### Computer Techniques in Vibration

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For example, type help help to get more information about help. There are three other commands for information: lookfor, helpwin, and helpdesk. Lookfor gives a list of functions with the keyword in their description. Helpwin gives a help window. Helpdesk is a web browser-based help. Given the early history of MATLAB as a matrix laboratory, it should not be surprising that one of its strengths is its ability to manipulate vectors and matrices very well. Printing is suppressed by ending the line with a semicolon. The best way to learn more about these commands is to type help and the command name.

This way, you get the most up-to-date information concerning the function. References Atkinson, K. An Introduction to Numerical Analysis, 2nd ed. Cheney, E.

Numerical Mathematics and Computing, 4th ed. Isaacson, E. Palm, W. Pratap, R.

Recktenwald, G. Strang, G. Thomson, W. Theory of Vibration with Applications, 5th ed. Natural vibration analysis and response analysis are discussed in detail. The basic procedure in using the commercial FEA software packages for vibration analysis is outlined also see Chapter 1 and Chapter 4. The vibration analysis of a gearbox housing is presented to illustrate the procedure.

But this phenomenon is not always unwanted; for example, vibration is needed in the operation of vibration screens. Thus, reducing or utilizing vibration is among the challenging tasks that mechanical or structural engineers have to face.

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## Computer techniques in vibration /

Vibration modeling has been used extensively for a better understanding of vibration phenomena. The vibration modeling here implies a process of converting an engineering vibration problem into a mathematical model, whereby the major vibration characteristics of the original problem can be accurately predicted. These four terms are represented in differential equations of motion for discrete or, lumped-parameter systems, or boundary value problems for continuous systems.

A damping term is included if damping effects are of concern. Depending on the nature of the vibration problem, the complexity of the mathematical model varies from simple springâ€” mass systems to multi-degree-of-freedom DoF systems; from a continuous system for a single structural member beam, rod, plate, or shell to a combined system for a built-up structure; from a linear system to a nonlinear system.

The success of the mathematical model heavily depends on whether or not the four terms mentioned before can represent the actual vibration problem. The construction of such a representative and simple mathematical model requires an in-depth understanding of vibration principles and techniques, extensive experience in vibration modeling, and ingenuity in using vibration software tools.

Its success is attributed to the development of sophisticated software packages and the rapid growth of computer technology. In this chapter, several aspects of the construction of mathematical models of linear vibration problems without damping will be addressed. The capabilities of the available software packages for vibration analysis are listed and the basic procedure for vibration analysis is summarized.

As an illustration, an engineering example is given. Consider a singleDoF springâ€”mass system, as shown in Figure 2. These two forces are essential for mechanical vibration to exist. From a mathematical standpoint, the differential equations and the integral equations are equivalent in that one can be derived from another. Denote T as the system kinetic energy, V the system potential energy, and dW the virtual work done by nonconservative forces.

## Computer Techniques in Vibration - Google Books

This procedure is called discretization. The deformation within each element is expressed by interpolating polynomials. The energy expressions for the entire continua can be obtained by adding the energy expressions of its elements. Based on the element displacement expression Equation 2.

The equivalent nodal force Fe corresponding to the force f e applied onto the element e is determined by equaling the work done by Fe to the work done force by f e along any virtual displacement. Let L be the transformation matrix from the global coordinate system to the local coordinate system.

The vector u is the global nodal displacement vector of the structure.

It is desirable to reduce this full matrix into a diagonal matrix. In practice, this is achieved by lumping the element mass at its nodes. In particular, when automatic meshing schemes are not properly applied, or threedimensional elements must be used, the number of elements created could become too great to be costeffectively handled with limited computer capabilities. To solve this problem, modelers have to pay close attention to how the meshing is done in commercial software packages. One way is to mesh both the beam and the block using three-dimensional elements; the other way is to mesh the beam with one-dimensional beam elements and treat the block as a lumped mass, zero-dimensional element.

Another technique for reducing the number of elements comes from deleting the detailed features. Generally these detailed features can be deleted without any visible effect on the results, if the global behavior of the vibration problem is of concern. Note that such detailed features may have to be kept if the localized behavior such as fatigue stress induced by vibration is to be evaluated. When further model reduction is necessary, Guyan reduction [3] is considered. It was proposed two decades ago when computer capabilities were much more limited than today.

In fact, Guyan reduction is still in use today and has been cast into many commercial software packages. In Guyan reduction, the model scale is reduced by removing those DoF called slave DoF that can be approximately expressed by the rest of the DoF called master DoF through a static relation. The DoF associated with zero mass or relatively small mass are likely candidates for slave DoF. Substituting Equation 2. The response analysis refers to the calculation of the response, which can be displacements, strain, or stress, when the system is subjected to time-varying excitation forces.

The response analysis can be further divided into any one a combination of harmonic response analysis, transient response analysis, and response spectrum analysis, depending on the nature of excitation forces.

### Computer Techniques In Vibration De Silva Clarence W

In other words, they are intrinsic characteristics of the vibration problem. Therefore, they constitute an important part of vibration theory and vibration engineering. When vibration engineers specify design requirements in terms of vibration, they normally do so by restricting natural frequencies, and sometimes restricting mode shapes as well. According to the theory of second-order ordinary differential equations, the solution of Equation 2.

The numerical methods for solving the matrix eigenvalue problem given by Equation 2.

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A typical output is a plot showing response usually displacement of a certain DoF versus frequency. This plot indicates how the response at a certain DoF, as a function of excitation frequency, changes with excitation frequency.

### Computer Techniques In Vibration De Silva Clarence W

The harmonic response can also be used to calculate the response to a general periodic excitation force, if it can be satisfactorily approximated by a summation of its major harmonic components. After solving Equation 2. The computed response usually includes the time-varying displacements, accelerations, strains, and stresses. Consider Equation 2. As in the harmonic response analysis, Equation 2.

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If the time step is too large, portions of the response such as spikes could be missed or truncated. On the other hand, if the time step is too small, the analysis will take an excessively long time or even a prohibitive amount of time. Such excitations are normally treated as random excitations.

Some excitation forces, like those resulting from an uneven road, could be measured to any desired accuracy, and thus they would become deterministic rather than random. But it is not cost-effective and not convenient to do so. Therefore, engineers prefer to characterize these excitation forces by a statistical description that can be easily measured on any particular representative length of time history. Of the statistical descriptions, the autocorrelation function and the power spectral density function are the most important.

For the single DoF system given in Figure 2. In the case of the multiple-DoF system given by Equation 2. Normally, FEA modeling software has the following three major components: a preprocessor, a solver, and a postprocessor. In fact, the vibration analysis capability is only a small portion of their total capabilities. These special capabilities are listed below. Due to its wide range of functionality, ABAQUS usage spans many industries, including automotive, aerospace and defense, consumer electronics, manufacturing, medical, and rubber sealing.