hw6soln

hw6soln - ME365 Homework Set 6 Out: September 29, 2011 Due:...

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: ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 Problem No. 1 Given: A shock absorber is described by the following differential equation: where m is the mass of the shock absorber, b is the damping coefficient of the dashpot, k is the spring constant, K is the static gain, x is the displacement input (the depth of a pothole or the height of a speed bump or curb) and y is the displacement output of the shock absorber (the displacement the occupant of the vehicle feels). For a particular shock absorber, m = 1250kg, k = 5000 N/m and K = 5000 N/m. Find: (a) Determine the natural frequency, , and damping ratio, , of this system symbolically. Find the value of the damping constant, b, needed to attain a damping ratio of 0.7. (b) Using your value from (a), find the total time response of the shock absorber (particular solution +homogeneous solution) for a pothole of depth 10 cm. Assume that the shock absorbed starts from rest and that the pothole is a step input. (c) A batch of shock absorbers has been assembled with the wrong dashpot. The actual damping coefficient is 1000 kg/s. At what frequency would a sinusoidal input generate the largest cabin displacement? ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 (a) √ 2√ 2 ⋅ 0.7 ⋅ √5000 ⋅ 1250 ⋅ 3500 (b) 0⇒ 5000 / 5000 / Underdamped response ⇒ From ICs, 0 0 0 1.4 ⋅ 1 ⋅ ⋅1 0.1 0.1 1 , ⋅0 ⋅1 (c) ⋅0 → 0.1 1.428 0.14 ⋅1 → 0.0980 1000 2√ 12 1.428 0.1 ⋅0 1⋅ . 1.400 . 0.1 0 0 0 0.1 1 ⋅ 0, 0.1 2√5000 ⋅ 1250 2 1 2 ⋅ 0.200 1.428 0.200 1.9183 / 0.0980 ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 Problem No. 2 Given: An unknown RC circuit in the laboratory has been experimentally determined to have the frequency response given by the Bode pl ot below. You have been tasked l with identification of the circuit. The input to the c ircuit is a voltage at Vin, the output of the circuit is a voltage at Vout. Bode Diagram 20 Magnitude (dB) 10 0 -10 Phase (deg) -20 0 -45 -90 -2 10 -1 10 0 10 1 10 2 10 Frequency (rad/sec) Find: (a) From your previous experience, you remember that an RC circuit can be represented by a first order ODE. Use Ohm’s Law and Kirchoff’s Current Law to determine the first order ODE. Use symbols, not values. (b) Use the following measured frequency resp onse data a the figure above to and determine the static sensitivity and time co nstant. If the resistor was measured to be R = 1 kΩ, what is the value of C? | 8 | 1.87 ∠8 82.9 ° (c) What is the frequency response of the syste to the following input? em 1 ∙ cos What is the frequency response of the syste m to the following input? 1 ∙ cos 2 2 20 ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 (a) (b) ∠ 8 | (c) tan 8| 8 8 1 1 82.9° → 10 . → 1 10 10 17 1 11 ∠1 tan 1 ⋅ 1 45° 10 | 2 20 | 21.85 11 1 ∠ 2 20 tan 1 ⋅ 2 20 89.54° | 1| ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 Problem No. 3 Given: A piezoelectric stack actuator is a compact, high bandwidth device capable of producing large forces but very small displacements. Many thin wafers of piezoelectric material are sandwiched between electrodes and stacked together. The result is that the wafers are electrically in parallel while mechanically in series. Find: (a) A first-cut model of the extension (in µm) of a piezoelectric stack actuator in response to an input voltage is given by 2 where K = 0.18 µm/V, ωn = 3 kHz and ζ = 0.1. Determine the magnitude and phase of the model frequency response as a function of the frequency, ω. (b) Using your answer from (a), draw the Bode plot of the system model using straight line approximations. Identify the magnitude and phase values at the resonant frequency on your plot. (c) Assume the extension of the piezoelectric stack actuator to be measured with a sensor having a static sensitivity of 0.5 V/µm. For data collection, a firstorder low-pass filter is to be included between the displacement sensor and the ADC. Two filters are available, having cutoff frequencies of 1 kHz or 10 kHz. Which filter is a better choice and why? For your selection, determine the frequency response function of the combined piezoelectric stack actuator, sensor and low pass filter. Draw the Bode plot of the combined piezoelectric stack actuator, sensor and low pass filter using straight line approximation. 1 ME365 Homework Set 6 Out: September 29, 2011 Due: October 14, 2011 (a) | 0.18 | 2 2 ⋅ 0.1 ⋅ 2 ⋅ 3000 1 2 ∠ tan 4 ⋅ 3000 2 ⋅ 0.1 ⋅ 2 ⋅ 3000 tan 1 1 1 4 ⋅ 3000 (b) Drawing will be a straight line approximation, should roughly match the plot given in this solution. Bode Diagram 0 Magnitude (dB) -20 -40 -60 -80 - 100 0 Phase (deg) -45 -90 -135 -180 3 4 10 5 10 6 10 10 Frequency (rad/sec) 1 2 2 ⋅ 3000 1 2 ⋅ 0.1 | ∠ 0.8715 84.2318° | 18660 2970 (c) 10 kHz is better choice to not roll off the peak. Drawing will be a straight line approximation, should roughly match one of the plots given in this solution. ME365 Homework Set 6 Bode Diagram Original System 0 Original System + 1 kHz LP -50 Magnitude (dB) Out: September 29, 2011 Due: October 14, 2011 Original System + 10 kHz LP -100 -150 Phase (deg) - 200 0 -90 -180 -270 2 10 3 10 4 10 5 10 Frequency (rad/sec) 6 10 7 10 ...
View Full Document

Ask a homework question - tutors are online