hw_3_solu

hw_3_solu - ME163 Mechanical Vibrations (Winter 2009) HW-3...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ( t + ), we get x = 2 sin (3 t + / 4) (b) Calculate the jerk by differenciating x three times x = 3 2 cos (3 t + / 4) x =- 9 2 sin (3 t + / 4) ... x =- 27 2 cos (3 t + / 4) ME163 Mechanical Vibrations 1 HW3 Now cos ( - ) =- cos ( ) and cos ( ) = cos (- ) or ... x = 27 2 cos (3 t- 3 / 4) Thus Jerk LAGS the force by 3 / 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....
View Full Document

Page1 / 7

hw_3_solu - ME163 Mechanical Vibrations (Winter 2009) HW-3...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online