L. 2 uestion 3 (20 marks) A rotating bar of length L and mass m stiffness k and a damper with dam...
Question 3 (24 marks) A uniform bar of length L= 1.1m and mass m = 4.2kg is connected with a spring with stiffness k = 2000N/m and a damper with damping ratio š = 0.3. The bar is rotating about a point that is 10cm from the left end. (1) Calculate the total kinetic energy and total potential energy. (2) Derive the equation of motion using energy based approach. (3) Determine the undamped natural frequency and damped natural frequency of...
2 with spring stiffness k 1000 N/m, Consider a mass-spring-damper system shown in Figure mass m = 10 kg, and damping constant c-150 N-s/m. If the initial displacement is xo-o and the initial velocity is 10 m/s (1) Find the damping ratio. (2) Is the system underdamped or overdamped? Why? (3) Calculate the damped natural frequency (4) Determine the free vibration response of the system.
1. A rod of length 3a is hinged at one end and supported by a spring of stiffness k at the other end. A mass m is attached 1/3 of the length from the hinge and a dashpot having a hinge. Ignore the mass of the rod, spring and damper (a) Derive the equation of motion for the system (101 1. A rod of length 3a is hinged at one end and supported by a spring of stiffness k at...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
Figure 1 shows a system comprising a bar with mass m=12 kg and the length of the bar L=2 m, two springs with stiffness k_t=1000 N-m/rad and k=2000 N/m, one damper with damping coefficient c=50 N-s/m and two additive masses at the end of the bar, where each mass (M) is equal to 50 kg. The rotation about the hinge A, measured with respect to the static equilibrium position of the system is θ(t). The system is excited by force...
Q1. For the system shown in Figure 1 where the beam with mass m and length L is connected to the fixed surfaces through three springs with same stiffness k, (i) Calculate the total kinetic energy and total potential energy of the system; (ii) Derive the equation of motion in terms of rotation angle 0; (iii) Find the natural frequency of the system; (iv) Calculate the natural period if the stiffness k of all springs is doubled; (v) If the...
The figure below shows a uniform slender bar supported by cantilevers at A and C. At B a linear spring with stiffness K' is connected to an additional point mass 'm'. Note the physical properties of the bar include cross sectional area A, Young's modulus E, second moment of area I, and, density ρ, and length AB-BC-L. 1. 2. Develop the matrix equation of motion for the FEM system in the model How many natural frequencies are in the system?...
F Fosin t m k 2 Figure Qla: System is subjected to a periodic force excitation (a) Derive the equation of motion of the system (state the concepts you use) (b) Write the characteristic equation of the system [4 marks 12 marks (c) What is the category of differential equations does the characteristic equation [2 marks fall into? (d) Prove that the steady state amplitude of vibration of the system is Xk Fo 25 + 0 marks (e) Prove that...
A 3 m rigid bar AB is supported with a vertical translational spring at A and a pin at B The bar is subjected to a linearly varying distributed load with maximum intensity g Calculate the vertical deformation of the spring if the spring constant is 700 kN/m. (ans: 21.43 mm) 2. A steel cable with a nominal diameter of 25 mm is used in a construction yard to lift a bridge section weighing 38 kN. The cable has an...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...