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.

Structural Engineers
Structural engineers have a unique and important responsibility for ensuring that structures are safe, as well as optimized for weight and load bearing capability. The importance of a structural engineer’s abilities and credentials are evident in all walks of life. This can be seen in the design of everyday structures, such as skyscrapers and bridges. It is also obvious in ‘high tech’ aerospace applications, such as airplanes, rockets, or space vehicles, such as the Space Shuttle. Structural engineers routinely provide safe structures that protect people, ensure the safety of astronauts, or simply ‘perform an intended function’. Structural engineers must provide structures that do not fail by overload or fatigue, regardless of the application. The structural engineer works closely with many other disciplines within and outside of engineering.

Education
The educational requirements for structural engineers are very similar to all engineering students. There is a strong emphasis on math skills, with science and the humanities being important secondary skills. Most structural engineers take all the rudimentary classes in statics, dynamics, and strength of materials. In addition, further study in engineering mechanics specialties such as vibration and advanced mechanics of materials is normally emphasized in both Bachelor’s and advanced degree programs. Emphasis on both theoretical and experimental verification of structural behavior is also an important part of the educational process that occurs at structural engineering schools. Once the educational process is completed, graduates can look forward to finding a variety of structural engineering jobs with engineering firms, the government, and private industry.

History
Structural engineering dates back to the days of the pyramids in Egypt. It is obvious that a masterful job of structural design and assembly was done to create these huge structures. These structures were commonly built during this time period because of their stability and the ability for engineers to scale their dimensions for new projects. It is likely that structural theory did not exist at this time, but pyramid designs were based on empirical information that had been gathered from previous successful projects. The Civil Engineering community can look with pride at these historic structures for the engineering ingenuity and resourcefulness in building these massive stone structures.

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.

History
The finite element method (FEM) originated in the 1940’s. At this time, there was a need to solve more complex problems in elasticity and structural analysis. As the method evolved, various approaches were recognized that involved dividing a continuous domain into sub domains, or elements. Mathematically, the method involved solving an array of partial differential equations. During the 1950’s, further development of FEA continued in airframe and structural analysis. Research at several universities culminated in more efficient methods for solving the stiffness matrices that were the basis of the method. This research was concentrated in civil engineering applications during the 1960’s, but applications to other engineering disciplines were also beginning to appear at this time. During the late 60’s, three of the more recognized finite element codes were introduced: ANSYS, NASTRAN, and STARDYNE. Advances in computing power and software development over the years have culminated in the efficient, multi-purpose finite element (FE) programs that are available today.

The Finite Element Method
The finite element method consists of three major tasks. The first is pre-processing, where the analyst develops a mesh which consists of nodes and elements. This mesh is the basis for the analysis, and is supplemented with appropriate material properties, element properties, and boundary conditions. The second task is the solution phase, where the elements are assembled into matrices. These matrices are then solved for basic parameters, such as displacements or temperatures. The final step is post-processing, where the analyst checks the results. Part of this process involves reviewing the magnitudes and distributions of the primary solution parameters (such as displacements and stresses).

The Finite Element Model
An FE model consists of a finite number of points (nodes) and elements. The nodes are actually points in space that are used to define the elements. The elements consist of various numbers of nodes (typically from 1 to 20). These elements are defined in space by the location and connectivity of the nodes. Finite elements are actually mini structures themselves, with displacement functions that are defined by the element types. Each element is further defined by element and material properties. The density of the finite element mesh may vary throughout the model, depending on the stress gradients within the structure.

Answers from FE Models
Each node in an FE model is characterized by a specific number of degrees of freedom (DOF). For three dimensional structural problems, the maximum number of DOF is 6 (three translations and three rotations in a global coordinate system). The minimum number of DOF is one (such as temperature for a thermal model). In simple terms, static structural FE models solve the equation F = k x, where F is the applied force, k is the stiffness, and x is the displacement. All of this is done in matrix format, where thousands of DOF are calculated in a single FE model. The most common results from FEA are displacements and stresses. Nonlinear design analyses include large deflection, elastic-plastic deformation, and contact between adjacent structures.

General Structural Design Process
Like many engineering design procedures, structural design is an iterative process. Historically, it has been synonymous with many successes and failures. However, man’s resolve has turned many of these failures into successes. Simply put, structural design is a decision-making process that involves translating the customer’s requirements into an acceptable engineering solution. Within each engineering discipline, the specifics of the structural design process vary, but the general principle of meeting design requirements is constant.

Basics of Structural Design
Structural design is based on the laws of physics, as they are applied to engineering problems. This includes application of new theory and tried and true methods that have been proven (oftentimes by testing) throughout the years. The principles of tensile, compressive, and bending strength are commonly examined during the structural design process. Most structures can be broken down into basic elements including beams, columns, and plates and shells. The introduction of finite element analysis allows present day designers to assemble the structure into a composite body for evaluation during the preliminary and detail design phases of a project.

Aerospace Engineering Structural Design
Structures are the main component of most airborne and space borne products. Failure of aerospace structures can lead to loss of life, or, as a minimum, loss of a very expensive vehicle. The design requirements for aerospace structures include acceleration, aerodynamic loading, temperature extremes, vibration, and pyrotechnic shock. Fatigue is a concern for commercial aerospace projects, while large dynamic loads are an important consideration for space borne structures. Software developments have created a variety of solution methods that help today’s aerospace engineer solve complex problems.

Civil Engineering Structures
Civil engineering structures are dominated by a wide variety of buildings, including commercial and residential variations. Besides buildings, projects in the civil realm include bridges, highways, dams and airports. These structures must be designed for a wide variety of physical and environmental loads, including wind, seismic loading, dead weight, temperature, and fatigue. One of the unique aspects of civil engineering structures is their materials. Steel, concrete, and wood comprise the materials of choice for many of these projects. Building design must also provide a structure that is easy to build and meets local building codes. Civil engineering structures are critical to the success of large-scale development projects that. These structures must be designed to safely protect both people and their possessions.

Mechanical Engineering Structural Design
Structural design in the mechanical engineering world involves a wide variety of applications. One aspect of mechanical structures is that they can be used in stationary, static applications; or they can be used in moving, dynamic situations. This can make the design process more complicated, but modern tools in the analytical regime make the engineering process easier. Since machine design is such a large part of mechanical engineering, structural design of each part is necessary for the success of the overall design process. Understanding loads and unique customer requirements is critical to the design of mechanical products. The design process can involve linear statically determinate structures, or it can involve non-linear indeterminate structures. In all cases, the structural design team must be equipped to ensure that each structural part meets the design requirements.

Prerequisites
Prerequisites for studying dynamics include a background in calculus, engineering physics, and engineering statics. Students will make use of the mathematical principles learned in calculus and analytical geometry, as well as the classical mechanics principles studied in physics and statics. In reality, much of dynamics is an extension of physics principles applied to engineering structures and machines.

Definition
Engineering mechanics dynamics is the engineering student’s first exposure to understanding bodies that are in motion. Dynamics is a departure from statics, where structures are in static equilibrium. The concept of dynamic equilibrium requires study of the variable forces that occur in rigid and elastic bodies that experience loads such as acceleration and vibration.

Topics of Study
The first topic of study is the application of Newton’s Laws to basic engineering systems. Learning about particle kinematics will re-acquaint the student with concepts of linear and angular motion. Developing equations of motion for single and multiple degree-of-freedom systems is an important skill for the dynamic analyst. As in statics analysis, vector mechanics will be necessary during the study of dynamics. Vibration of rigid and elastic structures and dynamic response are advanced topics of study.

Free Body Diagrams (FBD’s)
The concept of Free Body Diagrams (FBD’s) is just as critical in dynamic analysis, as it is in static analysis. The main difference in the two FBD’s is the difference in static and dynamic equilibrium. In static equilibrium, the sums of the forces and moments must be zero. In dynamic equilibrium, the sums of the forces and moments would equal the product of mass and acceleration. This leads to the determination of the appropriate equations of motion for the dynamic system being analyzed.

Prerequisites
Prerequisites for studying statics include a background in calculus and college-level physics. Students will make use of the mathematical principles learned in calculus and analytical geometry, as well as the classical mechanics principles studied in physics. In reality, much of statics is an extension of physics principles applied to engineering structures.

Definition
Engineering mechanics statics is a fundamental topic that is necessary to understanding the forces (and their effects) in real-world engineering applications. Statics consists of analyzing the interactions of forces that occur in rigid bodies. It is the analysis of how multi-dimensional simple structures react and support applied loads. It is imperative that the structure being analyzed be in static equilibrium

Topics of Study
During the study of statics, the student will learn how to resolve forces, moments, and couples to achieve equilibrium in stationary structures. The calculation of centers of gravity, properties of plane areas, and moments of inertia will form the basis for further study in dynamics and engineering mechanics of materials. Additional topics include vector analysis, particle equilibrium, distributed forces, and internal forces in structural members. Finally, the analysis of friction in structures, and the concepts of virtual work typically conclude the study of engineering statics.

Vector Analysis (Vector Mechanics)
Vector analysis is a diagnostic skill that is critical to understanding engineering statics. It is often necessary to analyze a single force by considering it as two separate forces. These unique forces are known as components, and have same effect on a structure as the original force (the resultant). Analyzing forces as vectors requires the analyst to differentiate between a resultant and its components. This can be accomplished graphically by using any technique that clearly shows the difference between the resultant and it components.

Free Body Diagrams (FBD’s)
Drawing Free Body Diagrams (FBD’s) is probably the most important process for correctly assessing forces, and for solving engineering mechanics problems. Without doing this, forces can easily be mis-calculated, and the resulting analysis can become erroneous. Additional study in dynamics and mechanics of materials cannot be done without using free body diagrams.

Simple Structures
In engineering mechanics, rigid members that are connected in specific patterns form the basis for simple structures. A bridge, a wind turbine tower, and the frame of a car all are all considered simple structures. Simple structures include trusses, beams, frames, and machines. These simple structures are the types that are studied in engineering statics. For instance, the method of joints is a method that is used to determine the forces in each member of a simple truss or space frame.

Vibration Testing Equipment
Vibration testing equipment includes accelerometers, controllers, analyzers, amplifiers, shakers, and vibration test fixtures. While each industry utilizes vibration testing in a unique manner, the most important components of a vibration test system are basically the same. Vibration is measured and controlled using displacement transducers or accelerometers. Vibration monitoring equipment includes PC-based controllers and analyzers. Shakers come in electrodynamic or hydraulic versions. Vibration is transmitted to the test product using stiff, lightweight test fixtures.

Vibration Analyzers & Controllers
Vibration analyzers and controllers are perhaps the most important component in a vibration test system. These data acquisition systems are independent of the other test equipment, allowing any analyzer or controller to operate within any vibration test system. PC-based analyzers and controllers provide extensive data acquisition, modal analysis, and data analysis capability. Random vibration testing is a popular method that is provided by today’s vibration controllers.

Vibration Shakers
Vibration shakers are available in electrodynamic or hydraulic versions. Electrodynamic shakers are used for smaller products that require smaller displacements and a larger frequency range. Electrodynamic shakers are powered by amplifiers that may rival a radio station for electrical power output. Hydraulic shakers are used for larger products that require larger displacement, lower frequency vibration. Vibration shaker systems are versatile enough so that shock and vibration testing can be performed on test products.

Vibration Test Fixtures
Vibration test fixtures are designed to transmit the specified vibration to the test product. Because a fixture can transmit undesirable vibration to the test item, vibration test fixtures are designed to be stiff and lightweight. Using vibration test fixtures, testing can be performed in both horizontal and vertical orientations, decreasing the cost of the test significantly.

Collecting Vibration Data
During vibration condition monitoring, the data from the rotating machinery is collected, stored, interpreted, and then compared with historical vibration measurements. Depending on the machine, the vibration data may be gathered using two different techniques. Vibration measurements on larger machines are taken using permanently attached transducers (accelerometers). For smaller machines (with hard-to-reach locations), vibration data is taken using handheld data collectors with portable transducers. The data is then stored and analyzed to determine the machine’s mechanical condition.

Analyzing Vibration Data
Vibration measurements can be expressed in terms of displacement, velocity, acceleration, and high frequency content (for bearing condition detection). Most data collectors use FFT (Fast Fourier Transform) to convert the data from the time domain to the frequency domain. Vibration data collectors have a built-in PC interface that allows transfer of the measurements to a PC for data management. Analysis software displays spectrum, trend, waterfall plots, and waveform for advanced analysis. To assess a machine, the vibration data is compared with historical profiles from the same machine.

Vibration Data Collector Features
Quick, accurate diagnosis of machine vibration conditions, including bearing vibration
Simplifies the task and increases the speed of collecting vibration monitoring data
Incorporates the latest software for vibration data analysis and performance prediction
Combines basic analysis functions and rotating machinery balancing capabilities
Lightweight, compact portable vibration data collectors are easy to use
Used with a portable computer, such as a laptop or notebook
Multiple channel models are available
Usually do not require any special training