\\\\ m, 0.5m 1m The figure shows a slender rod with a mass of 6 kg. Attached to the rod is a spring of stiffness k = 250 N /m .Amoveable point mass m1= 1 kg is position at a distance I from the pivot point. You can assume small angle oscillations from the equilibrium position shown in the figure.

k 0.5m 1m The figure shows a slender rod with a mass of 6 kg. Attached to the rod is a spring of stiffness k = 250 N/m.Amoveable point mass m1 = 1 kg is position at a distance X from the pivot point. You can assume small angle oscillations from the equilibrium position shown in the figure. Calculate the required position, X, of the moveable point mass m1 needed to give the system a natural frequency of 5.3 rad/s. Give your answer to two decimal places.

The figure shows a bar of mass M-4 Kg and length L-1 m, has a concentrated mass at the end of the bar of m= 4 kg. The bar is supported by a pin at O and three springs as shown. In the value k- 800 N/m Find the natural frequency of the system. 2k O 17.0 rad/s O 14.4 rad/s O 10.9 rad/s 20.3 rad/s elle 00000 + 4 1 2 k > bar of mass M mass m

A pendulum is made up of a spring with a mass attached to the end. The spring is arranged to lie in a straight line, by wrapping the spring around a massless bar. The equilibrium length of the bar is “l”. Using lagrange’s equations, find the equations of motion.

Problem 2. For the single-degree-of-freedom mechanical system shown in the figure, the bar has mass m and length l (so that Io spring is unstretched when 0 = 0: ml² 피). If the %3D k a) find the linearized governing equations of motion for small 0; (т,1) b) find the natural frequency, damping ration, and damped natural frequency of oscillation for the free response; Neglect gravity.

A mechanical system is depicted in the figure below. A pinion with fixed rotation axis passing through C2 meshes with a translatory rack of mass m = 0.8 kg. The rack, which is supported on two identical guides (bearings), each of damping coefficient c = 12 N s/m, meshes with another solid pinion that, as a result, rolls without slippage on another fixed rack. Derive the differential equation describing the movement of the system. Determine transfer function G(s) = X(s)/F(s) where X(s) is the Laplace transform of x(t) and F(s) is the Laplace transform of f(t). Find x(t) if the input is f(t) = 80(t) = step function. The system parameters are: J1 = 0.001 kg m² and J2 = 0.0003 kg m², m. = 0.4 kg, the radii R1 = 0.05 m, R2 = 0.07 m, R = 0.06 m, the stiffness k = 160 N/m, %3D a. b. C. Pinion J2 R Rack Guide RI Pinion R2 Fixed rack

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A homogeneous rod AB, of mass m and length 2a, hangs in a thread from a fixed point O. The thread is unstretchable and massless and of length da. See figure. The distance AC is 2a/3. The rod performs small oscillations in a vertical plane containing O. Determine lambda such that AB perpetually forms an angle with the vertical twice as big as OC does, if the system is started suitably.Treat the rod as one-dimensional, and assume that it never collides with the thread. 2a В

An undamped spring/mass system, in which m = 8 slugs and the spring constant k = 16 lb/ft has a driving force f(t) = t^pi*t on [0,pi] Extend f(t) into the negative t-axis to obtain an odd function. Find the particular solution ??(?). (Note: f(t) is given as half of the period, p = π.)

Problem (1): Derive the equation of motion of the system shown in Figure and find: • The natural frequency of vibration • The damping ratio • The free damped frequency of vibration If m= 10 kg, c = 50 N-s/m, k = 1000 N/m, L= 1 m 44 3k Uniform rigid bar, mass m 2 2