Consider the model of a turbine blade (Griffin and Hoosac, 1984) shown in Figure P5.2. Model parameters (SI units) are as follows: \(m_{1}=0.0114, m_{2}=0.0427, m_{3}=0.0299, k_{1}=430,300, k_{2}=\) \(17,350,000\), and \(k_{3}=7,521,000\).

a. Compute the natural frequencies and the mode shapes. Examine the orthogonality of the mode shapes.

b. Let \(f(t)=\sin \omega t\), where \(\omega\) is the excitation frequency. Using modal decomposition, find the amplitude and the phase of each mass as a function of the excitation frequencies near the first natural frequency. Assume the modal damping ratio to be 0.01 .

Figure P5.2

 

Transcribed Image Text:

x1 f(t) k m m2 x2 X3 k m3 k3