# TAMU MEEN 617 - HD 8 prop damped modal analysis 2008 (23 pages)

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## HD 8 prop damped modal analysis 2008

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Lecture Notes

- Pages:
- 23
- School:
- Texas A&M University
- Course:
- Meen 617 - Mechanical Vibration

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Handout 8 Modal Analysis of MDOF Systems with Proportional Damping The governing equations of motion for a n DOF linear mechanical system with viscous damping are D U K U F MU t t 1 and U are the vectors of generalized displacement where U U velocity and acceleration respectively and F t is the vector of generalized external forces acting on the system M D K represent the matrices of inertia viscous damping and stiffness coefficients respectively1 The solution of Eq 1 is uniquely determined once initial conditions are specified That is U at t 0 U 0 U o U 0 o 2 Consider the case in which the damping matrix D is of the form D M K 3 where are constants2 usually empirical This type of damping is known as PROPORTIONAL i e proportional to either the mass M of the system or the stiffness K of the system or both The matrices are square with n rows n columns while the vectors are nrows 2 These constants have physical units is given in 1 sec and in sec 1 MEEN 617 HD 8 Modal Analysis with Proportional Damping L San Andr s 2008 1 Proportional damping is rather unique since only one or two parameters appear to fully describe the complexity of damping irrespective of the system number of DOFs n This is clearly not realistic Hence proportional damping is not a rule but rather the exception Nonetheless the approximation of proportional damping is useful since most times damping is quite an elusive phenomenon i e difficult to model predict and hard to measure but for a few DOFs Next consider one already has found the natural frequencies and natural modes eigenvectors for the UNDAMPED case i e K U 0 given M U i i 1 2 n satisfying i M i2 K i 0 i 1 n with properties T M M T K K 4 5 As in the undamped modal analysis consider the modal U t q t transformation 6 q U q t then EOM 1 becomes And with U t t t D q K q F t M q 7 which offers no advantage in the analysis However premultiply T the equation above by to obtain M q D q K q F T T T T t 8 And using the modal properties Eq 5 and MEEN

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