##### Reference no: EM1337237

**Module FEA - FINITE ELEMENT ANALYSIS **

**Question 2/** Method to assess the value of your analysis output

a) What is the characteristic of the system matrix of a FEA analysis? Which information can you extract from the pivot ratio?

b) Through three reasons discuss the possible errors occurred by un-correct loading.

c) Give three reasons for which non-linear FEA is required when not expected.

d) Which tools does Ansys propose to estimate errors? Give brief explanation and provide the command line for the three of them?

e) Discuss linear and parabolic elements within their application. Which one of linear or parabolic elements would you use for highly curved geometries? Explain why.

f) What is a discretisation error? What is a formulation error? What is a numerical error?

**Question 3/ Thermal analysis **

Problem description

The heat exchanger shown below is used to cool a motor. The coolant coming from the motor (hot fluid) runs through the block near a cold fluid coming from a separate reservoir. The base of the heat exchanger is mounted on an insulated surface (no heat transfer occurs across the base). The heat exchanger also utilizes fins along its top surface to increase heat transfer to the surrounding air through convection.

As shown in the figure below, the heat exchanger block is 54 mm wide by 80 mm high, and is sufficiently long that it can be analysed using a 2D thermal analysis; i.e., heat flow in the third direction (into the paper) can be neglected. The five fins on the top surface are 6 mm thick, with 6 mm spacing between fins, rounded on the inner and outer edges with a 3 mm radius. The fins are 20 mm long. Both fluid channels have a diameter of 24 mm and are centred horizontally in the block.

The air temperature surrounding the heat exchanger is 300 K, and the convective heat transfer (film) coefficient can be assumed to be h = 400 W/m^{2}-K. This convection acts on all external surfaces except for the base, which is insulated. The temperature of the hot fluid is T hot = 600 K, and the temperature of the cold fluid is Tcold = 285 K. The convective heat transfer coefficient of both fluids is h = 5000 W/m^{2}-K.

The heat exchanger is made of an aluminium alloy with the following thermal properties (density, ρ = 2702 kg/m3):

Temperature (K) 200 300 400 600

Thermal Conductivity, k (W/m-K) 237 237 240 231

Specific Heat, c (J/kg-K) 798 903 949 1033

Assume an initial temperature for the heat exchanger of 300 K. Assume the thermal loads (convections) are applied suddenly (stepped).

a) Build an appropriate finite element model of the heat conduction problem using Ansys and provide a complete set of command lines. The text file should contain three set of commands: heading, pre-processor, and solve.

b) Justify the type and the dimension of element you have chosen.

Perform a transient thermal analysis of the heat exchanger from which you will determine the temperature distribution in the exchanger after 2 minutes, and generate temperature vs. time plots for the hottest and coolest nodes in the heat exchanger, as well as the node at the top of the centre fin.

c) Plot a nodal temperature plot after 2 minutes of the heat exchanger, and a temperature vs. time plot for the nodes listed above. Three plots should be provided.

d) Determine the length of time for the hottest node to reach 440 degree Kelvin.

e) Has the temperature of the heat exchanger reached steady-state conditions after 2 minutes?

f) List the nonlinear solution parameters used in the analysis (i.e.: time step size, and max and min values if automatic time stepping was used).

**Question 4 PZT actuator **

a) Using the code below model a PZT actuated composite beam.

b) Using the ANSYS help file Mechanical APDL-Command Reference lightly comment the supplied code

c) Solve the static model and extract the maximum tip deflection and plot stress

d) Using a suitable frequency range extract 3 resonant modes

e) Using the same geometry & inputs how could the tip deflection be increased

FINISH

/CLEAR

/PREP7

/UNITS, SI

um=1e-6

eps_0=8.854e-12

dens_p=7700

c11_p=1.57e11

c12_p=1.09e11

c13_p=9.77e10

c33_p=1.23e11

c44_p=2.57e10

e31_p=-1.93

e33_p=15.5

e24_p=10.81

eps11_p=1370*eps_0

eps33_p=1500*eps_0

damp_pzt=1.0e-3

loss_pzt=0.013

dens_s=2330

young_s=160e9

poiss_s=0.23

damp_si=1.0e-5

beam_dn=2

piez_dn=1

meshsize=25*um

beam_w=75*um

beam_l=350*um

beam_t=10*um

piez_t=2*um

ET, 1, PLANE42

ET, 2, SOLID186

ET, 3, SOLID5

MP, EX, 1, young_s

MP, DENS, 1, dens_s

MP, NUXY, 1, poiss_s

MP, DMPR,1,damp_si

MP, DENS, 2, dens_p

MP,DMPR,2,damp_pzt

TB, ANEL, 2

TBDATA, 1, c11_p

TBDATA, 2, c12_p

TBDATA, 3, c13_p

TBDATA, 7, c11_p

TBDATA, 8, c13_p

TBDATA, 12, c33_p

TBDATA, 16, (c11_p-c12_p)/2

TBDATA, 19, c44_p

TBDATA, 21, c44_p

TB, PIEZ, 2

TBDATA, 3, e31_p

TBDATA, 6, e31_p

TBDATA, 9, e33_p

TBDATA, 14, e24_p

TBDATA, 16, e24_p

MP, PERX, 2, eps11_p

MP, PERY, 2, eps11_p

MP, PERZ, 2, eps33_p

RECTNG, 0, beam_l, 0, beam_w

ESIZE, meshsize

MSHAPE,0,2D

AMESH, ALL

TYPE, 2

MAT,1

ESIZE, , beam_dn

VEXT, ALL, , , , , -beam_t

TYPE, 3

MAT, 2

ESIZE, , piez_dn

ASEL, , LOC, Z, 0,1e-11

VEXT, ALL, , , , , piez_t

ALLSEL, ALL

NSEL,S, LOC, z, 0,1E-11

*GET,node_o,NODE,,NUM,MIN

CM,out,NODE

CP, NEXT, VOLT, ALL

D,node_o, VOLT, 0

NSEL,S , LOC, z, piez_t, piez_t+1.0E-11

*GET,node_s,NODE,,NUM,MIN

CM,signal,NODE

CP, NEXT, VOLT, ALL

D,node_s, VOLT, 5

NSEL,S,LOC, X, 0, 1e-11

D, ALL, UX, 0

D, ALL, UY, 0

D, ALL, UZ, 0