Deriving 1 DOF Equations of Motion Worked-Out Examples. MCE371: Vibrations. Prof. Richter. Department of Mechanical Engineering. Handout 3 Fall 2017

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1 MCE371: Vibrations Prof. Richter Department of Mechanical Engineering Handout 3 Fall 2017

2 Masses with Rectilinear Motion Follow Palm, p.63, and Sect.2.6. Refine your skill in drawing correct free body diagrams (FBD). Use the methodology of Handout 2 to figure out the right expressions for the damper and spring forces. We consider that each mass can be affected by forces of 5 types: spring, damper, gravity, cable tension or applied external force. A fundamental equation is written for each mass: Fi = mẍ F i are the forces acting on mass m, x is the position of the mass relative to a non-accelerating frame of reference (usually fixed). Refresher from Handout 2: Study Example in Palm.

3 Geometric Compatibility When the 1-DOF system has many moving parts, the coordinates must be linked by geometric relationships For example, the center of a non-slipping rolling element of radius R displaces x = Rθ when the element rotates an angle θ. Pulleys introduce a geometric relationship between the displacement of their center and the amount of cable that must be fed/recovered to avoid slack. We ll learn this through several examples. Hinged bars introduce geometric relationships according to triangle similarities (small displacement case).

4 Pulley/Cable Kinematics Carefully study Example in Palm. Solve Prob. 2.5 Alternatively, (less fuss) picture this sequence: y x 3. How much cable must we reel in to remove slack? 1. Original position 2. Pulley center displaces y upwards. Cable remains, creating slack

5 When Are Pulley Tensions Equal? Palm, p.71 Newton s Law in the vertical direction: F 1 F 2 +F 3 = m p ẍ Newton s Law for rotation : (F 1 F 2 )R = I p ẇ If I p is negligible we get F 1 = F 2. If, in addition m p is negligible: F 3 = F 1 +F 2 = 2F 1. Apply the same ideas to cable-driven drums.

6 When Can We Ignore Gravity in 1DOF Systems? When the frame of reference is placed at the equilibrium position. Read page 97 in Palm.

7 Example Deriving 1 DOF Equations of Motion We solve Prob

8 Masses with Rotary Motion Refine your skill in drawing correct free body diagrams (FBD). We consider that each rotary mass can be affected by torques of 4 types: cable tension, contact force in gears, friction force in non-slipping rolling motion or external applied torque. A fundamental equation is written for each rotary mass: Ti = I θ = Iẇ T i are the forces acting on inertia I, θ is the angular position measured through the axis of rotation (a principal inertia axis).

9 Example: Rack-and-Pinion A torque T is applied to a pinion of radius r and inertia J. The pinion drives a rack of mass m. There s no friction. Find the equation of motion for the rack (draw FBDs for the pinion and the rack)

10 Equivalent Mass and Inertia Read Palm, Sect. 2.5 Suppose a rotating element with non-negligible mass is connected to a mass moving in a straight line If there s no friction at the connecting point, we can treat the assembly as a unit, avoiding an extra FBD If we are interested in the rectilinear coordinate, we find the equivalent mass If we are interested in the angular coordinate we find the equivalent inertia

11 Finding equivalent mass and inertia The sum of kinetic energies equals the energy of an element with the equivalent mass or inertia. The coordinate of the equivalent element matches the coordinate of interest. Rectilinear motion: K = 1 2 mẋ2 Rotary motion: K = 1 2 Iw2 Use equivalence to find the equation of motion for the pinion in the previous example.

12 Equivalent Inertia of Gear Trains We can use the kinetic energy method to reduce chains of gears (no friction) to a single rotary inertia with the coordinate of either input or output shaft. We solve Prob

13 Example Deriving 1 DOF Equations of Motion

14 Example Deriving 1 DOF Equations of Motion

15 Prob Deriving 1 DOF Equations of Motion

16 Recommended Problems 1 P P P P P P P. 3.69