The
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 r ...
of a non-offset
piston connected to a rotating
crank through a
connecting rod (as would be found in
internal combustion engine
An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal c ...
s) can be expressed by
equations of motion
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.''Encyclopaedia of Physics'' (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Ver ...
. This article shows how these equations of motion can be derived using
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
as functions of angle ''(
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
domain)'' and of time ''(
time domain
Time domain refers to the analysis of mathematical functions, physical signals or time series of economic or environmental data, with respect to time. In the time domain, the signal or function's value is known for all real numbers, for the c ...
)''.
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:
:
rod length (distance between
piston pin
In internal combustion engines, the gudgeon pin (UK, wrist pin or piston pin US) connects the piston to the connecting rod, and provides a bearing for the connecting rod to pivot upon as the piston moves.Nunney, Malcolm James (2007) "The Recipr ...
and
crank pin
A crankpin or crank pin, also known as a rod bearing journal, is a mechanical device in an engine which connects the crankshaft to the connecting rod for each cylinder. It has a cylindrical surface, to allow the crankpin to rotate relative to t ...
)
:
crank radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
(distance between crank center and crank pin, i.e. half
stroke)
:
crank
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
(from
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
bore centerline
Center line, centre line or centerline may refer to:
Sports
* Center line, marked in red on an ice hockey rink
* Centre line (football), a set of positions on an Australian rules football field
* Centerline, a line that separates the service cou ...
at
TDC)
:
piston pin
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
(distance upward from crank center along cylinder bore centerline)
The following variables are also defined:
:
piston pin
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
(upward from crank center along cylinder bore centerline)
:
piston pin
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
(upward from crank center along cylinder bore centerline)
:
crank
angular velocity (in the same direction/sense as crank angle
)
Angular velocity
The
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
(
Hz) of the
crankshaft's rotation is related to the engine's speed (
revolutions per minute
Revolutions per minute (abbreviated rpm, RPM, rev/min, r/min, or with the notation min−1) is a unit of rotational speed or rotational frequency for rotating machines.
Standards
ISO 80000-3:2019 defines a unit of rotation as the dimension ...
) as follows:
:
So the
angular velocity (
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. The unit was formerly an SI supplementary unit (before that ...
/s) of the crankshaft is:
:
Triangle relation
As shown in the diagram, the
crank pin
A crankpin or crank pin, also known as a rod bearing journal, is a mechanical device in an engine which connects the crankshaft to the connecting rod for each cylinder. It has a cylindrical surface, to allow the crankpin to rotate relative to t ...
, crank center and piston pin form triangle NOP.
By the
cosine law it is seen that:
:
where
and
are constant and
varies as
changes.
Equations with respect to angular position (Angle Domain)
Angle domain equations are expressed as functions of angle.
Deriving angle domain equations
The angle domain equations of the piston's reciprocating motion are derived from the system's geometry equations as follows.
Position
Position with respect to crank angle (from the triangle relation,
completing the square
:
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form
:ax^2 + bx + c
to the form
:a(x-h)^2 + k
for some values of ''h'' and ''k''.
In other words, completing the square places a perfe ...
, utilizing the
Pythagorean identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations be ...
, and rearranging):
:
Velocity
Velocity with respect to crank angle (take first
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
, using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
):
:
Acceleration
Acceleration with respect to crank angle (take second
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
, using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
and the
quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
):
:
Non Simple Harmonic Motion
The angle domain equations above show that the motion of the piston (connected to rod and crank) is not
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
, but is modified by the motion of the rod as it swings with the rotation of the crank. This is in contrast to the
Scotch Yoke
The Scotch Yoke (also known as slotted link mechanism) is a reciprocating motion mechanism, converting the linear motion of a slider into rotational motion, or vice versa. The piston or other reciprocating part is directly coupled to a sliding ...
which directly produces simple harmonic motion.
Example graphs
Example graphs of the angle domain equations are shown below.
Equations with respect to time (time domain)
Time domain equations are expressed as functions of time.
Angular velocity derivatives
Angle is related to time by angular velocity
as follows:
:
If angular velocity
is constant, then:
:
and:
:
Deriving time domain equations
The time domain equations of the piston's reciprocating motion are derived from the angle domain equations as follows.
Position
Position with respect to time is simply:
:
Velocity
Velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
with respect to time (using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
):
:
Acceleration
Acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...
with respect to time (using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
and
product rule
In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
, and the angular velocity
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s):
:
Scaling for angular velocity
From the foregoing, you can see that the time domain equations are simply ''scaled'' forms of the angle domain equations:
is unscaled,
is scaled by ''ω'', and
is scaled by ''ω²''.
To convert the angle domain equations to time domain, first replace ''A'' with ''ωt'', and then
scale for angular velocity as follows: multiply
by ''ω'', and multiply
by ''ω²''.
Velocity maxima and minima
By definition, the velocity
maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
occur at the acceleration zeros ''(crossings of the horizontal axis)''.
Crank angle not right-angled
The velocity maxima and minima ''(see the acceleration zero crossings in the graphs below)'' depend on rod length
and half stroke
and do not occur when the crank angle
is right angled.
Crank-rod angle not right angled
The velocity maxima and minima do not necessarily occur when the crank makes a right angle with the rod. Counter-examples exist to disprove the statement ''"velocity maxima and minima only occur when the crank-rod angle is right angled"''.
Example
For rod length 6" and crank radius 2" (as shown in the example graph below), numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°. Then, using the triangle
law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
\frac \,=\, \frac \,=\, \frac \,=\, 2R,
where , and ar ...
, it is found that the rod-vertical angle is 18.60647° and the crank-rod angle is 88.21738°. Clearly, in this example, the angle between the crank and the rod is not a right angle. Summing the angles of the triangle 88.21738° + 18.60647° + 73.17615° gives 180.00000°. A single counter-example is sufficient to disprove the statement ''"velocity maxima/minima occur when crank makes a right angle with rod"''.
Example graphs of piston motion
Angle Domain Graphs
The graphs below show the angle domain equations for a constant rod length
(6.0") and various values of half stroke
(1.8", 2.0", 2.2").
''Note in the graphs that ''L'' is rod length
and ''R'' is half stroke.
.''
Animation
Below is an animation of the piston motion equations with the same values of rod length and crank radius as in the graphs above
Units of Convenience
Note that for the
automotive/
hotrod
Hot rods are typically American cars that might be old, classic, or modern and that have been rebuilt or modified with large engines optimised for speed and acceleration. One definition is: "a car that's been stripped down, souped up and made ...
use-case the most convenient ''(used by enthusiasts)'' unit of length for the piston-rod-crank geometry is the
inch, with typical dimensions being 6" (inch) rod length and 2" (inch) crank radius. This article uses units of inch (") for position, velocity and acceleration, as shown in the graphs above.
See also
*
Connecting rod
*
Crankshaft
*
Internal combustion engine
An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal c ...
*
Piston
*
Reciprocating engine
A reciprocating engine, also often known as a piston engine, is typically a heat engine that uses one or more reciprocating pistons to convert high temperature and high pressure into a rotating motion. This article describes the common fea ...
*
Scotch yoke
The Scotch Yoke (also known as slotted link mechanism) is a reciprocating motion mechanism, converting the linear motion of a slider into rotational motion, or vice versa. The piston or other reciprocating part is directly coupled to a sliding ...
*
Simple Harmonic Motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
*
Slider-crank linkage
A slider-crank linkage is a four-link mechanism with three revolute joints and one prismatic, or sliding, joint. The rotation of the crank drives the linear movement the slider, or the expansion of gases against a sliding piston in a cylinder ...
References
*
*
*{{cite web, url=http://www.epi-eng.com/piston_engine_technology/piston_motion_basics.htm, title=Piston Motion Basics @ epi-eng.com
External links
animated enginesAnimated Otto Engine
desmosInteractive Stroke vs Rod Piston Position and Derivatives
desmosInteractive Crank Animation
codecogsPiston Velocity and Acceleration
youtubeRotating SBC 350 Engine
youtube3D Animation of V8 Engine
youtubeInside V8 Engine
Motion equations
Engine technology
Mechanical engineering
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