k, K2 m k3 : ks K6 K4
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For the single DOF mass-spring system shown in Fig. Q5.a, determine the equivalent spring
constant of the system. Assume k1=k2=35 N/m, k3=k4=55 N/m, k5=70 N/m, and k6=20 N/m.
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- (b) A system of mass rotation rod has been run at = 40 rad/s as shown in Figure Q1. The system has the following data, (i) (ii) Mass Mass A Mass B Mass C Weigh 1.2 kg 1.8 kg Mc Radius 1.135 m 0.822 m Re Angle 113.4° 48.8° вс Find counterweight mass-radius product and its angular location needed to balance the system. Distinguish the relationship between mass, radius and angle for counterweight mass system.2. A kilogram mass is attached to the end of the spring with spring constant 2 N/m. Find the equation of motion if the mass is initially released (set in motion) from rest from a point 1 meter above equilibrium position. (Use the convention that displacements measured below the equilibrium position are positive.) (a) Write the initial-value problem which describes the position of the mass. (b) Find the solution to your initial-value problem from part (a). (c) Graph the solution found in (b) on (0You 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) -TireA. A car's body has mass 100 slugs. This mass is supported by the car's 4 shock absorbers. The shock absorbers are identical vertical linear springs. Complete a FBD of the car's body (this would not include the car's wheels, axels etc. just the car's body) Each spring is compressed 1.25 inches. Determine the spring "k" value. Answer: 644 lb/inch. Find the compression of each spring if 4 people, each with mass 6 slugs, get into the car, Answer: 1.55" ii. iii.A spring system is shown here: k₁ 3 Ę k3 2 K₂ www ma 4 Ę₂ Part 1: For this specific system, develop the: a. Global stiffness matrix . b. Boundary condition vector • c. Load vector • d. Reduced system of equations • e. Reaction force equations (i.e., the equations eliminated by the boundary conditions) Part 2: Given: k1 = 70 N/mm, k2 = 110 N/mm, k3 = 165 N/mm, F1 = 150 N, F2 = 100 N, and nodes 1 and 3 are fixed; calculate the: a. Global stiffness matrix b. Displacements of nodes 2 and 4 c. Reaction forces at nodes 1 and 3 d. Spring force in each of the springsThree springs with different spring constants are connected as shown below. You are going to use spring elements to simulate this system. Suppose that the spring constants of the first, second and third elements are k1=3,410 N/m, k2=3,160 N/m and k3=3,380 N/m, respectively. Two horizontal forces are applied to the system (as shown) at nodes. 2 and 3. Find the displacement of node 3 and write your answer in mm (millimetre). Hint: Write your answer with 5 decimal places. For example if you calculated the value 1.2345678, then rounding off to 5 decimal places yields 1.23457 and that is the value you need to type in the answer box. U₁=0 (1) F₂ = 2N U₂ = ? F3 = -1N (2) M U3 = ? (3) U4 = 0igure 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 springs are 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. DE only 225 Your tasks: X k 0 3k a F a M y A Derive an equation of motion for the system in terms of the angular displacement 0, and its derivatives (you should not have y or its derivatives in this equation.) B Derive an equation of motion for the system in terms of the displacement y, and its derivatives (you should not have or its derivatives in this equation. C Assuming there is no external actuator force F acting on the system, write down the total energy H of the system in terms of 0,0 and element constants. Derive an expression for the time derivative H of the total energy. D Transform the equation from part B, which is in y, to another…(a) A body of mass m, controlled by an elastic system, is given a displacement x. Derive an expression for the periodic frequency, n, of linear motion of the elastic system 1 n = Hz 1 Hz where ő is the static deflection in metres under the load, mg. (b) Figure TQ3.3 shows a suspended pendulum from a fixed pivot at O. The pendulum consists of a bar B, of mass 1kg, and block C of mass 6kg. The centre of gravity G1 and G2 of B and C are at distance 150mm and 375mm from 0. The radius of gyration of B and C, each about its own centre of gravity, are respectively 100mm and 25mm. A light spring is attached to the pendulum at point P, 200mm from 0, and is anchored at a fixed point, Q. When the Pendulum is in equilibrium, the line OG,PG2 is at 45° from the vertical and the angle OPQ is 90°. The spring has a stiffness of 700N/m. Calculate the natural frequency of the pendulum for small oscillations about the equilibrium position. 45° 0-150 0-20 Ig N G2 f0-3750 Y6g N Figure TQ3.34.13. Derive the differential equation of motion of the system shown in Fig. P4.1. Obtain the steady state solution of the absolute motion of the mass. Also obtain the displacement of the mass with respect to the moving base. For this system, let m=3 kg, k1 =k2 = 1350 N/m, c= 40 N s/m, and y=0.04 sin 15t. The initial conditions are such that xo =5 mm and io =0. Determine the displacement, velocity, and acceleration of the mass after time t =1s. y = Y, sin @,1 k1 in k2 m Fig. P4.1Equation of motion of a suspension system is given as: Mä(t) + Cx(t) + ax² (t) + bx(t) = F(t), where the spring force is given with a non-linear function as K(x) = ax²(t) + bx(t). %3D a. Find the linearized equation of motion of the system for the motion that it makes around steady state equilibrium point x, under the effect of constant F, force. b. Find the natural frequency and damping ratio of the linearized system. - c. Find the step response of the system ( Numerical values: a=2, b=5, M=1kg, C=3Ns/m, Fo=1N, xo=0.05mThank you in advance! A certain mass-spring-damper system has the following equation of motion. x''+ cx' + 100x = f(t) Suppose that the initial conditions are zero and that the applied force f(t) is a step function of magnitude 100. Solve for x(t) for the following three cases: (a) c=10 and (b) c=20, (c) c=50.Q.5 A 6-kg uniform cylinder of radius 50 cm can roll without sliding on a horizontal surface and is attached by a pin at point C to the 4-kg horizontal bar AB. The bar is always horizontal and attached to two springs, each of constants k= 5 kN/m as shown. If the bar is moved 2 cm to the left side and released: a) Find the resulting period of oscillation, b) Find the maximum velocity and maximum acceleration of point C, and c) Find the maximum angular velocity and maximum angular acceleration of the cylinder.SEE MORE QUESTIONS