高等结构动力学大作业

Advanced Structural Dynamics Project

The dynamic response and stability analysis of

the beam under vertical excitation

Instructor： Dr. Li Wei

Name：

Student ID：

1. Problem description and the purpose of the project

1.1 calculation model

An Eular beam subjected to an axial force. Please build the

differential equation of motion and use a proper difference method to

solve this differential equation. Study the dynamic stability of the beam

related to the frequency and amplitude of the force. As shown in the Fig

1.1.

Fig1.1

1.2 purpose and process arrangement

a. learning how to create mathematical model of the continuous

system and select proper calculation method to solve it.

b. learning how to build beam vibration equation and solve Mathieu

equation.

c. using Floquet theory to judge vibration system’s stability and

analyze the relationship among the frequency and amplitude of the

force and dynamic response.

This project will introduce the establishment of the mathematical

model of the continuous system in section 2, the movement equation and

the numerical solution of using MATLAB in section 3, Applying

Floquent theory to study the dynamic stability of the beam related to the

frequency and amplitude of the force in section 4. In the last of the project, we get some conclusions in section 5. 2. the mathematical model of the system

The geometric model of the beam and force-simplified diagram is

shown in Fig.2.1.We assume that its stiffness(EI) is constant and the

deflection of the beam is small, and the boundary conditions is simply

support. Now the beam subjected to an axial force. We assume the

force is equal to0cosPt.

F=f0coswtyx

Fig.2.1

We select the length of x in any position of the beam, the

free-body diagram is shown in Fig.2.2.

FFM(x,t)S(x,t)VM(x+_x,t)S(x+_x,t)x

Fig.2.2

Using equations of movement equilibrium, that is to say:

+()yyFma （1）

0GM （2）

From equation (1), we will get:

22),(),(tyxAtxxStxS （3） Divide equation (3) by x and take the limit:

22xyAxS （4）

Then synthesize equation (2),we can get:

0),()],(),([),(),(xtxxStxytxxyFtxMtxxM （5）

Divide equation (5) by x and take the limit:

SxyFxM （6）

Combine equation (4) with (6):

0222222tyAxyFxM （7）

And 22),(xyEItxM （8）

Combine equation (7) with (8):

0)(22222222tyAxyFxyEIx （9）

We know EI is a constant, so

0)(222244tymxytFxyEI （10）

In equation (10), m is the mass of unit length.

Now we will use assumed-modes method. Named lxntTtxynsin)(),(,so:

0sin)(22244422lxnTlntFTlnEIdtTdmnnn （11）

0))(1(222nnonnTFtFPdtTd n=1,2,...... （12）

In the equation (12)

222222,lEInFmEIlnPnon And tFtFcos)( ,so

0)cos1(222nnonnTtFFPdtTd n=1,2,...... （13）

0)cos(22nnTtdtTd （14）

In the equation (12)

4)(LnAEI 22)(LnF

Equation (14) is the Mathieu equation. it is difficult to solve the

analytical solution directly, thus, we use the approximate derivative

namely an average acceleration method to get the numerical solution

from the reference.

3. Numerical solution

3.1 using MATLAB to solve equation

We will use the Newmark-β method [1] to solve equation (14). We

can use the initial condition 00uu和to integrate the move equation:

m0ucuku （15）

Fig.3.1 As shown in Fig.3.1

))(2(11iiiiiuutuu （16）

)(4121iiiiiiiuuttuuu （17）

0cosiiuwtu （18）

From equation (16), (17) and (18), we will get:

iiutwnucos （19）

iiiiuutu2)2( （20）

22]cos[44]cos[)(2ttwntutwnutuiii （21）

When applying the MATLAB, we need discrete the processing time

t, get time step02.0t.When solving the vibration stability interval,

there are three variables to participate in the discussion, namelywc,,. So

take a particular w first and discuss the remaining two parameters.

From Floquent theory[2]，we can use parameter A to judge stability.

Equation 0)()(22ytdtdytdtyd （22）