## Ritz Method Aeroelasticity

graduate level texts on structural dynamics.) In those cases, the Ritz and Galerkin's methods give the same results when used with the same assumed mode functions. As we see here, however, Galerkin's method provides a viable alternative to the Ritz method in cases where the equations of motion are not of the form presented earlier in this chapter.

### 2.5 Epilogue

In this chapter we have considered the free-vibration analysis and modal representation for flexible structures, along with methods for solving initial-value and forced response problems associated therewith. Moreover, modal approximation methods based on the Ritz and Galerkin methods were introduced. This sets the stage for consideration of aeroelastic problems in Chapters 3 and 4. The static aeroelasticity problem, treated in Chapter 3, results from interaction of structural and aerodynamic loads. These loads are a subset of those involved in dynamic aeroelasticity, which includes inertial effects. One aspect of dynamic aeroelasticity is flutter, which is treated in Chapter 4. It will be seen that both the modal representation and the modal approximation methods apply equally well to both types of problems.

### Problems

1. By evaluating the appropriate integrals, prove that each function in the following two sets of functions is orthogonal to all other functions in its set over the interval

Use of a table of integrals may be helpful.

2. Considering Eq. (2.59), plot the displacement at time t — 0 for a varying number of retained modes, showing that as more modes are kept the shape more closely resembles the initial shape of the string given in Fig. 2.4.

3. Compute the propagation speed of elastic torsional deflections along prismatic, homogeneous, isotropic beams with circular cross sections and made of

(a) aluminum

Hint: Compare the governing wave equation with that for the uniform string problem, noting that for beams with a circular cross section J = Ip.

Answers', (may vary slightly depending on properties used)

4. For a uniform string attached between two walls with no external loads, determine the total string deflection v(x, t) for an initial string deflection of zero and an initial transverse velocity distribution given by

Answer.

where nit

5. Consider a uniform string of length I and mass per unit length m that has been stretched between two walls with tension T. Transverse vibration of the string is restrained at its midpoint by a linear spring with spring constant k. The spring is unstretched when the string is undeflected. Write the generalized equation of motion for the i th mode, giving particular attention to the writing of the generalized force E, . As a check, derive the equation taking the spring into account through the potential energy instead of through the generalized force.

Answer: Letting ire / T

m one finds that the generalized equations of motion are

+0,2$ + _(-!)¥ £ (_1)¥|,=0 (/ = 1,3 oo), ml . , ,

Consider a uniform string of length i with mass per unit length m that has been stretched between two walls with tension T. Up until the time t — 0 the string is at rest. At time r — 0 concentrated loads of magnitude E0 sin Qt are applied at x — f /3 and x = 2f /3 in the positive (up) and negative (down) directions, respectively. In addition, a distributed force

is applied to the string. What is the total string displacement v(x. t ) for time t > 0?

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