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Dynamics Course

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- 1. GROUND EXCITED SYSTEMS Prof. A. Meher Prasad Department of Civil Engineering Indian Institute of Technology Madras email: prasadam@iitm.ac.in
- 2. Dynamic Equations of Motion Force excited system Ground excited system where is the relative displacement of the structure w.r.t ground. Non-moving reference Ground Acceleration vector where, are the ground accelerations in x,y,z directions respectively. are null vectors except that those elements are equal to 1, which corresponds to x,y,z translational DOF.
- 3. Let System equations reduce to following uncoupled equations where participation factors, Note: a j = b j = 0 since initial conditions are zero i.e Modal Superposition applied to GES
- 4. Solution to uncoupled equation of motion can be expressed as, In general , for design the response quantities of interest are: R = maximum values of (u , f s , Δ, V, M) Equivalent lateral loads Storey shears Storey Moments Storey drifts Relative displacements
- 5. <ul><li>uses mode shapes to reduce size, uncouple the equations of motion. </li></ul><ul><li>Summation of individual modal response in frequency domain. </li></ul>max deformation of spring Modal Frequency Response Analysis Damping m k
- 6. Ground Excited MDOF System = relative displacement of the structure w r t ground = Ground acceleration vector : where, are ground accelerations in x, y & z directions respectively Reference base x y z
- 7. 1) SRSS 2) CQC 3) Double Sum 4) Grouping Serious errors for closely spaced frequencies and for 3-D structures ,which include torsional contribution. SRSS : Square Root of Sum of Squares .It gives most probable maximum response. Modal combination rules ** Since the maximum response in each mode would not necessarily occur at the same instant of time, over conservative to add separate modal maximum responses.
- 8. CQC : Complete Quadratic Combination Rule (Wilson, Der Kiureghion & Baya 1981). It is based on random vibration theory. Note: All cross modal terms included very good agreement with full modal superposition extra computation minimal.
- 9. <ul><li>Structures subjected to time varying forces or enforced motions </li></ul><ul><li>Discretization in Space- Time </li></ul>Finite Element Method In Structural Dynamics <ul><li>Methods of solution </li></ul><ul><li>Time domain </li></ul><ul><li>Frequency domain </li></ul>
- 10. <ul><li>Explicit methods </li></ul><ul><li>Equation of motion at t n conditionally stable </li></ul><ul><li>Implicit methods </li></ul><ul><li>Equation of motion at t n+1 can be unconditionally stable. </li></ul><ul><li>Direct integration method </li></ul><ul><li>Response at discrete interval of time (usually equally spaced). Process of marching along time evaluated from values at previous time stations. </li></ul>Time Domain Methods
- 11. <ul><li>Unconditional stability when applied to linear problems </li></ul><ul><li>No more than one set of implicit equations </li></ul><ul><li>Second-order accuracy </li></ul><ul><li>Self-starting </li></ul><ul><li>Controllable algorithmic dissipation in the higher modes. </li></ul>Desirable attributes:
- 12. <ul><li>Modal Superposition Method </li></ul><ul><li>Transformation of co-ordinates results in a set of uncoupled SDOF equations in terms of modal co-ordinates. </li></ul><ul><li>Solve SDOF equations </li></ul><ul><li>Useful for many problems where the response can be approximated very well by using few eigen modes. </li></ul>Time Domain Methods
- 13. <ul><li>Suitable for linear problem subjected to sinusoidal or oscillatory forces - </li></ul>Frequency Domain Methods <ul><ul><li>Response {x}e i t is a complex number having magnitude and phase w.r.t the applied force. </li></ul></ul><ul><ul><li>Structural excitation computed at discrete excitation frequencies. </li></ul></ul><ul><ul><li>Solve coupled matrix equation using Complex Algebra. </li></ul></ul>Direct Frequency Response Analysis
- 14. Multiple support Excitation Super structure free Dof Support Dof
- 15. Decompose {u f } into pseudo static and dynamic parts {u f }= {u s } + {u d } Considering only static response ( i.e. stiffness matrix alone) Influence matrix Describes influence of support displacement on structural displacement j th column of [ i ]=structural displacements due to unit support displacement u rl only ( l th base displacement)
- 16. (By definition ) and i.e.
- 17. If assume light damping Uncoupled equations of motion are,
- 18. A big mass (much bigger than the total mass of the structure ( ~10 6 total mass ) is added to each degree of freedom at moving bases. As more big masses are applied, more low frequency modes have to be extracted.
- 19. The desired base motion is obtained by applying a point force to each degree of freedom at moving bases by Where M big =big mass and is the applied acceleration prescribed for degree of freedom N associated with moving supports The combined equation of motion is with Where is the diagonal matrix containing the big masses for moving base ‘i’ and is the base motion applied to this base
- 20. The mass matrix [M] now contains the mass of the structure as well as the big masses associated with the secondary base. The modal equations with
- 21. 1.000 1.000 6.7662 52.2836 0.0 4.7876 5.2909 10 8 0.9999 1.000 6.7661 52.2836 0.0 4.7876 5.2909 10 6 0.9995 1.0003 6.7641 52.2823 10 -10 4.7871 5.2910 10 4 0.9524 1.0335 6.5531 52.0552 10 -9 4.8011 5.3025 10 2 Response peaks (m/s 2 ) X 1 max X 2 max X 3 max X 4 max Natural frequency Ratio of large mass to structure

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