1. Consider the mass-damper system shown in Fig. 1. Velocity v(t) of the mass takes the following form v(t) = − (1 − j)e³(2πt) + jej (4πt) 1 - ej(6πt) +c.c. 2 (1) where j√-1 and c.c. represents complex conjugates of the three complex terms in (1). Moreover, the velocity v(t) and time t are both in MKS units. Answer the following questions. (a) What is the fundamental frequency wo of the velocity v(t)? Please include the unit of wo. (b) Compare the velocity v(t) in (1) with the following complex Fourier series 8 v(t): = Σ Cejnwot n=-∞ where wo is the fundamental frequency. Calculate the magnitude and phase of In can be positive or negative. Obtain and plot the complex spectrum of v(t). (c) Integrate the velocity v(t) in (1) to obtain displacement x(t) as (2) Сп where x(t) = =- (1-1) ej(2πt) + jej (4nt). - 2πj 4πj 1 ~ēj (6πt) + c.c. 2 (6πj) (3) Express the displacement x(t) in the form of trigonometric functions (e.g., sine and cosine functions). (d) For the velocity v(t) shown in (1), does its Fourier transform exist? Explain why. (e) When a Fourier series converges to a periodic function f(t), i.e., 8 f(t) = A0+(An cos nwot + B sin nwot) n=1 what is minimized when more terms are retained in the Fourier series? (4)

Consider the mechanical energy equation: P + VB? + gzB =PA + V? + g ZA + w – loss A +g ZA + W – loss where loss 2 0 Select ALL true statements below. a fluid particle will flow through point A before point B a fluid particle will flow through point B before point A points A and B do not lie on the same streamline if a turbine is present, w < 0 if a pump is present, w < 0 if the fluid is perfectly inviscid, loss can be 0

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X Your answer is incorrect. The figure below provides steady-state data for water vapor flowing through a piping configuration where V1 = 35 m/s and T1 = 200°C. At each exit, the volumetric flow rate, pressure, and temperature are equal. P2= 4.8 bar T = 320°C Water vapor (AV)2 = (AV)3 V, T A1 = 0.2 m? Pi = 5 bar 3 P3 = 4.8 bar T3 = 320°C Determine the mass flow rate at the inlet and exits, each in kg/s. i 25.78 kg/s = i 12.89 kg/s m3 = i 12.89 kg/s