for instance, flexural waves in beam 1 with torsional waves in beam 2). The power transmitted from the excited beam, for each waves coupling combination was calculated and normalized with respect to the input power. A unit magnitude harmonic force applied at the free end of one of the beams (beam 1) was used as excitation.

Figure 4 shows the results for flexural-longitudinal and flexural-flexural waves coupling of two square cross section (5 cm) equal length (1 m) Euler-Bernoulli beams, excited by an in-plane force. It is very noticeable the importance of the flexural waves, particularly at the lower frequencies. Longitudinal waves in beam 2 absorb a significant amount of energy at frequencies above 500 Hz, where the first longitudinal modes resonances are present.

Figure 5 shows the results of the power flow transmitted between two Euler-Bernoulli beams, excited by an out-of-plane harmonic force applied at the free end of beam 1. Because of the direction of the applied force energy can now be transmitted by the following types of wave couplings: flexural-flexural, flexural-torsional and torsional-flexural.

Calculations of the transferred power normalized with respect to the input power were carried out for two square cross section beams, of different length, joined in L. The flexural-torsional coupling type is responsible for most of the energy transferred to the second beam. One notices that the transversal force induces flexural waves in beam 1, which couples with torsional waves in beam 2. This shows the importance of the torsional waves on the analysis of grillages excited by transverse forces.

Figure 6 shows the results for a similar analysis, considering this time a pair of beams of larger cross section. Flexural waves were modeled by Timoshenko's theory. The conclusions are very similar to those shown in Fig. 5. One notices, once again, the importance of the flexural-torsional waves coupling in the transmission of energy.

A comparison of the results for flexural-flexural waves coupling presented in Figs. 5 and 6, indicates that larger ratios of energy are transmitted by this type of coupling between beams of large cross section and inertia moment.

Aplication to Two Grillages

In this section it is described the application of the mobility approach to two small grillage like structures. The first case consists of a square frame, as shown in Fig. 7. In this application the frame was separated into two L-shaped beams. The forces and moments considered at the joints are also shown in the same figure. A transversal force was applied in beam 1.

The expressions for the linear and angular velocities at the joints, for each component, are

The following physical conditions must be met at these points

Substituting these expressions into Eqs. (33) to (44), the remaining equations can be written in a matrix form, as follows

The solution of this system of equations leads to the determination of the forces and moments acting at the joints, with which one can, subsequently, determine the response at any point of any components, as well as the power flowing at the joints by each wave coupling type.

The four beams in the analyzed frame had same cross section (0,20 m x 0,20 m), loss factors (h = 0,01), mass density (r = 8000 kg/m3) and elasticity modulus (E = 2,1 x 1011 Pa). The length are L1 = L4 = 1,5 m and L2 = L3 = 2,0 m. A unit magnitude harmonic force was applied in beam 1, at 1,0 m from its origin, and the transversal velocity was calculated at 0,7 m in the same beam.

Figure 8 shows a comparison between the responses calculated by the mobility approach and by Finite Elements. In this case both methods considered flexural waves, modeled by Euler theory.

The software used in the numeric model was ANSYS, version 5.0. The mobility calculation is made by ANSYS through a harmonic analysis and the used method was the modal superposition, based on the sums of contributions of the eigenfunctions of the structure. It was chosen for being the method of better effectiveness in terms of precision, time of execution and computer capacity demanded. The use of the modal superposition requires, initially, a modal analysis of the structure to calculate the eigenfunctions and the eigenvalues, that later will be used in the harmonic analysis. The used method was the subspace, which uses the complete and symmetrical matrices, due to the better precision, in spite of the larger computer power required.

The harmonic analysis gives as result the nodal displacements. Multiplying the nodal displacement by jw and dividing the result by the external load, the mobility is obtained. The used element was BEAM4, an elastic 3D beam element. It is a uniaxial element, supporting axial, torsional and flexural loads. It has two nodes per element and six degrees of freedom: transverse displacements in the x, y and z directions; and angular displacements (rotation) in the same directions. The mesh had 560 elements, corresponding to 40 elements per wavelength The square was considered simply supported in the four corners (zero displacement in the x and y directions), but with free displacement in the z direction. Good agreement in the results is observed for all third-octave frequency bands.

The same analysis was repeated this time modeling flexural waves by Timoshenko's theory. In the numeric analysis, the same elements and mesh from the Euler analysis were used. The shear effect is considered using the command RMORE for SHEARZ. A comparison of the response results obtained by both methods is presented in Fig. 9, showing also good agreement Figs. 8 and 9, however, reveals larger response of beams modeled by Timoshenko's theory. This is attributed to shear effects, which increase the modal density of the beams, and, consequently, absorb larger amounts of power from the externally applied force.

The second frame analyzed, shown in Fig. 10, was separated into three components. All forces and moments acting at their joints, as well as their respective angular and linear displacements are also indicated in the figure.

The velocities at the joints can be calculated from the forces and moments, and from the components mobility functions, as written below

The following physical and equilibrium conditions must be obeyed

When substituted in Eqs. (47) to (67), one obtains a set of linear equations, expressed in matrix form

where [Y] is the mobility matrix, the vector {M} represents the unknown forces and moments acting at the joints and {F} is the vector related to the external load.

An external excitation was applied in beam 1 at 1,0 m from the origin, and the response calculated at 0,7 m of the origin of the same beam. The dimensions and material characteristics of the beams are as follows: E = 2,1 x 1011 Pa; r = 8000 kg / m3; h = 0,01; b = h = 0,05 m (square cross section); L1 = L3 = L5 = L6 = 1,5 m and L2 = L4 = L7 = 2,0 m.

Figure 11 shows the response results calculated by the Mobility approach and by Finite Elements. For FEM, the same element was used again, with a mesh of 960 elements, giving 40 elements per wavelength. The good agreement indicates that the mobility approach is a suitable technique for analyzing response of and power flow through grillages.

Conclusions

Response analysis of grillages excited by transversely applied forces must consider flexural-torsional waves couplings due to their efficient power flow mechanism between connected beams, when compared to flexural-flexural waves couplings.

It is essential to model larger cross sections beams by Timoshenko's theory, for taking into account their increased modal density, which is properly represented by the shear effects term. Response and power flow results calculated through Timoshenko's models yield to increased values when compared to Euler-Bernoulli's theory.

The mobility approach produces accurate results and is of easy application to grillage-like structures. For the two cases presented in this paper the computing time by the mobility approach is considerably lower than by Finite Elements.

The main advantage of the mobility approach lies in the possibility of calculating a deterministic response at any frequency, and at any point, including those in the high frequency range, which is quite cumbersome of being obtained by the numerical techniques currently available.

Article received: May, 2001. Technical Editor: Átila P. S. Freire

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Publication Dates

• Publication in this collection
  19 Aug 2002
• Date of issue
  2001

History

• Received
  May 2001