Motion Characteristic Equation of the Particle with Spatial Sliding
The particle
P in the crushing chamber was selected as the research object. When the motion characteristic of particle
P was spatial sliding, it slid along the mantle surface and moved with the mantle in the spatial compound motion. In this process, the particle remained in contact with the mantle. The force and acceleration components of the particle
P are shown in
Figure 4.
As shown in
Figure 4a, the absolute acceleration
ap of the particle
P was composed of the following components: the acceleration
ao of the origin
o of the dynamic coordinate system, the normal acceleration
an and tangential acceleration
aτ of the contact point between the particle and the mantle, the relative acceleration
ar of the particle sliding on the mantle, and the Coriolis acceleration
ac generated by the motion of the dynamic coordinate system, as shown in Equation (15).
Since the dynamic coordinate system
oxyz fixed on the mantle rotated around the central axis of the cone crusher, the acceleration
ao of the origin
o of the dynamic coordinate
oxyz can be expressed as Equation (16).
The second derivative of the particle position vector
nP in the dynamic coordinate system can be used to determine the relative acceleration
ar of the particle.
Since the mantle rotates around its central axis, the normal acceleration
an and tangential acceleration
aτ of the of contact point between the particle
P and the mantle can be expressed as Equations (18) and (19).
The Coriolis acceleration
ac of the contact point between the mantle and the particle can be expressed as Equation (20).
In combined Equations (15–20), the absolute acceleration
ap of particle
P with spatial sliding in the global coordinate system can be expressed as Equation (21).
As shown in
Figure 4b, the resultant force of particle
P consists of the following parts: the support force
FN of the mantle to particle
P, the friction force
Ff between the mantle and the particle
P, and gravity
G.
The support force
FN on the particle
P perpendicular to the mantle surface can be expressed as Equation (23).
where
denotes the equation of the mantle surface.
The friction force
Ff between the particle
P and the mantle surface is proportional to the support force
FN, and the direction is opposite to its motion relative to the mantle. As a result, the friction force
Ff of particle
P can be expressed as Equation (24).
where
μ denotes the friction coefficient between particle and mantle surface.
The gravity
G of particle
P can be expressed as Equation (25).
where
m is the weight of the particle
P,
g denotes the gravitational acceleration.
According to Newton’s second law, the motion characteristic equation of the particle with spatial sliding can be expressed as Equation (26).
Through the coordinate transformation matrix
K4 and
K3, the coordinates and vectors in the dynamic coordinate system are transformed into the global coordinate system, and the Equation (26) is decomposed along the
X-axis,
Y-axis, and
Z-axis of the global coordinate system.
The combining Equations (27)–(29) include not only algebraic terms such as x, y, z, and FN, but also first-order and second-order differential terms such as x’, y’, z’, x’’, y’’, and z’’. As a result, the combining equations of the particle with spatial sliding are nonlinear second-order differential-algebraic equations (DAEs). By solving the DAEs, the motion characteristics of particle with spatial sliding were obtained.
Motion Characteristic Equation of the Particle with Free-Falling
The particle
P in the crushing chamber was selected as the research object. When the particle
P was released by the mantle and concave, the motion characteristic of the particle
P was free-falling, as it was only affected by gravity in the falling process. The force and acceleration components of the particle
P are shown in
Figure 5.
During the free-falling of the particle, the size and direction of gravity
G are constant. Thus, the motion characteristic equation of particle
P with free-falling can be expressed as Equation (30).
where
X,
Y, and
Z represent the real-time position of the particle in the global coordinate system.
X0,
Y0, and
Z0 denote the initial position of the particle in the global coordinate system.
t is the time of free-falling.
At a certain time, the free-falling particle
P contacted with the mantle. The contact point was not only the end point of the free-falling of the particle
P, but also a point on the mantle surface with the spatial compound motion. According to the coordinate system transformation matrix
K4, the coordinates of the contact point between the particle
P and the mantle surface can be solved by the Equation (31).
where
x,
y, and
z represent the real-time position of the particle in the dynamic coordinate system.
By combining Equations (30) and (31), the free-falling time of particles P was solved, and the motion characteristic equation of particle with free-falling was developed.
Motion Characteristic Equation of the Particle with Spatial Compound Falling
The motion characteristic of the particle with the spatial compound falling both possessed the features of spatial sliding and free-falling. Particle P freely fell as it was released by the mantle and concave. At a certain time, the particle P contacted with the mantle and slid on the mantle surface. As a result, the spatial compound falling of particle P was composed of a free-falling part and a spatial sliding part. In the free-falling part of particle P, the free-falling time and the coordinates of the contact point were determined by the initial coordinates of particle P and the motion characteristic equation of free-falling. The coordinates of the contact point and the free-falling time are taken as the initial state of the spatial sliding part of particle P. According to the motion characteristic equation of spatial sliding, the motion characteristics of the spatial sliding of particle P can be solved. Thus, all the motion characteristics of the particle P with spatial compound falling were achieved.