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Dynamic response with respect to base movement of spring frame supported cantilever simple beam with concentrated mass
Abstract
In this paper, the principle of relative motion is used to investigate the dynamic response of a spring-frame-supported cantilever simple beam with concentrated mass. Both free vibration and forced vibration caused by the base movement are studied. The cantilever simple beam is discretized into a simple beam and a cantilever beam. Therefore, the general analytical solution of homogeneous beams with uniform cross-section can be used in each beam. The general analytical solution of the whole beam in terms of initial parameters is obtained. In the case of free vibration, the frequency equation can be obtained in an analytical form, and in the case of base movement, a final solution can also be obtained analytically by using the dynamic response of the supporting spring frame as its vibrating source. From our analysis one can see that the accuracy of the solution of the spring-mass of two degrees of freedom is quite satisfactory for engineering purposes in the case of considering only the movement of the point where the concentrated mass P stands.
In order to solve the issue of a heavy workload required in the traditional design of the tower crane jib, a parameterized design system for tower crane jib was developed by extended development of ANSYS using Visual Basic 6.0. The required parameters are entered by using man-machine user interface of this system to complete the design of initial parameters of tower crane jib. This system not only solves the problem of the operational complexity of the finite element software but also meet requirements of specific professional, and the design work is also simplified.
Two-dimensional mechanical model design is the main method to design crane in the domestic but this method can not reflect the dynamic characteristics of cranes accurately. To solve this problem, ANSYS is used to carry out mode analysis and transient dynamic analysis of the overall structure of the crane. The mode analysis is used to determine the dynamic stiffness and vibration characteristics of the overall structure. The transient dynamic analysis is used to determine responses of the displacement change with time of the main girder under the influence of the lifting impact load. Inhibitory effects of the damping on the vibration of crane and influence laws of the loading time on vibration system are obtained by using transient dynamic analysis.
In this paper, we discuss two analytic methods suggested by the author for solving structural optimization problems, viz. the “stepped reduction method [1, 3–5] and a method based on the principle of complementary energy [2]. In the former, we discretize first the Young’s modulus E(x) and the thickness of h(x) of structural elements. We use the initial parameter method and the Heaviside function for descriving results in analytic form and then we use a combination of homogeneous solutions to eliminate discontinuities of bending moment and shearing force at element boundaries. This means that we need to solve only simultaneous algebraic linear equations with a finite number of unknowns. The above methods will be illustrated on the following problems:
1.
Optimal design of thin elastic solid annular plate under an arbitrary load (stepped reduction method) [6]; and
2.
Uniform strength for statically indeterminate beams [2,7].
The effect of gusset plates on free and forced vibration and stability analyses of plane trusses is investigated. The gusset plates are considered to be finite joints possessing mass and rotational flexibility. The bars of the truss are assumed to be elastic Bernoulli-Euler beams with distributed mass. Axial deformation of the bars and the effect of a constant axial force on the bending stiffness are taken into account. On the basis of these assumptions element stiffness matrices are constructed and presented in detail. The general formulation and solution of stability and free and forced vibration problems of trusses is discussed. Examples are presented in detail which demonstrate the effect of the gusset plates on the behavior of trusses under static or dynamic loads.
In this paper, a new method, the step-reduction method, is proposed to investigate the dynamic response of the Bernoulli-Euler
beams with arbitrary nonhomogeneity and arbitrary variable cross-section under arbitrary loads. Both free vibration and forced
vibration of such beams are studied. The new method requires to discretize the space domain into a number of elements. Each
element can be treated as a homogeneous one with uniform thickness. Therefore, the general analytical solution of homogeneous
beams with uniform cross-section can be used in each element. Then, the general analytic solution of the whole beam in terms
of initial parameters can be obtained by satisfying the physical and geometric continuity conditions at the adjacent elements.
In the case of free vibration, the frequency equation in analytic form can be obtained, and in the case of forced vibration,
a final solution in analytical form can also be obtained which is involved in solving a set of simultaneous algebraic equations
with only two unknowns which are independent of the numbers of elements divided. The present analysis can also be extended
to the study of the vibration of such beams with viscous and hysteretic damping and other kinds of beams and other structural
elements with arbitrary nonhomogeneity and arbitrary variable thickness.
Calculating Structural Dynamics
- Lin Jiahao
Lin Jiahao, Calculating Structural Dynamics. Higher Education Press, China (1989).
Yeh Kai-Yuan, General analytic solution of dynamic response of beams with nonhomogeneity and variable cross section
- Yang Guitong
Yang Guitong, Elastic Dynamics. Chinese Railway Press. Beijing (1988). Yeh Kai-Yuan, General analytic solution of dynamic response of beams with nonhomogeneity and variable cross section. Appl. Math. Mech. 9, 753-779 (1992).
Dynamics in Engineering Structures
- V Kolousek
V. Kolousek, Dynamics in Engineering Structures. Butterworths, London (1973).