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I've been studying this topic for a couple of days now. I started by asking some questions regarding output Torque and gear ratios relations. Now I came with the whole picture and some calculations that I been working on. My goal is to find out if I'm in the right direction, and also point-out relations between equations that I'm not sure of. To make this more clear and clean, I divide the system into three sections; A, T and B. enter image description here

In section A its the electric motor and worm/spur torque transmission.

On another topic with a lot of help, I realise that the power provides by the motor is crucial to generate the output torque and angular velocity that your system requires. No matter the radius, ratios or N of teeth of the spur; if the power isn't enough, the gate won't move at the desired velocity.

So, the problem in question: My goal is to find out if this system will be able to move a gate of 600 kg with a velocity of 0.33 m/s, and how much time it will take to reach that velocity.

The technical data of the motor is:

P = 1/2 hp ; RPM = 1450 ; 220V/50Hz

SECTION A:

P = 0.5 hp = 373 (kg * m2 /s3) Pin = Tin * win

win = 1450 rpm = 151.84 (rad/seg)

Tin = 373 (kg * m2 /s3) / 151.84 (rad/seg) = 2.456 N*m

This, of course, is in an ideal situation, without considering losses. (how can I add an estimation of electric motor losses?)

The worm/spur system has a gear ratio of 23:1.

Considering the efficiency of the system of 80%, we have:

Tout = 23 * (0.8 * Pin) / win = 45.2 N*m

Now, Tout also can be calculated by Tout = 0.8 * Pin / wout

angular velocity applied on the spur is: wout = (0.8 * Pin) / Tout

wout = 0.8 * 373 (kg * m2 /s3) / 45.2 N*m = 6.6 rad/sec --> 63 rpm

To consider real circumstances, its the moment of inertia has something to do within this case? I mean, is this Tout enough to accelerate the spur to 63 rpm??

SECTION T:

In this section, I'm wondering what happens with force transmitted in the distance "d" to the pinion. I'm not sure if I need to consider losses in this section.

SECTION B:

Here I have a lot of question about what's happening. The Pinion has 17 teeth and is module 4.

From gear design, I understand that module, and axial pitch determines the Lead and lead angle of the gear. This has a direct relation with the surface contact (involute) in the system and frictional forces.

Is the torque Tout the same as calculated for the worm/spur system? What about the radius of the pinion? If I increase the radius, the Torque will increase as well? The radius of the pinion is = 0.025 (m)

Tout = F * r

F = 45.2 N*m / 0.025 m = 1808 N

Does this mean that the force applied to the rack is 1808 N? Is this force enough to accelerate until 0.33 m/s?

F = m * a = 600 kg * 9.8 m/s2 = 5880 N ; here I need to consider a friction factor according to the wheels and ground. (0.1)

F = 5880 N * 0.1 = 588 N; this means that I need to generate 588 N force to move the gate, but what about the acceleration? if the gate has 4 m, how much time will take to reach the 0.33 m/s

I need a last push to understand this system entirely.

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O.K. After your last question we still have a couple of things to figure out. let's dig into it.

Similarly to my last answer, let's start our analysis at the load section (the gate). Its free body diagram may look as follows:

enter image description here

The pinion, from the rack & pinion mechanism generates the pushing force. As you correctly noticed, there is a friction resisting force. Last time we decided to conservatively define the friction coefficient as 0.1. It means that the total friction force would be 600 [kg] times 9.8 [m/sec^2] * 0.1 = 588 [N]. The net force, i.e. the difference between the pinion pushing force to the friction force is what drives the gate and causes it to accelerate. This difference divided by the mass results in the gate acceleration. For instance, let's say you would like your gate to reach its final velocity (0.33 m/sec) after 4 seconds. Assuming constant acceleration, it would end up in 0.33 [m/sec] over 4 [sec] = 0.0825 [m/sec^2]. The net force for acceleration the gate should be 0.0825 [m/sec^2] times 600 [kg] = 49.5 [N]. This means that the pinion force is 49.5 [N] + 588 [N] = 637.5 [N].

Got it? Don't confuse between Mg which is the weight of the gate to Ma which is the force needed to drive at a acceleration.

Regarding the pinion radius - we already discussed it last time. The rack & pinion mechanism does not multiply the input torque by a factor (like the worm gear), it just converts the torque into a force. This conversion factor is the pinion radius.

Regarding the inertia - It should be consider only when dealing with acceleration. Your constant speed calculation shall not take it into account. I guess the pinion inertia would be negligible relatively to the reflected gate inertia. Try to compare between the two after you get all the basic ideas.

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  • $\begingroup$ Got it! I still have the last doubt, though. In the calculations convert the torque output from the reduction system, which is 45.2N*m and divided by the pinion radius, which is 0.025m. That gives me 1808N, Is this the tangential force of the pinion, which is transmitted to the rack? What's the meaning of that? You assumed 4 sec to achieve the 0.33 m/sec. If I somehow determined the pinion force to calculate the acceleration of the gate, and then the time at which the gate will achieve the 0.33 m/s. Is this 1808 N the tangential force to do that calculation? $\endgroup$ – Joaquin Osses Aug 2 '19 at 14:47
  • $\begingroup$ Correct. Take this force, subtract the friction force and then divide the result by the gate mass to determine the acceleration. $\endgroup$ – Yaniv Ben David Aug 2 '19 at 15:25
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  1. This, of course, is in an ideal situation, without considering losses. (how can I add an estimation of electric motor losses?)

Firstly you need to specify the motor output torque according to the motor output speed. In order to do that you need to calculate the performance of AC Motor:

Performance of the AC motor is variable. It depends on

  • full load speed (rpm),

  • full load torque (N*m),

  • phase type,

  • current output speed.

In order to determine the performance of an AC motor you usually need a torque-speed curve diagram.

In your case,

  • synchronous speed: 1450 rpm

  • full load torque: 2.456 N*m

  • 220V/50 Hz is a typical single phase AC motor. But you also clarify the type of this single phase AC motor according to your design specifications. (These types are: Shaded pole, permanent-split capacitor, split-phase and capacitor-start). Each is unique in its physical construction and provides different starting and running characteristics. This is an example diagram which indicates torque-speed curves.

Torque-Speed curves of different types of single phase AC motors

In order to get full load torque from the motor, roughly you run the motor 85% of its maximum speed. This is approximately 1200 rpm as motor output speed.

  1. To consider real circumstances, its the moment of inertia has something to do within this case? I mean, is this Tout enough to accelerate the spur to 63 rpm??

The important thing in order to run a mechanism at the first start not the inertia. It is static friction. For example: If you have zero-close static friction and high inertia for a machine, you need to wait longer in order to achieve required speed. In opposite, if you have high static friction and low inertia, you need to wait longer in order to first start but after that it starts to accelerate quickly. So, if you think your motor is weak for the first motion, using a capacitor may be a good idea. But in my opinion, it doesn't seem you face such a problem.

  1. In this section, I'm wondering what happens with force transmitted in the distance "d" to the pinion. I'm not sure if I need to consider losses in this section.

In this section T, you need to determine the specifications of bearings which holds the shaft. Only loss comes from bearings. But this loss quite negligible with respect to the gear meshes. Nevertheless, if you want to add this loss accurately, you can look at this website: https://www.amroll.com/friction-frequency-factors.html. The safest way in here is to consult the manufacturers for your design.

Other than these, your calculations seem correct.

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  • $\begingroup$ Thanks, Samet for your post, helps me a lot to understand the mechanism. In section T, when you consider that the friction losses from bearings are negligible respects to the gear meshes, I find 80% efficiency for the worm/spur mesh. Is it accurate to consider the efficiency % of the mesh as friction losses forces? Or do I need to take into account other parameters? $\endgroup$ – Joaquin Osses Aug 1 '19 at 18:03
  • $\begingroup$ Bearings are high efficiency mechanical elements. So, you don't need to deal with it. $\endgroup$ – Samet Baykul Aug 1 '19 at 19:00

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