Notes-MechV - Mechanical Vibrations A mass m is suspended...

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Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u ( t ) denote the displacement, as a function of time, of the mass relative to its equilibrium position. Recall that the textbook’s convention is that downward is positive Then, u > 0 means the spring is stretched beyond its equilibrium length, while u < 0 means that the spring is compressed. The mass is then set in motion (by any one of several means).
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The equations that govern a mass-spring system At equilibrium: (by Hooke’s Law ) mg = kL While in motion: m u ″ + γ u ′ + k u = F ( t ) This is a second order linear differential equation with constant coefficients. It usually comes with two initial conditions: u ( t 0 ) = u 0 , and u′ ( t 0 ) = u′ 0 . Summary of terms : u ( t ) = displacement of the mass relative to its equilibrium position. m = mass ( m > 0) γ = damping constant ( γ ≥ 0) k = spring (Hooke’s) constant ( k > 0) g = gravitational constant L = elongation of the spring caused by the weight F ( t ) = Externally applied forcing function, if any u ( t 0 ) = initial displacement of the mass u′ ( t 0 ) = initial velocity of the mass
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Undamped Free Vibration ( γ = 0, F ( t ) = 0) The simplest mechanical vibration equation occurs when γ = 0, F ( t ) = 0. This is the undamped free vibration. The motion equation is m u ″ + k u = 0. The characteristic equation is m r 2 + k = 0. Its solutions are i m k r ± = . The general solution is then u ( t ) = C 1 cos ω 0 t + C 2 sin ω 0 t . Where m k = 0 ω is called the natural frequency of the system. It is the frequency at which the system tends to oscillate in the absence of any damping. A motion of this type is called simple harmonic motion . Comment : Just like everywhere else in calculus, the angle is measured in radians, and the (angular) frequency is given in radians per second. The frequency is not given in hertz (which measures the number of cycles per second). Instead, their relation is: 2 π radians/sec = 1 hertz . The (natural) period of the oscillation is given by 0 2 π = T (seconds).
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To get a clearer picture of how this solution behaves, we can simplify it with trig identities and rewrite it as u ( t ) = R cos ( ω 0 t - δ ). The displacement is oscillating steadily with constant amplitude of oscillation 2 2 2 1 C C R + = . The angle δ is the phase or phase angle of displacement. It measures how much u ( t ) lags (when δ > 0), or leads (when δ < 0) relative to cos( ω 0 t ), which has a peak at t = 0. The phase angle satisfies the relation 1 2 tan C C = δ . More explicitly, it is calculated by: 1 2 1 tan C C - = , if C 1 > 0, π + = - 1 2 1 tan C C , if C 1 < 0, 2 = , if C 1 = 0 and C 2 > 0, 2 - = , if C 1 = 0 and C 2 < 0, The angle is undefined if C 1 = C 2 = 0.
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An example of simple harmonic motion: Graph of u ( t ) = cos( t ) - sin( t ) Amplitude: 2 = R Phase angle: δ = - π /4
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Notes-MechV - Mechanical Vibrations A mass m is suspended...

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