Problem 7 (FSI, closed book.) Consider the case of 1D motion of the body in line...

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Problem 7 (FSI, closed book.) Consider the case of 1D motion of the body in line with the time-varying velocity at infinity. You may use without a proof Morison's equation for the aerodynamic force F acting on the body: dx CD ma +p- dt 2 dx dx A, dt F = (pV + m.)dux - m + pop (1-2) | | |A dt dt where p is the fluid density, V is the body volume, ma is the added mass, u = (t) is the velocity at infinity, x is the body coordinate, t is time, CD is the drag coefficient, and A is the cross-sectional area of the body. The body has mass m and is elastically supported with the damping coefficient and elastic stiffness k. Assume that the the velocity at infinity is constant, u (t) = u = const, and that the body is in an oscillatiry motion of a small amplitude, so that do (a) Write down the equation of motion of the body in this case. Assume further that the amplitude of the oscillations is constant. dx dt [20%] (b) Do the oscillations increase or decrease the drag? Give the proof. [20%] (c) Explain what fluid damping is, and how it is affected by u. Prove your answer. [20%] (d) Write down the formula for the natural frequency as a function of u. [20%] (e) Derive the formula for the value of the free-stream velocity u above which the damping is super-critical. [20%] Problem 7 (FSI, closed book.) Consider the case of 1D motion of the body in line with the time-varying velocity at infinity. You may use without a proof Morison's equation for the aerodynamic force F acting on the body: dx CD ma +p- dt 2 dx dx A, dt F = (pV + m.)dux - m + pop (1-2) | | |A dt dt where p is the fluid density, V is the body volume, ma is the added mass, u = (t) is the velocity at infinity, x is the body coordinate, t is time, CD is the drag coefficient, and A is the cross-sectional area of the body. The body has mass m and is elastically supported with the damping coefficient and elastic stiffness k. Assume that the the velocity at infinity is constant, u (t) = u = const, and that the body is in an oscillatiry motion of a small amplitude, so that do (a) Write down the equation of motion of the body in this case. Assume further that the amplitude of the oscillations is constant. dx dt [20%] (b) Do the oscillations increase or decrease the drag? Give the proof. [20%] (c) Explain what fluid damping is, and how it is affected by u. Prove your answer. [20%] (d) Write down the formula for the natural frequency as a function of u. [20%] (e) Derive the formula for the value of the free-stream velocity u above which the damping is super-critical. [20%]