Chapter 3: Fundamentals of Mechanics and Heat. 1/11/00 Electromechanical Dynamics 1


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1 Chapter 3: Fundamentals of Mechanics and Heat 1/11/00 Electromechanical Dynamics 1
2 Force Linear acceleration of an object is proportional to the applied force: F = m a x(t) F = force acting on an object [N] m = mass of the object [kg] a = acceleration of the object [m/s 2 ] F m 1/11/00 Electromechanical Dynamics 2
3 Torque Torque is produced when a force exerts a twisting action on an object, tending to make it rotate Torque is the product of the force and the perpendicular distance to the axis of rotation: T = F r sinφ T = torque [Nm] axis of rotation F = applied force [N] φ r = radius [m] F r φ = angle of applied force T Example calculate the braking force needed for a motor with a 1 m diameter braking drum that develops a 150 Nm starting torque 1/11/00 Electromechanical Dynamics 3
4 Work Work is done whenever a force F moves an object a distance d in the direction of the force: W = F d W = work [J] F = force [N] d = distance [m] Example calculate the work done on a mass of 50 kg that is lifted to a height of 10 m 1/11/00 Electromechanical Dynamics 4
5 Work Work is performed on a rotating object by a torque when there is an angular rotation: W = T δ T = torque [N m] δ = angular displacement [radians] Example calculate the work performed by an electric motor that develops a 100 Nm torque at 1750 rpm on a pulley that lifts a mass in 25 s 1/11/00 Electromechanical Dynamics 5
6 Power Power is the rate of which work is performed P = power [W] W = work [J] t = time to do the work [s] Common units are kw and hp 1 hp = 746 W = kw Example calculate the power developed by an electric motor that lifts a mass of 500 kg to a height of 30 m in 12 s P = W t 1/11/00 Electromechanical Dynamics 6
7 Power The mechanical power output of a motor depends on the torque and rotational speed: 2 n P = π nt T P = mechanical power [W] T = torque [N m] n = speed [rpm] In more general terms: ω = speed [radian/s] Example P = ω T Calculate the power output on a motor rotating at 1700 rpm during a prony brake test the two spring scales indicate 25 N and 5 N, respectively 1/11/00 Electromechanical Dynamics 7
8 Transformation of Energy Forms of energy include: mechanical energy (potential and kinetic) thermal, chemical, and atomic energy electrical energy (electric and magnetic) Energy can be transformed from one form to another the term machine is the generic term for those devices that convert power from one form of energy into another conservation of energy: can not be created or destroyed conservation of power: power in plus stored released energy equals power out plus energy stored and power losses 1/11/00 Electromechanical Dynamics 8
9 Machine Efficiency Whenever energy is transformed, the output is always less than the input because all machines have losses: P i = Po + P P o = output power P i = input power P loss = power losses P The efficiency of a machine is defined as: η = Pi Po η = percent efficiency η = Po + Ploss Alternate forms of the definition: Pi Ploss η = P P i Machine 100% i 1/11/00 Electromechanical Dynamics 9 P o P Loss o 100% 100% loss
10 Kinetic Energy Kinetic energy is stored in moving objects energy must be added to an object to make it move 1 2 For objects with linear motion: E k = m v E k = kinetic energy [J] m = object s mass [kg] v = object s velocity [m/s] For objects with rotational motion: J = moment of inertia [kg m 2 ] ω = angular velocity [rad/s] 2 E k = J m ω J v ω 1/11/00 Electromechanical Dynamics 10
11 Inertia, Torque, and Speed To change the speed of a rotating object, a torque must be applied for a period of time The rate of change of the speed (angular acceleration) depends upon the inertia as well as the torque: tt ω = J ω = change in angular velocity tt n= 9.55 J t = time interval of applied torque T = torque J = moment of inertia Example a flywheel with an 10.6 kg m 2 inertia turns at 60 rpm. How long must a 20 Nm torque be applied to increase the speed to 600 rpm? 1/11/00 Electromechanical Dynamics 11
12 Speed of a Motor / Load System An electric motor applies a torque on the shaft A load applies a countertorque on the shaft The net torque will accelerates or decelerate the shaft: T net = T m T ld Motor Load T m ω T ld 1/11/00 Electromechanical Dynamics 12
13 Speed of a Motor / Load System Torquespeed characteristics of an electric induction motor and a fan load Max torque 20 kn m Torque 10 motor fan operating point zero net torque constant speed rpm Speed 1/11/00 Electromechanical Dynamics 13
14 Directional Flow of Power Power supplied to the mechanical system applied torque is in the same direction as rotation Power absorbed from the mechanical system applied torque is in the opposite direction of rotation Motor Load T m ω Power T ld Motor Load T m ω T ld Power 1/11/00 Electromechanical Dynamics 14
15 Heat Heat is a form of energy and the SI unit is the joule energy of vibrating atoms/molecules thermal potential is expressed as a temperature Thermal energy systems are analogy to DC circuits heat [J] electrical charge temperature [K, C] voltage heat flow [W] current thermal mass [J/ C] capacitance thermal conductivity [W/(m C)] conductance thermal insulation resistance 1/11/00 Electromechanical Dynamics 15
16 Temperature The temperature depends upon the received heat, mass, and material characteristics: Q = m c t Q = change in the quantity of heat [J] t m = mass of object [kg] c = specific heat [J/(kg C)] t = change in temperature [K, C] Example for a water heater, calculate the heat required to raise the temperature of 200 L of water from 10 C to 70 C assuming no losses (c H2O = 4180 J/kg C; 1 L H2O = 1 kg) Q m 1/11/00 Electromechanical Dynamics 16
17 Temperature Kelvin temperature scale is a measure of the absolute value Thermal mass is the mass of object times specific heat 1/11/00 Electromechanical Dynamics 17
18 Heat Transfer by Conduction By heating one end of a metal bar its temperature rises due to increase atomic vibrations the vibrations are transmitted down the bar the temperature at the other end of the bar rises Thermal conduction is similar to the flow of electric current: λ A P= ( t1 t P = heat transmitted [W] 2) d (t 1  t 2 ) = temperature difference across object [ C] d = thickness of object [m] d A A = cross sectional area [m 2 ] P λ = thermal conductivity [W/(m C)] λ P t 1 t 2 1/11/00 Electromechanical Dynamics 18
19 Thermal Convection A continual current of fluids that provide cooling is called natural convection Fluids, like air, oil, and water, in contact with hot surfaces warm up and become lighter lighter fluids rise cooler fluids replace the rising fluids the warm fluids cool and sink The convection process can be accelerated by employing a fan or pump to create a rapid circulation called force convection 1/11/00 Electromechanical Dynamics 19
20 Heat Loss by Convection The heat lost by natural air convection is: P = heat loss by convection [W] A = surface area of the object [m 2 ] t 1 = surface temperature of the object [ C] t 2 = ambient temperature of the surrounding air [ C] Example a totally enclosed motor has an external surface area of 1.2 m 2 when operating at fullload, the surface temperature rises to 60 C in an ambient of 20 C calculate the heat loss by natural convection ( ) P = 3 A t t 1/11/00 Electromechanical Dynamics
21 Heat Loss by Convection The heat loss by forced air convection is: P = heat loss by convection [W] V a = volume of cooling air [m 3 /s] t 1 = temperature of the incoming (cool) air [ C] t 2 = temperature of the outgoing (warm) air [ C] Example ( t ) P = 1280Va t a fan rated at 3.75 kw blows 240 m3/min of air through a 750 kw moter to carry away the heat if the inlet temperature is 22 C and the outlet temperature is 31 C, estimate the losses in the motor 1 2 1/11/00 Electromechanical Dynamics 21
22 Radiant Heat Radiant heat energy (electromagnetic wavesinfrared spectrum) can pass through empty space or vacuum All objects radiate heat energy as a function of temperature All objects absorb radiant energy from other surrounding objects An object reaches a temperature equilibrium point when it is the same temperature as that of its surroundings it radiates as much energy as it receives and the net radiation is zero 1/11/00 Electromechanical Dynamics 22
23 Radiant Heat Loss The heat that an object looses 4 by radiation: P = k A T 1 T2 P = heat radiated [W] A = surface area of object [m 2 ] T 1 = object s temperature [K] T 2 = temperature of surrounding objects [K] k = constant that depends on the nature of the object s surface Example ( 4 ) Type of surface k [W/(m 2 K 4 ) polished silver bright copper oxidized copper aluminum paint oxidized iron insulation enamel paint calculate the heat loss by radiation for the motor from the natural convection example, which has an enamel surface 1/11/00 Electromechanical Dynamics 23
24 Homework Problems: 35, 39, 312, 317, and /11/00 Electromechanical Dynamics 24
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