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Piston Motion Equations
The reciprocating motion of a non-offset piston connected to a rotating crank through a connecting rod (as would be found in internal combustion engines) can be expressed by equations of motion. This article shows how these equations of motion can be derived using calculus as functions of angle ''(angle domain)'' and of time ''(time domain)''. Crankshaft geometry The geometry of the system consisting of the piston, rod and crank is represented as shown in the following diagram: Definitions From the geometry shown in the diagram above, the following variables are defined: : l rod length (distance between piston pin and crank pin) : r crank radius (distance between crank center and crank pin, i.e. half stroke) : A crank angle (from cylinder bore centerline at TDC) : x piston pin position (distance upward from crank center along cylinder bore centerline) The following variables are also defined: : v piston pin velocity (upward from crank center along cylinder bore centerlin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Reciprocating Motion
Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single reciprocation cycle are called strokes. A crank can be used to convert into reciprocating motion, or conversely turn reciprocating motion into circular motion. For example, inside an internal combustion engine (a type of reciprocating engine), the expansion of burning fuel in the cylinders periodically pushes the piston down, which, through the connecting rod, turns the crankshaft. The continuing rotation of the crankshaft drives the piston back up, ready for the next cycle. The piston moves in a reciprocating motion, which is converted into the circular motion of the crankshaft, which ultimately propels the vehicle or does other useful work. The reciprocating motion of a pump piston is close to but different from, sinusoidal A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bore (engine)
In a piston engine, the bore (or cylinder bore) is the diameter of each cylinder. Engine displacement is calculated based on bore, stroke length and the number of cylinders: displacement = The stroke ratio, determined by dividing the bore by the stroke, traditionally indicated whether an engine was designed for power at high engine speeds ( rpm) or torque at lower engine speeds. The term "bore" can also be applied to the bore of a locomotive cylinder or steam engine pistons. In steam locomotives The term bore also applies to the cylinder of a steam locomotive or steam engine. Bore pitch Bore pitch is the distance between the centerline of a cylinder bore to the centerline of the next cylinder bore adjacent to it in an internal combustion engine. It's also referred to as the "mean cylinder width", "bore spacing", "bore center distance" and "cylinder spacing". The bore pitch is always larger than the inside diameter of the cylinder (the bore and piston diameter) sinc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Completing The Square
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form to the form for some values of and . In terms of a new quantity , this expression is a quadratic polynomial with no linear term. By subsequently isolating and taking the square root, a quadratic problem can be reduced to a linear problem. The name ''completing the square'' comes from a geometrical picture in which represents an unknown length. Then the quantity represents the area of a square of side and the quantity represents the area of a pair of Congruence (geometry), congruent rectangles with sides and . To this square and pair of rectangles one more square is added, of side length . This crucial step ''completes'' a larger square of side length . Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian Empire, Old Babylonian clay tablets dating from 1800–1600 BCE, and is still taught in elementary algebra c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Position (geometry)
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a Point (geometry), point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin (mathematics), origin ''O'', and its Direction (geometry), direction represents the angular orientation with respect to given reference axes. Usually denoted x, r, or s, it corresponds to the straight line segment from ''O'' to ''P''. In other words, it is the displacement (vector), displacement or translation (geometry), translation that maps the origin to ''P'': :\mathbf=\overrightarrow. The term position vector is used mostly in the fields of differential geometry, mechanics and occasionally vector calculus. Frequently this is used in two-dimensional or three-dimensional space, but can be easily generalized to Euclidean spaces and affine spaces of any dimension.Keller, F. J., Gettys, W. E. et al. (1993), p. 28–29. Relativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cosine Law
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see Fig. 1), the law of cosines states: \begin c^2 &= a^2 + b^2 - 2ab\cos\gamma, \\ mua^2 &= b^2+c^2-2bc\cos\alpha, \\ mub^2 &= a^2+c^2-2ac\cos\beta. \end The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if is a right angle then , and the law of cosines reduces to . The law of cosines is useful for solving a triangle when all three sides or two sides and their included angle are given. Use in solving triangles The theorem is used in solution of triangles, i.e., to find (see Figure 3): *the third side of a triangle if two sides and the angle between them is known: c = \sqrt\,; *the angles of a triangle if the three sides are known: \gamma = \arccos\left(\frac\right)\,; *the third side of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit,: "The CGPM decided to interpret the supplementary units in the SI, namely the radian and the steradian, as dimensionless derived units." defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as . Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing. Definition One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Revolutions Per Minute
Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or r⋅min−1) is a unit of rotational speed (or rotational frequency) for rotating machines. One revolution per minute is equivalent to hertz. Standards ISO 80000-3:2019 defines a physical quantity called ''rotation'' (or ''number of revolutions''), dimensionless, whose instantaneous rate of change is called ''rotational frequency'' (or ''rate of rotation''), with units of reciprocal seconds (s−1). A related but distinct quantity for describing rotation is ''angular frequency'' (or ''angular speed'', the magnitude of angular velocity), for which the SI unit is the radian per second (rad/s). Although they have the same dimensions (reciprocal time) and base unit (s−1), the hertz (Hz) and radians per second (rad/s) are special names used to express two different but proportional ISQ quantities: frequency and angular frequency, respectively. The conversions between a frequency and an angular frequency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Crankshaft
A crankshaft is a mechanical component used in a reciprocating engine, piston engine to convert the reciprocating motion into rotational motion. The crankshaft is a rotating Shaft (mechanical engineering), shaft containing one or more crankpins, that are driven by the pistons via the connecting rods. The crankpins are also called ''rod bearing journals'', and they rotate within the "big end" of the connecting rods. Most modern crankshafts are located in the engine block. They are made from steel or cast iron, using either a forging, casting (metalworking), casting or machining process. Design The crankshaft is located within the engine block and held in place via main bearings which allow the crankshaft to rotate within the block. The up-down motion of each piston is transferred to the crankshaft via connecting rods. A flywheel is often attached to one end of the crankshaft, in order to smoothen the power delivery and reduce vibration. A crankshaft is subjected to enormou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Frequency
Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light. The interval of time between events is called the period. It is the reciprocal of the frequency. For example, if a heart beats at a frequency of 120 times per minute (2 hertz), its period is one half of a second. Special definitions of frequency are used in certain contexts, such as the angular frequency in rotational or cyclical properties, when the rate of angular progress is measured. Spatial frequency is defined for properties that vary or cccur repeatedly in geometry or space. The unit of measurement of frequency in the International System of Units (SI) is the hertz, having the symbol Hz. Definitions and units For cyclical phenomena such as oscillations, waves, or for examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Angular Velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates (spins or revolves) around an axis of rotation and how fast the axis itself changes direction. The magnitude of the pseudovector, \omega=\, \boldsymbol\, , represents the '' angular speed'' (or ''angular frequency''), the angular rate at which the object rotates (spins or revolves). The pseudovector direction \hat\boldsymbol=\boldsymbol/\omega is normal to the instantaneous plane of rotation or angular displacement. There are two types of angular velocity: * Orbital angular velocity refers to how fast a point object revolves about a fixed origin, i.e. the time rate of change of its angular position relative to the origin. * Spin angular velocity refers to how fast a rigid body rotates around a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Acceleration
In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Acceleration is one of several components of kinematics, the study of motion. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction (geometry), direction). The orientation of an object's acceleration is given by the orientation of the ''net'' force acting on that object. The magnitude of an object's acceleration, as described by Newton's second law, is the combined effect of two causes: * the net balance of all external forces acting onto that object — magnitude is Direct proportionality, directly proportional to this net resulting force; * that object's mass, depending on the materials out of which it is made — magnitude is Inverse proportionality, inversely proportional to the object's mass. The International System of Units, SI unit for acceleration is metre per second squared (, \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical quantity, quantity, meaning that both magnitude and direction are needed to define it. The Scalar (physics), scalar absolute value (Magnitude (mathematics), magnitude) of velocity is called , being a coherent derived unit whose quantity is measured in the International System of Units, SI (metric system) as metres per second (m/s or m⋅s−1). For example, "5 metres per second" is a scalar, whereas "5 metres per second east" is a vector. If there is a change in speed, direction or both, then the object is said to be undergoing an ''acceleration''. Definition Average velocity The average velocity of an object over a period of time is its Displacement (geometry), change in position, \Delta s, divided by the duration of the period, \Delt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
