Unformatted text preview: odal Parameter Estimation [AWP #12] Frequency Damping • Half Power [TD 3.10, AWP #8] Mode Shape • Quadrature MDOF Systems [AWP #11] o Equations of Motion Matrix Solution o Natural Frequencies o Mode Shapes Transient Solution of Equation of Motion [TD 4, AWP #13] o Forced Vibrations Impulse Excitation [TD 4.1] Transient Excitation [TD 4.2] o Convolution [TMH 2.7] Properties: F[f(t)*g(t)] = F[f(t)] F[g(t)] o Numerical Solutions [AWP #14] Runge-Kutta MDOF Detailed Overview (provided for reference) [AWP #15] Lecture Notes -4- 06/16/06 12:25 PM Mechanical Vibrations I #1 – Introduction to Mechanical Vibrations
All mechanical systems vibrate or undergo oscillatory motion. However, in order to discuss the behavior of vibrating systems, it is first necessary to establish some nomenclature. There are several terms used to describe vibrations, among which are: oscillatory motion, periodic motion, harmonic motion, period, and frequency. Oscillatory motion is any pattern of motion where the system under observation moves back and forth across some equilibrium position, but does not necessarily have any particular repeating pattern. Periodic motion is a specific form of oscillatory motion where the motion pattern repeats itself with a uniform time interval. This uniform time interval is referred to as the period and has units of seconds per cycle. The reciprocal of the period is referred to as the frequency and has units of cycles per second. This unit combination has been given a special unit symbol and is referred to as Hertz (Hz). Harmonic motion is a specific form of periodic motion where the motion pattern can be describe by either a sine or cosine. This motion is also sometimes referred to as simple harmonic motion. Because the sine or cosine technically uses angles in radians, the frequency term expressed in the units radians per second ( rad sec ). This is sometimes referred to as the circular frequency. The relationship between the frequency in Hz and the frequency in rad sec is simply the relationship, 2π rad cycle . Natural frequency is the frequency at which an undamped system will tend to oscillate due to initial conditions in the absence of any external excitation. Because there is no damping, the system will oscillate indefinitely. Damped natural frequency is frequency that a damped system will tend to oscillate due to initial conditions in the absence of any external excitation. Because there is damping in the system, the system response will eventually decay to rest. Resonance is the condition of having an external excitation at the natural frequency of the system. In general, this is undesirable, potentially producing extremely large system response. Degrees-of-freedom is the number of independent coordinates necessary to describe them configuration (state) of a system. It can also be expressed as the number of coordinates used to describe the configuration of a system minus the number of independent constraint equations between those...
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- Fall '11
- homogeneous solution