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m2k_dfq_repeat - Repeated Real Roots of the Characteristic...

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Repeated Real Roots of the Characteristic Equation; Method of Reduction of Order Repeated Roots of Characteristic Equation Suppose a mass m is connected to a rigid support by means of a spring. We will denote the position of the mass by y , the static equilibrium position of the mass (where the spring is neither stretched nor compressed, by y = 0. We assume the spring is a linear spring , by which we mean that if it is stretched or compressed it exerts an opposing force proportional to the amount of stretching or compression (Hooke’s law): f 1 = - k y. In addition there may be a frictional force, typically proportional to velocity, tending to slow the motion of the mass: f 2 = - γ dy dt . Then Newton’s third law gives m d 2 y dt 2 = = - k y - γ dy dt m d 2 y dt 2 + γ dy dt + k y = 0 . The roots of the associated characteristic equation, m r 2 + γ r + k = 0 , can be obtained from the quadratic formula r = - γ ± γ 2 - 4 km 2 m . 1
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When the discriminant γ 2 - 4 km > 0 we obtain two real roots: r 1 = - γ - γ 2 - 4 km 2 m , r 2 = - γ + γ 2 - 4 km 2 m . The general solution then takes the form, as we saw in the pre- vious section of the notes, y ( t, c 1 , c 2 ) = c 1 e r 1 t + c 2 e r 2 t . In this case both r 1 and r 2 are negative but r 1 is closer to zero. In mechanics this is called the overdamped case because the e r 1 t term returns to zero fairly slowly. This is what happens, for example, if a screen door closer is set with damping too high. In some engineering situations, for example in the design of shock absorbers, the parameter γ is a design parameter and can be set at any desired value. In the overdamped case the exponential rate at which solutions decay to zero, assuming γ > 0, is equal to minus the smaller of the real parts of r 1 and r 2 . One can see fairly easily that this quantity reaches its maximum with the choice γ 2 - 4 km =
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