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

Abstract

The invention discloses a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, several local coordinate systems are set up on beam length direction, beam is divided into by several beam sections by local coordinate system；Then the condition of continuity equation of displacement, corner, moment of flexure and shearing is set up between two neighboring beam section, and according to the boundary types of beam, 4 boundary condition equations is set up at beam two ends；The sectional twisting angle for finally causing the lateral displacement of each beam section and bending corresponds to substitution condition of continuity equation respectively and 4 boundary condition equations obtain system of homogeneous linear equations, and each rank natural frequency of Timoshenko beams is solved in substituting into system of homogeneous linear equations by assignment circular frequency successively.The present invention can break through the bottleneck of existing calculating Timoshenko deck-molding rank natural frequency Exact Solutions, and greatly extension solves scope, realizes the calculating of Timoshenko deck-molding rank natural frequency Exact Solutions.

Description

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

Technical field

The invention discloses a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, and in particular to beam class Structural dynamic characteristic analysis field.

Background technology

Beam structure accurately obtains its dynamic characteristics as most common class basic structure, and especially natural frequency is More concern in engineering, while the high frequent vibration problem of beam is all most important in machinery, aerospace field. Timoshenko beam theories are a kind of beam theories calculated suitable for deck-molding rank natural frequency, and a seat is set up by beam one end Mark system, sets up 4 boundary condition equations, obtains 4 rank systems of homogeneous linear equations, can be calculated Timoshenko beam higher-orders The Exact Solutions of (typically smaller than 12 ranks) natural frequency.But due to the limitation of common computer numerical stability, it is impossible to obtain more The Exact Solutions of high-order natural frequency, can only obtain its approximate solution by other numerical methods.Based on such consideration, it is necessary to consider A kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions of design.

The content of the invention

The technical problems to be solved by the invention are：For the side of existing calculating Timoshenko beam natural frequency Exact Solutions In method, due to the limitation of numerical computations condition, it is impossible to obtain the problem of the Exact Solutions of higher order natural frequency, propose to set up a kind of The method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions.

The present invention uses following technical scheme to solve above-mentioned technical problem：

The present invention proposes a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, comprises the following steps：

A kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, comprises the following steps：

(1) n local coordinate system, is set up on beam length direction, beam is divided into by n beam section by local coordinate system, built The unified calculation expression formula of the sectional twisting angle that the lateral displacement and bending for founding each beam section cause, n is the natural number more than 1；

(2) condition of continuity equation of displacement, corner, moment of flexure and shearing, n, are set up respectively between two neighboring beam section Individual beam section is obtained 4n-4 condition of continuity equation；According to the boundary types of beam, 4 boundary condition sides are set up at beam two ends Journey；

(3), the sectional twisting angle for causing the lateral displacement of each beam section and bending corresponds to substitute into what step (2) was set up respectively 4n-4 condition of continuity equation and 4 boundary condition equations, obtain 4n rank systems of homogeneous linear equations, then by assignment successively Circular frequency simultaneously substitutes into each rank natural frequency that Timoshenko beams are solved in 4n rank systems of homogeneous linear equations.

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, step (1) In, if i-th length of beam section is S_i, beam lengthI=1,2 ..., n-1；It is horizontal in each beam section local coordinate system Coordinate is x_i, dimensionless abscissa ζ_i=x_i/S_i, 0≤ζ_i≤1；Then i-th lateral displacement W of beam section_iWhat (ζ) and bending caused Sectional twisting angle ψ_i(ζ) is respectively：

W_i(ζ_i)=A_i cosh(γ₁S_iζ_i/L)+B_i sinh(γ₁S_iζ_i/L)+C_i cos(γ₂S_iζ_i/L)+D_i sin(γ₂S_iζ_i/L)

ψ_i(ζ_i)=A_im₁ sinh(γ₁S_iζ_i/L)+B_im₁ cosh(γ₁S_iζ_i/L)+C_im₂ sin(γ₂S_iζ_i/L)-D_im₂ cos(γ₂S_iζ_i/L)

Wherein S=θ r,β=τ (τ rs-1), E are its elastic modelling quantity, and G is modulus of shearing, and I is section inertia Square, ρ is density of material, and A is area of section, and k is cross-sectional shear coefficient, and ω is circular frequency, A_i,B_i,C_i,D_iRespectively step (3) In unknown number to be asked, sinh and cosh is respectively hyperbolic sine and hyperbolic cosine function.

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, step (2) It is the condition of continuity side that displacement, corner, moment of flexure and shearing are set up in i-th adjacent beam section and i+1 beam section junction Journey is as follows respectively:

The condition of continuity equation of displacement is：

The condition of continuity equation of corner is：

The condition of continuity equation of moment of flexure is：

The condition of continuity equation of shearing is：

Wherein, W_i'(ζ_i)、ψ_i'(ζ_i) lateral displacement W is represented respectively_iThe slope and corner ψ of (ζ)_iThe slope of (ζ).

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, each beam End boundary condition is combined by any two equation in following 4 boundary condition equations and obtained：

W (ζ)=0

ψ (ζ)=0

ψ ' (ζ)=0

W'(ζ)-ψ (ζ)=0；

Wherein, ζ₁=0, ζ_n=1.

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, three kinds often The beam-ends boundary condition seen is as follows：Fixing end boundary condition be W (ζ)=0 and ψ (ζ)=0, free end boundary condition be ψ ' (ζ)= 0 and W'(ζ)-ψ (ζ)=0, simply supported end boundary condition is W (ζ)=0 and ψ ' (ζ)=0.

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, by step (2) lateral displacement W in_iThe sectional twisting angle ψ that (ζ) and bending cause_i(ζ) substitutes into 4n-4 continuity equation and 4 boundary condition sides Journey, obtains with A_i,B_i,C_i,D_iIt is the 4n rank systems of homogeneous linear equations of unknown number.

Further, the method for calculating Timoshenko deck-molding rank natural frequency Exact Solutions proposed by the present invention, step (3) It is middle solution natural frequency method be：

Assignment circular frequency ω and the coefficient matrix D (ω) of system of homogeneous linear equations is substituted into successively in the range of solution, by it Determinant | D (ω) | obtains Y (ω),

Make the value of Y (ω) null ω, as j-th natural frequency of beam for j-th.

The present invention uses above technical scheme compared with prior art, with following technique effect：

The invention discloses a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, the method can break through The bottleneck of existing calculating Timoshenko deck-molding rank natural frequency Exact Solutions, greatly extension solve scope, realize The calculating of Timoshenko deck-molding rank natural frequency Exact Solutions.

Brief description of the drawings

Fig. 1 is the beam schematic diagram under local coordinate of the invention.

Fig. 2 is the Timoshenko beam natural frequency solving result figures for using prior art.

Fig. 3 is to use Timoshenko beams natural frequency solving result figure of the invention.

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

Specific embodiment

Technical scheme is described in further detail below in conjunction with the accompanying drawings：

Those skilled in the art of the present technique it is understood that unless otherwise defined, all terms used herein (including skill Art term and scientific terminology) have with art of the present invention in those of ordinary skill general understanding identical meaning.Also It should be understood that those terms defined in such as general dictionary should be understood that with the context of prior art in The consistent meaning of meaning, and unless defined as here, will not be explained with idealization or excessively formal implication.

First, as shown in figure 4, the present invention proposes a kind of side of calculating Timoshenko deck-molding rank natural frequency Exact Solutions Method, comprises the following steps：

(1), as shown in figure 1, setting up n local coordinate system on beam length direction, beam is divided into by n by local coordinate system Individual beam section；I-th length of beam section is S_i, beam lengthI=1 ..., n-1；It is horizontal in each beam section local coordinate system Coordinate is x_i, dimensionless abscissa ζ_i=x_i/S_i, 0≤ζ_i≤1；Then i-th lateral displacement W of beam section_iWhat (ζ) and bending caused Sectional twisting angle ψ_i(ζ) is respectively：

W_i(ζ_i)=A_i cosh(γ₁S_iζ_i/L)+B_i sinh(γ₁S_iζ_i/L)+C_i cos(γ₂S_iζ_i/L)+D_i sin(γ₂S_iζ_i/ L),

ψ_i(ζ_i)=A_im₁ sinh(γ₁S_iζ_i/L)+B_im₁ cosh(γ₁S_iζ_i/L)+C_im₂ sin(γ₂S_iζ_i/L)-D_im₂ cos(γ₂S_iζ_i/ L),

Wherein S=θ r,β=τ (τ rs-1), E are its elastic modelling quantity, and G is modulus of shearing, and I is section inertia Square, ρ is density of material, and A is area of section, and k is cross-sectional shear coefficient, and ω is circular frequency, A_i,B_i,C_i,D_iRespectively step (3) In unknown number to be asked, sinh and cosh is respectively hyperbolic sine and hyperbolic cosine function.

(2) condition of continuity equation of displacement, corner, moment of flexure and shearing, is set up between two neighboring beam section, is obtained 4n-4 equation；Specifically, displacement, corner, moment of flexure and shearing are set up in i-th adjacent beam section and i+1 beam section junction Condition of continuity equation respectively it is as follows:

One end is clamped, and free 4 boundary condition equations in one end are specially：

One end is fixed, and 4 boundary condition equations of one end freely-supported are specially：

4 boundary condition equations of two ends freely-supported are specially：

(3), by lateral displacement W_iThe sectional twisting angle ψ that (ζ) and bending cause_i(ζ) substitutes into 4n-4 continuity equation and 4 Boundary condition equation, obtains with A_i,B_i,C_i,D_iIt is the 4n rank systems of homogeneous linear equations of unknown number, solves Timoshenko beams each Rank natural frequency.

System of homogeneous linear equations is D (ω) C=0

Wherein wait to seek unknown number vector C=(A₁,B₁,C₁,D₁,A₂,B₂,C₂,D₂,...,A_n,B_n,C_n,D_n)^T

Coefficient matrix

P_i=(P_i,1 P_i,2)

It is clamped for one end, the boundary condition of one end freely-supported：

It is clamped for one end, the boundary condition of one end freely-supported：

For the boundary condition of two ends freely-supported：

(3), assignment circular frequency ω and the coefficient matrix D (ω) of system of homogeneous linear equations is substituted into successively in the range of solution In, by its determinant, | D (ω) | obtains Y (ω),

Make the value of Y (ω) null ω, as j-th natural frequency of beam for j-th.

The technical scheme of concrete example explanation below, square-section cantilever beam used by the present embodiment (one end is clamped, One end freedom) elastic modelling quantity is 70Mpa, modulus of shearing is 26.3Mpa, and cross-sectional shear coefficient is 0.851, and density is 2700kg/ m³, length is 1m, and sectional dimension is 0.01m × 0.01m.

First, 4 local coordinate systems are set up on beam length direction according to step (1), is divided beam by local coordinate system Into 4 beam sections, each beam section length is 0.25m.

Secondly, built in the i-th adjacent (i=1,2,3) individual beam section and i+1 beam section junction according to step (2)-(3) Found 12 condition of continuity equations of displacement, corner, moment of flexure and shearing:

With 4 boundary condition equations：

Obtain 16 rank systems of homogeneous linear equations.

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

Make j-th natural frequency of the value of Y (ω) null ω, i.e. beam for j-th.

Fig. 2 is that Fig. 3 is using of the invention using the Timoshenko beam natural frequency solving result figures of prior art Timoshenko beam natural frequency solving result figures, the Timoshenko beam natures obtained using prior art and using the present invention As listed in table 1, comparing result to can be found that and can only calculate 11 rank natural frequencies before Timoshenko beams using prior art frequency, Numerical computations mistake just occurs afterwards, and uses a kind of calculating Timoshenko deck-molding ranks natural frequency proposed by the present invention accurate Really the method for solution, can effectively calculate at least preceding 18 rank natural frequency.

Table 1

The above is only some embodiments of the invention, it is noted that for the ordinary skill people of the art For member, under the premise without departing from the principles of the invention, some improvements and modifications can also be made, these improvements and modifications also should It is considered as protection scope of the present invention.

Claims (7)

1. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions, it is characterised in that comprise the following steps：

(1) n local coordinate system, is set up on beam length direction, beam is divided into by n beam section by local coordinate system, set up each The unified calculation expression formula of the sectional twisting angle that the lateral displacement of beam section and bending cause, n is the natural number more than 1；

(2) condition of continuity equation of displacement, corner, moment of flexure and shearing, n beam, are set up respectively between two neighboring beam section Section is obtained 4n-4 condition of continuity equation；According to the boundary types of beam, 4 boundary condition equations are set up at beam two ends；

(3), the sectional twisting angle for causing the lateral displacement of each beam section and bending corresponds to substitute into the 4n-4 that step (2) is set up respectively Individual condition of continuity equation and 4 boundary condition equations, obtain 4n rank systems of homogeneous linear equations, then justify frequency by assignment successively Rate simultaneously substitutes into each rank natural frequency that Timoshenko beams are solved in 4n rank systems of homogeneous linear equations.

2. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 1, its feature It is, in step (1), if i-th length of beam section is S_i, beam lengthI=1,2 ..., n-1；In each beam section office Abscissa is x in portion's coordinate system_i, dimensionless abscissa ζ_i=x_i/S_i, 0≤ζ_i≤1；Then i-th lateral displacement W of beam section_i(ζ) The sectional twisting angle ψ caused with bending_i(ζ) is respectively：

W_i(ζ_i)=A_i cosh(γ₁S_iζ_i/L)+B_i sinh(γ₁S_iζ_i/L)+C_i cos(γ₂S_iζ_i/L)+D_i sin(γ₂S_iζ_i/ L)

ψ_i(ζ_i)=A_im₁ sinh(γ₁S_iζ_i/L)+B_im₁ cosh(γ₁S_iζ_i/L)+C_im₂ sin(γ₂S_iζ_i/L)-D_im₂ cos (γ₂S_iζ_i/L)

WhereinS=θ R,β=τ (τ rs-1), E are its elastic modelling quantity, and G is modulus of shearing, and I is cross sectional moment of inertia, ρ It is density of material, A is area of section, and k is cross-sectional shear coefficient, and ω is circular frequency, A_i,B_i,C_i,D_iRespectively treated in step (3) Unknown number, sinh and cosh is asked to be respectively hyperbolic sine and hyperbolic cosine function.

3. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 2, its feature It is that step (2) is to set up displacement, corner, moment of flexure and shearing in i-th adjacent beam section and i+1 beam section junction Condition of continuity equation, it is as follows respectively:

The condition of continuity equation of displacement is：

The condition of continuity equation of corner is：

The condition of continuity equation of moment of flexure is：

The condition of continuity equation of shearing is： Wherein, Wc_i(ζ_i)、ψ′_i(ζ_i) lateral displacement W is represented respectively_iThe slope and corner ψ of (ζ)_iThe slope of (ζ).

4. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 2, its feature It is that each beam-ends boundary condition is combined by any two equation in following 4 boundary condition equations and obtained：

W (ζ)=0

ψ (ζ)=0

ψ ' (ζ)=0

W'(ζ)-ψ (ζ)=0；

Wherein, ζ₁=0, ζ_n=1.

5. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 4, its feature It is that three kinds of common beam-ends boundary conditions are as follows：Fixing end boundary condition is W (ζ)=0 and ψ (ζ)=0, free end perimeter strip Part is ψ ' (ζ)=0 and W'(ζ)-ψ (ζ)=0, simply supported end boundary condition is W (ζ)=0 and ψ ' (ζ)=0.

6. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 3, its feature It is, by lateral displacement W in step (2)_iThe sectional twisting angle ψ that (ζ) and bending cause_i(ζ) substitutes into 4n-4 continuity equation and 4 Individual boundary condition equation, obtains with A_i,B_i,C_i,D_iIt is the 4n rank systems of homogeneous linear equations of unknown number.

7. a kind of method of calculating Timoshenko deck-molding rank natural frequency Exact Solutions according to claim 6, its feature It is to solve natural frequency method in step (3) to be：

Assignment circular frequency ω and the coefficient matrix D (ω) of system of homogeneous linear equations is substituted into successively in the range of solution, by its ranks Formula | D (ω) | obtains Y (ω),

$Y\left(\mathrm{\ω}\right)={\sinh }^{-1}\left(\right|D\left(\mathrm{\ω}\right)\left|\right)=ln\left(\right|D\left(\mathrm{\ω}\right)|+\sqrt{|D\left(\mathrm{\ω}\right){|}^2+1});$

Make the value of Y (ω) null ω, as j-th natural frequency of beam for j-th.