RBFNN model for predicting nonlinear response of uniformly loaded paddle cantilever

American Journal of Applied Sciences, Jan, 2009 by Abdullah H. Abdullah

INTRODUCTION

Modeling and simulation are indispensable when dealing with complex engineering systems. It makes it possible to do essential assessment before systems are built, it can alleviate the need for expensive experiments and it can provide support in all stages of a project from conceptual design, through commissioning and operation. Cantilever sensors are the most important electric machinery in all the fields of industry. Cantilever sensors are based on relatively well known and simple transduction principle. A simple cantilever beam can be used as a sensor for biomedical, chemical and environmental applications. When micro-fabricated multilayered cantilever beam is exposed to sensing environment, it bends because of single or a combination of external forces like electrostatic, electric, magnetic, mass, nuclear radiation or mere mass. Similarly, it can bend because of intrinsic stresses generated due to chemical, physical or thermal means within the upper layer of cantilever itself.

As recent research efforts advance in several converging areas of science and technology, cantilever-based sensors have been proved to be quite versatile and sensitive devices and have been used mainly in the trace detection of bio-chemical materials. The cantilever method of bio-chemical sensing does not require any fluorescence tagging, therefore gets many attentions (1), (2). Micromachined silicon cantilever beams have been applied in fluid flow volume sensing (3), (4).

In addition, the actual mechanism for detection of the cantilever deflections is also very important. The amount of deflections of a cantilever beam can be detected by several read-out systems, including optical detection, capacitive detection, tunneling detection and interferometer detection. The optical level technique and the piezoresistive method are usually used to detect cantilever beam deflection. In general, the deflection is caused by its interaction with measured under circumstances of stress, a small force and a change of mass or temperature. However, for more complex structures, finite element modeling is useful to analyze and optimize these structures(5).

This research the present work will explore the use of Redial Base Function Neural Network (RBFNN) modeling of the paddle cantilevers in conjugation with finite element analysis (FEA) software. The model is constructed through the use of the neural network design (nntool) toolbox in MATLAB.

THEORETICAL MODEL ANALYSIS

In this section, the simple and paddle cantilever sensors as shown in Fig. 1-2 were modeled using the static equations of mechanics. To calculate the amount of deflection at the tip of a cantilever beam, the differential equation of a cantilever beam for a small deflection is given by(6).

EI [[d.sup.2]y(x)/d[x.sup.2]] = M (1)

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

where M is the bending moment, E is Young's modulus, y (x) is the deflection along the cantilever beam and I is the area moment of the cross section with respect to the neutral axis of the cantilever. M = Px when a single force P is applied on the free end of the cantilever. M = [q[x.sup.2]/2] under a flowing fluid situation, where q is a force element at the position x along the cantilever beam and is proportional to the surface area facing towards the flowing fluid and drag force. The drag force is proportional to the fluid density, the drag coefficient of the cantilever and the flow velocity squared in a turbulent flow or flow velocity in a laminar flow.

When the x-axis origin is selected at the free end of the cantilever beam, the boundary conditions are given by

x = L [dy/dx] = 0

x = L then y = 0

Now we integrate the differential equation for cantilever deflection and use the above mentioned boundary conditions.

[dy(x)/dx] = [1/EI][[P[x.sup.2]/2] + [K.sub.1]] (2)

Eventually, the deflection of the cantilever beam when a single force is applied at the free end of the cantilever is given as:

y(x) = [2P/[Ew[t.sup.3]]][[x.sup.3] - 3[L.sup.2]x + 2[L.sup.3]] (3)

RBFNN FOR PADDLE CANTILEVER DEFLECTION

From the examples ANN captures the domain knowledge. ANN can handle continuous as well as discrete data and have good generalization capability as with fuzzy expert systems. An ANN is a computational model of the brain. They assume that computation is distributed over several simple units called neurons, which are interconnected and operate in parallel thus known as parallel distributed processing systems or connectionist systems. Implicit knowledge is built into a neural network by training it. Several types of ANN structures and training algorithms have been proposed.

The basic form of RBF architecture involves entirely three different layers. The input layers is made n, of source nodes while the second layer is hidden layer of high enough dimension which senses a different purpose from that in a multilayer perception. The output layer supplies the response of the network to the activation patterns applied to the input layer. The tram formation from the input layer to hidden is nonlinear whereas the transformation from the hidden from unit to the output layer is linear.