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CU-Boulder ASEN 5022 - First Mid-Term Exam

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ASEN5022 (Spring, 2011) - First Mid-Term ExamClosed books and One Crib Sheet (Tuesday, 15 February 2011)Problem 1.0 General Concepts (30 points; 4 points for 1.1-1.5 and 5 pointsfor 1.6-1.7)1. The moment of force F about point O as the vector product of itsmoment arm r and F is defined as (a) MO= F × r, (b) MO= rF , (c)MO= r × F, (d) MO= −F × r. (Pick as many)2. The potential energy V (r) is defined as V (r) =Rrrefrf inalFc· dr. Theevaluation of the change in the potential energy 4V =Rr2r1Fc· dr is givenby (a) 4V = V1− V2, (b) 4V = −V1+ V2, (c) 4V = V1+ V2. (pick one)3. The principle of virtual work is stated as¯δWi=Pni=1Fi·δri= 0. In thatstatement, the virtual displacement δriinvolves (a) an independent variableri, (b) sometimes dependent variable ri, (c) any arbitrary variable ri. (pickone)4. The extended Hamilton’s principle states {Rt2t1(δT +¯δW )dt = 0, δri(t1) =δri(t2) = 0, i = 1, 2, ..., N}. The end conditions δri(t1) = δri(t2) = 0 signifies(a) ri(t) is specified at t = t1and t = t2, (b) ri(t) is arbitrary at t = t1andt = t2, (c) ri(t) is unknown at t = t1and t = t2.5. The strain energy stored in the string is proportional to: (a)RT dx,(b)RT (dwdx)dx, (c)RT (dwdx)2dx, where T is the tension acting on the string.(Pick one)6. The governing equations of motion for continuum systems are in general:(a) dependent on the boundary conditions, (b) independent of the boundaryconditions, (c) dependent both on the boundary conditions as well as on theinitial conditions. (Pick one letter)7. The equation of motion for the string is given byddx[T (x)dwdx] + f (x, t) = ρd2w(x, t)dt2which governs: (a) an in-plane(cord-wise stretching) motion, (b) a transversemotion, (c) both in-plane and transverse motions. (pick one)1θmgyOLmgLyF(t)xxoA‘B‘A, Bmassless guidemassless guideInitial Static Equilibrium StateConfiguration at time tafter F(t) is applied.mgkku(t)Fig. 1. Figure for Problem 02Problem 2: A spring-mass system moving along a massless pendu-lum guide. (40 points)A system consists of mass m that is attached to spring k. The mass and thespring are housed in a massless pendulum guide as shown in Figure 1. Theroller that guides mass m experiences no friction when moving along the guide.It is assumed that, at time t after a conservative force F (t)(as shown in Fig.1) is applied to the system, the mass-spring-massless guide has undergone arotation θ and a spring extension u(t) as indicated in the figure.2.1 (3 items, 10 points) Choose your coordinate system (and define unitvectors you will utilize for this problem) (4 points)Express the position vector of the mass point, B0. (3 points)Identify the appropriate generalized coordinates of the system. (3 points)2.2 Obtain the velocity vector of the mass point, B0, and express theangular velocity vector of the mass-spring-massless guide system. (5 points)2.3 Obtain the system kinetic energy. (5 points)2.4 Obtain the total potential energy (due to gravity and spring extension)(5 points)2.5 Obtain the generalized forces that correspond to the generalized co-ordinates that you have identified in Problem 2.1. (5 points)2.6 Obtain the equations of motion via Euler-Lagrange’s formalism. (10points)2Help: Generic form of Euler-Lagrange’s equation:ddt(∂L∂ ˙qk) −∂L∂qk= Qkδ¯W =nXk=1Qkδqk=nXk=1(fnoncons)k· δrkL = T − VT =12nXk=1m˙rk·˙rkV =nXk=1{ZrrefkrkFk· drk}(1)Alternate approaches for Problem 2.3 - 2.6 are accepted to arrive at2.6 and you will receive a full credit, viz., 20 points. However, youmust answer 2.1 and 2.2.3x(t)y(t)piezo-controllerK(EI, m)Solid beamLMCBase motion, x_0(t)Proof MassFig. 2. Figure for Problem 03Problem 3: Modeling of a 2-DOF spring-mass system with groundmotions. (30 points)Suppose you would like to model the vibration control of a wing tip (or abuilding or an automobile suspension system) by applying a proof-mass typeactuator as shown in Figure 2. The proof mass consists of its concentratedmass M, its damper C and spring K. Ignore the mass of the proof-massactuator housing for simplicity. The base motion is prescribed and assumedto be known as x0(t).It has been known that a simplified modeling of the vertical motion of thecantilever beam tip by a single degree of freedom mass and spring constantcan be approximated asMb¨x(t) + Kbx(t) = 0, Mb= 0.242ρAL, Kb= 3EI/L3where x(t) is the vertical displacement at the beam tip when there is no basemotion, and the mass Mbis assumed to be concentrated at the beam tip.3.1 Obtain the position vectors of the beam tip, and of the proof massM. Obtain the accelerations of the beam and the proof mass. (10 points)3.2 Sketch free-body diagrams indicating the forces acting on the proofmass M, and on the beam mass Mbat the beam tip. (10 points)3.3 Derive the coupled two-DOF equations of motion when the proof-mass is mounted as shown. You may employ any of the analytical methodswe have studies including Newton’s method, d’Alembert’s principle, Euler-Lagrange formalism, or Hamilton’s principle. (10


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