© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Static Analysis: Modal Superposition

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Objectives This module will present the equations and theory used to perform a Modal Superposition Analysis of a linear system.  The basis of Mode shapes will be discussed.  A system of single degree of freedom equations will be presented.  The individual modes and total response will be examined. Section II – Static Analysis Module 6 – Modal Superposition Page 2

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Equations of Motion  The equations of motion for a linear system developed in Module 5: Natural Frequency Analysis are  The mode shapes form a basis for the solution space of the undamped system and the displacements can be written as a linear combination of the mode shapes  is a matrix that has a mode shape in each column. Section II – Static Analysis Module 6 – Modal Superposition Page 3

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Time Derivative Equations  The velocities and accelerations written in terms of the mode shapes and coefficients are  These equations can be used to transform the equations of motion from a coupled set of equations to an uncoupled set. Note that the mode shapes are not a function of time and only the coefficients, x, have time derivatives. Section II – Static Analysis Module 6 – Modal Superposition Page 4

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Transformation Process  Substituting the time derivative equations into the equations of motion for an undamped system yields the equation Section II – Static Analysis Module 6 – Modal Superposition Page 5

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Mass Orthonormal Mode Shapes  If the mode shapes are orthonormal to the mass matrix, then the following relationships hold (reference Module 5: Natural Frequency Analysis). is the identity matrix Square matrix with natural frequencies on the diagonal elements and zero for all non- diagonal elements Section II – Static Analysis Module 6 – Modal Superposition Page 6

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Uncoupled Equations  Combining the equations on the previous two slides yields the equation  Each row of the this equation contains an equation of the form where is equal to the i th row of times. Section II – Static Analysis Module 6 – Modal Superposition Page 7

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Modal Participation Factors  The i th row of the forcing function term can be written as where each term is called a modal participation factor.  The subscripts are Section II – Static Analysis Module 6 – Modal Superposition Page 8 i th equation j th mode shape

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Physical Significance  If a force is applied at the end of a cantilevered beam, it has a large affect in exciting the first mode because the mode shape has a large value at this point.  If a force is applied at the point of zero displacement in the second mode, the force does not excite the second mode because the mode shape value at this point is zero. The physical significance of the modal participation factors is best illustrated using an example. Section II – Static Analysis Module 6 – Modal Superposition Page 9

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Integration  Each of the equations can be integrated separately to determine the variation of the coefficient as a function of time.  Once all of the equations have been integrated, the response of the total system can be obtained from the linear superposition equation Section II – Static Analysis Module 6 – Modal Superposition Page 10

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Practical Systems  The development up to this point has used all n-degrees of freedom of the system.  It is generally only necessary to include the response of a small subset of the total number of modes to accurately compute the response of the system.  The frequency content of the applied load and the points on the structure where loading is applied, play a large role in determining how many modes should be considered in determining the response.  It is not necessary to include mode shapes having natural frequencies that are significantly higher than the highest frequency in the excitation forces. Section II – Static Analysis Module 6 – Modal Superposition Page 11

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Damping  Damping was omitted in the preceding development but can be included.  Damping can be attributed to a variety of material properties (viscoelastic materials have good damping properties) or displacement dependent mechanisms (slipping of bolted joints).  Damping is difficult to quantify analytically and must be determined experimentally.  Damping is generally different for each mode and can be nonlinear (i.e. the more a structure deforms, the higher the damping). Section II – Static Analysis Module 6 – Modal Superposition Page 12

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example Problem Section II – Static Analysis Module 6 – Modal Superposition Page 13 5 lb. force distributed over the 17 nodes on the upper edge of the free end Fixed End 1 inch wide x 12 inch long x 1/8 inch thick. Material T6 aluminum. Brick elements with mid-side nodes are used to improve the bending accuracy through the thin section inch element size. Simulation is used to compute the step response of the cantilevered beam using the first five natural frequencies and mode shapes computed in Module 5.

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Section II – Static Analysis Module 6 – Modal Superposition Page 14 List of nodes on upper edge of free end Forces become active at 0.05 seconds Discussed on next slide  is equal to 0.5% for all modes Load curves are scaled by a factor of 1 in the y-direction Example Analysis Parameters

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example – Time Step  The time step is based on the natural frequencies in the system and the frequency content of the input.  The input force causes bending about the weak axis and will only excite modes that have bending about the weak axis.  The natural frequency of the third weak axis bending mode was computed in Module 5 to be 592 Hz.  The period of this frequency is seconds.  8-10 time steps per smallest period is generally used as a time step. Section II – Static Analysis Module 6 – Modal Superposition Page 15 Mode 4, 592 Hz, 3 rd bending mode about the weak axis

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example – Restart Data  A modal superposition analysis uses natural frequency and mode shape data from a Natural Frequency Analysis.  The Design Scenario containing this information must be specified.  In this problem, the natural frequency analysis was performed in Design Scenario 2 that has been renamed to Nat Freq and Mode Shapes. Section II – Static Analysis Module 6 – Modal Superposition Page 16

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example - Load Curves 5 lb. acting in the negative y- direction is divided among 17 nodes When activated after 0.05 seconds, the load will be constant Section II – Static Analysis Module 6 – Modal Superposition Page 17

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example – Fixed End Stress History Section II – Static Analysis Module 6 – Modal Superposition Page 18  The higher frequency response can be seen superimposed on the dominant first mode response. is plotted

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example – Tip Displacement Section II – Static Analysis Module 6 – Modal Superposition Page 19 Start of Step Input sec 27.8 Hz The response is dominated by the 1 st bending mode. This curve looks much smoother than the stress history curve. Why? The stresses are based on strains that depend on the derivatives of the displacements.

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Module Summary  The equations and theory used to perform a Modal Superposition Analysis of a linear system are presented.  Mode shapes form a basis and all solutions can be written in terms of them.  This results in a system of single degree of freedom equations that can be integrated separately.  The results from the individual modes are then combined to determine the total response.  Often, only a small subset of the total number of modes need to be used to obtain accurate results. Section II – Static Analysis Module 6 – Modal Superposition Page 20