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MECH 515 Midterm Review

Discrete systems: systems composed of an assembly of multiple parts, in which thesystems response is dependent on the combination of each individual components

behavior.

Componentso Springs

Proportional to displacement(linear range) store/release potentialenergy.

Assumed massless Spring force is conservative: independent of path Predicted behavior

o Viscous Damper Proportional to velocity (linear range) dissipate energy, Assumed massless Damper force is non-conservative

o Masses Proportional to acceleration store/release kinetic energy

1st Order Systems: no mass, so 2nd Order Systems: Equilibrium: point where the net forces equal zero. Stability

Asymptotically Stable Stable Unstable

Natural Frequency: frequency at which the system will resonate under free vibration.

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Damped 2nd Order Systemso Overdamped ( : two

different real roots, takes longer

to converge than critically

damped without oscillating.

o Critically damped ( : theimaginary part goes to 0, so there

is a double real root. Converges

to zero as fast as possible without oscillating.

o Underdamped ( 1): complex roots, oscillates at the damped naturalfrequency (), takes longer to converge.

Harmonic Excitationo Excitation frequency: frequency at which the force is exiting the systemo Frequency response (): measure of the system response to a harmonic

excitation of frequency (); usually complex.

o Magnification factor (||): the amplitude of forced vibration motion withrespect to the magnification of the static deflection as a function of the frequency

ratio (/ )

o Resonance condition: violent vibration when the excitation frequency equals tothe natural frequency ( )

o Harmonic balance: balances the response to ensure is steady state, by equatingeach coefficient creating 2 equations, which have to be satisfied.

Periodic Excitationo Periodic functions can be expressed as linear combinations of harmonic functions

by Fourier series.

o Superposition can be used to combine individual responses for each individualexcitation.

o The series can be either in exponential (complex form) or trigonometric (realform) functions.

o Fundam. frequency ( ): called 1st harmonic, wh. is the p harmonic.o For periodic function even with time, the Fourier series reduces to a cosine terms.o For periodic function oddwith time, the Fourier series reduces to asine terms.o Frequency ratio: determines the systems performance