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System Dynamics
Consider the mechanical system shown in the figure in the textbook. Both the input translational displacement x(t) and the output angular displacements theta(t) are measured from the equilibrium positions. The rod's mass m is not negligible, so the moment of inertia of the rod J=mL^2/3 has to be included in the equation of motion. The system parameters are m=15 kg, L=0.2 m, b=5 Ns/m, k=20 N/m. Plot the response theta(t) with zero initial conditions for 10s for the three inputs x(t) defined in the figures.
Express the given harmonic functions as Acos(wt-B) with lag phase B. (a) x(t)=-4coswt+7sinwt (b) y(t)=-4sinwt+7coswt .
A tank with a capacitance of 2 m^2 initially has a steady state head of 1 m and a 0.02 m^3/s steady inflow. The inflow is changed to 0.06 m^3/s at t=0, and it remains constant for t greater than 0. Find the time for the head in the tank to reach 2.5 m assuming the outflow throught the pipe is laminar.
Consider a second order mechanical system with the following transfer function: G(s). (a) Determine the damping ratio and the natural frequency wn so that the response of the syistem to a step input has a 5% maximum overshoot and a settling time of 2 s. (b) Find the rise time and peak time of the response.
Obtain the inverse Laplace transform of F(s)=(2s+10)/(s+1)^2(s+4).
Derive the equation of motion for the system shown using the downward displacement variable x, and then find the natural frequency of the motion.
A rod of mass m and length L is pinned at point O and subjected to a vertical force F as shown. The moment of inertia of the rod is Jo=mL^2/9. Derive the equation of motion for small angle of oscillation theta.
Consider the mechanical system shown in the figure. The system is at rest for t less than 0. The input force u is given at t=0. The displacement x is the output of the system and is measured from the equilibrium position. Obtain the transfer function X(s)/U(s).
Express the given harmonic functions as Acos(wt-B) with lag phase B. (a) x(t)=-4coswt+7sinwt (b) y(t)=-4sinwt+7coswt .
A tank with a capacitance of 2 m^2 initially has a steady state head of 1 m and a 0.02 m^3/s steady inflow. The inflow is changed to 0.06 m^3/s at t=0, and it remains constant for t greater than 0. Find the time for the head in the tank to reach 2.5 m assuming the outflow throught the pipe is laminar.
Consider a second order mechanical system with the following transfer function: G(s). (a) Determine the damping ratio and the natural frequency wn so that the response of the syistem to a step input has a 5% maximum overshoot and a settling time of 2 s. (b) Find the rise time and peak time of the response.
Obtain the inverse Laplace transform of F(s)=(2s+10)/(s+1)^2(s+4).
Derive the equation of motion for the system shown using the downward displacement variable x, and then find the natural frequency of the motion.
A rod of mass m and length L is pinned at point O and subjected to a vertical force F as shown. The moment of inertia of the rod is Jo=mL^2/9. Derive the equation of motion for small angle of oscillation theta.
Consider the mechanical system shown in the figure. The system is at rest for t less than 0. The input force u is given at t=0. The displacement x is the output of the system and is measured from the equilibrium position. Obtain the transfer function X(s)/U(s).
Obtain the state space representation of the mechanical system shown. The moment of inertia with respect to the center of the disk is J, and both the displacement z and the angle of rotation theta are measured from their respective equilibrium positions. The input force is u=F, and the outputs are y1=force in the spring and y2=force in the damper.
For the given transfer function G(s)=Y/U, (a) Obtain a state space representation (including both state equation and output equation) using the following state variables: x1=y, x2=ydot, x3=ydoubledot. (b) Obtain a state space representation using the partial fraction expansion approach. (c) Obtain a state space representation using tf2ss command in MATLAB.
Consider the mechanical system shown in the figure. The system is initially at rest. The displacements x1 and x2 are measured from their respective equilibrium positions before the input u is applied. Assume that b1=Ns/m, b2=10Ns/m, k1=4N/m, and k2=20N/m. Obtain the displacement x2(t) when u is a step force input of 2 N. Derive the transfer function and then plot the response for 5 s using step command. The step input force u has a magnitude of 2 N.
In the mechanical system shown, assume that the rod is massless, perfectly rigid, and pivoted at point P. The displacement x is measured from the equilibrium position. Assuming that x is small, that the weight mg at the end of the rod is 5 N, and that the spring constant k is 400 N/m, find the natural frequency of the system. Use the energy method to derive the equation of motion and then find the natural frequency.
The mechanical system shown consists of a uniform disk (mass M and radius R) and a block (mass m) connected to a spring and a damper. Derive the equation of motion of the system, and then find its natural frequency. The moment of inertia J is about the center of the disk, and the mass of the small pulley is negligible.
Derive the inverse Laplace transform of F(s)=1/(s^2(s^2+w^2))
Use symbolic MATLAB to verify your answers in problems 1, 2, and 3.
Consider the mechanical system shown in the figure in the textbook. Find the amplitude of the steady state displacement response using the following system parameters: m=5 kg, b2= 20 N/sm, k1=k2=k=360 n/m, and input parameters: p=180 N, and f=4 Kz for the excitation frequency.
For the given transfer function G(s)=Y/U, (a) Obtain a state space representation (including both state equation and output equation) using the following state variables: x1=y, x2=ydot, x3=ydoubledot. (b) Obtain a state space representation using the partial fraction expansion approach. (c) Obtain a state space representation using tf2ss command in MATLAB.
Consider the mechanical system shown in the figure. The system is initially at rest. The displacements x1 and x2 are measured from their respective equilibrium positions before the input u is applied. Assume that b1=Ns/m, b2=10Ns/m, k1=4N/m, and k2=20N/m. Obtain the displacement x2(t) when u is a step force input of 2 N. Derive the transfer function and then plot the response for 5 s using step command. The step input force u has a magnitude of 2 N.
In the mechanical system shown, assume that the rod is massless, perfectly rigid, and pivoted at point P. The displacement x is measured from the equilibrium position. Assuming that x is small, that the weight mg at the end of the rod is 5 N, and that the spring constant k is 400 N/m, find the natural frequency of the system. Use the energy method to derive the equation of motion and then find the natural frequency.
The mechanical system shown consists of a uniform disk (mass M and radius R) and a block (mass m) connected to a spring and a damper. Derive the equation of motion of the system, and then find its natural frequency. The moment of inertia J is about the center of the disk, and the mass of the small pulley is negligible.
Derive the inverse Laplace transform of F(s)=1/(s^2(s^2+w^2))
Use symbolic MATLAB to verify your answers in problems 1, 2, and 3.
Consider the mechanical system shown in the figure in the textbook. Find the amplitude of the steady state displacement response using the following system parameters: m=5 kg, b2= 20 N/sm, k1=k2=k=360 n/m, and input parameters: p=180 N, and f=4 Kz for the excitation frequency.
A free vibration of the mechanical system shown in the figure indicates that the amplitude of vibration decreases to 25% of the value at t=t0 after four consecutive cycles of motion, as the figure shows. Determine the viscous-friction coefficient b of the system if m=1 kg and k=500 N/m.
A tank of capacitance 2.5 m^2 is initially at steady state with a steady head of 4 m. The outflow through the valve is turbulent and it is related to the head by q=0.2h^1/2. If the inflow is increased by 0.005 m^3/s, determine the change in head using the linearized equations of motion.
Obtain the transfer functino Eo(s)/Ei(s) of the system shown in the figure.
Compute z for the following cases and express the answers in both rectangular form and polar (or Euler) form. Note: Show your detailed step-by-step work; writing the final answers directly from the calculator is not acceptable. (a) z=-6+15i-10e^i-130 degrees (b) z=((-3^1/2+3i)^4(e^i(-5pi/6)))/(2+2i)^2 .
In the given simplified dc servomotor the inductance in the armature circuit can be neglected. Also motor damping (c1) and motor spring constant (k1) are both small and negligible. Obtain the transfer function G(s) between the output rotation theta2 and input voltage ea using the shown set of parameters.
At steady state, the flow rate throughout the liquid-level system shown in the figure is Qbar, and the heads of tanks 1 and 2 are H1bar and H2bar, respectively. At t=0, the inflow rate is changed from Qbar to Qbar+q, where q is small. The resulting changes in the heads (h1 and h2) and flow rates (q1 and q2) are assumed to be small as well. The capacitances of tanks 1 and 2 are C1 and C2, respectively. The resistance of the outflow valve of tank 1 is R1 and that of tank 2 is R2. Obtain the transfer function for the system when q is the input and q2 the output.
Consider the mechanical vibratory system shown in the figure. Assume that the displacement x is measured from the equilibrium position in the absence of the sinusoidal excitation force. The initial conditions are x(0)=0 and xdot(0)=0, and the input force p(t)=Psinwt is applied at t=0. Assume that m=2 kg, b=24 Ns/m, k=200N/m, P=5N, and w=6 rad/s. Obtain only the steady state response Xss(t) of the system.
Find the Laplace transform of the given function: f(t)=5-2t^3-e^(-4t)+t^2e^-6t-4sin5t+tcost-9e^-7tsin8t .
Find the inverse Laplace transform for the given function F(s)=(s^2+7s+2)/((s+1)(s+3)(s+5))
A tank of capacitance 2.5 m^2 is initially at steady state with a steady head of 4 m. The outflow through the valve is turbulent and it is related to the head by q=0.2h^1/2. If the inflow is increased by 0.005 m^3/s, determine the change in head using the linearized equations of motion.
Obtain the transfer functino Eo(s)/Ei(s) of the system shown in the figure.
Compute z for the following cases and express the answers in both rectangular form and polar (or Euler) form. Note: Show your detailed step-by-step work; writing the final answers directly from the calculator is not acceptable. (a) z=-6+15i-10e^i-130 degrees (b) z=((-3^1/2+3i)^4(e^i(-5pi/6)))/(2+2i)^2 .
In the given simplified dc servomotor the inductance in the armature circuit can be neglected. Also motor damping (c1) and motor spring constant (k1) are both small and negligible. Obtain the transfer function G(s) between the output rotation theta2 and input voltage ea using the shown set of parameters.
At steady state, the flow rate throughout the liquid-level system shown in the figure is Qbar, and the heads of tanks 1 and 2 are H1bar and H2bar, respectively. At t=0, the inflow rate is changed from Qbar to Qbar+q, where q is small. The resulting changes in the heads (h1 and h2) and flow rates (q1 and q2) are assumed to be small as well. The capacitances of tanks 1 and 2 are C1 and C2, respectively. The resistance of the outflow valve of tank 1 is R1 and that of tank 2 is R2. Obtain the transfer function for the system when q is the input and q2 the output.
Consider the mechanical vibratory system shown in the figure. Assume that the displacement x is measured from the equilibrium position in the absence of the sinusoidal excitation force. The initial conditions are x(0)=0 and xdot(0)=0, and the input force p(t)=Psinwt is applied at t=0. Assume that m=2 kg, b=24 Ns/m, k=200N/m, P=5N, and w=6 rad/s. Obtain only the steady state response Xss(t) of the system.
Find the Laplace transform of the given function: f(t)=5-2t^3-e^(-4t)+t^2e^-6t-4sin5t+tcost-9e^-7tsin8t .
Find the inverse Laplace transform for the given function F(s)=(s^2+7s+2)/((s+1)(s+3)(s+5))
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