Fast calculation method for vibration response of nonlinear cantilever beam with fractional order damping

Abstract

The present application belongs to the field of mechanics, and relates to a kind of fast calculation method of nonlinear cantilever beam vibration response, comprising: 1) using Euler-Bernoulli beam theory to establish the transverse vibration equation of nonlinear cantilever beam with fractional order damping;2) using Galerkin discretization method to discretize the transverse vibration equation obtained in step 1), to obtain discrete equation;3) equivalent to the fractional order damping term of discrete equation, and the vibration equation after equivalent is solved iteratively, to obtain the vibration response value of each order mode;4) based on the vibration response value of each order mode calculated in step 3) iteration result, the vibration dynamic response of cantilever beam is calculated.The present application provides a kind of fast calculation method of nonlinear cantilever beam vibration response with low solving difficulty and high solving efficiency.

Description

Technical Field

[0001] This invention belongs to the field of mechanics and relates to a rapid calculation method for the vibration response of a nonlinear cantilever beam, and more particularly to a rapid calculation method for the vibration response of a fractionally damped nonlinear cantilever beam under a moving mass load. Background Technology

[0002] The minute vibrations of a structure caused by the movement of an object on a slender, flexible structure (such as a tower crane) can be simplified to the vibration process of a cantilever beam under a moving mass load. Studying the dynamic characteristics of time-varying systems can provide theoretical guidance for the optimal design, vibration control, and improvement of lateral movement accuracy of cantilever beam systems, possessing significant theoretical value and practical significance. With the advancement of materials science, many viscoelastic materials are widely used in beam structures. For viscoelastic materials, fractional-order models have a significant advantage, as they can reflect the more realistic mechanical properties of viscoelastic materials with a small number of parameters. The fractional order not only affects the damping characteristics of the structure but also its linear stiffness. The effect of the fractional derivative can usually be equated to linear restoring force and linear damping force for system analysis. Summary of the Invention

[0003] To address the aforementioned technical problems in the background art, this invention provides a rapid calculation method for the vibration response of a nonlinear cantilever beam that is easy to solve and has high calculation efficiency.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A rapid calculation method for the vibration response of a nonlinear cantilever beam, characterized by the following steps:

[0006] 1) The transverse vibration equation of a nonlinear cantilever beam with fractional-order damping is established using Euler-Bernoulli beam theory;

[0007] 2) The transverse vibration equation obtained in step 1) is discretized using the Galerkin discretization method to obtain the discrete equation;

[0008] 3) The fractional damping terms of the discrete equation are equivalent, and the equivalent discrete vibration equation is solved iteratively to obtain the vibration response values ​​of each mode.

[0009] 4) Calculate the dynamic vibration response of the cantilever beam based on the vibration response values ​​of each mode obtained in step 3).

[0010] The expression for the vibration equation of the cantilever beam with fractional-order damping in step 1) above is:

[0011]

[0012] in:

[0013] δ(·) represents the Dirac function;

[0014] The boundary condition for the beam is: w(0,t)=w x (0,t)=w xx (0,t)=w xxx (0,t)=0

[0015] w(x,t) represents the lateral vibration displacement of the cantilever beam; where x represents the position on the beam and t represents time;

[0016] w x This represents the first-order partial derivative of the lateral displacement w(x,t) with respect to x.

[0017] ε is the damping coefficient of the beam material;

[0018] α is a fractional derivative parameter, with a range of 0 < α < 1;

[0019] A is the cross-sectional area of ​​the cantilever beam;

[0020] ρ is the density of the cantilever beam;

[0021] E is the elastic modulus of the material of the cantilever beam;

[0022] I is the moment of inertia of the cross section;

[0023] M is the mass of the moving object;

[0024] v is the object's speed;

[0025] g is the acceleration due to gravity;

[0026] N x This represents the axial force generated due to nonlinear deformation, where N x The expression is:

[0027]

[0028] l is the length of the cantilever beam.

[0029] The expression for the discrete equation in step 2) above is:

[0030]

[0031] in:

[0032]

[0033] The expression is:

[0034]

[0035] in:

[0036] α is a fractional derivative parameter, with a range of 0 < α < 1;

[0037] Γ(·) represents the Gamma function;

[0038] t is time;

[0039] u is the integration variable;

[0040] w k It is the generalized coordinate of the k-th modal.

[0041] The specific implementation method of step 3) above is as follows:

[0042] 3.1) The fractional damping terms of the discrete equation are equivalently processed to obtain the equivalent equation;

[0043] 3.2) The Runge-Kutta numerical method is used to solve the equivalent equation obtained in step 3.1), and the first-order approximate solution of the equivalent equation is obtained.

[0044] 3.3) Substitute the first-order approximate solution of the equivalent equation obtained in step 3.2) into the equivalent equation obtained in step 3.1), and continue to solve the equivalent equation obtained in step 3.1) using the Runge-Kutta numerical method to obtain the solution result;

[0045] 3.4) Determine whether the solution obtained in step 3.3) has converged; if yes, use the solution as the vibration response value of each mode; if no, repeat step 3.3) until the solution converges.

[0046] The expression for the equivalent equation in step 3.1) above is:

[0047]

[0048] in:

[0049] The expressions for the equivalent natural frequency and damping coefficient are as follows:

[0050]

[0051] The specific implementation of step 3.2) above is as follows: Ignoring the coupling terms in the equivalent equation, the Runge-Kutta numerical method is used to solve the following decoupled ordinary differential equation, resulting in w. k The first approximation of (t) is denoted as The This is a first-order approximate solution to the equivalent equation; the expression for the decoupled ordinary differential equation is:

[0052]

[0053] The specific implementation of step 3.3) above is as follows: the right side of the equivalent equation is regarded as the excitation force, where... use Instead, the Runge-Kutta numerical method is used to solve the nonlinear vibration equation for the following explicit excitation, and the response is labeled as follows. The To obtain the solution, the expression for the nonlinear vibration equation of the explicit excitation is:

[0054]

[0055] The specific implementation method of step 3.4) above is as follows:

[0056] Based on step 3.3), a series of functions are recursively derived. Set the iteration cutoff error ε0, ε0, if If the solution process is converged, then let... The result is w k (t) is the iteration result, which is the vibration response value of each mode.

[0057] The advantages of this invention are:

[0058] This invention provides a rapid calculation method for the vibration response of a nonlinear cantilever beam. The method first establishes the transverse vibration equation of a nonlinear cantilever beam with fractional-order damping under the action of moving mass using Euler-Bernoulli beam theory. Then, the established equation is discretized using the Galerkin discretization method. Due to the presence of fractional-order damping, the discrete vibration differential equations are difficult to solve. This invention, based on the generalized harmonic function technique, approximates the fractional damping force as a damping force and a conservative force. Subsequently, a fast and efficient iterative algorithm is designed to calculate the vibration equations of each modal vibration to obtain the vibration response of the cantilever beam, effectively avoiding the disadvantages of high difficulty and low efficiency in solving coupled equations. The displacement at the ends of the cantilever beam is calculated, and the influence of the fractional damping coefficient is analyzed. Based on this, this invention uses Caputo-type fractional derivatives to describe the viscoelastic damping characteristics of cantilever beams. To facilitate analysis, it employs widely used generalized harmonic function techniques to handle fractional orders. Then, it proposes an efficient and accurate numerical iteration method and analyzes the vibration response laws of cantilever beams with different structural characteristics, providing a general basis for the structural design or vibration control of viscoelastic cantilever beams under moving loads. Attached Figure Description

[0059] Figure 1 This is a schematic diagram of a cantilever beam under the action of a moving mass;

[0060] Figure 2It is a flowchart for solving fractional-order vibrational differential equations;

[0061] Figure 3 The generalized coordinates w of each modality k Time history graph of (t);

[0062] Figure 4 These are the displacements of various points on the beam at different times;

[0063] Figure 5 This is a time history graph of the displacement at the beam's endpoints. Detailed Implementation

[0064] The core principle of this invention is as follows: This invention proposes a rapid calculation method for the dynamic response of a nonlinear cantilever beam with fractional damping under moving mass excitation. First, the transverse vibration equation of the nonlinear cantilever beam with fractional damping under moving mass is established using Euler-Bernoulli beam theory. Then, the established equation is discretized using the Galerkin discretization method. Due to the presence of fractional damping, the discrete vibration differential equations of each order are difficult to solve. This invention, based on the generalized harmonic function technique, approximates the fractional damping force as damping force and conservative force. Then, a fast and efficient iterative algorithm is designed to calculate the vibration equations of each modal vibration to obtain the vibration response of the cantilever beam, effectively avoiding the disadvantages of high difficulty and low efficiency in solving coupled equations. The displacement at the ends of the cantilever beam is calculated, and the influence of the fractional damping coefficient is analyzed.

[0065] The technical solution provided by the present invention will be described in detail below with reference to the accompanying drawings:

[0066] 1. Dynamic Model and Analysis

[0067] See Figure 1 This is a schematic diagram of a cantilever beam under the action of a moving mass. Consider, for example... Figure 1 A cantilever beam with fractional-order damping, according to vibration theory, has the following vibration equation:

[0068]

[0069] The boundary conditions of the beam are

[0070] w(0,t)=w x (0,t)=w xx (0,t)=w xxx (0,t)=0(2)

[0071] in:

[0072] w(x,t) represents the lateral vibration displacement of the beam; where x represents the position on the beam, t represents time, and w xThe first partial derivative of the lateral displacement w(x,t) with respect to x is given by ε, where ε is the damping coefficient of the beam material; α is a fractional derivative parameter, with a value range of 0 < α < 1; A is the cross-sectional area of ​​the cantilever beam; ρ is the density of the cantilever beam; E is the elastic modulus of the cantilever beam material; I is the moment of inertia of the cross section; M is the mass of the moving object; v is the velocity of the moving mass; g is the acceleration due to gravity; N x This represents the axial force generated by nonlinear deformation.

[0073] There are multiple definitions of fractional derivatives. Here, we adopt the most common Caputo type fractional derivative, as detailed below:

[0074]

[0075] Where Γ(·) represents the Gamma function,

[0076] N in the equation x The axial force generated by nonlinear deformation has the following form:

[0077]

[0078] l is the beam length;

[0079] The transverse vibration displacement w(x,t) of the beam in equation (1) can be expressed as a combination of different modal vibrations, with the following form:

[0080]

[0081] Where w i (t) is a generalized coordinate, X i (x) is the i-th mode function of a beam having the following form.

[0082]

[0083] Where: λ i It can be obtained through the following equation:

[0084] cos(λ i l)·ch(λ i l)=-1 (7)

[0085] By using the Galerkin discretization method, equation (5) is substituted into equation (1), and both sides are multiplied by X. i (x), and using the orthogonality condition of the modal functions, a generalized coordinate system w was established. i The equation that (t) satisfies.

[0086]

[0087] in:

[0088]

[0089] It has the following form

[0090]

[0091] Where α is a fractional-order parameter.

[0092] 2. Model simplification and iterative solution algorithm design

[0093] 2.1) Further simplification of the original equation

[0094] The fractional damping force in equation (8) will make the equation difficult to solve. Typically, fractional damping... It contributes to both damping force and restoring force. Therefore, the fractional damping force can be approximated by the modified damping force and restoring force.

[0095] The response of equation (8) is usually periodic with small damping. The amplitude does not change much within a very short period and can be assumed to be the form of a periodic solution of a corresponding conservative system.

[0096]

[0097] The solution to the above vibration equation has the following form:

[0098]

[0099] in:

[0100] a k θ represents the amplitude of the vibration response. k Indicates the phase angle;

[0101] Since fractional viscoelastic force can be expressed as the combined effect of equivalent damping force and viscoelastic force, the following approximate expression for equivalent fractional force can be obtained.

[0102]

[0103] To calculate C k (α) and K k (α), the following approximate formula needs to be used in the calculation process.

[0104]

[0105] Substitute expression (12) into expression (10) of fractional order damping force and perform integration. Ignore the related terms that decrease exponentially with time during the calculation. K k (α) and C kThe expression for (α) can be obtained through stepwise integration and has the following form.

[0106]

[0107] Based on the above derivation, the equivalent expression for fractional-order damping can be obtained.

[0108]

[0109] Furthermore, substituting (10) into the expression for nonlinear axial force, we can obtain

[0110]

[0111] in:

[0112]

[0113] Substituting equation (17) into equation (8), we obtain the equivalent system of the original system without fractional derivatives, which has the following expression:

[0114]

[0115] The equivalent natural frequency and damping coefficient are expressed as follows:

[0116]

[0117] 2.2) Iterative solution algorithm

[0118] Due to the vibration equation (20) regarding the generalized coordinate w k The equations for (t) are coupled together, and the equivalent equations are difficult to solve directly. Therefore, iterative calculations are required.

[0119] The first step is to ignore the coupling terms in the equations and use the Runge-Kutta numerical method to solve the following decoupled ordinary differential equations:

[0120]

[0121] We can get w k The first approximation of (t) is denoted as .

[0122] The second step is to consider the right side of equation (20) as the excitation force, where... It can be used Instead, equation (20) is transformed into a nonlinear vibration equation with explicit excitation, which can be used for the next iteration.

[0123]

[0124] The Runge-Kutta numerical method can be used to solve this. The exported response is then labeled as... Following the steps described above, a series of functions can be recursively derived. Equation (20) can be truncated to the required precision to obtain the final w. k (t).

[0125] The overall solution process is as follows: Figure 2 As shown:

[0126] 3. Example:

[0127] Given the parameters of the cantilever beam, the dynamic response can be obtained by solving the problem using the method described above. In this invention, aluminum is chosen as the material for the beam. The geometric parameters used in the calculation are: l = 3.2 m, ρ = 2700 kg / m. 3 E = 7.2 × 10 10 Pa, I = 2.04 × 10 -7 m 4 The fractional-order parameter α = 0.95. The mass and velocity of the moving object are M = 4.5 kg and v = 420 m / s, respectively.

[0128] This invention proposes a rapid calculation method for the dynamic response of a nonlinear cantilever beam with fractionally damped moving mass. First, the dynamic equations are established, and then discretized using the Galerkin method. By utilizing the periodic behavior of generalized coordinates in each modal vibration, the fractional damping force can be approximated as an equivalent restoring force and an equivalent damping force. Finally, the dynamic response of the beam is discussed, and the influence of parameters on the fractional damping force is examined.

Claims

1. A method for fast calculation of the response of a nonlinear cantilever beam, characterized in that: The fast calculation method of the nonlinear cantilever beam vibration response comprises the following steps: 1) a transverse vibration equation of a nonlinear cantilever beam with fractional order damping is established by using Euler-Bernoulli beam theory; the expression of the vibration equation of the cantilever beam with fractional order damping is: Wherein: δ(·) represents a Dirac function; The boundary conditions for the beam are: w(0, t) = w x (0, t) = w xx (0, t) = w xxx (0, t) = 0 w(x, t) represents a transverse vibration displacement of the cantilever beam; wherein x represents a position on the beam, and t represents time; w x denotes the first order partial derivative of the lateral displacement w(x,t) with respect to x; ε is a damping coefficient of the beam material; α is a fractional derivative parameter, and the value range is 0 < α < 1; A is a cross-sectional area of the cantilever beam; ρ is a density of the cantilever beam; E is an elastic modulus of the material of the cantilever beam; I is a sectional moment of inertia; M is a moving object mass; v is an object moving speed; g is a gravity acceleration; N x represents the axial force due to the nonlinear deformation, where N x The expression for N l is a length of the cantilever beam; 2) the transverse vibration equation obtained in step 1) is discretely processed by using a Galerkin discrete method to obtain a discrete equation; 3) the fractional order damping term of the discrete equation is equivalent, and the discrete vibration equation after the equivalent is iteratively solved to obtain vibration response values of each order mode; the specific implementation mode of step 3) is: 3.1) the fractional order damping term of the discrete equation is equivalent to obtain an equivalent equation; the expression of the equivalent equation is: Wherein: The expressions of the equivalent natural frequency and the damping coefficient are respectively: 3.2) the first order approximate solution of the equivalent equation obtained in step 3.1) is obtained by using a Runge-Kutta numerical method to solve the equivalent equation, and specifically is: Neglecting the coupling terms in the equivalent equation, the first approximation of w k (t) is obtained by solving the decoupled ordinary differential equation as follows using the Runge-Kutta numerical method, denoted as The is the first order approximate solution of the equivalent equation; the expression of the decoupled ordinary differential equation is: 3.3) the first order approximate solution of the equivalent equation obtained in step 3.2) is substituted into the equivalent equation obtained in step 3.1), and the Runge-Kutta numerical method is used to continue to solve the equivalent equation obtained in step 3.1) to obtain a solution result, and specifically is: The right side of the equivalent equation is considered as the excitation force, where Instead, the Runge-Kutta numerical method is used to solve the explicit excited nonlinear vibration equation as follows, to obtain the response marked as The To solve the result, the expression of the explicit excited nonlinear vibration equation is:​ 3.4) it is judged whether the solution result obtained in step 3.3) converges; if yes, the solution result is taken as the vibration response value of each order mode; if no, step 3.3) is repeated until the solution result converges; 4) the vibration dynamic response of the cantilever beam is calculated based on the vibration response values of each order mode calculated in step 3).

2. The method of claim 1, wherein: The expression of the discrete equation in step 2) is: Wherein: The expression is: Wherein: α is a fractional derivative parameter, and the value range is 0 < α < 1; Γ(·) represents a Gamma function; t is time; u is an integral variable; w k is the kth order modal generalized coordinate.

3. The method for fast calculation of the response of a nonlinear cantilever beam according to claim 2, characterized in that: The specific implementation mode of step 3.4) is: According to step 3.3), a series of functions are recursively derived Set an iteration stop error ε0, ε0, if Then determine that the solution process converges, and let The obtained w k (t) is the iteration result, which is the vibration response value of each order mode.

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