05-10-2016, 02:35 PM
1457891620-projectreport.docx (Size: 4.27 MB / Downloads: 4)
Abstract
Deflection is defined as structural displacement under loads. The deflection of various parts in a machine can cause severe problems. Deflection is property which we don’t want to have in our mechanical parts.
This project is oriented for analyzing various cross sections of beams of various materials using the MATLAB software. In MATLAB, beam element method is used for calculating the deflection of beams at various points or nodes. The programming done is capable of executing various cross section beams with different moment of inertia and various loading condition.
An experimental setup is also developed to measure the practical value of deflection using point laser and angular potentiometer. The basic principal behind the experimental setup is trigonometric identities.
The final concluding aim of project is to find out whether the simulating software and numerical methods give the results comparable to practical results or not, whether these techniques are giving correct idea of the properties or not.
Deflection
Deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance. The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work,direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.
Deflection of a Beam Deflected Symmetrically and Principle of Superposition
Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.
The classic formula for determining the bending stress in a beam under simple bending is:
where
is the bending stress
M - the moment about the neutral axis y - the perpendicular distance to the neutral axis Ix - the second moment of area about the neutral axis x.
1.3 Methods of Deflection Analysis
There are various methods to calculate deflection of the beam which are as follow:
1. Direct Integration of Beam
2. Castigliano's Method
3. Macaulay's Method
4. Direct Stiffness Method
1.3.1 Direct integration of a beam
Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam.
Direct stiffness method
As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a matrix method that makes use of the members' stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method (FEM). In applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of these elements are then, through matrix mathematics, compiled into a single matrix equation which governs the behavior of the entire idealized structure. The structure’s unknown displacements and forces can then be determined by solving this equation. The direct stiffness method forms the basis for most commercial and free source finite element software.
The direct stiffness method originated in the field of aerospace. Researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theory, energy principles in structural mechanics, flexibility method and matrix stiffness method. It was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation.
1.3.4.1 History
Between 1934 and 1938 A. R. Collar and W. J. Duncan published the first papers with the representation and terminology for matrix systems that are used today. Aeroelastic research continued through World War II but publication restrictions from 1938 to 1947 make this work difficult to trace. The second major breakthrough in matrix structural analysis occurred through 1954 and 1955 when professor John H. Argyris systemized the concept of assembling elemental components of a structure into a system of equations. Finally, on Nov. 6 1959, M. J.
Turner, head of Boeing’s Structural Dynamics Unit, published a paper outlining the direct stiffness method as an efficient model for computer implementation (Felippa 2001).
1.3.4.2 Member stiffness relations
A typical member stiffness relation has the following general form:
where, m = member number m.
= vector of member's characteristic forces, which are unknown internal forces.
= member stiffness matrix which characterizes the member's resistance against deformations.
= vector of member's characteristic displacements or deformations.
= vector of member's characteristic forces caused by external effects (such as known forces and temperature changes) applied to the member while qm=0).
If are member deformations rather than absolute displacements, then are independent member forces, and in such case (1) can be inverted to yield the so-called member flexibility matrix, which is used in the flexibility method.
•System stiffness relation
For a system with many members interconnected at points called nodes, the members' stiffness relations such as Eq. (1) can be integrated by making use of the following observations:
The member deformations can be expressed in terms of system nodal displacements r in order to ensure compatibility between members. This implies that r will be the primary unknowns.
The member forces help to the keep the nodes in equilibrium under the nodal forces R. This implies that the right-hand-side of (1) will be integrated into the right-hand-side of the following nodal equilibrium equations for the entire system:
where
= vector of nodal forces, representing external forces applied to the system's nodes.
= system stiffness matrix, which is established by assembling the members' stiffness matrices .
= vector of system's nodal displacements that can define all possible deformed configurations of the system subject to arbitrary nodal forces R.
= vector of equivalent nodal forces, representing all external effects other than the nodal forces which are already included in the preceding nodal force vector R. This vector is established by assembling the members' .
•Solution
The system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because is symmetric. Once the supports' constraints are accounted for in (2), the nodal displacements are found by solving the system of linearequations (2), symbolically:
Subsequently, the members' characteristic forces may be found from Eq.(1) where can be found from r by compatibility consideration.
Applications
The direct stiffness method was developed specifically to effectively and easily implement into computer software to evaluate complicated structures that contain a large number of elements. Today, nearly every finite element solver available is based on the direct stiffness method. While each program utilizes the same process, many have been streamlined to reduce computation time and reduce the required memory. In order to achieve this, shortcuts have been developed.
One of the largest areas to utilize the direct stiffness method is the field of structural analysis where this method has been incorporated into modeling software. The software allows users to model a structure and, after the user defines the material properties of the elements, the program automatically generates element and global stiffness relationships. When various loading conditions are applied the software evaluates the structure and generates the deflections for the user.
Finite Element Method
In mathematics, the finite element method (FEM) is a numerical technique for finding approximate solutions to boundary value problems for differential equations. It uses variational methods (the calculus of variations) to minimize an error function and produce a stable solution. Analogous to the idea that connecting many tiny straight lines can approximate a larger circle, FEM encompasses all the methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain.
2.1 History
While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by A. Hrennikoff and R. Courant. In China, in the later 1950s and early 1960s, based on the computations of dam constructions, K. Feng proposed a systematic numerical method for solving partial differential equations. The method was called the finite difference method based on variation principle, which was another independent invention of finite element method. Although the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.
2.2 General Principles
The subdivision of a whole domain into simpler parts has several advantages:
• Accurate representation of complex geometry
• Inclusion of dissimilar material properties
• Easy representation of the total solution •Capture of local effects.
A typical work out of the method involves (1) dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem, followed by (2) systematically recombining all sets of element equations into a global system of equations for the final calculation. The global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer.
In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often partial differential equations (PDE). The process, in mathematics language, is to construct an integral of the inner product of the residual and the weight functions and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
• A set of algebraic equations for steady state problems,
• A set of ordinary differential equations for transient problems.
In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the reference coordinate system. The process is often carried out by FEM software using coordinate data generated from the subdomains.
2.3 Applications
A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.
FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.
2.4 Beam Element
Beam deflections could be calculated by direct integration but it becomes cumbersome for more complex beams, especially if the problem is statically indeterminate. It is a method based on techniques from finite element analysis but since it has limits to point loads and constant cross-sections between joints, the solution will be exact.
Beam element has six degrees of freedom at each node
i) Beam element is a slender structure ii) It has a uniform cross section.
iii) The element is unsuitable for structures that have complex geometry, holes, and points of stress concentration.
iv) The stiffness constant of a beam element is derived by combining the stiffness constants of a beam under pure bending, a truss element, and a torsion bar.
v) A beam element can represent a beam in bending, a truss element, and a torsion bar.
vi) In FEA it’s a common practice to use beam elements to represent all or any of these three loads.
2.4.1 Review: Solving a Set of Linear Equations
A set of linear equations can be written in matrix form as shown below where K is a square matrix of constants, u is the vector of unknowns and F is a vector of constants.
Ku = F (Eq. 2.4.1)
In order to solve the equations we require:
i) As many equations as unknowns.
ii) All the equations to be independent (none of the rows of k can be made from linear combinations of some of the other rows, i.e. K is not a singular matrix)
Checking the first condition is straight forward; K must be square and the number of rows in
K, u, and F must be the same. Checking the second condition is by computing the determinant (det(K)) and ensuring it doesn’t equal zero.
. MATLAB
The name MATLAB stands for MATrix LABoratory. MATLAB was written originally to provide easy access to matrix software developed by the LINPACK (linear system package) and EISPACK (Eigen system package) projects.
MATLAB is a high-performance language for technical computing. It integrates computation, visualization, and programming environment. Furthermore, MATLAB is a modern programming language environment: it has sophisticated data structures, contains built-in editing and debugging tools, and supports object-oriented programming. These factors make MATLAB an excellent tool for teaching and research.
MATLAB has many advantages compared to conventional computer languages (e.g., FORTRAN) for solving technical problems. MATLAB is an interactive system whose basic data element is an array that does not require dimensioning. The software package has been commercially available since 1984 and is now considered as a standard tool at most universities and industries worldwide.
It has powerful built-in routines that enable a very wide variety of computations. It also has easy to use graphics commands that make the visualization of results immediately available. Specific applications are collected in packages referred to as toolbox. There are toolboxes for signal processing, symbolic computation, control theory, simulation, optimization, and several other fields of applied science and engineering.
3.1 Starting MATLAB
After logging into your account, you can enter MATLAB by double-clicking on the MATLAB shortcut icon (MATLAB 7.0.4) on your Windows desktop. When you start MATLAB, a special window called the MATLAB desktop appears. The desktop is a window that contains other windows. The major tools within or accessible from the desktop are:
• The Command Window
• The Command History
• The Workspace
• The Current Directory
• The Help Browser
• The Start button
1 Potentiometer Construction
Potentiometers comprise a resistive element, a sliding contact (wiper) that moves along the element, making good electrical contact with one part of it, electrical terminals at each end of the element, a mechanism that moves the wiper from one end to the other, and a housing containing the element and wiper.
Many inexpensive potentiometers are constructed with a resistive element formed into an arc of a circle usually a little less than a full turn and a wiper sliding on this element when rotated, making electrical contact. The resistive element, with a terminal at each end, is flat or angled. The wiper is connected to a third terminal, usually between the other two. On panel potentiometers, the wiper is usually the center terminal of three. For single-turn potentiometers, this wiper typically travels just under one revolution around the contact. The only point of ingress for contamination is the narrow space between the shaft and the housing it rotates in.
Another type is the linear slider potentiometer, which has a wiper which slides along a linear element instead of rotating. Contamination can potentially enter anywhere along the slot the slider moves in, making effective sealing more difficult and compromising long-term reliability. An advantage of the slider potentiometer is that the slider position gives a visual indication of its setting. While the setting of a rotary potentiometer can be seen by the position of a marking on the knob, an array of sliders can give a visual impression of, for example, the effect of a multi-band equalizer.
The resistive element of inexpensive potentiometers is often made of graphite. Other materials used include resistance wire, carbon particles in plastic, and a ceramic/metal mixture called cermet. Conductive track potentiometers use conductive polymer resistor pastes that contain hard-wearing resins and polymers, solvents, and lubricant, in addition to the carbon that provides the conductive properties. Others are enclosed within the equipment and are intended to be adjusted to calibrate equipment during manufacture or repair, and not otherwise touched.
Multiturn potentiometers are also operated by rotating a shaft, but by several turns rather than less than a full turn. Some multiturn potentiometers have a linear resistive element with a sliding contact moved by a lead screw; others have a helical resistive element and a wiper that turns through 10, 20, or more complete revolutions, moving along the helix as it rotates. Multiturn potentiometers, both user-accessible and preset, allow finer adjustments; rotation through the same angle changes the setting by typically a tenth as much as for a simple rotary potentiometer.
A string potentiometer is a multi-turn potentiometer operated by an attached reel of wire turning against a spring, enabling it to convert linear position to a variable resistance.
User-accessible rotary potentiometers can be fitted with a switch which operates usually at the anti-clockwise extreme of rotation. Before digital electronics became the norm such a component was used to allow radio and television receivers and other equipment to be switched on at minimum volume with an audible click, then the volume increased, by turning a knob.
Multiple resistance elements can be ganged together with their sliding contacts on the same shaft, for example, in stereo audio amplifiers for volume control.
4.2.3 Potentiometer Applications
Potentiometers are rarely used to directly control significant amounts of power (more than a watt or so). Instead they are used to adjust the level of analog signals (for example volume controls on audio equipment), and as control inputs for electronic circuits. For example, a light dimmer uses a potentiometer to control the switching of a TRIAC and so indirectly to control the brightness of lamps.
Preset potentiometers are widely used throughout electronics wherever adjustments must be made during manufacturing or servicing. User-actuated potentiometers are widely used as user controls, and may control a very wide variety of equipment functions. The widespread use of potentiometers in consumer electronics declined in the 1990s, with rotary encoders, up/down push-buttons, and other digital controls now more common. However they remain in many applications, such as volume controls and as position sensors. Other applications are
i. Audio control
ii. Television
iii. Motion control iv. Transducers
v. Computation