01 - Free Undamped
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SHEET #1- PART I - FREE UNDAMPED VIBRATIONS 1.1 The spring of Figure 1.2 is successively loaded with mass and the corresponding (static) displacement is recorded below. Plot the data and calculate the spring's stiffness. Note that the data contain some error. Also calculate the standard deviation. m(kg) x(m) 10 1.14 11 1.25 12 1.37 13 1.48 14 1.59 15 1.71 16 1.82 1.2 Derive the solution of mx + kx = 0 and plot the result for at least rum periods for t he ca se with Oln = 2 ra d/s, X o = mm, and V o = J 5 rnm/s. 1.3 Solve m x + kx =0 for k = 4 N/m, m = 1 kg, X o = 1 mm, and V o = O. Plot the solution. 1.4 The amplitude of vibration of an undamped system is measured to be 1 mm, T he p has e shift f rom t =0 is measured to be 2 rad and the frequency is found to be 5 rad/s. Calculate the initial conditions that caused this vibration to occur. Assume the response is of the form xCt) =A sinew) + (P). 1.5 Find the equation of motion for the hanging spring-mass system of Figure P1.5, and compute the natural frequency. In particular, using static equilibrium along with Newton's law, determine what effect gravity has on the equation of m tion and the system's natural frequency. t:t) ___~ F ig ur e P1.5
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