3. The angular position of an output link (heta4) of a...

3. The angular position of an output link ($ heta_4$) of a four-bar linkage corresponding to any specified angular position of the input link ($ heta_2$) can be computed using the Freudenstein's Equation: $k_1 cos heta_4 - k_2 cos heta_2 + k_3 = cos( heta_2 - heta_4)$ where $k_1 = frac{d}{a}, k_2 = frac{d}{c}, and k_3 = frac{a^2 - b^2 + c^2 + d^2}{2ac}$ a, b, c, and d are the link lengths shown in the figure. Find the value of $ heta_4$ given the following data: $ heta_2 = 30^circ$, a = 1, b = 2, c = 4, and d = 5. Use the Newton-Raphson's Method with a starting value for $ heta_4$ as $90^circ$ and $varepsilon = 0.001.$

Transcript

00:01 In this question, the equation is provided to us which is k1, pos theta 4 minus k2, cos theta 2 plus k3, that is equivalent to cos theta 2 minus theta 4.

00:13 Consider this as the first equation.

00:14 This is the equation for calculating angular position.

00:18 Now, where this k1 is equals to d divided by a, this is t divided by a, k2 here is t divided by c, and k3 here is a square minus b.

00:31 Square plus c square plus t squared divided by two ac so these are the value and it is also given the value of a which is equals to one value of b is two value of c is four value of b is five and value of theta two is thirty degrees so k1 is equivalent to five divided by one that is five k two here is five divided by four which is equals to one point two five and k three here is one square minus two square plus four square plus five square divided by two multiplied by two multiplied by 1 multiplied by 4 which is equals to 4 .75.

01:05 So these are the value.

01:06 Now we will substitute all these value.

01:08 The first equation, so we will get 5 pos theta 4 minus 1 .25 cos 30 degree plus 4 .75 equivalent to cos 30 degree minus theta.

01:19 So we know that posa minus b, plus a cosby plus sine a sine.

01:23 So we can write this as 5 pos theta 4 plus this value is 3 .667 plus after writing both of the we will get this and this cost this can be written as cos 30 degree cos theta 4 plus sine 30 degree sign theta 4 solving this out so we get we are going to take theta as equals to theta 4 divided by 2 so here we get this value as 4 .134 okay we know what we know that cos a this is equivalent to what 1 minus 10 square a divided by 2 and 1 plus 10 square a divided by 2 so theta power theta 4 divided by 2 so we can take this earlier as 4 .134 this is 1 minus 10 square theta plus theta power 4 divided by 2 is theta divided by 1 plus 10 square theta and minus we have 0 .5 multiplied by 210 theta and whole divided by 1 plus 10 square theta and we add here 3 .667 to get this all equivalent to this is it is in multiplying...

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