Rapid calculation method for vibration response of nonlinear cantilever beam containing fractional order damping

Abstract

The invention belongs to the field of mechanics, and relates to a quick calculation method for vibration response of a nonlinear cantilever beam, which comprises the following steps of: 1) establishing a transverse vibration equation of the nonlinear cantilever beam with fractional order damping by adopting an Euler-Bernoulli beam theory; 2) performing discretization processing on the transverse vibration equation obtained in the step 1) by using a Galerkin discretization method to obtain a discrete equation; 3) carrying out equivalence on a fractional order damping item of the discrete equation, and carrying out iterative solution on the equivalent discrete vibration equation to obtain a vibration response value of each order of modal; and 4) calculating the vibration dynamic response of the cantilever beam based on the vibration response value iteration result of each order mode obtained in the step 3). The rapid calculation method for the vibration response of the nonlinear cantilever beam is low in solving difficulty and high in solving efficiency.

Description

Technical Field

[0001] The invention belongs to the field of mechanics, and relates to a fast calculation method for a nonlinear cantilever beam vibration response, and in particular to a fast calculation method for a nonlinear cantilever beam vibration response with fractional-order damping under a moving mass load. Background Art

[0002] The tiny vibration of the structure (such as a tower crane) caused by the movement of an object on a slender flexible structure can be simplified as the vibration process of a cantilever beam under the action of a moving mass load. Studying the dynamic characteristics of a time-varying system can provide theoretical guidance for the optimal design of the cantilever beam system, controlling vibration, and improving the lateral movement accuracy, which has important theoretical value and practical significance. With the progress and development of materials science, many viscoelastic materials are widely used in beam structures. For viscoelastic materials, the fractional order model has great advantages, which can reflect the more realistic mechanical properties of viscoelastic materials through a small number of parameters. The fractional order not only affects the damping characteristics of the structure, but also affects the linear stiffness of the structure. Usually, the effect of the fractional derivative can be equivalent to the linear restoring force and the linear damping force to analyze the system. Summary of the invention

[0003] In order to solve the above technical problems existing in the background technology, the present invention provides a fast calculation method for the vibration response of a nonlinear cantilever beam with low solution difficulty and high solution efficiency.

[0004] In order to achieve the above object, the present invention adopts the following technical solution:

[0005] A fast calculation method for a nonlinear cantilever beam vibration response, characterized in that the fast calculation method for a nonlinear cantilever beam vibration response comprises the following steps:

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

[0007] 2) using the Galerkin discretization method to discretize the lateral vibration equation obtained in step 1) to obtain a discrete equation;

[0008] 3) Equivalent the fractional-order damping term of the discrete equation, iteratively solve the equivalent discrete vibration equation, and obtain the vibration response value of each mode;

[0009] 4) Based on the vibration response values ​​of each mode calculated in step 3), the vibration dynamic response of the cantilever beam is calculated.

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

[0011]

[0012] in:

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

[0014] The boundary condition of 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 the time;

[0016] w x 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, and its value range is 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 cantilever beam material;

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

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

[0024] v is the speed of the object;

[0025] g is the acceleration due to gravity;

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

[0027]

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

[0029] The expression of 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, and its value range is 0<α<1;

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

[0038] t is the time

[0039] u is the integration variable

[0040] w k are the k-th mode generalized coordinates.

[0041] The specific implementation of the above step 3) is:

[0042] 3.1) Perform equivalent treatment on the fractional-order damping term of the discrete equation to obtain the equivalent equation;

[0043] 3.2) Using the Runge-Kutta numerical method to solve the equivalent equation obtained in step 3.1), and obtain a first-order approximate solution of the equivalent equation;

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

[0045] 3.4) Determine whether the solution obtained in step 3.3) converges; if so, use the solution as the vibration response value of each mode; if not, repeat step 3.3) until the solution converges.

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

[0047]

[0048] in:

[0049] The expressions of the equivalent natural frequency and damping coefficient are:

[0050]

[0051] The specific implementation method of the above step 3.2) is: ignore the coupling terms in the equivalent equation, use the Runge-Kutta numerical method to solve the following decoupled ordinary differential equation, and obtain w k The first approximation of (t) is denoted by Said is the first-order approximate solution of the equivalent equation; the expression of the decoupled ordinary differential equation is:

[0052]

[0053] The specific implementation method of the above step 3.3) is: regard the right side of the equivalent equation as the excitation force, where use Instead, the Runge–Kutta numerical method is used to solve the following nonlinear vibration equations for explicit excitation, and the response is marked as Said To solve the result, the expression of the nonlinear vibration equation for explicit excitation is:

[0054]

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

[0056] According to step 3.3), a series of functions are recursively derived Set the iteration cutoff error ε0,ε0, if Then the solution process is judged to be convergent, then let Income k (t) is the iteration result, which is the vibration response value of each mode.

[0057] The advantages of the present invention are:

[0058] The present invention provides a method for fast calculation of the vibration response of a nonlinear cantilever beam. The method first uses the Euler-Bernoulli beam theory to establish the lateral vibration equation of a nonlinear cantilever beam with fractional-order damping under the action of a moving mass; then the established equation is discretized using the Galerkin discretization method. Due to the existence of fractional-order damping, the discrete vibration differential equations of each order are difficult to solve. Based on the generalized harmonic function technology, the present invention approximates the fractional damping force as a damping force and a conservative force. Subsequently, a fast and effective iterative algorithm is designed to calculate the vibration equations of each order of the modal state to calculate the vibration response of the cantilever beam, effectively avoiding the disadvantages of difficulty and low efficiency in solving the coupling equation. The displacement of the end of the cantilever beam is calculated, and the influence of the fractional damping coefficient is analyzed. Based on this, the present invention adopts Caputo-type fractional derivatives to describe the viscoelastic damping characteristics of the cantilever beam. In order to facilitate analysis, the widely used generalized harmonic function technology is used to deal with fractional orders. Then an efficient and accurate numerical iteration method is proposed, and the vibration response laws of different structural characteristics of the cantilever beam are analyzed, which provides a general basis for the structural design or structural vibration control of the viscoelastic cantilever beam under moving loads. BRIEF DESCRIPTION OF THE DRAWINGS

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

[0060] Figure 2 It is a flow chart for solving fractional-order vibration differential equations;

[0061] Figure 3 is the generalized coordinates w of each mode k (t) time history diagram;

[0062] Figure 4 is the displacement of each point on the beam at different times;

[0063] Figure 5 It is the time history diagram of the displacement of the beam end points. DETAILED DESCRIPTION

[0064] The core principle of the present invention is: the present invention proposes a fast calculation method for the dynamic response of a nonlinear cantilever beam with fractional damping under the excitation of a moving mass load. First, the lateral vibration equation of a nonlinear cantilever beam with fractional damping under the action of a moving mass is established using the Euler-Bernoulli beam theory. Then, the established equation is discretized using the Galerkin discretization method. Due to the existence of fractional damping, the discrete vibration differential equations of each order are difficult to solve. Based on the generalized harmonic function technology, the present invention approximates the fractional damping force as a damping force and a conservative force. Then, a fast and effective iterative algorithm is designed to calculate the modal vibration equations of each order to calculate the vibration response of the cantilever beam, effectively avoiding the disadvantages of difficulty and low efficiency in solving the coupling equation. The displacement of the end 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 is described in detail below in conjunction with the accompanying drawings:

[0066] 1. Kinetic model and analysis

[0067] See also Figure 1 , is a schematic diagram of a cantilever beam under the action of a moving mass. Consider Figure 1 According to the vibration theory, the vibration equation of the cantilever beam with fractional damping is

[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 the time, and w xrepresents the first-order partial derivative of the lateral displacement w(x,t) with respect to x, 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 material of the cantilever beam; I is the moment of inertia of the section; M is the mass of the moving object; v is the moving speed of the moving mass; g is the acceleration due to gravity; N x Represents the axial force due to nonlinear deformation.

[0073] There are many definitions of fractional derivatives. Here, the most commonly used Caputo-type fractional derivative is used, as follows:

[0074]

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

[0076] In the equation, N x represents the axial force due to nonlinear deformation and has the following form:

[0077]

[0078] l is the beam length;

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

[0080]

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

[0082]

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

[0084] cos(λ i l)·ch(λ i l)=-1(7)

[0085] By using the Galerkin discretization method, substituting equation (5) into equation (1) and multiplying both sides by X i (x), and using the orthogonality condition of the mode function, the generalized coordinate w is established i (t) satisfies the equation.

[0086]

[0087] in:

[0088]

[0089] Has the following form

[0090]

[0091] Where: α is the 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. Usually the fractional damping Contributes to both the damping force and the 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. In a very short period, the amplitude does not change much and can be assumed to be in the form of a periodic solution of a conservative system.

[0096]

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

[0098]

[0099] in:

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

[0101] Since the fractional-order viscoelastic force can be expressed as the combined effect of the equivalent damping force and the viscoelastic force, the following approximate expression of the equivalent fractional-order 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 damping force and perform integration operation. In the calculation process, ignore the relevant terms that decrease exponentially with time. K k (α) and C kThe expression of (α) can be obtained by stepwise integration and has the following form

[0106]

[0107]

[0108] According to the above derivation, the equivalent expression of fractional damping can be obtained:

[0109]

[0110] In addition, substituting (10) into the expression of nonlinear axial force, we can obtain

[0111]

[0112] in:

[0113]

[0114] Substituting equation (17) into equation (8), we obtain the equivalent system of the original system without fractional derivatives, which has the following expression:

[0115]

[0116] The equivalent natural frequency and damping coefficient have the following expressions:

[0117]

[0118]

[0119] 2.2) Iterative solution algorithm

[0120] Since the vibration equation (20) has the generalized coordinate w k The equations for (t) are coupled to each other, and the equivalent equations are difficult to solve directly. Therefore, iterative calculations are required.

[0121] In the first step, the coupling terms in the equation are ignored and the Runge-Kutta numerical method is used to solve the following decoupled ordinary differential equation:

[0122]

[0123] You can get w k The first approximation of (t) is denoted by

[0124] In the second step, the right side of equation (20) can be regarded as the excitation force, where Can be used Instead, Equation (20) is transformed into a nonlinear vibration equation with explicit excitation, and the next iteration can be performed

[0125]

[0126] It can be further solved using the Runge–Kutta numerical method. The derived response is then marked as Following the above steps, a series of functions can be recursively derived Equation (20) can be truncated to the required accuracy to obtain the final w k (t).

[0127] The overall solution process is as follows Figure 2 As shown:

[0128] 3. Embodiment:

[0129] By giving the parameters of the cantilever beam, the above method can be used to solve and obtain the dynamic response. In the present invention, the material of the beam is aluminum. The geometric parameters used in the calculation are: l = 3.2m, ρ = 2700kg / 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.

[0130] This paper proposes a fast calculation method for the dynamic response of a nonlinear cantilever beam with a moving mass with fractional damping. First, the dynamic equations are established and then discretized according to the Galerkin method. By exploiting 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 effect of parameters on the fractional damping force is examined.

Claims

1. A fast calculation method for the vibration 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) The lateral vibration equation of a nonlinear cantilever beam with fractional-order damping is established using the Euler-Bernoulli beam theory; 2) using the Galerkin discretization method to discretize the lateral vibration equation obtained in step 1) to obtain a discrete equation; 3) Equivalent the fractional-order damping term of the discrete equation, iteratively solve the equivalent discrete vibration equation, and obtain the vibration response value of each mode; 4) Based on the vibration response values ​​of each mode calculated in step 3), the vibration dynamic response of the cantilever beam is calculated.

2. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 1 is characterized in that: The expression of the vibration equation of the cantilever beam with fractional damping in step 1) is: in: δ(·) represents the Dirac function; The boundary condition of the beam is: w(0,t)=w x (0,t)=w xx (0,t)=w xxx (0,t)=0 w(x,t) represents the lateral vibration displacement of the cantilever beam; where x represents the position on the beam and t represents the time; w x represents the first-order partial derivative of the lateral displacement w(x,t) with respect to x, ε is the damping coefficient of the beam material; α is a fractional derivative parameter, and its value range is 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 section; M is the mass of the moving object; v is the speed of the object; g is the acceleration due to gravity; N x represents the axial force due to nonlinear deformation, where N x The expression is: l is the length of the cantilever beam.

3. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 2 is characterized in that: The expression of the discrete equation in step 2) is: in: The expression is: in: α is a fractional derivative parameter, and its value range is 0<α<1; Γ(·) represents the Gamma function; t is the time u is the integration variable w k are the k-th mode generalized coordinates.

4. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 3 is characterized in that: The specific implementation method of step 3) is: 3.1) Perform equivalent treatment on the fractional-order damping term of the discrete equation to obtain the equivalent equation; 3.2) Using the Runge-Kutta numerical method to solve the equivalent equation obtained in step 3.1), and obtain a first-order approximate solution of the equivalent equation; 3.3) Substituting the first-order approximate solution of the equivalent equation obtained in step 3.2) into the equivalent equation obtained in step 3.1), and using the Runge-Kutta numerical method to continue solving the equivalent equation obtained in step 3.1) to obtain a solution result; 3.4) Determine whether the solution obtained in step 3.3) converges; if so, use the solution as the vibration response value of each mode; if not, repeat step 3.3) until the solution converges.

5. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 4 is characterized in that: The expression of the equivalent equation in step 3.1) is: in: The expressions of the equivalent natural frequency and damping coefficient are:

6. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 5 is characterized in that: The specific implementation method of step 3.2) is: ignoring the coupling terms in the equivalent equation, using the Runge-Kutta numerical method to solve the following decoupled ordinary differential equation, and obtaining w k The first approximation of (t) is denoted by Said is the first-order approximate solution of the equivalent equation; the expression of the decoupled ordinary differential equation is:

7. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 6, characterized in that: The specific implementation method of step 3.3) is: regard the right side of the equivalent equation as the excitation force, where use Instead, the Runge–Kutta numerical method is used to solve the following nonlinear vibration equations for explicit excitation, and the response is marked as Said (t) To solve the result, the expression of the nonlinear vibration equation of explicit excitation is:

8. The fast calculation method of the vibration response of a nonlinear cantilever beam according to claim 7, characterized in that: The specific implementation method of step 3.4) is: According to step 3.3), a series of functions are recursively derived Set the iteration cutoff error ε0,ε0, if Then the solution process is judged to be convergent, then let Income k (t) is the iteration result, which is the vibration response value of each mode.

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