Model 2 - Mechanical-Electrical Analogies [35%] Constant-coefficient second-order ordinary differential equations are a

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Model 2 Mechanical Electrical Analogies 35 Constant Coefficient Second Order Ordinary Differential Equations Are A 1 (117.41 KiB) Viewed 36 times

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Model 2 - Mechanical-Electrical Analogies [35%] Constant-coefficient second-order ordinary differential equations are a class of equations that are frequently used to study systems that combine inertial, dissipative and elastic (restorative) components. They are an informative and simple way to analyse harmonic oscillators. For example, the position x(t) of a mass-spring-damp system can be modelled as: 2.1 d²x dx m + c · + kx = 0, dt2 dt Where m is the mass [kg], k is the elastic constant and c is the damping coefficient as depicted in Figure 2. This figure shows a schematic of how the elastic kx and the d²x damping forces are opposed to the inertial force m acting on the mass m. dx dt dt2 kx cx k elle с 0 m m + x +x Figure 2. Schematic of a mechanical oscillator. Conveniently, the second order constant-coefficient equation used to model the vibration of a particle can also model the oscillations in the current i(t) in an electric circuit: 2.2 dai di 1 L +R+=i = 0, dt с dt2 CONTINUED
Where L is the inductance, R is the resistance and C is the capacitance of this system. Since the variable i is usually saved in electrical engineering for the current, the complex unit is denoted as j = V-1. In Figure 3, when the switch is in contact with B, the potential difference around the circuit must be zero. R L AL <B с NE Figure 3. Schematic of a simple electric circuit. a) [20 marks] Find all possible values of y such that x(t) = XeWt is a solution of equation 2.1 and i(t) = 1.e4t is a solution of equation 2.2. Then, consider the discriminant of the characteristic equations in this question and discuss the relationship between y and all constant parameters in these equations. A general representation for a harmonic oscillator is given by 2.3 daz dz + 2ζωο + wêz = 0. dt2 dt In some specific scenarios, the solution of these types of equations yield oscillatory solutions, that is, a solution that involves sine and/or cosine functions with frequency wo. When this is the case, the angular frequency of these trigonometric functions is called the natural frequency of the system. In the first 30 seconds of this video, you can see two structures vibrating at their natural frequency. CONTINUED
b) [15 marks] Find the expression for and wo in mechanical oscillators and electric circuits, then show that the solution of both systems is of the form .(-3+√3²-1) wot (-5-132-1)wot z(t) = Zied + Zzed = c) [20 marks] Solve and outline all possible expressions for z(t) = Z, eWt in the cases where is real and V is complex. Then, discuss the mathematical behaviour of both electrical and mechanical systems according to the value of If there is an oscillatory external force F = F, cos at acting on the mechanical system, the differential equation becomes d²z dz m + c + kz = Ao cost. dt2 dt Alternatively, if a voltage V = V cos Nt is applied to the electrical circuit, we can use the experimental relation V = iR to show that the differential equation becomes dai di L +R dt2 dt 1 + i = A cost. C In both these cases, the particular integral of the general differential equation is given by A(12) cos(Nt - 8) where 2ζωoΩ 8 = tan-1 wo-122 And A(2) is the amplitude function defined by A A(N2) = [(wă - 122)2 + (23W02)] A resonant system is a particular case where I = wo. This is a system that is excited at its natural frequency, and the synergy between internal and external forces makes this system resonate, which results in a response of higher amplitude. However, this type of knowledge is not a mere curiosity. It is a key element in understanding both
CONTINUED why the Tacoma Narrows Bridge collapsed, and how to design earthquake-resistant buildings. A(Ω) d) [20 marks] Plot the nondimensional amplitude for both electrical and A(12=0) Ω mechanical systems as a function of the ratio - for 3 = {0.1, 0.25, 1, 2}. Then, explain wo the effects of s on the amplitude of the forced system. How is this representation of the data related to non-dimensionalisation? e) [25 marks] Find the value of the amplitude ratio that maximises the Wo amplitude A(12).