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Degree of freedom
Degree of freedom(Jiyuudo,English: degree of freedom) Is, in general,variableIt is the number of independently selectable ones, that is, the number of all variables minus the number of relational expressions (binding conditions, constraint conditions) that hold between them.Mathematically speakingManifoldIs the dimension of. Expressed as "1 degree of freedom", "1 degree of freedom", etc.
- 1 mass: If translation in 3D space is allowed, the degrees of freedom are 3.
- 2Mass system: If each mass moves independently, the degree of freedom is 6.With the center of mass of both as the representative coordinates of the system,Center of gravityTranslation 3 degrees of freedom, rotation around the center of gravity 2 degrees of freedom, change in relative distance between 2 mass points sandwiching the center of gravity, that isvibrationOften expressed by one degree of freedom.
- Rigid body : n Mass system (however n In ≥ 3), an algebraic constraint that all mass points are invariant is imposed.Therefore, the total number of degrees of freedom of the system is 3 translational degrees of freedom and 3 rotational degrees of freedom, for a total of 6 degrees of freedom.If the motion is constrained on a plane, there are a total of 2 degrees of freedom, 1 translational degrees of freedom and 3 rotation degree of freedom.
MechanicsRefers to the number of movable variables that determine the structure of the entire mechanism.Make up the mechanismLinks TheRigid bodySince it is regarded as, it follows the definition in mechanics and is the contact point of the link.Even numberSince the constraint condition is expressed algebraically by, it is relatively easy to formulate the degree of freedom calculation.
For example, the total degrees of freedom of a mechanism consisting of one-degree-of-freedom kinematic pair and two-degree-of-freedom kinematic pair on a plane is expressed by the following equation.
However, f Is the degree of freedom,n Is the number of links,n1 Is the total number of kinematic pairs with one degree of freedom,n2 Is the total number of pairs with two degrees of freedom.Further, it is assumed that one of the links forming the mechanism is fixed in the space.
For example, the degree of freedom of a plane 5-node closed link system with 1 links and 5 total kinematic pairs with 5 degree of freedom is
The formula expressing the degree of freedom of the mechanism having a three-dimensional structure is as follows.
However, the degree of freedom i Total number of kinematic pairs ni And
Movement mechanism, that is,Artificial satelliteIn such cases, one base link is not fixed in space, and is virtually 3 degrees of freedom kinematic pair in the case of a plane and 6 degrees of freedom kinematic pair in the case of a solid.Inertial systemConsidered to be bound to.In addition, an inertial frame is added to one of the links.For example, all joints consist of 1 degree of freedom joints, 7 joints with 7 links and 6 joints on one arm and 6 links and 26 joints on one leg.Humanoid robotIn the case of, the total number of links is 28 including the fuselage link and inertial system, so the total degree of freedom is
Further, for example, in the case of a wheel-type moving mechanism, if the wheels do not slip on the road surface, a constraint condition () that cannot be expressed due to an algebraic relationship is imposed.Therefore, the calculation of degrees of freedom is not simple.
In general,C ingredient P phaseDegrees of freedom if is present in equilibrium F The
To sayGibbs phase ruleIt is expressed by an expression called.In this case, the state can be described by the ratio of each component (minus the number of phases) in addition to the two state variables.
For example, when pure water exists only in the liquid phase, the degree of freedom is 1 because it is a one-component one-phase system.That is, two state variables (temperatureとpressure, Temperaturevolume, Etc.) can describe the state.
statisticsDefines degrees of freedom for various statistics.
大 き さ n Observation data in the specimen ofx1, x2..., xn) Degrees of freedom n And.Specimens obtained from themaverage x The same is true for.
The relational expression (here x Is the population mean μ The degree of freedom is one less because there is an estimator of n It becomes -1.Therefore, in the denominator n -1 is used.
- Yasuo Nishioka "Mathematics Tutorial Easy Talking Probability Statistics"Ohmsha, 2013.ISBN 9784274214073.
- Koji Fushimi""Kawade Shobo, 1942.ISBN 9784874720127.
- Mathematical Society of Japan"Mathematical Dictionary"Iwanami Shoten, 2007.ISBN 9784000803090.
- Japanese Standards Association, JIS Z 8101-1: 1999 Statistics-Terms and Symbols-Part 1: Probability and General Statistical Terms