Question

3.7 Underdamped free vibrations (e.g. a vibrating beam or slinky in damp air A system whose response is governed by the following constant-coefficient, is performing underdamped free vibrations if oc 1 (underdamped ). (Seetion 6.7). linear, 2td order, ODE ) and 1t)- (freei (a) After assuming a solution t)et where C and p are conustants, show every step that proves yit) can be expressed in terms of the yet-to-be-determined constants A and B and he constant wa (defined above) called damped natural frequency (b) Initial value problem: Write A. B, and u(0) in terms of the initinl values v(o) and g(O), form. Express C and φ in terms of A and B when y(t) is expressed in am,plitude-phase Result: B y(0) = uld atan2(-A, B) y(t) C cs(ua t + 4) ε..く.nt φ = 卫0) cos( at)}e-c n' (c) with U(0)=0, 1-sin(ua t) + splotted below versus un t. Label each curve with ζ 0.1 or ζ=0.2. 0 for 0 S wnt s 20 v(0) The period of a vibration Tperiod is de fined in Section 6.1 and for an un/underdamped constant-coefcient 2nd order ode (0 sS 1), is 04 0.2 Tperiod Mark Tperiod On your sketch by first marking the 2 local maxima (or minima). In other words, mark 4 the points on the curve where it) and ý(t) <0 and label the time interval between two successive local maxima (or minima) as Tperiod- 15 (d) In view of your sketch, complete the following statements about slightly more damping: Increasing ζ frorn 0.1 to 0.2 causes y(t)-0 (for large t) slower/faster. (e) Knowing wn = 1 , find wd and Teri d for the values of S in the following table. 2.0 0.0 imaginary imaginary period (sec)6.28 (f) In view of your table, complete the following statements: Increasing the darnping ratio from ζ-0 to 0.5 corresponds to a smaller/slightly smaller/no change/slightly larger/larger damped natural frequency and a smaller/slightly smaller/no change/slightly larger/larger period of vibration.

Answer 1

4 od pond 6,29 63 6 6 7,2 6 O:94 6-6 2 6 sincre aina danpin CorresPondo tSinSmalles dampe 0 he perio oscillahi on