Mechanical Systems

Abstract

A mechanical system is a system in which the dominant forms of energy storage, transfer, and dissipation are described by Newton’s laws. Fluid systems are described by Newton’s laws, but use different variables due to fluid mass. They will be addressed in Chap. 5.Mechanical energy is stored as kinetic energy and as strain energy. Gravitational potential energy is presented as a force source. Mechanical energy is dissipated due to shear of a fluid, a material in its plastic state, or between two solid surfaces. In order to work with scalar equations, motion is restricted to single axes. Translational and rotational motions are separate energy storage modes. The parameter values of the mechanical elements can be calculated from the geometry and material properties of mechanical components in many instances. Otherwise, the parameters are experimentally determined by dynamic tests. The mechanical properties of viscoelastic materials, which include natural and synthetic polymers and biological tissues, are described by the dynamic response.

Problems

Reminders

• 1. Draw a linear graph by identifying nodes of distinct values of the across variable, velocity for mechanical systems, on the schematic. A rigid object has a single velocity. Ideal rods, bars, and shafts are rigid and massless. Identify the two nodes of the across variable associated with each element in the schematic by identifying nodes on either end of the schematic symbol, except for translational masses and rotational inertias,Tables 4.6 and 4.7. Mass and inertia are rigid. They have a velocity node and an inertial (ground) reference. After identifying two nodes for each element, eliminate the redundant nodes, and then label the distinct nodes with numbers, except ground, which is identified with a “g.” All ground nodes are the same velocity.

• 2. Use nodal notation for the across variable drop and indicate the positive direction of the through variable in the direction of the drop in the across variable, except for sources. Sources increase the across variable in the direction of the through variable “flow.”

• 3. Check the units of the system equation in terms of the power variables of the system and time, not fundamental units or conventional units.

• 4. Use superposition to create a pulse input from scaled and time shifted Heaviside step functions. Use superposition to create the response function, by scaling and time-shifting the unit step response with the same factors and time shifts used to create the input function. Each step response in the response function must be multiplied by the corresponding time shifted Heaviside unit step function to zero-out the response of that term unit its corresponding input acts on the system.

• 5. A dynamic system has only one characteristic equation, independent of our choice of the output variable. Consequently, there is only one time constant of a first-order system. All of the power variables in the system vary at the same rate, although their step responses will differ. Some variables will grow, while others will decay, but they do so in synchronization. Second-order systems have two eigenvalues. If the eigenvalues are real, then the system is overdamped and does not have an oscillatory homogeneous or step response. If the eigenvalues are complex conjugates, then the system is underdamped and the homogeneous and step responses are oscillatory.

Problem 4.1 A rotational mechanical which consists of a angular velocity source, a drag cup, with damping constant, $b=8\,\,\text{N}\cdot \text{m}\cdot \text{sec/rad}$, and a torsion spring constant, $K=60\,\,\text{N}\cdot \text{m/rad}$ is shown in the schematic Fig. P4.1a. The system had reached steady-state under the input angular velocity − 20 rad/sec, applied at an unknown time, $t<0$, before a step input of$\text{ }\!\!\Delta\!\!\text{ }\Omega ( t )=50\,\,\text{rad/sec}\,{{u}_{s}}( t )$ was applied at time, $t=0$, as plotted in Fig. P4.1b.

• 4.1.a Derive the system equation that relates the applied velocity input to

  • i The torque acting through the torsion spring.

  • ii The velocity drop across the drag cup.

  • iii The velocity drop across the torsion spring.

Check the units of the system equations in terms of the power variables and time.

• 4.1.b The input velocity Ω(t) shown in the plot. Determine the unit step responses of the system equations derived in part a. Use superposition to determine the responses to the angular velocity input plotted in Fig. 4.1b.

• 4.1.c Plot the responses using Mathcad or MATLAB.

Fig. P4.1

a System with an angular velocity source, drag cup (fluid coupling) and torsion spring. b Angular velocity input applied to the system

Problem 4.2 A rotational mechanical system consisting of an angular velocity source, a drag cup (fluid coupling) with damping b and a flywheel with mass moment of inertia J is shown in Fig. P4.2a. The damping constant is $b=80\,\,\text{N}\cdot \text{m}\cdot \text{sec/rad}$. The mass moment of inertia is $J=12\,\,\text{kg}\cdot {{\text{m}}^{\text{2}}}$. The system had reached steady-state under a step input in angular velocity of 40 rad/sec applied at an unknown time, $t<0$, before a step input was applied at time, $t=0$, reducing the angular velocity input to − 20 rad/sec, as shown in Fig. P4.2b.

• 4.2.a Derive the system equation that relates the applied velocity input to

  • i The torque acting to accelerate the flywheel.

  • ii The angular velocity of the flywheel.

Check the units of the system equations in terms of the power variables and time.

• 4.2.b Determine the unit step responses of the system equations derived in part a. Use superposition to solve the system equations for the angular velocity input shown in Fig. P4.2b.

• 4.2.c Plot the responses using Mathcad or MATLAB.

Fig. P4.2

a System with an angular velocity source, drag cup (fluid coupling), and flywheel, b Angular velocity input applied to the system

Problem 4.3 A rotational mechanical which consists of a torque source T(t), a rotational inertia, $J=8\,\,\text{kg}\cdot {{\text{m}}^{\text{2}}}$, and a hydrodynamic bearing with $b=5\,\,\text{N}\cdot \text{m}\cdot \text{sec/rad}$ is shown in Fig. P4.3a. The system had reached steady-state under the torque of $50\,\,\text{N}\cdot \text{m}$applied at a unknown time, $t<0$, when at time, t = 0, the input torque is increased to $150\,\,\text{N}\cdot \text{m}$, as shown in Fig. P4.3b.

• 4.3.a Reduce the equation list to the differential system equation which relates the input torque T(t) to

  • i The angular velocity of inertia J.

  • ii The torque TJ acting to accelerate the inertia J.

  • iii The torque acting shear the fluid of the hydrodynamic bearing, T_b.

Check the units of the system equations in terms of the power variables and time.

• 4.3.b Solve the system equations of part 4.3a for the input plotted in Fig. P4.3b.

• 4.3.c Plot the response using Mathcad or MATLAB.

Fig. P4.3

a Rotational mechanical system comprised of a torque source, mass moment of inertia J and damping b, b Torque input applied to the system

Problem 4.4 A rotational mechanical system consisting of an angular velocity source, a drag cup (fluid coupling) with damping coefficient, $b=8\,\,\text{N}\cdot \text{m}\cdot \text{sec/rad}$, and a torsional spring with spring constant, $K=60\,\,\text{N}\cdot \text{m/rad}$, is shown in Fig. P4.4a. The system has reached steady-state under the input of $\Omega ( t )=-20\,\text{rad/s}$ applied at an unknown time, $t<0$, when at time, $t=0$, the input angular velocity is increased to $\Omega (t)=30\,\text{rad/s}$, as shown in Fig. P4.4b.

• 4.4.a Derive the system equation that relates the input angular velocity to

  • i The angular velocity of the across the spring, Ω_2g.

  • ii The torque acting through the spring, T_K.

  • iii The velocity drop across the drag cup (fluid coupling), Ω₁₂.

Check the units of the system equations in terms of the power variables and time.

• 4.4.b Determine the unit step response and the use superposition to solve the system equations of part a for the velocity input Ω(t) shown in Fig. P4.4b.

• 4.4.c Plot the response using Mathcad or MATLAB.

Fig. P4.4

a System with an angular velocity source, drag cup (fluid coupling) and torsion spring. b Angular velocity input applied to the system

Problem 4.5 A rotational mechanical which consists of a torque source T(t), a rotational inertia, $J=5\,\,\text{kg}\cdot {{\text{m}}^{\text{2}}}$, and a hydrodynamic bearing with $b=8\,\,\text{N}\cdot \text{m}\cdot \text{s/rad}$is shown in Fig. P4.5a below. The system had reached steady-state under the torque of $50\,\,\text{N}\cdot \text{m}$applied at a unknown time, $t<0$, when at time, $t=0$, the input torque pulse shown in Fig. P4.5b is applied to the system.

• 4.5.a Reduce the equation list to the differential system equation which relates the input torque T(t) to

  • i The angular velocity of inertia J.

  • ii The torque TJ acting to accelerate the inertia J.

  • iii The torque acting shear the fluid of the hydrodynamic bearing, T_b.

Check the units of the system equations in terms of the power variables and time.

• 4.5.b Determine the unit step response and the use superposition to solve the system equations of part a for the input shown in Fig. P4.5b.

• 4.5.c Plot the response using Mathcad or MATLAB.

Fig. P4.5

a Rotational mechanical system comprised of a torque source, mass moment of inertia, J, and damping, b, b Torque input applied to the system

Problem 4.6 A rotational mechanical system modeled as consisting of an angular velocity source, two dampers, ${{b}_{\,1}}=60\,\,\text{N}\cdot \text{m}\cdot \text{sec}$ and ${{b}_{\,2}}=5\,\,\text{N}\cdot \text{m}\cdot \text{sec}$, and torsional spring, $K=40\,\,\text{N}\cdot \text{m/rad}$, is shown schematically in Fig. P4.6a.

• 4.6.a Derive the system equations for

  • i The difference in the angular velocity of the two ends of spring K.

  • ii The torque in damper b₁.

  • iii The angular velocity across damper b₂.

and check their units.

A step input of angular velocity $\Omega ( t )=-20\,{{u}_{s}}( t )$ was applied to the system at a time, $t<0$, sufficient for the system to reach steady-state prior to when a second step input is applied at time, $t=0$, as plotted in Fig. P4.6b.

• 4.6.b Solve the system equations.

• 4.6.c Plot the responses using Mathcad or MATLAB

• 4.6.d Plot the power flow from the source using Mathcad or MATLAB

Fig. P4.6

a Angular velocity source Ω(t) is connected to the input shaft of a fluid coupling with damping, b ₁. The output shaft of the fluid coupling connects to the torsion spring, K, the end of which is free to rotate on bearings with damping, b ₂, b Angular velocity applied to the system by the source Ω(t)

Problem 4.7 A translational mechanical system is modeled as consisting of a force source, two dampers, ${{b}_{\,1}}=5.0\,\,\text{kN}\cdot \rm{sec}\text{/m}$ and ${{b}_{\,2}}=3.0\,\,\text{kN}\cdot \rm{sec}\,/\text{m}$, and a spring, $K=4.0\,\,\text{kN/m}$, is shown in the schematic, Fig. P4.7a. The bar is modeled as rigid, massless, and constrained to horizontal motion.

• 4.7.a Derive the system equations for

  • i The force in the spring.

  • ii The force in damper b1.

  • iii The velocity of the bar.

and check their units.

The system de-energized when the pulse input of force shown in Fig. P4.7b is applied at time, $t=0$.

• 4.7.b Solve the system equations.

• 4.7.c Plot the responses using Mathcad or MATLAB.

• 4.7.d Plot the power flow from the source using Mathcad or MATLAB

Fig. P4.7

a Force source F(t) acts the bar which is attached to dampers, b ₁ and b ₂. Damper b ₁ is connected directly to ground. Damper b ₂is connected to spring K which is attached to ground. b Force pulse applied to the system

Problem 4.8 A rotational mechanical system modeled as consisting of an angular velocity source, two dampers, ${{b}_{\,1}}=60\,\,\text{N}\cdot \text{m}\cdot \text{sec}$ and ${{b}_{\,2}}=5\,\,\text{N}\cdot \text{m}\cdot \text{sec}$, and a rotational inertia, $J=12\,\,\text{kg}\cdot {{\text{m}}^{2}}$, is shown schematically in Fig. P4.8a below.

Fig. P4.8

a Angular velocity source Ω(t) is connected to the input shaft of a fluid coupling with damping b ₁. The output shaft of the fluid coupling connects to the mass moment of inertia J which is supported on bearings with damping b ₂

• 4.8.a Derive the system equations for

  • i The angular velocity of the inertia.

  • ii The torque in damper b₁.

  • iii The torque in damper b₂.

and check their units.

The system was in steady-state at time, t = 0, under a previously applied input when the angular velocity pulses shown Fig. 4.8b were applied to the system.

• 4.8.b Solve the system equations.

• 4.8.c Plot the responses using Mathcad or MATLAB.

• 4.8.d Plot the power flow from the source using Mathcad MATLAB.

Fig. P4.8

b Velocity pulses applied to the system by the angular velocity source Ω(t)

Problem 4.9 A translational mechanical system is modeled as consisting of a velocity source, two dampers, ${{b}_{\,1}}=5.0\,\,\text{N}\cdot \rm{sec}\text{/m}$ and ${{b}_{\,2}}=3.0\,\,\text{N}\cdot \rm{sec}\text{/m}$, a spring, $K=4.0\,\,\text{N/m}$, and a rigid, massless bar (a force spreader) is shown in Fig. P4.9a. The bar is constrained to horizontal translation.

• 4.9.a Derive the system equations for

  • i The force in the spring.

  • ii The force in damper b ₁.

  • iii The velocity of the force spreader bar.

and check their units.

The system was at rest and relaxed before the velocity pulse plotted in Fig. P4.9b acted on the system.

• 4.9.b Solve the system equations.

• 4.9.c Plot the responses using Mathcad or MATLAB.

• 4.9.d Plot the power flow from the source using Mathcad or MATLAB

Fig. P4.9

a Velocity source v(t) acts on damper b ₁ which is attached to the force spreader bar. Attached between the bar and ground are spring K and damper b ₂. b Velocity pulse applied to the system

Problem 4.10 The translational mechanical system shown in Fig. P4.10a consists of a velocity source which acts on damper b ₁ which is connected to mass M. The mass slides on a fluid film with damping b ₂. The parameter values are ${{b}_{\,1}}=300\,\,\text{N}\cdot \rm{sec}\text{/m}$, ${{b}_{\,2}}=10\,\,\text{N}\cdot \rm{sec}\text{/m}$, and $M=100\,\,\text{kg}$.

• 4.10.a Derive the system equations for

  • i The force from the velocity source.

  • ii The velocity of the mass.

  • iii The force acting to accelerate the mass.

and check their units.

The system was in steady-state under the previously applied step of 100 m/sec at time, t = 0 when the velocity input plotted in Fig. P4.10b was applied to the system.

• 4.10.b Solve the system equations.

• 4.10.c Plot the responses using Mathcad or MATLAB.

Fig. P4.10

a Schematic of the translational mechanical system, b Velocity input applied to the system

Problem 4.11 A rotational mechanical system is shown in Fig. P4.11a.

• 4.11.a Derive the system equations for

  • i The velocity of the flywheel J.

  • ii The torque acting through spring K.

  • iii The velocity difference across damper b.

and check their units.

The system was in steady-state under the torque input of − 200 N·m before the torque was increased to + 200 N·m at time, t = 0, and then to + 400 N·m at t = 10 sec, as shown in Fig. P4.11b.

• 4.11.b Solve the system equations for the parameter values:

	$b,\,\,\frac{\text{N}\cdot \text{m}\cdot \rm{sec}}{\text{rad}}$	$K,\frac{\text{N}\cdot \text{m}}{\text{rad}}$	$J,\text{kg}\cdot {{\text{m}}^{\text{2}}}$
i	10	1,500	25
ii	1	1,500	30
iii	5	3,000	10

• 4.11.c Plot the responses using Mathcad or MATLAB.

• 4.11.d Plot the power flow from the source using Mathcad or MATLAB.

Fig. P4.11

a A rotational system consisting of torque source T(t), a flywheel with mass moment of inertia J, a compliant shaft represented schematically as a torsion spring K. The shaft turns a drag cup with its output shaft rigidly fixed. b Torque applied by the source to the rotational mechanical system

Problem 4.12 A rotational mechanical system is shown in Fig. P4.12a.

• 4.12.a Derive the system equations for

  • i The torque acting through spring K.

  • ii The velocity difference across damper b ₁.

  • iii The velocity of the Flywheel J.

  • iv The torque acting to accelerate the flywheel.

and check their units.

The system was in steady-state under the angular velocity input of − 100 rad/sec before the velocity was changed to + 100 rad/sec at time, t = 0, then to + 400 rad/sec at t = 5 sec, and then to 0 rad/sec at t = 10 sec. as shown in Fig. P4.12b.

• 4.12.b Solve the system equations for the parameter values:

	${{b}_{\,1}},\,\,\frac{\text{N}\cdot \text{m}\cdot \rm{sec}}{\text{rad}}$	${{b}_{\,2}},\,\,\frac{\text{N}\cdot \text{m}\cdot \rm{sec}}{\text{rad}}$	$K,\frac{\text{N}\cdot \text{m}}{\text{rad}}$	$J,\text{kg}\cdot {{\text{m}}^{\text{2}}}$
i	1,000	0.5	3,000	50
ii	100	0.5	3,000	50
iii	1,000	0.5	1,500	500

• 4.12.c Plot the responses using Mathcad or MATLAB.

• 4.12.d Plot the power flow from the source using Mathcad or MATLAB.

Fig. P4.12

a A rotational system consisting of velocity source Ω(t) driving a fluid coupling with damping b _1, a torsion spring K, and a flywheel supported on a bearing with damping b ₂. The hydrodynamic bearing is represented schematically as a drag cup. b Angular velocity applied by the source Ω(t) driving the rotational mechanical system

Problem 4.13 A translational mechanical system is modeled as consisting of a force source, two masses, $M=400\,\,\text{kg}$ and $M=800\,\,\text{kg}$, and two dampers, ${{b}_{\,1}}=1,000\,\,\text{N}\cdot \rm{sec}\text{/m}$ and ${{b}_{\,2}}=2,000\,\,\text{N}\cdot \rm{sec}\text{/m}$, as shown schematically in Fig. P4.13a.

• 4.13.a Derive the system equations for

  • i The force acting through damper b ₁.

  • ii The force acting through damper b ₂.

  • iii The velocity of mass M ₁.

  • iv The velocity of mass M ₂.

  • v The force acting to accelerate mass M ₂.

and check their units.

The system was at rest when the force pulse shown in Fig. P4.13b was applied to the system. Note that the units of force are kilonewtons.

• 4.13.b Solve the system equations.

• 4.13.c Plot the responses using Mathcad or MATLAB.

• 4.13.d Plot the power flow from the source using Mathcad or MATLAB.

Fig. P4.13

a Force source $F( t )$ acts on a systems of two masses and two dampers. b Force pulse in kilonewtons applied to the system

Problem 4.14 A translational mechanical system is shown schematically in Fig. P4.14a. Parameter values are M = 10 kg, $b=0.4\,\,\text{N}\cdot \sec \text{/m}$, and K = 3,000 N/m.

• 4.14.a Derive the system equations for

  • i The force acting through spring K.

  • ii The force acting through the lubricating film damping b.

  • iii The velocity of mass M.

  • iv The force acting to accelerate the mass M.

and check their units.

The system was at rest when the velocity pulse shown in Fig. P4.14b was applied to the system.

• 4.14.b Solve the system equations.

• 4.14.c Plot the responses using Mathcad or MATLAB.

• 4.14.d Plot the power flow from the source using Mathcad or MATLAB.

Fig. P4.14

a Velocity source $v( t )$ acts on a system of two masses and two dampers, b Velocity pulse applied to the system

Problem 4.15 A translational mechanical system is shown schematically in Fig. P4.15a.

• 4.15.a Derive the system equations for

  • i The force acting through spring K.

  • ii The velocity drop across spring K.

  • iii The velocity drop across damper b.

  • iv The velocity of mass M.

and check their units.

The system was at rest when the velocity pulse shown in Fig. P4.15b was applied to the system.

• 4.15.b Solve the system equations for the parameter values:

  • i i M = 500 kg, $b=2,000\,\,\text{N}\cdot \rm{sec}\text{/m}$, and K = 1,000 N/m.

  • ii ii M = 300 kg, $b=400\,\,\text{N}\cdot \rm{sec}\text{/m}$, and K = 4,000 N/m.

  • iii iii M = 500 kg, $b=1,000\,\,\text{N}\cdot \rm{sec}\text{/m}$, and K = 6,000 N/m.

• 4.15.c Plot the responses using Mathcad or MATLAB.

• 4.15.d Plot the power flow from the source using Mathcad or MATLAB.

Fig. P4.15

a Velocity source $v( t )$ acts on a systems of two masses and two dampers. b Velocity pulse applied to the system

Problem 4.16 A cantilevered beam with rectangular cross-section is shown in Fig. P4.16. Attached to the beam is a cylinder of mass M. The diameter of the cylinder equals the width of the beam b. The height of the cylinder is twice its diameter.

• 4.16.a Determine the undamped angular frequency of the beam and mass system when they are carbon steel and L = 40 mm, b = 5 mm, and h = 1 mm.

• 4.16.b Determine the undamped angular frequency of the beam and mass system when they are carbon steel and L = 50 mm, b = 10 mm, and h = 1 mm.

• 4.16.c Determine the undamped angular frequency of the beam and mass system when they are aluminum 6061 and L = 50 mm, b = 10 mm, and h = 1 mm.

Fig. P4.16

A cantilevered beam with a cylindrical mass attached at distance L from the support

Problem 4.17 A rotational mechanical system consisting of a torque source driving a shaft with a flywheel is shown in Fig. P4.17. The shaft and flywheel are carbon steel. The shaft is solid and has a circular cross-section. It is supported by three rolling contact bearings, each with damping $b=0.5\,\,\text{N}\cdot \text{m}\cdot \text{s/rad}$.

• 4.17.a Calculate the torsion spring constant K of the shaft and the mass moment of inertia J of the flywheel for the following dimensions.

Table P4.17 Flywheel and shaft dimensions

• 4.17.b Determine the angular velocity of the flywheel when a step input of 200 N-m of torque is applied to the de-energized system.

• 4.17.c Plot the response of the system in Mathcad or MATLAB.

Fig. P4.17

a Rotational mechanical system. b Plan and cross-section of the flywheel

Problem 4.18 A test was performed to determine the energetic element parameters of translational mechanical system shown in Fig. P4.18a. The system was de-energized before it was subjected to a step input of force, $F(t)=100.0\,\,N\,{{u}_{s}}(t)$. The velocity of its point of application of the force on the ideal massless and rigid bar was measured. The data are presented in Fig. P4.18b and Table P4.18.

Table P4.18 Velocity data

Determine the damping coefficient b and the spring constant K. Report your results in the correct SI units.

Fig. P4.18

a Translational mechanical system, b Velocity of the ideal massless and rigid bar

Problem 4.19 A machine’s energetic attributes are modeled as a mass moment of inertia J supported on an ideal, rotational damper b, Fig. P4.19a. A test was performed in which a step input torque of 1,000 N-m was applied to the system and the angular velocity of the inertia measured, Fig. P4.19b and Table P4.19. Determine the magnitudes of the inertia J and the damping coefficient b. Interpolate if necessary. Report your results in the correct SI units.

Table P4.19 Angular velocity data

Fig. P4.19

a Rotational mechanical system, b Angular velocity of the mass moment of inertia J