ME 4053 Mechanical Engineering Computation Fall 2022 Homework #4

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ME 4053

Mechanical Engineering Computation

Fall 2022

Homework #4

Kineto-Static Analysis of a Snow Thrower Engine

(due 5 PM Tuesday 11/29/22)

Executive Summary

A slider-crank linkage used in a snow thrower engine is illustrated on the back of this page.  A lywheel is rigidly attached to the crank link (link 2). The crank and lywheel are balanced, so their center of gravity falls on top of ground pivot OA . The crank rotates at a nearly constant angular velocity of 4000 RPM1 . The critical dimensions and mass properties of the piston, the connecting rod (coupler link), and the crank and lywheel assembly are provided below:

OA —   A = 20 mm    OA —   G2  = 0 mm   m2 = 1.5 kg     IG2  = 4.00x10—3  kg-m2

A —   B = 80 mm      A —    G3  = 33 mm    m3 = 0.11 kg   IG3  = 0.134x10—3  kg-m2

B —    G4  = 16 mm    m4 = 0.13 kg

The force applied to the top of the piston by the gas contained in the piston cylinder is FP . This force varies greatly, depending on the piston’s position and whether the piston is on its power stroke or its purge stroke. FP  does not include the inertia force attributable to the piston acceleration2 . You may disregard the friction force between the piston and the cylinder wall in the scope of this assignment.

Please calculate the torque at the crank, T2, and the forces on pins“OA”,“A”and“B”for a crank angle,  2 , and a piston force, FP, assigned to you on a separate list posted on Canvas3 .

Note  1 : The side force, Fside, balances the x —force applied to the piston by pin B .

Note 2 : The equations for the angular acceleration of the connecting rod, α3 , and the magnitude of the linear acceleration of the slider, , given the angular velocity, ω2 , and angular acceleration, α2 , of the crank, for an in-line slider-crank, are:

D [ω32 sin(2 —    S ) —   α2 cos(2 —    S )]+ Cω sin(3 —    S )

C cos(3 —    S )

 = —    D [ω22 cos(2 —    3 ) + α2 sin(2 —    3 )]+ Cω cos(3 —    S )

where C is the length of coupler (connecting rod), D is the length of the crank,  2  is the angle of the crank,  3  is the angle of the coupler,  S  is the angle of the in-line slider, and ω3  is the angular velocity of the coupler.

FP

1

G4

Fside

A

2

G2

 

(all dimensions in mm)

Assignment Guidelines & Requirements

1. This is an individual assignment, not a team assignment.

2. Please write a MATLAB program to calculate the requested forces and torque. Please use a top-down design chart and input/output charts to properly structure your program. Please document your program to comply with the class documentation standards.

3. You need only compute the forces and torque for the single value of crank angle and piston force assigned to you.  However, please write your functions so that you can compute the forces and torque for any crank angle and piston force.  Structuring your functions in this manner will enable you to carry these functions over for use in Project 3.

4. Please print the calculated forces and torque to the MATLAB command window.  Please state each forces as a magnitude and an angle (in degrees), instead of x —    and y —components. Label each force and the torque so that the grader can clearly interpret them.

5. Please draw free body diagrams of the crank, connecting rod and piston. Show the derivation of the dynamic equations implemented in your program from the forces illustrated on the free body diagrams.

6. The mechanism is assumed to operate in the vertical plane, so please consider the weight of the links in your analysis.

7. We recommend using the matrix left-division capability of MATLAB to solve the set of simultaneous linear equations resulting from your dynamic analysis.

8. Please validate your MATLAB analysis with the following hand computations:

(a) Compute the angular velocity of the connecting rod (link 3) and the velocity of the piston (link 4).  You may use either graphical velocity analysis or apply the analytical equations for an in-line slider-crank.

(b) Compute the acceleration of the center of gravity of the connecting rod, the angular acceleration of the connecting rod, and the acceleration of the center of gravity of the piston. You may use either graphical acceleration analysis or apply the analytical equa- tions for an in-line slider-crank.

(c) Manually sum all forces applied to the connecting rod and check that the resultant equals the mass times the acceleration of the center of gravity of the connecting rod. (You may sum the forces either graphically or analytically.)

(d) Manually sum the torques created by all forces applied to the connecting rod about its center of gravity and check that the resultant torque equals the mass moment of inertia times the angular acceleration of the connecting rod.

(e) Manually sum all forces applied to the piston and check that the resultant equals the mass times the acceleration of the piston.  (You may sum the forces either graphically or analytically.)

9. Please package your work for Items 5 and 8 into a single .pdf ile.

10. Please combine the MATLAB code for your main program and all of your functions into a single“ .m”ile for grading.

Note that while it is generally preferable to keep each MATLAB function in its own“.m”ile, this makes uploading your program for grading awkward.

We will provide two links for uploading your assignment to Canvas.  The irst will be for the “ .m”ile containing your MATLAB code. That link will be through Gradescope. The second will be for submitting your documentation for Items 5 and 8. That link will be in Canvas.


2022-11-28

Kineto-Static Analysis of a Snow Thrower Engine