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Unformatted text preview: ME163 Mechanical Vibrations (Winter 2009) HW-3 Due 1.29.09 in class-25% if late within 24 hours-50% if later than that 1. This problem aims to give you a flavor into the process of designing cars for an adventurous roller coaster ride. We can model the car as a single degree of freedom system subjected to a forcing function (this is due to the design of the roller coaster rails). As an example assuming that the governing equation is as follows Figure 1: Roller coaster design ¨ x + 9 ˙ x + 36 x = 54cos 3 t (a) Find the steady state response (otherwise known as the asymptotic or the particular solution) of the coaster car analytically i.e x ( t ). (b) While designing such rides, one of the important parameters is the rate of acceleration i.e ˙ a = ... x the person is subjected to. This rate is called jerk . Find the steady state jerk and compute its phase with respect to the force. (a) Take x = Asin (3 t ) + Bcos (3 t ) substituting in the differential equation given (- 9 A + 27 B + 36 A ) cos (3 t ) + (- 9 B- 27 A + 36 B ) sin (3 t ) = 54( cos (3 t )) or (- 9 A + 27 B + 36 A ) cos (3 t ) = 54 cos (3 t ) (- 9 B- 27 A + 36 B ) sin (3 t ) = 0 Hence A = 1 and B = 1,Thus x = ( cos (3 t ) + sin (3 t )) Expressing it as Asin ( ωt + φ ), we get x = √ 2 sin (3 t + π/ 4) (b) Calculate the jerk by differenciating x three times ˙ x = 3 √ 2 cos (3 t + π/ 4) ¨ x =- 9 √ 2 sin (3 t + π/ 4) ... x =- 27 √ 2 cos (3 t + π/ 4) ME163 Mechanical Vibrations 1 HW3 Now cos ( π- θ ) =- cos ( θ ) and cos ( θ ) = cos (- θ ) or ... x = 27 √ 2 cos (3 t- 3 π/ 4) Thus Jerk LAGS the force by 3 π/ 4 2. Consider a disk of mass m and radius r connected to a linear spring and to an arrangement of three dashpots (shown in Figure 2). The stiffness of the spring is k , its un-stretched length is L and the damping coefficient of each damper is C . Further assume that the disk rolls without slipping i.e the relative velocity of the contact point P of the disk with respect to the ground is always zero....
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- Winter '08
- Frequency, Ωn