Is a circle a curve
Circles of curvature of plane curves
Circles of curvature on a trajectory
The circle of curvature hugs a curve in the vicinity of a point of contact. The figure on the right shows the (blue, black or green) circles of curvature for three points of the curve (drawn in red) (the curve is the trajectory of a point on an oscillating crank loop, see the animation below). As a rule, there is a different circle of curvature for each point on a curve with its own center of curvature and its own radius of curvature.
What do you need that for?
In technical mechanics, the radius of curvature plays an important role in various areas, for example:
- In the case of a bending beam, the bending moment is inversely proportional to the radius of curvature of the deformed beam. On this basis, the non-linear differential equation of the bending line is derived in the chapter "Deformations due to bending moments".
- The normal acceleration when moving on a curved path is calculated according to the formula an= v2/ ρ calculated (with the path velocity v and the radius of curvature of the path curve ρ). This is usually the most important (because it is by far the largest) component of the overall acceleration.
The formulation of "optimal snuggling" used above can be put into mathematically usable definitions in different ways. The first definition is based on the definition of the tangent on a curve: Two points clearly define a straight line, which is called a secant if the two points lie on a curve. If these two points come closer and closer together until they finally coincide at the border crossing, then the secant becomes the tangent to the curve at this point. The following is defined in the same way:
The following definition is completely equivalent to this:
Definition 2 is used below to derive the required formulas.
Formulas for the radius of curvature
A function in parametric representation is given
This is not a restriction of generality, because every function y = f (x) can also be represented as x = t & nbsp, & nbsp y = f (t). A circle is searched for (the parameter representation is also used for this)
which also corresponds to this function in a point given by the coordinates x and y in the two derivatives y 'and y' '. These derivatives are formed for the circle:
t can be eliminated, and one obtains a formula for ρ:
In a corresponding way you get from
a formula for the parametric representation of the function. Because a radius must of course have a positive value, the absolute value is taken for the second derivative, which can be positive or negative, and the following result ...
Formulas for the radius of curvature:
Coordinates of the center of curvature
The function of the circle of curvature, written down in general form above, is converted according to the coordinates of the center point:
In order to avoid the sign problem with the root, we refer to the intermediate results achieved above
are used, which can be used to obtain the following formulas, for example:
For the function given in the parametric representation, analog formulas are calculated in a corresponding way;
Coordinates of the center of curvature:
Evolute and involute
The geometric location of all centers of curvature of a curve is called the evolute, the curve itself is the involute of this evolute. These curves play an important role in the gear geometry.
Calculate for the function y & nbsp = & nbspax2 generally the radius of curvature ρ (x) and the coordinates of the centers of curvature xM.(x) and yM.(x). For a = 0.5, the radii of curvature that belong to points (0; 0) or (1; 0.5) and the parametric representation of the evolute must be specified.
Function (drawn in red) with two circles of curvature and Evolute
The function and its first two derivatives
are used in the formulas given above:
The calculated formulas for the coordinates of the centers of curvature xM.(x) and yM.(x) can be viewed as a parametric representation (parameter is x) of the Evolute. For a = 0.5 one obtains with
the parametric representation of the Evolute. In the graphic above, the function y = x2/ 2 red, the two special circles of curvature blue and black and the evolute green.
The mechanism sketched opposite is a so-called "swinging crank loop". The crank rotates with the constant angular velocity ω0.
Given: R; ω0 ; a / R = 1.5; l / R = 2.
- For a complete revolution of the crank, the radius of curvature ρ (φ), the path speed of the point Bv (φ), its tangential acceleration are at(φ) and normal acceleration an(φ) graphically (φ = ω0t).
- The trajectory of point B should be shown and it should be made clear that the radius of curvature changes constantly with such a curve.
The determination of the trajectory is required as a question of a task from the chapter "Kinematics of the point". The result can be found on the Internet in the section "Solutions to the exercises". The curve can only be given meaningfully in the parametric representation. With a Cartesian coordinate system with the origin at point C, if point A of the crank is at point D at time t = 0:
The formation of the first two derivatives of these functions, which are required for writing down the radius of curvature and the center of curvature, is quite laborious (but can still be mastered "by hand"). It is well advised to get help from the computer.
The (tangentially directed) path speed is calculated from the speed components, and the tangential acceleration is the derivation of the path speed in terms of time:
For the normal acceleration (perpendicular to the tangent to the trajectory)
the radius of curvature is also required, which is calculated using the formulas given above. The graphic representation of the functions can be seen on the right. It can be seen that the normal acceleration (shown in red) takes on greater values than the tangential acceleration. Their maximum value is not in the area of particularly small radii of curvature, because the square of the speed in the numerator of the fraction has a greater influence. It is typical that the tangential acceleration can have positive and negative values, the normal acceleration, on the other hand, is always given as positive because it is always directed towards the center of curvature.
In the animation below you can see the red trajectory of point B and the constantly changing (blue drawn) circles of curvature:
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