Department of Mechanical Engineering and Materials Science

Rice University

 

MECH 343:  Modeling Dynamic Systems

Homework #1

 

Modeling Mechanical Systems

Due: Thursday, September 7, 2006 (midnight)

 

 

1.   A schematic for a mass-spring-damper system with an additional spring is shown below. The ¡'s represent frictionless linear bearings so the mass slides freely in the horizontal direction (no viscous damping or Coulomb friction). Assign a coordinate x to this system, pointing to the left and attached to the mass. Let x = 0 at equilibrium (when the applied force f = 0).  [20 points total]

 

 

  1. Draw a Free Body Diagram (FBD) of the mass, showing all horizontal forces. State whether you will be using Newton's Second Law or D'Alembert's Law. [6 points]

 

  1. Write each of the forces acting on the mass in terms of x and/or its derivatives. [4 points]

 

  1. Write Newton's or D'Alembert's Law and substitute your answers from the previous question to obtain an equation of motion for the mass. [6 points]

 

  1. Show that, as far as the mass and its equation are concerned, it makes no difference whether the force acts directly on the mass or spring k2. [4 points]

 

2.     Solve problem 2.15 in the textbook (page 43). [20 points total]

 

3.     See Figure P5.9 from the textbook (page 129). The input is the force fa(t) applied at the top of the lever, and the output is the support force on the lever, taking the positive sense to the right. The lever can be considered ideal, and the motion from the vertical position small, so the motion of the top and midpoint can be regarded as horizontal. [30 points total]

 

  1. Draw Free Body Diagrams (FBDs) of the bodies in this system, showing all forces. State whether you will be using Newton's Second Law or D'Alembert's Law to develop the Equations of Motion (EOM). [10 points]

 

  1. Write each of the forces acting on the objects in terms of x, y and/or their derivatives. [10 points]

 

  1. Write Newton's or D'Alembert's Law and substitute your answers from the previous question to obtain the equations of motion. [10 points]

 

4.     Consider the schematic below, which is similar to the one reviewed in recent lecture notes. The difference between these systems is that, in the figure below, the shaft connecting the pulley (radius r1) and the gear (radius r2) has a torsional stiffness coefficient k3. [30 points total]

 

  1. Draw the necessary Free Body Diagrams (FBDs) for this system, showing all torques/forces.  State whether you will be using Newton's Second Law or D'Alembert's Law. [10 points]

 

  1. Write each of the forces and/or torques in terms of x and/or y and/or their derivatives.  [10 points]

 

  1. Write Newton's or D'Alembert's Law and substitute your answers from the previous question to obtain two coupled equations of motion. [10 points]

 

 

 

Good practice problems from the book include: