In Figure P4.23, assume that the cylinder rolls without slipping. The spring is at its free length when x and y are zero. (a) Derive the equation of motion in terms of x, with
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System Dynamics
- + b 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent…arrow_forwardFor this problem, take a look at Figure 2 below. A disk with uniformly distributed mass m, radius R, and center of mass at point O is connected to a combination of springs at point P, which are then connected to a fixed wall. The disk rolls without slipping at point Q along an inclined plane that is at an angle a from the horizontal. Gravity acts in the vertical direction (towards the bottom of the page). ₁ is the linear coordinate of the point O along the inclined plane. The positive direction of ₁ is as shown. When the springs are undeflected, *₁ = 0. An angle , about the instant center of rotation, is shown. You may assume that the motion (and therefore angle ) is small. puny m Massless structure between springs R Figure 2: System schematic. Your tasks: A Draw the FBD for the disk. Don't forget the forces at point Q B Derive the equation of motion with as the dynamic variable. Be sure to put it in input-output standard form (inputs and constant forces on the right, things related to…arrow_forwardA uniform rod of mass m is pivoted at a point O as shown in Figure 2. The rod is constrained by three identical linear springs, and a point mass M is attached at the end of the rod (also shown in Figure 2). The parameters of the system are as follows: L= 1 m, k = 1000 N/m, m= 50 kg, M = 25 kg. i. Derive the equation of motion of the system in terms of the angular displacement (0). Choose O as the clockwise angular displacement of the rod from the system's static equilibrium position.arrow_forward
- Two carts with negligible rolling friction are connected as shown in Figure (1b). An input force u(t) is applied. The masses of the two carts are M, and M, and their displacements are x(t) and q(t), respectively. The carts are connected by a spring k and a damper b. Answer the following questions: (b) By using Newton's Second Law, derive two mathematical equations that describe the motion of the two carts. Hence derive the following two transfer functions: G,(s) = S) U(s) (c) x(1) 9(t) k M, M2 u(t) Figure (1b)arrow_forward4.1. Some problems. 1. A spherical pendulum consists of a bob of mass m suspended from an inextensible string of length l. Determine the Lagrangian and the equations of motion. See Figure |1.7. Note that 0, the polar angle, is measured from the positive z axis, and p, the azimuthal angle, is measured from the x axis in the x – y plane.? Solution:arrow_forwardProblem (6) A 25 kg mass is resting on a spring of 4900 N/m and dashpot of 147 N-se/m in parallel. If a velocity of 0.10 m/sec is applied to the mass at the rest position, what will be its displacement from the equilibrium position at the end of first second?arrow_forward
- QUESTION 3. Consider the mass-spring oscillator without friction: y" + y = 0. Suppose we add a force x(t) which corresponds to a push (to the left) of the mass as it oscillates. We will suppose the push is described by the function x(t) = -u(t – 2n) + u(t – (2n + a)) for some a > 2n which we are allowed to vary. (A small a will correspond to a short duration push and a large a to a long duration push). Here, u(t) is the unit step function. We are interested in solving the initial value problem y" + y = x(t), y(0) = 1, y'(0) = 0.arrow_forward3.32. A single degree of freedom mass-spring system has a mass m=9 kg, a spring stiffness coefficient k = 8 x 10° N/m, a coefficient of dry friction u = 0.15, an initial displacement xo = 0.02 m, and an initial velocity io = 0. Determine the number of cycles of oscillations of the mass before it comes to rest.arrow_forwardThe following mass-and-spring system has stiffness matrix K. The system is set in motion from rest (x, '(0) = x2'(0) = 0) in its equilibrium position (x, (0) = x2(0) = 0) with the given external forces F, (t) = 0 and F, (t) = 270 cos 4t acting on the masses m, and m,, respectively. Find the resulting motion of the system and describe it as a %3D superposition of oscillations at three different frequencies. k2 mi ww m2 k3 - (k, + k2) k2 K= k2 - (k2 + k3) m, = 1, m, = 2; k, = 1, k, = 6, k3 = 2 %3D Find the resulting motion of the system. X4 (t) = X2(t) (Type exact answers, using radicals as needed.)arrow_forward
- A simple single degree of freedom model of a wheel mounted of a spring as shown in figure below .the friction in the system is such that the wheel rolls without slipping. Calculate the natural frequency of oscillation using the energy method .Assume that no energy is lost during the contact x(t) k m,J The rotation of the wheel (of radius r) is given by 0(t) and the linear displacement is denoted by x(t). The wheel has mass ( m) and moment of inertia (J) ,and the spring has stiffness (k ).arrow_forwardQ5 A sphere with mass m and radius r is released with no initial velocity, and it rolls without slipping on the incline as shown in Figure Q5. The angle 9 is 30°. The mass moment of inertia for a sphere is (2/5) m r2. Go Figure Q5 (a) In this case, what is the relationship between linear and angular accelerations? Explain. (b) Draw the free-body diagram and the kinetic diagram for the sphere. (c) Calculate the minimum friction coefficient required between sphere and incline. (d) What is the mass moment of inertia and what is its role in second Newton's law, compared to the role of massarrow_forward4.13. 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. P4.1arrow_forward
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