LECTURE #5 System Modeling& Responses MECHATRONIC SYSTEM DESIGN [ENT 473] LECTURE #5 System Modeling& Responses HASIMAH ALI Programme of Mechatronic Engineering,. EXT: 5205, hashimah@unimap.edu.my

Outlines Plant Modeling Mechanical System Electrical System Hydraulic System Pneumatic System Thermal System

Modeling And Simulation Plant Modeling Modeling And Simulation Modeling is the process of representing the behavior of real system (physical system) by collection of mathematical equation. The term mathematical model, in the control engineering perspective, implies a set of differential equations that describe the dynamic behavior of a process.

Plant Modeling The set of differential equations that describe the behavior of physical systems are typically obtained by utilizing the physical laws of the process.

Plant Modeling How to obtain the mathematical models? Physical laws of the process  Differential equations Electrical system  Kirchhoff's laws Mechanical system  Newton's laws Fluid system  ? Thermal System  ?

Plant Modeling Example: Gantry Crane System

Mechanical System Basic building blocks representing mechanical systems are: Spring Dashpots/ damper Masses All these blocks have a force as an input and a displacement as output.

Mechanical System For mechanical system, the basic building block are springs, dashpots and masses.

State Space Representation Mechanical System Example: Spring-Mass-Damper State Space Representation

Mechanical System Exercise 1 A spring-mass-damper is shown in figure below, obtain the: transfer function, state space representation and block diagram representation

Mechanical System Exercise 1 Derive a model for the mechanical system represented by the system of mass, spring and dashpot given in Figure below. The input to the system is the force F and the output is the displacement y.

Mechanical System Example: Suspension System A quarter car model x1 m1 b k1 k2 W x2 x1 A quarter car model

Mechanical System Example: Rotational system For rotationals system, the basic building block are a torsional-spring, a rotary damper and the moment of inertia.

Rotational System

Rotational System Rotational system

Rotational System Example: Develop a model of a system shown in Figure, of the rotation of the disk as a result of twisting a shaft.

Electrical System For electrical system, the basic building block are a resistor, inductor and capacitor.

Electrical System Capacitor For a capacitor, the potential difference v across it depends on the charge q on the capacitor plates with v = q/C, where C is the capacitance. Thus: Since current / is the rate of movement of charge:

Electrical System Example: RLC Circuit.

Electrical System Exercise 2 Develop a model for a d.c. permanent magnet motor relating the current through the armature to the applied voltage.

Analogies: Electrical and Mechanical The mechanical analogue of the resistor is the dashpot. In this case the current can be analogues to the force , then the potential difference is analogues to the velocity and the dashpot constant c to the inverse of R. These analogies between current and force, potential difference and velocity, hold also for the other building blocks. Spring is analogues to inductance, mass is analogues to capacitance.

Hydraulic System In fluid systems there are 3 basic building blocks are resistance, capacitance and inductance. Volumetric rate of flow q is equivalent to current. Pressure difference (p1-p2) is equivalent to electrical potential difference. Fluid systems can be considered to fall into 2 categories: Hydraulic Pneumatic

Hydraulic System Hydraulic Resistance Hydraulic resistance R is the resistance to flow which occurs when a liquid flows from one diameter pipe to another and is defined as being given by the hydraulic equivalent of Ohm's law:

Hydraulic System Hydraulic Capacitance Hydraulic capacitance C is the term used to describe energy storage where the hydraulic liquid is stored in the form of potential energy. The rate of change of volume V of liquid stored is equal to the difference between the volumetric rate at which liquid enters the container q1 and the rate at which it leaves q2, i.e.

Hydraulic System Hydraulic Capacitance p1-p2=ρgh The pressure difference between the input and output is: p1-p2=ρgh Hence, substituting for h gives: The hydraulic capacitance C is defined as:

Hydraulic System Hydraulic inertance Consider a block of liquid of mass. The net force acting of the liquid will cause the mass to accelerate with an acceleration a,

Hydraulic System Example 3 Develop a model for the hydraulic system shown in Figure below where there is a liquid entering a container at one rate q1 and leaving through a valve at another rate q2. For capacitance: For resistance:

Pneumatic System With pneumatic systems the three basic building blocks are as with hydraulic system; resistance, capacitance and inertance. However, gasses differ from liquids in being compressible. A change in volume causes a change in pressure and a change in density.

Pneumatic System p1-p2 =R dm/dt Pneumatic resistance R is defined in terms of the mass rate flow dm/dt and the pressure difference p1-p2. p1-p2 =R dm/dt Pneumatic capacitance is due to the compressibility of the gas and is comparable to the way in which the compression of a spring stores energy. If there is a mass rate of flow dm1/dt entering the container of volume V and a mass rate of flow of dm2/dt leaving it, the rate at which the mass in the container is changing is (dm1/dt-dm2/dt)

for idea gas, with consequent Pneumatic System Rate of change of mass in container: Since, and, for idea gas, with consequent

Pneumatic System thus, Then, rate of change of mass in container: Pneumatic capacitance due to change in volume. Pneumatic capacitance due to compressibility of the gas

Pneumatic System hence , or,

Pneumatic System Analysis of Pneumatic System Mass flow rate : All the gas remains in the container:

Thermal System There are only two basic building blocks for thermal systems; Resistance Capacitance.

Thermal System Thermal Resistance If q is the rate of flow of heat and (T1-T2) the temperature difference, then: (a) Thermal resistance, (b) thermal capacitance

Thermal System Thermal Resistance The value of resistance depends on the mode of transfer: Unidirectional conduction: Hence Convection: ,hence

Thermal System Thermal Capacitance Is the measure of internal energy storage. Rate of change of internal energy = q1-q2. Where m is the mass and c is the specific heat capacity. Hence, where C=mc

Thermal System Example of Thermal Systems Consider a thermometer at temperature T which has just been inserted into a liquid at temperature TL. If the thermal resistance to heat flow from the liquid to the thermometer is R,

Thermal System Example of Thermal Systems Since there is only a net flow of heat from the liquid to the thermometer, q1=q and q2=0. Thus,

Further Readings Godfrey C. Onwubolu, “Mechatronics: Principles and Applications”. Chapter 7 D. G. Alciatore and M. B. Histand, "Introduction to Mechatronics and Measurement Systems Chapter 5