notes on strain energy

uni/mmme/2047_thermodynamics_and_fluid_dynamics/gas_mixtures.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2047 // Gas Mixtures and Combustion
+tags: [ combution, gas_mixtures ]
+uuid: 5ed55ce0-e6b1-423c-90ac-96e1c8cbe43f
+---
+
+# Law of Partial _____
+
+Given that there are $i$ gas components completely mixed and occupying a
+volume $V$ at temperature $T$.
+
+## Pressure
+
+$$p = \sum_i p_i$$
+
+given that the volume $p$ and all $p_i$ are acting in are equal.
+
+## Internal Energy
+
+$$U = \sum_i U_i$$
+
+## Volumes
+
+$$V = \sum_i V_i$$
+
+given that the pressures of $V$ and all values of $V_i$ are equal.
+
+This is useful as $V \propto n$ meaning that we can use volumes to
+achieve stoichiometric (ideal) mixtures of gases for combustion reactions.
+
+# Avogadro's Number and the Mole
+
+Avogadro's Number is roughly $6.022\times10^{23}$.
+The mass in grams of this many particles of an element is pretty much equal to:
+
+- its atomic mass number
+- the number of nucleons in the nucleus

uni/mmme/2049_engineering_management_1/prviate_enterprises.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2049 // Private Enterprises
+tags: [ business ]
+uuid: 1b37fd6a-5255-43a5-b19f-a427880b46aa
+lecture_slides: ./lecture_slides/P002_Presentation_2023.pdf
+---
+
+# What is a Private Enterprise
+
+- fancy word for 'business'
+- they are a means to 'create' wealth
+- it's often helpful to distinguish between manufacturing and service industries, but not always
+  possible
+
+  - manufacturing businesses bring in materials, add value, and sell physical goods at a profit
+  - service enterprises are like hairdressers, bank, and taxis where they do something for you
+
+# Business Models
+
+> a plan for the successful operation of a business,
+> identifying sources of revenue, the intended customer base, products [or services], and details of
+> finance
+
+# Definitions
+
+- sales = financial inflow due to sales
+- cost of goods sold (COGS) = input cost + process cost
+
+  where input cost = cost of raw materials; process cost = cost of converting input into sold goods
+
+- gross profit = sales - cogs
+- added value = sales - input
+- operating profit = gross profit - operating costs
+
+![](./images/vimscrot-2023-02-08T18:02:49,145102193+00:00.png)
+
+# Prices, Costs
+
+$$\text{price} \neq \text{cost}$$
+
+- cost is how much needs to be given up to obtain something
+- price is the amount the market is prepared to pay for it and is determined by interaction between
+  buyer and seller
+- sometimes price and cost are entirely unrelated
+- price is not always greater than cost
+
+
+# This Diagram of 'the financial structure of a business' Doesn't Make Sense to Me
+
+![](./images/vimscrot-2023-02-08T18:02:12,873108922+00:00.png)
+
+# The Value Chain
+
+> A value chain is the set of activities that a business performs with the objective of delivering a
+> valuable product, which can be a good, a service, or both, to its customers.
+
+~ Michael Porter
+
+![](./images/vimscrot-2023-02-08T18:04:39,286950278+00:00.png)
+
+# Legal Form
+
+- every business must select a legal structure to follow - this is the legal form of ownership
+- this decision is made by the founders before the business operates but it can be changed later
+- it determines how resources are structured, management roles are administered, taxes are paid, and
+  financial information is reported
+- legal forms vary from country to country
+
+## Non-exhaustive List of Types
+
+- sole trader
+- partnership
+- limited company
+
+  - private limited company
+  - public limited company
+
+  - day to day management is separated from ownership
+  - liability is limited to the money the owners put in (shareholders are not liable for company debt)
+
+- and more!
+
+## Shares
+
+Limited companies must have at least one owner who owns a fraction of the company called a share.
+Collectively, shares are also known as equity.
+
+This normally gives shareholders the ability to vote on important decision which affect the business.
+
+# Businesses as Organisation of People
+
+- every business is also an organisation comprising people to achieve a collective goal (make the
+  shareholders richer)
+- this perspective of businesses deals with who is and isn't in the organisation and who has authority
+- generally as business gets older it gets bigger:
+
+  - micro --- single individual or small group based locally
+  - small --- has a small management team working nationally
+  - medium --- central organisation with limbs and possibly international
+  - large --- global with large, semi-autonomous groups
+
+  - the micro, small, and medium sizes are known a SMEs (<250 people)
+
+# Lines of Reporting
+
+![](./images/vimscrot-2023-02-08T18:16:39,109473929+00:00.png)
+
+Companies may also be organised by product divisions:
+
+![](./images/vimscrot-2023-02-08T18:17:40,633851867+00:00.png)
+
+# What Do Managers DO
+
+Two types of managers are needed:
+
+- Strategists ---- set objectives and overall strategy
+- Doers --- manage resources to achieve objects and implement strategy
+
+## Management By Objective (MBO)
+
+MBO is a strategic model in which managers and employees agree to a set of objective.
+
+Peter Drucker proposed the eight criteria:
+
+1. market standing
+1. innovation --- there are costs and risks but lack of innovation creates opportunity for competitors
+1. productivity --- all assets must contribute to productivity:
+
+   - labour sales productivity = sales revenue / number of employees
+   - labour profit productivity = operating profit / number of emloyees
+
+1. physical and financial reserves
+1. profitability
+1. worker performance and attitude
+1. manager performance and attitude
+1. public responsibility
+
+## General objectives
+
+- continuous growth
+- profits
+- gain market share
+- show dividend for shareholders
+
+## Market Standing
+
+- a basic measure of this would the market share
+- price competitions squeeze small companies out
+- small companies win in niches that larger companies overlook
+- restrictive regulation
+- complacency
+- inability to react to technological changes
+
+## Physical and Financial Reserves
+
+- possession of necessary resources
+
+
+i give up just read the lecture slides lol

uni/mmme/2053_mechanics_of_solids/strain_energy.md

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+---
+author: Akbar Rahman
+date: \today
+title: MMME2053 // Strain Energy
+tags: [ strain_energy ]
+uuid: 9cbe6fab-c733-45d0-b8e7-b3d949b3eeb7
+lecture_slides: [ ./lecture_slides/MMME2053 SE L1 Slides.pdf, ./lecture_slides/MMME2053 SE L2 Slides.pdf, ./lecture_slides/MMME2053 SE L3 Slides.pdf ]
+lecture_notes: [ ./lecture_notes/Strain Energy Methods Notes.pdf ]
+worked_examples: [ ./worked_examples/MMME2053 SE WE Slides.pdf ]
+exercise_sheets: [ ./exercise_sheets/Strain Energy Methods Exercise Sheet.pdf, ./exercise_sheets/Strain Energy Methods Exercise Sheet Solutions.pdf ]
+---
+
+# Strain Energy Definition
+
+This section refers to the first two slide sets.
+
+Strain energy in a body is equal to the work done on the body by the applies loads:
+
+$$U = \int_0^u P\mathrm du$$
+
+![](./images/vimscrot-2023-02-16T14:19:48,580021845+00:00.png)
+
+## Bending
+
+\begin{equation}
+U = \int_0^\Phi M\mathrm d\Phi = \int^L_0\frac{M^2}{2EI}\delta s
+\label{eqn:bendingeqn1}
+\end{equation}
+
+![](./images/vimscrot-2023-02-16T14:22:41,358964129+00:00.png)
+
+If this material represents an element of a larger beam of length $L$ and curvature of radius $R$
+
+![](./images/vimscrot-2023-02-16T14:41:32,526245370+00:00.png)
+
+The strain energy within this element will be:
+
+$$\delta U = \frac12 M\delta \Phi$$
+
+From the elastic beam bending equation we know:
+
+$$\frac MI = \frac ER$$
+
+and as the angle subtended by the element is equal to the change in slope, the expression for the
+arc created by the element is:
+
+$$\delta s = R\delta \Phi$$
+
+Eliminating $R$ from the above two equations and rearranging gives
+
+$$\delta \Phi = \frac{M}{EI}\delta s$$
+
+This can be substituted into \label{eqn:bendingeqn1} (left and middle) to get
+
+$$\delta U = \frac{M^2}{2EI}\delta s$$
+
+Integrate over full length of beam to get total strain energy:
+
+$$U = \int^L_0\frac{M^2}{2EI}\delta s$$
+
+## Torsion
+
+$$U = \int_0^\theta T\mathrm d\theta = \int^L_0 \frac{T^2}{2GJ} \delta s$$
+
+Derivations for the equation above is analogous to those for bending and axial loads, and can be
+found in the second lecture slide set (p9-p11).
+
+![](./images/vimscrot-2023-02-16T14:23:06,415583742+00:00.png)
+
+## Elastic Axial Loading
+
+\begin{equation}
+U = \frac12 Pu = \int^L_0\frac{P^2}{2EA}\delta s
+\label{eqn:elasticaxialloading1}
+\end{equation}
+
+![](./images/vimscrot-2023-02-16T14:26:48,809335151+00:00.png)
+
+If this material represents an element, of length $\delta s$, of a larger beam of length $L$, and
+the change in length of this element due to the applied load $P$ is $\delta u$ ten the strain energy
+within the element is
+
+$$\delta U = \frac12 P\delta u$$
+
+
+![](./images/vimscrot-2023-02-16T14:40:43,750030500+00:00.png)
+
+> Note that there are transverse strains/displacements due to Poisson's effects but there are no
+> transverse stresses/loads, thus there is no work done in the transverse direction.
+
+Axial strain is:
+
+$$\epsilon = \frac{\delta u}{\delta s}$$
+
+Equating this to Hooke's law yields:
+
+$$\delta u = \frac{P}{EA}\delta s$$
+
+which can be substituted into equation \ref{eqn:elasticaxialloading1} (left and middle) to get
+
+$$\delta U = \frac{P^2}{2EA}\delta s$$
+
+Integrate over full length of beam to get total strain energy:
+
+$$U = \int^L_0\frac{P^2}{2EA}\delta s$$
+
+# Castigliano's Theorem
+
+This section refers to the third slide set.
+
+