28.2 PLANAR

---

À  p

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.

Consider the dynamic effects on the four bar linkage below,

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.)

28.2.1 Measuring Mass Properties

(|

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.

The basic procedure is,

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.