Propagation & Mixed Systems

Propagation Problems

• Finite difference methods, step size extrapolation, and recurrence formulae with higher-order truncation errors are solutions for propagation problems

Mixed Systems

• Physical systems with only a single type of constitutive relationship has been considered
• Mechanical (and structural) system have a relationship of ∆ & 𝐹
• Electrical systems have a relationship of ∆𝑉 & 𝑖
• Hydraulic system have a relationship of ∆𝑃 & 𝑞
• Thermal Systems have a relationship of ∆𝑇 & ℎ
• Mixed systems have combinations of elements
• Mixed Systems convert energy through special elements between different parts of the combined system
• Equilibrium models node variables represent flows
• These flows go through the element
• Loop variables act across the elements, according to the constitutive relationships
• Different systems, Effort (e), and Flow (f) are;
  

Energy loss from elements

• Some elements dissipate energy, Electrical Resistor, and Mechanical Damper
• Energy can leave an element by a force source with the ∆ in the same direction, providing energy to other parts of the system
  

Fluid Resistance

• Pressure drop from friction is expressed as ∆𝑃𝑓 = 𝑅𝑓 𝑞
• Rate of energy dissipation is expressed as 𝑑𝐸𝑑/𝑑𝑡 = 𝑞 𝑑∆ 𝑃𝑓 = 1/(2𝑅𝑓) ∆𝑃𝑓 2
• Power is positive at the front end of the element and is expressed as as 𝑃𝑖𝑖𝑖𝑖 𝑞 > 0
• Power is negative at the back end of the element and is expressed as 𝑃𝑜𝑢𝑡 𝑞 < 0
• Friction force always acts in the opposite direction to the flow (dissipative)
• Power flows in the direction of flow

Ideal System Elements

• An A-type energy equation is written in terms of Across variable
• A T-type element energy equation is written in terms of Through variable
• A D-type element dissipates energy

Portality

• Energy storage and dissipation devices are one-port devices
• They accept an effort or flow as input, and they output the complementary effort or flow
• Current is input to a capacitor, causing charge buildup and a change in output voltage
• Force is input to a mass, causing it to accelerate, with a change in velocity

Transducers and Transformers

• Transducers & Transformers are two-port devices that interconnect two effort/flow pairs
• A Transformer transforms an energy flow from one effort/flow ratio to another, in the same medium, with little loss of energy
• Gears, levers, and electrical transformers are examples of Transformers

System

• Subsystem 1 & Subsystem 2
• Loop Var
• & Sensor/Actuator & Sensor/Actuator …
• Node Variable, Constitutive Relationship, Node Variable, Constitutive Relationship
• Physical, Var 1 Var 2, Physical Var 2 Var 3
• Domain 1, Domain 2
  

Generic elements

• Generic Elements have common characteristics of elements regardless of the type of subsystem
• Equilibrium: There is no change in energy storage or release, and no dissipation
• These diagrams show the efforts and flows on the element, not the effort on the environment (nodes)

Generic Elements (dynamic conditions)

• Power out = power in, ideal transfer, no losses
• Power out > power in, instantaneous release of energy
• Power out < power in, instantaneous storage of energy
• Power out + dissipation = power in, energy lost from system

Power

• Power goes in the same direction as the through variable
• Power out is a load when the dot product of through & across vectors is negative

Mechanical Transformers

• Kinematic relationships convert one type of motion to another
• For example linear to rotational (rack & pinion)
• In ideal circumstances, Ω = 𝑛𝑣 ; 𝑛τ = 𝐹 ; where 𝑛 is the coupling ratio
• Viscous friction causes efficiency loss (proportional to velocity)
  • Ω = 𝑛∗ 𝑣 ; 𝑛∗ τ = 𝐹 ; 𝑛∗ < 𝑛
• If there is slip, then Ω = 𝑛(𝑣) 𝑣
• In Mechancial systems, Force are Through variables (𝐹) -> 1/𝑛 (τ)
• In Mechancial systems, Velocity are Across variables (𝑣) <- 1/𝑛 (Ω)

Transducers

• Transducers convert between different types of power: electric motors, piezoelectric actuators, propellors/impellers, hydraulic cylinders
• Energy Flow Relations for a Hydraulic Cylinder

Electromechanical Elements

• Transmit power and convert it to another form
• Through and Across variables convert
• Rotary motors, and generators
• Power for an ideal motor/generator is Pelect = 𝑃mech, which means 𝑖 𝑒𝑖 = Ω τ

Ideal Electric Motor

• Most electric motor designs are inherently bidirectional and can function as motors or generators
• Each type of electric motor has a specific governing equation

Flow and Effort Relationships

• Relationship between the flow (through) and effort (across)
• Can be expressed as 𝛀 = 𝜶τ 𝒆, where Mechanical and Electrical
• 𝜶τ is the coupling coefficient that converts an electrical “across” variable to a mechanical one
• Be careful to use the correct system to for correct transformations
• Across: 𝝉𝝉 𝛀𝛀 = 𝒆𝒆 𝒊𝒊 ;
• Through: 𝝉𝝉 = (1/𝜶τ) 𝒊

Actuators & Sensors

• Sensors & Actuators use Transformers & Transducers
• Sensors are transducers that transform a variable that is measureable based on sensor constitutive relationships
• Have minimal consumption of power
• can be modeled as the ff:
  • They accept one effort or flow as input (e.g., electrical voltage)
  • They transform it to a different effort or flow (e.g., angular velocity)
  • The load attached to the output determines the complementary effort or flow (e.g., torque)
  • The actuator or sensor transforms the complementary effort or flow that is attached to its output into the complementary input effort or flow (e.g., motor current)

Amplifiers & Modulators

• 3-port devices Operational Amplifiers, Electric Motor controllers and Hydraulic Servovalves
• Modulating input is not affected by the power consumption of the output, impedance is not affected by the impedance of the load connected to the output
  

Hydraulic Servovalves

• Servovalves are variable resistance 3-port
• Regulate the flow or output pressure by varying the resistance to flow
• Servovalve includes a control spool that is moved by an electro-magnetic actuator

Connecting Subsystems

• One physical system is connected by transducer element(s) to another type of physical system
• State equations can be expressed in terms of a single set of variable types

Mixed systems

• Have different subsystems that have particular loop and node relationships
• Interacting subsystems connected by elements that describe the interaction. These may be transformations of energy in the case of physical subsystems
• May have equations relating to system variables, without physical relationships, but rather technological attributes such as duration, resource allocation, and cost