##
24. Mechanism Acceleration

• The amount that a point on a mechanism speeds up or slows down.

• The acceleration of a point ‘P’ is the first derivative of velocity, or second derivative of position,

• To analyze linear acceleration we must first separate the velocity into a magnitude, and direction.

• We can also consider angular acceleration,

• If we consider the relative velocity and acceleration between two points on a rigid body we get,

• For planar calculations we can write a complex equivalent,

Problem 24.1 If the link lengths are AB=18”, BC=12”, CD=8”, DA=6”, AE=12”, EB=10” and the angular velocity of the crank about D is 200 rad/sec. Find the acceleration of point E in this system.

• there are a couple of other relationships of use,

• Apparent acceleration is found by first separating velocity into magnitude and direction.

• Note: the coriolis component above happens in systems with moving frames of reference. This can create a whip effect.

Problem 24.2 If the crank EA has been rotated 30° counterclockwise about E and has a rotational velocity of 36 rad/sec, what is the acceleration of the rod BD?

• Apparent Angular acceleration is found using,

• For rolling contact the centripetal and normal accelerations are zero, therefore we can simplify the apparent acceleration equation to,

Problem 24.3 If the crank AB has a rotational velocity of 90°/sec, what is the angular acceleration of the angled rod BC?

• We can also solve these problems using simple derivatives.

Problem 24.4 The crank is 20cm long and 20° below the horizon, and the driver is 80cm long. If the crank is rotating at 60rpm, what are the accelerations of each of the links?

Problem 24.5 AB is being turned with an angular velocity of 100 rad/sec, and angular acceleration of 5 rad/sec.

24.1 Instant Centers of Acceleration
• We can find this center using normals to acceleration vectors applied at joints.

• This point will generally not the same as the instantaneous center of velocity.

• Generally there are few uses for this value.

24.2 Problems
Problem 24.6 Find the positions, velocities, and accelerations of all of the members in the mechanism below. The link lengths are AB=14.1”, BC=20”, CD=10”, DA=10”, and the angular velocity of the crank about D is 20 rad/sec.

24.3 References
24.1 Erdman, A.G. and Sandor, G.N., Mechanism Design Analysis and Synthesis, Vol. 1, 3rd Edition, Prentice Hall, 1997.

24.2 Shigley, J.E., Uicker, J.J., “Theory of Machines and Mechanisms, Second Edition, McGraw-Hill, 1995.