本次Python代写是完成机械工程中的物理定律计算、铁饼飞行轨道等

MECH2700 Engineering Analysis I (2020)

Assignment 1: Computational Mechanics

Task 1 Read Section 14.6 Numerical Solutions of an old version of Bedford & Fowler’s Engineering

Mechanics text. You may have this book from your Applied Mechanics course or find it

in the library. In case the numbering is different in your copy of the text, Chapter 14 is

labelled “Force, Mass and Acceleration”.

Task 2 Show that the state of motion of the object at t = t0 + ∆t (i.e. a short time after some

arbitrary time t0) can be approximated by

x(t0 + ∆t) = x0 + vx0∆t +

1

2

ax0∆t

2

(3)

y(t0 + ∆t) = y0 + vy0∆t +

1

2

ay0∆t

2

(4)

vx(t0 + ∆t) = vx0 + ax0∆t (5)

vy(t0 + ∆t) = vy0 + ay0∆t (6)

1Possibly because you do not have the forcing functions in analytic form or because they are too complicated

to integrate.

1

where the subscript 0 indicates a value at t = t0.

The simulation

We will begin the simulation of the discus flight from when it leaves the thrower’s hand. We will

only consider motion in the x − y plane, and so we ignore any sidewards curving of the discus

flight. A depiction of the simulation is shown in Figure 1. The thrower converts the energy of

her swing into motion of the discus. A large fraction of the energy goes towards translational

energy of the discus. A small fraction goes towards imparting spin on the discus. The spin gives

the discus stability during its flight. We will be concerned with the translational energy as this

gives an initial velocity, v0, of the discus in its flight.

Figure 1: Cartoon of discus flight. The initial flight path angle, γ0, is the angle between the

horizontal and the direction of discus flight as it leaves the hand.

At t = 0, the discus leaves the thrower’s hand. We will assume that the thrower’s hand is

1.8 m above the ground. The discus leaves with a speed of v0 m/s at a flight-path angle, γ0,

with respect to the horizontal. The women’s discus has a mass of 1 kg.

While in flight, we will consider three forces acting on the discus: self-weight due to the

Earth’s gravity (g = 9.81 m/s); aerodynamic lift; and aerodynamic drag. The magnitude of the

lift force is given by

Flift =

1

2

ρv2ArefCL (7)

where ρ = 1.2 kg/m3

is the density of the air, v is the speed of the discus, Aref is a reference

area for the lift coefficient, and CL is a dimensionless lift coefficient that has been determined

by experiment. Similarly, the magnitude of the drag force is

Fdrag =

1

2

ρv2ArefCD (8)

where CD is a dimenionless drag coefficient, also determined by experiment. If we specify all

of our quantities in metres, kilograms and seconds, then the lift and drag forces will be in

Newtons. The direction of the aerodynamic lift force is perpendicular to the direction of the

discus’ velocity, while the aerodynamic drag force is in the opposite direction to that of the

discus’ velocity. The reference area is taken as the area of ths discus when viewed from above.

For a women’s discus, that is Aref = 0.025 m2

.

In practice, the lift and drag coefficients change during flight as the discus changes its angle

of attack. As a simplifying assumption, we will assume that the discus has a constant angle of

attack of 30◦

.

2 This means that we can take the lift and drag coefficients as constant during the

flight. Those constant values are CL = 0.87 and CD = 0.54.

2Note, do not confuse angle of attack with the flight-path angle. In our model of the flight, angle of attack is

taken as constant, but the flight-path angle will vary.

2

Task 3 Draw a free-body diagram showing the forces acting on the discus while it is in flight.

Task 4 Write expressions for the acceleration components ax and ay. Each of these may be built

up as a set of expressions.

Task 5 Write a Python function that computes the trajectory of the discus once it has left the

thrower’s hands and continues until it hits the ground. Your program should start with

the initial state as discussed above and with x0 = 0.0, y0 = 1.8, v0 = 30 m/s and a guessed

value for γ0. It should then integrate the equations of motion with small increments of

time and save the discus’ position and velocity at discrete points in time. Terminate the

integration process when the discus has hit the ground.

Task 6 For CL = 0.0 and CD = 0.0, integrate the equations of motion analytically to determine an

exact solution for the ideal case of no aerodynamic forces. Demonstrate that your program

agrees with your analytic solution.

Task 7 Determine a time-step that gives a sufficiently accurate estimate of the trajectory for the

case of nonzero aerodynamic forces. Use the time of flight and distance of the throw as

the test criteria. Explain why you consider this to be good enough.

Task 8 Use your program to determine the optimum initial flight angle so that the competitor

can beat the world record by the largest possible margin. Assume they are limited to

a throwing speed of 30 m/s. This could be considered a computational experiment and,

although the search for the optimum initial angle could be automated, doing a manual

search is fine for this exercise.

Task 9 Compute and plot the flight path for your optimum initial angle.

Task 10 The men’s discus has a mass of 2 kg, and is larger in size, so its reference area is 0.038 m2

.

Assume the initial throwing speed is 25 m/s for a top male competitor. Use your program

to determine the optimum initial angle for a competitor in the men’s discus event.

**程序代写代做C/C++/JAVA/安卓/PYTHON/留学生/PHP/APP开发/MATLAB**

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