Mathematics For Machine Technology
Mathematics For Machine Technology
8th Edition
ISBN: 9781337798310
Author: Peterson, John.
Publisher: Cengage Learning,
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Textbook Question
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Chapter 21, Problem 16A

Figure 21−9 shows a compound gear train. Gears B and C are keyed to the same shaft; therefore, they turn at the same speed. Gear A and gear C are driving gears. Gear B and gear D are driven gears. Set up a proportion for each problem and determine the unknown values, x, y, and z in the table. Round the answers to 1 decimal place where necessary.

Chapter 21, Problem 16A, Figure 219 shows a compound gear train. Gears B and C are keyed to the same shaft; therefore, they

Expert Solution
Check Mark
To determine

(a)

To find the speed of the different gears given in the problem.

Answer to Problem 16A

Speed of gear B is x= 320 rpm.

Speed of gear C is y= 320 rpm.

Speed of gear D is z= 800 rpm.

Explanation of Solution

Given information:

A gear and pinion arrangement is given as below.

  Mathematics For Machine Technology, Chapter 21, Problem 16A , additional homework tip  1

Number of teeth on gear A=80, Number of teeth on gear B=30,Number of teeth on gear C=50, Number of teeth on gear D=20, Y1 speed of gear A=120.0 rpm, speed of gear B=xrpm,speed of gear C=yrpm,speed of gear D=zrpm,

Calculation:

We have been given below information,

  Number of teeth on gear A=80, Number of teeth on gear B=30,Number of teeth on gear C=50, Number of teeth on gear D=20, y speed of gear A=120.0 rpm, speed of gear B=xrpm,speed of gear C=yrpm,speed of gear D=zrpm,

As we know that in a gear arrangement,

13 Teeth in driving gear Teeth in driven gear =Revolutions in driven gear Revolution in driving gear

So,

  Teeth in driving gear A Teeth in driven gear B=Revolutions in driven gear B Revolution in driving gearA8030=x rpm120.0Applying cross multiplication,30x=80×120.0 rpmx=80×120.030rpmx=320 rpm

Hence, speed of gear B is

  320 rpm.

Since, gear B and C are on same shaft. So, speed of both gear is equal.

Therefore, speed of gear C is y= 320 rpm.

Again,

  Teeth in driving gear C Teeth in driven gear D=Revolutions in driven gear D Revolution in driving gearC5020=z rpm320.0Applying cross multiplication,20z=50×320.0 rpmz=50×320.020rpmz=800.0 rpm

Hence, speed of gear D is 800 rpm.

Thus, speed of gear B is x= 320 rpm.

Speed of gear C is y= 320 rpm.

Speed of gear D is z= 800 rpm.

Expert Solution
Check Mark
To determine

(b)

To find the number of teeth in the gear and the speed of the gear.

Answer to Problem 16A

Number of teeth in gear B is x= 20.

Number of teeth in gear D is y= 30.

Speed of gear C is z= 300.0 rpm.

Explanation of Solution

Given information:

A gear and pinion arrangement is given as below.

  Mathematics For Machine Technology, Chapter 21, Problem 16A , additional homework tip  2

Number of teeth on gear A=60, Number of teeth on gear B=x,Number of teeth on gear C=45, Number of teeth on gear D=y, speed of gear A=100.0 rpm, speed of gear B=300.0rpm,speed of gear C=zrpm,speed of gear D=450.0rpm,

Calculation:

We have been given below information,

  Number of teeth on gear A=60, Number of teeth on gear B=x,Number of teeth on gear C=45, Number of teeth on gear D=y, speed of gear A=100.0 rpm, speed of gear B=300.0rpm,speed of gear C=zrpm,speed of gear D=450.0rpm,

As we know that in a gear arrangement,

  Teeth in driving gear Teeth in driven gear =Revolutions in driven gear Revolution in driving gear

So,

  Teeth in driving gear A Teeth in driven gear B=Revolutions in driven gear B Revolution in driving gearA60x=300.0 100.0Applying cross multiplication,300.0x=60×100.0x=60×100.0300.0x=20

Hence, number of teeth in gear B is x= 20.

Since, gear B and C are on same shaft. So, speed of both gear is equal.

Therefore, speed of gear C is z= 300.0 rpm.

Again,

  Teeth in driving gear C Teeth in driven gear D=Revolutions in driven gear D Revolution in driving gearC45y=450.0300.0Applying cross multiplication,450.0y=45×300.0y=45×300.0450.0y=30

Hence, number of teeth in gear D is y= 30.

Thus, number of teeth in gear B is x= 20.

Number of teeth in gear D is y= 30.

Speed of gear C is z= 300.0 rpm.

Expert Solution
Check Mark
To determine

(c)

To find the number of teeth as well as speed of the gear.

Answer to Problem 16A

Number of teeth in gear A is x= 20.57.

Speed of gear B is y= 168.0 rpm.

Speed of gear C is z= 168.0 rpm.

Explanation of Solution

Given information:

A gear and pinion arrangement is given as below.

  Mathematics For Machine Technology, Chapter 21, Problem 16A , additional homework tip  3

Number of teeth on gear A=x, Number of teeth on gear B=24,Number of teeth on gear C=60, Number of teeth on gear D=36, speed of gear A=144.0 rpm, speed of gear B=yrpm,speed of gear C=zrpm,speed of gear D=280.0rpm,

Calculation:

We have been given below information,

  Number of teeth on gear A=x, Number of teeth on gear B=24,Number of teeth on gear C=60, Number of teeth on gear D=36, speed of gear A=144.0 rpm, speed of gear B=yrpm,speed of gear C=zrpm,speed of gear D=280.0rpm,

As we know that in a gear arrangement,

  Teeth in driving gear Teeth in driven gear =Revolutions in driven gear Revolution in driving gear

So,

  Teeth in driving gear A Teeth in driven gear B=Revolutions in driven gear B Revolution in driving gearAx24=144.0 yApplying cross multiplication,xy=24×144.0      ...eq(i)

Since, gear B and C are on same shaft. So, speed of both gear is equal. i.e. y=z.

Again,

  Teeth in driving gear C Teeth in driven gear D=Revolutions in driven gear D Revolution in driving gearC6036=280.0zApplying cross multiplication,60z=36×280.0z=36×280.060z=168

Hence, speed of gear B and gear C is 168 rpm. i.e. y=168 and z=168.

Putting, y=168 in eq(i),

  xy=24×144.0168x=24×144.0       {Put y=168}x=24×144.0168x=20.57

Thus, number of teeth in gear A is x= 20.57.

Speed of gear B is y= 168.0 rpm.

Speed of gear C is z= 168.0 rpm.

Expert Solution
Check Mark
To determine

(d)

To find the speed of the different gears and the number of teeth.

Answer to Problem 16A

Number of teeth in gear C is x= 30.

Speed of gear A is y=79.55 rpm.

Speed of gear B is z=175.0 rpm.

Explanation of Solution

Given information:

A gear and pinion arrangement is given as below.

  Mathematics For Machine Technology, Chapter 21, Problem 16A , additional homework tip  4

Number of teeth on gear A=55, Number of teeth on gear B=25,Number of teeth on gear C=x, Number of teeth on gear D=15, speed of gear A=y rpm, speed of gear B=zrpm,speed of gear C=175.0rpm,speed of gear D=350.0rpm,

Calculation:

We have been given below information,

  Number of teeth on gear A=55, Number of teeth on gear B=25,Number of teeth on gear C=x, Number of teeth on gear D=15, speed of gear A=y rpm, speed of gear B=zrpm,speed of gear C=175.0rpm,speed of gear D=350.0rpm,

As we know that in a gear arrangement,

  Teeth in driving gear Teeth in driven gear =Revolutions in driven gear Revolution in driving gear

So,

  Teeth in driving gear A Teeth in driven gear B=Revolutions in driven gear B Revolution in driving gearA5525=z yApplying cross multiplication,55y=25z      ...eq(i)

Since, gear B and C are on same shaft. So, speed of both gear is equal. i.e. z=175.0.

Putting x=175.0 in above eq(i),

  55y=25×175.0y=25×175.055y=79.55

Hence, speed of gear A is y=79.55 rpm.

Again,

  Teeth in driving gear C Teeth in driven gear D=Revolutions in driven gear D Revolution in driving gearCx15=350.0175.0Applying cross multiplication,175.0x=15×350.0x=15×350.0175.0x=30

Hence, number of teeth in gear C is x= 30.

Thus, number of teeth in gear C is x= 30.

Speed of gear A is y=79.55 rpm.

Speed of gear B is z=175.0 rpm.

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