Lecture 4 Machine Construction - Most Important Physical Quantities PDF

Introduction to Machine Construction - Most important physical quantities in machine construction 1/13 Most important physical quantities in machine construction Physical quantities: we deal with two types of quantities (variables): scalar and vector variables. Scalar variables have only magnitud...

Introduction to Machine Construction - Most important physical quantities in machine construction 1/13 Most important physical quantities in machine construction Physical quantities: we deal with two types of quantities (variables): scalar and vector variables. Scalar variables have only magnitude, for example: length, mass, temperature, time. Vector variables have magnitude and direction, for example: speed, force, torque. The direction of the vector is defined by the angles of the force witch each axis. The vector variables are usually represented using bold symbols with arrows on top. Several forces can act on a body or point, each force having different direction and magnitude. In engineering the focus is on the resultant force acting on the body. The resultant of concurrent forces (acting in the same plane) can be found using the parallelogram law, the triangle rule or the polygon rule. Two or more vectors (e.g. velocities or forces) are concurrent if their direction crosses through a common point. For example, two concurrent forces F1 and F2 are acting on the same point P. In order to find their resultant R, we can apply either the parallelogram law or triangle rule. The resultant force is the vector sum between the components: 𝑅⃗ = 𝐹⃗ + 𝐹⃗ Introduction to Machine Construction - Most important physical quantities in machine construction 2/13 Forces - a force is a push or pull upon an object resulting from the object's interaction with another object. Whenever there is an interaction between two objects, there is a force upon each of the objects. When the interaction ceases, the two objects no longer experience the force. Forces only exist as a result of an interaction. For simplicity sake, all forces (interactions) between objects can be placed into two broad categories: 1. contact forces, and 2. forces resulting from action-at-a-distance Contact forces are those types of forces that result when the two interacting objects are perceived to be physically contacting each other. Examples of contact forces include frictional forces, tensional forces, normal forces, air resistance forces, hydrodynamic forces, buoyancy forces. Contact forces are always associated with a surface and they are often called surface forces. Action-at-a-distance forces are those types of forces that result even when the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation. Examples of action-at-a-distance forces include gravitational forces and very important inertial forces (both are called mass forces). Electric forces are action-at-a-distance forces. For example, the protons in the nucleus of an atom and the electrons outside the nucleus experience an electrical pull towards each other despite their small spatial separation. And magnetic forces are action-at-a-distance forces. Forces and resultants or components F – force, Fy, Fx - components or resultants, ∝ − angle 𝐹 = 𝐹 ∙ cos ∝, 𝐹 = 𝐹 ∙ sin ∝, |𝐹| = 𝐹 +𝐹 , 𝛼 = tan Force is a quantity that is measured using the standard metric unit known as the Newton: ∙ 1 Newton = Introduction to Machine Construction - Most important physical quantities in machine construction 3/13 Law of forces according to Newtonian mechanics 1. First law: objects continue to move in a state of constant velocity unless acted upon by an external net force (resultant force). Remember that zero velocity is include in the term “constant velocity” 2. Second Law: the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force. 𝑑𝑝 𝑑(𝑚𝑣) 𝐹= = 𝑑𝑡 𝑑𝑡 The second law can also be stated in terms of an object's acceleration a. Since Newton's second law is valid only for constant-mass systems, m can be taken outside the differentiation operator by the constant factor rule in differentiation: 𝑑𝑣 𝐹=𝑚 = 𝑚𝑎 𝑑𝑡 3. The third law: all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force FA on a second object B, then B simultaneously exerts a force FB on A, and the two forces are equal in magnitude and opposite in direction: 𝐹 = −𝐹 Introduction to Machine Construction - Most important physical quantities in machine construction 4/13 Centrifugal force: In Newtonian mechanics, the centrifugal force is an inertial force that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is parallel to the axis of rotation and passing through the coordinate system's origin. If the axis of rotation passes through the coordinate system's origin, the centrifugal force is directed radially outwards from that axis. The magnitude of centrifugal force F on an object of mass m at the distance r from the origin of a frame of reference rotating with angular velocity ω is: 𝐹 = 𝑚𝜔 𝑟 where: m – mass of the object [kg], 𝜔 – angular velocity [1/s] Normal force: - Force acting as a reaction to contact. Direction is normal to the surface of contact. Magnitude enough to cancel the weight so object doesn't go through the plane. Here: 𝐹 ⃗ = 𝑊⃗ cos(𝛼) = 𝑚𝑔⃗ cos(𝛼) Object slides down plane with remaining force: 𝑚𝑔⃗ sin(𝛼) Friction force: f in the above picture is a friction force - The friction force is the force exerted by a surface as an object moves across it or makes an effort to move across it. There are at least two types of friction force - sliding and static friction. Though it is not always the case, the friction force most frequently opposes the motion of an object. For example, if a book slides across the surface of a desk, then the desk exerts a friction force in the opposite direction of its motion. Friction results from the two surfaces being pressed together closely, causing intermolecular attractive forces between molecules of Introduction to Machine Construction - Most important physical quantities in machine construction 5/13 different surfaces. As such, friction depends upon the nature of the two surfaces and upon the degree to which they are pressed together: 𝑓 = 𝜇 ∙𝐹 ⃗ where: µ - is the coefficient of friction, which is an empirical property of the contacting materials Moment of force – generally a moment is an expression involving the product of a distance and physical quantity, the moment of force acting on an object, often called torque, is the product of the force and the distance to the object (i.e., the reference point). Moment = Force x Perpendicular Distance : 𝑀 = 𝐹 ∙ 𝑑 Unit: [𝑁] ∙ [𝑚] = [𝑁𝑚] The force is not always perpendicular to the given lever arm. In this case, the correct perpendicular distance must be determined. (By the way, the perpendicular distance is also the SHORTEST distance between the force and the pivot point.) Calculating the perpendicular (shortest) distance; Ship application – righting moment Introduction to Machine Construction - Most important physical quantities in machine construction 6/13 A couple of forces - a couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment. The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. The forces have a turning effect or moment called a torque about an axis which is normal (perpendicular) to the plane of the forces. The SI unit for the torque of the couple is newton metre. If the two forces are F and −F, then the magnitude of the torque is given by the following formula: 𝑀 =𝐹∙𝑑 d is the perpendicular distance between the forces, sometimes called the arm of the couple Introduction to Machine Construction - Most important physical quantities in machine construction 7/13 Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed: 𝐹 𝑘𝑔 ∙ 𝑚 1 𝑝= ∙ = [𝑃𝑎]𝑝𝑎𝑠𝑐𝑎𝑙 𝐴 𝑠 𝑚 Hydrostatic pressure: 𝑝 = 𝜌∙𝑔∙ℎ+𝑝  - density of fluid [kg/m3] g – gravity acceleration h - is the depth (or height) of the liquid pa – atmospheric pressure (boundary pressure) Distribution of hydrostatic pressure on ship hull effects: 1. hull deflection or damage 2. buoyancy force Archimedes principle: upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces: 𝐹⃗ = 𝜌 ∙ 𝑔 ∙ 𝑉 where: FB – buoyancy force, [N];  - fluid density, (e.g. water) [kg/m3]; V – volume of the body submerged [m3]. Explanation where buoyancy force comes from: Introduction to Machine Construction - Most important physical quantities in machine construction 8/13 Equilibrium of gravitational and buoyancy forces: 1. Archimedes principle and 2. the Newton’s first law of mechanics – a ship can float. Introduction to Machine Construction - Most important physical quantities in machine construction 9/13 Dynamic pressure If fluid is static and not flowing, the measured pressure is the same in all directions (e.g. hydrostatic pressure). If the fluid is moving, the measured pressure depends on the direction of motion. The dynamic pressure is the kinetic energy per unit volume of a fluid, and can be calculated as follows: 1 𝑝= ·𝜌·𝑣 2 v – is the velocity of flow [m/s]. Lift and drag forces – they are contact type forces, that are created thanks to the dynamic pressure (precisely changes of the dynamic pressure). 𝟏 𝐿 (𝑙𝑖𝑓𝑡 𝑓𝑜𝑟𝑐𝑒) = 𝐶 ∙ ∙ 𝝆 ∙ 𝑽𝟐 ∙ 𝐴 𝟐 𝟏 𝐷 (𝑑𝑟𝑎𝑔 𝑓𝑜𝑟𝑐𝑒) = 𝐶 ∙ ∙ 𝝆 ∙ 𝑽𝟐 ∙ 𝐴 𝟐 where: CL, CD – coefficients of drag and lift, A – area of the wing, foil, blade, element, etc. Hydro / aerodynamic foil, aviation foil - the shape of the cross-section of wings, propeller blades and airplane tail blades, helicopter rotor blades, turbines, etc. flow machines, propeller wings, car spoiler, yacht sail, etc., all these elements work using the lift and drag forces: Introduction to Machine Construction - Most important physical quantities in machine construction 10/13 Work, Power and Energy Work - when a force F acting on an object displaces it by a certain distance d, the force F is said to have done work W ∙ 𝑊 =𝐹∙𝑑 ∙ 𝑚 = [𝑁𝑚] = [𝐽] be careful the same unit as moment of force Work in rotational motion: 𝑊 =𝑀∙𝜑 where: M – moment of force [Nm], φ – angular displacement [rad] Power is the rate at which work is performed: ∆𝑊 𝑃= [𝑊] ∆𝑡 where: ∆𝑊 – change of work, ∆𝑡 – change of time And for linear changes: ∙ = = 𝐹 ∙ 𝑣 – important relation for a ship powering performance, v – speed [m/s] Power in rotational motion – if moment of force is transmitted at an angular speed  [rad/s] : 𝑃 =𝑀∙𝜔 In the situation when power is transmitted in rotation motion we refer to moment of force as torque and denote it by T or Q 𝑃 = 𝑇 ∙ 𝜔 = 𝑻 ∙ 2𝜋𝑛 𝑜𝑟 𝑃 = 𝑸 ∙ 2𝜋𝑛 where: n – number of revolutions per second [1/s], if rotational speed is expressed in rpm (rev per minute), then: 2𝜋 ∙ 𝑟𝑝𝑚 2𝜋 ∙ 𝑟𝑝𝑚 𝑃 =𝑻∙ 𝑜𝑟 𝑃 = 𝑻 ∙ 60 60 Introduction to Machine Construction - Most important physical quantities in machine construction 11/13 Efficiency: The ratio of power output to power input in known as the efficiency of a machine. It is always less than unity and usually is represented as percentage. It is denoted by a Greek letter η : 𝑃𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦, 𝜂= 𝑃𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 Since a machine does not contain a source of energy, nor can it store energy, from conservation of energy the power output of a machine can never be greater than its input, so the efficiency can never be greater than 1. All real machines lose energy to friction; the energy is dissipated usually as heat. Therefore their power output is less than their power input 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 − 𝑙𝑜𝑠𝑠 Introduction to Machine Construction - Most important physical quantities in machine construction 12/13 Introduction to Machine Construction - Most important physical quantities in machine construction 13/13 Energy - the capacity for doing work. It may exist in potential, kinetic, thermal, electrical, chemical, nuclear or other various forms. Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. Gravitational Potential Energy: Energy stored by an object as it gains elevation within a gravitational field 𝐸 = 𝑚𝑔ℎ m – mass of an object, g – gravitational acceleration (9.83 m/s 2), h – elevation of an object Elastic Potential Energy: Energy stored by an object when it is stretched or bent: 1 𝐸= 𝑘𝑥 2 k – spring constant [N/m], x – spring stretch/compression/distance [m] Kinetic Energy: Energy associated with an object’s motion 1 𝐸= 𝑚𝑣 2 v – speed of object Kinetic energy in rotational motion: 1 𝐸= 𝐼𝜔 2 where:  - rotational speed, I – mass moment of inertia around the axis of rotation, 𝐼 = 𝑚𝑟 ; r – distance from the center of gravity of a body from the axis of rotation, [kg·m2] energy unit – the same as work, electricity - The energy unit used for everyday electricity, particularly for utility bills, is the kilowatt-hour (kWh); one kWh is equivalent to 3.6×106 J (3600 kJ or 3.6 MJ). Electricity usage is often given in units of kilowatt-hours per year (kWh/yr). This is actually a measurement of average power consumption, i.e., the average rate at which energy is transferred. One kWh/yr is about 0.11 watts.