Question

Solve a,b and c

The vibratory movement of the engineering system shown in Figure 3 can be described by two generalised coordinates, x, a Cartesian coordinate, and 6, a polar coordinate systems. The mass m and its mass moment of inertia about an axis that goes through its centre of gravity G is J. When the system is slightly pushed down from the top comer at the right hand edge of mass m, the induced vibrational motion is found to sway the system in the direction of x and θ in a concurrent manner. It is observed that the generalized coordinate, x restricts the motion of the system to a vertical upward/downward displacement at the mass cente of gravity G, and θ, restricts the clockwise/counter clockwise rotational motion about an axis that goes through point G (a) Derive the equations of motion of this system using Lagrange's formalisms, and [14 marks (b) If this system satisfies the condition Ka- K2b, determine its characteristics present these equations in matix fom. equation for m 2000kg, J5x106kg.m2, K1000 N/m, K2 500 N/m, a= 10 m, and b=20m. What type ofcoordinates x and θ are? [8 marks] (c Compute the natural frequencies of this system. [8 marks] m.J Ki K2 Figure 3

Answer 1