5 ProblemConsider the single degree-of-freedom rotational system shown below with two masses m con-centrated at each end. An upward force, u(t), is applied at the leftmost point, and two springswith stiffness k are attached at distances away from the center rotation point O. The springs areunstretched when the angle of the rod is 0 = 0. Assume small angles and account for weight ofmasses.• Derive an expression for the natural frequency in terms of the parameters given.• Show that the transfer function for this system isⒸ(s)U(s)-24mLs²+kL mL/2Tu(t) jL/2kL/2Hint: Recall that for N masses, the inertia around point O is:NIo = Σmil|ri/o||²i=1L/2kwhere ri/o is the vector that points to the ith mass from point O.0m

To findDerive an expression for the natural frequency in terms of the parameters given.Show that the…

Answered: 5 Problem Consider the single…

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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6. It is desired to design a valve spring of I.C. engine for the following details: (a) Spring load when valve is closed 80 N (b) Spring load when valve is open = 100 N (c) Space constraints for the fitment of spring are : Inside guide bush diameter = 24 mm Outside recess diameter= 36 mm (d) Valve lift 5 mm (e) Spring stcel has the following properties: Maximum permissible shear stress = 350 MPa Modulus of rigidity 84 kN/mm? ried: 1 Wire diameter; 2. Spring index; 3. Total number of coils; 4. Solid length of sprine: 5. Free
6. It is desired to design a valve spring of I.C. engine for the following details : (a) Spring load when valve is closed = 80 N (b) Spring load when valve is open = 100 N (c) Space constraints for the fitment of spring are : Inside guide bush diameter = 24 mm Outside recess diameter = 36 mm (d) Valve lift = 5 mm (e) Spring steel has the following properties: Maximum permissible shear stress = 350 MPa Modulus of rigidity = 84 kN/mm? Find : 1. Wire diameter; 2. Spring index; 3. Total number of coils; 4. Solid length of spring; 5. Free Тop
140 lb 12 lb/in. 8 in. Problem 5-54 what are the values of the reaction forces O r1=150KNM r2=177KN O r1=177KN, r2=150KN O r1=177.9167N, r2=157.0833KN O r1=171.9KN, r2=150KN
Problem 2: Design a helical compression spring using round steel wire, ASTM A227. The spring will actuate a clutch and must withstand several million cycles of operation. When the clutch discs are in contact, the spring will have a length of 60 mm and must exert a force of 90 N. When the clutch is disengaged, the spring will be 50 mm long and must exert a force of 155 N. The spring will be installed around a round shaft having a diameter of 38 mm.
Figure Q2 shows two configurations of a part sorting machine in a production process. Since the process involve high speed movement, the design engineer have proposed two configurations of design as shown in Figure Q2 (a) – Configuration 1 and Figure Q2 (b) – Configuration 2. The mass m1 and m2 are 180 kg and 70 kg respectively and the spring coefficient, k = k1 = k2 = k3 = k4 = 300 N/m.(a) Sketch the Free Body Diagram for each configuration (Configuration 1 and Configuration 2).(b)Construct the Equation of Motion for each configuration (Configuration 1 and Configuration 2).(c)Predict which configuration that would lead to the highest first natural frequency, ωn1. State the reason for your answer.(d)Validate your answer in Q2(c) by providing natural frequency ωn calculation for each configuration.
Under some circumstances, when two parallel springswith constants k1 and k2 support a single mass, theeffective spring constant of the system is given byk 4k1k2/(k1 + k2). A mass weighing 20 pounds stretchesone spring 6 inches and another spring 2 inches. Thesprings are first attached to a common rigid supportand then to a metal plate. As the figure illustrates(see image), the mass is attached to the center of the plate in the distributiondouble spring. Determine the constant of theeffective spring for this system. find the equationof motion if the mass is initially released fromequilibrium position with downward speed of2 ft/sec.
For this problem, take a look at Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y = 0, and the spring is in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. k Schen a 0 a F The equation of motion for the system can be derived to be: a 4aM0+ ak0 =-F-2Mg T y M Your tasks: A. Transform the rotational equation of motion, which is in 0, given above, to another variable, y, which is zero at the static equilibrium position. onical system in sta bace form. Using MATLAB or a calculator lues of INI. W 16 [N/m], and C. Derive the response of the system in the Laplace (s) domain. Use the static equilibrium value found in part A (Ost) as the initial value, (0), for the problem. Assume (0) and the force, F, are both zero. You may treat gravity as g = 10 [m/s²] for ease of…
For this problem, take a look at Figure 2. Assume that the rod is massless, perfectly rigid, and pivoted at point P. When the rod is perfectly horizontal, the angle 0 = 0, the displacement y = 0, and the spring is in neither tension nor compression. Gravity acts on the system (e.g. on mass M). We assume that y is a small displacement. A mass M is attached at the end of the rod. k Schen a 0 a F The equation of motion for the system can be derived to be: a 4aM0+ ak0 =-F-2Mg T y M Your tasks: A. Transform the rotational equation of motion, which is in 0, given above, to another variable, y, which is zero at the static equilibrium position. B. Represent the mechanical system in state space form. Using MATLAB or a calculator (just say what you used), calculate the eigenvalues of your A matrix for the following system parameters: a = 0.25 [m], M = 1 [kg], k = 16 [N/m], and F = 0 [N]. What do the eigenvalues say about the stability of the system? C. Derive the response of the system in the…
7. A compound lever as shown below. Find the effort P. if the system is used to lift a mass of 800kg. (10) S00 mam 400 mm 350 mm S0mm 800 kg 8. An oil drum 600mm in diameter and length Im is to be rolled over a step 120mm in height. Find the least pull through the centre of the drum required just to turn the drum over the step. Also find the reaction on the step. Assuming all the surfaces to be smooth. Take density of oil as 1 kg/litre and neglecting weight of the drum. (10) 50 mm
Problem 7. We have a line of two equal masses m connected by three springs with spring constants c₁ = 1, c₂ = 1, and c3 = S. Spring 1 is fixed at the top and spring 3 at the bottom, so xo = x3 = 0. (a) Find the stiffness matrix K in the equation Kx = f for the mass displacements. (b) Solve for the displacements x₁ and x2. (c) If the third spring constant S becomes very large or very small (S→∞ and S→ 0) what are the limiting values of the displacements x₁ and .x₂? (d) If the third spring constant S becomes very large or very small (S→∞ and S→ 0) what are the limiting values of the spring forces y1, y2, and y3? 91 www m 3-WW www C₁ X₁ "Xo = 0 C2₂ X2 C3 X3 = 0
Problem. Consider the following problem. Find the displacement of node 4. Assume F2=200 N, F3=300 N, F4=400 N and Kı= 100, K2= 200, and K3= 300 N/mm. F. 3 2 3 4 Note: The stiffness matrix of individual springs should be first written, then the equilibrium equations at each node should be written, expanded in terms of nodal displacement, and used to perform the assembly of the global stiffness matrix of the problem. Then, the appropriate boundary conditions should be applied and the nodal displacement vector should be found. Every step should be shown; missing any step would result in point deduction even if the final answer is correct.
Three springs are connected to a mass and to fixed supports. When the coordinate y is equal to zero, there is no spring force. Your tasks: Tw m k3 assume Staur C équilibrition Fload Figure 1: System schematic for Problem 1. A Draw the free body diagram for the mass m including all 3 spring forces and the load force Fload). B Draw the free body diagram for the mass m, but combine the springs into one equivalent spring with spring constant keq. Write the equivalent stiffness of the spring keq in terms of the other spring constants. (3 C Given that k₁ = 300 N/m, k₂ = 100 N/m, and k3 = 200 N/m, and Fload = 300 N, calculate the loaded deflection SA Yloaded- D Compute the amount of potential energy stored in each spring. E Compute the forces, in Newtons, in each spring and denote them as F₁, F2, and F3 when the load from Part D is applied. www

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