A punching machine carries out of 5 holes per minute each hole of 30 mm diameter in 40 mm thick plate requires 10 Nm of energy/mm2 of the sheared area.The punch has a stroke of 90 mm.Find the power of the motor required if the mean speed of the fly wheel is 20 m/sec.If the total fluctuation of the speed is not to exceed 2 % of the mean speed,determine the mass of the fly wheel.
Well, this is quite a math-heavy question! It seems like you're dealing with some engineering calculations. Let me put it in a way that's a little more fun:
So, we have a punching machine that's like a superhero, punching holes through a plate with its mighty fists! The machine can punch out 5 holes per minute, and each hole has a diameter of 30mm. That's some serious power!
Now, punching through a thick plate requires energy. In this case, it's 10 Nm of energy per mm2 of the sheared area. That's a lot of power packed into a small space!
The punch has a stroke of 90 mm, which means it can travel 90 mm with each punch. That's quite a workout for our superhero machine!
To determine the power of the motor required, we need to consider the mean speed of the flywheel. If the mean speed is 20 m/s, that's a pretty fast flywheel! We want to make sure the total fluctuation of the speed doesn't exceed 2% of the mean speed. We don't want our punching machine going off balance!
Now, to determine the mass of the flywheel, we need to know a little more. Is it a lightweight flywheel, or a heavyweight champion? The mass of the flywheel will depend on its size and material.
In conclusion, this punching machine needs a powerful motor to keep its flywheel spinning at a mean speed of 20 m/s. And we need to make sure the flywheel's mass is just right to prevent any wobbling. It's a balancing act, just like life!
To find the power of the motor required, we need to calculate the total energy required to punch all the holes in one minute and then convert it into power. Let's break down the steps:
Step 1: Calculate the number of holes punched in one minute:
Given: A punching machine carries out 5 holes per minute.
Number of holes punched in one minute = 5 holes.
Step 2: Calculate the total sheared area:
Given: Diameter of each hole = 30 mm.
Given: Thickness of the plate = 40 mm.
Area of each hole = (π/4) × (diameter)^2
= (π/4) × (30 mm)^2
= 706.86 mm^2
Total sheared area in one minute = Number of holes × Area of each hole
= 5 holes × 706.86 mm^2
= 3534.3 mm^2
Step 3: Calculate the total energy required:
Given: Energy required per mm^2 of the sheared area = 10 Nm/mm^2
Total energy required in one minute = Energy required per mm^2 × Total sheared area
= 10 Nm/mm^2 × 3534.3 mm^2
= 35343 Nm
Step 4: Calculate the power of the motor required:
Given: Mean speed of the flywheel = 20 m/s
Power = Total energy / Time
The time required to punch one hole can be calculated using the mean speed of the flywheel and the stroke of the punch:
Given: Stroke of the punch = 90 mm
Time to punch one hole = Stroke / Mean speed
= 90 mm / 20 m/s
= 0.0045 s
Total time required to punch 5 holes = Time to punch one hole × Number of holes
= 0.0045 s × 5 holes
= 0.0225 s
Power = Total energy / Total time
= 35343 Nm / 0.0225 s
= 1,569,200 W
Therefore, the power of the motor required is 1,569,200 W.
Step 5: Determine the mass of the flywheel:
Given: Total fluctuation of the speed should not exceed 2% of the mean speed.
Total fluctuation of the speed = 2% of the mean speed
= 0.02 × 20 m/s
= 0.4 m/s
The formula to calculate the kinetic energy of the flywheel is:
Kinetic energy = (1/2) × (mass of the flywheel) × (fluctuation speed)^2
Let's assume the mass of the flywheel as 'm' kg.
Kinetic energy = (1/2) × m × (0.4 m/s)^2
Since kinetic energy is equal to the total energy required to punch the holes:
35343 Nm = (1/2) × m × (0.4 m/s)^2
Solving this equation for 'm', we can find the mass of the flywheel.
I hope this helps! Let me know if you need further assistance.
To find the power of the motor required, we need to first calculate the energy required to punch each hole and then determine the total energy required per minute.
Step 1: Calculate the energy required to punch each hole:
The area of each hole can be calculated using the formula for the area of a circle: A = πr², where r is the radius (diameter/2).
Given that the diameter of the hole is 30 mm, the radius (r) is 30 mm/2 = 15 mm = 0.015 m.
So, the area of each hole is A = π(0.015)² = 0.00070686 m².
The energy required for 1 mm² of sheared area is given as 10 Nm/mm².
Therefore, the energy required to punch each hole is 0.00070686 m² × 10 Nm/mm² = 0.0070686 Nm.
Step 2: Calculate the total energy required per minute:
The punching machine carries out 5 holes per minute. So, the total energy required per minute is 5 holes × 0.0070686 Nm/hole = 0.035343 Nm.
Step 3: Determine the power of the motor required:
Power is defined as the amount of work done or energy transferred per unit time. So, the power required can be calculated using the formula: Power (P) = Energy (E) / time (t).
Given that the mean speed of the flywheel is 20 m/sec, we can calculate the time it takes for one revolution of the flywheel using the formula: time = Distance (d) / Speed (v).
The distance covered in one revolution of the flywheel is equal to the circumference of the flywheel, which can be calculated using the formula: Circumference = π × diameter.
Given that the punch has a stroke of 90 mm, the diameter of the flywheel is twice the stroke = 90 mm × 2 = 180 mm = 0.18 m.
So, the circumference of the flywheel is 0.18 m × π = 0.56548 m.
Now, we can calculate the time taken for one revolution as follows: time = 0.56548 m / 20 m/sec = 0.028274 s.
Using this time value, we can calculate the power required: Power = Energy / time = 0.035343 Nm / 0.028274 s = 1.2499 W.
Step 4: Determine the mass of the flywheel:
Given that the total fluctuation of the speed should not exceed 2% of the mean speed, we can calculate the maximum fluctuation in speed.
The maximum fluctuation in speed is 2% of the mean speed = 0.02 × 20 m/sec = 0.4 m/sec.
The maximum fluctuation in speed can be related to the kinetic energy of the flywheel using the formula: ΔK.E = 0.5 × I × (Δω)², where I is the moment of inertia of the flywheel and Δω is the change in angular velocity.
Since the change in speed is Δω = 0.4 m/sec, we can assume that the angular velocity is ω = 2πf, where f is the frequency of revolution.
Since one revolution takes 0.028274 seconds (as calculated earlier), the frequency is f = 1 / (0.028274 s) = 35.371 Hz.
So, the angular velocity is ω = 2π × 35.371 Hz = 222.666 rad/sec.
Rearranging the formula, we can calculate the moment of inertia of the flywheel: I = (ΔK.E) / (0.5 × (Δω)²).
Given that the change in kinetic energy ΔK.E = 1.2499 W × 1 s = 1.2499 J (since power = energy / time), we can calculate the moment of inertia: I = 1.2499 J / (0.5 × (0.4 m/sec)²) = 7.81125 kg.m².
The moment of inertia of a flywheel can be calculated using the formula: I = 0.5 × mass × radius², where mass is the mass of the flywheel and radius is the radius of the flywheel.
Since the diameter of the flywheel is 0.18 m, the radius is 0.18 m / 2 = 0.09 m.
Substituting the values into the formula, we can solve for the mass of the flywheel: 7.81125 kg.m² = 0.5 × mass × (0.09 m)².
Simplifying the equation, we have: mass = (7.81125 kg.m²) / (0.5 × (0.09 m)²) = 193.344 kg.
Therefore, the power of the motor required is 1.2499 W, and the mass of the flywheel is 193.344 kg.