CN106776483A - A kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions - Google Patents

Abstract

The invention discloses a method for calculating the exact solution of the high-order natural frequency of a Timoshenko beam. Several local coordinate systems are established in the length direction of the beam, and the beam is divided into several beam sections through the local coordinate system; and then between two adjacent beam sections Establish the continuity condition equations of displacement, rotation angle, bending moment and shear force, and establish four boundary condition equations at both ends of the beam according to the boundary type of the beam; finally, the lateral displacement of each beam segment and the section rotation angle caused by bending correspond to The homogeneous linear equations were obtained by substituting the continuity condition equation and four boundary condition equations, and the natural frequencies of each order of the Timoshenko beam were solved by substituting the circular frequency in sequence into the homogeneous linear equations. The invention can break through the existing bottleneck of calculating the accurate solution of the high-order natural frequency of the Timoshenko beam, greatly expand the solution range, and realize the calculation of the accurate solution of the high-order natural frequency of the Timoshenko beam.

Description

A Method for Calculating the Exact Solution of Timoshenko Beam Higher-Order Natural Frequency

technical field

The invention discloses a method for calculating the exact solution of the high-order natural frequency of a Timoshenko beam, and specifically relates to the field of dynamic characteristic analysis of beam structures.

Background technique

Beam structure is the most common type of basic structure. Accurately obtaining its dynamic characteristics, especially the natural frequency, is a concern in engineering. At the same time, the high-order vibration of beams is very important in the fields of machinery and aerospace. The Timoshenko beam theory is a beam theory suitable for the calculation of high-order natural frequencies of beams. By establishing a coordinate system at one end of the beam and establishing four boundary condition equations, the fourth-order homogeneous linear equations can be obtained, and the higher-order Timoshenko beam can be calculated ( Usually less than the 12th order) the exact solution of the natural frequency. However, due to the limitation of the numerical calculation accuracy of ordinary computers, the exact solution of higher order natural frequencies cannot be obtained, and the approximate solution can only be obtained by other numerical methods. Based on this consideration, it is necessary to consider designing a method to calculate the exact solution of the high-order natural frequencies of Timoshenko beams.

Contents of the invention

The technical problem to be solved by the present invention is: in view of the existing method for calculating the exact solution of the natural frequency of the Timoshenko beam, due to the limitation of numerical calculation conditions, it is impossible to obtain the exact solution of the higher-order natural frequency, and it is proposed to establish a method for calculating the high-order A method for the exact solution of natural frequencies.

The present invention adopts the following technical solutions for solving the problems of the technologies described above:

The present invention proposes a method for calculating the exact solution of Timoshenko beam high-order natural frequency, comprising the following steps:

A method for calculating the exact solution of high-order natural frequencies of Timoshenko beams, comprising the following steps:

(1) Establish n local coordinate systems in the length direction of the beam, divide the beam into n beam segments through the local coordinate system, and establish a unified calculation expression for the lateral displacement of each beam segment and the section rotation angle caused by bending, n is greater than the natural number of 1;

(2), respectively establish the continuity condition equations of displacement, rotation angle, bending moment and shear force between two adjacent beam sections, n beam sections get 4n-4 continuity condition equations in total; according to the boundary type of the beam , establish four boundary condition equations at both ends of the beam;

(3), the lateral displacement of each beam segment and the section rotation angle caused by bending are respectively substituted into the 4n-4 continuity condition equations and 4 boundary condition equations established in step (2), to obtain a 4n order homogeneous linear equation group, Then, the natural frequencies of each order of the Timoshenko beam are solved by assigning the circular frequencies in turn and substituting them into the 4n order homogeneous linear equations.

Further, in the method for calculating the exact solution of Timoshenko beam high-order natural frequency proposed by the present invention, in step (1), the length of the i-th beam segment is set to S _i , and the beam length i=1,2,...,n-1; in the local coordinate system of each beam segment, the abscissa is x _i , the dimensionless abscissa ζ _i =x _i /S _i , 0≤ζ _i ≤1; then The lateral displacement W _i (ζ) of the i-th beam segment and the section rotation angle ψ _i (ζ) caused by bending are respectively:

W _i (ζ _i )＝A _i cosh(γ ₁ S _i ζ _i /L)+B _i sinh(γ ₁ S _i ζ _i /L)+C _i cos(γ ₂ S _i ζ _i /L)+D _i sin(γ ₂ S _i ζ _i /L)

ψ _i (ζ _i )＝A _i m ₁ sinh(γ ₁ S _i ζ _i /L)+B _i m ₁ cosh(γ ₁ S _i ζ _i /L)+C _i m ₂ sin(γ ₂ S _i ζ _i /L)-D _i m ₂ cos(γ ₂ S _i ζ _i /L)

in s = θr, β=τ(τrs-1), E is the elastic modulus, G is the shear modulus, I is the section moment of inertia, ρ is the material density, A is the section area, k is the section shear coefficient, and ω is the circular frequency , A _i , B _i , C _i , D _i are the unknowns to be sought in step (3) respectively, sinh and cosh are hyperbolic sine and hyperbolic cosine functions respectively.

Further, in the method for calculating the exact solution of Timoshenko beam high-order natural frequency proposed by the present invention, step (2) is to establish displacement, rotation angle, bending moment and shear at the junction of adjacent i-th beam section and i+1-th beam section The force continuity conditional equations are as follows:

The continuity condition equation of displacement is:

The continuity condition equation of the rotation angle is:

The continuity condition equation of bending moment is:

The continuity condition equation of shear force is:

Wherein, W _i '(ζ _i ), ψ _i '(ζ _i ) respectively represent the slope of the lateral displacement W _i (ζ) and the slope of the rotation angle ψ _i (ζ).

Further, in the method for calculating the exact solution of Timoshenko beam high-order natural frequency proposed by the present invention, each beam end boundary condition is obtained by combining any two equations in the following 4 boundary condition equations:

W(ζ)=0

ψ(ζ)=0

ψ'(ζ)=0

W'(ζ)-ψ(ζ)=0;

Wherein, ζ ₁ =0, ζ _n =1.

Further, in the method for calculating the exact solution of Timoshenko beam high-order natural frequencies proposed by the present invention, three common beam end boundary conditions are as follows: the fixed end boundary conditions are W(ζ)=0 and ψ(ζ)=0, the free end boundary conditions ψ'(ζ)=0 and W'(ζ)-ψ(ζ)=0, the simply supported boundary conditions are W(ζ)=0 and ψ'(ζ)=0.

Further, the method for calculating the exact solution of the high-order natural frequency of the Timoshenko beam proposed by the present invention is to substitute the lateral displacement W _i (ζ) and the section rotation angle ψ _i (ζ) caused by bending into 4n-4 continuity equations and Four boundary condition equations are used to obtain a 4nth-order homogeneous linear equation system with A _i , B _i , C _i , and D _i as unknowns.

Further, the method for calculating the exact solution of Timoshenko beam high-order natural frequency proposed by the present invention, the method for solving natural frequency in step (3) is:

Assign the circular frequency ω sequentially within the solution range and substitute it into the coefficient matrix D(ω) of the homogeneous linear equation system, and obtain Y(ω) from its determinant |D(ω)|,

The jth value of ω that makes Y(ω) equal to zero is the jth natural frequency of the beam.

Compared with the prior art, the present invention adopts the above technical scheme and has the following technical effects:

The invention discloses a method for calculating the accurate solution of the high-order natural frequency of the Timoshenko beam. The method can break through the existing bottleneck of calculating the accurate solution of the high-order natural frequency of the Timoshenko beam, greatly expand the solution range, and realize the calculation of the accurate solution of the high-order natural frequency of the Timoshenko beam.

Description of drawings

Fig. 1 is a schematic diagram of a beam in local coordinates of the present invention.

Fig. 2 is a result diagram of solving the natural frequency of a Timoshenko beam using the prior art.

Fig. 3 is a result diagram of solving the natural frequency of the Timoshenko beam of the present invention.

Fig. 4 is a flow chart of the method of the present invention.

detailed description

Below in conjunction with accompanying drawing, technical scheme of the present invention is described in further detail:

Those skilled in the art can understand that, unless otherwise defined, all terms (including technical terms and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. It should also be understood that terms such as those defined in commonly used dictionaries should be understood to have a meaning consistent with the meaning in the context of the prior art, and will not be interpreted in an idealized or overly formal sense unless defined as herein Explanation.

At first, as shown in Figure 4, the present invention proposes a kind of method for calculating the exact solution of Timoshenko beam high-order natural frequency, comprising the following steps:

(1) As shown in Figure 1, n local coordinate systems are established in the beam length direction, and the beam is divided into n beam segments through the local coordinate system; the length of the i-th beam segment is S _i , and the beam length i=1,...,n-1; in the local coordinate system of each beam segment, the abscissa is x _i , and the dimensionless abscissa ζ _i =x _i /S _i , 0≤ζ _i ≤1; then i The transverse displacement W _i (ζ) of each beam segment and the section rotation angle ψ _i (ζ) caused by bending are respectively:

W _i (ζ _i )＝A _i cosh(γ ₁ S _i ζ _i /L)+B _i sinh(γ ₁ S _i ζ _i /L)+C _i cos(γ ₂ S _i ζ _i /L)+D _i sin(γ ₂ S _i ζ _i /L),

ψ _i (ζ _i )＝A _i m ₁ sinh(γ ₁ S _i ζ _i /L)+B _i m ₁ cosh(γ ₁ S _i ζ _i /L)+C _i m ₂ sin(γ ₂ S _i ζ _i /L)-D _i m ₂ cos(γ ₂ S _i ζ _i /L),

in s = θr, β=τ(τrs-1), E is the elastic modulus, G is the shear modulus, I is the section moment of inertia, ρ is the material density, A is the section area, k is the section shear coefficient, and ω is the circular frequency , A _i , B _i , C _i , D _i are the unknowns to be sought in step (3) respectively, sinh and cosh are hyperbolic sine and hyperbolic cosine functions respectively.

(2), establish the continuity conditional equations of displacement, rotation angle, bending moment and shear force between two adjacent beam sections, and obtain 4n-4 equations; specifically, the adjacent i-th beam section and the i-th beam section The continuity conditional equations of displacement, rotation angle, bending moment and shear force established at the connection of +1 beam segments are as follows:

The four boundary condition equations with one end fixed and one end free are as follows:

The four boundary condition equations with one end fixed and one end simply supported are as follows:

The four boundary condition equations simply supported at both ends are as follows:

(3) Substituting the lateral displacement W _i (ζ) and the section rotation angle ψ _i (ζ) caused by bending into 4n-4 continuity equations and 4 boundary condition equations, A _i , B _i , C _i , D _i is a 4n order homogeneous linear equation system of unknowns, and solve the natural frequency of each order of Timoshenko beam.

The homogeneous linear equation system is D(ω)C=0

Among them, the unknown vector C=(A ₁ ,B ₁ ,C ₁ ,D ₁ ,A ₂ ,B ₂ ,C ₂ ,D ₂ ,...,A _n ,B _n ,C _n ,D _n ) ^T

coefficient matrix

P _i ＝(P _i,1 P _i,2 )

For boundary conditions that are fixed at one end and simply supported at one end:

For boundary conditions that are fixed at one end and simply supported at one end:

For boundary conditions simply supported at both ends:

(3) Assign the circular frequency ω sequentially within the solution range and substitute it into the coefficient matrix D(ω) of the homogeneous linear equation system, and obtain Y(ω) from its determinant |D(ω)|,

The jth value of ω that makes Y(ω) equal to zero is the jth natural frequency of the beam.

The technical scheme of the present invention is illustrated in detail below. The used rectangular cross-section cantilever beam (one end is fixedly supported, and the other end is free) elastic modulus of the present embodiment is 70Mpa, and the shear modulus is 26.3Mpa, and the section shear coefficient is 0.851, and the density is 2700kg /m ³ , the length is 1m, and the cross-sectional size is 0.01m×0.01m.

First, according to step (1), four local coordinate systems are established in the beam length direction, and the beam is divided into four beam segments through the local coordinate system, and the length of each beam segment is 0.25m.

Secondly, according to steps (2)-(3), establish 12 displacement, rotation angle, bending moment and shear Force continuity condition equation:

and 4 boundary condition equations:

A system of homogeneous linear equations of order 16 is obtained.

Finally, the circular frequency ω is assigned sequentially from 0 to 500rad/s and substituted into the coefficient matrix D(ω), and Y(ω) is obtained from its determinant |D(ω)|,

The jth value of ω that sets Y(ω) equal to zero, i.e. the jth natural frequency of the beam.

Fig. 2 is to use the Timoshenko beam natural frequency solution result figure of prior art, Fig. 3 is to use the Timoshenko beam natural frequency solution result figure of the present invention, use prior art and use the Timoshenko beam natural frequency that the present invention obtains as listed in table 1 , the comparison results show that using the existing technology can only calculate the first 11 natural frequencies of the Timoshenko beam, and then there will be numerical calculation errors, but using a method for calculating the exact solution of the high-order natural frequencies of the Timoshenko beam proposed by the present invention can effectively calculate at least The first 18 natural frequencies.

Table 1

The above descriptions are only part of the embodiments of the present invention. It should be pointed out that those skilled in the art can make some improvements and modifications without departing from the principles of the present invention. It should be regarded as the protection scope of the present invention.

Claims (7)

1. a method for calculating the exact solution of Timoshenko beam high-order natural frequency, is characterized in that, comprises the steps: (1) Establish n local coordinate systems in the length direction of the beam, divide the beam into n beam segments through the local coordinate system, and establish a unified calculation expression for the lateral displacement of each beam segment and the section rotation angle caused by bending, n is greater than the natural number of 1; (2), respectively establish the continuity condition equations of displacement, rotation angle, bending moment and shear force between two adjacent beam sections, n beam sections get 4n-4 continuity condition equations in total; according to the boundary type of the beam , establish four boundary condition equations at both ends of the beam; (3), the lateral displacement of each beam segment and the section rotation angle caused by bending are respectively substituted into the 4n-4 continuity condition equations and 4 boundary condition equations established in step (2), to obtain a 4n order homogeneous linear equation group, Then, the natural frequencies of each order of the Timoshenko beam are solved by assigning the circular frequencies in turn and substituting them into the 4n order homogeneous linear equations. 2. a kind of method for calculating the exact solution of Timoshenko beam high-order natural frequency according to claim 1 is characterized in that, in step (1), the length of the i beam section is set as S _i , the beam length i=1,2,...,n-1; in the local coordinate system of each beam segment, the abscissa is x _i , the dimensionless abscissa ζ _i =x _i /S _i , 0≤ζ _i ≤1; then The lateral displacement W _i (ζ) of the i-th beam segment and the section rotation angle ψ _i (ζ) caused by bending are respectively: W _i (ζ _i )＝A _i cosh(γ ₁ S _i ζ _i /L)+B _i sinh(γ ₁ S _i ζ _i /L)+C _i cos(γ ₂ S _i ζ _i /L)+D _i sin(γ ₂ S _i ζ _i /L) ψ _i (ζ _i )＝A _i m ₁ sinh(γ ₁ S _i ζ _i /L)+B _i m ₁ cosh(γ ₁ S _i ζ _i /L)+C _i m ₂ sin(γ ₂ S _i ζ _i /L)-D _i m ₂ cos(γ ₂ S _i ζ _i /L) in s = θr, β=τ(τrs-1), E is the elastic modulus, G is the shear modulus, I is the section moment of inertia, ρ is the material density, A is the section area, k is the section shear coefficient, and ω is the circular frequency , A _i , B _i , C _i , D _i are the unknowns to be sought in step (3) respectively, sinh and cosh are hyperbolic sine and hyperbolic cosine functions respectively. 3. a kind of method for calculating the accurate solution of Timoshenko beam high-order natural frequency according to claim 2, is characterized in that, step (2) is to set up at adjacent i beam section and i+1 beam section connection The continuity conditional equations of displacement, rotation angle, bending moment and shear force are as follows: The continuity condition equation of displacement is: The continuity condition equation of the rotation angle is: The continuity condition equation of bending moment is: The continuity condition equation of shear force is: Among them, Wc _i (ζ _i ), ψ′ _i (ζ _i ) represent the slope of lateral displacement W _i (ζ) and the slope of rotation angle ψ _i (ζ) respectively. 4. a kind of method for calculating the accurate solution of Timoshenko beam high-order natural frequency according to claim 2, is characterized in that, each beam end boundary condition is obtained by any two equation combinations in following 4 boundary condition equations: W(ζ)=0 ψ(ζ)=0 ψ'(ζ)=0 W'(ζ)-ψ(ζ)=0; Wherein, ζ ₁ =0, ζ _n =1. 5. a kind of method according to claim 4 calculating the accurate solution of Timoshenko beam high-order natural frequency is characterized in that, three kinds of common beam end boundary conditions are as follows: fixed end boundary condition is W (ζ)=0 and ψ (ζ )=0, the free end boundary conditions are ψ'(ζ)=0 and W'(ζ)-ψ(ζ)=0, the simply supported end boundary conditions are W(ζ)=0 and ψ'(ζ)=0 . 6. a kind of method according to claim 3 calculating the accurate solution of Timoshenko beam high-order natural frequency, is characterized in that, lateral displacement W _i (ζ) and section angle ψ _i (ζ) caused by bending in step (2) are substituted into 4n-4 continuity equations and 4 boundary condition equations, get 4n order homogeneous linear equations with A _i , B _i , C _i , D _i as unknowns. 7. a kind of method for calculating the exact solution of Timoshenko beam high-order natural frequency according to claim 6, is characterized in that, in the step (3), the natural frequency method for solving is: Assign the circular frequency ω sequentially within the solution range and substitute it into the coefficient matrix D(ω) of the homogeneous linear equation system, and obtain Y(ω) from its determinant |D(ω)|, $\mathrm{<span\; class="notranslate">\; Y</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; sinh</span>}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; l</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>\sqrt{<span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; D.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; \&omega;</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$ The jth value of ω that makes Y(ω) equal to zero is the jth natural frequency of the beam.