System Models Mathematical Models Mechanical System Building Blocks

System Models Mathematical Models Mechanical System Building Blocks Electrical System Building Blocks Fluid System Building Blocks Thermal Systems Building Blocks

Mathematical Models • Think how systems behave with time when subject to some disturbances. • In order to understand the behaviour of systems, mathematical models are required. • Mathematical models are equations which describe the relationship between the input and output of a system. • The basis for any mathematical model is provided by the fundamental physical laws that govern the behaviour of the system.

Building Blocks • Systems can be made up from a range of building blocks. • Each building block is considered to have a single property or function. • Example: an electric circuit system which is made up from blocks which represent the behaviour of resistance, capacitance, and inductor, respectively. • By combining these building blocks a variety of electrical circuit systems can be built up and the overall inputoutput relationship can be obtained. • A system built in this way is called a lumped parameter system.

Mechanical System Building Blocks • Basic building block: spring, dashpots, and masses. • Springs represent the stiffness of a system • Dashpots represent the forces opposing motion, for example frictional or damping effects. • Masses represent the inertia or resistance to acceleration. • Mechanical systems does not have to be really made up of springs, dashpots, and masses but have the properties of stiffness, damping, and inertia. • All these building blocks may be considered to have a force as an input and displacement as an output.

Rotational Systems • The mass, spring, and dashpot are the basic building blocks for mechanical systems where forces and straight line displacements are involved without any rotation. • If rotation is involved, then the equivalent three building blocks are a torsional spring, a rotary damper and the moment of inertia (i. e. the inertia of a rotating mass). • With a torsional spring the angle rotated is proportional to the torque: T = k. • With a rotary damper a disc is rotated in a fluid and the resistive torque T is proportional to the angular velocity . • The moment of inertia block exhibit the property that the greater the moment of inertia J the greater the torque needed to produce an angular acceleration

Stiffness of a Spring • Stiffness of a spring is described as the relationship between the force F used to extend or compress a spring and the resulting extension or compression x. • In the case of spring where the extension or compression is proportional to the force (linear spring): F = kx, where k is a constant, the bigger the value of k the greater the forces have to be to stretch or compress the spring and so the greater the stiffness. F Spring x

Translational Spring, k (N) x(t) Fa(t)

Rotational Spring, ks (N-m-sec/rad) (t) Fa(t) ks

Dashpot • The dashpot block represents the types of forces experienced when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces. • The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder. • Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.

Translational Damper, Bv (N-sec) x(t) Fa(t) Bm

Rotational Damper, Bm (N-m-sec/rad) (t) Fa(t) Bm

Mass • The mass exhibits the property that the bigger the mass the greater the force required to give it a specific acceleration. • The relationship between the force F and acceleration a is Newton’s second law as shown below. • Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot. Force Mass Acceleration

Mechanical Building Blocks Building Block Spring Dashpot Mass Spring Damper Moment of inertia Equation Translational F = kx F = c dx/dt F = m d 2 x/dt 2 Rotational T = k T = c d /dt T = J d 2 /dt 2 Energy representation E = 0. 5 F 2/k P = cv 2 E = 0. 5 mv 2 E = 0. 5 T 2/k P = c 2 P = 0. 5 J 2

Building Mechanical Blocks Output, displacement • Mathematical model of a machine mounted on the ground Mass Ground Input, force

Building Mechanical Blocks Moment of inertia Torque • Mathematical model rotating a mass Torsional resistance Block model Shaft Physical situation of a

Electromechanical Analogies • From Newton’s law or using Lagrange equations of motions, the secondorder differential equations of translational-dynamics and torsional-dynamics are found as

Electrical System Building Blocks • The basic building blocks of electrical systems are resistance, inductance and capacitance.

Resistance, R (ohm) i(t) v(t) R

Inductance, L (H) i(t) v(t) L

Capacitance, C (F) i(t) v(t) C

For a series RLC circuit, find the characteristic equation and define the analytical relationships between the characteristic roots and circuitry parameters.

Fluid System Building Blocks • The basic building blocks of fluid systems are the volumetric rate of flow q and the pressure difference. Input Output Volumetric rate of flow Pressure difference Fluid system can be divided into two types: hydraulic and pneumatic. Hydraulic resistance is the resistance to flow of liquid as the liquid flow through valves or changes in pipe diameter takes place. p 1 - p 2 is pressure difference R is the hydraulic resistance q is the volumetric rate of flow

• Hydraulic capacitance is the term used to describe energy storage with a liquid where it is stored in the form of potential energy. A height of liquid in a container is one form of such a storage. For such capacitance, the rate of change of volume V in the container (d. V / dt) is equal to the difference between the volumetric rate at which liquid enters the container q 1 and the rate at which it leaves q 2.

• Hydraulic inertance is the equivalent of inductance in electrical systems or a spring in mechanical systems. To accelerate a fluid and so increase its velocity a force is required. F 1=p 1 A Mass m F 2=p 2 A L

• With pneumatic systems the three basic buildings blocks are as with hydraulic systems, resistance, capacitance, and inertance. However, gasses differ from liquids in being compressible.

A fluid system q 1 h q 2

Thermal System Building Blocks • There are only two basic building blocks for thermal systems: resistance and capacitance. • There is a net flow of heat between two points if there is a temperature difference between them. • The value of the resistance depends on the mode of heat transfer.

Thermal System T TL q