CN112214868A - Method for researching amplitude variation vibration characteristic of aerial work platform arm support - Google Patents

Method for researching amplitude variation vibration characteristic of aerial work platform arm support Download PDF

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CN112214868A
CN112214868A CN202010894245.9A CN202010894245A CN112214868A CN 112214868 A CN112214868 A CN 112214868A CN 202010894245 A CN202010894245 A CN 202010894245A CN 112214868 A CN112214868 A CN 112214868A
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vibration
arm support
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cylinder
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邓铭
纪爱敏
王豪
赵仲航
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Changzhou Campus of Hohai University
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Abstract

The invention discloses a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform, which comprises the following steps: s01, carrying out spring equivalence on the amplitude-variable hydraulic oil cylinder to obtain an equivalent post-stiffness calculation model; s02, calculating boundary conditions, continuity conditions and dynamic differential equations by using a Hamiltonian principle; s03, homogenizing the inhomogeneous vibration differential equation obtained in the S02 to obtain a vibration mode preliminary solution; s04, obtaining a homogeneous differential equation set of vibration and obtaining a vibration mode characteristic value; s05, obtaining a state space equation of vibration in the Hilbert space by combining orthogonality among mode shapes; and S06, performing dynamic simulation in a Matlab/Simulink environment to obtain the vibration response of the arm support head along with the change of the elevation angle. The invention provides a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform, which can solve the problems that the vibration response calculation of the head part of the arm support is inaccurate and does not accord with the engineering practice and the like along with the change of an amplitude elevation angle in the amplitude variation motion process of the existing aerial work platform.

Description

Method for researching amplitude variation vibration characteristic of aerial work platform arm support
Technical Field
The invention relates to a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform, and belongs to the technical field of the vibration control of the arm support of the aerial work platform.
Background
An aerial work platform is equipment for lifting personnel and equipment to a certain height for operation. With the increase of the working height in the actual engineering, the length of the arm support of the overhead working truck is larger and larger, even exceeds more than one hundred meters, so the requirements on the safety and comfort of people on the arm support are further improved, and in order to ensure the requirements, the vibration of the arm support must be strictly controlled and reduced. In the vibration research aiming at variable-amplitude motion, when a boom model is established, an amplitude-variable system of the boom is simplified into a driving hub with rotational inertia, the distance between a root hinge point of the boom and a hinge point of an amplitude-variable oil cylinder is ignored, namely a triangular area formed by the tail of the boom and the amplitude-variable oil cylinder is regarded as a rigid area, and therefore the boom is changed into a cantilever beam structure with a variable cross section. The simplification ignores the actual situation of the aerial work platform and may bring errors to the inhibition and control of the vibration of the arm support.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform, which can solve the problems that the vibration response calculation of the head of the arm support is inaccurate and does not accord with the engineering practice and the like along with the change of the amplitude elevation angle in the amplitude variation motion process of the aerial work platform.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows:
a method for researching the amplitude variation vibration characteristic of an aerial work platform arm support comprises the following steps:
s01, carrying out spring equivalence on the amplitude-variable hydraulic oil cylinder, enabling hydraulic oil on two sides of the piston rod to be equivalent to hydraulic oil springs connected in parallel, enabling the piston rod to be equivalent to be connected with the hydraulic oil springs in series, and obtaining an equivalent post-stiffness calculation model;
s02, calculating boundary conditions, continuity conditions and dynamic differential equations by using a Hamiltonian principle;
s03, homogenizing the non-homogeneous vibration differential equation obtained in S02, and obtaining a vibration mode preliminary solution by utilizing Laplace transform and inverse Laplace transform in combination with the vibration characteristic of an Euler Bernoulli beam;
s04, preliminarily solving a homogeneous differential equation set of vibration by using boundary conditions, continuity conditions, a homogeneous differential equation and a vibration mode to obtain a vibration characteristic value;
s05, obtaining a state space equation of vibration in the Hilbert space by combining orthogonality among mode shapes;
and S06, performing dynamic simulation in a Matlab/Simulink environment to obtain the vibration response of the arm support head along with the change of the elevation angle.
In S01, the equivalent stiffness K of the amplitude variation cylinder of the elastic support of the amplitude variation hydraulic oil cylinder is equal to the spring stiffness of the hydraulic oil, and the calculation mode is as follows:
Figure BDA0002657932600000021
wherein x represents the displacement distance of the piston of the hydraulic cylinder, L represents the stroke, AK、ARRespectively representing the working areas of the left and right sides of the hydraulic cylinder, VK、VRRespectively representing the volumes of the left and right sides of the cylinder, VLK、 VLRRespectively representing dead volumes of oil in pipelines at two sides of the hydraulic cylinder, betaeThe volume modulus of the hydraulic oil.
In S02, the whole arm support is divided into n sections with concentrated parameters and mutually continuous arm sections, and the Hamilton principle formula is utilized:
Figure 14
wherein, t1、t2Respectively starting time and ending time of movement, T is the kinetic energy of the arm support, W is the virtual work of the non-conservative force, and V is the potential energy of the arm support;
Figure BDA0002657932600000023
in the formula:
Figure BDA0002657932600000024
respectively representing the right deviation and the left deviation of the joint of the arm joint i and the i-1, z represents the distance between a point on the arm support and the original point along the z axis, and rho AiRepresents the linear density of the arm segment i,
Figure BDA0002657932600000025
represents the derivative of the amplitude variation motion angle theta (t) to the time t,
Figure BDA0002657932600000031
represents the derivation of the neutral axis deflection omega (z, t) of the arm joint to the time t, mc、JcRespectively represents the concentrated mass and the moment of inertia of the top of the arm support, l represents the total length of the arm support,
Figure BDA0002657932600000032
represents the derivative of the deflection omega (l, t) of the top of the arm support to the time t,
Figure 15
representing the derivative of the top deflection omega (l, t) of the arm support to the time t and the coordinate z;
when the arm support potential energy V is considered, the obtained equivalent spring potential energy is considered,
Figure RE-GDA0002829229940000034
W＝Mθ(t)
in the formula: EI (El)iRepresenting the bending density of an arm section i, omega' (z, t) representing the neutral axis deflection omega (z, t) of the arm support and solving a quadratic derivative of a coordinate z, g representing the inertia acceleration, K representing the equivalent rigidity of the amplitude variation cylinder, and l0Represents the length of the amplitude variation cylinder in the initial state of the arm support, l1The length of the current amplitude variation cylinder is represented, a represents the distance from the hinge point of the amplitude variation cylinder to the bottom of the arm support, omega (a, t) represents the neutral axis deflection of the arm support at the hinge point of the hydraulic cylinder, alpha represents the included angle between the amplitude variation cylinder and the arm support, and M represents the driving moment of the amplitude variation cylinder to the arm support.
The kinetic differential equation, boundary conditions, and continuity conditions are respectively as follows:
Figure RE-GDA0002829229940000035
Figure 9
Figure 10
in the formula: v (z, t) represents the arc from a point on the boom to the horizontal, the subscript i represents the ith arm, and has: vi(z,t)＝zθ+ω(z,t),z∈(zi-1 +,zi -)，Vi Ⅳ(z, t) represents the fourth derivative of the displacement coordinate z,
Figure RE-GDA0002829229940000038
representing the second derivative over time t, KαEquivalent rigidity K and (sin alpha) of variable amplitude cylinder2Product of δ1The (z-a) functions each represent a dicke function, which reflects the bearing force at the luffing cylinder only at z ═ a, H (z-z)1 +) The unit of a step function is represented,
Figure RE-GDA0002829229940000039
representing the second derivative of the boom tip arc with respect to time t and the first derivative of the displacement coordinate z, Vn″(l,t)、Vn"(l, t) respectively represents the second and third derivatives of the radian of the tail end of the arm support to the displacement coordinate z,
Figure RE-GDA0002829229940000041
representing the second derivative of the boom tip arc with respect to time t,
Figure RE-GDA0002829229940000042
and
Figure RE-GDA0002829229940000043
the second and third derivatives of the radian measure at successive arm segments i and i +1, respectively, with respect to the displacement coordinate z.
In S03, the Euler Bernoulli beam has a vibration characteristic of w2＝EIiγi 4/ρAiWhere w is the natural frequency of beam vibration, γiIn order to be a characteristic value of the frequency,
and dynamic response Vi(z, t) can be derived from the generalized mode shape basis function
Figure BDA0002657932600000043
And generalized coordinates q (t) representing:
Figure BDA0002657932600000044
wherein the subscript b denotes the b-th order mode,
Figure BDA0002657932600000045
and b-order mode base functions representing any arm joint i are substituted into a kinetic differential equation to obtain:
Figure RE-GDA0002829229940000047
Figure RE-GDA0002829229940000048
wherein
Figure BDA0002657932600000048
Is a mode shape basis function
Figure BDA0002657932600000049
The fourth derivative of the displacement coordinate z.
Carrying out Laplace transformation and inverse Laplace transformation on the obtained object to obtain a preliminary solution of the vibration mode;
Figure RE-GDA00028292299400000411
wherein:
Figure BDA00026579326000000411
cosh and sinh are hyperbolic cosine and hyperbolic sine functions respectively,
Figure RE-GDA00028292299400000413
Figure RE-GDA00028292299400000414
Figure RE-GDA00028292299400000415
Figure RE-GDA0002829229940000051
in S04, the homogeneous equation set is MnPn0, wherein
Figure RE-GDA0002829229940000052
Figure RE-GDA0002829229940000053
Figure RE-GDA0002829229940000054
Figure BDA0002657932600000055
In the formula: mnCoefficient matrix, B, representing homogeneous equation1And BnRespectively representing the boundary condition coefficients of the bottom and the top of the arm support, Ci(z) coefficient matrix, P, representing the continuous conditions of the boomnIs a coefficient vector consisting of a mode shape function and a derivative thereof;
and has the following components: e ═ gamman(l-zn-1)、
Figure BDA0002657932600000056
x1＝γi(z-zi-1 +)、x3＝γi(z-a)、H2＝cT(γia)H(zi-a)[1-H(zi-z1)]、 H3＝cV(γia)H(zi-a)[1-H(zi-z1)]。
Utilizing the general solution V (z, t) of the boom vibration differential equation in S04, and according to the orthogonality, taking the first three-order vibration mode at the tail end of the boom to solve V1(t)、V2(t)、V3(t) the first three principal coordinate vibration differential equations in S05 can be obtained,
Figure BDA0002657932600000061
Figure BDA0002657932600000062
Figure BDA0002657932600000063
in the formula:
Figure BDA0002657932600000064
and
Figure BDA0002657932600000065
representing the second derivative of the first three mode solutions,
Figure BDA0002657932600000066
and
Figure BDA0002657932600000067
first derivative, xi, representing the first third order mode solution1、ξ2And xi3Damping ratio of the first three vibration modes, w1、w2And w3The method is divided into the first three-order natural frequency, f is inhomogeneous modal dynamic load, and for establishing a state space equation of vibration in a Hilbert space, a state vector is defined as follows:
Figure BDA0002657932600000068
then there is
Figure BDA0002657932600000069
Wherein: g ═ g1(t) g2(t) g3(t) g4(t) g5(t) g6(t)]T， u＝θ，B＝[0 w1 2f 0 w2 2f 0 w3 2f]T，D＝0
Figure RE-GDA00028292299400000612
Figure 3
In the formula:
Figure 100002_1
and
Figure 2
and the vibration mode basis function of the tail end of the arm frame of the first three steps is divided.
The invention has the beneficial effects that: the invention provides a method for researching the amplitude-variable vibration characteristic of an arm support of an aerial work platform, which mainly aims at carrying out spring equivalence on an amplitude-variable hydraulic cylinder in an amplitude-variable mechanism, and the amplitude-variable hydraulic cylinder is regarded as an elastic support, so that a triangular part consisting of an amplitude-variable oil cylinder, the arm support and a rotary table is prevented from being directly simplified into a rigid area, and the vibration mode and the frequency of the head part (namely a working platform) of the arm support in amplitude-variable motion can be better reflected; meanwhile, the vibration mode and the natural frequency obtained by calculating the equivalent spring stiffness of the luffing hydraulic cylinder are substituted into a state space equation of the vibration of the arm support, and dynamic simulation is carried out in a Matlab/Simulink environment to obtain the vibration response of the head of the arm support. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform can improve the accuracy of calculating the amplitude of the head of the boom and provide more accurate theoretical reference for vibration control of the aerial work platform.
Drawings
FIG. 1 is a process schematic diagram of a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform;
FIG. 2 is an equivalent schematic diagram of an aerial work platform model according to the present invention;
FIG. 3 is an equivalent schematic diagram of a hydraulic spring of the luffing hydraulic cylinder of the invention;
FIG. 4 is a graph of the vibration mode of the arm support according to the present invention;
FIG. 5 is a response curve of the vibration speed of the boom head according to the present invention;
FIG. 6 is a graph of boom displacement arc length response in accordance with the present invention;
fig. 7 is a comparison of the spring stiffness of the luffing cylinder with the vibration response of the rigid zone in consideration of the present invention.
Detailed Description
The present invention is further described with reference to the accompanying drawings, and the following examples are only for clearly illustrating the technical solutions of the present invention, and should not be taken as limiting the scope of the present invention.
The following describes a research method provided by the present invention, with a certain type of aerial work platform with a total length of 35.35 meters as a research object (specific parameters are shown in table 1):
TABLE 1 detailed boom parameters
Figure BDA0002657932600000071
As shown in fig. 1, the invention discloses a method for researching the amplitude variation vibration characteristic of an arm support of an aerial work platform, which comprises the following steps:
step one, performing spring equivalence on the amplitude-variable hydraulic oil cylinder, enabling hydraulic oil on two sides of the piston rod to be equivalent to hydraulic oil springs connected in parallel, enabling the piston rod to be equivalent to be connected in series with the hydraulic oil springs, and obtaining an equivalent post-stiffness calculation model, which is specifically shown in fig. 3.
Because the rigidity of the piston rod is far greater than the rigidity of the hydraulic oil, the rigidity of the piston rod can be ignored when calculating the equivalent rigidity of the elastic support of the amplitude-variable hydraulic oil cylinder, the equivalent rigidity K of the amplitude-variable cylinder of the elastic support of the amplitude-variable hydraulic oil cylinder is equal to the rigidity of the hydraulic oil spring, and the calculation method is as follows:
Figure BDA0002657932600000081
wherein x represents the displacement distance of the piston of the hydraulic cylinder, L represents the stroke, AK、ARRespectively representing the working areas of the left and right sides of the hydraulic cylinder, VK、VRRespectively representing the volumes of the left and right sides of the cylinder, VLK、VLRRespectively representing dead volumes of oil in pipelines at two sides of the hydraulic cylinder, betaeThe volume modulus of the hydraulic oil.
And step two, calculating by using a Hamilton principle to obtain a boundary condition, a continuity condition and a kinetic differential equation. The Hamilton principle formula is:
Figure 5
wherein, t1、t2The motion start time and the motion end time are respectively, T is the arm frame kinetic energy, W is the non-conservative force virtual work, V is the arm frame potential energy, and the equivalent schematic diagram of the aerial work platform model is shown in FIG. 2.
Figure 6
In the formula:
Figure BDA0002657932600000084
which represents the derivation of the time t,
Figure BDA0002657932600000085
representing the derivation of the neutral axis deflection of the boom over time t,
Figure BDA0002657932600000086
representing the derivative of the deflection of the top of the arm support to the time t,
Figure BDA0002657932600000087
Respectively represent the right deviation and the left deviation of the joint of the arm section i and the arm section i-1,
Figure 7
representing the derivative of the deflection of the top of the arm support to the time t and the coordinate z;
when the arm support potential energy V is considered, the obtained equivalent spring potential energy is considered,
Figure RE-GDA0002829229940000091
W＝Mθ(t)
in the formula: m represents the driving moment of the luffing cylinder to the arm support, rho Ai、EIi、viRespectively representing the linear density, flexural density and velocity, z, of the arm segment iiRepresenting the distance between the arm joint i and the origin along the z-axis, omega (z, t) representing the neutral axis deflection of the arm support, the superscript omega' representing the derivation of the coordinate z, g representing the inertial acceleration, theta representing the amplitude variation angle of the arm support, K representing the equivalent stiffness of the amplitude variation cylinder, and l0Represents the length of the amplitude variation cylinder in the initial state of the arm support, l1Representing the length of the current amplitude variation cylinder, l representing the total length of the arm support, a representing the distance from the hinge point of the amplitude variation cylinder to the bottom of the arm support, omega (a, t) representing the neutral axis deflection of the arm support at the hinge point of the hydraulic cylinder, alpha representing the included angle between the amplitude variation cylinder and the arm support, mc、JcRespectively representing the concentrated mass and the moment of inertia of the top of the arm support, omega (l, t) representing the neutral axis deflection of the tail end of the arm support, and omega (z, t) representing the second derivative of the neutral axis deflection of the arm support to a coordinate z.
The dynamic differential equation, the boundary condition and the continuous condition can be obtained through the formula as follows;
Figure RE-GDA0002829229940000092
Figure 11
Figure 12
in the formula: kαEquivalent rigidity K and (sin alpha) of variable amplitude cylinder2Product of (a), H (z-z)1 +) Representing unit step function, δ1The (z-a) functions each represent a dicke function which reflects the bearing force at the luffing cylinder only when z ═ a, Vi(z, t) represents the radian from a certain point on the arm support to the horizontal line, and comprises: vi(z,t)＝zθ+ω(z,t),z∈(zi-1 +,zi -)；
Wherein, Vi Ⅳ(z,t)、
Figure RE-GDA0002829229940000095
Respectively representing four, three and two derivatives of Vi (z, t) to the displacement coordinate z;
Figure RE-GDA0002829229940000096
represents the second derivative of Vi (z, t) over time t;
Figure RE-GDA0002829229940000097
representing the second derivative of the boom tip Vi (z, t) over time t and the first derivative of the displacement coordinate z.
And step three, homogenizing the inhomogeneous vibration differential equation obtained in the step two, and combining the vibration characteristics of the Euler Bernoulli beam and obtaining the initial solution of the vibration mode by utilizing Laplace transformation and inverse Laplace transformation.
The Euler Bernoulli beam has vibration characteristics of omega2＝EIiγi 4/ρAiWhere w is the natural frequency of beam vibration, γiIn order to be a characteristic value of the frequency,
and dynamic response Vi(z, t) can be determined by mode shape function
Figure BDA0002657932600000101
And generalized coordinates q (t) are expressed as:
Figure BDA0002657932600000102
wherein the subscript b denotes the b-th order mode,
Figure BDA0002657932600000103
and b-order mode-shape function representing any arm joint i is substituted into a kinetic differential equation to obtain:
Figure RE-GDA0002829229940000104
Figure RE-GDA0002829229940000105
carrying out Laplace transformation and inverse Laplace transformation on the obtained object to obtain a preliminary solution of the vibration mode;
Figure RE-GDA0002829229940000106
wherein:
Figure RE-GDA0002829229940000107
cosh and sinh are hyperbolic cosine and hyperbolic sine functions respectively,
Figure RE-GDA0002829229940000108
Figure RE-GDA0002829229940000109
Figure RE-GDA00028292299400001010
Figure RE-GDA00028292299400001011
step four, preliminarily solving the homogeneous differential equation set M of the vibration by utilizing the boundary condition, the continuity condition and the homogeneous differential equation in combination with the vibration modenPn0, conditional detM with non-zero solution according to its equation setnObtaining a vibration characteristic value and then obtaining the vibration mode and the vibration frequency of each section of arm support;
Figure RE-GDA0002829229940000111
Figure RE-GDA0002829229940000112
Figure RE-GDA0002829229940000113
Figure RE-GDA0002829229940000114
in the formula: e ═ gamman(l-zn-1)、
Figure BDA0002657932600000115
x1＝γi(z-zi-1 +)、 x3＝γi(z-a)、PnIs a coefficient vector consisting of the mode shape function and its derivative, H2＝cT(γia)H(zi-a)[1-H(zi-z1)]，H3＝cV(γia)H(zi-a)[1-H(zi-z1)]。
And step five, combining orthogonality among modal shapes, and considering a vibration differential equation in the first three-order main coordinate space as follows:
Figure RE-GDA0002829229940000121
Figure RE-GDA0002829229940000122
Figure RE-GDA0002829229940000123
in the formula: the damping ratio of the first three-order vibration mode is xi1＝0.05、ξ2When the value is 0.005, xi 3 is 0.005, w is divided into the first three natural frequencies, f is the inhomogeneous modal dynamic load, and the state vector is defined as:
Figure BDA0002657932600000121
obtaining a state space equation for vibration in Hilbert space
Figure BDA0002657932600000122
Wherein: g ═ g1(t) g2(t) g3(t) g4(t) g5(t) g6(t)]T，u＝θ， B＝[0 w,1 2f1 0 w,2 2f2 0 w,3 2f3]T，D＝0
Figure RE-GDA0002829229940000128
Figure 13
In the formula:
Figure BDA0002657932600000125
and the vibration mode function of the tail end of the arm frame of the first three steps is divided.
And sixthly, performing dynamic simulation in a Matlab/Simulink environment to obtain the vibration response of the arm support head along with the change of the elevation angle. Wherein the movement of the arm support is set as follows: 0 to 10s stop at the horizontal position, 10s to 30.4s rotate the arm support at the speed of 2 degrees/s, stop for 20s after reaching 50.8 degrees, then move to 72 degrees and stop at the speed of 1.53 degrees/s, the mode shape diagram of the arm support head of the aerial work platform is shown as the following figure 4, the speed response is shown as the figure 5, the response displacement is shown as the figure 6, and the comparison diagram is shown as the figure 7.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.

Claims (7)

1. A method for researching the amplitude variation vibration characteristic of an aerial work platform arm support is characterized by comprising the following steps: the method comprises the following steps:
s01, carrying out spring equivalence on the amplitude-variable hydraulic oil cylinder, enabling hydraulic oil on two sides of the piston rod to be equivalent to hydraulic oil springs connected in parallel, enabling the piston rod to be equivalent to be connected with the hydraulic oil springs in series, and obtaining an equivalent post-stiffness calculation model;
s02, calculating boundary conditions, continuity conditions and dynamic differential equations by using a Hamiltonian principle;
s03, homogenizing the non-homogeneous vibration differential equation obtained in S02, and obtaining a vibration mode preliminary solution by utilizing Laplace transform and inverse Laplace transform in combination with the vibration characteristic of an Euler Bernoulli beam;
s04, preliminarily solving a homogeneous differential equation set of vibration by using boundary conditions, continuity conditions, a homogeneous differential equation and a vibration mode to obtain a vibration characteristic value;
s05, obtaining a state space equation of vibration in the Hilbert space by combining orthogonality among mode shapes;
and S06, performing dynamic simulation in a Matlab/Simulink environment to obtain the vibration response of the arm support head along with the change of the elevation angle.
2. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform as claimed in claim 1, wherein the method comprises the following steps: in S01, the equivalent stiffness K of the amplitude variation cylinder of the elastic support of the amplitude variation hydraulic oil cylinder is equal to the spring stiffness of the hydraulic oil, and the calculation mode is as follows:
Figure RE-FDA0002829229930000011
wherein x represents the displacement distance of the piston of the hydraulic cylinder, L represents the stroke, AK、ARRespectively representing the working areas of the left and right sides of the hydraulic cylinder, VK、VRRespectively representing the volumes of the left and right sides of the cylinder, VLK、VLRRespectively representing dead volumes of oil in pipelines at two sides of the hydraulic cylinder, betaeThe volume modulus of the hydraulic oil.
3. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform as claimed in claim 2, wherein the method comprises the following steps: in S02, the whole arm support is divided into n sections with concentrated parameters and mutually continuous arm sections, and the Hamilton principle formula is utilized:
Figure RE-FDA0002829229930000021
wherein, t1、t2Respectively starting time and ending time of movement, T is the kinetic energy of the arm support, W is the virtual work of the non-conservative force, and V is the potential energy of the arm support;
Figure RE-FDA0002829229930000022
in the formula:
Figure RE-FDA0002829229930000023
respectively representing the right deviation and the left deviation of the joint of the arm joint i and the i-1, z represents the distance between a point on the arm support and the original point along the z axis, and rho AiRepresents the linear density of the arm segment i,
Figure RE-FDA0002829229930000024
represents the derivative of the amplitude variation motion angle theta (t) to the time t,
Figure RE-FDA0002829229930000025
represents the derivation of the neutral axis deflection omega (z, t) of the arm joint to the time t, mc、JcRespectively represents the concentrated mass and the moment of inertia of the top of the arm support, l represents the total length of the arm support,
Figure RE-FDA0002829229930000026
represents the derivative of the deflection omega (l, t) of the top of the arm support to the time t,
Figure RE-FDA0002829229930000027
representing the derivative of the top deflection omega (l, t) of the arm support to the time t and the coordinate z;
when the arm support potential energy V is considered, the obtained equivalent spring potential energy is considered,
Figure RE-FDA0002829229930000028
W＝Mθ(t)
in the formula: EI (El)iRepresenting the bending density of an arm section i, omega' (z, t) representing the neutral axis deflection omega (z, t) of the arm support and solving a quadratic derivative of a coordinate z, g representing the inertia acceleration, K representing the equivalent rigidity of the amplitude variation cylinder, and l0Represents the length of the amplitude variation cylinder in the initial state of the arm support, l1Representing the length of the current amplitude variation cylinder, a representing the distance from the hinge point of the amplitude variation cylinder to the bottom of the arm support, omega (a, t) representing the neutral axis deflection of the arm support at the hinge point of the hydraulic cylinder, alpha representing the included angle between the amplitude variation cylinder and the arm support, M representing amplitude variation oilThe driving moment of the cylinder to the boom.
4. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform as claimed in claim 3, wherein the method comprises the following steps: the kinetic differential equation, boundary conditions, and continuity conditions are respectively as follows:
Figure RE-FDA0002829229930000031
Figure RE-FDA0002829229930000032
Figure RE-FDA0002829229930000033
in the formula: v (z, t) represents the arc from a point on the boom to the horizontal, the subscript i represents the ith arm, and has:
Figure RE-FDA0002829229930000034
Vi Ⅳ(z, t) represents the fourth derivative of the displacement coordinate z,
Figure RE-FDA0002829229930000035
representing the second derivative over time t, KαEquivalent rigidity K and (sin alpha) of variable amplitude cylinder2Product of δ1The (z-a) functions each represent a dicke function, which reflects the bearing force at the luffing cylinder only at z ═ a, H (z-z)1 +) The unit of a step function is represented,
Figure RE-FDA0002829229930000036
representing the second derivative of the boom tip arc with respect to time t and the first derivative of the displacement coordinate z, Vn″(l,t)、Vn"(l, t) respectively represents the second and third derivatives of the radian of the tail end of the arm support to the displacement coordinate z,
Figure RE-FDA0002829229930000037
representing the second derivative of the boom tip arc with respect to time t,
Figure RE-FDA0002829229930000038
and
Figure RE-FDA0002829229930000039
the second and third derivatives of the radian measure at successive arm segments i and i +1, respectively, with respect to the displacement coordinate z.
5. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform as claimed in claim 4, wherein the method comprises the following steps: in S03, the Euler Bernoulli beam has a vibration characteristic of w2＝EIiγi 4/ρAiWhere w is the natural frequency of beam vibration, γiIn order to be a characteristic value of the frequency,
and dynamic response Vi(z, t) can be derived from the generalized mode shape basis function
Figure RE-FDA00028292299300000310
And generalized coordinates q (t) representing:
Figure RE-FDA00028292299300000311
wherein the subscript b denotes the b-th order mode,
Figure RE-FDA00028292299300000312
and b-order mode base functions representing any arm joint i are substituted into a kinetic differential equation to obtain:
Figure RE-FDA00028292299300000313
Figure RE-FDA00028292299300000314
wherein
Figure RE-FDA00028292299300000315
Is a mode shape basis function
Figure RE-FDA00028292299300000316
The fourth derivative of the displacement coordinate z.
Carrying out Laplace transformation and inverse Laplace transformation on the obtained object to obtain a preliminary solution of the vibration mode;
Figure RE-FDA0002829229930000041
wherein:
Figure RE-FDA0002829229930000042
cosh and sinh are hyperbolic cosine and hyperbolic sine functions respectively,
Figure RE-FDA0002829229930000043
Figure RE-FDA0002829229930000044
Figure RE-FDA0002829229930000045
Figure RE-FDA0002829229930000046
6. the aerial work platform boom of claim 5 with variable amplitudeThe method for researching the vibration characteristics is characterized in that: in S04, the homogeneous equation set is MnPn0, wherein
Figure RE-FDA0002829229930000047
Figure RE-FDA0002829229930000048
Figure RE-FDA0002829229930000049
Figure RE-FDA0002829229930000051
In the formula: mnCoefficient matrix, B, representing homogeneous equation1And BnRespectively representing the boundary condition coefficients of the bottom and the top of the arm support, Ci(z) coefficient matrix, P, representing the continuous conditions of the boomnIs a coefficient vector consisting of a mode shape function and a derivative thereof;
and has the following components: e ═ gamman(l-zn-1)、
Figure RE-FDA0002829229930000052
x1＝γi(z-zi-1 +)、x3＝γi(z-a)、H2＝cT(γia)H(zi-a)[1-H(zi-z1)]、
H3＝cV(γia)H(zi-a)[1-H(zi-z1)]。
7. The method for researching the amplitude variation vibration characteristic of the boom of the aerial work platform as claimed in claim 6, wherein the method comprises the following steps: using S04, taking the first three-order vibration mode of the tail end of the arm support to solve V (z, t) according to the orthogonality of the general solution V (z, t) of the arm support vibration differential equation1(t)、V2(t)、V3(t) the first three principal coordinate vibration differential equations in S05 can be obtained,
Figure RE-FDA0002829229930000053
Figure RE-FDA0002829229930000054
Figure RE-FDA0002829229930000055
in the formula:
Figure RE-FDA0002829229930000056
and
Figure RE-FDA0002829229930000057
representing the second derivative of the first three mode solutions,
Figure RE-FDA0002829229930000058
and
Figure RE-FDA0002829229930000059
first derivative, xi, representing the first third order mode solution1、ξ2And xi3Damping ratio of the first three vibration modes, w1、w2And w3The method is divided into the first three-order natural frequency, f is inhomogeneous modal dynamic load, and for establishing a state space equation of vibration in a Hilbert space, a state vector is defined as follows: g1(t)＝V1(t),
Figure RE-FDA00028292299300000510
g3(t)＝V2(t),
Figure RE-FDA00028292299300000511
g5(t)＝V3(t),
Figure RE-FDA00028292299300000512
Then there is
Figure RE-FDA00028292299300000513
y＝Cg+Du；
Wherein: g ═ g1(t) g2(t) g3(t) g4(t) g5(t) g6(t)]T，u＝θ，B＝[0 w1 2f 0 w2 2f 0 w3 2f]T，D＝0
Figure RE-FDA0002829229930000061
Figure 1
In the formula:
Figure RE-FDA0002829229930000063
and
Figure RE-FDA0002829229930000064
and the vibration mode basis function of the tail end of the arm frame of the first three steps is divided.
CN202010894245.9A 2020-08-31 2020-08-31 Method for researching amplitude variation vibration characteristic of aerial work platform arm support Withdrawn CN112214868A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116772944A (en) * 2023-08-25 2023-09-19 克拉玛依市燃气有限责任公司 Intelligent monitoring system and method for gas distribution station
CN117444991A (en) * 2023-12-26 2024-01-26 南通中远海运船务工程有限公司 Dynamic path planning method, multi-mechanical arm platform structure and related device

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116772944A (en) * 2023-08-25 2023-09-19 克拉玛依市燃气有限责任公司 Intelligent monitoring system and method for gas distribution station
CN116772944B (en) * 2023-08-25 2023-12-01 克拉玛依市燃气有限责任公司 Intelligent monitoring system and method for gas distribution station
CN117444991A (en) * 2023-12-26 2024-01-26 南通中远海运船务工程有限公司 Dynamic path planning method, multi-mechanical arm platform structure and related device
CN117444991B (en) * 2023-12-26 2024-04-19 南通中远海运船务工程有限公司 Dynamic path planning method, multi-mechanical arm platform structure and related device

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