Question

Unter Berücksichtigung eines kubischen Verschiebungsansatzes ermittle man die Eigenfrequenzen eines einseitig eingespannten Stabes (Dichte \(\rho\), Querschnitt A , Länge \(\mathrm{L}\) ) aus linear-elastischem , isotropem Material (Elastizitätsmodul E). Die Ergebnisse vergleiche man mit den exakten Werten und den Werten, die man auf der Basis eines linearen (Ü3.2.9) und quadratischen (Ü3.2.14) Verschiebungsansatzes gefunden hat.

Short answer

Find eigenfrequencies using cubic displacement, apply boundary conditions, solve characteristic equation, and compare results with exact and lower-order approaches.

Step-by-step solution

Introduction to the problem

Determine the natural frequencies of a cantilever beam (density \( \rho \), cross-sectional area \( A \), length \( L \)) made of linear-elastic, isotropic material (Elasticity modulus \( E \)) using a cubic displacement approach.

Formulate the displacement function

Consider the cubic displacement function for the beam, \( u(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \). Use boundary conditions to determine the coefficients.

Apply boundary conditions

For a cantilever beam, the deflection and slope at the fixed end are zero. This gives two boundary conditions: \( u(0) = 0 \) and \( u'(0) = 0 \). At the free end, the moment and shear force are zero: \( u''(L) = 0 \) and \( u'''(L) = 0 \). Apply these to the displacement function.

Derive the characteristic equation

Using the derived displacement function with boundary conditions, find the characteristic equation for the system. This will involve solving the resulting polynomial equation to find \( \omega \).

Solve for the eigenfrequencies

Solve the characteristic equation to find the eigenfrequencies \( \omega_n \). The eigenfrequencies are given by \( \omega_n = \sqrt{\frac{E I}{\rho A}} \beta_n \), where \( \beta_n \) are the roots of the characteristic equation.

Compare with exact and lower-order displacement approaches

Compare the obtained eigenfrequencies with the exact values and those determined using linear and quadratic displacement approaches. Ensure to highlight the accuracy improvement.

Key concepts

cubic displacement approach

Starting with the problem, we use the cubic displacement approach to approximate the deflection of a cantilever beam. A cubic displacement function is a polynomial of degree three, and it's written as: \( u(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 \).
This approach can capture more complex behaviors of the beam compared to linear or quadratic approaches, as it adds more flexibility to the shape of the deflection curve.
By adjusting the coefficients \( a_0, a_1, a_2, a_3 \), we can fit the displacement profile more accurately. This is especially useful for engineering applications where precision is crucial.

characteristic equation

The heart of solving for eigenfrequencies lies in deriving the characteristic equation.
After applying the boundary conditions to our cubic displacement function, we get a system of equations to determine the coefficients.
By substituting these into our equation for motion and setting the determinant to zero, we get the characteristic equation.
The characteristic equation relates the natural frequencies of the system to its physical properties (elasticity modulus, density, and geometry).

boundary conditions analysis

For a cantilever beam, boundary conditions are critical to obtaining accurate results.
At the fixed end, both the deflection and slope are zero, meaning \( u(0) = 0 \) and \( u'(0) = 0 \). This reflects the fact that the beam is rigidly supported.
At the free end, the shear force and bending moment are also zero, giving us \( u''(L) = 0 \) and \( u'''(L) = 0 \). These conditions ensure continuity and equilibrium at the free end.
By applying these conditions to our cubic displacement function, we systematically solve for the coefficients.

linear-elastic materials

In this problem, the material of the cantilever beam is assumed to be linear-elastic and isotropic.
This means the material follows Hooke's Law, where the stress is directly proportional to the strain: \( \sigma = E \varepsilon \).
The elasticity modulus (E) measures the material's stiffness and is a critical parameter in determining the beam's deflection under load.
Isotropic materials have identical properties in all directions, simplifying the analysis without needing to account for directional variations.

eigenfrequency comparison

Once we have the eigenfrequencies from the cubic displacement approach, it’s essential to compare these with known exact values and those obtained from simpler linear and quadratic approaches.
This allows us to evaluate the improvement in accuracy provided by the cubic approach.
Typically, as the displacement function becomes more complex (from linear to quadratic to cubic), the approximation of the natural frequencies becomes more accurate.
By comparing these, students can understand how higher-order approximations lead to better results, although at the cost of increased computational complexity.