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Mechanical System A dynamic system shown in the Fig

Mechanical Engineering

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Mechanical System

A dynamic system shown in the Fig. I below consists of a lever, constrained by the spring with stiffness coefficient k and damper with damping coefficient c. The lever is subject to the external force f(t) and rotates in the direction shown. Angular displacement of the lever is 0;

its Mass Moment of inertia about its pivot is /₀; center of mass of the lever is at the pivot. When θ =0 spring is at its free length.

 (a) Derive the governing equation

(b) Derive the transfer function, considering that the input is the applied external force

f(t), and the output is the damper force Fc

 

Problem 2

Mechanical System

The dynamic system shown in Fig. 2 starts motion from the equilibrium position that corresponds to x=0 and y=0

The mass moment of inertia of the rotating pulley about the stationary pivot is I₀ and its radius is r ; pulley in planar motion is considered massless.

The solid homogenous disk of mass m₂ and radius R rolls without slipping. Its mass moment of inertia about its center of mass (geometric center of the disk) is I_g = M₂ R². The coefficient of friction µ is not known. The angle θ of the incline is also not known. Motion of the disk is constrained by a spring with the stiffness coefficient k, and a damper damping coefficient d.

While moving, block m₁, slides along the vertical surface, experiencing viscous friction force proportional to block velocity. Viscous friction coefficient is c.

Derive the governing equation(s) in terms of the displacement of the block x.

Problem 3

A dynamic system is described with its transfer function:

G(s) = 6s/(s+1) (s+5) (s² +3s+5)

(a) Prove that the system is stable

(b) Assuming that this system converges to a steady-state when subject to a ramp inputs determine its steady-state response to the input 10tu(t)

A dynamic system is described with the following differential equation:

4y + 6ky + 36y = 18u(t)

where k is a constant coefficient.

a) Find the value of k when system maximum overshoot is 15%. At the derived value of k find system damping ratio and natural frequency, as well as the following parameters of the response to the indicated above input considering zero initial conditions: rise time, settling time at 2% error, and steady-state response

b) For k = 4 and the initial conditions y(0) = 1, y(0) =2 find system response and maximum overshoot in % (if exists)

Problem 4

Passive Circuit Analysis

Derive governing equations for the electric circuit, shown in Fig. 3 below. Inputs are the current i_s(t) and voltage e_m(t). and output is the shown current l_o.

Part 2

Derive the state-space model for the electric circuit shown in Fig. 4. Inputs are voltages e₁ (t) and e₂ (t) and e,(t). and the output is the shown current i_0.

Problem 5

Active Circuit Analysis

Derive the governing equation expressing the relationship between the input voltage e_n output voltage e_out inverting op-amp circuit, shown in Fig 5 below

Problem 6

Electromechanical System

Electromechanical system shown in Fig. 6. consists of the standard armature-controlled DC motor and load assembly. Inertia of the motor mechanical subsystem is J_m; it is supported by bearings with damping coefficient b_m Voltage input to the armature circuit is v_a(t). Motor speed Is @,, = θ_m  The motor constant is Km and the back-emf constant kb,,.

Inertia of the load is J1 . The load is connected to the motor with a flexible shaft that is 3 modeled as a torsional spring with the stiffness coefficient k. Load is supported by bearings a with damping coefficient b1. System output is the load speed, @, = θ L . -

a) Derive the governing equations, clearly showing equations for each subsystem and the coupling equations

b) Draw a modular block diagram, clearly show all the transfer functions and indicate all the pertinent variables: _

input voltage va armature current ia, back emf vb, motor torque T_M , motor speed m, spring torque TK, load sped

 

 

For every problem show all your work and clearly state your final results:

(a) for mechanical system problems state number of degrees of freedom and all the necessary assumptions, show free-body diagrams with all the forces/torques acting on each element of the system, indicate all the relevant relationships between system variables such as, for example, angular and linear displacements;

(b) for electrical circuit problems show loops/ assumed loop currents or nodes/ node voltages,

KVL/ KCL. equations, clement currents/ voltages (if appropriate), element equations, choice of state variables, op-amp equations, elc:

(c) for response-related problems clearly show appropriate expression for free and forced responses as Well as for every computed response parameter together with numerical results (state Where you used software for computations; if you used software to generate graphic solution, sketch the generated graph and show the relevant numerical value(s) you Were (driving to derive:

(d) for electromechanical system problem clearly show governing equations for each subsystem and coupling equations.