
Engineers struggle to understand magnetic coupling technology. Traditional explanations are too complex or too simple. You need clear technical details to make informed design decisions.
A magnetic rodless cylinder works by using powerful permanent magnets to transfer force through the cylinder wall, with internal magnets attached to the piston and external magnets mounted on a carriage, creating synchronized motion without physical connection through magnetic field coupling.
Last month, I helped David, a design engineer at a German automation company, solve a critical contamination problem. Their traditional rod cylinder kept failing in a dusty environment. We replaced it with a magnetic rodless cylinder that eliminated seal contamination and increased their system reliability by 300%.
Table of Contents
- What Are the Core Components of a Magnetic Rodless Cylinder?
- How Does Magnetic Coupling Transfer Force Through the Cylinder Wall?
- What Types of Magnets Are Used in Magnetic Rodless Cylinders?
- How Do Sealing Systems Work in Magnetic Rodless Cylinders?
- What Factors Affect Magnetic Coupling Performance?
- How Do You Calculate Force and Performance Parameters?
- What Are Common Problems and Solutions for Magnetic Rodless Cylinders?
- Conclusion
- FAQs About Magnetic Rodless Cylinders
What Are the Core Components of a Magnetic Rodless Cylinder?
Understanding component functions helps engineers troubleshoot problems and optimize performance. I explain the technical details that matter for practical applications.
Core components of a magnetic rodless cylinder include the cylinder tube, internal piston with magnets, external carriage with magnets, sealing system, end caps, and mounting hardware, all designed to work together for reliable magnetic force transfer.
Cylinder Tube Construction
The cylinder tube houses the internal piston and provides the pressure boundary. Non-magnetic materials like aluminum or stainless steel are essential to allow magnetic field penetration.
Wall thickness must be optimized for magnetic coupling efficiency. Thinner walls allow stronger magnetic coupling but reduce pressure capacity. Typical wall thickness ranges from 2-6mm depending on bore size and pressure rating.
Surface finish inside the tube affects seal performance and piston movement. Honed surfaces provide smooth operation and long seal life. Surface roughness typically ranges from 0.4-0.8 Ra.
Tube ends include mounting features and port connections. Precision machining ensures proper alignment and sealing. End cap attachment methods include threaded, flanged, or tie-rod designs.
Internal Piston Assembly
The internal piston contains permanent magnets and sealing elements. Piston design must balance magnetic coupling strength with sealing effectiveness.
Magnet mounting methods include adhesive bonding, mechanical retention, or molded-in designs. Secure mounting prevents magnet displacement during high-acceleration operations.
Piston seals maintain pressure while allowing smooth movement. Seal selection affects friction, leakage, and service life. Common seal materials include nitrile, polyurethane, and PTFE.
Piston weight affects dynamic performance. Lighter pistons enable higher acceleration and speed. Material selection balances weight, strength, and magnetic properties.
External Carriage System
The external carriage carries the external magnets and provides load attachment points. Carriage design affects coupling strength and mechanical performance.
Magnet positioning in the carriage must align precisely with internal magnets. Misalignment reduces coupling force and causes uneven wear.
Carriage materials must be non-magnetic to prevent field distortion. Aluminum alloys provide good strength-to-weight ratios for most applications.
Load attachment methods include threaded holes, T-slots1, or custom brackets. Proper load distribution prevents carriage distortion and maintains alignment.
Magnetic Assembly Design
Magnet assemblies in both piston and carriage must be precisely matched for optimal coupling. Magnet orientation and spacing are critical parameters.
Magnetic circuit design optimizes field strength and distribution. Pole piece design concentrates magnetic flux for maximum coupling force.
Temperature compensation may be needed for applications with wide temperature ranges. Magnet selection and circuit design affect temperature stability.
Protective coatings prevent magnet corrosion and damage. Nickel plating is common for neodymium magnets in industrial applications.
Component | Material Options | Key Functions | Design Considerations |
---|---|---|---|
Cylinder Tube | Aluminum, Stainless Steel | Pressure Boundary | Wall Thickness, Surface Finish |
Internal Piston | Aluminum, Steel | Magnet Carrier | Weight, Seal Compatibility |
External Carriage | Aluminum Alloy | Load Interface | Stiffness, Alignment |
Magnets | Neodymium, Ferrite | Force Transfer | Temperature Rating, Coating |
Sealing System Components
Primary seals on the piston maintain pressure separation between cylinder chambers. These seals must operate with minimal friction while preventing leakage.
Secondary seals at cylinder ends prevent external leakage. These static seals are easier to design but must handle thermal expansion.
Wiper seals prevent contamination entry while allowing carriage movement. Seal design must balance sealing effectiveness with friction.
Seal materials must be compatible with operating fluids and temperatures. Chemical compatibility charts guide material selection for specific applications.
Mounting and Connection Hardware
Cylinder mounting hardware must handle operating loads and forces. Mounting methods include flange, foot, or trunnion designs.
Port connections provide compressed air supply and exhaust. Port sizing affects flow capacity and operating speed.
Position sensing provisions may include sensor mounting brackets or integrated sensor systems. Sensor selection affects positioning accuracy and system cost.
Protective covers or boots may be needed in contaminated environments. Protection level must balance contamination exclusion with heat dissipation.
How Does Magnetic Coupling Transfer Force Through the Cylinder Wall?
Magnetic coupling is the key technology that enables rodless operation. Understanding the physics helps optimize performance and troubleshoot problems.
Magnetic coupling transfers force through attractive forces between internal and external permanent magnets, with magnetic field lines passing through the non-magnetic cylinder wall to create synchronized movement without physical contact.
Magnetic Field Physics
Permanent magnets create magnetic fields that extend beyond the magnet boundaries. Field strength decreases with distance according to inverse square law2 relationships.
Magnetic field lines form closed loops from north to south poles. Field concentration and direction determine coupling force magnitude and direction.
Non-magnetic materials like aluminum allow magnetic fields to pass through with minimal attenuation. Magnetic materials would distort or block the field.
Field strength measurement uses gaussmeters or hall effect sensors. Typical field strengths range from 1000-5000 gauss at the coupling interface.
Force Transfer Mechanism
Attractive forces between opposite magnetic poles create the coupling force. North poles attract south poles while like poles repel each other.
Force magnitude depends on magnet strength, air gap distance, and magnetic circuit design. Closer spacing increases force but may cause mechanical interference.
Force direction follows magnetic field lines. Proper magnet orientation ensures force acts in the desired direction for load movement.
Coupling efficiency depends on magnetic circuit design and air gap uniformity. Well-designed systems achieve 85-95% force transfer efficiency.
Air Gap Considerations
Air gap distance between internal and external magnets affects coupling strength significantly. Doubling the gap typically reduces force by 75%.
Cylinder wall thickness contributes to total air gap. Thinner walls allow stronger coupling but may reduce pressure capacity.
Manufacturing tolerances affect air gap uniformity. Tight tolerances maintain consistent coupling force throughout the stroke.
Thermal expansion can change air gap dimensions. Design must account for temperature effects on coupling performance.
Magnetic Circuit Optimization
Pole piece design concentrates magnetic flux for maximum coupling force. Iron or steel pole pieces focus magnetic fields effectively.
Magnet arrangement affects field distribution and coupling uniformity. Multiple magnet pairs provide more uniform coupling along the stroke.
Back iron or return paths complete the magnetic circuit. Proper design minimizes flux leakage and maximizes coupling efficiency.
Finite element analysis3 tools help optimize magnetic circuit design. Computer modeling predicts performance before prototype testing.
What Types of Magnets Are Used in Magnetic Rodless Cylinders?
Magnet selection significantly affects performance, cost, and service life. Different magnet types suit different applications and operating conditions.
Magnetic rodless cylinders primarily use neodymium rare-earth magnets for high performance applications, ferrite magnets for cost-sensitive applications, and samarium cobalt magnets for high-temperature environments.
Neodymium Rare-Earth Magnets
Neodymium magnets provide the highest magnetic strength available commercially. Energy products range from 35-52 MGOe4 for different grades.
Temperature ratings vary by grade from 80°C to 200°C maximum operating temperature. Higher temperature grades cost more but handle demanding applications.
Corrosion protection is essential for neodymium magnets. Nickel plating is standard, with additional coatings available for harsh environments.
Cost is higher than other magnet types but performance advantages often justify the expense. Price varies with grade, size, and market conditions.
Ferrite Ceramic Magnets
Ferrite magnets cost less than rare-earth types but provide lower magnetic strength. Energy products typically range from 3-5 MGOe.
Temperature stability is excellent with operating ranges from -40°C to +250°C. This makes ferrite suitable for high-temperature applications.
Corrosion resistance is inherently good due to ceramic construction. No protective coatings are typically needed.
Applications include cost-sensitive designs where lower forces are acceptable. Larger magnet sizes compensate for lower strength.
Samarium Cobalt Magnets
Samarium cobalt magnets provide excellent high-temperature performance with operating temperatures up to 350°C.
Corrosion resistance is superior to neodymium without protective coatings. This suits harsh chemical environments.
Magnetic strength is high but less than neodymium. Energy products range from 16-32 MGOe depending on grade.
Cost is the highest among common magnet types. Applications justify cost through superior environmental performance.
Magnet Grade Selection
Temperature requirements determine minimum magnet grade needed. Higher grades cost more but handle demanding conditions.
Force requirements determine magnet size and grade combination. Optimization balances cost with performance needs.
Environmental conditions affect magnet selection and protective requirements. Chemical compatibility must be verified.
Service life expectations influence magnet grade selection. Higher grades typically provide longer service life.
Magnet Type | Energy Product (MGOe) | Temperature Range (°C) | Relative Cost | Best Applications |
---|---|---|---|---|
Neodymium | 35-52 | -40 to +200 | High | High Performance |
Ferrite | 3-5 | -40 to +250 | Low | Cost Sensitive |
Samarium Cobalt | 16-32 | -40 to +350 | Highest | High Temperature |
Magnet Mounting Methods
Adhesive bonding uses structural adhesives to secure magnets. Bond strength must exceed operating forces with appropriate safety factors.
Mechanical retention uses clips, bands, or housings to secure magnets. This method allows magnet replacement during maintenance.
Molded-in mounting encapsulates magnets in plastic or metal housings. This provides excellent retention but prevents magnet replacement.
Mounting method selection depends on force levels, maintenance requirements, and manufacturing considerations.
Magnet Safety Considerations
Strong magnets can cause injury during handling and installation. Proper training and tools prevent accidents.
Magnetic fields affect pacemakers and other medical devices. Warning labels and restricted access may be required.
Magnet fragments can cause injury if magnets break. Quality magnets and proper handling reduce this risk.
Storage and shipping require special precautions. Magnetic shielding prevents interference with other equipment.
How Do Sealing Systems Work in Magnetic Rodless Cylinders?
Sealing systems maintain pressure while allowing smooth operation. Proper seal design and selection are critical for reliable performance.
Magnetic rodless cylinder sealing systems use static seals at cylinder ends and dynamic seals on the internal piston, with no seals needed between internal and external components due to magnetic coupling through the cylinder wall.
Static Sealing Systems
End cap seals prevent external leakage at cylinder ends. These O-ring seals operate in static applications with minimal stress.
Port seals prevent leakage at air connections. Thread sealants or O-rings provide reliable sealing for standard fittings.
Mounting seals may be needed for some mounting configurations. Gaskets or O-rings prevent leakage at mounting interfaces.
Static seal selection is straightforward with standard O-ring materials suitable for most applications.
Dynamic Piston Sealing
Primary piston seals maintain pressure separation between cylinder chambers. These seals must operate with minimal friction while preventing leakage.
Seal design affects friction, leakage, and service life. Single-acting seals work in one direction while double-acting seals work bidirectionally.
Seal materials must be compatible with operating fluids and temperatures. Nitrile rubber suits most pneumatic applications.
Seal groove design affects seal performance and installation. Proper groove dimensions ensure optimal seal function.
Contamination Prevention
Wiper seals prevent contamination entry while allowing carriage movement. Seal design must balance sealing effectiveness with friction.
Protective boots or covers provide additional contamination protection. These flexible covers move with the carriage.
Breather filters allow pressure equalization while preventing contamination entry. Filter selection depends on contamination levels.
Environmental sealing requirements vary by application. Clean environments need minimal protection while harsh conditions require comprehensive sealing.
Seal Material Selection
Nitrile rubber (NBR) suits most pneumatic applications with good oil resistance and moderate temperature range.
Polyurethane provides excellent wear resistance and low friction. This material suits high-cycle applications.
PTFE offers chemical resistance and low friction but requires careful installation. Composite seals combine PTFE with elastomer backup.
Fluorocarbon (FKM) provides excellent chemical and temperature resistance for demanding applications.
Lubrication Considerations
Some seal materials require lubrication for optimal performance. Oil-free air systems may need special seal materials.
Lubrication methods include oil injection into compressed air or grease application during assembly.
Over-lubrication can cause problems in clean environments. Minimal lubrication maintains seal performance without contamination.
Lubrication intervals depend on operating conditions and seal materials. Regular maintenance extends seal life.
What Factors Affect Magnetic Coupling Performance?
Multiple factors influence magnetic coupling effectiveness. Understanding these factors helps optimize performance and prevent problems.
Magnetic coupling performance is affected by air gap distance, magnet strength and alignment, temperature variations, contamination between magnets, cylinder wall thickness, and external magnetic interference.
Air Gap Distance Effects
Air gap distance has the greatest impact on coupling force. Force decreases rapidly with increasing gap distance.
Typical air gaps range from 1-5mm total including cylinder wall thickness. Smaller gaps provide higher forces but may cause mechanical interference.
Gap uniformity affects coupling consistency. Manufacturing tolerances and thermal expansion influence gap variations.
Gap measurement requires precision instruments. Feeler gauges or dial indicators verify gap dimensions during assembly.
Temperature Impact on Performance
Magnet strength decreases with increasing temperature. Neodymium magnets lose about 0.12% strength per degree Celsius.
Thermal expansion affects air gap dimensions. Different materials expand at different rates, changing gap uniformity.
Temperature cycling can cause fatigue in magnet mounting systems. Proper design accommodates thermal stresses.
Operating temperature limits depend on magnet grade selection. Higher grade magnets handle higher temperatures.
Contamination and Interference
Metal particles between magnets reduce coupling force and may cause binding. Regular cleaning maintains performance.
External magnetic fields can interfere with coupling. Motors, transformers, and other magnets may cause problems.
Non-magnetic contamination has minimal effect on coupling but may cause mechanical problems.
Contamination prevention through proper sealing and filtration maintains coupling performance.
Mechanical Alignment Factors
Magnet alignment affects coupling uniformity and efficiency. Misalignment causes uneven forces and premature wear.
Carriage stiffness affects alignment maintenance under load. Flexible carriages may deflect and reduce coupling effectiveness.
Guide system accuracy influences alignment consistency. Precision guides maintain proper magnet positioning.
Assembly tolerances accumulate to affect final alignment. Tight tolerances improve coupling performance.
Load and Dynamic Effects
High acceleration forces can overcome magnetic coupling. Maximum acceleration depends on coupling strength and load mass.
Shock loads may cause temporary coupling loss. Proper design includes adequate coupling safety factors.
Vibration can affect coupling stability. Resonant frequencies should be avoided in system design.
Side loads on the carriage can cause misalignment and reduce coupling effectiveness.
Performance Factor | Effect on Coupling | Typical Range | Optimization Methods |
---|---|---|---|
Air Gap Distance | Inverse Square Law | 1-5mm | Minimize Wall Thickness |
Temperature | -0.12%/°C | -40 to +150°C | High Grade Magnets |
Contamination | Force Reduction | Variable | Sealing, Cleaning |
Alignment | Uniformity Loss | ±0.1mm | Precision Assembly |
Safety Factor Considerations
Coupling force safety factors account for performance variations and degradation over time. Typical safety factors range from 2-4.
Peak force requirements may exceed steady-state forces. Acceleration and shock loads require higher coupling forces.
Magnet aging causes gradual strength reduction. Quality magnets maintain 95% strength after 10 years.
Environmental degradation affects long-term performance. Proper protection maintains coupling effectiveness.
How Do You Calculate Force and Performance Parameters?
Accurate calculations ensure proper cylinder sizing and reliable operation. I provide practical calculation methods for real-world applications.
Calculate magnetic rodless cylinder performance using magnetic coupling force equations, load analysis, acceleration forces, and safety factors to determine required cylinder size and magnet specifications.
Basic Force Calculations
Magnetic coupling force depends on magnet strength, air gap, and magnetic circuit design. Manufacturer specifications provide coupling force data.
Available cylinder force equals coupling force minus friction losses. Friction typically consumes 5-15% of coupling force.
Load force requirements include static weight, friction, and dynamic forces. Each component must be calculated separately.
Safety factors account for performance variations and ensure reliable operation. Apply factors of 2-4 depending on application criticality.
Magnetic Field Strength Calculations
Magnetic field strength decreases with distance according to inverse relationships. Field strength at distance d: B = B₀ × (r/d)²
Coupling force relates to magnetic field strength and magnet area. Force equations require detailed magnetic circuit analysis.
Computer modeling tools simplify complex magnetic calculations. Finite element analysis provides accurate predictions.
Empirical testing validates calculated predictions. Prototype testing confirms performance under actual operating conditions.
Dynamic Performance Analysis
Acceleration forces use Newton’s second law: F = ma, where m is total moving mass and a is acceleration.
Maximum acceleration depends on available coupling force minus load forces. Higher coupling forces enable faster operation.
Deceleration forces may exceed acceleration forces due to momentum effects. Proper calculation prevents coupling failure.
Cycle time calculations consider acceleration, constant velocity, and deceleration phases. Total cycle time affects productivity.
Pressure and Flow Requirements
Cylinder force relates to air pressure and piston area: F = P × A, where P is pressure and A is piston area.
Flow requirements depend on cylinder volume and cycle speed. Higher speeds need greater flow rates.
Pressure drop calculations account for valve restrictions and line losses. Adequate pressure ensures proper operation.
Air consumption calculations help size compressor systems. Total consumption includes all cylinders and losses.
Load Analysis Methods
Static loads include part weight and constant external forces. These loads act continuously during operation.
Dynamic loads result from acceleration and deceleration. These forces vary with motion profile and timing.
Friction forces depend on guide systems and seal types. Coefficient of friction5 values guide calculations.
External forces may include springs, gravity, or process forces. All forces must be considered in sizing calculations.
Calculation Type | Formula | Key Variables | Typical Values |
---|---|---|---|
Coupling Force | Fc = K × B² × A | Magnetic Field, Area | 100-5000N |
Acceleration Force | Fa = m × a | Mass, Acceleration | Variable |
Friction Force | Ff = μ × N | Friction Coefficient | 5-15% of Load |
Safety Factor | SF = Fc / (Fl + Ff + Fa) | All Forces | 2-4 |
Performance Optimization
Magnet selection optimizes coupling force for specific applications. Higher grade magnets provide more force but cost more.
Air gap minimization increases coupling force significantly. Design optimization balances force with manufacturing tolerances.
Load reduction through design changes improves performance. Lighter loads require less coupling force.
Guide system optimization reduces friction and improves efficiency. Proper lubrication maintains low friction operation.
What Are Common Problems and Solutions for Magnetic Rodless Cylinders?
Understanding common problems helps prevent failures and reduce downtime. I see similar issues across different applications and provide proven solutions.
Common magnetic rodless cylinder problems include reduced coupling force, position drift, contamination between magnets, temperature effects, and alignment issues, most preventable through proper installation and maintenance.
Coupling Force Reduction
Coupling force reduction indicates magnet degradation, increased air gap, or contamination. Symptoms include slower operation and position drift.
Magnet aging causes gradual strength reduction over time. Quality magnets maintain 95% strength after 10 years of normal operation.
Air gap increases due to wear or thermal expansion. Measure gaps regularly and adjust as needed.
Contamination between magnets reduces coupling effectiveness. Metal particles are particularly problematic.
Solutions include magnet replacement, gap adjustment, contamination removal, and improved environmental protection.
Position Drift Problems
Position drift indicates coupling slippage or external force changes. Monitor position accuracy over time to identify drift patterns.
Insufficient coupling force allows load forces to overcome magnetic coupling. Increase coupling force or reduce loads.
External force variations affect position stability. Identify and control variable forces in the system.
Temperature changes affect magnet strength and mechanical dimensions. Compensate for temperature effects in critical applications.
Solutions include coupling force increase, load reduction, force stabilization, and temperature compensation.
Contamination Issues
Metal particles between magnets cause binding and force reduction. Regular inspection and cleaning prevent problems.
Magnetic particles are attracted to magnet surfaces and accumulate over time. Establish cleaning schedules based on contamination rates.
Non-magnetic contamination may cause mechanical interference. Proper sealing prevents most contamination entry.
Contamination sources include machining operations, wear particles, and environmental exposure. Identify and control sources.
Solutions include improved sealing, regular cleaning, contamination source control, and protective covers.
Temperature-Related Problems
High temperatures reduce magnet strength and may cause permanent damage. Monitor operating temperatures in critical applications.
Thermal expansion changes air gaps and mechanical alignment. Design must accommodate thermal effects.
Temperature cycling causes fatigue in mounting systems. Use appropriate materials and design for thermal stresses.
Low temperatures may cause condensation and icing problems. Provide heating or insulation as needed.
Solutions include temperature monitoring, thermal protection, expansion compensation, and environmental control.
Alignment and Mechanical Issues
Misalignment causes uneven coupling forces and premature wear. Check alignment regularly using precision instruments.
Guide system problems affect carriage alignment and coupling effectiveness. Maintain guides according to manufacturer recommendations.
Mounting system flexibility allows misalignment under load. Use rigid mounting and proper support structures.
Wear in mechanical components gradually degrades alignment. Replace worn components before alignment becomes critical.
Solutions include precision alignment, guide maintenance, rigid mounting, and component replacement schedules.
Problem Type | Common Causes | Symptoms | Solutions |
---|---|---|---|
Force Reduction | Magnet Aging, Gap Increase | Slow Operation | Magnet Replacement |
Position Drift | Coupling Slippage | Accuracy Loss | Force Increase |
Contamination | Metal Particles | Binding, Noise | Regular Cleaning |
Temperature Effects | Heat Exposure | Performance Loss | Thermal Protection |
Misalignment | Mounting Issues | Uneven Wear | Precision Assembly |
Preventive Maintenance Strategies
Regular inspection schedules prevent most problems before they cause failures. Monthly inspections catch problems early.
Cleaning procedures remove contamination before it causes problems. Use appropriate cleaning methods for magnet types.
Performance monitoring tracks coupling effectiveness over time. Trending data predicts maintenance needs.
Component replacement schedules ensure reliable operation. Replace wear items before failure occurs.
Documentation helps identify problem patterns and optimize maintenance procedures. Keep detailed maintenance records.
Conclusion
Magnetic rodless cylinders use sophisticated magnetic coupling technology to provide space-efficient linear motion. Understanding the working principles, components, and performance factors enables optimal application and reliable operation.
FAQs About Magnetic Rodless Cylinders
How does a magnetic rodless cylinder work internally?
A magnetic rodless cylinder works by using permanent magnets attached to an internal piston and external carriage, with magnetic fields passing through the non-magnetic cylinder wall to create synchronized movement without physical connection.
What types of magnets are used in magnetic rodless cylinders?
Magnetic rodless cylinders primarily use neodymium rare-earth magnets for high performance, ferrite magnets for cost-sensitive applications, and samarium cobalt magnets for high-temperature environments up to 350°C.
How does magnetic coupling transfer force through the cylinder wall?
Magnetic coupling transfers force through attractive forces between internal and external permanent magnets, with magnetic field lines passing through the non-magnetic aluminum or stainless steel cylinder wall.
What factors affect magnetic coupling performance?
Key factors include air gap distance (most critical), magnet strength and alignment, temperature variations, contamination between magnets, cylinder wall thickness, and external magnetic interference.
How do you calculate the force output of a magnetic rodless cylinder?
Calculate force using magnetic coupling specifications from manufacturers, subtract friction losses (5-15%), add safety factors (2-4), and consider dynamic forces from acceleration using F = ma.
What are common problems with magnetic rodless cylinders?
Common problems include reduced coupling force from magnet aging, position drift from insufficient coupling, contamination between magnets, temperature effects on performance, and alignment issues.
How do you maintain magnetic rodless cylinders properly?
Maintenance includes regular cleaning of magnetic surfaces, monitoring air gap dimensions, checking alignment, replacing worn seals, and protecting from contamination through proper environmental sealing.
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View the standard profiles and dimensions for T-slot systems used in industrial automation and framing. ↩
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Explore the fundamental physics of the inverse square law and how it applies to forces like magnetism and gravity. ↩
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Learn about the principles of Finite Element Analysis (FEA) and its use as a computational tool in engineering design. ↩
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Understand the definition of MegaGauss-Oersted (MGOe) and its significance as a measure of a permanent magnet’s strength. ↩
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Review the definition of the coefficient of friction and the distinction between static and kinetic friction in mechanical systems. ↩