SolSec1.10

# SolSec1.10 - Problems and Solutions Section 1.10(1.91...

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Problems and Solutions Section 1.10 (1.91 through 1.103) 1.91 A 2-kg mass connected to a spring of stiffness 10 3 N/m has a dry sliding friction force ( F d ) of 3 N. As the mass oscillates, its amplitude decreases 20 cm in 15 cycles. How long does this take? Solution : With m = 2kg, and k = 1000 N/m the natural frequency is just ω n = = 1000 2 22 36 . rad/s From equation (1.101): slope = = = 2 2 µ ω π ω π mg k f k x t n c n Solving the last equality for t yields: t x k f c n = = = π ω π 2 0 20 10 2 3 22 36 4 68 3 ( . )( )( ) ( )( . ) . s 1.92 Consider the system of Figure 1.40 with m = 5 kg and k = 9 × 10 3 N/m with a friction force of magnitude 6 N. If the initial amplitude is 4 cm, determine the amplitude one cycle later as well as the damped frequency. Solution: Given m k f x c = = × = = 5 9 10 6 0 04 3 0 kg N/m, N, m , . , the amplitude after one cycle is x x f k c 1 0 3 4 0 04 4 6 9 10 0 0373 = = × = . ( )( ) . m Note that the damped natural frequency is the same as the natural frequency in the case of Coulomb damping, hence ω n k m = = × = 9 10 5 42 43 3 . rad/s

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1.93* Compute and plot the response of the system of Figure P1.93 for the case where x 0 = 0.1 m, v 0 = 0.1 m/s, µ = 0.05, m = 250 kg, θ = 20 ° and k =3000 N/m. How long does it take for the vibration to die out? m k u m Figure P1.93 Solution: The equation of motion for this system is 0 cos = + + kx x x mg x m & & & & θ µ Answer: The oscillation dies out after about 0.9 second. M ATLAB Code: xo=[0.1; 0.1]; ts=[0 10]; [t,x]=ode45('f',ts,xo); plot(t,x(:,1)) title('problem 9.44');xlabel('time(s)');ylabel('displacement (m)') %--------------------------------------------- function v=f(t,x) m=250; k=3000; u=0.05;g=9.81; v=[x(2); x(1).*-k/m-x(2).*u*g*cos(20*pi/180)/abs(x(2))];
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 problem 9.44 time(s) displacement (m)

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1.94* Compute and plot the response of a system with Coulomb damping of equation (1.88) for the case where x 0 = 0.5 m, v 0 = 0, µ = 0.1, m = 100 kg and k =1500 N/m. How long does it take for the vibration to die out?
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