Running Machinery Too Fast Suppose that a piston is moving...

Running Machinery Too Fast Suppose that a piston is moving straight up and down and that its position at time $t$ seconds is $$s=A \cos (2 \pi b t)$$ with $A$ and $b$ positive. The value of $A$ is the amplitude of the motion, and $b$ is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why machinery breaks when you run it too fast.)

Running Machinery Too Fast Suppose that a piston is moving straight up and down and that its position at time $t$ seconds is
$$s=A \cos (2 \pi b t)$$
with $A$ and $b$ positive. The value of $A$ is the amplitude of the motion, and $b$ is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston's velocity, acceleration, and jerk? (Once you find out, you will know why machinery breaks when you run it too fast.)

Key Concepts

Sinusoidal Motion

Sinusoidal motion describes periodic movement modeled by sine or cosine functions. This type of motion is characteristic of many physical systems and provides a framework for analyzing oscillatory behavior, such as that exhibited by a piston moving up and down.

Frequency

Frequency indicates the number of complete cycles per unit time and determines how quickly the motion oscillates. In the context of periodic motion, variations in frequency directly impact the speed of the cycle and, subsequently, the dynamic responses of the system.

Derivative Relationships in Kinematics

In kinematics, key quantities such as velocity, acceleration, and jerk correspond to the first, second, and third derivatives of the position function with respect to time. Each differentiation step introduces additional factors that scale with the frequency, thereby magnifying the respective rates of change.

Scaling Laws in Harmonic Motion

The scaling laws in harmonic motion reveal that while velocity is linearly proportional to the frequency, acceleration is proportional to the square of the frequency and jerk is proportional to the cube. Consequently, doubling the frequency results in a doubling of velocity, a quadrupling of acceleration, and an eightfold increase in jerk, which explains why machinery can experience dramatically increased dynamic stresses at higher operational speeds.

Transcript

00:01 In this question, we are given a situation where there is a machine with a piston whose vertical position is given here in terms of the time t.

00:16 So because this is a periodic function, we know the piston is going to be going up and down and up and down, and it might be turning a wheel or something like that.

00:29 In this formula, the positive constants a and b represent the amplitude and frequency of this.

00:37 Oscillation.

00:40 Now, we want to see what happens if we double the frequency.

00:48 What effect is that going to have on the machine? and we're going to do that specifically by considering what happens to the velocity, acceleration, and a jerk of the piston, given that the frequency is doubled.

01:04 So because of velocity acceleration and jerk are derivatives of the position s with respect to t, we want to do some differentiation.

01:19 First we find the velocity by differentiating with respect to t once.

01:28 We know the derivative of course is negative sign, but we have to make sure that we multiply by the derivative of the inside because of chain rule.

01:54 Chain rule tells us that the derivative of this composite function is the derivative of the outside function, multiplied by the derivative of the function that is placed inside it.

02:13 So here is the velocity, now the acceleration, which is the derivative of velocity, or the second derivative of the position.

02:28 Again, we must use chain rule.

02:31 This time we have the outside function with a derivative of cos.

02:38 The derivative of sine is cos, and then the derivative of the inside is 2 pi b, again, because it's the same inside as before.

02:51 Now we have 2 pi b multiplied by 2 pi b.

02:58 At this point, you probably see where we're going with this.

03:03 Next, the jerk is the first derivative of the acceleration with respect to time, or the second derivative of the velocity, or the third derivative of the position...

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