Introduction
Vibration analysis is an important and popular discipline within mechanical engineering and engineering mechanics. It typically involves predicting vibration in new equipment, or analyzing the vibration that occurs in existing machinery. Vibration analysis is a learned skill that becomes easier with experience. While there is much credence given to classroom study, there is much to be gained from actually predicting and measuring vibratory motion. The process of vibration analysis can occur in all types of products, from airplanes and missiles to microelectronics and mini-structures. In a theoretical sense, vibration analysis is the study of deformable structures vibrating about a neutral position of equilibrium. The main tasks in vibration analysis include modal analysis, FEA, random vibration, and sine vibration. Today’s modern engineering tools and software advancements make the practice of vibration analysis a much more accurate process than in years past.

Machinery Vibration Analysis
Analyzing the vibration that occurs in existing machinery is an important aspect of condition monitoring. It is a predictive maintenance technique that is used to determine when failure of a machine is likely to occur. This allows the plant engineering staff to schedule maintenance and downtime when it is least likely to affect the overall operational scenario of the equipment being monitored. This process typically involves measuring the vibratory frequencies and amplitudes that occur over time, while the equipment is operating. Analyzing trends in the vibratory behavior of rotating machinery can predict problems with misaligned shafts, worn bearings, or rotating parts that are out of balance. Suspect parts can then be replaced before more costly equipment failures and resulting downtime occur.

Modal Analysis
Modal analysis is the most important aspect of vibration analysis. If algebra is the study of X, then vibration is the study of resonance. Modal analysis is the process of predicting a structure’s natural (resonant) frequencies (eigenvalues) and corresponding mode shapes (eigenvectors). Modal analysis characterizes a structure’s response to dynamic or vibratory excitation. Any structure will respond to dynamic excitation at a specific number of ‘natural frequencies’, and their corresponding deformation shapes. Modal analysis assumes that the structure vibrates in the absence of damping. Each natural frequency and mode shape is called a mode. Even though modal analysis can (in theory) predict an infinite number of modes, the corresponding dynamic response and stresses will occur at a much smaller number, usually three to six.

Once modal analysis is completed, the analyst can proceed to determining the effects of various dynamic excitations, such as sine or random vibration. The resulting displacements and stresses can now be predicted and assessed. In important consideration in modal analysis is the fact that the mode shapes are only shapes, not deflections. These results can be used, however, for comparisons of movement within the same mode. Typically, mode shapes are normalized so that the maximum movement is one. However, the addition of damping and a vibration input to a mode shape makes it a deflection shape.

Superimposing a structure’s modes is an effective way to characterize its dynamic response. This allows the analyst to perform analysis in the time domain, as well as the frequency domain. Time domain dynamic analysis is necessary for performing transient dynamic response. In summary, modal analysis is the most important step in performing vibration analysis.

FEA Vibration Analysis
Finite Element Analysis (FEA) is a popular and important method used in vibration analysis. Advances in FEA technology allow the analyst to solve more complex vibration problems and make more accurate predictions of vibratory response. Today’s popular FE programs (such as ANSYS) provide solutions for modal analysis, harmonic and random vibration response, and transient dynamic behavior. Most of today’s FE programs allow the analyst to import solid models from CAD programs to begin the process of analyzing a specific structure. As with any vibration analysis, the process begins with modal analysis. The FE mesh for this process does not necessarily have to be as refined as one that would be used to predict dynamic stresses. However, there must be enough nodal density to accurately describe the critical modes (natural frequencies) and their associated mode shapes. Also, a coarse mesh will overestimate the stiffness, and the calculated natural frequencies will be artificially high.

Introduction
Mechanics of materials is an important topic in the study of engineering mechanics. It is normally an undergraduate subject that is taught to engineering students who have previously completed prerequisite courses in statics and dynamics. This subject is also known by other names, such as strength of materials and mechanics of deformable bodies. In simple terms, mechanics of materials is the study of how forces produce stresses in solid bodies. It is an essential topic for engineers and students who are solving problems in aerospace, aeronautical, civil, and mechanical engineering.

Static and Dynamic Loading
To determine the deformations and resulting stresses in a solid body, it is imperative that the analyst understand the applied loads. There are two types of loads that are commonly encountered in the analysis of structural members. The first load type is static which includes constant forces, pressures, moments, and dead weights. The second load type is dynamic which includes varying forces, accelerations, vibration, shock, and transient dynamic forces. Real world application of dynamic loads includes impact forces, wind loads, snow loads, and earthquake loads. In the application of any type of loading, it is essential that the analyst construct free body diagrams to determine the resolution of the applied loads. When a solid body experiences combined loading, the principle of superposition can be used to add the effect of the applied loads.

Material Properties-Stress & Strain
A critical aspect of mechanics of materials is the relationship between stress and strain. Materials that obey Hooke’s law exhibit a linear relationship between stress and strain. The resulting deformation is considered elastic when the structure retains its original shape after the applied load has been removed. When the applied load is increased, and the deformation is permanent, the material behavior is called plastic. The relationship between a material’s stress and strain determines the likelihood for a structure to survive in the real world. Stresses that occur at a point are analyzed using Mohr’s circle for two or three dimensional loading.

Types of Structural Members
Mechanics of materials provides the analyst with closed form solutions for simple structural members that experience a variety of loading conditions. Prismatic bars can be loaded axially, or in torsion. A special case of a prismatic bar is a circular shaft that is loaded in bending and torsion. Beams can be loaded axially, in pure bending, or in torsion. The beam bending case is analyzed using shear and moment diagrams. As with any axial loading, it can be either tensile or compressive. Columns are loaded axially in compression, and are normally assessed for buckling stability. Thin walled pressure vessels are typically analyzed for hoop stresses.

Preparation for Advanced Topics
Understanding the basic concepts and simplified methods in mechanics of materials prepares the student for advanced study in engineering mechanics. Analyzing real world engineering structure can be done with a background in statics, dynamics, and strength of materials. Advanced topics include machine design, vibration, advanced mechanics of materials, plates and shells, elasticity, and plasticity.