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 0 ? x ? ? :

1. sin for n = 1, 2, 3,...

2. cos for n = 0, 1, 2,...

Use of a table of integrals may be helpful.

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.

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

1. aluminum

2. steel

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

Answers: (may vary slightly depending on properties used)

1. 3140 m/s

2. 3110 m/s

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

Consider a uniform string of length ? 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