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MECHANICAL SYSTEMS CALCULATIONS OF THE MARINE FACILITIES
MECHANICAL SYSTEMS CALCULATIONS OF THE MARINE FACILITIES



Mechanical systems calculations of the marine facilities. Lessons notes. Textbook.


Introduction

Chapter 1. Forces, moments of forces, force systems, force pairs, coordinate systems, system equilibrium, mechanical systems in marine structures, connections in mechanical systems

1.1. Mechanical bodies, the action of forces and the interaction of bodies, axioms of the properties of force vectors
1.2. Finding force vector projections
1.3. Vector of the moment of force relative to a point, magnitude of the moment of force relative to the axis
1.4. Vector transformation of a system of forces acting on a mechanical system
1.5. The action of a pair of forces and the transformation of a pair of forces to the moment
1.6. Equilibrium of a plane system of forces and forms of balance equations
1.7. Hinged joints in mechanical systems and estimation of degree of freedom
1.8. Static indefinability of a mechanical system, geometric invariability of the system
1.9. Types of connections in mechanical systems
1.10. Rigid embedding in the mechanics and its replacement by force vectors
1.11. Finding the reaction of supports using balance conditions for elements of mechanical systems
1.12. The balance and degrees of freedom of instantly variable mechanical systems

Chapter 2. Static calculations of elements of mechanical systems (frames and trusses), finding reactions and creating force diagrams, equation of three moments, moving load and lines of influence
2.1. The balance of a system of converging forces
2.2. Considering the balance of converging forces when analyzing statics issues
2.3. Using the three–force theorem in solving statics problems
2.4. Calculation of flat trusses using the section method
2.5. Calculation of statically definable mechanical systems using the method of sections
2.6. Determination of forces in truss rods using the Ritter method
2.7. Finding the reactions of the supports of statically definable frame, when it is loaded by a system of forces
2.8. Determination of the magnitude and point of application of the resultant distributed load
2.9. Creating diagrams of forces and moments in the calculations of rod structures
2.10. Calculation of forces in multi–span statically definable beams and determining the diagrams of forces and moments
2.11. Using the forces method in the form of three–moment equation to determine the forces acting in continuous beams
2.12. Impact of a moving load on a beam and creation of a line of influence of a moving load
2.13. Lines of influence during nodal transmission of moving loads
2.14. Lines of influence of forces acting in the truss rods
2.15. Conditions for the balance of a spatial system of forces
2.16. Reducing an arbitrary system of forces to its simplest form
2.17. Varignon's theorem on the moment of the resultant force
2.18. The balance of the rod elements of a flat statically indefinable frame
2.19. Calculation of forces in truss rods by cutting out the truss nodes


Chapter 3. Static indefinability of mechanical systems. Canonical equations of force and displacements methods used to determine reactions under static indefinability
3.1. Statically indefinable systems and the degree of indefinability
3.2. Using forces method finding the forces in statically indefinable mechanical systems, choosing the main system of the forces method
3.3. Canonical equations of the method of forces when finding forces in the elements of statically indefinable system
3.4. Coefficients and free terms of canonical equations of the forces method when calculating forces in the elements of statically indefinable system
3.5. The use of force diagrams in the elements of a mechanical system when determining the coefficients of the canonical equations of the forces method
3.6. Simplification of the canonical equations of the forces method under the condition of symmetry of the mechanical system
3.7. Application of Mohr’s integral method to determine the displacements of nodes during deformation of an elastic mechanical system
3.8. Finding the Mohr’s integral using the Vereshchagin’s method when determining the displacements of nodes in deformed elastic system
3.9. Determination of the Mohr’s integral using Simpson's formula and the method of trapezoid
3.10. Using the forces method to determine the efforts in a statically indefinable system due to temperature effects on it
3.11. Finding temperature deformations and displacements of mechanical system nodes using Mohr’s integral
3.12. Determination of displacements due to settlement of supports of a mechanical system using the principle of possible displacements
3.13. Using forces method finding the forces in statically indefinable frame
3.14. Determination of forces in elements of a statically indefinable mechanical system, using the displacements method
3.15. The main system and canonical equations of the displacements method
3.16. Finding the coefficients and free terms of canonical equations of the displacements method
3.17. The formulas for bending moments and reactions finding in statically indefinable beams under standard external influences caused the unit displacements

Chapter 4. The use of computer algebra Mathcad and the Microsoft EXCEL program for mathematical operations on matrices and vectors, to determine the displacements of nodes of a mechanical system using Mohr’s integral, to find reactions, forces and displacements using the finite element method
4.1. Rules for multiplying matrices, finding the determinant of a matrix and the inverse of a matrix, rules for multiplying vectors used in calculations of mechanical systems
4.2. Operators of computer algebra Mathcad and EXCEL programs used for operations on matrices and vectors in calculations of mechanical systems
4.3. Application of computer algebra system Mathcad to determine forces in a statically indefinable mechanical system, using the displacements method. Use of the Villiot’s diagram to find displacements of nodes taking into account the angles of hinge connection of incompressible rods
4.4. Calculation of forces in the elements of a flat truss using the finite element method in the Mathcad, specifying the geometric characteristics of the truss and nodal load, finding the stiffness matrices of finite elements, the incidence matrix and the stiffness matrix of a discrete model, determining the displacement vectors and end forces
4.5. The concept of finite element method in calculations of mechanical systems
4.6. Determination of the stiffness matrix of a mechanical system when using the finite element method in the EXCEL
4.7. Calculation of the Mohr’s integral and displacements of frame nodes in the Mathcad
4.8. The use of the Mathcad computer algebra system to calculate forces in the nodes of a flat truss using the method of cutting nodes
4.9. Calculation of displacements of nodes of a statically definable truss in the computer algebra system Mathcad
4.10. Calculation of statically indefinable frame using the finite element method in the Mathcad. Setting the geometric characteristics of the frame in matrix and vector forms
4.11. Determination of external forces in finite element nodes in the Mathcad
4.12. Stiffness matrix of rod finite elements in the global coordinate system, determined in the Mathcad
4.13. Determination in the Mathcad of the incidence matrix and stiffness matrix of a discrete model when calculating frame using the finite element method
4.14. Finding nodal displacements when calculating a frame in the Mathcad using the finite element method, converting the displacement matrix to local coordinates
4.15. Determination in the Mathcad of forces in finite elements in a local coordinate system under end reactions and under external load, transformation of the displacement matrix to local coordinates

Chapter 5. Mechanical sustainability of marine structures elastic elements, the influence of material plasticity on sustainability
5.1. Sustainability of a rod compressed along the axis, finding the critical force using the Euler’s formula
5.2. The influence of material plasticity on the sustainability of compressed elements of mechanical systems, the limits of applicability of Euler’s formula for finding critical forces
5.3. Assessing the sustainability of a compressed rod using stress reduction factors
5.4. Mechanical sustainability of an axially compressed rod made of standard profile parts
5.5. Assessment of the mechanical sustainability of the shaft, the cross–sectional shape of which is specified geometrically

Chapter 6. Shear and torsion of mechanical system elements
6.1. Mechanical shear under the influence of transverse force
6.2. Calculation of the shear strength of rows of rivets when joining sheets
6.3. Geometric and physical characteristics of torsion of a mechanical system element
6.4. Evaluation of shaft strength and torsional rigidity
6.5. Determining the torque diagram and assessing the strength and rigidity of shaft

Chapter 7. Stress–strain state of elastic elements of mechanical systems, hypotheses of plasticity and destruction of elements, main areas and main stresses
7.1. Stresses on inclined areas of sections of mechanical system elements
7.2. Main areas and main stresses in the three–dimensional stress state of a mechanical system material
7.3. Generalized Hooke's law for the three–dimensional stress state of an elastic material
7.4. Specific potential energy of deformation of an elastic material in its stress–strain state
7.5. Hypotheses of plasticity and destruction of elastic material of a mechanical system under stress–strain state
7.6. Finding the main stresses and deformations of the elastic element and checking the strength using fracture hypotheses for plane stress state
7.7. Testing the strength of a material under a complex stress state – with the combined action of bending and shear on a mechanical system
7.8. Determination of load when assessing the overall longitudinal strength of a ship
7.9. Checking the strength and determining the strong cross–sectional sizes of the structural elements of the ship, taking into account the general loads acting on the hull

Chapter 8. Methods for specifying the motion of point and kinematics of elements in mechanical systems, the coordinate systems, curvature of motion trajectory, instantaneous centers of velocities and accelerations
8.1. A natural way to specify the motion of a material point
8.2. The coordinate planes and axes associated with the trajectory of a material point motion
8.3. Evaluation of trajectory curvature using the natural method of specifying the motion of a material poin
8.4. Determination of instantaneous center of acceleration in plane motion
8.5. Determining the instantaneous center of velocities and performing kinematic calculations for plane motion of a rigid body
8.6. Finding elements of the trajectory of a material point using the Mathcad computer algebra system
8.7. Kinematic calculation of a flat mechanism

Chapter 9. Forces acting on elements during the movement of mechanical system (the mechanism), the laws of motion under influence of forces
9.1. Characteristics of body motion when rotating around a fixed axis
9.2. Kinematic elements of mechanism motion
9.3. The action of sliding friction on solids and mechanical elements
9.4. Determination of rolling friction force on a plane elastic surface
9.5. Finding the Coriolis acceleration for complex motion of a material point
9.6. Equilibrium conditions taking into account active forces in accordance with d'Alembert's principle
9.7. Direct issue of dynamics, determination of acting forces by analyzing the elements of motion of a mechanical system
9.8. Inverse issue of dynamics, determination of the elements of motion of a mechanical system under the influence of efforts, applied to it
9.9. Solving the equation of motion of a load under the action of external forces, weight and friction

Chapter 10. The amount of translational and rotational motion, the momentum and moment of momentum, the energy of mechanical system and the inertia characteristics of it elements
10.1. The use of mechanical system Momentum Change Theorem to determine elements of the system motion
10.2. Determination of patterns of changes in the coordinates of the center of mass and the support reaction of a mechanical system using the Theorem on the motion of the center of mass
10.3. Finding the forces acting on a rotor turning with eccentricity using the theorem on the motion of the center of mass
10.4. Indicators (the moments) of inertia of rotational motion of solid bodies, total, proper and transfer moments of inertia, the Huygens–Steiner theorem
10.5. Determining the position of the center of gravity by coordinates of body shape using computer algebra Mathcad
10.6. The use the addition method to find center of the area of a plane figure
10.7. The Momentum Change Theorem of a material point
10.8. Momentum of mechanical system and the Momentum Change Theorem
10.9. The Moment of Momentum Change Theorem of a mechanical system
10.10. Kinetic energy of a mechanical system
10.11. The use of the Kinetic Energy Change Theorem to determine the elements of motion of a mechanical system

Chapter 11. Vibrations of mechanical systems under the influence of external forces, weight, inertia and elasticity
11.1. Basic characteristics of free vibrations of a material point under the influence of weight and elastic forces
11.2. The influence of the mass and stiffness of elements of mechanical systems on the characteristics of their vibrations, the influence of parallel and series connection of springs on the characteristics of vibrations
11.3. Oscillations of a mechanical system with one degree of freedom (the single–mass system)
11.4. Determination of the natural characteristics of free vibrations of a multi–mass mechanical system
11.5. Spectrum of natural frequencies and main modes of free vibrations of a frame with two concentrated masses, use of the Mathcad computer algebra system to find the spectrum and main modes

Chapter 12. The linear calculations of ship rolling and motion in water
12.1. Classification of forces and moments acting on a ship during it motion and roll, the coordinate systems and the systems of equations of motion, the elements of ship rolling
12.2. Determination of own and associated inertia characteristics using empirical formulas for the main kinds of ship rocking
12.3. Finding the roll elements of ship on regular waves using the complex amplitudes method
12.4. The characteristics of sea waves. Determination of rocking spectrum of ship in irregular waves
12.5. The controllability of ship in water and equations of ship motion
12.6. Self–oscillations of a moving ship without control
12.7. The numerical integration of the equations of ship motion
12.8. Numerical solution of ship rolling nonlinear equations on regular waves
12.9. Keeping the ship on course using controls. The particular solution of ship motion equations
12.10. Finding the elements of ship motion when replacing the control function with its expansion into a trigonometric series


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Key words: mechanical system; mechanism; element of the system; link in the system; marine engineering; marine transport; ship design; theoretical mechanics; theory of elasticity; coordinate system; force; moment of forces; couple of forces; system of forces; converging forces; resultant of forces; distributed load; mechanical structure; frame; truss; rod; beam; shaft; multi-span beam; support reaction; mechanical hinge; rigid embedding; mechanical diagram; moving load; influence line; equation of three moments; instantaneous variability; method of cutting out nodes; method of sections; Ritter's method; Varignon's theorem; compatibility of deformations; equilibrium condition; degree of freedom; static indeterminacy; degree of indeterminacy; deformation of the system; method of forces; canonical equations; system of equations; main system; displacement method; Mohr's integral; Vereshchagin's method; Simpson's method; trapezoidal method; unit displacement; unit force; temperature deformation; support settlement; nodal load; mathematical modeling; vector; matrix; vector transformation; computer algebra; finite element; discrete model; stiffness matrix; local coordinate system; global coordinate system; compressive stability; Euler's formula; Hooke's law; effect of plasticity on stability; shear strain; torsional strain; torsional strength; stresses in volume; material failure; failure hypotheses; strength criteria; equivalent stress; main sites; main stresses; energy of deformation; ship strength; general longitudinal strength; plane section; critical points; load on structural material; stress diagram; element of motion; kinematic calculation; trajectory of motion; trajectory elements; instantaneous center of velocity; dynamics issues; direct problem; inverse problem; translational motion; rotational motion; dry friction; rolling resistance; instantaneous center of acceleration; inertial force; normal acceleration; tangential acceleration; Coriolis acceleration; d'Alembert's principle; momentum; angular momentum; kinetic energy; mechanical work; impulse; angular impulse; mechanical inertia; theorem on the motion of the center of mass; theorem on the change of momentum; theorem on the change of kinetic energy; characteristics of inertia; Huygens-Steiner theorem; mechanical vibrations; parallel connection of springs; series connection of springs; multi-mass system; free vibrations; natural characteristics of oscillations; frequency spectrum; vibration modes; main modes; elements of ship motion; elements of ship roll; forces and moments acting on the ship; coordinate systems associated with the ship; ship motion without control; ship instability on course; ship keeping on course; associated characteristics of ship inertia; roll in calm water; roll in regular waves; roll in irregular waves; spectral characteristics of ship roll; nonlinear roll equation; linearized roll equation; complex amplitude method; numerical solution of the roll equation; general and particular solution of the equation; ship maneuver; ship control system; feedback system; control function; trigonometric Fourier series