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1、精选优质文档-倾情为你奉上Advanced Structural Dynamics ProjectThe dynamic response and stability analysis of the beam under vertical excitationInstructor：Dr. Li WeiName：Student ID：1. Problem description and the purpose of the project1.1 calculation modelAn Eular beam subjected to an axial force. Please build the
2、 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.Fig1.11.2 purpose and process arrangementa. learning how to create mathematica
3、l 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 systems stability and analyze the relationship among the frequency and amplitude of the force and d
4、ynamic 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 a
5、nd 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 systemThe 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
6、 is small, and the boundary conditions is simply support. Now the beam subjected to an axial force. We assume the force is equal to. Fig.2.1We select the length of in any position of the beam, the free-body diagram is shown in Fig.2.2.Fig.2.2Using equations of movement equilibrium, that is to say: +
7、 （1） （2）From equation (1), we will get: （3）Divide equation (3) by and take the limit: （4）Then synthesize equation (2),we can get: （5）Divide equation (5) by and take the limit: （6）Combine equation (4) with (6): （7）And （8）Combine equation (7) with (8): （9）We know EI is a constant, so （10）In equation (
8、10), m is the mass of unit length.Now we will use assumed-modes method. Named ,so: （11） n=1,2,. （12）In the equation (12)And ,so n=1,2,. （13） （14）In the equation (12) Equation (14) is the Mathieu equation. it is difficult to solve the analytical solution directly, thus, we use the approximate derivat
9、ive namely an average acceleration method to get the numerical solution from the reference.3. Numerical solution3.1 using MATLAB to solve equationWe will use the Newmark- method 1 to solve equation (14). We can use the initial condition to integrate the move equation: （15）Fig.3.1As shown in Fig.3.1
10、（16） （17） （18）From equation (16), (17) and (18), we will get: （19） （20） （21）When applying the MATLAB, we need discrete the processing time t, get time step.When solving the vibration stability interval, there are three variables to participate in the discussion, namely. So take a particular w first
11、and discuss the remaining two parameters.From Floquent theory2，we can use parameter A to judge stability.Equation （22）Take two sets of special solution: （20）Parameter2 （21）If abs (A) is less than 1, the system is stability. And if abs (A) is greater than 1, the system is instability. When abs(A) is
12、equal to 1, the system is critical state.We use MATLAB Codes to solve equations. We use =2 Mathieu Equation to judge the validity of the codes. From Fig.3.2 and Fig.3.3, we can consider the codes are correct.In these follow figures, =2, the horizontal axis is , vertical axis is .Fig.3.2. stable doma
13、in in reference3and2Fig.3.3. stable domain in MATLAB solutionCompared Fig.3.2 with Fig.3.3, we can see that the stability domain of numerical solutions applying average acceleration method are consistent with the standard solutions. it can concluded that when the system have solution whose cycle is
14、equal to or 2, This chapter discusses the accuracy of the vibration stability determination with Floquent theory. The next chapter will discuss the numerical solution and stable domain and two parameters influences on the stability for this question.4. Parameters influenceIn this part, we only consi
15、der two parameters, namely the frequency and amplitude of the force.4.1the influence of the forces frequency4.1.1the stability of the systemWhen we discuss the stability of the system related to the frequency of the force, we should select some different frequencies, so we choose =1,2,4,6,8 and10. U
16、sing MATLAB codes, we can obtain the figs of the stability. We can know the stable region is bigger with the increase of the frequency in Fig.4.1.=1 =2=4 =6=8 =10Fig.4.1 stable domain with different 4.1.2the response of the systemWhen we discuss the response of the system, the system should be stabl
17、e. So we choose ，=2,4 and 6. In Fig.4.2, the cycle of the response increase and the range of the reactive amplitude is smaller with the increase of the frequency.Fig.4.2 responses of the system with different 4.2the influence of the forces amplitudeThe is related to the forces amplitude P. The cycle
18、 of the response a little increase and the range of the reactive amplitude is bigger with the increase of the forces amplitude, in Fig.4.3.and Fig4.4.Fig.4.3 vibration response curve with different The red curve is w=2, =12, =1; the blue curve is w=2, =14, =1.This figure state that the vibration cyc
19、le is smaller and the amplitude have a little change with the increase of the .Fig.4.3 vibration response curve with different The red curve is w=2, =12, =1; the blue curve is w=2, =12, =5. This figure state that the amplitude is smaller and the vibration cycle have a little change with the increase
20、 of the .5. Conclusion(1) With the increase of the frequency, the stable region and the cycle of the response are bigger, but the range of the reactive amplitude is smaller.(2)With the increase of the forces amplitude, the cycle of the response a little increase and the range of the reactive amplitu
21、de is much bigger.(3)Vibration in the stable region, the vibration cycle is smaller with the increase of the ; the amplitude is smaller with the increase of the .AcknowledgementsThe author is grateful for upperclassman Li Yong, he give me much assistance. And the author is also grateful for Doctor L
22、i Wei. In his classes, I felt very happy and can understand his class effectively. At last, the author is also grateful for classmates in the same laboratory, they give me much guidance and encourage. I gain a lot of knowledge through this study and I will work harder in the future.References1、ROY R.CRSIG, Jr. STRUCTURAL DYNAMICS. New York: John Wiley & Sons.2、王海期. 非线性振动. 北京: 高等教育出版社, 19923、顾志平. 非线性振动. 北京: 中国电力出版社, 2012专心-专注-专业