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Chapter: Mechanical : Finite Element Analysis : Dynamic Analysis Using Element Method

Transient Response Analysis

(Dynamic Response/Time-History Analysis) Structure response to arbitrary, time-dependent loading.



(Dynamic Response/Time-History Analysis)


Structure response to arbitrary, time-dependent loading.


Compute responses by integrating through time:


B. Modal Method


First, do the transformation of the dynamic equations using the modal matrix before the time marching:


Then, solve the uncoupled equations using an integration method. Can use, e.g., 10%, of the total modes (m= n/10).


Uncoupled system, Fewer equations,


No inverse of matrices,

More efficient for large problems.



1Cautions in Dynamic Analysis


Symmetry: It should not be used in the dynamic analysis (normal modes, etc.) because symmetric structures can have antisymmetric modes.


Mechanism, rigid body motion means = 0. Can use this to check FEA models to see if they are properly connected and/or supported.


Input for FEA: loading F(t) or F( ) can be very complicated in real applications and often needs to be filtered first before used as input for FEA.



Impact, drop test, etc.




In the spring structure shown k1 = 10 lb./in., k2 = 15 lb./in., k3 = 20 lb./in., P= 5 lb. Determine the deflection at nodes 2 and 3.





Again apply the three steps outlined previously.


Step 1: Find the Element Stiffness Equations


Element 1:



Step 2: Find the Global stiffness matrix


Now the global structural equation can be written as above.



Step 3: Solve for Deflections


The known boundary conditions are: u1 = u4 = 0, F3 = P = 3lb. Thus, rows and columns 1 and 4 will drop out, resulting in t following matrix equation,


Solving, we get     u2 = 0.0692 & u3 = 0.1154    




In the spring structure shown, k1 = 10 N/mm, k2 = 15 N/mm, k3 = 20 N/mm, k4 = 25 N/mm, k5 = 30 N/mm, k6 = 35 N/mm. F2 = 100 N. Find the deflections in all springs.




Here again, we follow the three-step approach described earlier, without specifically mentioning at each step.


Now, apply the boundary conditions, u1 = u4 = 0, F2 = 100 N. This is carried out by deleting the rows 1 and 4, columns 1 and 4, and replacing F2 by 100N. The final matrix equation is,




Spring 1:              u4 – u1       = 0                       

Spring 2:              u2 – u1       = 1.54590            

Spring 3:              u3 – u2       = -0.6763            

Spring 4:              u3      – u2   = -0.6763            

Spring 5:              u4      – u2   = -1.5459            

Spring 6:              u4      – u3   = -0.8696            



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Mechanical : Finite Element Analysis : Dynamic Analysis Using Element Method : Transient Response Analysis |

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