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physics
classical dynamics of particles

System Dynamics 3rd edition William Palm III - Solutions

For the systems shown in Figure, assume that the resulting motion is small enough to be only horizontal and determine the expression for the equivalent damping coefficient ce that relates the applied force f to the resulting velocity v.a.b.
Refer to Figure a, which shows a ship's propeller, drive train, engine, and flywheel. The diameter ratio of the gears is D1/D2 = 1.5. The inertias in kg-m2 of gear 1 and gear 2 are 500 and 100, respectively. The flywheel, engine, and propeller inertias are 104, 103, and 2500, respectively. The
In this problem, we make all the same assumptions as in Problem 4.75, but we do not discount the flywheel inertia, so our model consists of three inertias, as shown in Figure. Obtain the natural frequencies of the system.
Refer to Figure, which shows a turbine driving an electrical generator through a gear pair. The diameter ratio of the gears is D2/D1 = 1.5. The inertias in kg-m2 of gear 1 and gear 2 are 100 and 500, respectively. The turbine and generator inertias are 2000 and 1000, respectively. The torsion
Refer to Figure, which is a simplified representation of a vehicle striking a bump. The vertical displacement x is 0 when the tire first meets the bump. Assuming that the vehicle's horizontal speed v remains constant and that the system is critically damped, obtain the expression for x(t).
Refer to Figure a, which shows a water tank subjected to a blast force f(t). We will model the tank and it's supporting columns as the mass-spring system shown in part (b) of the figure. The blast force as a function of time is shown in part (c) of the figure. Assuming zero initial conditions,
The beam shown in Figure has been stiffened by the addition of a spring support. The steel beam is 3 ft long, 1 in thick, and 1 ft wide, and its mass is 3.8 slugs. The mass m is 40 slugs. Neglecting the mass of the beam,a. Compute the spring constant k necessary to reduce the static deflection to
The "sky crane" shown on the text cover was a novel solution to the problem of landing the 2000 lb Curiosity rover on the surface of Mars. Curiosity hangs from the descent stage by 60-ft long nylon tethers (Figure a). The descent stage uses its thrusters to hover as the rover is lowered to the
Obtain the equations of motion for the system shown in Figure for the case where m1 = m2 and m2 = 2m. The cylinder is solid and rolls without slipping. The platform translates without friction on the horizontal surface.
Obtain the equations of motion for the system shown in Figure for the case where m1 = m2 = m. The cylinder is solid and rolls without slipping. The platform translates without friction on the horizontal surface.
In Figure a tractor and a trailer is used to carry objects, such as a large paper roll or pipes. Assuming the cylindrical load m3 rolls without slipping, obtain the equations of motion of the system.
Suppose a mass m moving with a speed v1 becomes embedded in mass m2 after striking it (Figure 4.6.1). Suppose m2 = 5m. Determine the expression for the displacement x(t) after the collision.
Consider the system shown in Figure 4.6.3. Suppose that the mass m moving with a speed v\ rebounds from the mass m2 = 5m after striking it. Assume that the collision is perfectly elastic. Determine the expression for the displacement x(t) after the collision.
The mass m1 is dropped from rest a distance h onto the mass m2, which is initially resting on the spring support (Figure). Assume that the impact is inelastic so that mi sticks to m2. Calculate the maximum spring deflection caused by the impact. The given values are m1 = 0.5 kg, m2 = 4 kg,k = 400
Figure shows a mass m with an attached stiffness, such as that due to protective packaging. The mass drops a distance h, at which time the stiffness element contacts the ground. Let x denote the displacement of m after contact with the ground. Determine the maximum required stiffness of the
Figure represents a drop forging process. The anvil mass is m1 = 1000 kg, and the hammer mass is m2 = 200 kg. The support stiffness is k = 107 N/m, and the damping constant is c = 1 N-s/m. The anvil is at rest when the hammer is dropped from a height of h= 1 m. Obtain the expression for the
Refer to Figure. A mass m drops from a height h and hits and sticks to a simply supported beam of equal mass. Obtain an expression for the maximum deflection of the center of the beam. Your answer should be a function of h, g, m, and the beam stiffness k.
Determine the equivalent spring constant of the arrangement shown in Figure. All the springs have the same spring constant k.
(a) Obtain the equations of motion of the system shown in Figure. The masses are m1 = 20 kg and m2 = 60 kg. The spring constants are k1 = 3 x 104 N/m and k2 = 6 x 104 N/m.(b) Obtain the transfer functions X1(s)/F(s) and X2(s)/F(s).(c) Obtain a plot of the unit-step response of x1 for zero initial
(a) Obtain the equations of motion of the system shown in Figure.(b) Suppose the inertias are I1 = I and I2 = 21 and the torsional spring constants are k1 = k2 = h = k. Obtain the transfer functions ÆŸ1(s)/T2(s) and ÆŸ2(s) / T2(s) in terms of I and k.(c) Suppose that I = 10
Refer to part (a) of Problem 4.90. Use MATLAB to obtain the transfer functions X1(s)/F(s) and X2(s)/F(s). Compare your answers with those found in part (b) of Problem 4.90.In Question 90(a) Obtain the equations of motion of the system shown in Figure. The masses are m1 = 20 kg and m2 = 60 kg. The
Refer to Problem 4.91. Use MATLAB to obtain the transfer functions Ɵ1(s)/T2(s) and Ɵ2(s)/T2(s) for the values I1 = 10, I2 = 20, and k1 = k2 = k3 = 60. Compare your answers with those found in part (b) of Problem 4.91. In Problem 91 Suppose the inertias are I1 = I and I2 = 21 and the torsional
(a) Obtain the equations of motion of the system shown in Figure. Assume small angles. The spring is at its free length when Î¸1 = Î¸2 = 0.(b) For the values m1 = 1 kg, m2 = 4 kg, k = 10 N/m, L1 = 2 m, and L2 = 5 m, use MATLAB to plot the free response of Î¸1 if Î¸1(0) = 0.1 rad and Î¸1
(a) Obtain the equations of motion of the system shown in Figure.(b) Suppose that the masses are m1 = 1 kg, m2 = 2 kg, and the spring constants are k1 = k2 = k3 = 1.6 x 104 N/m. Use MATLAB to obtain the plot of the free response of x1. Use x1 (0) = 0.1 m, x2(0) = 1 (0) = 2(0) = 0.
Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure.
Use the MATLAB series and feedback functions to obtain the transfer functions C(s)/R(s) and C(s)/D(s) for the block diagram shown in Figure.
Obtain the state model for the reduced-form model 5 + 7 + 4x = f(t).
Obtain the state model for the reduced-form model2 d3y/dt3 + 5 d2y/dt2 + 4dy/dt + 7y = f(t)
Obtain the state model for the reduced-form model 2 + 5 + 4x = 4y(t).
Obtain the state model for the transfer-function modelY(s)/F(s) = 6/(3s2 + 6s + 10)
Obtain the state model for the two-mass system whose equations of motion arem1(1 + k1(x1 - x2) = f(t)m22 - k1(x1 - x2) + k2x2 = 0
Obtain the state model for the two-mass system whose equations of motion for specific values of the spring and damping constants are101 + 81 - 52 + 40x1 - 25x2 = 052 - 25x1 + 25x2 -5x1+52 = f(t)
Put the following model in standard state-variable form and obtain the expressions for the matrices A, B, C, and D. The output is x.2 + 5 + 4x = 4y(t)
Given the state-variable model 1 = - 5x1 + 3x2 + 2u1 2 = - 4x2 + 6u2 and the output equations y1=x1 + 3x2 + 2u1 y2 = x2
Given the following state-variable models, obtain the expressions for the matrices A, B, C, and D for the given inputs and outputs. a. The outputs are x1 and x2; the input is u. 1 = - 5x1 + 3x2 2 = x1 - 4x2 + 5u b. The output is x1; the inputs are u1 and u2. 1 = -5x1 + 3x2 + 4u1 2 = x1 - 4x2 +
Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure.
Obtain the expressions for the matrices A, B, C, and D for the state-variable model you obtained in Problem 16. The outputs are xi and x2.In Problem 16101 + 81 - 52 + 40x1 - 25x2 = 052 - 25x1 + 25x2 -5x1+52 = f(t)
The transfer function of a certain system isY(s)/F(s) = (6s + 7)/(s + 3)Use two methods to obtain a state-variable model in standard form. For each model, relate the initial value of the state-variable to the given initial value y(0).
The transfer function of a certain system isY(s)/ F(s) = (s + 2)/(s2 + 4s + 3)Use two methods to obtain a state-variable model in standard form. For each model, relate the initial values of the state variables to the given initial values y(0) and (0).
Use MATLAB to create a state-variable model; obtain the expressions for the matrices A, B, C, and D, and then find the transfer functions of the following models, for the given inputs and outputs. a. The outputs are x1 and x2; the input is u. 1 = -5x1 + 3x2 2 = x1 - 4x2 + 5u b. The output is xi;
Use MATLAB to obtain a state model for the following equations; obtain the expressions for the matrices A, B. C. and D. In both cases, the input is f(t); the output is y.a.b.Y(s)/F(s) = 6/(3s2+ 6s + 10)
Use MATLAB to obtain a state-variable model for the following transfer functions, a.a. Y(s)/F(s) = (6s + 7)/(s + 3)b. Y(s)/ F(s) = (s + 2)/(s2 + 4s + 3)
For the following model the output is x1 and the input is f(t).ẋ1 = - 5x1 + 3x2ẋ2 = x1 - 4x2 + 5f(t)a. Use MATLAB to compute and plot the free response for x1 (0) = 3, and x2(0) = 5.b. Use MATLAB to compute and plot the unit-step response for zero initial conditions.c. Use MATLAB to compute and
Given the state-variable model x1 = -5x1 + 3x2 + 2u1 x2 = -4x2 + 6u2 And the output equations y1 = x1 + 3x2 + 2u1 y2 = x2 Use MATLAB to find the characteristic polynomial and the characteristic roots.
The equations of motion for a two-mass, quarter-car model of a suspension s\ stem arem1ẍ1 = c1(ẋ2 – ẋ1) +k1(x2 –x1)m2ẍ2 = -c1(ẋ2 – ẋ1) – k1(x2 – x1) + k2(y - x2)Suppose the coefficient values are: m1 = 240 kg, m2 = 36 kg, k1 = 1.6 x 104 N/m, k2 = 1.6 x 105 N/m, c1 = 98 N
A representation of a car's suspension suitable for modeling the bounce and pitch motions is shown in Figure, which is a side view of the vehicle's body showing the front and rear suspensions. Assume that the car's motion is constrained to a vertical translation x of the mass center and rotation θ
Obtain the transfer function X(s)/F(s) from the block diagram shown in Figure.
a. Use a MATLAB ode function to solve the following equation for 0 ≤ / ≤ 12. Plot the solution. = costy(0) = 6b. Use the closed-form solution to check the accuracy of the numerical method.
a. Use a MATLAB ode function to solve the following equation for 0 ≤ / ≤ 1. Plot the solution. = 5e-4t y(0) = 2 b. Use the closed-form solution to check the accuracy of the numerical method.
a. Use a MATLAB ode function to solve the following equation for 0 ≤ t ≤ l. Plot the solution. + 3y = 5e4t y(0) = 10 b. Use the closed-form solution to check the accuracy of the numerical method.
a. Use a MATLAB ode function to solve the following nonlinear equation for 0 ≤ t ≤ 4. Plot the solution. + sin y = 0 y(0) = 0.1 b. For small angles, sin y ( y. Use this fact to obtain a linear equation that approximates equation (1). Use the closed-form solution of this linear equation to
Sometimes it is tedious to obtain a solution of a linear equation, especially if all we need is a plot of the solution. In such cases, a numerical method might be preferred. Use a MATLAB ode function to solve the following equation for 0 ≤ t ≤ 7. Plot the solution.y + 2y = f(t)y(0) = 2Where
A certain jet-powered ground vehicle is subjected to a nonlinear drag force. Its equation of motion, in British units, is 50 = f - (20u + 0.05u2) Use a numerical method to solve for and plot the vehicle's speed as a function of time if the jet's force is constant at 8000 lb and the vehicle starts
The following model describes a mass supported by a nonlinear, hardening spring. The units are SI. Use g = 9.81 m/s2. 5 = 5g - (900y+ 1700y3)
Van der Pol's equation is a nonlinear model for some oscillatory processes. It isÿ-b(1 - y2) ẏ + y = 0Use a numerical method to solve for and plot the solution for the following cases:1. b = 0.1,y(0) = ẏ (0) = 1,0 ≤ f ≤ 252. b = 0.1,y(0) = ẏ (0) = 3,0 ≤ t ≤ 253. b = 3,y(0) = ẏ (0)
Van der Pol's equation isÿ-b(1- y2) ẏ + y = ()This equation can be difficult to solve for large values of the parameter b. Use b = 1000 and 0 ≤ f ≤ 3000, with the initial conditions y(0) = 2 and ẏ(0) = 0. Use ode45 to plot the response.
The equation of motion for a pendulum whose base is accelerating horizontally with an acceleration a(t) isLθ + g sin θ = a(t)cosθSuppose that g = 9.81 m/s2, L = 1 m, and θ (0) = 0. Solve for and plot θ (f) for 0 ≤ / ≤ 10 s for the following three cases.a. The acceleration is constant: a =
Draw a block diagram for the following equation. The output is X(s) the inputs are F(s) and G(s).5ẍ + 3 ẋ + 7x = 10/(0 – 4g(f)
Suppose the spring in Figure is nonlinear and is described by the cubic force-displacement relation. The equation of motion ism = c( - ) + k1(y - x) + k2(y - x)3Where m = 100, c = 600, k1 = 8000, and k2 = 24000. Approximate the unit-step input y(t) with y(t) = 1 - e-t/Ï„, where r is
Create a Simulink model to plot the solution of the following equation for 0 ≤ t ≤ 6.10ÿ = 7 sin 4t + 5 cos 3ty (0) = 4 (0) = I
A projectile is launched with a velocity of 100 m/s at an angle of 30° above the horizontal. Create a Simulink model to solve the projectile's equations of motion, where x and y are the horizontal and vertical displacements of the projectile.ẍ = 0x(0) = 0ẋ (0) = 100 cos 30° ÿ =
In Chapter 2 we obtained an approximate solution of the following problem, which has no analytical solution even though it is linear. +x = tan t................... x(0) = 0 The approximate solution, which is less accurate for large values of t, is x(t) = 1/3 t3 -r2 + 3t -3 + 3e-t Create a
Construct a Simulink model to plot the solution of the following equation for 0 < t < 10. 15 + 5x = Aus, (t) - 4us (t - 2)........................x(0) = 2
Use Simulink to solve Problem 18 for zero initial conditions, u1 a unit-step input, and u2: = 0.In Problem 18Given the state-variable modelẋ 1 = - 5x1 + 3x2 + 2u1ẋ 2 = - 4x2 + 6u2and the output equations y1=x1 + 3x2 + 2u1y2 = x2
Use Simulink to solve Problem 18 for the initial conditions x1(0) = 4, x2(0) = 3, and u1 = u2 = 0. In problem 18 Given the state-variable model 1 = - 5x1 + 3x2 + 2u1 2 = - 4x2 + 6u2 and the output equations y1=x1 + 3x2 + 2u1 y2 = x2
Use Simulink to solve Problem 19a for zero initial conditions and u = 3 sin 2t. In Problem 19a The outputs are x1 and x2; the input is u. 1 = - 5x1 + 3x2 2 = x1 - 4x2 + 5u
Use Simulink to solve Problem 26c. In Problem 26c Use MATLAB to compute and plot the response for zero initial conditions with the input f(t) = 3sin 10πt, for 0 ≤ / ≤ 2.
Use the Transfer Function block to construct a Simulink model to plot the solution of the following equation for 0 ≤ t ≤ 4. 2 + I2 + 10x2 = 5us(t) - 5us(t - 2) x(0) = (0) = 0
Draw a block diagram for the following model. The output is X(s) the inputs are F(s) and G(.v). Indicate the location of Y(s) on the diagram. = y - 5x + g(t) = I0f(t) - 30x
Construct a Simulink model to plot the solution of the following equation for 0 ≤ t ≤ 4. 2 + 12 + 10x2 = 5 sin 0.8t x(0) = (0) = 0
Use the Saturation block to create a Simulink model to plot the solution of the following equation for 0 ≤ t ≤ 6. 3 + y = f(t) y(0) = 2 Where
Construct a Simulink model of the following problem.5 + sin x = f(t)x(0) = 0The forcing function is
Create a Simulink model to plot the solution of the following equation for 0 ≤ t ≤ 3. ẋ + 10x2 = 2 sin 4tx(0) = 1
Construct a Simulink model of the following problem. 10 + sin x = f(t) x(0) = 0 The forcing function is f(t) = sin 2t. The system has the dead-zone nonlinearity shown in Figure 5.6.6.
The following model describes a mass supported by a nonlinear, hardening spring. The units are SI. Use g = 9.81 m/s2.5ÿ = 5g - (900y + 1700y3)y(0) = 0.5ẏ(0) = 0.Create a Simulink model to plot the solution for 0 ≤ t ≤ 2.
Consider the system for lifting a mast, discussed in Chapter 3 and shown again in Figure. The 70-ft-long mast weighs 500 lb. The winch applies a force f = 380 lb to the cable. The mast is supported initially at an angle of Î¸ = 30° and the cable at A is initially horizontal. The
A certain mass, m = 2 kg, moves on a surface inclined at an angle ( = 30° above the horizontal. Its initial velocity is v(0) = 3 m/s up the incline. An external force of f1 = 5 N acts on it parallel to and up the incline. The coefficient of dynamic friction is μd = 0.5. Use the Coulomb Friction
If a mass-spring system has Coulomb friction on the horizontal surface rather than viscous friction, its equation of motion ismÿ = -ky +f/(t) – μmg if ẏ > 0mÿ = - ky + f(t) + μmg if ẏ < 0Where μ is the coefficient of friction develop a Simulink model for the case where
Redo the Simulink suspension model developed in subsection 5.6.2, using the spring relation and input function shown in Figure, and the following damper relation.a.b.Use the simulation to plot the response. Evaluate the overshoot and undershoot. Figure
Referring to Figure, derive the expressions for the variables C(s), E(s), and M(s) in terms of R(s) and D(s).
Consider the system shown in Figure P5.60. The equations of motion arem11 + (c1 + c2) 1] + (k1 + k2)x1 - c22 - k2x2 = 0m22 + c22 + k2x2 _ c21 - k2x1 = f(t)Suppose that m1 = m2 = I, k1 = 3, c2 = l, k1 = 1, and k2 = 4.a. Develop a Simulink simulation of this system. In doing this, consider whether to
Referring to Figure, derive the expressions for the variables C(s), E(s), and M(s) in terms of R(s) and D(s).
Use the MATLAB series and feedback functions to obtain the transfer functions X(s)/F(s) and X(s)/G(s) for the block diagram shown in Figure.
Use the MATLAB series and feedback functions to obtain the transfer functions C(s)/R(s) and C(s)/D(s) for the block diagram shown in Figure.
Determine the equivalent resistance Rc of the circuit shown in Figure, such that vs = Rei. All the resistors are identical and have the resistance R.
The resistance of a telegraph line is R = 10 Ω, and the solenoid inductance is L = 5 H. Assume that when sending a "dash," a voltage of 12 V is applied while the key is closed for 0.3 s. Obtain the expression for the current i(t) passing through the solenoid.
Obtain the model of the voltage vo, given the supply voltage vs, for the circuit shown in Figure.
Obtain the model of the voltage vo, given the supply voltage vs, for the circuit shown in Figure.
Obtain the model of the current i, given the supply voltage vs, for the circuit shown in Figure.
Obtain the model of the voltage vo given the supply current is, for the circuit shown in Figure.
Obtain the model of the currents i1, i2, and i3, given the input voltages v1 and v2, for the circuit shown in Figure.
Obtain the model of the currents i1, i2, and the voltage i3, given the input voltages v1 and v2, for the circuit shown in Figure.
For the circuit shown in Figure, determine a suitable set of state variables, and obtain the state equations.
For the circuit shown in Figure, determine a suitable set of state variables, and obtain the state equations.
For the circuit shown in Figure, determine a suitable set of state variables, and obtain the state equations.
Determine the voltage vi in terms of the supply voltage vs for the circuit shown in Figure.
Use the impedance method to obtain the transfer function Vo(s)/ Vs(s) for the circuit shown in Figure.
Use the impedance method to obtain the transfer function i(s)/Vs(s) for the circuit shown in Figure.
Use the impedance method to obtain the transfer function Vo(s)/Vs(s) for the circuit shown in Figure.
Use the impedance method to obtain the transfer function Vo(s)/Is(s) for the circuit shown in Figure.

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