The spring-held follower A B has a mass of 0.5 kg and moves back and forth as its end rolls on t

The spring-held follower A B has a mass of 0.5 kg and moves...

The spring-held follower $A B$ has a mass of $0.5 \mathrm{kg}$ and moves back and forth as its end rolls on the contoured surface of the cam, where $r=0.15 \mathrm{mand} z=(0.02 \cos 2 \theta) \mathrm{m}$ If the cam is rotating at a constant rate of 30 rad/s, determine the force component $F_{z}$ at the end $A$ of the follower when $\theta=30^{\circ} .$ The spring is uncompressed when $\theta=90^{\circ} .$ Neglect friction at the bearing $C$.

Key Concepts

Cam-Follower Mechanism

A cam-follower mechanism is a system where a rotating cam with a specific profile imparts a prescribed motion to a follower. The follower's motion is directly determined by the shape of the cam's surface, which can be defined mathematically, ensuring that the output motion meets desired timing, displacement, and acceleration specifications.

Kinematics of Rotational to Linear Motion

This concept involves translating the rotational motion of the cam into linear motion of the follower. By defining the follower’s displacement as a function of the cam’s rotation angle, one can use differentiation with respect to time (via the chain rule) to determine the follower’s velocity and acceleration, which are essential for subsequent force calculations.

Inertial Forces in Rotating Systems

In systems where components accelerate due to rotational motion, inertial forces arise. These forces, which include centripetal and tangential components, must be considered when analyzing dynamic behavior. In the context of a follower mass moving under a rotating cam, calculating these forces is crucial for determining the net load transmitted through the mechanism.

Spring Force and Equilibrium

A spring attached to a follower exerts a force proportional to its deformation from the equilibrium position, according to Hooke's law. In dynamic systems, knowing the point of zero spring compression (or equilibrium) is essential to determine the effective spring force acting as the follower is displaced from this position.

Differential Kinematics and Dynamic Analysis

Analyzing the follower's motion involves computing the derivatives of its displacement function with respect to the cam's rotation. This process yields the velocity and acceleration of the follower. These time derivatives are fundamental for dynamic analyses, as they link the geometric profile of the cam to the forces experienced by the follower due to its mass and the spring.

Transcript

00:01 Hi everyone.

00:01 So what we have here is a spring held follower that has a mass of 0 .5 kilograms, and it moves back and forth on a contoured surface that has an r equal to 0 .5 meters, and the contour is given by z equal to 0 .02 cosine of 2 theta.

00:18 We know that the cam is rotating at a constant rate of 30 radiance per second, and that the spring has a spring constant of 1 ,000 nons per meter.

00:27 What we're asked to solve for is the force in the z -directors.

00:31 At the end of the follower when theta is equal to 30 degrees.

00:36 And how we're going to be doing that is using our knowledge of acceleration in the linear directions, linear and rotational directions.

00:46 And we're going to be using that with newton's second law to develop our equation of motion to solve for this force.

00:53 So what we're going to do first is write down our motion in z.

01:00 So we know that z is equal to 0 .02, cosine 2 theta.

01:06 Next, what we can do is we can take the derivative twice to get our acceleration in the z direction.

01:11 So we get that z dot is equal to negative 0 .04, or 0 .04, 0 .04 times a quantity cosine of 2 theta times 2 theta dot plus sine of 2 theta times theta dot, plus sine of 2 theta times theta double dot and for theta dot equal to 30 radiance per second and for theta double dot to equal zero this gives us a z double dot that's equal to negative 72 coast 2 theta so we can go ahead and plug in for our theta we know our theta which is equal to 30 degrees and this gives us a z double dot that's equal to negative 36 meters per second squared...

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