DESIGN METHOD FOR INERTER WITH ADAPTIVELY ADJUSTED INERTIA RATIO

Abstract

A design method for an inerter with adaptively adjusted inertia ratio is based on a lead screw-flywheel inerter, which is to change the positions of mass blocks on a flywheel along the radial direction of the flywheel, so as to change of the moment of inertia of the flywheel, and thus to realize adaptive adjustment of the inertia ratio of the inerter. Specifically, the change of angular velocity of the flywheel is caused by the change of an external force load on a lead screw, a centrifugal force on the mass blocks in spring-mass block structures is changed by the angular velocity, and the positions of the mass blocks in the radial direction of the flywheel is determined by the balanced relation of the centrifugal force and a spring restore force, so that the design purpose is achieved.

Claims

1. A design method for inerter with adaptively adjusted inertia ratio, wherein the design method is to design the structure of a flywheel based on a lead screw-flywheel inerter in order to reduce the basic moment of inertia of the flywheel, and at the same time, spring-mass block structures are added to a disk of the flywheel; when a lead screw is subjected to an external force load to make the flywheel rotate, mass blocks are moved along the radial direction of the flywheel under the combined action of a centrifugal force and a spring restore force, and the change of the positions of the mass blocks makes the moment of inertia of the flywheel changed accordingly, so that the inertia ratio of the inerter is adaptively adjusted along with the change of the external force load; the design method specifically comprises steps of: step 1: theoretical design of inerter with adaptively adjusted inertia ratio the moment of inertia of the flywheel is related to the mass distribution thereof, the spring-mass block structures are designed on the disk of the flywheel, the following relation between the positions of the mass blocks in the radial direction of the flywheel and the angular velocity of the flywheel is obtained by the force balance between the centrifugal force on the mass blocks and the spring restore force during the rotation of the flywheel: $\begin{array}{cc}m\frac{d^{2}R\left(t\right)}{d{t}^2}=m{\omega }^2\left(t\right)R\left(t\right)-k\left[R\left(t\right)-R_0\right]& \left(1\right)\end{array}$ wherein the left side of the equal sign is an inertial force on the mass blocks; the first term on the right side of the equal sign is the centrifugal force on the mass blocks, and the second term on the right side of the equal sign is the spring restore force; m is mass of the mass blocks, ω(t) is angular velocity of the flywheel, R(t) is distance between the centroid of the mass blocks and the center of rotation of the flywheel, k is stiffness coefficient of springs, R.sub.0 is minimum distance between the centroid of the mass blocks and the center of rotation of the flywheel, and t is a time variable; based on the movement principle of a lead screw-nut sport pair, the movement equation between the angular velocity ω(t) of the flywheel and the load on the lead screw is expressed as the following differential equation: $\begin{array}{cc}J\left(t\right)\frac{d\omega \left(t\right)}{dt}=\frac{p}{2\pi }{F}_l\left(t\right)& \left(2\right)\end{array}$ wherein p is pitch of the lead screw; F.sub.l(t) is load on the lead screw; J(t) is moment of inertia of the flywheel and is expressed as:
J(t)=J.sub.0+nmR.sup.2(t)  (3) wherein J.sub.0 is fixed moment of inertia of the flywheel; the second term on the right side of the equation represents an adjustable moment of inertia part of the flywheel, n is quantity of the spring-mass block structures, and n is an integral multiple of 2; it is obtained from equations (1)-(3) that the adaptive inertia ratio of the inerter is: $\begin{array}{cc}{b}_v\left(t\right)={\left(\frac{2\pi }{p}\right)}^2\left[J_0+nm{R}^2\left(t\right)\right]& \left(4\right)\end{array}$ step 2: theoretical design of the structure of the flywheel with adjustable moment of inertia it can be known from equation (4) that the inertia ratio of the inerter can be adjusted in a relatively large range by reducing the fixed moment of inertia J.sub.0 of the flywheel and increasing the mass m of the mass blocks; taking the flywheel as a homogeneous disc making fixed axis rotation around the center, then the fixed moment of inertia of the flywheel is expressed as:
J.sub.0=½ρVr.sup.2  (5) wherein r is the radius of the flywheel, ρ is the density of the flywheel, and V is the total volume of the flywheel; it can be known in combination with equations (4) and (5) that J.sub.0 of the flywheel can be reduced by reducing the density and the volume of the flywheel in the condition that the radius of the flywheel is constant; therefore, the main body of the flywheel is made of aluminum alloy, and part of the material on the disk of the flywheel is removed so as to further reduce the fixed moment of inertia of the flywheel; according to equation (3), and fully considering the limitation of the size of the inerter and the constraint to the radius of gyration of the flywheel, the mass m and the quantity n of the mass blocks are increased, and the material of the mass blocks is copper; step 3: structural design of the adaptive adjustable flywheel of the inerter the inerter comprises a shell (3), the lead screw (4), the flywheel (5) and a nut (6); the nut (6) is in solid connection to the flywheel (5), and the axis of the nut (6) is coaxial with the center of rotation of the flywheel (5); the top of the lead screw (4) is an inerter endpoint B (2); the lead screw (4) is connected to the nut (6), and the lead screw-nut sport pair is composed by the lead screw (4) and the nut (6) to convert the rectilinear motion of the inerter endpoint B (2) into the rotational motion of the flywheel; the flywheel (5) comprises a flywheel basic part (9), the springs (7) and the mass blocks (8); the flywheel basic part (9) is formed by removing part of the material on the disk of the flywheel (5); the flywheel (5) is provided with a plurality of guide grooves along the radial direction, and the mass blocks (8) are installed in the guide grooves and can make rectilinear motion along the radial direction of the flywheel; one end of each spring (7) is connected to each mass block (8), and the other end thereof is connected to one side of each guide groove near the center of rotation of the flywheel; the shell (3) has a hollow structure, which packs a structure composed of the flywheel (5), the nut (6) and the lead screw (4) therein; step 4: simulation verification a MATLAB/Simulink simulation model of the inerter with adaptively adjusted inertia ratio is built, and the inerter is verified by simulated load signals.

Description

DESCRIPTION OF DRAWINGS

[0025] FIG. 1 is a structural schematic diagram of an inerter with adaptively adjusted inertia ratio.

[0026] FIG. 2 is a structural schematic diagram of a flywheel with adaptively adjusted moment of inertia.

[0027] FIG. 3 shows harmonic signals of simulated external load in simulation.

[0028] FIG. 4 shows change of moment of inertia of an inerter designed by the present invention under simulated external load.

[0029] FIG. 5 shows change of inertia ratio of an inerter designed by the present invention under simulated external load.

[0030] In the figures: 1 inerter endpoint A; 2 inerter endpoint B; 3 shell; 4 lead screw; 5 flywheel; 6 nut; 7 spring; 8 mass block; and 9 flywheel basic part.

DETAILED DESCRIPTION

[0031] The specific design and embodiments of the present invention are described below in detail in combination with the drawings and the derivation process of the theoretical basis of the adaptive inertia ratio.

[0032] The specific process of this embodiment is conducted in view of the structure of the adaptive inerter shown in FIG. 1, and the detailed design steps are as follows:

[0033] Step 1: building a mathematical model of an inertia ratio adaptive adjustable structure of the adaptive inerter. For the physical structure of the adaptive inerter shown in FIG. 1, when the inerter endpoint A 1 and the inerter endpoint B 2 are subjected to an external force load, the external force load is converted into the torque of the flywheel under the action of the lead screw-nut sport pair, and thus to drive the flywheel to rotate. When the flywheel rotates at angular velocity ω(t), the centrifugal force F.sub.c(t) on a mass block m is expressed as

F.sub.c(t)=mω.sup.2(t)R(t)  (1)

Considering that when the mass block m moves along the radial direction of the flywheel, the spring restore force F.sub.k (i) on the mass block is

F.sub.k(t)=k[R(t)−R.sub.0]  (2)

According to Newton's law of motion, a resultant external force on the mass block m when the flywheel rotates is as follows

[00004]$\begin{array}{cc}ma=m\frac{d^{2}R\left(t\right)}{d{t}^2}=F_c\left(t\right)-F_k\left(t\right)& \left(3\right)\end{array}$

Wherein a is the acceleration of the mass block m in the radial direction of the flywheel.

[0034] Equations (1) and (2) are substituted into equation (3) to obtain a movement equation of the flywheel rotating in a horizontal plane

[00005]$\begin{array}{cc}m\frac{d^{2}R\left(t\right)}{d{t}^2}=m{\omega }^2\left(t\right)R\left(t\right)-k\left[R\left(t\right)-R_0\right]& \left(4\right)\end{array}$

[0035] In order to obtain the angular velocity ω(t) of the flywheel, it is necessary to obtain the movement equation of the flywheel under the external force load. The following relation exists for the flywheel

J(t)β=N(t)  (5)

Wherein β is the angular acceleration of the flywheel and

[00006]$\beta =\frac{d\omega \left(t\right)}{dt},$

N(t) is a resultant external torque, and considering the characteristics of the lead screw-nut sport pair, it can be obtained that N(t) is

[00007]$\begin{array}{cc}N\left(t\right)=\frac{p}{2\pi }{F}_l\left(t\right)& \left(6\right)\end{array}$

Wherein F.sub.l(t) is the external force load on the lead screw. By substituting equation (6) and β into equation (5), the movement equation of the flywheel under the action of the external force is

[00008]$\begin{array}{cc}J\left(t\right)\frac{d\omega \left(t\right)}{dt}=\frac{p}{2\pi }{F}_l\left(t\right)& \left(7\right)\end{array}$

In equation (6), the moment of inertia of the flywheel J(t) is composed of the fixed moment of inertia part and the adjustable moment of inertia part of the flywheel, i.e.,

J(t)=J.sub.0+nmR.sup.2(t)  (8)

[0036] Equations (4), (7) and (8) are the basic principle and theoretical basis of the adaptive adjustable moment of inertia of the inerter designed by the present invention, and then the inertia ratio of the inerter can be obtained as

[00009]$\begin{array}{cc}{b}_v\left(t\right)={\left(\frac{2\pi }{p}\right)}^2\left[J_0+nm{R}^2\left(t\right)\right]& \left(9\right)\end{array}$

Thus the basic principle of adaptive adjustment of the inertia ratio of the inerter designed by the present invention along with load is obtained.

[0037] Step 2: considering that the main purpose of the present invention is to realize adaptive adjustment of the inertia ratio of the inerter in a relatively large range, i.e., to realize the adaptive adjustment of the moment of inertia of the flywheel. The flywheel rotating around the center can be considered as a disc having uniform density distribution and rotating around the center, then the fixed moment of inertia part thereof (the flywheel basic part 9 as shown in FIG. 2) can be expressed as

J.sub.0=½ρVr.sup.2  (10)

In combination with equation (8), it can be known that reducing the basic moment of inertia J.sub.0 is favorable for increasing the adjustable range of the overall moment of inertia J(t) of the flywheel. Considering equation (10), the basic moment of inertia J.sub.0 of the flywheel can be reduced by reducing the volume V of the flywheel and using a material with a smaller density ρ when the radius of gyration r is given.

[0038] Therefore, the flywheel basic part 9 in the inerter of the present invention is made of aluminum alloy with a density of 2.7 g/cm.sup.3. In order to further reduce the basic moment of inertia of the flywheel, part of the material on the disk of the flywheel is removed (comparing 5 in FIG. 1 with 9 in FIG. 2) on the premise of ensuring that the structure of the flywheel has sufficient strength. Based on the above parameters and conditions, the basic moment of inertia J.sub.0=2×10.sup.−4 kg.Math.m.sup.2 of the flywheel in the designed inerter is determined.

[0039] It can be known from the second term on the right side of equation (8) that in the condition that the diameter of the flywheel is fixed, increasing the mass m of the mass block and increasing the quantity n of the mass blocks is an important means to increase the adjustable range of the moment of inertia of the flywheel, therefore the mass block is made of copper with relatively large density and self-lubricating property, and is a cube with the length, the width and the height being all 20 mm, and both ends of the mass block are provided with slide blocks matched with slide rails on the flywheel to realize that the mass block moves along the radial direction of the flywheel under the action of the centrifugal force F.sub.c(t) and the spring restore force F.sub.k(t). It is determined that the mass of the mass block is 0.072 kg, and the quantity is 4. As shown in FIG. 2, each mass block 8 is connected to the flywheel 5 through each spring 7, so as to realize the change of the position R(t)=10˜96 mm of the centroid of the mass block and the center of rotation of the flywheel.

[0040] Step 3: structural design of the adaptive adjustable flywheel of the inerter based on the above-mentioned theory

[0041] The inerter designed mainly comprises the following core components: a shell 3, the lead screw 4, the flywheel 5 and a nut 6.

[0042] The nut 6 is in solid connection to the flywheel 5, the axis of the nut is coaxial with the center of rotation of the flywheel (as shown at the position of the nut 6 in FIG. 2), and the lead screw-nut sport pair is composed by the nut 6 and the lead screw 4 in order to convert the rectilinear motion of the inerter endpoint B into the rotational motion of the flywheel. The lead screw 4 and the nut 6 matched with the lead screw can be selected as required, and the pitch of the lead screw 4 selected to be p=10 mm.

[0043] The flywheel 5 is composed of a flywheel basic part 9, the springs 7 and the mass blocks 8. The mass blocks 8 are installed on the flywheel 5 which is provided with guide grooves along the radial direction, and the mass blocks 8 can make rectilinear motion along the radial direction of the flywheel. One end of each spring 7 is connected to each mass block 8, and the other end thereof is connected to one side of each guide groove near the center of rotation of the flywheel in order to provide a pulling force directing to the center of rotation of the flywheel for the mass block.

[0044] The shell 3 has a hollow structure in order to provide a relatively sealed and clean environment for the flywheel of the inerter and reduce the external interference during the operation of the inerter. Another function of the shell 3 is to facilitate the installation of the inerter, therefore the structure of the shell is not constant, can be specially designed according to the actual use requirements and installation conditions, and has no uniform requirements on style, material and the like, and the design of the shell is not explained too much in the present invention.

[0045] Step 4: simulation verification

[0046] The response of the inerter as shown in FIG. 1 under the external force load is verified and analyzed by simulation. According to the properties of the inerter, the external force load thereon is

F.sub.l(t)=F.sub.2(t)−F.sub.1(t)  (11)

Wherein F.sub.1(t) and F.sub.2 (t) are respectively the external force loads on the endpoint A 1 and the endpoint B 2 of the inerter designed by the present invention. Considering the actual condition that the inerter endpoint A 1 is often used as a fixed point in the application of the inerter, the inerter endpoint A 1 is set to be a fixed end in the simulation verification of this step, i.e., F.sub.1(t)=0. According to the actual condition, the external force load is set to be a harmonic load as shown in FIG. 3, and the harmonic amplitude is 10N. Relevant parameters of the inerter designed by the present invention are set to be n=4, m=0.28 kg, k=500 N/m, J.sub.0=0.0002 kg.Math.m.sup.2, R.sub.0=0.01m, p=0.01m. The change of the moment of inertia and the inertia ratio along with the load is verified by numerical simulation.

[0047] FIG. 4 shows the change of the moment of inertia of the inerter designed by the present invention along with the external load. It can be seen from the figure that when the external force load begins to change at t=4 s (as shown in FIG. 3), the moment of inertia of the inerter also begins to increase and become greater. At t=5 s (as shown in FIG. 3, when the external force load changes to the half cycle of the maximum change rate), the moment of inertia of the inerter increases rapidly and reaches the maximum value (4.8×10.sup.−4 kg.Math.m.sup.2) in a cycle; after that, the moment of inertia of the inerter decreases rapidly and restores to 2.8×10.sup.−4 kg.Math.m.sup.2 (the basic moment of inertia plus the moment of inertia of the mass blocks in closest positions) near t=6 s. The later period in FIG. 4 further illustrates that the moment of inertia of the inerter designed by the present invention can be changed according to the change of the external force load.

[0048] In order to further verify the adaptive inerter designed by the present invention, the inertia ratio b.sub.v of the inerter is simulated as shown in FIG. 5. In combination with FIG. 3, it can be seen that along with the change of the external force load, the inertia ratio of the adaptive inerter designed by the present invention can be adaptively adjusted in a large range (89.8 kg-279.6 kg), which indicates that the inerter can meet the design purpose of adaptively adjusting the inertia ratio according to the external force load.

[0049] The above embodiments only express the implementation of the present invention, and shall not be interpreted as a limitation to the scope of the patent for the present invention. It should be noted that, for those skilled in the art, several variations and improvements can also be made without departing from the concept of the present invention, all of which belong to the protection scope of the present invention.