Please complete the following tasks to obtain the equations

Question

Project Description-

The purpose of this project is for you to develop an algorithm that can be used to estimate the payload (amount of material) for an electric mining shovel. An electric mining shovel is a large digging machine that is used to mine the world's commodities (copper, gold, coal, etc.) A picture of an electric mining shovel is shown in Figure 1. The mining shovel has a dipper (Fig. 1) that holds all of the required material. Your job will be to determine how much material is in the dipper at a given instant in time, based on the equilibrium equations you have learned in statics. There are two main motions that allow this machine to operate properly. There is a crowd motion that is used to move the handle in and out based on a rack and pinion system. There is also a hoist motion which is controlled by a pulley system. The hoist ropes are connected directly to the dipper through the bail pin.

Shown in Fig. 2 is a dipper/handle combination. There is an assumed handle center of mass. There is also an assumed dipper/payload center of mass. The rack and pinion system is also shown. The saddle pinion is controlled by a large motor which causes the pinion to rotate and there the handle as a result of the rack and pinion system.

In your project, you will assume that the system geometry, hoist force, and crowd force are all known from sensors that are on the shovel. Your job will be to then back calculate the payload in the dipper given the known geometry as well as hoist and crowd forces.

The system geometry is shown below:

Assumptions

In any mathematical derivation, it is important to specify all assumptions. We will assume the following.

1. All bodies are rigid.
2. Gravity is included.
3. There exists no friction in the joints.
4. There is no slacking in the hoist ropes.

Project Tasks

Please complete the following tasks to obtain the equations of motion of this machine.

1. Draw a free body diagram of the handle/dipper combination. You may assume that this is one rigid body.

2. Assign a coordinate system of your choosing.

3. Write out the appropriate equilibrium equations.

4. Solve for the value of the unknown mass of material of the dipper in terms of the known values.

m_payload= ??(F_h????????,??₁,??₂,??₃,??₄,??₅,??₆,??₇, ??, m_????????????, m_h??????????, ??, ??)

Note that m_?????? = m_payload+ m_dipper.

Please also note that the payload value is not dependent on the crowd force.