Lecture 8(Hydraulic , Pneumatic and Thermal Systems)Mathematical Modeling

Hydraulic Systems Works due to liquid pressure difference Fluid used is relatively incompressible such as petroleum base oils or certain non-inflammable synthetic fluids Advantages: (i)The hydraulic motors can be made much smaller in physical size than an electric motor for the same power output.

(ii)Hydraulic components are rapid acting and more rugged compared to the corresponding electrical components.

Disadvantages:(i)Have the inherent problems of leak and sealing them against foreign particles,

(ii)Have operating noise and the tendency to become sluggish at low temperatures because of increase viscosity of the fluid

(iii) Hydraulic lines are not as flexible as electric cables.

Applications: Power steering and brakes in automobiles, the steering mechanisms of large ships, the control of large machine tools etc.

Hydraulic Systems

Types of Hydraulic output devices: Those whose output is rotary motion , known as hydraulic motors. Those whose output is translational motion, known as linear hydraulic actuators.

Hydraulic Linear Actuator: A piston device To find Transfer Function

Let,q= volumetric flow rate in the power pistonp= differential pressure across the pistonx= small spool displacement

The relationship between q, x and p may be written as:q=f(x , p)Expanding the above equation into Taylors series about the normal operating point (q o, x o , p o ) and neglecting all the terms of second and higher derivatives, we get:

But , here at normal operating point: q o = 0, x o = 0, p o = 0 Therefore the above equation may be written as:

q = K1 x K2 p

Where, ,

Or, K2 p = K1 x q -----------(i)

The rate of oil flow is given by: -----------(ii)Where A is the area of the pistonApplying force balance on the piston we have:-----------(iii)

Applying Laplace transformation and then solving equations (i), (ii) and (iii) we have: is the required transfer function of the hydraulic linear actuator.

Pneumatic Systems works due to pressure difference of air or any other gasAdvantages:Air is non-inflammable Have almost negligible viscosity compared to the high viscosity of hydraulic fluids

Disadvantages: Pneumatic Systems have considerable amount of compressibility flow so that such systems are characterized by longer time delays while the high incompressibility of a hydraulic fluid causes the force wave to travel faster and therefore hydraulic systems have a shorter response time.Applications: In process control field, in guided missiles, aircraft systems etc.

Pneumatic Bellows: A device to measure pressureIts action is similar to that of a spring. An increase in the pressure within the bellows result in an increase in the separation between the input and output surfaces.

To find Transfer FunctionPneumatic BellowsLet, Pressure difference inside the bellow and the surrounding Area of each flat surfaceStiffness of the bellows = K

Force opposing the separation Force acting to separate the two surfacesTherefore in equilibriumTaking Laplace both sidesis the required transfer function of the Bellows.

Thermal Systems: E g. Water Heating SystemTo find Transfer Function

Assumptions:The temperature of the medium is uniform. The tank is insulated from the surrounding atmosphere.Let, Steady state heat input rate from the electric heater = H Heat storage rate of the liquid = Heat outflow rate = Steady liquid flow rate (mass/time)= Q Specific heat of the liquid = s Mass of liquid in the tank = M Rise of temperature of the liquid in the tank = Rise of temperature of the out flowing liquid =

Applying heat balance equation we have

Heat input rate from the electric heater =Heat storage rate of the liquid in the tank +Heat outflow rate from the tank

Taking Laplace Transformation of both sides we have, called thermal resistance, called thermal capacitance

is the required transfer function of the thermal system

***