hw_8_solu

hw_8_solu - ME163 Mechanical Vibrations (Winter 2009) HW-8...

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-8 Due Wednesday 4.04.09 7pm (in the box or to a TA) Answers on web by 9pm, midterm the day after 1. 2DOF Modeling. Consider the system shown in Figure 1 which is a crude model of a coal cart with a scoop attached to its side. The cart has a mass M and is propelled along a horizontal track by a force f . The scoop hangs off the side and is actuated by a torque T (the scoop has a total mass m and moment of inertia I about the center of mass C ). Obtain the equations of motion using x 1 and as coordinates and f and T as the inputs. Consider gravity. T M f g m C a y x 1 x 2 Figure 1: Coal cart with attached scoop. One choice of coordinates is x 1 and . The free body diagrams are shown in Figure 2. F x and F y are the reaction forces in the x and y directions due to the physical connection between f F M M F R R g 1 2 x y T x F F y mg C Figure 2: Free body diagrams the pendulum and the mass M . R 1 and R 2 are the vertical reaction forces due to the wheels. Summing forces on M in the horizontal and vertical directions gives M x 1 = F x + f ME163 Mechanical Vibrations 1 HW8 M y 1 = F y + R 1 + R 2- Mg For mass m , summing forces in the horizontal and vertical directions gives m x 2 =- F x m y 2 =- F y- mg Summing moments about the center of mass gives I = aF x cos + aF y sin + T. Using the previous equations to eliminate F x and F y we obtain M x 1 + m x 2 = f (1) I + m x 2 a cos + ma ( y 2 + g )sin = T (2) The displacements are related as follows: x 2 = x 1 + a sin y 2 =- a cos Thus x 2 = x 1- a 2 sin + a cos y 2 = a 2 cos + a sin Substituting x 2 and y 2 in (1) and (2) we obtain ( I + ma 2 ) + ma x 1 cos + mga sin = T ( M + m ) x 1- ma 2 sin + ma cos = f....
View Full Document

Page1 / 9

hw_8_solu - ME163 Mechanical Vibrations (Winter 2009) HW-8...

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