For a dynamical system governed by the equation,
$\ddot{x}(t)+2 ζ \omega_n \dot{x}(t)+\omega_n^2x(t)=0$
the damping ratio ζ is equal to $\frac{1}{2\pi}\log_e2$. The displacement x of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be 4 mm. Neglecting higher powers (> 1) of the damping ratio, the displacement at the next peak in the positive direction will be _______ mm (in integer).
A spring mass damper system (mass m, stiffness k, and damping coefficient c) excited by a force F(t) = B sin ωt, where B, ω and t are the amplitude, frequency and time, respectively, is shown in the figure. Four different responses of the system (marked as (i) to (iv)) are shown just to the right of the system figure. In the figures of the responses, A is the amplitude of response shown in red color and the dashed lines indicate its envelope. The responses represent only the qualitative trend and those are not drawn to any specific scale.
Four different parameter and forcing conditions are mentioned below.
(P) c > 0 and $ω=\sqrt{k/m}$
(Q) c < 0 and ω ≠ 0
(R) c = 0 and $\omega=\sqrt{k/m}$
(S) c = 0 and $\omega \cong\sqrt{k/m}$
Which one of the following options gives correct match (indicated by arrow →) of the parameter and forcing conditions to the responses?
In the configuration of the planar four-bar mechanism at a certain instant as shown in the figure, the angular velocity of the 2 cm long link is ω2 = 5 rad/s. Given the dimensions as shown, the magnitude of the angular velocity ω4 of the 4 cm long link is given by rad/s (round off to 2 decimal places).
