A single-pole oil cylinder valve contains a spool that regulates hydraulic pressure, which is then applied to a piston that drives a load. The transfer function relating piston displacement, Xp(s) to spool displacement from equilibrium, Xv(s), is given by (Qu, 2010):
where A1=effective area of a the valve’s chamber, Kq=rate of change of the load flow rate with a change in displacement, and
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CONTROL SYSTEMS ENGINEERING
- 0.052 J A block attached to a spring, oscillates on a frictionless horizontal surface with a period of 0.3 s. The time needed by the block to move (for the first time) from position x = -A to x = -A/2 is: %3D 0.3 sec 0.1 sec O 0.2 sec 0.15 sec 0.05 sec A traveling wave on a taut string with a tension force T, is given by the wavearrow_forwardHarmonic oscillators. One of the simplest yet most important second-order, linear, constant- coefficient differential equations is the equation for a harmonic oscilator. This equation models the motion of a mass attached to a spring. The spring is attached to a vertical wall and the mass is allowed to slide along a horizontal track. We let z denote the displacement of the mass from its natural resting place (with x > 0 if the spring is stretched and x 0 is the damping constant, and k> 0 is the spring constant. Newton's law states that the force acting on the oscillator is equal to mass times acceleration. Therefore the differential equation for the damped harmonic oscillator is mx" + bx' + kr = 0. (1) k Lui Assume the mass m = 1. (a) Transform Equation (1) into a system of first-order equations. (b) For which values of k, b does this system have complex eigenvalues? Repeated eigenvalues? Real and distinct eigenvalues? (c) Find the general solution of this system in each case. (d)…arrow_forwardQ1 X Determine: k (i) (ii) m CA Figure Q1 Figure Q1 shows a forced spring-mass system with damping, where mass m = 1 kg, spring constant k = 0.2 N/m, and damping coefficient c = 0.3 N-s/m. F (a) This forced spring-mass system with damping can be described by the following differential equation: d²x(t) c dx(t), k - + − x(t): dt² m dt m + 1 - F(t) == m Laplace Transfer Function of this system, This system's steady state gain, damping ratio and natural frequency. (b) Sketch frequency response of the system from part (a) in the form of Bode plot as accurately as you can.arrow_forward
- 3. The relationship between arterial blood flow and blood pressure in a single artery satisfies the following first-order differential equation: dP(t) + dt RC mmHg (cm³/s) P(t) = where Qin is the volumetric blood flow, R is the peripheral resistance, and C is arterial compliance (all constant). Qin-60 cm³/s and the initial arterial pressure is 6 mmHg. Also, assume R = 4 and C= 0.4- Oin cm³ mmHg (a) Find the transient solution Ptran(t) for the arterial pressure. The unit for P(t) is mmHg. (b) Determine the steady-state solution Pss(t) for the arterial pressure. (c) Determine the total solution P(t) assuming that the initial arterial pressure is 0.arrow_forwardA mass-spring-dashpot system with constants k = 0,6, c = 0,7 and m = 100 g makes mechanical vibration. The system is exposed to no external force. If the system is at rest, initial position and velocity are 2 m and 6 m/sn, respectively. Under these circumstances, find the general equation of certain motion using the "Method of Laplace Transforms". 1. %3Darrow_forwardConsider the following system mä(t) + cx(t) + kx(t) = F,8(t) + F28(t – t1) Assume zero initial conditions. The units are in Newtons. If m = 42.7334 kg, c = 44.7394 Ns/m, k = 6776.8559 N/m, F1 = 47.5525 N.s, F2 = -5.3708 N.s and t = 3.9737 s, the velocity of the system at t2 = 2.3387 s isarrow_forward
- Figure 1 below depicts the popular Spring-Mass-Damper system in which m is the mass, c represented by the dashpot symbol in the figure is called the damping factor, and k is the spring constant. The Spring-Mass-Damper system is with m = 20 kg, c = 20 Ns/m, k = 4000 Px k în с Li m Figure 1: The Spring-Mass-Damper system N/m. Moreover, denote by x(t) the displacement of the spring (from its equilibrium position). The system is acted on by a periodic harmonic force F(t) = Fo sin(wt) where Fo and w are the amplitude and frequency of the harmonic force, respectively. Given that Fo= 100 N and w = 20 rad/s. The Spring-Mass-Damper in Figure 1 is modelled by the following ODE: d²x dx m. +c- + kx = F(t). dt² dt F(t) dx (0) dt (1) Assume that x(0) = 0.01 m and = 0.0. Your duty as an engineer is to analyse this Spring- Mass-Damper system by fulfilling the following requirements: a) Establish the spring displacement trajectory x(t) by solving analytically the ODE equation (1) modelling the…arrow_forwardFind the differential equation of the mechanical system in Figure 1(a) To obtain the differential equation of motion of the mass and spring system given in Fig. 1. (a) one may utilize the Newton's law for mass and spring relations defined as shown in Fig. 1. (b) and (c) use f = cv for viscous friction, where v is the velocity of the motion and c is a constant. Z/////// k M F. F, F F F, F, k EF=ma F = k(x, - x,) = kx (b) (c) Figure 1: Mass-spring system (a), Force relations of mass (b) and spring (c)arrow_forward3. Consider the mechanical system shown in Fig. 3. Let V3(t) be the input and the acceleration of the mass be the output. Derive the state equations and the output equation using linear graphs and normal trees. V,(t) В m k2 101 k, Figure 3: A mechanical system with an across-variable sourcearrow_forward
- a) A vehicle circulates on a road as shown in Figure Q2a. The road profile can be modelled as the input u(t). The vehicle is modelled as a quarter car of mass m, and the suspension has a spring stiffness coefficient k = 2 Nm and a damper of coefficient c = 2 Nm s. Find the position of the mass, y(t) and any time t if the road profile is a unit step. m 一 Road Figure Q2a b) In Figure Q2b, a disk flywheel J of mass m 32 kg and radius r = 0.5 m is driven by an electric motor that when it is working produces an oscillating torque of Tin = 10sin(wt) N- m. The shaft bearings may be modelled as viscous rotary dampers with a damping coefficient of BR = 0.4 N-m-s/rad and stiffness coefficient KR = 2 Nm If the flywheel is at rest at t 0 and the power is suddenly applied to the motor, do the following [Hint:J = mr²/2]: (i) Find the natural frequency of oscillation of the disk expressed in Hz. (ii) Find the damping ratio for this system. (iii) Describe in no more than 3-4 lines the behaviour of the…arrow_forward6. The electro-mechanical system shown below consists of an electric motor with input voltage V which drives inertia I in the mechanical system (see torque T). Find the governing differential equations of motion for this electro-mechanical system in terms of the input voltage to the motor and output displacement y. Electrical System puthiy C V V₁ R bac (0) T bac T Motor - Motor Input Voltage - Motor Back EMF = Kbac ( - Motor Angular Velocity - Motor Output Torque = K₂ i Kbacs K₁ - Motor Constants Mechanical System M T Frictionless Supportarrow_forwardEquation 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=0.05marrow_forward
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