The crank A B has a constant angular velocity ω. Determine the velocity and acceleration of the

The crank A B has a constant angular velocity ω. Determine...

Key Concepts

Trigonometric Relationships in Mechanism Analysis

Trigonometric relationships are essential for relating angular parameters to linear displacements in mechanical systems. When the motion of a slider is linked to the rotation of a crank, sines and cosines of the involved angles appear in the position equations. These relationships are crucial for expressing the slider's position in terms of the crank angle and for subsequently differentiating to find velocity and acceleration.

Differentiation and the Chain Rule in Kinematics

In analyzing mechanisms, differentiation is employed to obtain velocity and acceleration from a known position function with respect to time. The chain rule is particularly important because the position variable is often a function of an angle (? or ?) that is itself a function of time. Thus, obtaining time derivatives requires careful application of the chain rule to capture the rate of change accurately.

Angular Velocity

Angular velocity is the rate at which an object rotates about an axis, expressed in radians per second. In many problems, especially in mechanisms like the crank-slider, a constant angular velocity simplifies the kinematic analysis by providing a fixed rate of change for the rotational coordinate, which is used to relate rotational motion to linear motion in the connected parts.

Slider-Crank Mechanism

The slider-crank mechanism is a common mechanical assembly that converts rotary motion into linear motion. It consists of a rotating crank and a connecting rod (or coupler) that ultimately drives a slider along a straight path. Analyzing this mechanism involves relating the angular displacement of the crank to the linear displacement of the slider, often using geometric relations and trigonometric identities.

Kinematic Relationships in Mechanisms

Kinematic relationships describe how the displacement, velocity, and acceleration of different parts of a mechanism are interrelated. For the slider in a slider-crank mechanism, the position can be described as a function of the crank angle. Differentiating this positional relationship with respect to time yields the velocity expression, and differentiating again gives the acceleration, making it crucial to understand how geometric constraints dictate motion.

Transcript

00:01 In this question, we have a crank ab, which rotates with a constant angular velocity omega.

00:07 We need to determine the velocity and acceleration of slider at c as a function of theta.

00:15 Okay, and we are given a hints that we should be using the x coordinate to express the motion of c and the phi coordinate for cb.

00:24 And then x equals to 0 is when 5 equals to 0.

00:27 Okay, so we are given that x is equal to 0 and when 5 is equal to 0 degrees, which means that x coordinate of a point a is actually l plus b.

00:58 And then so in that case, and at any point x a is, x plus from the diagram plus l cosine 5 plus b cosine data okay so based on this relationship we can find that x is equal to l plus b minus l cosine 5 minus b b cosine data.

01:41 Okay, then we'll be taking time derivative.

01:50 Okay, we are going to do two time derivatives because we are asked to find the velocity and acceleration in terms of data.

02:00 Okay, but note that we have five in this question, so we need to do something with the five dots and five other dots.

02:09 Okay, so x dot is equal to l, sine phi, five dot, five dot, plus plus b sine theta data dot okay then we have x double dots is equal to our cosine phi five five dot square plus l sine five five double dots plus b cosine data data dot square plus b side data double dot okay and then from the diagram okay from the diagram here the y coordinates are the same or at is the y coordinate for b is the same which is l sine 5 is equal to b sine data okay so which means that sign 5 is equal to b over l sign theta okay so just remember that we are we need to find velocity and acceleration as a function of data.

03:29 So here we need to get rid of sine five, we also need to get rid of phi.

03:37 Okay.

03:39 Yeah, so here we can also take time derivative.

03:43 So you are going to have l cosine five, five dot is equal to b cosine data data dot.

03:56 And then so five dot is equal to b cosine data data dot divide by l cosine phi and then cosine phi using the pettagoras theorem we have okay using pettagoras theorem is going to be square root of 1 minus sine square 5 here we can replace our sign 5 here and get b over l cosine data data dot divide by b over square root 1 minus b over l sign data the whole thing square so now we are ready to so now we are ready to replace our sign 5 and 5 dot in the x dot equation.

05:20 Okay, so we have so v equals to x dot, this is equal to just to copy down our sign 5 times 5 dot plus b side data data dot now this will be replacing our sign 5 to be l, b over l, 5 .5, side data.

05:57 Phi dot would be b over l, cosine data, data, divided by the square term, 1 minus, b over l, sine data, square, plus b, sine data, data, data dot okay okay so you'll be so we know that data dot is omega okay so we are going to factor out be omega sign data okay so the l here cancers out okay so there will be omega sign data so the second time for here okay so the second term will just become 1 and the first term will become b over l cosine data divide by the square return 1 minus b over l like data square okay so this is our v me equals to the omega sine data there's a rewrite just to make it clear okay, so this is our v.

07:39 Okay, we are done with the first thing we need to do.

07:43 Now we proceed to find the acceleration.

07:49 Okay, now acceleration.

07:56 Okay, so just copy down what we have last time.

08:00 So x double dots is equal to l cosine 5, 5 dot square plus l sign 5, 5, 5 double dot, plus b, b cosine theta, data, data dot square, plus b sign data, data double dots.

08:22 Okay, so we are given that omega is constant.

08:28 This is data dot, so data double dots is zero.

08:34 Okay, and then, so looking at this, we have cosine five, we have five dot, we have five dot.

08:46 Now we need to try to replace five double dots.

08:51 Okay, right.

08:55 So, okay, so this is, this is one equation that we have.

09:06 And then we also have another equation from the y coordinates, which is l cosine 5, l sine 5 is equal to b signed data from previous parts.

09:20 When we did this, this is the y coordinate of b, of point b.

09:25 Okay, so we take time derivative, we get l cosine 5, 5 dot is equal to b cosine data, data.

09:41 If we take time derivative again, we are able to get five double dots.

09:46 So let's do that, minus l, sign 5, five, five dot square plus l cosine five, five double dot is equal to minus b sign data data dot square plus e cosine data data double dot so this is zero okay as mentioned here it's mentioned here then um then we are going to replace sci -fi i dot and cosine five and then we are able to get our 5.

10:33 Okay.

10:36 So, sign 5 is b over l, 9 data.

10:43 K5 .5.

10:43 Square is something that we obtained before, which is here...

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