Problem

Problems deal further with the car of Example. Its upward displacement function satisfies...

Problems deal further with the car of Example. Its upward displacement function satisfies the equation mx″ + cx? + kx = cy′ + ky when the shock absorber is connected (so that c > 0) With y = a sin ωt for the road surface, this differential equation becomesmx″ + cx′ +kx = E0 cosωt + F0 sin ωtwhere E0 = cωa and F0 = ka.

Figure shows the graph of the amplitude function C(ω) using the numerical data given in Example(including c = 3000 N.s/m). It indicates that, as the car accelerates gradually from rest, it initially oscillates with amplitude slightly over 5 cm. Maximum resonance vibrations with amplitude about 14 cm occur around 32 mi/h, but then subside to more tolerable levels at high speeds. Verify these graphically based conclusions by analyzing the function C(ω). In particular, find the practical resonance frequency and the corresponding amplitude