Static Analysis: Direct Integration

1. Static Analysis: Direct Integration

2. Objectives Section II – Static Analysis Module 7 – Direct Integration Page 2 This module will present the equations and numerical methods used to solve the equations of motion directly. Although more computationally intensive, this method can be used to solve problems that are not characterized by constant mode shapes. • In Module 6, the Modal Superposition method of solving the equations of motion was presented. This method required the determination of the mode shapes and natural frequencies of the system and then used them to transform the coupled equations into uncoupled modal equations of motion. • Problems having gaps, surface contact, and non-linearities can be solved using the method presented in this module.

3. Governing Equations Section II – Static Analysis Module 7 – Direct Integration Page 3 • The governing equations developed for static problems in Module 4 are • Inertial forces and viscous damping forces can be introduced as external force terms, resulting in • Note that the displacement increment {Du} in going from time, t, to time, t+Dt, and the acceleration and velocity at t+Dt are unknowns.

4. Equations of Motion Section II – Static Analysis Module 7 – Direct Integration Page 4 • The previous equation can be rewritten as • One of the most commonly used numerical methods for solving this set of equations is the Newmark-b method. • The Newmark-b method assumes a linear variation of acceleration during the time interval, Dt, and uses two interpolation parameters to select the acceleration used in the solution.

5. First Acceleration Approximation Section II – Static Analysis Module 7 – Direct Integration Page 5 • The acceleration during the time interval, t+Dt, can be estimated using the equation • The parameter, g, is used to select the acceleration used in the numerical integration procedure. • The selected value of the parameter, g, affects the accuracy and stability of the resulting numerical integration scheme. • The Newmark-b method is stable, provided .

6. Graphical Illustration Section II – Static Analysis Module 7 – Direct Integration Page 6 • If g is equal to zero, then the acceleration at time, t, is used. • If g is equal to one, then the acceleration at time, t+Dt, is used. • If g is equal to ½, then the acceleration at the middle of the time interval is used. t

7. Kinematic Relationships Section II – Static Analysis Module 7 – Direct Integration Page 7 • The kinematic equations for acceleration are • If a is a constant, this equation can be integrated to yield where and are initial conditions.

8. Second Acceleration Approximation Section II – Static Analysis Module 7 – Direct Integration Page 8 • Newmark based the second acceleration approximation on this kinematic relationship, via the following equation: where • is an interpolation parameter that, like g, is used to select the acceleration used in the numerical integration procedure. • The Newmark- b method uses two parameters for accelerations used in the procedure and

9. Governing Approximation Equations Section II – Static Analysis Module 7 – Direct Integration Page 9 • The Newmark-b method is based on the two equations • The second of these equations can be rearranged to yield and

10. Governing Approximation Equations Section II – Static Analysis Module 7 – Direct Integration Page 10 • Substituting the last equation on the previous slide into the top equation on the previous slide yields • These last two equations provide equations for and in terms of the displacement increment and the velocity and accelerations at the beginning of the time interval. • The velocity and acceleration at the beginning of the time interval are known. • The only unknown is the displacement increment, .

11. Combination of Equations Section II – Static Analysis Module 7 – Direct Integration Page 11 • The three equations used to determine the displacement increment using the Newmark-b method are: • Equations of Motion • Acceleration at the end of the time step • Velocity at end of the time step

12. Combined Equations Section II – Static Analysis Module 7 – Direct Integration Page 12 • These three equations can be combined to yield the following equation • The right hand side of the equation yields an effective load vector based on quantities at time, t, that are known. • The left hand side of the equation is an effective tangent stiffness matrix that includes mass and viscous damping terms.

13. Equivalent Static Problem Section II – Static Analysis Module 7 – Direct Integration Page 13 • The equation on the previous slide can be written as • These show that finding the displacement increment in a dynamic analysis is equivalent to solving a static problem using an effective tangent stiffness matrix and internal restoring force vector. where

14. Stability and Accuracy Section II – Static Analysis Module 7 – Direct Integration Page 14 • The Newmark-b method is unconditionally stable for linear problems when g and b satisfy the equations • Values of g=1/2 and b=1/4 are frequently used. • The method is generally stable for nonlinear problems if these same criteria for g and b are used and equilibrium iterations are used to improve accuracy. and

15. Time Step Size Section II – Static Analysis Module 7 – Direct Integration Page 15 • A sufficiently small time step must be used to ensure solution accuracy. • A Dt of around one-tenth of the period of the highest natural frequency of interest is commonly used. • The time step does not have to be constant for all time steps and it is common for variable time step methods to be used. • Autodesk Simulation 2012 uses a variable time step in the Mechanical Event Simulation module.

16. Rayleigh Damping Section II – Static Analysis Module 7 – Direct Integration Page 16 • Rayleigh damping is a mathematically convenient way of describing viscous damping. • Rayleigh damping is defined by the equation • The constants a and b must be determined from experimental data. • This is a convenient form because the damping matrix can be uncoupled along with the mass and stiffness matrices using the mode shapes.

17. Rayleigh Damping Section II – Static Analysis Module 7 – Direct Integration Page 17 • The transformation of the Rayleigh damping equation to the mode shape domain takes the form • The i th equation can be written as where zi is the critical damping ratio for the i th mode.

18. Finding a and b Section II – Static Analysis Module 7 – Direct Integration Page 18 • a and b can be found from this equation if z is known for two modes. • A least squares approximation to a and b can be found if z is known for more than two modes.

19. 1 inch wide x 12 inch long x 1/8 inch thick. Material - 6061-T6 aluminum. Example Problem Section II – Static Analysis Module 7 – Direct Integration Page 19 Simulation is used to compute the step response of the cantilevered beam shown in the figure. This is the same beam used in Module 6: Modal Superposition. Fixed End Brick elements with mid-side nodes are used to improve the bending accuracy through the thin section. 0.0625 inch element size. 5 lb. force distributed over the 17 nodes on the upper edge of the free end

20. Example – Analysis Parameters Section II – Static Analysis Module 7 – Direct Integration Page 20 Values for the Rayleigh damping factors are presented on a following slide. Same as in Module 6 Forces can be applied here or through the FE Editor. The FE Editor was used in this example.

21. Example – Load Curve Factor Section II – Static Analysis Module 7 – Direct Integration Page 21 The load curve is zero until 0.05 seconds. At that time, it goes to one in 0.0001 seconds to simulate a step input.

22. Example – Force Magnitude Section II – Static Analysis Module 7 – Direct Integration Page 22 5 lb./17 nodes acting in negative y-direction Nodes selected along upper edge Load Curve 1 is defined in Analysis Parameters

23. Example – Load Summary Section II – Static Analysis Module 7 – Direct Integration Page 23 F(t) = Load Curve Factor * Magnitude Load Curve Factor 1 0.05 seconds Time F(t) -0.294 lb. 0.05 seconds Time

24. Example – Rayleigh Damping Factors Section II – Static Analysis Module 7 – Direct Integration Page 24 • Damping for each mode is estimated to be 0.5 percent of critical. • Modes associated with bending about the weak axis will be used to determine a and b. • The first three weak axis bending modes were computed in Module 5. They are: • Mode 1  28 Hz = 176 rad/sec, • Mode 2  175 Hz = 1100 rad/sec, • Mode 4  492 Hz = 3091 rad/sec.

25. Example – MATLAB Program Section II – Static Analysis Module 7 – Direct Integration Page 25 • This MATLAB program finds the Rayleigh damping coefficients for this example. The critical damping ratio for each mode is 0.005 or 0.5%.

26. Example – Clamped End Stress Section II – Static Analysis Module 7 – Direct Integration Page 26 is plotted Note that this curve is the same as that computed using modal superposition in Module 6.

27. Example – Free End Tip Displacement Section II – Static Analysis Module 7 – Direct Integration Page 27 This curve is much smoother than the stress curve. The stress curve is based on strains that are computed from the derivatives of the displacements.

28. Module Summary Section II – Static Analysis Module 7 – Direct Integration Page 28 • This module has presented the equations used to perform a direct integration of the equations of motion used for a linear or non-linear dynamic analysis. • It was shown that the Newmark-b method for integrating the equations of motion reduces the dynamic problem to a sequence of static analyses that uses an effective tangent stiffness matrix and internal restoring force vector. • The Newmark-b method is unconditionally stable for linear problems and generally stable for non-linear problems that use equilibrium iterations. • A sufficiently small time step must be used to ensure accurate results. • Results from an example were the same as those obtained using the modal superposition method in Module 6.