Modeling and Simulation

A model is a simplified representation of a real part, machine or process that keeps the features important to the design question and ignores the rest. Models let the designer predict performance, compare alternatives and find faults before any metal is cut.

Role of models in design

• Predict stress, deflection, temperature, life or cost before manufacture.
• Reduce cost and risk: a failed model is cheaper than a failed prototype.
• Allow comparison of many concepts quickly and support optimisation.
• Help communication between designer, manufacturer and customer.

Types of models

Mathematical modelling

Steps: define the problem and variables, make assumptions, write governing equations (equilibrium, material law, geometry), solve, then validate against experiment or handbook data. Always state the assumptions, because the model is valid only inside them.

Computer simulation and parameter variation

The mathematical model is programmed and run many times while one parameter (diameter, load, material, speed) is varied at a time. The results give a sensitivity of the performance to each parameter, show which parameters are critical, and give a safe operating range. Example: varying wire diameter and number of coils of a spring on a spreadsheet shows how stiffness and stress change, so the designer chooses a feasible design quickly.

(a) Similitude

Similitude is the set of conditions under which results measured on a model can be used to predict the behaviour of the full-size prototype.

• Geometric similarity: all linear dimensions have the same ratio $\lambda = L_p/L_m$, and angles are equal.
• Kinematic similarity: motions are similar; velocities at corresponding points have the same ratio and direction.
• Dynamic similarity: forces at corresponding points have the same ratio (for example equal Reynolds, Froude or Mach number); this includes similar load and stress patterns in solids.

(b) Numerical

Scale ratio $\lambda = L_p/L_m = 4$. Same material, so $E$ is the same.

Bending stress at the root: $\sigma = \dfrac{M c}{I} \propto \dfrac{F L \cdot L}{L^4} = \dfrac{F}{L^2}$.

Tip deflection: $\delta = \dfrac{F L^3}{3EI} \propto \dfrac{F L^3}{L^4} = \dfrac{F}{L}$ (E constant).

1. Prototype load for equal stress

$$\begin{aligned} \frac{F_p}{F_m} &= \lambda^2 = 16 \\ F_p &= 16 \times 2 = 32\ \text{kN} \end{aligned}$$

2. Prototype deflection

$$\begin{aligned} \frac{\delta_p}{\delta_m} &= \frac{F_p/F_m}{\lambda} = \frac{16}{4} = 4 \\ \delta_p &= 4 \times 1.2 = 4.8\ \text{mm} \end{aligned}$$

(Deflection scales with $\lambda$, as in geometric similarity at equal stress.)

3. Stress when the prototype carries only 2 kN

$$\sigma_p = \sigma_m \times \frac{F_p}{F_m}\times\frac{1}{\lambda^2} = 75 \times \frac{1}{16} = 4.69\ \text{MPa}$$

Answer: prototype load = 32 kN, prototype tip deflection = 4.8 mm, stress under 2 kN = 4.69 MPa.

(a) Steps of the finite element method

The finite element method (FEM) divides a body into small elements joined at nodes, solves a simple equation on each element and assembles them to get the behaviour of the whole body.

1. Preprocessing: create the geometry, choose element type (bar, beam, triangle, quadrilateral, tetrahedron, brick), assign material properties ($E$, $\nu$) and section data.
2. Discretisation (meshing): divide the body into elements. Use a finer mesh where stress changes quickly (holes, fillets).
3. Element equations: for each element write $[k]^e\{u\}^e = \{f\}^e$ using shape functions. For a bar element: $[k]=\dfrac{AE}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$.
4. Assembly: combine the element matrices into the global equation $[K]\{U\}=\{F\}$.
5. Boundary conditions and loads: fix the supports and apply forces.
6. Solution: solve for nodal displacements $\{U\}$.
7. Postprocessing: calculate strains, stresses and reactions, plot contours, and check convergence by refining the mesh.

  Geometry -> Mesh -> Element k -> Assemble K
     -> Apply loads/supports -> Solve U
     -> Stress, strain -> Check, refine mesh

(b) Wireframe, surface and solid models