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Solution HW 4

# Solution HW 4 - Advanced Vibration Solution Assignment 4...

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Advanced Vibration: Solution Assignment # 4 Due Date: 11/18/2010 Problem 1: Consider the axially vibrating bar connected with the discrete spring as shown in the figure below. Assume that the bar is made of Aluminum with the following parameters: A =0.001 m 2 , L =1 m , 3 , E = 7 10 10 Pa and a spring stiffness of 1 10 6 N/m. Figure 1. (a) Model the bar with two elements and calculate its first two natural frequencies. Compare it with the natural frequency obtained analytically in class notes. (20 Points) Solution: The finite element model for the two-element bar is ( ) ( ) t t Mu Ku 0 where 1 2 3 ( ) ( ) ( ) ( ) T t u t u t u t u such that the elemental mass and stiffness matrices are: 1 2 1 2 2 1 2 1 , 1 2 1 2 6 6 1 1 1 1 , 1 1 1 1 Ah Ah M M EA EA K K h h The elemental stiffness matrix associated with the discrete spring element is: 0 0 0 spring K k as the spring force only apply force to 3 rd node of freedom 3 ku The global system of equation can be written as follows, x , , E A L k h k h 1 2 3 1 u 2 u 3 u

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Advanced Vibration: Solution Assignment # 4 1 1 2 2 3 3 2 1 0 1 1 0 0 1 2 2 1 + 1 1 1 1 0 6 0 1 2 0 0 1 1 u u Ah EA u u h u u h k EA   By applying the boundary conditions 1 0 u we obtain the following system of equation, 2 2 3 3 2 1 4 1 0 + 1 1 1 2 0 6 u u Ah EA h u k u h EA . The eigenvalue problem associated with the above system can be computed and natural frequencies can thus be obtained. (b) Now develop the finite element code for any arbitrary n number of equal size elements and by choosing n =20, compute the natural frequencies and compare them with the associated analytical solutions. (30 Points) Similar formulation can be obtained for any number of finite elements. As shown in the class the natural frequencies can be obtained analytically from the roots of the following frequency equation: tan 0 L EA k E   i i (Analytically, rad/sec) i (FEM n =2, rad/sec) i (FEM n =20, rad/sec) 1 8044.14 8254.69 8046.24 2 24009.74 28693.01 24065.38 3 39999.77 ----- 40257.38 4 55993.34 ----- 56701.09 5 71988.07 ----- 73494.80
Advanced Vibration: Solution Assignment # 4 Problem 2: Consider a transversely vibrating beam connected with a discrete mass spring system as shown in the figure below. Assume that the bar is made of material with the following parameters: A =0.01 m 2 , L =1 m , 7800kg/m 3 , E = 2 10 11 Pa and I = 10 -6 m 4 .

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Solution HW 4 - Advanced Vibration Solution Assignment 4...

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