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:
- m
-
- d
- 2
-
- R
-
- (
- t
- )
- d
-
- t
- 2
- =
- m
-
- ω
- 2
-
- (
- t
- )
-
- R
-
- (
- t
- )
- -
- k
-
- [
- R
-
- (
- t
- )
- -
- R
- 0
- ]
- (
- 1
- )
- 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, R0 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:
- J
-
- (
- t
- )
-
- d
-
- ω
-
- (
- t
- )
- d
-
- t
- =
- p
- 2
-
- π
-
- F
- l
-
- (
- t
- )
- (
- 2
- )
- wherein p is pitch of the lead screw;
- Fl(t) is load on the lead screw;
- J(t) is moment of inertia of the flywheel and is expressed as:
- J(t)=J0+nmR2(t) (3)
- wherein J0 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:
- b
- v
-
- (
- t
- )
- =
- (
- 2
-
- π
- p
- )
- 2
-
- [
- J
- 0
- +
- n
-
- m
-
- R
- 2
-
- (
- t
- )
- ]
- (
- 4
- )
- 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 J0 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:
- J0=½ρVr2 (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 J0 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.