Vector Spaces and OPC Standards

A subspace is a subset of a vector space that is also a vector space itself. It must satisfy three properties: contain the zero vector, closed under addition, and closed under scalar multiplication.

A non-empty subset of a vector space is a subspace if, along with every pair of vectors, any linear combination formed with those vectors is also included in the subset.

The zero vector is the additive identity element in a vector space. It's the vector that when added does not change the other vector.

Closure under addition means that if you add any two vectors in a subspace, the resulting vector is also in the subspace.

Closure under scalar multiplication means that if you multiply any vector in a subspace by a scalar, the resulting vector is also in the subspace.

OPC stands for Open Platform Communication. It's a standard that allows systems from different manufacturers to communicate with each other, enabling automated processes involving various machines.

The OPC standard defines how clients and servers communicate with each other, both for direct connections and server-to-server communication. It ensures consistent access to real-time data, alarm and event monitoring, and historical data.

OPC was released in 1996 to standardize PLC-specific protocols like Modbus and Profibus. This enabled HMI/SCADA systems to communicate with different PLCs through a common interface that translates requests between devices.

Originally, OPC was only available on Windows OS and utilized OLE (Object Linking & Embedding) for process control.

OPC UA (Unified Architecture) was developed to address challenges related to service-oriented architectures in manufacturing systems. It aims to provide a feature-rich, scalable, and extensible open platform technology that is future-proof.

Normal stress acts perpendicular to the cut surface of a material.

Tensile stress occurs when an object is stretched, pulling the material apart.

Compressive stress occurs when an object is squeezed or compressed, pushing the material together.

The equation for normal stress is σ = F/A, where σ is the normal stress, F is the force acting perpendicular to the surface, and A is the cross-sectional area.

The equation for normal strain is ε = ΔL/L₀, where ε is the normal strain, ΔL is the change in length, and L₀ is the original length.

The modulus of elasticity is the slope of the initial linear portion of the stress-strain curve. It represents the material's stiffness and is denoted by E. It is defined as E = σ/ε.

The proportionality limit is the point on the stress-strain curve beyond which the material no longer exhibits a linear relationship between stress and strain. It marks the end of the elastic region.

Yielding is when a material undergoes significant plastic deformation without any further increase in stress.

The yield point is the point on the stress-strain curve where yielding begins, indicating the start of permanent deformation.

Strain hardening is a phenomenon where a material undergoing plastic deformation becomes stronger and more resistant to further deformation. It's due to changes in the material's internal structure.

Ultimate tensile stress (UTS) is the maximum stress a material can withstand before it starts to fail.

Necking is a phenomenon that occurs during tensile loading beyond UTS where the material starts to narrow significantly in a localized region.

To obtain a true stress-strain curve, you need to consider the actual cross-sectional area of the material, especially during necking, where the area changes significantly.

The reduction in load after UTS is primarily because of the decreasing cross-sectional area of the material, not a loss of stress strength.

Ductility is the ability of a material to deform plastically under tensile stress without breaking. It's a measure of how much a material can be stretched before fracturing.

Brittleness is a property where a material fails in tension at a relatively low value of strain, meaning it breaks easily under a small amount of stretching.

A header tank is a container that holds a 'head' of water in a hydraulic circuit. It serves as a reservoir for fluid and allows for expansion and contraction of the fluid due to temperature changes.

A header tank is installed at the highest point in the hydraulic circuit to provide space for fluid expansion and to allow for easy refilling.

A swirl tank is an alternative to a header tank that allows air or gas to escape due to its swirling fluid design. Air bubbles are forced upwards in the center.

Elasticity is a material's ability to return to its original dimensions after unloading. It's like a spring that returns to its original shape.

Creep is the tendency of a material to undergo a slow, continuous deformation under sustained stress, even at constant load. It's like a gradual stretching or sagging over time.

Poisson's ratio is a measure of the lateral strain (change in width) to the axial strain (change in length) of a material under tensile or compressive stress. It reflects how much a material contracts in width when stretched in length.

Shear stress is the stress that occurs when a force acts parallel to the surface of a material. It's like sliding one layer of a material over another.

Shear strain is the angle of deformation caused by shear stress. It's a measure of how much a material distorts when forces act parallel to its surface.

Hooke's law for shear states that the shear stress (τ) is proportional to the shear strain (γ) and the constant of proportionality is the shear modulus of elasticity (G). The equation is τ = Gγ.

Pure bending refers to the bending of a beam under a constant bending moment, where the shear force is zero. This means the bending moment remains the same along the beam's axis, resulting in a uniform curvature.

A fluid is a substance that continuously deforms under shear stress, no matter how small the stress. It cannot resist deformation.

Liquids are fluids with a fixed volume that changes minimally with temperature and pressure. They occupy a definite volume and retain their shape.

Gases are fluids with no definite volume and expand to fill the space they are in. They have no free surface due to their ability to completely fill a container.

Density is a measure of mass per unit of volume. It is denoted by ρ and is found using the formula ρ = m/V, where m is mass and V is volume.

Pressure is the force applied perpendicularly to a surface per unit area. In fluids, pressure is caused by the collisions of molecules with each other and surrounding surfaces.

Static pressure is the pressure exerted by a stationary fluid at a point. It can be measured as either absolute pressure or gauge pressure. The difference between them is the pressure of the surrounding atmosphere.

Absolute pressure is the pressure measured relative to a perfect vacuum.

Gauge pressure is the difference between the absolute pressure and the atmospheric pressure.

Vapour pressure is the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. It represents the tendency of a liquid to evaporate.