Chapter 4, Problem 4.58P

(a)

To determine

The ratio of the $R_a$ and $R_q$ for sine wave.

Answer to Problem 4.58P

The ratio of $R_a$ and $R_q$ for the sine wave is $0.90$ .

Explanation of Solution

Formula used:

The formula for the arithmetic mean value of surface roughness is given as,

  $R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx$ …… (1)

Here, $R_a$ is the arithmetic mean value of the roughness, $L$ is the total measured profile length and $n$ is the number of observations.

The formula for the root mean square value of surface roughness is given as,

  $R_q=\sqrt{\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx}$ …… (2)

Here, $R_q$ is the root mean square value of surface roughness.

The equation of the sine wave is given as

  $y=a\sin \left(\frac{2\pi x}{L}\right)$ …… (3)

Here, $a$ is the amplitude of the function.

Calculation:

Thearithmetic mean value $\left(R_a\right)$ of surface roughness is calculated by substituting the value of equation 3 in equation 1 as,

  $\begin{array}{l}{R}_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx\\ R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|a\sin \left(\frac{2\pi x}{L}\right)\right|}dx\\ R_a=\frac{2a}{L}{\displaystyle \underset{0}{\overset{\frac{l}{2}}{\int }}\sin \left(\frac{2\pi x}{L}\right)dx}\end{array}$

Further solving the above equation,

  $\begin{array}{l}{R}_a=\frac{2a}{L}{\left[-\cos \left(\frac{2\pi x}{L}\right)\right]}_0^{\frac{l}{2}}\times \frac{L}{2\pi }\\ R_a=\left[-\left(-1\right)+1\right]\\ R_a=\frac{2a}{\pi }\end{array}$ …… (4)

The root mean square value $\left(R_q\right)$ of surface roughness is calculated by substituting the value of equation 3 in the equation 2 as,

  $\begin{array}{c}{R}_q=\sqrt{\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx}\\ R_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx\\ R_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|{\left(a\sin \left(\frac{2\pi x}{L}\right)\right)}^2\right|}dx\\ R_q{}^2=\frac{a^2}{L}{\displaystyle \underset{0}{\overset{L}{\int }}{\sin }^2\left(\frac{2\pi x}{L}\right)}dx\end{array}$

Further solving the above equation,

  $\begin{array}{l}{R}_q{}^2=\frac{a^2}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\frac{1}{2}\left[1-\cos \left(\frac{4\pi x}{L}\right)\right]}dx\\ R_q{}^2=\frac{a^2}{2L}{\left[x-\sin \left(\frac{4\pi x}{L}\right)\times \frac{L}{4\pi }\right]}_0^L\\ R_q{}^2=\frac{a^2}{2}\\ R_q=\frac{a}{\sqrt{2}}\end{array}$ …… (5)

Take the ratio of equation 4 and equation 5 as,

  $\begin{array}{l}\frac{R_a}{R_q}=\frac{\frac{2a}{\pi }}{\frac{a}{\sqrt{2}}}\\ \frac{R_a}{R_q}=\frac{2\sqrt{2}}{\pi }\\ \frac{R_a}{R_q}=0.90\end{array}$

Conclusion:

Thus, the ratio of $R_a$ and $R_q$ for the sine wave is $0.90$ .

(b)

To determine

The ratio of the $R_a$ and $R_q$ for a saw-tooth profile.

Answer to Problem 4.58P

The ratio of $R_a$ and the $R_q$ for the saw tooth is $0.86$ .

Explanation of Solution

Formula used:

The formula for the arithmetic mean value of surface roughness is given as,

  $R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx$ …… (6)

Here, $R_a$ is the arithmetic mean value of the roughness, $L$ is the total measured profile length and $n$ is the number of observations.

The formula for the root mean square value of surface roughness is given as,

  $R_q=\sqrt{\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx}$ …… (7)

Here, $R_q$ is the root mean square value of surface roughness.

The equation of the sine wave is given as

  $y=\frac{4ax}{L}$ …… (8)

Here, $a$ is the amplitude of the function.

Calculation:

Thearithmetic mean value $\left(R_a\right)$ of surface roughness is calculated by substituting the value of equation 8 in equation 6 as,

  $\begin{array}{c}{R}_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx\\ R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\frac{4ax}{L}}dx\\ R_a=\frac{4}{L}{\displaystyle \underset{0}{\overset{\frac{l}{4}}{\int }}\frac{4ax}{L}dx}\end{array}$

Further solving the above equation,

  $\begin{array}{l}{R}_a=\frac{16a}{L^2}{\left[\frac{x^2}{2}\right]}_0^{\frac{l}{4}}\\ R_a=\frac{a}{2}\end{array}$ …… (9)

The root mean square value $\left(R_q\right)$ of surface roughness is calculated by substituting the value of equation 8 in the equation 7 as,

  $\begin{array}{l}{R}_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx\\ R_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(\frac{4ax}{L}\right)}^2}dx\\ R_q{}^2=\frac{4}{L}{\displaystyle \underset{0}{\overset{\frac{L}{4}}{\int }}{\left(\frac{4ax}{L}\right)}^2}dx\end{array}$

Solving the above equation as,

  $\begin{array}{l}{R}_q{}^2=\frac{64a^2}{L^3}\times \frac{1}{3}{\left(\frac{L}{4}\right)}^3\\ R_q{}^2=\frac{a^2}{3}\\ R_q=\frac{a}{\sqrt{3}}\end{array}$ …… (10)

Taking the ratio of equation 9 and 10 as,

  $\begin{array}{c}\frac{R_a}{R_q}=\frac{\frac{a}{2}}{\frac{a}{\sqrt{3}}}\\ R_q=\frac{\sqrt{3}}{2}\\ \frac{R_a}{R_q}=0.86\end{array}$

Conclusion:

Thus, the ratio of $R_a$ and the $R_q$ for the saw tooth is $0.86$ .

(c)

To determine

The ratio of the $R_a$ and $R_q$ for square wave.

Answer to Problem 4.58P

The ratio of $R_a$ and $R_q$ for the square wave is $1$ .

Explanation of Solution

Formula used:

The formula for the arithmetic mean value of surface roughness is given as,

  $R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx$ …… (11)

Here, $R_a$ is the arithmetic mean value of the roughness, $L$ is the total measured profile length and $n$ is the number of observations.

The formula for the root mean square value of surface roughness is given as,

  $R_q=\sqrt{\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx}$ …… (12)

Here, $R_q$ is the root mean square value of surface roughness.

The equation of the square wave is given as,

  $y=a$ …… (13)

Here, $a$ is the amplitude of the function.

Calculation:

Thearithmetic mean value $\left(R_a\right)$ of surface roughness is calculated by substituting the value of equation 13 in equation 11 as,

  $\begin{array}{c}{R}_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y\right|}dx\\ R_a=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}a}dx\\ R_a=a\end{array}$ …… (14)

The root mean square value $\left(R_q\right)$ of surface roughness is calculated by substituting the value of equation 13 in the equation 12 as,

  $\begin{array}{c}{R}_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}\left|y^2\right|}dx\\ R_q{}^2=\frac{1}{L}{\displaystyle \underset{0}{\overset{L}{\int }}{\left(a\right)}^2}dx\\ R_q{}^2=a^2\\ R_q=a\end{array}$ …… (15)

Taking the ratio of equation 14 and equation 15 as,

  $\begin{array}{c}\frac{R_a}{R_q}=\frac{a}{a}\\ \frac{R_a}{R_q}=1\end{array}$

Conclusion:

Thus, the ratio of $R_a$ and $R_q$ for the square wave is $1$ .