Method for calculating first-order natural cyclic frequency of bending vibration of stepped beam

Abstract

The invention relates to a calculation method of the first-order natural circular frequency of bending vibration of a stepped beam, which belongs to the calculation method of mechanical vibration. (1) Based on the bending free vibration theory of equal straight bars, the bending free vibration equations of each section of the stepped beam are listed; (2) The modal solution of the stepped beam is assumed by the method of separation of variables; (3) The boundary conditions and analysis of the whole beam are listed. (4) Substituting the hypothetical modal solution into the bending free vibration equation of each segment of the stepped beam to obtain the mode shape function of each segment; The state solution is brought into the boundary conditions and continuity conditions, and the corresponding boundary condition equations and continuity equations are obtained; (6) Solving the boundary condition equations and continuity equations, and solving the equations simultaneously, the elimination of elements only contains two undetermined coefficients and (7) Find the frequency equation of the stepped beam according to the condition that the equation system has a non-zero solution; (8) Use Matcad software to solve the frequency equation to obtain the first-order natural circular frequency value.

Description

Technical field

[0001] The invention relates to a method for calculating mechanical vibration, in particular to a method for calculating the first-order natural circular frequency of bending vibration of a stepped beam. Background technique

[0002] Stepped components are very common in engineering, such as stepped shafts that support rotating parts and transmit motion and power in power machinery, stepped drill strings in petroleum drilling engineering, stepped horns in ultrasonic machining, and steps in pumping systems Shaped sucker rod and stepped piston rod in the engine, workpiece in the turning process, etc. Stepped beams with bending as the main deformation can be called stepped beams. The vibration of stepped beams is a basic subject of mechanical vibration, and the first-order frequency is generally paid special attention to in engineering. The natural frequency of a stepped beam is affected by the cross-sectional shape, constraint conditions, material and segmented rod...

Examples

Embodiment 1

[0089] Example 1: Take figure 1 The shown cantilever stepped beam is taken as an example, and the natural circular frequency equation of its bending vibration is given. This equation is suitable for cantilever stepped beams with different section properties, different section sizes and segment lengths.

[0090] Such as figure 1 As shown, let the total length of the stepped beam be L, and the left beam L 1 The mass per unit volume is ρ 1 , Cross-sectional area A 1 , The shaft moment of inertia is I 1 , The elastic modulus of the material is E 1 ; Right beam L 2 The mass per unit volume is ρ 2 , The cross-sectional area is A 2 , The shaft moment of inertia is I 2 , The elastic modulus of the material is E 2 , That is, the bending stiffness of the two beams is E 1 I 1 , E 2 I 2 .

[0091] Let y(x,t) be the lateral vibration displacement of the cross section of the stepped beam from the coordinate origin x at time t, according to the lateral forced vibration equation of Bernoulli-Euler...

Technical field

[0001] The invention relates to a method for calculating mechanical vibration, in particular to a method for calculating the first-order natural circular frequency of bending vibration of a stepped beam. Background technique

[0002] Stepped components are very common in engineering, such as stepped shafts that support rotating parts and transmit motion and power in power machinery, stepped drill strings in petroleum drilling engineering, stepped horns in ultrasonic machining, and steps in pumping systems Shaped sucker rod and stepped piston rod in the engine, workpiece in the turning process, etc. Stepped beams with bending as the main deformation can be called stepped beams. The vibration of stepped beams is a basic subject of mechanical vibration, and the first-order frequency is generally paid special attention to in engineering. The natural frequency of a stepped beam is affected by the cross-sectional shape, constraint conditions, material and segmented rod...

Examples

Embodiment 1

[0089] Example 1: Take figure 1 The shown cantilever stepped beam is taken as an example, and the natural circular frequency equation of its bending vibration is given. This equation is suitable for cantilever stepped beams with different section properties, different section sizes and segment lengths.

[0090] Such as figure 1 As shown, let the total length of the stepped beam be L, and the left beam L 1 The mass per unit volume is ρ 1 , Cross-sectional area A 1 , The shaft moment of inertia is I 1 , The elastic modulus of the material is E 1 ; Right beam L 2 The mass per unit volume is ρ 2 , The cross-sectional area is A 2 , The shaft moment of inertia is I 2 , The elastic modulus of the material is E 2 , That is, the bending stiffness of the two beams is E 1 I 1 , E 2 I 2 .

[0091] Let y(x,t) be the lateral vibration displacement of the cross section of the stepped beam from the coordinate origin x at time t, according to the lateral forced vibration equation of Bernoulli-Euler...

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