In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t=0 no external torques act on the wheel. -22(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity fto. Solve the differential equation for response (1). c) Normally, we plot the response as (1) versus t. In- stead, suppose we plot the response as log, ((t)/no) versus t. What type of curve is this plot? If you mea- sured experimentally log, ((t)/to) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and the damping B?

In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t=0 no external torques act on the wheel. -22(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity fto. Solve the differential equation for response (1). c) Normally, we plot the response as (1) versus t. In- stead, suppose we plot the response as log, ((t)/no) versus t. What type of curve is this plot? If you mea- sured experimentally log, ((t)/to) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and the damping B?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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In this lab, we will observe the free response of a first order mechanical system: a rotating bicycle wheel. As shown in the figure, suppose that the bicycle wheel is suspended above the ground, and is free to rotate on its axle. Assume that the only source of damping is the axle bearing, and that after time t= 0 no external torques act on the wheel. ✈0(1) a) Derive the differential equation describing the angu- lar velocity of the wheel (t), in terms of the moment of inertia J and the damping B. b) Assume that a time t= 0 the wheel has initial an- gular velocity o. Solve the differential equation for response (1). c) Normally, we plot the response as (t) versus t. In- stead, suppose we plot the response as log (N(t)/Ng) versus t. What type of curve is this plot? If you mea- sured experimentally loge (n(t)/no) as a function of t, could you determine the system time constant 7 from the curve? d) What is the time constant 7 of the system expressed in terms of the moment of inertia J and…
A system consisting of a single degree of freedom mass-spring and damper that vibrates in the vertical direction of an automobile as were l taken into consideration. The car is driven on a road whose surface changes sinusoidally, as shown in the figure. Way the distance between the highest point on its surface and the lowest point is decimated by decimation of 0.2 m between two consecutive vertices along the path distance is 85 m. If the car's natural frequency is 1 Hz and the damping rate of the shock dampers is 0.1, (a) the car's speed at 70 km/h determine the amplitude of vibration. (b) when the speed of the car is changed, the most negative (maximum amplitude vibration) speed for passengers find its value. T0,2 m = 2Y %3D - 85 m - Please select one: A.X=0.144 m; V=124.782km /h B.X=0.103 m; V=409, 998km / h C.X=0.107 m; V=267, 390km /h D.X=0.105 m; V=303, 042km /h E.X=0.104 m; V=338, 694km /h F.X=0.123 m; V=160, 434km /h
My question and answer is in the image. Can you please check my work? A 2 kg mass is attached to a spring with spring constant 50 N/m. The mass is driven by an external force equal tof(t) = 2 sin(5t). The mass is initially released from rest from a point 1 m below the equilibrium position. (Use theconvention that displacements measured below the equilibrium position are positive.)(a) Write the initial-value problem which describes the position of the mass. 2y"+50y=2cos(5t) (b) Find the solution to your initial-value problem from part (a). (1+(1/2)tcos(t))cos(5t)-(1/2)tcos(t) (c) Circle the letter of the graph below that could correspond to the solution. B (d) What is the name for the phenomena this system displays? Resonance
(2) Please write out the Equation of Movements of the following system using matrix. (Figure 2) 1 Note: Lagrange's approach is recommended. „P₂(t) ks ww P₁(t) k₁ wwm, www k₂ m₂ k6 www P3(t) k₁ k3 www m3 www Figure 2: N Degree Vibration System
You are requested to design an automotive suspension or shock absorber system. In order to simplify the problem to one dimensional multiple mass-spring-damper system, a quarter vehicle model is used. The system parameters and free-body diagram of such system is shown below. M₁: Automobile body mass M₂: Wheel and suspension mass K₁: Spring constant of suspension system K2: Spring constant of wheel and tire B: Damping constant of shock absorber (a) Obtain the transfer function of X₁ (s) F(s) T₁(s) = = 2500 kg = 320 kg and T₂(s) = = 80,000 N/m = 500,000 N/m = 350 N-s/m Automobile- Suspension system Wheel- M₁ M₂ X₁ (s) - X₂ (S) F(s) in terms of the parameters of mass, damper and elastance (M, B and K). (b) Express the T₁ (s) and T₂ (s) with numerical values. c) Plot the x₁ (t) and x₁ (t) = x₂(t) outputs of this passive suspension system for the input torque f(t) = 2,000 N. fit) K₂ x₂(t) -Tire
Ex. A vehicle wheel, tire and suspension assembly can be modeled crudely as a single degree of freedom spring mass system. The mass of the assembly is measured to be about 300kg. its frequency of oscillation is observed to be π rad/sec. What is the approximate stiffness of the tire, wheel and suspension assembly? ilippines) Accessibility: Investigate
1. Suppose you are riding your bicycle on a bumpy road having a surface profile that varies harmonically with +/- 6 cm undulations. The distance between consecutive peaks of these undulations is 2 m. When you sit on the seat of your bike for your casual ride, the springs deflect 5 cm. When you are seated, the damper under the seat provides an equivalent linear viscous damping of 10% of the critical damping. A simple representation of your ride on the "never-ending" rough road is shown below. (a) If you are riding your bicycle at a horizontal speed of 2.5 m/sec, how much bumping up and down will you experience? (b) Next day, you are carrying a backpack which increases your on-seat weight by 20%. Assuming that you are still able to ride with same speed, will this "loaded" ride be more or less comfortable than your previous, "no backpack" ride? M k/23 k/2 6 cm 2 m
A disk of radius R = 2 m, whose mass is distributed homogeneously and equals 6 kg, is suspended at its center by a cable with torsion constant K = 0.5 Nm/rad and length 1 = 1 m. The system is immersed in a fluid in such a way that it experiences a damping force of constant b Ns/m. find: a) The value of b for which the system is critically damped. b) Make a graph of angular position v/s time where the period of the system can be seen explicitly, for a value of b equal to half the critical value. Choose an arbitrary value for the amplitude and initial phase of the system.
The slender bar of Figure 3.9(a) has a mass of 31 kg and a length of 2.6 m. A 50 N force is statically applied to the bar at P then removed. The ensuing oscillations of Pare moni- tored, and the acceleration data is shown in Figure 3.9(b) where the time scale is calibrated but the acceleration scale is not. (a) Use the data to find the spring stiffness k and the damping cocfficient c. (b) Calibrate the acceleration scale.
The mass of a single degree of freedom damped vibrating system is 75 kg and vertically attached to the spring of a stiffness 10.5 kN/m as shown in Figure 2. The damping coefficient of the system is 600 Ns/m and the system is excited by a force: F=700sin25t in Newtons and t-is the time in seconds 2.1 Calculate the natural frequency of the system in rad/s 2.2. Calculate the periodic time of the system if at free vibrating mode and that of exciting force.2.3. Using the damping ratio criteria, state the damping type of the system. (5)
Consider the double mass/double spring system shown below. - click to expand. Both springs have spring constants k, and both masses have mass m; each spring is subject to a damping force of Ffriction -cz' (friction proportional to velocity). We can write the resulting system of second-order DEs as a first-order system, t' (t) = Au(t), with = (₁, 21, 22, 2₂) I For values of k = 4, m = 1 and c = 1, the resulting eigenvalues and eigenvectors of A are -0.039-0.248i 0.813 A₁2=-0.5±3.2i, v₁ = 0.024 +0.153i -0.502 -0.134-0.302i 0.409 -0.2160.489 0.661 (a) Find a set of initial displacements (0), 2(0) that will lead to the fast mode of oscillation for this sytem. Assume that the initial velocities wil be zero. A3,4 -0.5± 1.13i, z = and (2₁ (0), ₂(0)) = Enter your answer using angle braces, (and). (b) At what frequency will the masses be oscillating in this mode? Frequency rad/s
4. Consider a mass of 500 g placed at the end of a spring with stiffness constant 100 N/m and hanging at rest. The mass is displaced downward by 1.0 cm and released from rest. When the motion sets in, a time-varying force F(t) = cos³ 2t is applied to the spring-mass system. Solve the nonhomogeneous 2nd order differential equation representing the motion. That is, solve for x(t). [Note: Take the downward direction as the negative x-axis.]