Moment of Inertia, isrigid bodyInertia when rotating around an axis (the rotating object maintains itsUniform circular motionOr static characteristics), represented by the letter I or J.[1]In classical mechanics, the moment of inertia (also known as mass moment of inertia, referred to as moment of inertia) is usuallyIorJIndicates that the SI unit is kg · m²。For aparticle，I=mr²，Where m is its mass,rAre particles andRotating shaftThe vertical distance of.
Its magnitude depends on theshape、mass distributionandRotating shaftLocation of.The moment of inertia of rigid bodies has important physical significance, and is also an important parameter in scientific experiments, engineering technology, aerospace, power, machinery, instrumentation and other industrial fields.Electromagnetic instrumentDue to the different moment of inertia of coils, the indication system of（Galvanometer）Or electricity quantity (impact galvanometer).It is necessary to accurately measure the moment of inertia in the shape design of engine blades, flywheels, gyroscopes and artificial satellites.
The moment of inertia only depends on the shape of the rigid body, the mass distribution and the position of the rotation axis, and is the same as the rotation state of the rigid body around the axis (such asangular velocitySize).The moment of inertia of a uniform rigid body with regular shape can be directly calculated by formula.The moment of inertia of irregular rigid bodies or heterogeneous rigid bodies is generally measured by experimental methods, so experimental methods are very important.Moment of inertia applied to various motions of rigid bodydynamicsCalculation in progress.
The expression of moment of inertia is
If the mass of the rigid body is continuously distributed, the formula for calculating the moment of inertia can be written as
(where
Represents the mass of a certain element of the rigid body,rRepresents the vertical distance from the particle to the rotating shaft,ρIt indicates the density at that place, and the summation sign (or integration sign) covers the entire rigid body.)[2]
Parallel axis theorem: set the mass of rigid body as
, the moment of inertia about the axis passing through the center of mass is
, move this axis parallel in any direction for a distance
, the moment of inertia around the new axis
For:
Parallel axis theorem
This theorem is called the parallel axis theorem.[3]
An object at angular velocityωAround fixed shaftzThe rotation of the shaft can also be regarded as being parallel tozRotation of a fixed axis passing through the center of mass.That is to say, windingzThe rotation of the axis is equal to the superposition of the rotation of the parallel axis around the center of mass and the rotation of the center of mass.
According to the parallel axis theorem, among a group of parallel rotating axes, the corresponding moment of inertia of the axis passing through the center of mass is the smallest.
Vertical axis theorem
Vertical axis theorem: a planar rigid bodysheetThe moment of inertia of the axis perpendicular to its plane is equal to the sum of the moment of inertia of any two orthogonal axes intersecting with the vertical axis in the plane.
expression:
WhereIx,Iy,IzRespectively represent rigid body pairsx,y,zMoment of inertia of three axles
The following vertical axis theorem also holds for non planar thin plate rigid bodies[3]:
The theorem of vertical axis can be used to calculate the moment of inertia of some rigid bodies to a specific axis
Theorem of vertical axis of thin plates
The moment of inertia of a rigid body to an axis can be converted into the moment of inertia of a single particle whose mass is equal to the mass of the rigid body to the axis.The distance from the particle to the axis of rotation obtained from this conversion is called theRadius of gyrationκ, the formula is
, whereMIs the mass of the rigid body;I is the moment of inertia.
In addition to the above two theorems, the extension rule is also commonly used.The stretching rule states that if any point of an object is displaced along a straight axis of any size in parallel, the moment of inertia of the object on this axis will not change.We can imagine pulling an object parallel to the straight axis towards both ends.When the object is extended, keep the vertical distance from any point of the object to the straight axis unchanged, then the extension rule states that the moment of inertia of the object on this axis is unchanged.The stretching rule can be simply obtained through the definition of the moment of inertia.
Kinetic formula
Announce
edit
The above is the definition and calculation formula of the moment of inertia.Here are some (fixed axis rotation)Rigid body dynamicsFormula.[1]
Relationship between angular acceleration and external torque:
Note that this is only the rotational kinetic energy of the rigid body around the fixed axis, and its total kinetic energy should be added to the translational kinetic energy of the center of mass.From this formula, the problem of rigid body dynamics can be analyzed from the angle of energy.
Tensor definition
Announce
edit
The inertia of a rigid body rotating around a point can be determined by the more generalInertial tensorDescription.Inertia tensor is second-order symmetrictensorIt completely depicts the magnitude of the moment of inertia of the rigid body around any axis passing through the point.For the sake of simplicity, here only the definition of the moment of inertia tensor around the center of mass and its expression in the moment equation are given.
With a rigid bodyA,Its centroid isC, rigid bodyAAround its centroidCMoment of inertia tensor of
Defined as[2]
This integral covers the entire rigid bodyA，Among them,
, is the center of mass of the rigid bodyCTo any point on the rigid bodyBOfVector diameter；expression
Is the dyadic of two vectors, and
Unit tensor
Is a typical unit orthogonal curve frame;
Is the density of the rigid body.
Moment equation of moment inertia tensor
Set rigid bodyAAround its center of massCThe resultant torque vector of is
, rigid bodyAThe angular velocity vector in the inertial system is
, angular acceleration vector is
，AThe tensor of moment of inertia about its center of mass is
, the moment equation is as follows:
By projecting the above vector form moment equation onto each coordinate axis (or, more precisely, multiplying it by the unit direction vector of each coordinate axis), we can obtain the scalar form moment equation in the component direction of each coordinate axis.
inertia tensor
Is a second-order tensor, although in the frame
It has nine components, but because it is a symmetric tensor, its actual independent components are only six.[4]
Experimental determination
Announce
edit
Trilinear pendulum[2]
In fact, the moment of inertia of irregular rigid bodies is often difficult to calculate accurately and needs to be measured by experiments.There are many methods to measure the moment of inertia of rigid bodies, including three wire pendulumTorsion pendulum、Compound pendulumEtc.The three wire pendulum is used to measure the moment of inertia of an object through torsional motion. It is characterized by clear physical images, easy operation, and being suitable for objects of various shapes, such as mechanical parts, motor rotors, gun projectiles, and fan blades.This experimental method has certain practical significance in theory and technology.
Experimental principle
Experimental schematic diagram
As shown on the right, the hanging wall of the three line pendulum is symmetrically connected to the triangle vertex at the edge of a larger uniform disk along the vertex of an equilateral triangle.In the experiment, the lower disk is unloaded, and the upper disk rotates a small angle. At this time, the lower disk begins to swing, and at the same time, the center of mass of the lower disk
Will rise and fall along the axis of rotation.Record the vibration period as
, the quality of the footwall is
。Next, set the mass as
Place the object to be measured on the footwall so that its center of mass is just on the central axis of the footwall, and then make the footwall swing again, and record its period as
。
It is calculated that when
Very small, the moment of inertia of the footwall meets the formula[5]
Is the vertical distance between the center of the upper and lower discs;
Is the rising height of the lower disc during vibration;
Is the radius of the upper disk;
Is the radius of the lower disk;
Is the twist angle.
Experiment content
1. Determine the instrument constant.
Properly select measuring instruments and tools to reduceMeasurement uncertainty。The experiment steps are designed to ensure the levelness of the upper and lower discs of the three wire pendulum, so that the instrument can reach the best measurement state.
2. Measure the moment of inertia of the lower disc and calculate its uncertainty.
Rotate the small disk above the three wire pendulum to make it rotate an angle around its axisα，With the help of the tension of the line, the lower disc makes a torsional movement to avoid left and right shaking.Make your own measurement methodcycleMeasurement ofUncertaintyIt is less than the uncertainty of other measurement quantities.Using the formula, the uncertainty transfer formula is derived, and the uncertainty is calculated.
3. Measure the moment of inertia of the ring
Place the ring to be measured on the lower disk, and make sure that the center of mass of the ring is exactly atRotating shaftMeasure the moment of inertia of the system.Measure the mass, inner and outer diameters of the ring.Calculate the moment of inertia of the ring by using the formula.And compare with the theoretical value to findrelative error。
4. Verify the parallel axis theorem
Overlap two metal cylinders with the same mass, shape and size on the lower disk, and make sure that the center of mass coincides with the center of mass of the lower disk.Measure the rotation shaft passing through the cylindercentroidIs the moment of inertia of the system.Then place the two cylinders symmetrically on both sides of the center of the lower disc.Measure the moment of inertia of the system at this time.The distance from the center of mass of the measured cylinder to the central axis is calculated and compared with the measured value.[5]
Calculation formula
Announce
edit
For thin rods
When the rotation axis passes through the midpoint (centroid) of the rod and is perpendicular to the rod,
；
When the rotation axis passes through the end point of the rod and is perpendicular to the rod,
。
WheremIs the mass of the rod,LIs the length of the rod.
For cylinders
Moment of inertia of partially homogeneous geometry
When the rotation axis is the axis of the cylinder,
。
WheremIs the mass of the cylinder,rIs the radius of the cylinder.
For thin torus
When the rotation axis passes through the ring center and is perpendicular to the ring surface,
；
When the rotation axis passes through the ring edge and is perpendicular to the ring surface,
；
When the rotary shaft is along a certain diameter of the ring,
。
WheremIs the mass of the thin ring,RIs the radius of the thin ring.
For thin disks
When the rotary shaft is perpendicular to the disk surface through the center,
；
When the rotary shaft is perpendicular to the disk surface through the edge,
。
WheremIs the mass of the thin disk,RIs the radius of the thin disk.
For hollow cylinders
When the rotation axis is the axis of symmetry of the hollow cylinder,
。
(Note that here isplusNot a minus sign[1], easy to remember wrong.Can be substituted
And the cylinder degenerates to cylindrical surface.)
WheremIs the mass of the hollow cylinder,RoneandRtwoThe inner and outer radii of the hollow cylinder.
For spherical shells
When the rotary shaft is the central shaft of the spherical shell,
；
When the rotation axis is tangent to the spherical shell,
。
WheremIs the mass of the spherical shell,RIs the radius of the spherical shell.
For a solid sphere
When the rotation axis is the central axis of the sphere,
；
When the rotation axis is tangent to the sphere,
。
WheremIs the mass of the sphere,RIs the radius of the sphere.
For cubes
When the rotation axis is the central axis of the cube,
；
When the rotation axis is the edge of the cube,
；
When the rotation axis is the body diagonal of the cube,
。
WheremIs the mass of the cube,LIs the side length of the cube.[3]
For a box
When the rotation axis is the center axis of the box,
。
WheremIs the mass of the box,loneandltwoIt is the length of two sides of a rectangle perpendicular to the rotation axis.
Example
It is known that a motor shaft with a diameter of 80mm and a length of 500mm is made of steel.Calculate the required speed when it reaches 500 rpm in 0.10 secondsmomentsize.
The motor shaft can be considered as the axis of the cylinder, and the moment of inertia is set asJ, quality ism, radius isr, the torque isM, angular acceleration isβ, with:
