Question

EXam 2 Name: 1. A n undamped vertical system consists of a mass weighing 100 N and a spring of stiffness 5000 N/m. It is acted on by a harmonic force of amplitude 80 N and frequency 5 Hz. Find i) The displacement of the spring due to the weight of the mass, The static displacement of the spring due to the maximum applied force, and The amplitude of forced motion of the mass for zero initial conditions iii)

Answer 1

Given that weight w = 100N, m = w/g = 100/ 9.81 = 10.193 kg k = 5000 N/m f_0 = soN f = 5HZ Rightarrow w = 2nf = 31.415 rad/sec i) Displacement of spring due to weight of the mass we know that spring constant k = Force/defflection Rightarrow 5000 = 100/spring displacement Rightarrow spring displacement = 0.02m ii) static displacement of spring due to max applied force Rightarrow k = f_0/delta Rightarrow delta = f_0/k = 80/5000 = 0.016m iii) Amplitued of force motion x = f_0/k /sqrt(1-r^2)^2 + (2xir)^2 for undamped Rightarrow xi = 0. r = w/w_n Rightarrow w_n = squareroot k/m = squareroot 5000/10.193 = 22.147 rad/s therefore r = 31.415/22.147 = 1.418 therefore x = 80/5000/ squareroot (1-1.418^2)^2 = 0.016/1.0107 = 0.015m amplitude = 15.83mm.