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System Dynamics

consider the mechanical system shown in the figure in the textbook. Ignore the mass of the bar and assume the motion is small (theta is small) with zero initial conditions. For an input displacement given as x(t)=psinwt, find the steady state response.

Find the inverse Laplace transform for the given function F(s)=(s^2-2-22)/((s+1)(s^2+4s+13)).

Obtain the inverse Labplace transform of F(s)=s/(s^2+2s+10).

The input of the mechanical system shown is the displacement z, and the output displacement is x, both measured from their respective equilibrium positions. Obtain the transfer function of the system G(s)=X(s)/Z(s).

The mechanical system shown consists of a cylinder and a block. The moment of inertia of the cylinder is J with respect to its center bearing, and both the displacement z and the angle of rotation theta are measured from their respective equilibrium positions. At its bearing (point O) the cylinder experiences a viscous moment Md and a restorning moment Mr (not shown in the figure), both in the opposite direction of the roational motion. The inputs of the system are the driving force and moment u1=F and u2=M. The required outputs are y1=displacement of the block, y2=angle of rotation of the rotor, y3=force in the spring, y4=viscous moment at the bearing, and y5=acceleration of the block. (a) Obtain the state space representation of the system. (b) For the given inputs: F=15S(t) N and M(t)=-50u(t) Nm with zero initial conditions, plot the outputs for 10 s. (c) For the given inputs shown in the figure with zero initial conditions, plot the outputs for 10s. (d) For the given inputs plot the outputs for 12 s: F(t)=10sin12t N and M(t)=-40cos8t Nm.

A block is connected to a light rigid rod that is pinned at O. Derive the equation of motion for the system using the displacement x shown in the figure, and then find the natural frequency.

In the liquid-level system shown in the figure, the heat is kept at 1 m for t less than 0. The inflow valve opening is changed at t=0, and the inflow rate is 0.05 m^3/s for t greater than 0. Determine the time needed to fill the tank to a 2.5-m level. Assume that the outflow rate Q m^3/s and heat H are related by Q=0.02H^1/2. The capacitance of the tank is 2 m^2.

Find the Laplace transform of f(t) defined in the figure shown.

Find the inverse Laplace transform for the given function F(s)=(s^2-2-22)/((s+1)(s^2+4s+13)).

Obtain the inverse Labplace transform of F(s)=s/(s^2+2s+10).

The input of the mechanical system shown is the displacement z, and the output displacement is x, both measured from their respective equilibrium positions. Obtain the transfer function of the system G(s)=X(s)/Z(s).

The mechanical system shown consists of a cylinder and a block. The moment of inertia of the cylinder is J with respect to its center bearing, and both the displacement z and the angle of rotation theta are measured from their respective equilibrium positions. At its bearing (point O) the cylinder experiences a viscous moment Md and a restorning moment Mr (not shown in the figure), both in the opposite direction of the roational motion. The inputs of the system are the driving force and moment u1=F and u2=M. The required outputs are y1=displacement of the block, y2=angle of rotation of the rotor, y3=force in the spring, y4=viscous moment at the bearing, and y5=acceleration of the block. (a) Obtain the state space representation of the system. (b) For the given inputs: F=15S(t) N and M(t)=-50u(t) Nm with zero initial conditions, plot the outputs for 10 s. (c) For the given inputs shown in the figure with zero initial conditions, plot the outputs for 10s. (d) For the given inputs plot the outputs for 12 s: F(t)=10sin12t N and M(t)=-40cos8t Nm.

A block is connected to a light rigid rod that is pinned at O. Derive the equation of motion for the system using the displacement x shown in the figure, and then find the natural frequency.

In the liquid-level system shown in the figure, the heat is kept at 1 m for t less than 0. The inflow valve opening is changed at t=0, and the inflow rate is 0.05 m^3/s for t greater than 0. Determine the time needed to fill the tank to a 2.5-m level. Assume that the outflow rate Q m^3/s and heat H are related by Q=0.02H^1/2. The capacitance of the tank is 2 m^2.

Find the Laplace transform of f(t) defined in the figure shown.

The pendulum shown consists of a sphere of mass M connected to a rigid rod of length L and negligible mass. Pin O is free to slide along the horizontal rail and its motion is described by the input displacement u. At equilibrium position, u=0, the rod is vertical, and the spring is unstretched. Derive the equation of motion of the rotational motion of the pendulum for small angle theta.

For the given first order ordinary differential equation xdot+3x=f(t), x(0)=-1, (a) Obtain the solution x(t) by hand using Laplace transform. (b) Plot the solution x(t) and input f(t) on the same graph for the first 3 seconds using MATLAB.

Obtain a mathematical model of the circuit shown in the figure. Obtain the dynamic equations for the electrical circuit shown in the figure. The variables are loop currents i1 and i2.

Obtain the transfer function Eo(s)/Ei(s) of the electrical circuit shown in the figure.

A mass of 20 kg is supported by a spring and damper as shown. The system is at rest for t<0. At t=0, a mass of 2 kg is added to the 20-kg mass. The system vibrates as shown. Determine the spring constant k and the viscous-friction coefficient b.

Two identical uniform bars, each of mass m and length L, are welded together, and the assembly is pinned at O and supported by a spring and a damper as shown. The moment of inertia of the assembly is J0=5mL^2/12. Derive the equation of motion for small angle theta, and then find the natural frequency.

Find the transfer function X0(s)/Xi(s) of the mechanical system shown. The displacements xi and x0 are measured from their respective equilibrium positions.

Derive the equations of motion for the system shown for small angle theta, and then express the equations of motion in the matrix form.

For the given first order ordinary differential equation xdot+3x=f(t), x(0)=-1, (a) Obtain the solution x(t) by hand using Laplace transform. (b) Plot the solution x(t) and input f(t) on the same graph for the first 3 seconds using MATLAB.

Obtain a mathematical model of the circuit shown in the figure. Obtain the dynamic equations for the electrical circuit shown in the figure. The variables are loop currents i1 and i2.

Obtain the transfer function Eo(s)/Ei(s) of the electrical circuit shown in the figure.

A mass of 20 kg is supported by a spring and damper as shown. The system is at rest for t<0. At t=0, a mass of 2 kg is added to the 20-kg mass. The system vibrates as shown. Determine the spring constant k and the viscous-friction coefficient b.

Two identical uniform bars, each of mass m and length L, are welded together, and the assembly is pinned at O and supported by a spring and a damper as shown. The moment of inertia of the assembly is J0=5mL^2/12. Derive the equation of motion for small angle theta, and then find the natural frequency.

Find the transfer function X0(s)/Xi(s) of the mechanical system shown. The displacements xi and x0 are measured from their respective equilibrium positions.

Derive the equations of motion for the system shown for small angle theta, and then express the equations of motion in the matrix form.

The vertical motion of the mechanical system, x, is caused by the oscillating base, z=psinwt. Given the following system and input parameters: m=5 kg, c=20 Ns/m, k=250 N/m, p=0.15 m, and frequency f=1.5 Hz, find the amplitude of the steady state displacement response.

Express the given harmonic functions as Asin(wt+alpha) with lead phase alpha. (a) x(t)=-4cos(wt)+7sin(wt) (b) y(t)=-4sin(wt)+7cos(wt).

Linearize the following function y=e^(x-1)x^1/2 about the point x0=1.

Derive the transfer function of the electrical circuit shown in the figure. Draw a schematic diagram of an analogous mechanical system.

For the spring-damper system shown, the input and output displacemetns, u and x, are measured from the equilibrium positions, and k=250 N/m. The system is given a step input u=0.05 m and a portion of the output curve is shown. Estimate the damping coefficient c.

Find the Laplace transform of the given function: f(t)=(2t^2+t-3)u(t-2).

Use MATLAB's symbolic inverse Laplace transform and we can easily verify our answers obtained by hand in problems 1, 3, 5, and 7.

A uniform disc connected to two springs as shown rolls without slipping horizontally. The displacement variable x is measured from the equilibrium position at the center of the disk, and the moment of inertia J is about the center of the disk. Use energy method to derive the equation of motion, and then find the natural frequency.

Consider the mechanical system shown in the figure. The system is at rest for t greater 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) with the input variable u as the displacement end of the node.

Express the given harmonic functions as Asin(wt+alpha) with lead phase alpha. (a) x(t)=-4cos(wt)+7sin(wt) (b) y(t)=-4sin(wt)+7cos(wt).

Linearize the following function y=e^(x-1)x^1/2 about the point x0=1.

Derive the transfer function of the electrical circuit shown in the figure. Draw a schematic diagram of an analogous mechanical system.

For the spring-damper system shown, the input and output displacemetns, u and x, are measured from the equilibrium positions, and k=250 N/m. The system is given a step input u=0.05 m and a portion of the output curve is shown. Estimate the damping coefficient c.

Find the Laplace transform of the given function: f(t)=(2t^2+t-3)u(t-2).

Use MATLAB's symbolic inverse Laplace transform and we can easily verify our answers obtained by hand in problems 1, 3, 5, and 7.

A uniform disc connected to two springs as shown rolls without slipping horizontally. The displacement variable x is measured from the equilibrium position at the center of the disk, and the moment of inertia J is about the center of the disk. Use energy method to derive the equation of motion, and then find the natural frequency.

Consider the mechanical system shown in the figure. The system is at rest for t greater 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) with the input variable u as the displacement end of the node.

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