Reference no: EM133365007

Mechanical Vibrations

Assignment Brief

You will develop numerical solutions to single degree of freedom problems, using schemes presented in MEC3075F and MEC4047F. You will be comparing your numerical solutions against analytical solutions for known problems, as well as developing analytical and numerical solutions for a real-world scenario. The real-world application we are considering is a domestic, front-loading washing machine (Figure 1a), which we will approximate as a vertically translating 1-DOF system in this assignment. The primary loading is due to uneven loading of washing causing rotating unbalance during the spin cycle (Figure 1c)

1 - Analytical Solution
(a) Draw the Free Body Diagram for the sprung components of the washing machine and determine the differential equation of motion. The "sprung components" include the ro- tating drum, the shroud which contains the water and the motor which is rigidly coupled to the shroud. Following the approach presented in class where we assume a harmonic steady-state displacement response of the form xp(t) = X1 cos(ωt) + X2 sin(ωt), deter- mine the dynamic magnification ratio and the phase angle. Plot the phase angle as a function of frequency ratio.

(b) Assuming zero initial displacement and velocity, find the analytical solution for the total vertical displacement response (i.e. transient and steady state). Plot the the total displacement response and establish how quickly (in natural periods) the transient dies away, for drum speeds of 400 rpm, 800 rpm and 1200 rpm.

(c) Derive a relationship for the total force exerted on the floor by the drum suspension (springs and dampers). You should plot the magnitude of force as a function of frequency ratio.

You will be given unique input values for the following input variables, via Vula Post 'Em. Some typical values are given in Table 1. Your submission must be based on your unique inputs. Note the table values are "effective" i.e. combining the effect of any springs into a single spring stiffness.

2- Central Difference Approximation
Implement a Central Difference (CD) scheme in Python, to find an approximate numerical solution for 1-DOF vibration problems, that can accommodate damping and external forces.

(a) You must first verify that your numerical scheme is producing reasonably accurate answers by comparing to a problem with a known analytical solution. We advise you to check your scheme first against the simplest possible free vibration problem before building to more complex forced, damped vibrations. For free vibrations, what size of time step (as a fraction of natural period) is necessary to predict the first peak and rebound displacements to better than 1% accuracy?

(b) You should then apply your CD approximation to the unbalanced washing machine. Using the time step criteria established earlier, investigate drum vertical displacement at fre- quency ratios of 0.2, 0.5, and 0.8. For each simulation, pick a drum speed appropriate for the frequency ratio (i.e. don't vary the forcing frequency with time).

(c) Compare the magnification ratio from your numerical results to that predicted by the analytical solution. Does your CD implementation produce accurate displacements - both from the perspective of magnitude of the peaks and timing of the peaks? How many harmonic functions (sine waves of single wavelength) appear in your numerical solution?

(d) Using the same time step, simulate the displacement response for frequency ratios of 2.0,
4.0 and 8.0. How accurate are these magnification ratios? What does this imply for time step choice when forcing functions are present?

3 - Alternative ODE Solvers
The CD method is one of the second-order accurate schemes for numerical approximate solu- tions for second-order differential equations and is an "explicit solver". There are alternative schemes such as the Runge-Kutta or Newmark methods, some of which are already available

in Python libraries such as SciPy. Choose ONE alternative scheme and use it to repeat the analysis of Sec 2.2(a). How does the minimum time step required for a stable solution for the chosen scheme compare to that for CD? What trade-offs are involved in choosing a numerical solver?

4 - Time-Varying Frequency

The analytical solutions and numerical approximations thus far have kept a constant drum speed (forcing frequency) within any given analysis. However, a real-world washing machine increases the drum speed from rest until it reaches the desired steady-state spin speed. Using your CD scheme, implement forcing functions for the following scenarios:

• The drum starts at rest and increases speed linearly to 1000 rpm over 30 seconds.
• The drum starts at rest and increases speed linearly to 1000 rpm over 90 seconds. Which scenario is better from the perspective of vibration amplitude?

Attachment:- Mechanical Vibrations.rar