CN102880807A - Transfer matrix computing method applicable to beam bending vibration analysis - Google Patents

Abstract

The invention discloses a transfer matrix computing method applicable to beam bending vibration analysis. The transfer matrix computing method includes the following steps of constructing a non-wellknown field matrix (7) of a beam unit by wellknown beam deflection curve equation formula (6); multiplying the field matrix (7) and a spot matrix (8) of the beam unit to obtain a transfer matrix (9) of the beam unit; setting a transfer equation formula (10) of the beam unit; dividing the beam into n units, combining transfer equation formula of the n units into a transfer equation formula (11) of the beam; and obtaining inherent frequency and vibration mode of the beam by introducing the boundary conditions at two ends of the beam into the transfer equation formula (11). The transfer matrix is set directly from the deflection function of the beam, the boundary conditions are introduced to compute, and compared with the prior art, the transfer matrix computing method has the advantages of wide application range and simple computing method.

Description

Be suitable for the transfer matrix computing method that Beam Vibration is analyzed

Technical field

The present invention relates to the computing method that Beam Vibration is analyzed, relate in particular to a kind of transfer matrix computing method that Beam Vibration is analyzed that are suitable for.

Background technology

Existing transfer matrix computing method for the Beam Vibration analysis have two kinds of diverse ways, first method adopts spring mass system, second method then directly adopts the beam segmentation of distributed mass to calculate, therefore, be identical with the overall calculation that adopts distributed parameter system, lost the meaning of transfer matrix method.

The beam element field matrix of the first transfer matrix method is:

${\left[T_f\right]}_i=\left[\begin{array}{cc}1& 1/k_i\\ 0& 1\end{array}\right]---\left(1\right)$

K wherein _iBe the bending stiffness of i beam element, need to calculate according to the different distortion form of beam, and can only calculate an end and fix freely structure of an end, such as building structure, and can only calculate two kinds of distortion patterns---cut curved and flexure type.

In the second method, the Rigidity Calculation of beam element flexural vibrations depends on the model function of vibration of beam:

φ(x)＝D ₁chax+D ₂shax+D ₃cosax+D ₄sinax (2)

In the formula:

$a={\left(\frac{\stackrel{\‾}{m}{\mathrm{\ω}}^2}{\mathrm{EI}}\right)}^{1/4}---\left(3\right)$

Wherein:

ω, EI are respectively quality, natural frequency and the cross section bendind rigidity of beam unit length.

Coefficient in the formula (2) then depends on the constraint condition at beam two ends.During Analytical Solution, except the free beam of pin-ended, the calculating of other constraint condition all needs to find the solution transcendental equation.Therefore, it is identical directly calculating as distributed parameter system with whole beam.The appearance of transfer matrix method is exactly for fear of the solution transcendental equation, and therefore, second method has lost the meaning that the method exists.

The field transfer matrix of the second transfer matrix method is:

${\left[T_f\right]}_i=\left[\begin{array}{cccc}{A}_i& a_i^{-1}{B}_i& a_i^{-2}{C}_i& -a_i^{-3}{D}_i\\ a_i{D}_i& A_i& a_i^{-1}{B}_i& -a_i^{-2}{C}_i\\ -a_i^2{C}_i& -a_i{D}_i& -A_i& a_i^{-1}{B}_i\\ a_i^3{B}_i& a_i^2{C}_i& a_i{D}_i& -A_i\end{array}\right]---\left(4\right)$

Wherein:

$A_i=\frac{1}{2}({\mathrm{cha}}_i{l}_i+\cos a_i{l}_i)$

$B_i=\frac{1}{2}(\mathrm{sh}{a}_i{l}_i+\sin a_i{l}_i)$ (5)

$C_i=\frac{1}{2}(\mathrm{ch}{a}_i{l}_i-\cos a_i{l}_i)$

$D_i=\frac{1}{2}(\mathrm{sh}{a}_i{l}_i-\sin a_i{l}_i)$

Can see that from formula (4), formula (5) and formula (2) the appearance matrix has comprised all the model function of vibration, therefore, also must face the calculating of transcendental equation when finding the solution.

The deficiencies in the prior art part is as follows:

1, the limitation of the first transfer matrix method is that mechanical model is the structure of semi-girder, and can only calculate the structure of two kinds of form of distortion.

2, there are same problem in second method and analytic method---and find the solution transcendental equation, therefore, lost the meaning that the method exists.

Summary of the invention

The technical problem to be solved in the present invention provides a kind of transfer matrix computing method that Beam Vibration is analyzed that are suitable for.The method simply and not reduces computational accuracy.

In order to solve the problems of the technologies described above, the technical solution used in the present invention is: be suitable for the transfer matrix computing method that Beam Vibration is analyzed, may further comprise the steps:

Deflection curve equation formula (6) with known beam:

$y\left(x\right)=y^L+{\mathrm{\θ}}^{L}x+\frac{1}{2}{\stackrel{\‾}{M}}^L{x}^2+\frac{1}{6}{\stackrel{\‾}{N}}^L{x}^3---\left(6\right)$

Construct non-known beam element field matrix form (7):

${\left[T_f\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ 0& 0& 0& 1\end{array}\right]---\left(7\right)$

In the formula (6): y ^L, θ ^L,

Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of beam left end, and Equivalent Moment and equivalent shearing divide in addition and are

Wherein: EI is the beam section bending stiffness;

In the formula (7): subscript i is the numbering of beam element, l _iThe length of i beam element;

Then with the field matrix form (7) of beam element and dot matrix type (8)

${\left[T_p\right]}_i=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ {\stackrel{\‾}{a}}_i& 0& 0& 1\end{array}\right]---\left(8\right)$

The transfer matrix formula (9) that multiplies each other and obtain beam element:

${\left[T\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ {\stackrel{\‾}{a}}_i& {\stackrel{\‾}{a}}_i{l}_i& {\stackrel{\‾}{a}}_i{l}_i^2/2& {\stackrel{\‾}{a}}_i{l}_i^3/6+1\end{array}\right]---\left(9\right)$

Thereby can set up the transfer equation (10) of beam element:

$\left\{\begin{array}{c}{y}_{i+1}^L\\ {\mathrm{\θ}}_{i+1}^L\\ {\stackrel{\‾}{M}}_{i+1}^L\\ {\stackrel{\‾}{N}}_{i+1}^L\end{array}\right\}={\left[T\right]}_i\left\{\begin{array}{c}{y}_i^L\\ {\mathrm{\θ}}_i^L\\ {\stackrel{\‾}{M}}_i^L\\ {\stackrel{\‾}{N}}_i^L\end{array}\right\}---\left(10\right)$

In formula (8):

Wherein:

EI _iBe respectively linear mass and the cross section bendind rigidity of beam element, ω is the natural frequency of whole beam, in formula (10): With

Be respectively left end amount of deflection, corner, Equivalent Moment and the equivalent shearing of i unit and i+1 unit.

If beam is divided into n unit, then the equation of transfer of this n unit is combined into the transfer equation (11) of beam:

$\left\{\begin{array}{c}{y}_n^R\\ {\mathrm{\θ}}_n^R\\ {\stackrel{\‾}{M}}_n^R\\ {\stackrel{\‾}{N}}_n^R\end{array}\right\}={\left[T\right]}_n{\left[T\right]}_{n-1}...{\left[T\right]}_2{\left[T\right]}_1\left\{\begin{array}{c}{y}_1^L\\ {\mathrm{\θ}}_1^L\\ {\stackrel{\‾}{M}}_1^L\\ {\stackrel{\‾}{N}}_1^L\end{array}\right\}---\left(11\right)$

In the formula: Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of n unit right-hand member;

By the boundary condition introduction-type (11) at beam two ends being tried to achieve natural frequency and the vibration shape of beam.

The invention has the beneficial effects as follows:

Directly set up transfer matrix from the line of deflection function of beam, then the boundary condition substitution is calculated, need not to change transfer matrix, also need not to consider the form of distortion of beam, the modular design of the program of being more convenient for.Compare with existing method and to have the wide and simple advantage of computing method of range of application.

Description of drawings

The present invention is further detailed explanation below in conjunction with the drawings and specific embodiments.

Fig. 1 is dividing elements and the lumped mass schematic diagram that the present invention is suitable for the transfer matrix computing method embodiment of Beam Vibration analysis, and element number is 1,2 from left to right ..., n.

Fig. 2 is the calculating schematic diagram that the present invention is suitable for the 1st and n-1 the beam element of the transfer matrix computing method embodiment that Beam Vibration analyzes.

Fig. 3 is that n (low order end) beam element that the present invention is suitable for the transfer matrix computing method embodiment that Beam Vibration analyzes calculates schematic diagram.

Among the figure:

m ₀, m ₁, m ₂..., m _i..., m _N-1, m _nLumped mass for each unit of beam;

l ₁, l ₂..., l _i..., l _N-1, l _nLength for each unit of beam;

With

Be respectively left end amount of deflection, corner, Equivalent Moment and the equivalent shearing of i unit and i+1 unit;

Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of n unit right-hand member.

Embodiment

The calculation procedure of the present embodiment is as follows:

1, at first according to the requirement of computational accuracy beam is divided into n unit, the quality of each unit is averagely allocated to two nodes of unit, as shown in Figure 1, wherein, m ₀, m ₁, m ₂..., m _i..., m _N-1, m _nLumped mass for each unit of beam; l ₁, l ₂..., l _i..., l _N-1, l _nLength for each unit of beam.

2, each unit is adopted the deflection curve equation formula (6) of beam:

$y\left(x\right)=y^L+{\mathrm{\θ}}^{L}x+\frac{1}{2}{\stackrel{\‾}{M}}^L{x}^2+\frac{1}{6}{\stackrel{\‾}{N}}^L{x}^3---\left(6\right)$

Construct non-known beam element (seeing a Fig. 2) matrix form (7):

${\left[T_f\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ 0& 0& 0& 1\end{array}\right]---\left(7\right)$

In the formula (6): y ^L, θ ^L,

Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of beam left end, and Equivalent Moment and equivalent shearing are respectively

Wherein: EI is the beam section bending stiffness; In the formula (7): subscript i is the numbering of beam element, l _iThe length of i beam element;

Then with the field matrix form (7) of beam element and dot matrix type (8)

${\left[T_p\right]}_i=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ {\stackrel{\‾}{a}}_i& 0& 0& 1\end{array}\right]---\left(8\right)$

The transfer matrix formula (9) that multiplies each other and obtain beam element (seeing Fig. 2):

${\left[T\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ {\stackrel{\‾}{a}}_i& {\stackrel{\‾}{a}}_i{l}_i& {\stackrel{\‾}{a}}_i{l}_i^2/2& {\stackrel{\‾}{a}}_i{l}_i^3/6+1\end{array}\right]---\left(9\right)$

Thereby can set up the transfer equation (10) of beam element:

$\left\{\begin{array}{c}{y}_{i+1}^L\\ {\mathrm{\θ}}_{i+1}^L\\ {\stackrel{\‾}{M}}_{i+1}^L\\ {\stackrel{\‾}{N}}_{i+1}^L\end{array}\right\}={\left[T\right]}_i\left\{\begin{array}{c}{y}_i^L\\ {\mathrm{\θ}}_i^L\\ {\stackrel{\‾}{M}}_i^L\\ {\stackrel{\‾}{N}}_i^L\end{array}\right\}---\left(10\right)$

In formula (8):

Wherein:

EI _iBe respectively linear mass and the cross section bendind rigidity of beam element, ω is the natural frequency of whole beam, in formula (10):

With

Be respectively left end amount of deflection, corner, Equivalent Moment and the equivalent shearing of i unit and i+1 unit.

3, the equation of transfer of all n beam element is combined into the transfer equation (I1) of beam:

$\left\{\begin{array}{c}{y}_n^R\\ {\mathrm{\θ}}_n^R\\ {\stackrel{\‾}{M}}_n^R\\ {\stackrel{\‾}{N}}_n^R\end{array}\right\}={\left[T\right]}_n{\left[T\right]}_{n-1}...{\left[T\right]}_2{\left[T\right]}_1\left\{\begin{array}{c}{y}_1^L\\ {\mathrm{\θ}}_1^L\\ {\stackrel{\‾}{M}}_1^L\\ {\stackrel{\‾}{N}}_1^L\end{array}\right\}---\left(11\right)$

In the formula:

Be respectively n unit (seeing Fig. 3) right-hand member, i.e. the amount of deflection in the right-hand member cross section of beam, corner, Equivalent Moment and equivalent shearing;

4, by the constraint condition introduction-type (11) of beam being tried to achieve natural frequency and the vibration shape of beam.These methods are known technology, repeat no more herein.

Above-described embodiment of the present invention does not consist of the restriction to protection domain of the present invention.Any modification of doing within the spirit and principles in the present invention, be equal to and replace and improvement etc., all should be included within the claim protection domain of the present invention.

Claims (1)

1. be suitable for the transfer matrix computing method that Beam Vibration is analyzed, it is characterized in that, may further comprise the steps:

Deflection curve equation formula (6) with known beam:

$y\left(x\right)=y^L+{\mathrm{\θ}}^{L}x+\frac{1}{2}{\stackrel{\‾}{M}}^L{x}^2+\frac{1}{6}{\stackrel{\‾}{N}}^L{x}^3---\left(6\right)$

Construct non-known beam element field matrix form (7):

${\left[T_f\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ 0& 0& 0& 1\end{array}\right]---\left(7\right)$

In the formula (6): y ^L, θ ^L,

Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of beam left end, and Equivalent Moment and equivalent shearing are respectively

Wherein: EI is the beam section bending stiffness;

In the formula (7): i is the numbering of beam element, l _iLength for beam element;

Then with the field matrix form (7) of beam element and dot matrix type (8)

${\left[T_p\right]}_i=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ {\stackrel{\‾}{a}}_i& 0& 0& 1\end{array}\right]---\left(8\right)$

The transfer matrix formula (9) that multiplies each other and obtain beam element:

${\left[T\right]}_i=\left[\begin{array}{cccc}1& l_i& l_i^2/2& l_i^3/6\\ 0& 1& l_i& l_i^2/2\\ 0& 0& 1& l_i\\ {\stackrel{\‾}{a}}_i& {\stackrel{\‾}{a}}_i{l}_i& {\stackrel{\‾}{a}}_i{l}_i^2/2& {\stackrel{\‾}{a}}_i{l}_i^3/6+1\end{array}\right]---\left(9\right)$

Thereby can set up the transfer equation (10) of beam element:

$\left\{\begin{array}{c}{y}_{i+1}^L\\ {\mathrm{\θ}}_{i+1}^L\\ {\stackrel{\‾}{M}}_{i+1}^L\\ {\stackrel{\‾}{N}}_{i+1}^L\end{array}\right\}={\left[T\right]}_i\left\{\begin{array}{c}{y}_i^L\\ {\mathrm{\θ}}_i^L\\ {\stackrel{\‾}{M}}_i^L\\ {\stackrel{\‾}{N}}_i^L\end{array}\right\}---\left(10\right)$

In formula (8): Wherein:

EI _iBe respectively linear mass and the cross section bendind rigidity of beam element, ω is the natural frequency of whole beam, in formula (10): With

Be respectively left end amount of deflection, corner, Equivalent Moment and the equivalent shearing of i unit and i+1 unit;

If beam is divided into n unit, then the equation of transfer of this n unit is combined into the transfer equation (11) of beam:

$\left\{\begin{array}{c}{y}_n^R\\ {\mathrm{\θ}}_n^R\\ {\stackrel{\‾}{M}}_n^R\\ {\stackrel{\‾}{N}}_n^R\end{array}\right\}={\left[T\right]}_n{\left[T\right]}_{n-1}...{\left[T\right]}_2{\left[T\right]}_1\left\{\begin{array}{c}{y}_1^L\\ {\mathrm{\θ}}_1^L\\ {\stackrel{\‾}{M}}_1^L\\ {\stackrel{\‾}{N}}_1^L\end{array}\right\}---\left(11\right)$

In the formula:

Be respectively amount of deflection, corner, Equivalent Moment and the equivalent shearing of n unit right-hand member;

By the boundary condition introduction-type (11) at beam two ends being tried to achieve natural frequency and the vibration shape of beam.

Images