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Various machine components used in mechanical engineering
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Mechanical engineering is the study of physical machines and mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and mathematics principles with materials science, to design, analyze, manufacture, and maintain mechanical systems.[1] It is one of the oldest and broadest of the engineering branches.

Mechanical engineering requires an understanding of core areas including mechanics, dynamics, thermodynamics, materials science, design, structural analysis, and electricity. In addition to these core principles, mechanical engineers use tools such as computer-aided design (CAD), computer-aided manufacturing (CAM), computer-aided engineering (CAE), and product lifecycle management to design and analyze manufacturing plants, industrial equipment and machinery, heating and cooling systems, transport systems, motor vehicles, aircraft, watercraft, robotics, medical devices, weapons, and others.[2][3]

Mechanical engineering emerged as a field during the Industrial Revolution in Europe in the 18th century; however, its development can be traced back several thousand years around the world. In the 19th century, developments in physics led to the development of mechanical engineering science. The field has continually evolved to incorporate advancements; today mechanical engineers are pursuing developments in such areas as composites, mechatronics, and nanotechnology. It also overlaps with aerospace engineering, metallurgical engineering, civil engineering, structural engineering, electrical engineering, manufacturing engineering, chemical engineering, industrial engineering, and other engineering disciplines to varying amounts. Mechanical engineers may also work in the field of biomedical engineering, specifically with biomechanics, transport phenomena, biomechatronics, bionanotechnology, and modelling of biological systems.

History

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Main article: History of mechanical engineering

The application of mechanical engineering can be seen in the archives of various ancient and medieval societies. The six classic simple machines were known in the ancient Near East. The wedge and the inclined plane (ramp) were known since prehistoric times.[4] Mesopotamian civilization is credited with the invention of the wheel by several, mainly old sources.[5][6][7] However, some recent sources either suggest that it was invented independently in both Mesopotamia and Eastern Europe or credit prehistoric Eastern Europeans with the invention of the wheel[8][9][10][11] The lever mechanism first appeared around 5,000 years ago in the Near East, where it was used in a simple balance scale,[12] and to move large objects in ancient Egyptian technology.[13] The lever was also used in the shadoof water-lifting device, the first crane machine, which appeared in Mesopotamia circa 3000 BC.[12] The earliest evidence of pulleys date back to Mesopotamia in the early 2nd millennium BC.[14]

The Saqiyah was developed in the Kingdom of Kush during the 4th century BC. It relied on animal power reducing the tow on the requirement of human energy.[15] Reservoirs in the form of Hafirs were developed in Kush to store water and boost irrigation.[16] Bloomeries and blast furnaces were developed during the seventh century BC in Meroe.[17][18][19][20] Kushite sundials applied mathematics in the form of advanced trigonometry.[21][22]

The earliest practical water-powered machines, the water wheel and watermill, first appeared in the Persian Empire, in what are now Iraq and Iran, by the early 4th century BC.[23] In ancient Greece, the works of Archimedes (287–212 BC) influenced mechanics in the Western tradition. The geared Antikythera mechanisms was an Analog computer invented around the 2nd century BC.[24]

In Roman Egypt, Heron of Alexandria (c. 10–70 AD) created the first steam-powered device (Aeolipile).[25] In China, Zhang Heng (78–139 AD) improved a water clock and invented a seismometer, and Ma Jun (200–265 AD) invented a chariot with differential gears. The medieval Chinese horologist and engineer Su Song (1020–1101 AD) incorporated an escapement mechanism into his astronomical clock tower two centuries before escapement devices were found in medieval European clocks. He also invented the world's first known endless power-transmitting chain drive.[26]

The cotton gin was invented in India by the 6th century AD,[27] and the spinning wheel was invented in the Islamic world by the early 11th century,[28] Dual-roller gins appeared in India and China between the 12th and 14th centuries.[29] The worm gear roller gin appeared in the Indian subcontinent during the early Delhi Sultanate era of the 13th to 14th centuries.[30]

During the Islamic Golden Age (7th to 15th century), Muslim inventors made remarkable contributions in the field of mechanical technology. Al-Jazari, who was one of them, wrote his famous Book of Knowledge of Ingenious Mechanical Devices in 1206 and presented many mechanical designs.[citation needed]

In the 17th century, important breakthroughs in the foundations of mechanical engineering occurred in England and the Continent. The Dutch mathematician and physicist Christiaan Huygens invented the pendulum clock in 1657, which was the first reliable timekeeper for almost 300 years, and published a work dedicated to clock designs and the theory behind them.[31][32] In England, Isaac Newton formulated his laws of motion and developed calculus, which would become the mathematical basis of physics. Newton was reluctant to publish his works for years, but he was finally persuaded to do so by his colleagues, such as Edmond Halley. Gottfried Wilhelm Leibniz, who earlier designed a mechanical calculator, is also credited with developing the calculus during the same time period.[33]

During the early 19th century Industrial Revolution, machine tools were developed in England, Germany, and Scotland. This allowed mechanical engineering to develop as a separate field within engineering. They brought with them manufacturing machines and the engines to power them.[34] The first British professional society of mechanical engineers was formed in 1847 Institution of Mechanical Engineers, thirty years after the civil engineers formed the first such professional society Institution of Civil Engineers.[35] On the European continent, Johann von Zimmermann (1820–1901) founded the first factory for grinding machines in Chemnitz, Germany in 1848.[citation needed]

In the United States, the American Society of Mechanical Engineers (ASME) was formed in 1880, becoming the third such professional engineering society, after the American Society of Civil Engineers (1852) and the American Institute of Mining Engineers (1871).[36] The first schools in the United States to offer an engineering education were the United States Military Academy in 1817, an institution now known as Norwich University in 1819, and Rensselaer Polytechnic Institute in 1825. Education in mechanical engineering has historically been based on a strong foundation in mathematics and science.[37]

Education

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A mechanical engineering teaching lab at Ohio State University, c. 1900

Degrees in mechanical engineering are offered at various universities worldwide. Mechanical engineering programs typically take four to five years of study depending on the place and university and result in a Bachelor of Engineering (B.Eng. or B.E.), Bachelor of Science (B.Sc. or B.S.), Bachelor of Science Engineering (B.Sc.Eng.), Bachelor of Technology (B.Tech.), Bachelor of Mechanical Engineering (B.M.E.), or Bachelor of Applied Science (B.A.Sc.) degree, in or with emphasis in mechanical engineering. In Spain, Portugal and most of South America, where neither B.S. nor B.Tech. programs have been adopted, the formal name for the degree is "Mechanical Engineer", and the course work is based on five or six years of training. In Italy the course work is based on five years of education, and training, but in order to qualify as an Engineer one has to pass a state exam at the end of the course. In Greece, the coursework is based on a five-year curriculum.[38]

In the United States, most undergraduate mechanical engineering programs are accredited by the Accreditation Board for Engineering and Technology (ABET) to ensure similar course requirements and standards among universities. The ABET web site lists 302 accredited mechanical engineering programs as of 11 March 2014.[39] Mechanical engineering programs in Canada are accredited by the Canadian Engineering Accreditation Board (CEAB),[40] and most other countries offering engineering degrees have similar accreditation societies.

In Australia, mechanical engineering degrees are awarded as Bachelor of Engineering (Mechanical) or similar nomenclature, although there are an increasing number of specialisations. The degree takes four years of full-time study to achieve. To ensure quality in engineering degrees, Engineers Australia accredits engineering degrees awarded by Australian universities in accordance with the global Washington Accord. Before the degree can be awarded, the student must complete at least 3 months of on the job work experience in an engineering firm.[41] Similar systems are also present in South Africa and are overseen by the Engineering Council of South Africa (ECSA).

In India, to become an engineer, one needs to have an engineering degree like a B.Tech. or B.E., have a diploma in engineering, or by completing a course in an engineering trade like fitter from the Industrial Training Institute (ITIs) to receive a "ITI Trade Certificate" and also pass the All India Trade Test (AITT) with an engineering trade conducted by the National Council of Vocational Training (NCVT) by which one is awarded a "National Trade Certificate". A similar system is used in Nepal.[42]

Some mechanical engineers go on to pursue a postgraduate degree such as a Master of Engineering, Master of Technology, Master of Science, Master of Engineering Management (M.Eng.Mgt. or M.E.M.), a Doctor of Philosophy in engineering (Eng.D. or Ph.D.) or an engineer's degree. The master's and engineer's degrees may or may not include research. The Doctor of Philosophy includes a significant research component and is often viewed as the entry point to academia.[43] The Engineer's degree exists at a few institutions at an intermediate level between the master's degree and the doctorate.[citation needed]

Coursework

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Standards set by each country's accreditation society are intended to provide uniformity in fundamental subject material, promote competence among graduating engineers, and to maintain confidence in the engineering profession as a whole. Engineering programs in the U.S., for example, are required by ABET to show that their students can "work professionally in both thermal and mechanical systems areas."[44] The specific courses required to graduate, however, may differ from program to program. Universities and institutes of technology will often combine multiple subjects into a single class or split a subject into multiple classes, depending on the faculty available and the university's major area(s) of research.

The fundamental subjects required for mechanical engineering usually include:

Mechanical engineers are also expected to understand and be able to apply basic concepts from chemistry, physics, tribology, chemical engineering, civil engineering, and electrical engineering. All mechanical engineering programs include multiple semesters of mathematical classes including calculus, and advanced mathematical concepts including differential equations, partial differential equations, linear algebra, differential geometry, and statistics, among others.

In addition to the core mechanical engineering curriculum, many mechanical engineering programs offer more specialized programs and classes, such as control systems, robotics, transport and logistics, cryogenics, fuel technology, automotive engineering, biomechanics, vibration, optics and others, if a separate department does not exist for these subjects.[47]

Most mechanical engineering programs also require varying amounts of research or community projects to gain practical problem-solving experience. In the United States it is common for mechanical engineering students to complete one or more internships while studying, though this is not typically mandated by the university. Cooperative education is another option. Research puts demand on study components that feed student's creativity and innovation.[48]

Job duties

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Mechanical engineers research, design, develop, build, and test mechanical and thermal devices, including tools, engines, and machines.

Mechanical engineers typically do the following:

  • Analyze problems to see how mechanical and thermal devices might help solve the problem.
  • Design or redesign mechanical and thermal devices using analysis and computer-aided design.
  • Develop and test prototypes of devices they design.
  • Analyze the test results and change the design as needed.
  • Oversee the manufacturing process for the device.
  • Manage a team of professionals in specialized fields like mechanical drafting and designing, prototyping, 3D printing or/and CNC Machines specialists.

Mechanical engineers design and oversee the manufacturing of many products ranging from medical devices to new batteries. They also design power-producing machines such as electric generators, internal combustion engines, and steam and gas turbines as well as power-using machines, such as refrigeration and air-conditioning systems.

Like other engineers, mechanical engineers use computers to help create and analyze designs, run simulations and test how a machine is likely to work.

License and regulation

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Engineers may seek license by a state, provincial, or national government. The purpose of this process is to ensure that engineers possess the necessary technical knowledge, real-world experience, and knowledge of the local legal system to practice engineering at a professional level. Once certified, the engineer is given the title of Professional Engineer United States, Canada, Japan, South Korea, Bangladesh and South Africa), Chartered Engineer (in the United Kingdom, Ireland, India and Zimbabwe), Chartered Professional Engineer (in Australia and New Zealand) or European Engineer (much of the European Union).

In the U.S., to become a licensed Professional Engineer (PE), an engineer must pass the comprehensive FE (Fundamentals of Engineering) exam, work a minimum of 4 years as an Engineering Intern (EI) or Engineer-in-Training (EIT), and pass the "Principles and Practice" or PE (Practicing Engineer or Professional Engineer) exams. The requirements and steps of this process are set forth by the National Council of Examiners for Engineering and Surveying (NCEES), composed of engineering and land surveying licensing boards representing all U.S. states and territories.

In Australia (Queensland and Victoria) an engineer must be registered as a Professional Engineer within the State in which they practice, for example Registered Professional Engineer of Queensland or Victoria, RPEQ or RPEV. respectively.

In the UK, current graduates require a BEng plus an appropriate master's degree or an integrated MEng degree, a minimum of 4 years post graduate on the job competency development and a peer-reviewed project report to become a Chartered Mechanical Engineer (CEng, MIMechE) through the Institution of Mechanical Engineers. CEng MIMechE can also be obtained via an examination route administered by the City and Guilds of London Institute.[49]

In most developed countries, certain engineering tasks, such as the design of bridges, electric power plants, and chemical plants, must be approved by a professional engineer or a chartered engineer. "Only a licensed engineer, for instance, may prepare, sign, seal and submit engineering plans and drawings to a public authority for approval, or to seal engineering work for public and private clients."[50] This requirement can be written into state and provincial legislation, such as in the Canadian provinces, for example the Ontario or Quebec's Engineer Act.[51]

In other countries, such as the UK, no such legislation exists; however, practically all certifying bodies maintain a code of ethics independent of legislation, that they expect all members to abide by or risk expulsion.[52]

Further information: FE Exam, Professional Engineer, Incorporated Engineer, Washington Accord (credentials), and Regulation and licensure in engineering

Salaries and workforce statistics

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The total number of engineers employed in the U.S. in 2015 was roughly 1.6 million. Of these, 278,340 were mechanical engineers (17.28%), the largest discipline by size.[53] In 2012, the median annual income of mechanical engineers in the U.S. workforce was $80,580. The median income was highest when working for the government ($92,030), and lowest in education ($57,090).[54] In 2014, the total number of mechanical engineering jobs was projected to grow 5% over the next decade.[55] As of 2009, the average starting salary was $58,800 with a bachelor's degree.[56]

Subdisciplines

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The field of mechanical engineering can be thought of as a collection of many mechanical engineering science disciplines. Several of these subdisciplines which are typically taught at the undergraduate level are listed below, with a brief explanation and the most common application of each. Some of these subdisciplines are unique to mechanical engineering, while others are a combination of mechanical engineering and one or more other disciplines. Most work that a mechanical engineer does uses skills and techniques from several of these subdisciplines, as well as specialized subdisciplines. Specialized subdisciplines, as used in this article, are more likely to be the subject of graduate studies or on-the-job training than undergraduate research. Several specialized subdisciplines are discussed in this section.

Mechanics

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Mohr's circle, a common tool to study stresses in a mechanical element
Main article: Mechanics

Mechanics is, in the most general sense, the study of forces and their effect upon matter. Typically, engineering mechanics is used to analyze and predict the acceleration and deformation (both elastic and plastic) of objects under known forces (also called loads) or stresses. Subdisciplines of mechanics include

  • Statics, the study of non-moving bodies under known loads, how forces affect static bodies
  • Dynamics, the study of how forces affect moving bodies. Dynamics includes kinematics (about movement, velocity, and acceleration) and kinetics (about forces and resulting accelerations).
  • Mechanics of materials, the study of how different materials deform under various types of stress
  • Fluid mechanics, the study of how fluids react to forces[57]
  • Kinematics, the study of the motion of bodies (objects) and systems (groups of objects), while ignoring the forces that cause the motion. Kinematics is often used in the design and analysis of mechanisms.
  • Continuum mechanics, a method of applying mechanics that assumes that objects are continuous (rather than discrete)

Mechanical engineers typically use mechanics in the design or analysis phases of engineering. If the engineering project were the design of a vehicle, statics might be employed to design the frame of the vehicle, in order to evaluate where the stresses will be most intense. Dynamics might be used when designing the car's engine, to evaluate the forces in the pistons and cams as the engine cycles. Mechanics of materials might be used to choose appropriate materials for the frame and engine. Fluid mechanics might be used to design a ventilation system for the vehicle (see HVAC), or to design the intake system for the engine.

Mechatronics and robotics

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Main articles: Mechatronics and Robotics

Mechatronics is a combination of mechanics and electronics. It is an interdisciplinary branch of mechanical engineering, electrical engineering and software engineering that is concerned with integrating electrical and mechanical engineering to create hybrid automation systems. In this way, machines can be automated through the use of electric motors, servo-mechanisms, and other electrical systems in conjunction with special software. A common example of a mechatronics system is a CD-ROM drive. Mechanical systems open and close the drive, spin the CD and move the laser, while an optical system reads the data on the CD and converts it to bits. Integrated software controls the process and communicates the contents of the CD to the computer.

Robotics is the application of mechatronics to create robots, which are often used in industry to perform tasks that are dangerous, unpleasant, or repetitive. These robots may be of any shape and size, but all are preprogrammed and interact physically with the world. To create a robot, an engineer typically employs kinematics (to determine the robot's range of motion) and mechanics (to determine the stresses within the robot).

Robots are used extensively in industrial automation engineering. They allow businesses to save money on labor, perform tasks that are either too dangerous or too precise for humans to perform them economically, and to ensure better quality. Many companies employ assembly lines of robots, especially in Automotive Industries and some factories are so robotized that they can run by themselves. Outside the factory, robots have been employed in bomb disposal, space exploration, and many other fields. Robots are also sold for various residential applications, from recreation to domestic applications.[58]

Structural analysis

[edit]
Main articles: Structural analysis and Failure analysis

Structural analysis is the branch of mechanical engineering (and also civil engineering) devoted to examining why and how objects fail and to fix the objects and their performance. Structural failures occur in two general modes: static failure, and fatigue failure. Static structural failure occurs when, upon being loaded (having a force applied) the object being analyzed either breaks or is deformed plastically, depending on the criterion for failure. Fatigue failure occurs when an object fails after a number of repeated loading and unloading cycles. Fatigue failure occurs because of imperfections in the object: a microscopic crack on the surface of the object, for instance, will grow slightly with each cycle (propagation) until the crack is large enough to cause ultimate failure.[59]

Failure is not simply defined as when a part breaks, however; it is defined as when a part does not operate as intended. Some systems, such as the perforated top sections of some plastic bags, are designed to break. If these systems do not break, failure analysis might be employed to determine the cause.

Structural analysis is often used by mechanical engineers after a failure has occurred, or when designing to prevent failure. Engineers often use online documents and books such as those published by ASM[60] to aid them in determining the type of failure and possible causes.

Once theory is applied to a mechanical design, physical testing is often performed to verify calculated results. Structural analysis may be used in an office when designing parts, in the field to analyze failed parts, or in laboratories where parts might undergo controlled failure tests.

Thermodynamics and thermo-science

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Main article: Thermodynamics

Thermodynamics is an applied science used in several branches of engineering, including mechanical and chemical engineering. At its simplest, thermodynamics is the study of energy, its use and transformation through a system.[61] Typically, engineering thermodynamics is concerned with changing energy from one form to another. As an example, automotive engines convert chemical energy (enthalpy) from the fuel into heat, and then into mechanical work that eventually turns the wheels.

Thermodynamics principles are used by mechanical engineers in the fields of heat transfer, thermofluids, and energy conversion. Mechanical engineers use thermo-science to design engines and power plants, heating, ventilation, and air-conditioning (HVAC) systems, heat exchangers, heat sinks, radiators, refrigeration, insulation, and others.[62]

Design and drafting

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A CAD model of a mechanical double seal
Main articles: Technical drawing and CNC

Drafting or technical drawing is the means by which mechanical engineers design products and create instructions for manufacturing parts. A technical drawing can be a computer model or hand-drawn schematic showing all the dimensions necessary to manufacture a part, as well as assembly notes, a list of required materials, and other pertinent information.[63] A U.S. mechanical engineer or skilled worker who creates technical drawings may be referred to as a drafter or draftsman. Drafting has historically been a two-dimensional process, but computer-aided design (CAD) programs now allow the designer to create in three dimensions.

Instructions for manufacturing a part must be fed to the necessary machinery, either manually, through programmed instructions, or through the use of a computer-aided manufacturing (CAM) or combined CAD/CAM program. Optionally, an engineer may also manually manufacture a part using the technical drawings. However, with the advent of computer numerically controlled (CNC) manufacturing, parts can now be fabricated without the need for constant technician input. Manually manufactured parts generally consist of spray coatings, surface finishes, and other processes that cannot economically or practically be done by a machine.

Drafting is used in nearly every subdiscipline of mechanical engineering, and by many other branches of engineering and architecture. Three-dimensional models created using CAD software are also commonly used in finite element analysis (FEA) and computational fluid dynamics (CFD).

Modern tools

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An oblique view of a four-cylinder inline crankshaft with pistons

Computer aided software suites

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Many mechanical engineering companies, especially those in industrialized nations, have incorporated computer-aided engineering (CAE) programs into their existing design and analysis processes, including 2D and 3D solid modeling computer-aided design (CAD). This method has many benefits, including easier and more exhaustive visualization of products, the ability to create virtual assemblies of parts, and the ease of use in designing mating interfaces and tolerances.

Other CAE programs commonly used by mechanical engineers include product lifecycle management (PLM) tools and analysis tools used to perform complex simulations. Analysis tools may be used to predict product response to expected loads, including fatigue life and manufacturability. These tools include finite element analysis (FEA), computational fluid dynamics (CFD), and computer-aided manufacturing (CAM).

Using CAE programs, a mechanical design team can quickly and cheaply iterate the design process to develop a product that better meets cost, performance, and other constraints. No physical prototype need be created until the design nears completion, allowing hundreds or thousands of designs to be evaluated, instead of a relative few. In addition, CAE analysis programs can model complicated physical phenomena which cannot be solved by hand, such as viscoelasticity, complex contact between mating parts, or non-Newtonian flows.

As mechanical engineering begins to merge with other disciplines, as seen in mechatronics, multidisciplinary design optimization (MDO) is being used with other CAE programs to automate and improve the iterative design process. MDO tools wrap around existing CAE processes, allowing product evaluation to continue even after the analyst goes home for the day. They also use sophisticated optimization algorithms to more intelligently explore possible designs, often finding better, innovative solutions to difficult multidisciplinary design problems.

On‑demand platforms for external FEA expertise

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Engineering teams can access external finite‑element analysis (FEA) expertise through on‑demand platforms. By submitting structured project inputs—such as CAD or FEA input files, load cases, boundary conditions and desired deliverables - users receive instant quotes and connect with pre‑qualified simulation engineers. Providers manage the entire workflow remotely, often running simulations on cloud and GPU‑based infrastructure to accelerate turnaround, which can be significantly faster than in‑house CPU‑based systems.[64][65] This model offers scalability, access to niche simulation domains like CFD or multiphysics, and avoids the delays, licensing burden and overhead of traditional procurement cycles. It is particularly valuable for smaller teams operating under tight deadlines or limited in‑house capacity.

Areas of research

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Mechanical engineers are constantly pushing the boundaries of what is physically possible in order to produce safer, cheaper, and more efficient machines and mechanical systems. Some technologies at the cutting edge of mechanical engineering are listed below (see also exploratory engineering).

Micro electro-mechanical systems (MEMS)

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See also: MEMS

Micron-scale mechanical components such as springs, gears, fluidic and heat transfer devices are fabricated from a variety of substrate materials such as silicon, glass and polymers like SU8. Examples of MEMS components are the accelerometers that are used as car airbag sensors, modern cell phones, gyroscopes for precise positioning and microfluidic devices used in biomedical applications.

Friction stir welding (FSW)

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Main article: Friction stir welding

Friction stir welding, a new type of welding, was discovered in 1991 by The Welding Institute (TWI). The innovative steady state (non-fusion) welding technique joins materials previously un-weldable, including several aluminum alloys. It plays an important role in the future construction of airplanes, potentially replacing rivets. Current uses of this technology to date include welding the seams of the aluminum main Space Shuttle external tank, Orion Crew Vehicle, Boeing Delta II and Delta IV Expendable Launch Vehicles and the SpaceX Falcon 1 rocket, armor plating for amphibious assault ships, and welding the wings and fuselage panels of the new Eclipse 500 aircraft from Eclipse Aviation among an increasingly growing pool of uses.[66][67][68]

Composites

[edit]
Composite cloth consisting of woven carbon fiber
Main article: Composite material

Composites or composite materials are a combination of materials which provide different physical characteristics than either material separately. Composite material research within mechanical engineering typically focuses on designing (and, subsequently, finding applications for) stronger or more rigid materials while attempting to reduce weight, susceptibility to corrosion, and other undesirable factors. Carbon fiber reinforced composites, for instance, have been used in such diverse applications as spacecraft and fishing rods.

Mechatronics

[edit]

Mechatronics is the synergistic combination of mechanical engineering, electronic engineering, and software engineering. The discipline of mechatronics began as a way to combine mechanical principles with electrical engineering. Mechatronic concepts are used in the majority of electro-mechanical systems.[69] Typical electro-mechanical sensors used in mechatronics are strain gauges, thermocouples, and pressure transducers.

Nanotechnology

[edit]
Main article: Nanotechnology

At the smallest scales, mechanical engineering becomes nanotechnology—one speculative goal of which is to create a molecular assembler to build molecules and materials via mechanosynthesis. For now that goal remains within exploratory engineering. Areas of current mechanical engineering research in nanotechnology include nanofilters,[70] nanofilms,[71] and nanostructures,[72] among others.

See also: Picotechnology

Finite element analysis

[edit]
Main article: Finite element analysis

Finite Element Analysis is a computational tool used to estimate stress, strain, and deflection of solid bodies. It uses a mesh setup with user-defined sizes to measure physical quantities at a node. The more nodes there are, the higher the precision.[73] This field is not new, as the basis of Finite Element Analysis (FEA) or Finite Element Method (FEM) dates back to 1941. But the evolution of computers has made FEA/FEM a viable option for analysis of structural problems. Many commercial software applications such as NASTRAN, ANSYS, and ABAQUS are widely used in industry for research and the design of components. Some 3D modeling and CAD software packages have added FEA modules. In the recent times, cloud simulation platforms like SimScale are becoming more common.

Other techniques such as finite difference method (FDM) and finite-volume method (FVM) are employed to solve problems relating heat and mass transfer, fluid flows, fluid surface interaction, etc.

Biomechanics

[edit]
Main article: Biomechanics

Biomechanics is the application of mechanical principles to biological systems, such as humans, animals, plants, organs, and cells.[74] Biomechanics also aids in creating prosthetic limbs and artificial organs for humans.[75] Biomechanics is closely related to engineering, because it often uses traditional engineering sciences to analyze biological systems. Some simple applications of Newtonian mechanics and/or materials sciences can supply correct approximations to the mechanics of many biological systems.

In the past decade, reverse engineering of materials found in nature such as bone matter has gained funding in academia. The structure of bone matter is optimized for its purpose of bearing a large amount of compressive stress per unit weight.[76] The goal is to replace crude steel with bio-material for structural design.

Over the past decade the Finite element method (FEM) has also entered the Biomedical sector highlighting further engineering aspects of Biomechanics. FEM has since then established itself as an alternative to in vivo surgical assessment and gained the wide acceptance of academia. The main advantage of Computational Biomechanics lies in its ability to determine the endo-anatomical response of an anatomy, without being subject to ethical restrictions.[77] This has led FE modelling to the point of becoming ubiquitous in several fields of Biomechanics while several projects have even adopted an open source philosophy (e.g. BioSpine).

Computational fluid dynamics

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Main article: Computational fluid dynamics

Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions.[78] With high-speed supercomputers, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as turbulent flows. Initial validation of such software is performed using a wind tunnel with the final validation coming in full-scale testing, e.g. flight tests.

Acoustical engineering

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Main article: Acoustical engineering

Acoustical engineering is one of many other sub-disciplines of mechanical engineering and is the application of acoustics. Acoustical engineering is the study of Sound and Vibration. These engineers work effectively to reduce noise pollution in mechanical devices and in buildings by soundproofing or removing sources of unwanted noise. The study of acoustics can range from designing a more efficient hearing aid, microphone, headphone, or recording studio to enhancing the sound quality of an orchestra hall. Acoustical engineering also deals with the vibration of different mechanical systems.[79]

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Manufacturing engineering, aerospace engineering, automotive engineering and marine engineering are grouped with mechanical engineering at times. A bachelor's degree in these areas will typically have a difference of a few specialized classes.

See also

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Lists
Associations
Wikibooks

References

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Further reading

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Mechanical engineering

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from Grokipedia
Mechanical engineering is the broadest branch of engineering, focusing on the design, analysis, manufacturing, and maintenance of mechanical systems that involve motion, energy, and force, applying principles from physics, mathematics, and materials science to create everything from microscale sensors to large-scale spacecraft. It encompasses the study of objects and systems in motion, including the human body as a complex mechanical entity, and addresses challenges across scales from atomic particles to global infrastructure. Rooted in the Industrial Revolution with innovations in engines and machinery, the field has evolved to integrate modern technologies such as robotics, artificial intelligence, and sustainable energy systems. Mechanical engineers play a pivotal role in advancing society by developing solutions that enhance , such as household appliances including washing machines and dishwashers, automobiles, medical devices for prosthetics, efficient , and autonomous vehicles for safer transportation. They design, build, and test components, machines, and processes, ensuring functionality, durability, and cost-effectiveness while considering thermal, fluid, and . The discipline's versatility spans numerous subfields, including for aircraft and satellites, for healthcare innovations, and for industrial efficiency, and energy systems for environmental sustainability, with over 36 technical divisions recognized by professional organizations. Beyond technical expertise, mechanical engineering demands creative problem-solving, interdisciplinary collaboration, and ethical consideration of global issues like and resource , positioning engineers as innovators who bridge theoretical with practical applications across industries from healthcare to . With a strong foundation in core areas like , , and materials, professionals in this field contribute to and technological progress, often leading to diverse career paths in , , and .

Overview

Definition and scope

Mechanical engineering is the branch of engineering that focuses on the , , production, and of mechanical systems to address real-world challenges. These systems encompass a wide scope, including machines, devices, structural elements, and conversion apparatus such as internal combustion engines, (HVAC) units, and robotic mechanisms. The field is fundamentally interdisciplinary, drawing on core principles from physics, mathematics, and materials science to create efficient and innovative solutions across scales, from nanoscale components to large infrastructure. This integration enables mechanical engineers to serve as versatile problem-solvers, collaborating with other disciplines to develop technologies that enhance energy efficiency, transportation, and processes. The term "mechanical engineering" originated in the early 19th century, stemming from the establishment of specialized technical institutions in around 1837, amid the broader evolution of the discipline during the that began in the late . Over time, the field has expanded from its roots in steam-powered machinery to encompass modern applications in and advanced , while remaining grounded in foundational concepts like .

Fundamental principles

Mechanical engineering relies on foundational physical principles to analyze and design systems involving motion, forces, and materials. At the core are , which describe the relationship between forces and the resulting motion of bodies. The first , also known as the of , states that an object at rest remains at rest, and an object in uniform motion remains in uniform motion unless acted upon by an external force. The second quantifies the produced by a , expressed as F=ma\mathbf{F} = m \mathbf{a}, where F\mathbf{F} is the force vector, mm is the , and a\mathbf{a} is the vector. The third asserts that for every action, there is an equal and opposite reaction, meaning forces always occur in pairs of equal magnitude but opposite direction. Complementing Newton's laws are the conservation principles, which govern the behavior of systems without external influences. The states that the mass of a remains constant over time, a principle essential for analyzing material flows in engineering processes. Conservation of momentum implies that the total momentum of an is constant, leading to the equation p=mv\mathbf{p} = m \mathbf{v} for linear momentum, where v\mathbf{v} is . The first law of thermodynamics, a form of , is given by ΔU=Q−W\Delta U = Q - W, where ΔU\Delta U is the change in , QQ is added to the , and WW is work done by the ; this law ensures energy balance in thermal-mechanical interactions. Basic and kinetics provide the framework for describing and predicting motion under forces. focuses on geometric aspects such as position, (v=drdt\mathbf{v} = \frac{d\mathbf{r}}{dt}), and (a=dvdt\mathbf{a} = \frac{d\mathbf{v}}{dt}), without considering causes. Kinetics incorporates forces to explain these motions, including , defined as τ=r×F\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}, where r\mathbf{r} is the position vector from the pivot to the force application point, enabling analysis of rotational effects. Principles of statics and dynamics extend these concepts to rigid bodies, which are idealized as non-deforming structures. Statics deals with bodies in equilibrium, where the sum of forces and moments is zero (∑F=0\sum \mathbf{F} = 0, ∑τ=0\sum \boldsymbol{\tau} = 0), ensuring no net translation or rotation. Dynamics applies Newton's laws to accelerating rigid bodies, using equations like ∑F=macm\sum \mathbf{F} = m \mathbf{a}_{cm} for the center of mass and ∑τ=Iα\sum \boldsymbol{\tau} = I \boldsymbol{\alpha} for rotation, where II is the moment of inertia and α\boldsymbol{\alpha} is angular acceleration. In materials, the stress-strain relationship under elastic deformation is captured by , which states that stress (σ\sigma) is proportional to strain (ε\varepsilon): σ=Eε\sigma = E \varepsilon, where EE is the modulus of elasticity, describing linear in the elastic regime before yielding. This principle underpins the analysis of how materials respond to loads without permanent deformation.

History

Ancient and pre-industrial developments

The roots of mechanical engineering trace back to ancient civilizations, where early innovators developed foundational devices and principles that harnessed natural forces and simple mechanics. In the 3rd century BCE, Archimedes of Syracuse made seminal contributions, including the invention of the Archimedes screw, a helical device used for lifting water in irrigation and drainage systems. He also formulated the law of the lever, demonstrating how a small force applied at a distance could balance a larger load, and established the principle of buoyancy, which explains the upward force exerted on an object immersed in a fluid equal to the weight of the displaced fluid. These concepts, derived from his treatises On the Equilibrium of Planes and On Floating Bodies, laid groundwork for later mechanical analysis. Greek and Roman engineers expanded these ideas into practical infrastructure and weaponry. The Romans constructed extensive aqueducts, such as the Aqua Appia completed in 312 BCE, which transported water over long distances using gravity-fed channels, siphons, and inverted arches to maintain flow and structural integrity. In military applications, they refined catapults like the and , employing torsion from twisted sinew or ropes to propel projectiles with calculated trajectories. , a 1st-century CE Greek mathematician and engineer, advanced gear systems and in his Pneumatica and Mechanica, describing devices such as automated doors powered by steam or , gear-driven automata, and force-multiplying mechanisms using epicyclic . His work on differential and programmable water organs demonstrated early control systems integrating with . During the medieval , engineers built upon Greco-Roman knowledge, introducing sophisticated automata and transmission mechanisms. Isma'il al-Jazari, a 12th-century from Artuqid , documented over 100 inventions in his Book of Knowledge of Ingenious Mechanical Devices (1206 CE), including humanoid automata like a programmable boat and that used cams, levers, and floats for timed movements. He is credited with refining the , a rotating shaft with offset throws that converts linear motion to rotary, enabling more efficient pumps and engines, as seen in his double-acting suction pump. These innovations, often incorporating feedback mechanisms like float valves, influenced later European designs. The Renaissance marked a revival of mechanical ingenuity, exemplified by Leonardo da Vinci's conceptual designs in the late 15th and early 16th centuries. Da Vinci sketched ornithopters—inspired by bird and bat wings—for human flight, featuring flapping mechanisms driven by pulleys, , and cranks to mimic avian motion, though never built to scale. He also devised spring-driven clocks with mechanisms for precise timekeeping, integrating helical springs and gear trains to regulate motion without pendulums. His notebooks reveal studies in , such as screw jacks and ball bearings, blending art with engineering principles. Pre-industrial societies relied on mechanical tools powered by natural energy sources and basic metallurgy to support agriculture and craft. Water wheels, dating to the Hellenistic period but widespread by the Roman era, converted the kinetic energy of flowing water into rotational power for grinding grain or forging; overshot designs, where water fell onto vanes from above, achieved efficiencies up to 60-70% by leveraging gravitational potential. Windmills emerged in Persia around 500-900 CE, using vertical-axis sails to pump water and mill grain in arid regions, with horizontal-axis variants spreading to Europe by the 12th century for similar tasks. Early metallurgy techniques, beginning with copper smelting around 5000 BCE in the Near East, involved bloomery furnaces heating ores with charcoal to produce wrought iron or bronze via reduction; annealing—reheating and slow-cooling metals—enhanced ductility for tools like hammers and plows. These developments set the stage for the mechanized innovations of the Industrial Revolution.

Industrial Revolution to modern era

The , beginning in the late in Britain, marked a pivotal shift in mechanical engineering, driven by innovations that mechanized production and harnessed energy sources beyond human and animal power. James Watt's improvements to the in 1769, including the addition of a separate condenser, dramatically increased from about 1% to around 5%, enabling widespread application in mining, manufacturing, and transportation. This breakthrough facilitated the development of textile machinery, such as Richard Arkwright's in 1769 and Samuel Crompton's in 1779, which automated spinning and weaving processes, boosting productivity in cotton mills by factors of up to 100 times compared to hand methods. systems emerged as a consequence, with centralized via belts and pulleys allowing multiple machines to operate from a single steam or water source, laying the groundwork for and urban industrialization. In the , mechanical engineering advanced further with innovations in energy conversion and materials processing. Nikolaus Otto's four-stroke , patented in 1876, provided a more efficient and portable alternative to steam power, achieving a of about 12% and powering early vehicles and generators. Concurrently, Henry Bessemer's 1856 process for production converted into in under 30 minutes using a blast of air, reducing costs by 80% and enabling stronger, more durable components for machinery, bridges, and railways. These developments expanded mechanical engineering's scope, influencing subdisciplines like through the application of principles such as the first law of thermodynamics in engine cycles. The 20th century saw mechanical engineering transform transportation and design methodologies. The ' 1903 powered flight of the relied on mechanically engineered propellers and control surfaces, achieving sustained flight through a 12-horsepower gasoline engine and mechanisms. Henry Ford's introduction of the moving in 1913 for the Model T automobile reduced production time from 12 hours to about 1.5 hours per vehicle, increasing output to over 1 million units annually by 1919 and revolutionizing efficiency. The advent of in the mid-20th century, particularly with the development of (CAD) systems like in 1963, enabled precise 3D modeling and simulation, reducing design errors by up to 50% in complex assemblies. Post-World War II advancements propelled mechanical engineering into aerospace and energy sectors. Frank Whittle's turbojet engine, first tested in 1941 but scaled post-war, powered aircraft like the and achieved speeds exceeding 600 mph, fundamentally altering propulsion. Mechanical engineers contributed to nuclear reactors, such as the of pressurized reactors in the 1950s at facilities like Shippingport, where robust and systems managed high-pressure generation for production. In space exploration, the Apollo program's mechanical systems, including the rocket's F-1 engines developed by Rocketdyne, delivered over 7.5 million pounds of thrust, enabling the 1969 through precision and structural integrity under extreme conditions. Entering the , mechanical engineering has integrated computational and paradigms. Digital twins, first conceptualized in around 2002 by and Michael Grieves, create virtual replicas of physical systems for real-time simulation and , improving efficiency in industries like automotive by reducing by 20-30%. Sustainable design practices post-2000 emphasize lifecycle analysis and eco-materials. In electric vehicles, Tesla's advancements, including the 4680 battery cells in the Cybertruck with integrated structural packs, have improved to approximately 240 Wh/kg (as of 2025) and reduced battery pack weight through structural integration, advancing mechanical integration of powertrains for zero-emission mobility.

Education and training

Academic programs and degrees

Mechanical engineering education follows a structured progression of degree levels, starting with undergraduate programs that lay the foundational knowledge for professional practice. The , typically awarded as a (B.S.) or (B.Eng.), equips students with essential principles in , , and application of mechanical systems. This entry-level qualification usually requires completion of general , , sciences, and engineering-specific courses, preparing graduates for entry-level roles or further study. Graduate programs build on this base, with master's degrees such as the (M.S.) or (M.Eng.) focusing on advanced topics, research methods, or specialized applications, often taking 1-2 years. For those pursuing academia or high-level research, the (Ph.D.) program emphasizes original contributions through dissertation work, typically spanning 3-5 years post-bachelor's. Accreditation ensures that mechanical engineering programs meet rigorous standards for quality and relevance. , the Board for Engineering and Technology () accredits programs under criteria that mandate at least 30 semester credit hours in and basic sciences, including and physics with experimental components, and 45 hours in engineering topics such as and modern tools. also evaluates student outcomes, including the ability to solve complex problems, apply principles considering societal impacts, communicate effectively, and demonstrate ethical responsibility. , the EUR-ACE label, administered by the European Network for Accreditation of Engineering Education (), certifies bachelor and master programs that align with the EUR-ACE Framework Standards and Guidelines, verifying competencies for ethical and safe engineering practice across disciplines like mechanical engineering. This system promotes program diversity while facilitating graduate mobility and employability. Bachelor's programs in mechanical engineering generally span 4 years of full-time study, integrating theoretical learning with and project-based experiences to develop practical skills. Many institutions offer co-operative (co-op) options, extending the duration to 5 years by alternating academic terms with paid industry placements, allowing students to apply concepts in real-world settings and gain professional experience. Educational approaches vary globally, reflecting cultural and industrial priorities. In , the combines university-level study with vocational training, where students alternate between academic coursework and on-the-job apprenticeships, such as in industrial mechanics, to emphasize hands-on expertise from the outset. By contrast, certain Asian programs, particularly in institutions like National Yang-Ming Chiao-Tung University in , prioritize theoretical research and deep foundational knowledge in physical and chemical sciences to foster analytical rigor. By 2025, mechanical engineering curricula have evolved to integrate emerging priorities, with dedicated modules on —focusing on energy-efficient design and renewable systems—and , including applications for optimization and simulation. These additions address global challenges like and , enhancing program relevance without altering core coursework in areas such as and .

Core coursework and skills

Mechanical engineering curricula at accredited institutions emphasize a strong foundation in and physical sciences to equip students with the analytical tools necessary for engineering analysis and design. Typical foundational courses include single- and multi-variable , linear algebra, differential equations, and introductory physics covering , , , and , often accompanied by components to reinforce experimental methods. These subjects provide the mathematical rigor required for modeling physical phenomena, with and differential equations enabling the solution of dynamic systems and optimization problems, while physics establishes principles of , , and motion. Building on this base, core engineering courses focus on applying these principles to mechanical systems. Key subjects include , which analyzes forces in equilibrium; dynamics, examining motion and acceleration; , assessing stress and deformation in solids; , governing liquid and gas behavior; and , studying energy conversion and . These courses, typically comprising at least 45 semester hours of engineering topics, integrate computational tools and emphasize thermal and mechanical system analysis, preparing students for real-world applications in , , and transportation. Hands-on skills are developed through and , where students engage in prototyping, (CAD) modeling, and experimental testing. Laboratories often involve building and testing prototypes using tools like 3D printers, CNC machines, and wind tunnels for experiments, fostering practical proficiency in fabrication and . CAD software, such as or , is taught for creating detailed models and simulations, while prototyping labs emphasize and . These experiences cultivate technical competence in integrating theory with practice. Complementing technical training, programs instill essential through collaborative activities. Problem-solving is honed via analytical assignments and challenges, while and are emphasized in group projects requiring reports, presentations, and peer feedback. Capstone requirements typically culminate in senior projects, where multidisciplinary teams address client-sponsored problems by integrating , dynamics, fluids, and into functional prototypes, often spanning two semesters and incorporating ethical considerations and . These projects simulate professional roles, enhancing adaptability and innovation.

Professional practice

Roles and responsibilities

Mechanical engineers fulfill a variety of primary roles across industries, including design engineers who conceptualize and develop mechanical systems and components using tools like CAD software; manufacturing specialists who optimize production processes to ensure efficiency and quality; project managers who coordinate teams, timelines, and budgets for engineering initiatives; and R&D engineers who innovate new technologies through experimentation and analysis. These roles often overlap, with engineers applying principles of , , and to solve practical problems. Key responsibilities encompass system design, where engineers create prototypes and models to meet performance specifications; prototyping and testing to validate functionality under real-world conditions; to identify root causes of malfunctions using techniques like finite element analysis; and maintenance planning to develop strategies for equipment longevity and reliability. For instance, in system design, engineers evaluate material properties and simulate stresses to prevent breakdowns, while in testing, they conduct iterative experiments to refine designs based on empirical data. Mechanical engineers operate in diverse work environments, such as industrial settings like factories and manufacturing plants where they oversee production lines; consulting firms providing expert advice on project feasibility and optimization; government laboratories focusing on defense and energy applications; and academia, involving research and teaching roles. In these contexts, engineers may spend time in offices performing simulations or on-site inspecting installations, adapting to collaborative and sometimes high-stakes settings. Certain mechanical engineering roles involve significant travel to customer or project sites. These include field service engineers, who perform on-site installation, maintenance, troubleshooting, and repair of machinery and equipment at customer locations, often requiring domestic or international travel; technical sales engineers or applications engineers, who visit clients, attend trade shows, and demonstrate products, involving frequent travel; project engineers or commissioning engineers, particularly in industries like oil and gas, manufacturing, or construction, who oversee project implementation at various sites, potentially involving extended travel; and consulting engineers, who travel to different locations to assess sites, collaborate on projects, or provide on-site expertise. These roles are common in industries such as energy, manufacturing, aerospace, and heavy equipment. Travel frequency varies from occasional to extensive, with some field roles requiring 50-75% travel time. Ethical duties are central to the profession, requiring adherence to safety standards outlined in codes like those from the (ASME), which mandate prioritizing public safety, health, and welfare in all professional activities. Engineers must integrate by considering environmental impacts and promoting resource-efficient designs, as per ASME's emphasis on . Additionally, they handle responsibly, respecting proprietary information and avoiding unauthorized use to uphold professional integrity. Career progression typically begins as a junior or graduate , focusing on foundational tasks like supporting designs and conducting tests (0-4 years); advances to mid-level roles involving independent project contributions (4-7 years); and progresses to senior or lead positions, where individuals mentor teams and oversee complex initiatives, often with specialization in areas like or energy systems. This path emphasizes continuous , with opportunities for leadership through demonstrated technical expertise and project success. A typical workday for mechanical engineers in 2024-2025 varies significantly depending on industry, role, and project stage, but commonly includes reviewing tasks and planning, researching and designing solutions using CAD software, attending team meetings, building and testing prototypes or models, troubleshooting issues, and documenting progress. Work often splits between desk-based activities such as research, design, and simulation and hands-on work in laboratories or shops, with standard eight-hour workdays and possible overtime to meet project deadlines. For example, in hospital maintenance roles, engineers focus on routine equipment checks, repairs, and regulatory compliance; in product development, they blend CAD-based design with physical prototyping; and in manufacturing, they engage in design work, team meetings, and shop floor support.

Licensing, ethics, and regulation

In the United States, mechanical engineers seeking professional licensure typically must pass the Fundamentals of Engineering (FE) exam administered by the National Council of Examiners for Engineering and Surveying (NCEES), followed by at least four years of progressive engineering experience under a licensed professional engineer, and then the Principles and Practice of Engineering (PE) exam specific to mechanical engineering. This two-stage process ensures competence in applying engineering principles to protect public health, safety, and welfare, with state licensing boards overseeing final approval and renewal requirements, including continuing education. Internationally, similar pathways exist, such as the Chartered Engineer (CEng) status in the , regulated by the through institutions like the (IMechE). To achieve CEng, candidates need an accredited master's-level degree or equivalent, combined with substantial professional competence demonstrated via a professional review interview and competency report, emphasizing ethical practice and lifelong learning. Ethical guidelines for mechanical engineers are codified by organizations like the National Society of Professional Engineers (NSPE) and the (ASME), which mandate prioritizing public safety, maintaining competence, avoiding conflicts of interest, and upholding honesty in professional dealings. The NSPE Code of Ethics, for instance, requires engineers to issue public statements only in an objective and truthful manner while holding paramount the safety, health, and welfare of the public. Similarly, ASME's Code of Ethics reinforces these principles, obligating members to advance the integrity of the profession and report violations that endanger safety. Regulatory frameworks enforce compliance through standards like ISO 9001, which outlines requirements for systems to ensure consistent product and service quality in mechanical engineering applications, including risk-based thinking and continual improvement. In the U.S., the (OSHA) mandates safeguards such as under 29 CFR 1910.212 to prevent hazards like mechanical injuries in workplaces involving mechanical systems. Internationally, the European Union's Machinery Directive 2006/42/EC sets essential health and safety requirements for machinery design and construction, requiring conformity assessments and to facilitate free market circulation while minimizing risks. By 2025, regulatory emphases have evolved to address emerging challenges in mechanical engineering, including heightened data privacy obligations for smart, connected systems under frameworks like the EU's (GDPR) and new U.S. state laws, which demand secure handling of user data in IoT-enabled machinery to prevent breaches. Additionally, green regulations, such as the impending Machinery Regulation (EU) 2023/1230 replacing the 2006 directive by 2027, prioritize through requirements for energy-efficient designs and lifecycle environmental assessments, compelling engineers to integrate eco-friendly practices without compromising safety.

Core subdisciplines

Mechanics of solids and structures

Mechanics of solids and structures is a fundamental subdiscipline of mechanical engineering that focuses on the behavior of solid materials and structures subjected to various loads, encompassing the analysis of deformation, stress distribution, and potential failure mechanisms. This field provides the analytical foundation for designing safe and efficient components that withstand mechanical forces without excessive deformation or rupture. Key concepts include the internal forces and deformations within solids, which are essential for predicting how materials respond to tensile, compressive, torsional, or loads in engineering applications. Stress refers to the internal resistance of a material to external forces, quantified as force per unit area. Normal stress acts perpendicular to a surface, causing extension or compression, while shear stress acts parallel to the surface, leading to sliding or angular distortion. Strain, the complementary measure, describes the deformation or relative displacement within the material, defined as the change in length or angle per unit original dimension. In elastic behavior, materials return to their original shape upon load removal, following Hooke's law where stress is linearly proportional to strain within the elastic limit, characterized by the modulus of elasticity EE. Beyond this limit, plasticity occurs, where permanent deformation takes place as the material yields and undergoes irreversible changes, often modeled by stress-strain curves that delineate the transition from elastic to plastic regimes. To analyze complex stress states, engineers use tools like , a graphical method for transforming stresses between coordinate systems and identifying principal stresses and maximum . The circle is constructed using the normal and shear stress components on two perpendicular planes, with the center at the average normal stress (σx+σy)/2(\sigma_x + \sigma_y)/2 and radius ((σx−σy)/2)2+τxy2\sqrt{((\sigma_x - \sigma_y)/2)^2 + \tau_{xy}^2}
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