Bodies, Inertias, Loads & Other Functions (Docstrings)

Bodies

class sympy.physics.mechanics.particle.Particle(name, point=None, mass=None)

一个粒子。

参数:

name ：结构

粒子名称

点 ：点

表示粒子位置、速度和加速度的物理/力学点

mass ：可解释

表示粒子质量的同调表达式

potential_energy : Sympifyable

粒子的势能。

解释

粒子的质量不为零，缺乏空间扩展；它们不占用空间。

初始化时需要提供值，但可以稍后更改。

实例

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import Symbol
>>> po = Point('po')
>>> m = Symbol('m')
>>> pa = Particle('pa', po, m)
>>> # Or you could change these later
>>> pa.mass = m
>>> pa.point = po

angular_momentum(point, frame)

质点的角动量。

参数:

点 ：点

粒子的角动量被期望的点。

框架 ：参考帧

需要角动量的帧。

解释

关于粒子P的某个点O的角动量H由下式给出：

H = cross(r, m * v)

其中r是从点O到质点P的位置矢量，m是质点的质量，v是质点在惯性系中的速度，N。

实例

>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v, r = dynamicsymbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> A = O.locatenew('A', r * N.x)
>>> P = Particle('P', A, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.angular_momentum(O, N)
m*r*v*N.z

kinetic_energy(frame)

Kinetic energy of the particle.

参数:

框架 ：参考帧

粒子的速度通常是相对于惯性系来定义的，但是任何已知速度的相关坐标系都可以提供。

解释

The kinetic energy, T, of a particle, P, is given by:

T = 1/2 (dot(m * v, v))

其中m是粒子P的质量，v是粒子在所提供的参考帧中的速度。

实例

>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy import symbols
>>> m, v, r = symbols('m v r')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.point.set_vel(N, v * N.y)
>>> P.kinetic_energy(N)
m*v**2/2

linear_momentum(frame)

粒子的线性动量。

参数:

框架 ：参考帧

需要线性动量的帧。

解释

The linear momentum L, of a particle P, with respect to frame N is given by:

L=m*v

其中m是粒子的质量，v是粒子在N帧中的速度。

实例

>>> from sympy.physics.mechanics import Particle, Point, ReferenceFrame
>>> from sympy.physics.mechanics import dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> A = Particle('A', P, m)
>>> P.set_vel(N, v * N.x)
>>> A.linear_momentum(N)
m*v*N.x

property mass

The body's mass.

property masscenter

The body's center of mass.

property name

The name of the body.

parallel_axis(point, frame)

返回粒子相对于另一个点和帧的惯性并矢。

参数:

点 : sympy.物理.矢量.点

表示惯性并矢的点。

框架 : sympy.物理.矢量.参考框架

用来构造并矢的参考坐标系。

返回:

惯性 : sympy.物理.矢量.并矢

质点的惯量并矢表示在给定的点和坐标系上。

property point

The body's center of mass.

property potential_energy

The potential energy of the body.

实例

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m

class sympy.physics.mechanics.rigidbody.RigidBody(name, masscenter=None, frame=None, mass=None, inertia=None)

理想化的刚体。

解释

本质上，这是一个容器，其中包含描述刚体的各种组件：名称、质量、质心、参考系和惯性。

所有这些都需要在创建时提供，但可以在以后更改。

实例

>>> from sympy import Symbol
>>> from sympy.physics.mechanics import ReferenceFrame, Point, RigidBody
>>> from sympy.physics.mechanics import outer
>>> m = Symbol('m')
>>> A = ReferenceFrame('A')
>>> P = Point('P')
>>> I = outer (A.x, A.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, A, m, inertia_tuple)
>>> # Or you could change them afterwards
>>> m2 = Symbol('m2')
>>> B.mass = m2

属性

名称	（字符串）实体的名称。
质量中心	（点）表示刚体质心的点。
框架	（ReferenceFrame）刚体固定在其中的ReferenceFrame。
群众	（值得同情的）身体的质量。
惯性	（并矢，点）物体对一点的惯性；如上图所示存储在元组中。
potential_energy	(Sympifyable) The potential energy of the RigidBody.

angular_momentum(point, frame)

返回刚体关于给定帧中某一点的角动量。

参数:

点 ：点

角动量需要角动量的点。

框架 ：参考帧

需要角动量的帧。

解释

刚体B关于框架N中某点O的角动量H由下式给出：

H = dot(I, w) + cross(r, m * v)

where I and m are the central inertia dyadic and mass of rigid body B, w is the angular velocity of body B in the frame N, r is the position vector from point O to the mass center of B, and v is the velocity of the mass center in the frame N.

实例

>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v, r, omega = dynamicsymbols('m v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, 1 * N.x)
>>> I = outer(b.x, b.x)
>>> B = RigidBody('B', P, b, m, (I, P))
>>> B.angular_momentum(P, N)
omega*b.x

property central_inertia

物体的中心惯性并矢。

property frame

The ReferenceFrame fixed to the body.

property inertia

The body's inertia about a point; stored as (Dyadic, Point).

kinetic_energy(frame)

Kinetic energy of the rigid body.

参数:

框架 ：参考帧

刚体的角速度和质心速度通常是相对于惯性系来定义的，但是任何已知速度的相关坐标系都可以提供。

解释

The kinetic energy, T, of a rigid body, B, is given by:

T = 1/2 * (dot(dot(I, w), w) + dot(m * v, v))

where I and m are the central inertia dyadic and mass of rigid body B respectively, w is the body's angular velocity, and v is the velocity of the body's mass center in the supplied ReferenceFrame.

实例

>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody
>>> from sympy import symbols
>>> m, v, r, omega = symbols('m v r omega')
>>> N = ReferenceFrame('N')
>>> b = ReferenceFrame('b')
>>> b.set_ang_vel(N, omega * b.x)
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (b.x, b.x)
>>> inertia_tuple = (I, P)
>>> B = RigidBody('B', P, b, m, inertia_tuple)
>>> B.kinetic_energy(N)
m*v**2/2 + omega**2/2

linear_momentum(frame)

刚体的线动量。

参数:

框架 ：参考帧

需要线性动量的帧。

解释

The linear momentum L, of a rigid body B, with respect to frame N is given by:

L = m * v

where m is the mass of the rigid body, and v is the velocity of the mass center of B in the frame N.

实例

>>> from sympy.physics.mechanics import Point, ReferenceFrame, outer
>>> from sympy.physics.mechanics import RigidBody, dynamicsymbols
>>> from sympy.physics.vector import init_vprinting
>>> init_vprinting(pretty_print=False)
>>> m, v = dynamicsymbols('m v')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, v * N.x)
>>> I = outer (N.x, N.x)
>>> Inertia_tuple = (I, P)
>>> B = RigidBody('B', P, N, m, Inertia_tuple)
>>> B.linear_momentum(N)
m*v*N.x

property mass

The body's mass.

property masscenter

The body's center of mass.

property name

The name of the body.

parallel_axis(point, frame=None)

返回物体相对于另一点的惯性并矢。

参数:

点 : sympy.物理.矢量.点

表示惯性并矢的点。

框架 : sympy.物理.矢量.参考框架

用来构造并矢的参考坐标系。

返回:

惯性 : sympy.物理.矢量.并矢

刚体的惯量并矢表示在给定点附近。

property potential_energy

The potential energy of the body.

实例

>>> from sympy.physics.mechanics import Particle, Point
>>> from sympy import symbols
>>> m, g, h = symbols('m g h')
>>> O = Point('O')
>>> P = Particle('P', O, m)
>>> P.potential_energy = m * g * h
>>> P.potential_energy
g*h*m

property x

The basis Vector for the body, in the x direction.

property y

The basis Vector for the body, in the y direction.

property z

The basis Vector for the body, in the z direction.

Inertias

class sympy.physics.mechanics.inertia.Inertia(dyadic, point)

Inertia object consisting of a Dyadic and a Point of reference.

解释

This is a simple class to store the Point and Dyadic, belonging to an inertia.

实例

>>> from sympy.physics.mechanics import ReferenceFrame, Point, Inertia
>>> N = ReferenceFrame('N')
>>> Po = Point('Po')
>>> Inertia(N.x.outer(N.x) + N.y.outer(N.y) + N.z.outer(N.z), Po)
((N.x|N.x) + (N.y|N.y) + (N.z|N.z), Po)

In the example above the Dyadic was created manually, one can however also use the inertia function for this or the class method from_tensor as shown below.

>>> Inertia.from_inertia_scalars(Po, N, 1, 1, 1)
((N.x|N.x) + (N.y|N.y) + (N.z|N.z), Po)

属性

dyadic	(Dyadic) The dyadic of the inertia.
point	(Point) The reference point of the inertia.

classmethod from_inertia_scalars(point, frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0)

Simple way to create an Inertia object based on the tensor values.

参数:

点 ：点

The reference point of the inertia.

框架 ：参考帧

The frame the inertia is defined in.

ixx ：可解释

The xx element in the inertia dyadic.

iyy ：可解释

The yy element in the inertia dyadic.

izz ：可解释

The zz element in the inertia dyadic.

ixy ：可解释

The xy element in the inertia dyadic.

iyz ：可解释

The yz element in the inertia dyadic.

izx ：可解释

The zx element in the inertia dyadic.

解释

This class method uses the :func`~.inertia` to create the Dyadic based on the tensor values.

实例

>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, Point, Inertia
>>> ixx, iyy, izz, ixy, iyz, izx = symbols('ixx iyy izz ixy iyz izx')
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> I = Inertia.from_inertia_scalars(P, N, ixx, iyy, izz, ixy, iyz, izx)

The tensor values can easily be seen when converting the dyadic to a matrix.

>>> I.dyadic.to_matrix(N)
Matrix([
[ixx, ixy, izx],
[ixy, iyy, iyz],
[izx, iyz, izz]])

sympy.physics.mechanics.inertia.inertia(frame, ixx, iyy, izz, ixy=0, iyz=0, izx=0)

创建惯性并矢物体的简单方法。

参数:

框架 ：参考帧

The frame the inertia is defined in.

ixx ：可解释

The xx element in the inertia dyadic.

iyy ：可解释

The yy element in the inertia dyadic.

izz ：可解释

The zz element in the inertia dyadic.

ixy ：可解释

The xy element in the inertia dyadic.

iyz ：可解释

The yz element in the inertia dyadic.

izx ：可解释

The zx element in the inertia dyadic.

解释

Creates an inertia Dyadic based on the given tensor values and a body-fixed reference frame.

实例

>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')
>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)

sympy.physics.mechanics.inertia.inertia_of_point_mass(mass, pos_vec, frame)

点质量相对于点O的惯性并矢。

参数:

mass ：可解释

点质量的质量

pos_vec ：矢量

从点O到点质量的位置

框架 ：参考帧

表示并矢的参考坐标系

实例

>>> from sympy import symbols
>>> from sympy.physics.mechanics import ReferenceFrame, inertia_of_point_mass
>>> N = ReferenceFrame('N')
>>> r, m = symbols('r m')
>>> px = r * N.x
>>> inertia_of_point_mass(m, px, N)
m*r**2*(N.y|N.y) + m*r**2*(N.z|N.z)

Loads

class sympy.physics.mechanics.loads.Force(point, force)

Force acting upon a point.

解释

A force is a vector that is bound to a line of action. This class stores both a point, which lies on the line of action, and the vector. A tuple can also be used, with the location as the first entry and the vector as second entry.

实例

A force of magnitude 2 along N.x acting on a point Po can be created as follows:

>>> from sympy.physics.mechanics import Point, ReferenceFrame, Force
>>> N = ReferenceFrame('N')
>>> Po = Point('Po')
>>> Force(Po, 2 * N.x)
(Po, 2*N.x)

If a body is supplied, then the center of mass of that body is used.

>>> from sympy.physics.mechanics import Particle
>>> P = Particle('P', point=Po)
>>> Force(P, 2 * N.x)
(Po, 2*N.x)

class sympy.physics.mechanics.loads.Torque(frame, torque)

Torque acting upon a frame.

解释

A torque is a free vector that is acting on a reference frame, which is associated with a rigid body. This class stores both the frame and the vector. A tuple can also be used, with the location as the first item and the vector as second item.

实例

A torque of magnitude 2 about N.x acting on a frame N can be created as follows:

>>> from sympy.physics.mechanics import ReferenceFrame, Torque
>>> N = ReferenceFrame('N')
>>> Torque(N, 2 * N.x)
(N, 2*N.x)

If a body is supplied, then the frame fixed to that body is used.

>>> from sympy.physics.mechanics import RigidBody
>>> rb = RigidBody('rb', frame=N)
>>> Torque(rb, 2 * N.x)
(N, 2*N.x)

Other Functions

sympy.physics.mechanics.functions.center_of_mass(point, *bodies)

Returns the position vector from the given point to the center of mass of the given bodies(particles or rigidbodies).

例子

>>> from sympy import symbols, S
>>> from sympy.physics.vector import Point
>>> from sympy.physics.mechanics import Particle, ReferenceFrame, RigidBody, outer
>>> from sympy.physics.mechanics.functions import center_of_mass
>>> a = ReferenceFrame('a')
>>> m = symbols('m', real=True)
>>> p1 = Particle('p1', Point('p1_pt'), S(1))
>>> p2 = Particle('p2', Point('p2_pt'), S(2))
>>> p3 = Particle('p3', Point('p3_pt'), S(3))
>>> p4 = Particle('p4', Point('p4_pt'), m)
>>> b_f = ReferenceFrame('b_f')
>>> b_cm = Point('b_cm')
>>> mb = symbols('mb')
>>> b = RigidBody('b', b_cm, b_f, mb, (outer(b_f.x, b_f.x), b_cm))
>>> p2.point.set_pos(p1.point, a.x)
>>> p3.point.set_pos(p1.point, a.x + a.y)
>>> p4.point.set_pos(p1.point, a.y)
>>> b.masscenter.set_pos(p1.point, a.y + a.z)
>>> point_o=Point('o')
>>> point_o.set_pos(p1.point, center_of_mass(p1.point, p1, p2, p3, p4, b))
>>> expr = 5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z
>>> point_o.pos_from(p1.point)
5/(m + mb + 6)*a.x + (m + mb + 3)/(m + mb + 6)*a.y + mb/(m + mb + 6)*a.z

sympy.physics.mechanics.functions.linear_momentum(frame, *body)

系统的线性动量。

参数:

框架 ：参考帧

需要线性动量的帧。

身体1，身体2，身体3。。。 ：粒子和/或刚体

需要线性动量的物体。

解释

此函数返回粒子和/或刚体系统的线性动量。系统的线性动量等于其组成部分的线性动量的矢量和。考虑一个由刚体a和质点P组成的系统S。系统的线动量L等于质点的线动量L1和刚体的线动量L2的矢量和，即。

L=L1+L2

实例

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, linear_momentum
>>> N = ReferenceFrame('N')
>>> P = Point('P')
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = Point('Ac')
>>> Ac.set_vel(N, 25 * N.y)
>>> I = outer(N.x, N.x)
>>> A = RigidBody('A', Ac, N, 20, (I, Ac))
>>> linear_momentum(N, A, Pa)
10*N.x + 500*N.y

sympy.physics.mechanics.functions.angular_momentum(point, frame, *body)

Angular momentum of a system.

参数:

点 ：点

角动量系统的角动量所期望的点。

框架 ：参考帧

需要角动量的帧。

身体1，身体2，身体3。。。 ：粒子和/或刚体

需要角动量的物体。

解释

此函数返回粒子和/或刚体系统的角动量。该系统的角动量等于其组成部分的角动量的矢量和。考虑一个由刚体a和质点P组成的系统S。系统的角动量H等于质点角动量H1和刚体角动量H2的矢量和，即。

H=H1+H2

实例

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, angular_momentum
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> angular_momentum(O, N, Pa, A)
10*N.z

sympy.physics.mechanics.functions.kinetic_energy(frame, *body)

多体系统的动能。

参数:

框架 ：参考帧

定义物体速度或角速度的帧。

身体1，身体2，身体3。。。 ：粒子和/或刚体

需要动能的物体。

解释

这个函数返回粒子和/或刚体系统的动能。这个系统的动能等于其组成部分的动能之和。考虑一个系统，S，包括一个刚体a和一个粒子P。系统的动能T等于粒子动能T1和刚体动能T2的矢量和，即。

T=T1+T2

动能是标量。

实例

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, kinetic_energy
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> kinetic_energy(N, Pa, A)
350

sympy.physics.mechanics.functions.potential_energy(*body)

多体系统的势能。

参数:

身体1，身体2，身体3。。。 ：粒子和/或刚体

需要势能的物体。

解释

这个函数返回粒子和/或刚体系统的势能。这样一个系统的势能等于其组成部分的势能之和。考虑一个系统，S，包括一个刚体a和一个粒子P。系统的势能V等于粒子势能V1和刚体势能V2的矢量和，即。

V=V1+V2

势能是标量。

实例

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, potential_energy
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> Pa = Particle('Pa', P, m)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> a = ReferenceFrame('a')
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, M, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> potential_energy(Pa, A)
M*g*h + g*h*m

sympy.physics.mechanics.functions.Lagrangian(frame, *body)

多体系统的拉格朗日。

参数:

框架 ：参考帧

定义物体速度或角速度以确定动能的框架。

身体1，身体2，身体3。。。 ：粒子和/或刚体

需要拉格朗日的物体。

解释

这个函数返回粒子和/或刚体系统的拉格朗日。这样一个系统的拉格朗日等于其组成部分的动能和势能之差。如果T和V是系统的动能和势能，那么它的拉格朗日，L，定义为

L=T-V

拉格朗日是标量。

实例

>>> from sympy.physics.mechanics import Point, Particle, ReferenceFrame
>>> from sympy.physics.mechanics import RigidBody, outer, Lagrangian
>>> from sympy import symbols
>>> M, m, g, h = symbols('M m g h')
>>> N = ReferenceFrame('N')
>>> O = Point('O')
>>> O.set_vel(N, 0 * N.x)
>>> P = O.locatenew('P', 1 * N.x)
>>> P.set_vel(N, 10 * N.x)
>>> Pa = Particle('Pa', P, 1)
>>> Ac = O.locatenew('Ac', 2 * N.y)
>>> Ac.set_vel(N, 5 * N.y)
>>> a = ReferenceFrame('a')
>>> a.set_ang_vel(N, 10 * N.z)
>>> I = outer(N.z, N.z)
>>> A = RigidBody('A', Ac, a, 20, (I, Ac))
>>> Pa.potential_energy = m * g * h
>>> A.potential_energy = M * g * h
>>> Lagrangian(N, Pa, A)
-M*g*h - g*h*m + 350

sympy.physics.mechanics.functions.find_dynamicsymbols(expression, exclude=None, reference_frame=None)

在表达式中查找所有动态符号。

参数:

expression : SymPy expression

排除 ：iterable of dynamicsymbols，可选

reference_frame ：ReferenceFrame，可选

确定给定向量的动态符号的帧。

解释

If the optional exclude kwarg is used, only dynamicsymbols not in the iterable exclude are returned. If we intend to apply this function on a vector, the optional reference_frame is also used to inform about the corresponding frame with respect to which the dynamic symbols of the given vector is to be determined.

实例

>>> from sympy.physics.mechanics import dynamicsymbols, find_dynamicsymbols
>>> from sympy.physics.mechanics import ReferenceFrame
>>> x, y = dynamicsymbols('x, y')
>>> expr = x + x.diff()*y
>>> find_dynamicsymbols(expr)
{x(t), y(t), Derivative(x(t), t)}
>>> find_dynamicsymbols(expr, exclude=[x, y])
{Derivative(x(t), t)}
>>> a, b, c = dynamicsymbols('a, b, c')
>>> A = ReferenceFrame('A')
>>> v = a * A.x + b * A.y + c * A.z
>>> find_dynamicsymbols(v, reference_frame=A)
{a(t), b(t), c(t)}