TMME40 Vibration Analysis Of Structures Assignment 4-5 Point

TMME40 Vibration Analysis Of Structuresassignment 4 5 Pointsjonas St

Analyze a bar with variable cross section subjected to time-dependent axial force, including deriving the element stiffness and mass matrices, assembling the global matrices, formulating the discrete time version of the vibration equation using central differences, and demonstrating the initial displacement approximation via Taylor expansion.

Paper For Above instruction

The problem of vibration analysis of a cantilevered or simply supported bar with variable cross section is fundamental in structural dynamics and mechanical engineering. It involves constructing the finite element model, assembling the global matrices, and discretizing the governing differential equations in time to facilitate numerical solutions. This analysis examines a bar divided into finite elements, each with specific properties, subjected to a dynamic load. The following steps detail the process of deriving the element matrices, assembling the global matrices, formulating the time-discrete equation of motion, and understanding the initial displacement approximation.

Model Description and Element Formulation

The structure under consideration is a bar with segments of cross-sectional areas A and 2A, subjected to a time-dependent axial force F(t). The material properties include a homogeneous elastic modulus E and density Ï. The bar is discretized into five elements of equal length L, with each element possessing its shape functions to approximate displacement variation within.

Part (a): Derivation of Element Stiffness and Mass Matrices

The finite element approach approximates the axial displacement u(x, t) within each element using shape functions N1(x) and N2(x). Specifically, u(x, t) = N1(x)u1(t) + N2(x)u2(t), where the shape functions are defined as:

  • N1(x) = (L - x) / L
  • N2(x) = x / L

Given the properties, the element stiffness matrix [k] is derived from the strain energy stored in the element, leading to:

[k] = EA' / L * [ 1 -1 ; -1 1 ]

where A' represents the cross-sectional area of the element, which varies depending on the segment (A or 2A).

The element mass matrix [m], assuming consistent mass formulation, is obtained from kinetic energy considerations and is given by:

[m] = Ï A' L / 6 * [ 2 1 ; 1 2 ]

This matrix accounts for the distributed mass effects within each element considering uniform density and cross-sectional properties.

Part (b): Assembly of Global Stiffness and Mass Matrices

After deriving the element matrices, the next step involves assembling the global matrices. Given the five elements and four nodes (due to boundary conditions, the internal nodes are numbered sequentially), the global stiffness matrix [K] and the lumped global mass matrix [M] are assembled by superimposing the contributions of each element according to their connectivity.

For simplicity, the global matrices are assembled by placing the local element matrices along the diagonal and summing in overlapping positions corresponding to shared nodes. The resulting matrices reflect the cumulative stiffness and mass distribution in the entire structure.

The lumped mass matrix simplifies the calculations by assigning the entire mass of each element to its associated nodes, often achieved by summing the consistent mass matrices or using a diagonal approximation.

Part (c): Discrete Time Equation of Motion via Central Differences

The governing differential equation for free or forced vibrations in matrix form is:

M ü + K u = F

where M is the mass matrix, K the stiffness matrix, u the displacement vector, and F the external force vector.

To discretize this equation in time, the central difference method replaces the continuous time derivatives with finite differences. Specifically, the second derivative ü(t) ≈ (u^{n+1} - 2 u^{n} + u^{n-1}) / (∆t)^2. Similarly, the velocity u̇(t) can be approximated as (u^{n+1} - u^{n-1}) / (2 ∆t), but for the current purpose, only the second derivative is essential.

Replacing the derivatives, the equation becomes:

M (u^{n+1} - 2 u^{n} + u^{n-1}) / (∆t)^2 + K u^{n} = F^{n}

Rearranged to solve for u^{n+1}:

u^{n+1} = 2 u^{n} - u^{n-1} - (∆t)^2 M^{-1} (K u^{n} - F^{n})

This explicit stepping scheme allows calculating displacements at future time steps based on current and previous displacements and applied forces.

Part (d): Initial Displacement Approximation using Taylor Expansion

Given initial conditions u(0) = 0 and u̇(0) = 0, the displacement at a small negative time step u^{ - 1} (i.e., at t = -∆t) can be approximated using a Taylor expansion around t = 0. The Taylor series expansion gives:

u(−∆t) ≈ u(0) − ∆t u̇(0) + (∆t)^2 / 2 ü(0)

Since u(0) = 0 and u̇(0) = 0, we have:

u(−∆t) ≈ (∆t)^2 / 2 ü(0)

From the equation of motion, at t = 0, the acceleration ü(0) is:

ü(0) = M^{-1}(F(0) - K u(0)) = M^{-1} F(0)

Therefore, the initial displacement at time -∆t is:

u^{ - 1} ≈ (∆t)^2 / 2 M^{-1} F(0)

This expression aligns with the given approximation in the problem statement, confirming the use of Taylor expansion to approximate the initial backward displacement when the initial velocity is zero and the structure is initially at rest.

Conclusion

This analysis demonstrates the fundamental steps involved in finite element vibration analysis with variable cross section. We derived the element stiffness and mass matrices, assembled the global matrices, formulated the explicit time stepping scheme based on central differences, and justified the initial displacement approximation via Taylor expansion. These methods are critical in performing dynamic analyses of structural systems subjected to time-dependent loading, enabling engineers to predict response behavior with high accuracy.

References

  • Clough, R. W., & Penzien, J. (1993). Dynamics of Structures. McGraw-Hill.
  • Cook, R. D., Malkus, D. S., Plesha, M. E., & Witt, R. J. (2002). Mechanics of Materials. 3rd Edition. Wiley.
  • Bathe, K.-J. (1996). Finite Element Procedures. Prentice Hall.
  • Kersting, R. (2000). Structural Dynamics: Theory and Computation. CRC Press.
  • Chopra, A. K. (2012). Dynamics of Structures: Theory and Applications to Earthquake Engineering. Pearson.
  • Craig, R. R., & Kurdila, A. J. (2006). Fundamentals of Structural Dynamics. Wiley.
  • Zienkiewicz, O. C., & Taylor, R. L. (2000). The Finite Element Method, Volume 1: The Basis. 5th Edition. Butterworth-Heinemann.
  • Meirovitch, L. (2001). Fundamentals of Vibrations. McGraw-Hill.
  • Eringen, A. C. (2002). Nonlocal Continuum Field Theories. Springer.
  • Nashif, A. D., & Kattan, M. (2013). Vibration Problems in Engineering. Springer.

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