4 Student 1:42 PM Mon Aug 8 webassignhet '7 6'0) 82%E} 5. [-/8 Points] DETAILS 27! (Dd = t2 t1 X2 A (0.74.1.72) (390,092) (2.32,-1.25) (5.48,-0.67) x(t) = R e'ga'n tsin(wdt) Engineers often describe damped harmonic motion with the formula because both C and and can be measured in a straightforward way. T = t2 fl is called the quasi-period, and is the damped natural frequency or quasi-frequency A X A = |n(1) is called the logarithmic decrement and There is no phase shift (p because we have chosen an initial time t = 0, to be a zero of x(t). If you measure the times and displacements, (t1,x1) and (t2,x2) , at two consecutive peaks, then, MY NOTES PRACTICE ANOTHER Student 1:42 PM Mo . . . 1 82% webassign.net a c'n (5.48,-0.67) -1 (2.32,-1.25) 1.5 -20 Engineers often describe damped harmonic motion with the formula x(t) = Re-50on tsin(wat) because both & and wd can be measured in a straightforward way. There is no phase shift q because we have chosen an initial time t = 0, to be a zero of x(t). If you measure the times and displacements, (t1,x1) and (t2,X2) , at two consecutive peaks, then, T = t2 - t1 is called the quasi-period, and 2 nt Wd = . - is the damped natural frequency or quasi-frequency t2 - t1 A = In(-) is called the logarithmic decrement and A A 5 = VA2 + 412 2 n is called the damping ratio. wd wn = V 1-2 2 is the (undamped) natural frequency. Use the measured values of (t1,x1) and (t2,X2) from the graph above to find: Wod =[ Wn = Now write the differential equation that x(t) satisfies in the form x " + x' + x = 0 Finally, use the original formula and your measured values of (t1,X1) to estimate R and the initial conditions. R= x(0) = x'(0)=4 Student 1:42 PM Mon Aug 8 no. '7 6' (I) 82% E} webassignnet ii 4. [-/7 Points] DETAILS MY NOTES PRACTICE ANOTHER 1. Suppose that a car weighing 3000 pounds is supported by four shock absorbers Each shock absorber has a spring constant of 6500 lbs/foot, so the effective spring constant for the system of 4 shock absorbers is 26000 lbs/foot. 1. Assume no damping and determine the period of oscillation of the vertical motion of the car. Hint: 9: 32 ftlsec2 , T: [: seconds. 2. After 10 seconds the car body is 1/2 foot above its equilibrium position and at the high point in its cycle. What were the initial conditions ? y(0)=:] ft. and y'(0)=:] ftlsec. 3. Now assume that oil is added to each the four shock absorbers so that, together, they produce an effective damping force of 7.13 lb-sec/ft times the vertical velocity of the car body. Find the displacement y(t) from equilibrium if y(0)=0 ft and y'(0)= -10 ftlsec. y(t) = 2. Suppose that you are designing a new shock absorber for an automobile. The car has a mass of 1000 kg (kilograms) and the combined effect of the springs in the suspension system is that of a spring constant of 4000 N/m (i.e.each of the four springs has a spring constant of 1000 N/m). 1. Before a damping mechanism is installed in the car, when the car hits a bump it will bounce up and down' How may bounces will a rider experience in the minute right after the car hits a bump? Alternatively, what is the frequency in cycles per minute? The car will bounce C] times per minute. 2. Yourjob is to design a damping mechanism which eliminates oscillations when the car hits a bump. What is the minimum value of the effective damping constant that can be used? 7 = [j kg/sec. 3. Suppose that at time i=0 the car hits a bump. Immediately before that time the car was not moving up and down and the effect of the bump is to add a vertical component to the speed of the car of 1.0 meter/sec. How high will the car rise above its equilibrium position if you design the system with the damping constant you found in part (b)? It will rise :J meters above the equilibrium position. 5. [I8Points] l DETAILS l l MYNOTES ll PRACTICEANOTHER l