ASEN 5022 - Second Mid-Term Quiz - 17 April 2008Your name:Problem 1: Classical theory of vibrations of a bar (40 points)We have studied that Hamilton’s principle for a continuum bar can be stated as shown below:Zt2t1[δT − δV + δW]dt = 0T =ZL012ρ(x) ˙u2(x, t) dx, 0 ≤ x ≤ LV =ZL012EAu2x(x, t) dxδW =ZL0f (x, t)δu(x, t) dx(1)1.1 Carry out the necessary variations. (10 points)1.2 Obtain the governing differential equation of motion from the variational equation that you havederived in Problem 1.1 and identify the terms that allow us to obtain the two boundary conditionsfor general cases. (10 points)1.3 Specialize the boundary conditions you obtained in Problem 1.2 for a bar with fixed-fixed ends,that is, at x = 0 and at x = L. (10 points)1.4 Utilizing the solution of u(x, t) in the formu(x, t) = U(x)ejωt, with β2= mω2/EAU(x) = C1cosβx +C2sinβx(2)Obtain the equation for determining the vibration frequency and mode shapes for the casse offixed-fixed ends, viz., a 2x2 matrix equation of the form Ac = 0. It is not necessary to computateβ L. (10 points)Problem 2: Understanding Beam Vibrations (30 points)We have studied that Hamilton’s action integral for a beam shown below can be expressed asZt2t1{[EI wxxx(x, t)] δw(x, t)}|x=Lx=0dt −Zt2t1{kw2w(L, t)δw(L, t) + kw1w(0, t)δw(0, t)} dt−Zt2t1{[EI wxx(x, t)]δwx(x, t)}|x=Lx=0dt −Zt2t1{kθ2wx(L, t)δwx(L, t) + kθ1wx(0, t)δwx(0, t)} dt−Zt2t1ZL0{ m(x)wtt(x, t) + [EI wxxxx(x, t)] − f (x, t) } δw(x, t) dxdt = 0(3)kw1kw2kθ1kθ2EI, m(x)xzwLBeam with unknown boundary conditionsMMVVSign convention of moment and shear force for continuum beamFigure for Problem 12.1 Identify the boundary condition(s) when kw16= 0, kw26= 0, kθ16= 0, and kθ26= 0. Are theboundary conditions identified natural or essential ones? (15 pts)2.2 For the case of {kw16= 0, kw2→∞, kθ1= 0, kθ26= 0}, sketch the mode shape of the firstvibration mode and explain why you believe your prediction makes sense. (Hint: Try to draw forlimiting cases of {kw1→∞, kw1→ 0} and {kθ2→∞, kθ2→ 0} then draw for finite cases of{kw1, kθ2}). (10 pts)2.3 An experimental data of a beam indicates that the fundamental frequency of the beam is far higherthan that of the corresponding cantilever beam (β L = 1.875) but somewhat lower than that of thesimply supported beam (β L = π) with a deflection observed at its right end during test. Can youspeculate its bounday conditions? (5 points)Problem 3: Design for Vibration of Bars (30 points)Consider a unifrom bar and two bars with different cross sections as shown below. The areas of Bar Band Bar C are related to that of the unifrom bar, A, byA1+ A2= 2A, A1= γ A, A2= (2 − γ)A, 0 <γ <2 (4)The fundamental frequencies of Bar A (uniform), Bar B and Bar C are designated as (ωA,ωB,ωC),respectively.Assume the following material properties:ρ = ρ1= ρ2, E = E1= E2(5)Clamped BaseBar (1)(EA, m, L)1Bar (2)(EA, m, L)2u1u2L/2L/2Clamped BaseBar (1)Bar (2)(EA, m, L)1(EA, m, L)2u1u2L/2L/2Clamped Base(EA, m, L)Uniform Baru1u2L/2L/2Bar ABar BBar CFigure for Problem 33.1 Can you prove that ωA<ωBalways holds, provided (1 <γ <2) ? (15 points)3.2 Can you find the range of the area ratio γ such that ωC<ωA<ωBholds? (15 points)OVERUseful generic formulas:1. When you utilize FEM modeling tools, use the following.lumped element mass matrix: [m] =ρ A2·1001¸element stiffness matrix: [k] =EA`·1 −1−11¸(6)Hint: Assemble the two-element mass and stiffness matrices and perform eigenvalue analysis whosefrequencies are a function of the area ratio γ .2. When you employ an analytical approach, one has the following equations for Bar B and Bar C:C1sin(β L/2) − C2= 0C2sin(β L/2) − C3cos(β L/2) = 0C1cos(β L/2) −A2A1C3= 0,β2= ω2ρ/E⇓tan2(β L/2) =A1A2=γ2 − γ(7)and the fundamental frequency for Bar A is given by(β L)2= ω2ρL2/E = (π/2)2(8)Hint: Solve for (βL) and determine γ satisfying the inequality