S1-1 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SECTION 1 REVIEW OF FUNDAMENTALS

S1-2 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation

S1-3 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n This section will introduce the basics of Dynamic Analysis by considering a Single Degree of Freedom (SDOF) problem n Initially a free vibration model is used to describe the natural frequency n Damping is then introduced and the concept of critical damping and the undamped solution is shown n Finally a Forcing function is applied and the response of the SDOF is explored in terms of time dependency and frequency dependency and compared to the terms found in the equations of motion SINGLE DOF SYSTEM

S1-4 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n Dynamic equation of motion for single degree-of-freedom linear system m= mass (inertia) b = damping (energy dissipation) k = stiffness (restoring force) P = applied force = displacement of mass = velocity of mass = acceleration of mass u, u, u, and p are time varying in general. m, b, and k are constants. SINGLE DOF SYSTEM (Cont.)...

S1-5 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM UNDAMPED FREE VIBRATIONS n Dynamic equation with no forcing n Solution to homogeneous equation n Initial conditions n Solution with initial conditions

S1-6 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n The response of the spring will be harmonic, but the actual form of the response through time will be affected by the initial conditions n If there is no response n If response is a sine function with magnitude n If response is a cosine function (180 0 phase change), with magnitude n If response is phase and magnitude dependent on the initial values SINGLE DOF SYSTEM UNDAMPED FREE VIBRATIONS (Cont.)

S1-7 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n The graph is from a transient analysis of a spring mass system with initial velocity conditions only Time Displacement k = 100 m = 1 TnTn Amp SINGLE DOF SYSTEM UNDAMPED FREE VIBRATIONS (Cont.)

S1-8 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FREE VIBRATIONS n Dynamic equation with no forcing n Critical damping n Fraction of critical damping n The amount of damping determines the form of the solution u Solution to homogeneous equation with underdamping l Damped natural frequency,

S1-9 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FREE VIBRATIONS (Cont.) u Solution to homogeneous equation with critical damping; no oscillation. u Overdamping No oscillation occurs. The system gradually returns to its equilibrium position (undisplaced position). The usual analysis case is underdamped. Structures have viscous damping in the 0%-10% range.

S1-10 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FREE VIBRATIONS (Cont.) n A swing door with a dashpot closing mechanism is a good example of the three types of damping. u If the door oscillates through the closed position it is underdamped u If it creeps slowly to the closed position it is overdamped u If it closes in the minimum possible time, with no overswing, it is critically damped.

S1-11 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n The graph is from a transient analysis of the previous spring mass system with damping applied Frequency and period as before Amplitude is a function of damping SINGLE DOF SYSTEM DAMPED FREE VIBRATIONS (Cont.)

S1-12 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM UNDAMPED FORCED VIBRATIONS n Dynamic equation with sinusoidal forcing where = forcing frequency Solution where Steady-state solution P/k is the static response. is the dynamic magnification factor.

S1-13 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM UNDAMPED FORCED VIBRATIONS (Cont.)

S1-14 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS n Dynamic equation with sinusoidal forcing n The transient solution decays rapidly, and for this case is ignored n Steady-state solution n These equations are described on the next page

S1-15 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.) n These equations need inspection as they show several important characteristics of the dynamic response. Solution to static loading for = 0 Term that causes the amplitude to increase considerably as n At = n this term is equal to 4 and controls the amplitude Phase-lead of the response relative to the input

S1-16 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.) is defined as the phase-lead for MSC.Nastran

S1-17 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n For u Magnification factor 1 (static solution) u Phase angle (response is in phase with the force) n For u Magnification factor 0 (no response) u Phase angle (response has opposite sign of force) n For Magnification factor ½ u Phase angle SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-18 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n A frequency response analysis can be used to explore the response of the mass/spring system to the forcing function. n This method allows the comparison of the response of the mass/spring system to the input force applied to the mass/spring system over a wide domain of input frequencies n It is more convenient in this case to do a frequency response analysis than performing multiple transient analyses, each with different input frequencies n Apply the input load as a 1 unit of force over a frequency domain from 0.1 Hz to 5.0 Hz n Damping is 1% of Critical SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-19 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation Magnification Factor = 1/2 = 1/G = 50 Static Response = P/k = 0.01 Peak Response = 0.5 at 1.59 Hz Note: Use of a Log scale helps identify low order response Displacement Frequency (Hz) SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-20 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n There are many important factors in setting-up a frequency response analysis that will be covered in a later section n For now, note the response is as predicted by the equation of motion u At 0 Hz result is P/k u At 1.59 Hz result is P/k factored by dynamic magnification u At 5 Hz the result is low and becoming insignificant n The phase change is shown here u In phase up to 1.59 Hz u Out of phase180 Degrees after 1.59 Hz SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-21 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n Perform a transient analysis with a unit force applied to the mass/spring system at 1.59 Hz n Again damping of 1% of critical damping is used n The result is shown on the next page u The response takes about 32 seconds to reach a steady-state solution u After this time the displacement response magnitude stays constant at 0.45 units u The theoretical value of 0.5 is not reached due to numerical inaccuracy (see later) and the difficulty of being at the sharp peak SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-22 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n Transient analysis with a unit force applied to the mass/spring at 1.59 Hz Displacement Time SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-23 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation n When plotting an input and output at the steady-state frequency/period, the input signal is not very accurate, hence there is a problem finding the exact magnification factor n It is seen that the phase between the input and output is about 90 degrees at resonance, as expected Lead = 0.18 sec (approx), 103 degrees (approx 90) T input = 1/1.59 = sec input output SINGLE DOF SYSTEM DAMPED FORCED VIBRATIONS (Cont.)

S1-24 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation MSC.NASTRAN DOCUMENTATION n Manuals u MSC.NASTRAN Quick Reference Guide u MSC.NASTRAN Reference Manuals n Users Guides u Getting Started with MSC.NASTRAN u MSC.NASTRAN Linear Static Analysis u MSC.NASTRAN Basic Dynamic Analysis u MSC.NASTRAN Advanced Dynamic Analysis u MSC.NASTRAN Design Sensitivity and Optimization u MSC.NASTRAN DMAP Module Dictionary u MSC.NASTRAN Numerical Methods u MSC.NASTRAN Aeroelastic Analysis u MSC.NASTRAN Thermal Analysis

S1-25 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation MSC.NASTRAN DOCUMENTATION (Cont.) n Other Documentation u MSC.NASTRAN Common Questions and Answers u MSC.NASTRAN Bibliography n Documentation available in online form for workstations and PCs

S1-26 NAS122, Section 1, August 2005 Copyright 2005 MSC.Software Corporation TEXT REFERENCES ON DYNAMIC ANALYSIS 1. W. C. Hurty and M. F. Rubinstein, Dynamics of Structures, Prentice-Hall, R. W. Clough and J. Penzien, Dynamics of Structures, McGraw-Hill, S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering, 4th Ed., John Wiley & Sons, K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice- Hall, J. S. Przemieniecki, Theory of Matrix Structural Analysis, McGraw-Hill, C. M. Harris and C. E. Crede, Shock and Vibration Handbook, 2nd Ed., McGraw-Hill, L. Meirovitch, Analytical Methods in Vibrations, MacMillan, L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, M. Paz, Structural Dynamics Theory and Computation, Prentice-Hall, W. T. Thomson, Theory of Vibrations with Applications, Prentice-Hall, R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, John Wiley & Sons, S. H. Crandall and W. D. Mark, Random Vibration in Mechanical Systems, Academic Press, J. S. Bendat and A. G. Piersel, Random Data Analysis and Measurement Techniques, 2nd Ed., John Wiley & Sons, 1986.