hwk09-1 - ME563 – Fall 2006 Purdue University West...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ME563 – Fall 2006 Purdue University West Lafayette, IN Homework Set No. 9 Assignment date: Wednesday, November 1 Due date: Wednesday, November 8 Please attach this cover sheet to your completed homework assignment. • If you choose to scan in your completed solution, please email this solution to vagrawal@purdue.edu by midnight of the due date given above. Please give your homework file the following name: “name_hwk”. For example, if your last name is Smith and you are submitting Homework Set No. 3, then you should name your file: “smith_03”. • Otherwise, if on campus, please bring to lecture on the due date. If off campus, mail to the address below with a postmark no later than midnight of the due date given above: Ms. Donna Cackley 585 Purdue Mall Purdue University West Lafayette, IN 47907-2088 Name Location _________________________________________ Problem 9.1 __________________ Problem 9.2 __________________ Problem 9.3 __________________ TOTAL __________________ ME 563 – Fall 2006 Homework Prob. 9.1 Here we consider a two-DOF model of an automobile traveling on a roadway as shown below. The automobile is traveling with a constant speed (z = vt) on a roadway whose roughness is modeled by a sinusoidal function y ( z) = y0 cos( 2"z / #) where λ is the wavelength of the roughness. The absolute coordinate x is used to describe the vertical displacement of the center of mass G of the automobile body. The rotation of the automobile body is described by the rotational coordinate θ. The body is to be modeled as ! a homogeneous rigid thin body of length L. Assume that the tires A and B remain in contact with the roadway; otherwise, ignore gravity in your analysis. λ a) Use the Lagrangian formulation to derive the EOM’s for this small oscillations of the coordinates x and θ. Use L/λ = 0.5. # x ( t ) & # X () ) & b) Solve for the particular solution of the EOM’s: $ P '=$ ' cos )t % L " P ( t )( % L * () )( c) Determine the natural frequencies and mode shapes. Use these to write down the uncoupled modal EOM’s. Solve these modal EOM’s. d) Using the results from either b) or c) above, make a sketch of X(Ω) and LΘ(Ω) ! vs. " / k / m . e) From your plots in d), identify speeds v corresponding resonances and antiresonances for the coordinates. ! ME 563 – Fall 2006 Homework Prob. 9.2 An undamped, single-DOF system has an EOM of the form: m˙˙ + kx = f ( t ) = f 0 sin "t cos3 "t x ! a) Observe that the “forcing” f ( t ) is a periodic function of time. Using trig identities, determine the Fourier series for f ( t ) . Identify the fundamental period T of the forcing (the fundamental period is the shortest time interval over which the function will repeat, f ( t ) = f ( t + T ) ). ! b) At what frequencies " are resonances expected in the response of the system? ! c) Determine the total response (homogeneous + particular) of the system ˙ corresponding to initial conditions of x (0) = x 0 and x (0) = 0 . ! ! ! ! ME 563 – Fall 2006 Homework Prob. 9.3 An undamped, single-DOF system is given a base motion of ˙˙ ( t ) = a0 f ( t ) where y f ( t ) = f ( t + T ) is the T-periodic function shown below. x y ! ! m k f(t) 1 t -1 T T T T T T From lecture, we know that f ( t ) = f ( t + T ) can be written in terms of its sine-cosine Fourier series as: # # f f ( t ) = 0 + $ f cj cos+ $ f sj sin j"t 2 j =1 ! j =1 where " = 2# / T . ! ! ! ! a) Considering the Fourier series for f(t) above, provide arguments that: f0 = 0 f cj = 0 ; j = 1, 2, 3,... and T /2 f sj = 2 # f (t ) sin j"t dt ; j = 1, 2, 3,... 0 b) Considering the results of a) above, determine the Fourier series for f(t). You will likely need to use integration by parts to evaluate these integrals. ! ...
View Full Document

This document was uploaded on 12/23/2011.

Ask a homework question - tutors are online