Problem 5 - Concepts and Applications of Finite Element Analysis

Question

The stiffness, consistent mass, and HRZ-lumped mass matrices for the three- node bar element shown in Problem 11.3-4 are respectively $$ \frac{A E}{3 L}\left[\begin{array}{rrr} 7 & -8 & 1 \\ -8 & 16 & -8 \\ 1 & -8 & 7 \end{array}\right] \quad \frac{m}{30}\left[\begin{array}{rrr} 4 & 2 & -1 \\ 2 & 16 & 2 \\ -1 & 2 & 4 \end{array}\right] \quad \frac{m}{6}\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ where \(m\) is the total element mass. Exact natural frequencies of axial vibration of an unsupported uniform bar of length \(L\) are \(\omega_{1}=0, \omega_{2}=\pi \sqrt{A E / m L}\), and \(\omega_{3}=2 \omega_{2}\) (a) Determine the mode shapes and percentage errors of calculated frequencies provided by the consistent mass matrix. (b) Repeat part (a) using the HRZ-lumped mass matrix.

Step-by-Step Solution

Answer

The mode shapes and frequencies will be determined by solving the generalized eigenvalue problem using both the consistent and the HRZ-lumped mass matrices. The percentage errors in each case will be calculated comparing the calculated and exact frequencies. No specific numerical values are given here as they will depend on the particular matrices \(M\) and \(K\) and the exact frequencies.

Step 1: Finding Eigenvalues and Eigenvectors using Consistent Mass Matrix

Begin by finding the eigenvalues (frequencies) and eigenvectors (mode shapes) of the mass matrix \( M \) and the stiffness matrix \( K \). Let's consider the system \( KX = \omega^2 MX \), where \( X \) is the mode shape, \( \omega \) is the frequency, \( K \) the stiffness matrix and \( M \) the consistent mass matrix. One could equivalently solve for \( \lambda = \omega^2 \) in the generalized eigenvalue problem \( (K - \lambda M)X = 0 \) to find the frequencies \( \omega = \sqrt{\lambda} \) and associated mode shapes.

Step 2: Calculating the Percentage Errors of Frequencies with Consistent Mass Matrix

After obtaining the calculated frequencies, compare them with the exact natural frequencies given in the problem. The percentage error can be calculated using the formula : \((\text{{Calculated frequency}} - \text{{Exact frequency}})/\text{{Exact frequency}}) \times 100\%.\) Calculate this for each of the frequencies \( \omega_1, \omega_2, \omega_3\).

Step 3: Finding Eigenvalues and Eigenvectors using HRZ-Lumped Mass Matrix

Repeat the process in steps 1 and 2, but this time use the HRZ-lumped mass matrix in place of the consistent mass matrix. Solve for the eigenvalues and eigenvectors of the mass and stiffness matrices. Then calculate the natural frequencies \( \omega = \sqrt{\lambda} \) and their respective mode shapes.

Step 4: Calculating the Percentage Errors of Frequencies with HRZ-Lumped Mass Matrix

Calculate the percentage errors for the frequencies obtained in step 3. Use the same formula as in step 2, but now with the exact frequencies and the frequencies calculated with the HRZ-lumped mass matrix. Resist from rounding the values in-between steps for more accurate results.

Key Concepts

Stiffness Matrix

In Finite Element Analysis, the **stiffness matrix** is a crucial component that represents how a structure resists forces. It is derived from the material properties and the geometry of the structure.

The stiffness matrix for a bar element can be imagined as a grid that maps loads to displacements. It essentially tells us how much a structure will bend or compress in response to forces.

• Each entry in the matrix corresponds to a force-displacement relationship between nodes in the element.
• For a three-node bar, the stiffness matrix will define how each nodal displacement affects others.

The stiffness matrix specific to our problem is given as:\[\frac{A E}{3 L}\left[\begin{array}{rrr} 7 & -8 & 1 \ -8 & 16 & -8 \ 1 & -8 & 7 \\end{array}\right]\]This equation uses properties like elasticity (AE) and length (L) to provide a dimensionless form that engineers can use to analyze or simulate structural behavior.

Mass Matrix

The **mass matrix** is another critical aspect in structural analysis as it represents the mass distribution within the structure its impacts on dynamic behavior.

• In this problem, you have two types of mass matrices: consistent and HRZ-lumped.
• The **consistent mass matrix** considers a more realistic (or distributed) mass via system matrix formulation.
• The **HRZ-lumped mass matrix**, on the contrary, approximates the mass as being concentrated at the nodes instead of being distributed along the elements.

The matrices given are:

• Consistent: \[ \frac{m}{30}\left[\begin{array}{rrr} 4 & 2 & -1 \ 2 & 16 & 2 \ -1 & 2 & 4 \end{array}\right] \]
• HRZ-Lumped:\[ \frac{m}{6}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array}\right] \]

The applications of these matrices are crucial in determining the dynamic responses, such as natural frequencies and mode shapes.

Natural Frequencies

Natural frequencies are fundamental properties of structures, indicating their propensity to vibrate at specific frequencies. These frequencies are inherent to the system and depend on mass and stiffness.

• The natural frequencies denote the spectrum of vibrations that could resonate if not properly managed.
• In this exercise, exact natural frequencies for a bar are given by: \( \omega_{1} = 0, \omega_{2} = \pi \sqrt{\frac{A E}{m L}}, \omega_{3} = 2 \omega_{2} \).
• These values are used as reference to estimate errors within the calculated frequencies derived from both mass allocation tactics (consistent and lumped).

Understanding and managing these values are essential to ensure stability and integrity of structural systems, preventing unwanted oscillations.

Mode Shapes

**Mode shapes** are specific patterns of deformation that structures undergo when oscillating at natural frequencies. They describe the way the structure moves or displaces in space when excited.

• Each natural frequency is associated with a distinct mode shape, further elucidating how different portions of the structure move relative to one another during vibration.
• In this exercise, identifying mode shapes involves solving the eigenvalue problem derived from the mass and stiffness matrices.
• The harmonics of the system, as visual depictions of mode shapes, are fundamental for designing against resonant conditions or assessing potential failure modes.

Solving for and understanding mode shapes allow engineers to predict how structures will respond under dynamic loads and thereby inform necessary design alterations or reinforcements.