• Basically, things in motion required forces to accelerate to a velocity, and will apply forces to other objects as they slow down.
• Some basic topics to review,
• The basic topics of statics that should be well understood are,
free body diagrams/rigid body assumption
moment components in 2D and 3D
equilibrium of forces and moments
mass properties: center of mass, centroids
moments and product of inertia, polar moment of inertia
• The basic inertial functions are,
• D’Alembert’s principle states that when we have a dynamic system the sum of all applied forces/moments including inertia should equal zero.
• Consider the simple link below that is known to have the forces applied, at the given velocities. What are the reactions at the pin joint.
• we can extend these principles to more complicated mechanisms.
Problem 17.1 • Consider the dynamic effects on the four bar linkage. Each of the rods weighs 1 lb/in., and AB is being turned with an angular velocity of 100 rad/sec, and angular acceleration of 5 rad/sec. a) Do a kinematic analysis. b) Do a static analysis. c) Do a dynamic analysis. d) What are the total forces on the bodies?
• When doing analysis of more complicated mechanisms we can use a technique called Superposition. This involves the basic steps,
1. do a kinematic analysis to find positions, velocities, accelerations.
2. do a static analysis of the mechanism to find the static forces at the pins
3. using the results of the kinematic analysis find the dynamic forces at the pins
4. add the results of the static and dynamic analyses for the total forces.
• Superposition assumes that the structure is linear. In other words the dynamic forces should not change the static forces, and vice versa. This will occur if the mechanism is flexible, has friction, etc. (We will not use this technique in this course because computational tools available (eg Mathcad) reduce its value.)
• This section will discuss how to find Moments of Inertia for complex solids using experimental techniques.
• The value of this material is somewhat limited, as most modern CAD systems will use the geometry to calculate centroids, moments of inertia, volume, etc.
• This method can be valuable when reverse engineering a part (i.e. no detailed designs available), or when testing a completed part.
• Basically these techniques use the part as a pendulum mass, and use the period to calculate the moment of inertia.
1. Locate the center of mass: this can be done by looking for x, y, & z planes of balance.
2. Measure the mass of the object.
3. Secure the object so that it rotates about some point other than the center of mass.
4. Introduce a very small displacement less than 1 degree, and allow it to oscillate.
5. Record the period of oscillation.
6. Calculate the moment of inertia.
• The differential equation and manipulated final form used to describe a pendulum is given below,
• other techniques employ similar approaches,
torsional rod: the object is hung from a rod of known torsional stiffness. The object is then twisted slightly and released. The period of oscillation is used to find the moment of inertia.
trifilar pendulum: a platform with three wires has the object placed on the platform. The platform is then twisted in plane, and released. It then oscillates in rotation. The period is used to calculate the combined moments of inertia of the object and the platform.
Problem 17.2 A Geneva mechanism is shown below with both components rotating.
a) Write the closed loop position equation for the configuration shown.
b) Write the closed loop velocity equation.
c) Write the acceleration closed loop equation.
d) Draw free body diagrams (FBDs) for both links.
e) Write D’Alembert’s dynamic equations for link 3, assume the link 3 has a reaction torque.
g) Expand the vector equations for link 3 to show vector components, or into individual equations, so that they could be solved. (Do not solve for numerical values)
17.1 Erdman, A.G. and Sandor, G.N., Mechanism Design Analysis and Synthesis, Vol. 1, 3rd Edition, Prentice Hall, 1997.
17.2 Shigley, J.E., Uicker, J.J., Theory of Machines and Mechanisms, 2nd Edition, McGraw-Hill, 1995.