You work as a Test Engineer for a global manufacturer of electrical and mechanical components and systems. Your Line Manager is responsible for delegating to you and your colleagues the testing of theory, princ

Task 1

Note: You need to have your calculator in radians (RAD) mode for this task (since the angles are given in radians – i.e., π is featured).

1. A current waveform in a robotic arm may be described by…

π

is = 13 cos (2πft − 4)  [𝐴]

where frequency, f = 1Hz and t represents time.

Make time (t) the subject of this formula and hence determine a point in time when the current waveform has a magnitude of +10A.

2. The instantaneous value of a power signal may be described by;

3π

12∠ ( )  [W]

8

Find the magnitude of the vertical and horizontal components of this signal. Note that the symbol ∠ indicates ‘angle’.

Task 2

1. A resistor, R, is connected in series with an inductor, L. An a.c. current, 𝑖, flows through this RL combination, causing a voltage (𝑉𝑅 ) of 30V to be developed across the resistor, and a voltage (𝑉𝐿) of 40V to be developed across the inductor.

Assuming that 𝑉𝑅 is in-phase with 𝑖, and 𝑉𝐿 leads 𝑉𝑅 by 90𝑜, draw a vector diagram for this arrangement and then calculate the magnitude of the resultant voltage across the whole RL combination.

2. A current-carrying filament is subjected to a strong magnetic field within an experimental chamber. It is required to find the force on the filament. The current may be modelled in three-dimensional space as:

𝐼 = 2𝑖 + 3𝑗𝑗 − 4𝑘

and the magnetic field as:

𝐵 = 3𝑖 − 2𝑗𝑗 + 6𝑘

Find the Cross Product of these two vectors to ascertain the characteristics of the force on the filament (i.e., find 𝐼 × 𝐵).

Sketch this Cross Product (or use software to do so).

Task 3

PART 1

The two signals below are sensed by a signal processor;

𝑣1 = 40 sin(4𝑡)

𝑣2 = 𝐴 cos(4𝑡)

The signal processor adds the signals to form a third signal, which must be described as a distinct signal in the following form;

𝑣𝑜 = 50 sin(4𝑡 + 𝛼)

Use a compound angle identity to determine the value of A (the amplitude of 𝑣2).

Ensure that you have your calculator in Radians (RAD) mode when determining your answer.

PART 2

The third harmonic of a sound wave is given by;

4 cos(3𝜃𝜃) − 6 sin(3𝜃𝜃)

Express this sound wave in the form;

𝑅 sin(3𝜃𝜃 + 𝛽)

Task 4

1. Use Graph software to plot the current signal given in Task 1 (a). Provide a screenshot of your waveform.

2. Use Graph software to plot the two voltage signals (𝑣1 𝑎𝑛𝑑 𝑣2) given in Task 3 Part 1. Provide a screenshot of your waveforms.

3. Use Geogebra to plot the vector given in Task 1 (b). Provide a screenshot of this vector function.

Sources of information to support you with this Assignment

Bird J. (2021) Higher Engineering Mathematics. 9th Ed. Routledge.

Bird J. (2019) Science and Mathematics for Engineering. 6th Ed. Routledge.

Glyn J. and Dyke P. (2020) Modern Engineering Mathematics. 6th edition. Pearson. Made Easy Editorial Board (2022) Engineering Mathematics for GATE 2023 and ESE 2023 (Prelims) – Theory and Previous Year Solved Papers. India: Made Easy Publications

Pvt Ltd.

Rattan K.S., Klingbeil N.W., and Baudendistel C.M. (2021) Introductory Mathematics for Engineering Applications. 2nd Ed. Wiley.

Ram M. (2021) Recent Advances in Mathematics for Engineering. CRC Press. Teodorescu P., Stanescu N., and Pandrea N. (2013) Numerical Analysis with Applications in Mechanics and Engineering. Wiley-IEEE Press.

Ram M. (2020) Mathematics in Engineering Sciences: Novel Theories, Technologies, and Applications. 1st Edition. CRC Press.

Sobot, R. (2022) Engineering Mathematics by Example. 1st Ed. Springer.

Stroud, K.A. and Booth, D.J. (2020) Engineering Mathematics. 8th Ed. Bloomsbury Publishing

Urbano M. (2019) Introductory Electrical Engineering with Math Explained in Accessible Language. Wiley.

Vick B. (2020) Applied Engineering Mathematics. CRC Press.

Revant Learning Outcomes and Assessment Criteria

Pass

Merit

Distinction

L03

Use analytical and computational methods for solving engineering and manufacturing sector problems by relating sinusoidal wave and vector functions to their respective applications

P6

Solve engineering/ manufacturing problems relating to sinusoidal functions.

Use compound angle identities to combine individual sine waves into a single wave, and illustrate graphically.

Apply engineering mathematical software to confirm the analytical solutions for at least three engineering/ manufacturing problems involving sinusoidal and vector functions.