What is Spatial Kinematics?

Kinematics, a branch of mechanics that describes and studies the law of the change of the position of an object over time from a geometric perspective (referring to the physical properties of the object itself and the force applied to the object). Based on studying the motion of two simplified models of particle and rigid body, and further studying the motion of deformable body (elastic body, fluid, etc.). To study the latter motion, the rigid displacement and strain of the micelles in the deformed body must be separated. The kinematics of points studies the kinematics, trajectories, displacements, velocities, accelerations and other motion characteristics of points, all of which vary with the selected reference frame. Rigid body kinematics also studies the rotation process, angular velocity, and angular acceleration of the rigid body itself. More complex motion characteristics.

Chinese name: Kinematics
Foreign name: kinematics

Side: Geometric angle
Features: Tracing studies the change of object position with time

Kinematics concept

A branch of theoretical mechanics that studies the motion of objects from the perspective of geometry. "Motion" here refers to mechanical movement, that is, the change in the position of the object; the so-called "geometrical angle" refers to the physical properties (such as mass) of the object and the force applied to the object. Mechanical motion is a generalized motion-one of the simplest basic motions of all changes in the universe.

Kinematics

The kinematics of points studies the kinematics, trajectories, displacements, speeds, accelerations and other motion characteristics of points, all of which vary with the selected reference frame. Rigid body motion can be divided into translational motion, rotation around a fixed axis, plane parallel motion, rotation around a fixed point, and general motion according to the characteristics of the motion. Kinematics provides the theoretical basis for dynamics and mechanics, and is also the basic knowledge necessary for natural sciences and engineering technology. Kinematics is a branch of theoretical mechanics. It uses geometric methods to study the motion of objects.

Introduction to Kinematics

Kinematics is a branch of theoretical mechanics.

Kinematics Methods to study the motion of objects usually do not consider the influence of factors such as force and mass. As for the relationship between the motion and force of an object, it is a research topic of dynamics.

To describe the motion of an object with a geometric method, a frame of reference must be determined. Therefore, from the perspective of kinematics, the description of any motion is relative. Here, the relativity of motion refers to the category of classical mechanics, that is, the measurement of time and space is the same in different frames of reference, and has nothing to do with the motion of frames of reference. But when the speed of an object approaches the speed of light, the measurement of time and space is related to the frame of reference. "Motion" here refers to mechanical movement, that is, the change of the position of the object; the so-called "geometrical angle" refers to the physical properties (such as mass) of the object and the force applied to the object.

Any object, such as a car, rocket, planet, etc., regardless of its size, if it can ignore the relative motion inside it, if every part of it is moving in the same direction and at the same speed, then , You can simply treat this object as a mass point and the position of the centroid of this object as the position of the mass point. In kinematics, this kind of particle motion, whether it is linear motion or curved motion, is the most basic research object.

Kinematics research project

Kinematics mainly studies points and laws of rigid body movement. Dots mean no size and

Kinematics Mass, geometric points occupying a certain position in space. A rigid body is a body that has no mass and does not deform, but has a certain shape and occupies a certain position in space. Kinematics includes point kinematics and rigid body kinematics. After mastering these two types of movements, it is possible to further study the movements of deformed bodies (elastomers, fluids, etc.).

In the study of deformed bodies, the rigid displacements and strains of micelles must be separated. These vary with the selected reference frame; rigid body kinematics also needs to study more complex motion characteristics such as the rotation process, angular velocity, and angular acceleration of the rigid body itself. According to the characteristics of motion, rigid body motion can be further divided into: translation of rigid body, fixed-axis rotation of rigid body, plane motion of rigid body, fixed-point rotation of rigid body, and general motion of rigid body.

Kinematic classification

Kinematics is divided into particle kinematics, rigid body kinematics, and kinematic constraints. It provides the theoretical basis for dynamics and mechanical principles (mechanics). It also contains the basic knowledge necessary for many disciplines of natural science and engineering technology.

Development history of kinematics

Early kinematics

Kinematics is subordinate to dynamics in the early stages of development and develops with dynamics. In ancient times, by observing the movement of ground objects and celestial bodies, people gradually formed the concept of the change of the position of objects in space and time. Chinese Warring States Period

Kinematics Period has been described in the "Mo Jing" about movement and time. Aristotle discussed falling and circular motions in Physics, and he already had the concept of speed.

Galileo

Galileo discovered the law that the distance is proportional to the square of time in the linear motion of constant acceleration, and established the concept of acceleration. In the study of the projectile's motion, he derived the parabolic trajectory and established the parallelogram rule of motion (or velocity) synthesis. Galileo laid the foundation for the kinematics of the point. On this basis, Huygens independently proposed the concept of centrifugal force in the study of pendulum motion and Newton's study of celestial body motion, and found that the centripetal acceleration is proportional to the quadratic square of velocity and the same radius. The law of inverse proportion.

Kinematic Euler

In the late 18th century, due to the development and needs of astronomy, shipbuilding, and machinery, Euler systematically studied the fixed-axis rotation of a rigid body and the fixed-point motion of a rigid body using geometric methods, and proposed the Euler named by his descendants. The concept of angle establishes Euler's kinematics equation and rigid body's finite rotational displacement theorem. From this, the concepts of instantaneous rotation axis and instantaneous angular velocity vector of rigid body are obtained, which profoundly reveals the basic motion characteristics of this complex form of motion. So Euler can be called the founder of rigid body kinematics.

Kinematics is described by geometric methods

Since then, Lagrange and Hamilton have introduced generalized coordinates, generalized velocity, and generalized momentum, respectively, which has opened up new ways for geometrically describing the motion of multi-degree-of-freedom particle systems in multidimensional configuration space and phase space, and facilitated the analysis. Development of dynamics.

Kinematic mechanism

Since the end of the 19th century, in order to meet different production needs and complete different actions,

Kinematics Various machines have appeared one after another and are widely used. Therefore, mechanism science came into being. The task of mechanism science is to analyze the movement laws of the mechanism, design a new mechanism and integrate the mechanism according to the movement required. The development of modern instruments and automation technology has promoted the further development of mechanism science, and proposed the analysis and synthesis of various plane and space mechanism motions. As the theoretical basis of mechanism science, kinematics has gradually departed from dynamics and became one of the classic mechanics. Independent branch.

Kinematic fluid

A branch of fluid mechanics that studies the geometric properties of fluid motion without involving specific effects of forces.

There are two methods to describe the description of fluid motion, namely Lagrangian method and Euler method. The Lagrangian method focuses on fluid particles and tries to describe the position of each fluid particle over time. Normally, rectangular coordinates or curvilinear coordinates a, b, and c of a fluid particle at the initial time are used as indicators for distinguishing between different fluid particles. The motion law of the fluid particle can be expressed as r = r (a, b, c, t), where r is the vector diameter of the fluid particle; t is time; a, b, c, and t are collectively called Lagrange variables. Euler's method looks at points in space and tries to describe the changes in fluid motion over time at each point in space. The motion law of the fluid particle can be expressed by the velocity vector v = v (r, t), where r and t are called Euler variables. The Euler method is widely used, and the Lagrangian method is rarely used, because the velocity function determined by Euler variables is defined at time and space points, so it is a velocity field, called a flow field, which can be solved using field theory knowledge Secondly, in the Euler method, since acceleration is a first derivative, the system of motion equations is a first-order partial differential equation system, which is easier to handle than the second-order partial differential equation system in Lagrangian method.

The geometric description of the flow describes the curve drawn by the fluid particle when it moves in space. It is called a trace; the curve that is tangent to the velocity vector at each point in the flow field is called a streamline. A trace is a curve formed by the same fluid particle at different times. It is a geometric representation of the movement law of the fluid particle in the Lagrangian method. A streamline is a curve composed of different fluid particles at the same time. It is the Euler method. Geometrical representation of the law of fluid particle motion. Only in stationary movements do the two overlap.

Flow analysis Fluid motion is more complex than rigid body motion. In addition to translation and rotation, deformation also occurs. The Helmholtz velocity decomposition theorem states that the movement of fluid micelles can be decomposed into the sum of translation, rotation and deformation

Kinematics (See Mechanical Movement). The important differences between the fluid velocity decomposition theorem and the rigid body velocity decomposition theorem are: the fluid micelle motion has more deformation velocity parts than the rigid body; the rigid velocity decomposition theorem holds for the entire rigid body, so it is a wholeness theorem, while the fluid velocity decomposition theorem This is true in fluid micelles and is therefore a local theorem.

Kinematics

Flow classification From the perspective of motion form, fluid motion can be divided into non-rotating motion and rotating motion. From the perspective of time, it can be divided into steady motion (all physical quantities do not change with time) and unsteady motion. From a spatial perspective, fluid motion can be divided into one-dimensional, two-dimensional, and three-dimensional motions depending on the physical quantities that depend on one, two, and three coordinates. Plane motion and axisymmetric motion are two important examples of two-dimensional motion.

The kinematics of a vortex In a vortex, the curve that is tangent to the vortex vector everywhere is called a vortex. Each fluid micelle on the vortex line rotates around the tangential direction of the vortex line. In the vortex field, a closed curve that is not vortex and does not self-intersect is taken. All vortices through it form a tubular curved surface, called a vortex tube. The kinematics of the scroll are: the vortex flux is equal to the same constant in all cross-sections of the scroll, which is called the scroll strength. A vortex tube cannot be created or terminated in a fluid. If it does not exist as a vortex ring, it can only extend to the boundary.

Mathematical expression of continuity equation for fluid mass conservation law. Suppose that any fluid with a volume of is taken in the flow field, and the peripheral interface of is . From the law of mass conservation, the increase rate of fluid mass in is equal to the mass of fluid flowing out of interface per unit time.