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ECCOMAS Course on Computational
Structural Dynamics (Summer, Prague
2018)
What we will learn in this short course (mylectures and lectures by others)
1. Fundamentals of mechanical dynamical systems.
2. Derivations of the equations of motion via various approaches.
3. Physics of structural vibrations, and modeling and solutionmethods.
4. Modeling of structure-medium interaction problems and theirefficient solution methods
5. An introduction to multiphysics problems: modeling, analysisand solution methods
Above All, Become Excellent Computational Mechanicians!
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Lecture 1: An Overview on Principlesin Dynamics
1.1 Newton’s Definitions, Three Laws, Four Rules,and Inertial Frame
In the Author’s Preface (p. XVII) of Philosophiae Naturalis Prin-cipia Mathematica (Mathematical Principles of Natural Philoso-phy) or simply Newton’s Principia (1687), he states . . . ”There-fore I offer this work as the mathematical principles ofphilosophy, for the whole burden of philosophy seems toconsist in this – from phenomena of motions to investi-gate the forces of nature, and then from these forces todemonstrate the other phenomena; and to this end thegeneral propositions in the first and second Book aredirected.” . .
His description of gravity (Principia, p.547): ”But hitherto I havenot been able to discover the cause of those properties of grav-ity from phenomena, and I frame no hypothesis; for whatever isnot deduced from the phenomena is to be called a(an) hypothe-sis; and hypothesis, whether metaphysical or physical, whetherof occult qualities or mechanical, have no place in experimen-tal philosophy. In this philosophy particular propositions are in-ferred from the phenomena, and afterwards rendered general byinduction.”
Newton’s Eight Definitions (Principia, pp.1-6)
DEFINITION I: The quantity of matter[mass] is the measureof the same, arising from its density and bulk conjointly.
DEFINITION II: The quantity of motion[linear momentum] isthe measure of the same, arising from the velocity and quantityof matter conjointly.
DEFINITION III: The vis insita, or the innate force of mat-
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ter[inertia force], is a power of resisting, by which every body,as much as in it lies, continues in its present state, whether it berest, or of moving uniformly forwards in a right[straight] line.
DEFINITION IV: An impressed force is an action exertedupon a body, in order to change its state, either of rest, or ofuniform motion in a right line.
DEFINITION V: A centripetal force is that by which bodiesare drawn or impelled, or any way tend, towards a point as toa centre.
DEFINITION VI: An absolute centripetal force is the mea-sure of the same, proportional to the efficacy of the cause thatpropagates it from the centre, through the spaces round about.
DEFINITION VII: The accelerative quantity of a centripetalforce is the measure of the same, proportional to the velocitywhich it generates in a given time.
DEFINITION VIII: The motive[motion-causing] quantity of acentripetal force is the measure of the same, proportional to themotion which it generates in a given time.
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Newton’s Three Laws (Principia, pp.13-14)
Law I: Every body continues in its state of rest, or ofuniform motion in a right line, unless it is compelled to changethat state by forces impressed upon it.
Law II: The change of motion is proportional to themotive force impressed; and is made in the direction of the rightline in which the force is impressed.
Law III: To every action there is always opposed an equalreaction; or, the mutual actions of two bodies upon each otherare always equal, and directed to contrary parts.
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Newton’s Four Rules for Scientific Discoveries(Principia, pp.398-400)
Rule 1. “We are to admit no more causes of natural things thansuch as are both true and sufficient to explain their appearance.”(Do not assume other causes than those which are necessary toexplain the phenomena.)
Rule 2. “Therefore to the same natural effects we must, as faras possible, assign the same causes.” (Relate as completely aspossible analogous effects to the same cause.)
Rule 3. “The qualities of bodies, which admit neither intensifica-tion nor remission of degrees, and which are found to belong toall bodies within the reach of our experiments, are to be esteemedthe universal qualities of all bodies whatsoever.” (Extend to allbodies the properties which are associated with those on whichit is possible to make experiments.)
Rule 4. “In experimental philosophy we are to look upon propo-sitions inferred by general induction from phenomena as accu-rately or very nearly true, till such time as other phenomena oc-cur, by which they may either be made more accurate, or liable toexceptions.” (Consider every proposition obtained by inductionfrom observed phenomena to be valid until a new phenomenonoccurs and contradicts the proposition or limits its validity.)
These four rules provide a guide for how to conduct scientific re-search for subsequent generations. In particular, it is Rule 3 thatNewton relied on to formulate the law of universal gravitation.
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Fig. 1. Reference Frame ( V is a constant translational velocity)
The origin of a reference frame is either fixed (V = 0) or movingwith a constant translational velocity. Hence, no rotational mo-tion is permitted for reference frames. In addition, there are aninfinite number of reference frames whose origins are (O1, O2, O3, ..., On)as long as their origins are either fixed or moving with constantvelocities (V1, V2, V3, ..., Vn). A choice of an reference frame froman infinite possible number of reference frames therefore leadsto the same equations of motion that could have been obtainedby employing other reference frames.
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Transition from Statics to Dynamics
Statics Dynamics (vis D’Alembert’s principle)
Force equilibrium
for a free body: ∑j f ij = 0
∑j f ij −miai = 0§
Moment equilibrium
around point ”P”: ∑j MPj = 0
∑MPj − rPj × (mjaj) = 0§
(1.1)
§ Observation: A critical aspect of dynamics is the need tocompute the acceleration vector (a) for every mass in the system.If there is no mass, no equations of motion is needed (This doesnot state one does not need static equilibrium equations at adiscrete point or the conditions of constraints).
Here we note that the inertia force (f inertiai = −miai) is associ-ated with the negative sign as defined by Newton as a power ofresting. ( Recall Newton’s Definition III: The vis insita, or theinnate force of matter[inertia force], is a power of resisting, bywhich every body, as much as in it lies, continues in its presentstate, whether it be rest, or of moving uniformly forwards in aright[straight] line.
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Fig. 2. Example of a Spring Mass System
m
k
f
Free-free state
Frame Ko
o'
(mass center)Frame K'
Static displacement
Assumed dynamic state
X X'
d
x'x
h
mg k(x-d) = kx'
ma
The position vectors and accelerations and inertia forces for O-frame are given, respectively, by
rO = h− d + x ⇒ a = rO = x ⇒ f inertiaO = −mx
rO′ = x′ ⇒ a′ = rO′ = x′ = (x− d) = x ⇒ f inertiaO′ = −mx
(1.2)
The equations of motion for the two inertial reference frames aregiven by
For x-frame: applied force + spring force− inertia force = 0
⇓(f −mg)− k(x− d)−mx = 0
For x′-frame:
(f −mg) + kx′ −mx′ = 0
(1.3)
via x′ = x− d and x′ = x, the two equations are the same.
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1.2 D’Alembert’s Principle
Before we begin with the mathematical description of D’Alembert’sprinciple, let’s recall the principle of virtual work which can bestated as
N∑i=1
f i · δri = 0 (1.4)
where f i is the generalized force at the spatial location ri whichis measured in an inertial reframe frame, and δri is its associatedvirtual displacement. To appreciate the nature of the virtual dis-placement or specifically the terminology virtuel, let’s considera vector r expressed in terms of two different representations:
r = xi + yj
= rer, r =√x2 + y2 er
et
=
cos θ sin θ
− sin θ cos θ
i
j
(1.5)
where the unit vectors (i, j) are attached on an inertial two-dimensional frame, (er, rt) are attached at the tip of the positionvector (r).
The virtual displacement of the position vector (r) takes on thefollowing two expressions:
δr = δxi + δyj
= δrer + rδer, δer = (− sin θi + cos θj)δθ(1.6)
As an example, if f i is given by f i = fxi + fyj, we have:
f i · δr = fxδx+ fyδy
=δr rδθ
cos θ sin θ
− sin θ cos θ
fxfy
= frδr +mrδθ,
fr = cos θfx + sin θfy,
mr = r(− sin θfx + cos θfy)
(1.7)
which shows that the different kinematic descriptions give riseto different (yet equivalent) virtual work expressions.
With the preceding short introduction of virtual displacementand virtual work, D’Alembert’s principle can be stated as
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∑j
δrj · f ij −miai+∑i
δθj · (MPj − rPj ×mjaj) = 0(1.8)
The resulting equations of motion that are derived from theabove D’Alembert’s principle do not contain any reaction forces(or alternatively constraint forces) that do not produce anywork. It is a distinct difference from the equations of motionobtained by applying Newton’s laws.
a. Virtual Displacements:
Referring to Fig. 3, virtual displacement for mass m:
δrA′ = δx i (1.9)
Virtual displacement for bar M :
δrC′ = δrA′ + δrA′C′
δrA′C′ =L
2(cos θi + sin θj) δθ
(1.10)
Virtual rotation for bar M :
Here, one utilizes the angular velocity ω = θk to obtain thevirtual rotation
δθ = δθk (1.11)
In general one has
δθ = δθxi + δθyj + δθzk (1.12)
b. Acceleration Vectors:
Acceleration vector for the sliding mass m:
aA′ = vA′ = xi( ⇐ rA′ = xi)
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Fig. 3. A Spring-Mass System
Fig. 4. Free-body diagram for mass
Acceleration vectors for the pendulum bar M :
aC′ = aA′ + aA′C′( ⇐ rA′C′ = 12L(cos θi− sin θj) )
aA′C′ = Lθ2
(cos θi + sin θj)− Lθ2
2(sin θi− cos θj)
c. Equilibrium equation for the sliding mass m:
f ′A = −kxi−mxi−mgj +NAj +XACi + YACj = 0
d. Equilibrium equation for the pendulum bar M :
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Fig. 5. Free-body diagram for the bar
f ′C = F i−MaC′ −Mgj−XACi− YACj = 0
MC′ = −ML2θ12
k + rC′A′ × (−XACi− YACj) + rC′B′ × F i = 0
1.3 Application of D’Alembert’s Principle forthe spring-mass-bar problem
∑(Fi −miri) · δri =f ′A · δr′A + f ′C · δr′C + M′
C · δθ′C = 0
⇓
−kxi−mxi−mgj +NAj
+XACi + YACj · δr′A
+ F i−MaC′ −Mgj
−XACi− YACj · δr′C
−ML2θ
12k + rC′B′ × F i
+ rC′A′ × (−XACi− YACj) · δθ(1.13)
Note that f ′A · δr′A 6= 0 and so other two terms!
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The first term in (1.13) becomes
f ′A · δr′A = −kxi−mxi−mgj +NAj +XACi + YACj · (δxi)
⇓f ′A · δr′A = −kx−mx+XAC δx
(1.14)
Remark: Observe that the reactions forces YAC and NA haveplayed no role in the resulting virtual work!
The second term in (1.13) becomes
f ′C · δr′C = F i−MaC′ −Mgj−XACi− YACj · δr′C
= F i−M [xi +Lθ
2(cos θi + sin θj)− Lθ2
2(sin θi− cos θj)]
−Mgj−XACi− YACj · δxi +L
2(cos θi + sin θj) δθ
⇓
f ′C · δr′C = F −M [x+Lθ
2cos θ − Lθ2
2sin θ]−XAC δx
+ F −M [x+Lθ
2cos θ − Lθ2
2sin θ]−XAC
L
2cos θ δθ
+ −M [Lθ
2sin θ − Lθ2
2cos θ]−Mg − YAC
L
2sin θδθ
⇓
f ′C · δr′C = F −M [x+Lθ
2cos θ − Lθ2
2sin θ]−XAC δx
+ L cos θ
2F − ML
2cos θ x− ML2
4θ − MLg
2sin θ
− L
2cos θ XAC −
L
2sin θYAC δθ
(1.15)
In evaluating the third term of (1.13), first, carry out:
rC′B′ × F i =L
2(sin θi− cos θj)× F i
=FL
2cos θk
rC′A′ × (−XACi− YACj) = L2
cos θXAC +L
2sin θYACk
(1.16)
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so that we obtain
MC′ · δθC′ = −ML2θ
12+FL
2cos θ + [
L
2cos θXAC +
L
2sin θYAC ]δθ
(1.17)
Substituting (1.14), (1.15) and (1.17) into (1.13), we obtain
N∑i=1
(Fi −miri) · δri = (M +m)x+ kx+ML
2cos θ θ − ML
2sin θ θ2 − F δx
+ ML2
3θ +
ML
2cos θ x+
MgL
2sin θ − L cos θFδθ = 0
(1.18)
Coupled equations for x and θ via d’Alembert’sprinciple
Since δx and δθ are arbitrary, we obtain:
(M +m)x+ kx+ML
2cos θ θ − ML
2sin θ θ2 = F
ML2
3θ +
ML
2cos θ x+
MgL
2sin θ = L cos θF
(1.19)
Observations:
1. When using Newton’s second law, (1.19) are obtained byeliminating the reaction forces XAC and YAC.
2. On the other hand, one does not need to consider reactionforces in applying d’Alembert’s principle! Only apparent forcesincluding inertia forces need to be considered. This is a majoradvantage of applying energy principles over Newton’s method.
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1.4 Hamilton’s Principle
Let’s begin with d’Alembert’s principle:
N∑i=1
(Fi −miri) · δri = 0
⇓
δT + δW −N∑i=1
d
dt(miri · δri) = 0,
¯δW =N∑i=1
Fi · δri, δT =N∑i
δTi, δTi = δ(12miri · ri)
(1.20)
Let’s integrate (1.20) in time:
∫ t2
t1δT + δW −
N∑i=1
d
dt(miri · δri) dt = 0
⇓∫ t2
t1δT + δW dt =
∫ t2
t1
N∑i=1
d
dt(miri · δri) dt
(1.21)
Noting that we have
∫ t2
t1
N∑i=1
d
dt(miri · δri) dt =
N∑i=1
(miri · δri)|t2t1 (1.22)
so that, if ri(t1) and ri(t2) are specified, we have
δri(t1) = δri(t2) = 0 (1.23)
Hence, equation (1.20) becomes
∫ t2
t1δT + δW dt = 0 (1.24)
which is known as extended Hamilton’s principle.
In general the work done, δW , consists of two parts:
15
δW = δWcons + δWnoncons (1.25)
where subscripts cons and noncons designate conservative andnonconservative systems, respectively.
1.4.1 Hamilton’s Principle for Conservative Systems
For conservative systems, we have have from (1.24) and (1.25)with δWnoncons = 0:
∫ t2
t1δT + δWcons dt = 0 (1.26)
which is known as Hamilton’s principle for conservative systems.
1.4.2 Action Integral for Conservative Systems
Observe that the work done on the conservative systems can beexpressed in terms of the corresponding potential energy:
δWcons = −δV, V =N∑i=1
(∫ rref
rFi · dri) (1.27)
Substituting (1.27) into (1.26), one obtains:∫ t2
t1δT − δV dt = 0
⇓∫ t2
t1δT − δV dt = 0
⇓
δS =∫ t2
t1δL dt = 0, L = T − V
(1.28)
where L is called system Lagrangian.
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1.5 Euler-Lagrange Equation’s of Motion
Let’s substitute the general work expression (1.25) into the ex-tended Hamilton’s principle (1.24) to obtain:∫ t2
t1δT + δWcons + δWnoncons dt = 0 (1.29)
which, with (1.27) and the definition of the system LagrangianL = T − V , becomes
∫ t2
t1δL+ δWnoncons dt = 0 (1.30)
Since δL can be expressed in terms of the generalized coordinatesas
δL =n∑k=1
∂L∂qk
δqk +∂L
∂qkδqk (1.31)
one obtains∫ t2
t1δL dt =
∫ t2
t1
n∑k=1
∂L∂qk
δqk +∂L
∂qkδqk dt (1.32)
Integrating by part the first term of the preceding integral, weobtain∫ t2
t1
n∑k=1
∂L∂qk
δqk dt =n∑k=1
∂L∂qk
δqk|t2t1 −∫ t2
t1
n∑k=1
ddt
(∂L
∂qk)δqk dt
(1.33)
Since δqi(t1) = δq(t2) = 0 as we discussed in deriving (1.23), wehave∫ t2
t1
n∑k=1
∂L∂qk
δqk dt =∫ t2
t1
n∑k=1
− d
dt(∂L
∂qk)δqk dt (1.34)
Substituting (1.34) into (1.32), then introducing the resulting
17
expression into (1.30), we finally obtain:
∫ t2
t1
n∑k=1
− d
dt(∂L
∂qk) +
∂L
∂qk+Qkδqk dt = 0, δWnoncons =
n∑k=1
Qkδqk
(1.35)
Since δqk are arbitrary, we obtain:
d
dt(∂L
∂qk)− ∂L
∂qk= Qk (1.36)
which is called Euler-Lagrange’s equations of motion.
An Example: the Spring-Mass-Pendulum Problem - Re-visited
For the example problem used in in the application of d’Alembert’sprinciple shown in Fig. 3, we have
Velocities: vA′ = xi, vA′C′ = −12Lθ(sin θi + cos θj),
vC′ = vA′ + vA′C′ , Θ = θk
Kinetic energy: T = 12mvA′ · vA′ + 1
2MvC′ · vC′ + 1
2Iz θ
2, Iz =1
4ML2
Potential energy: U = Vspring + Vgravity = 12kx2 +Mg
L
2(1− cos θ)
Vspring =∫ 0
x(−kxi) · drA′ =
1
2kx2
Vgravity =∫ rC
rC′(−Mgj) · drC′
External work: δW = FB′ · δrB′
= F i · [δxi + L(cos θi + sin θj)δθ]
= F (δx+ L cos θ δθ)
Qx = F, Qθ = FL cos θ
(1.37)
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The equations of motion derived via Euler-Lagrange formalism:
L = T − Ud
dt(∂L
∂x)− ∂L
∂x= Qx = F
d
dt(∂L
∂θ)− ∂L
∂θ= Qθ = FL cos θ
⇓
d
dt[(M +m)x+ 1
2MLθ cos θ] + kx = F
ML
6
d
dt(2Lθ + 3x cos θ) + 1
2ML(xθ + g) sin θ = FL cos θ
(1.38)
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1.6 Hamilton’s Equations
We recall the action integral from equation(1.28)) for conserva-tive systems as :
δS =∫ t2
t1δL dt = 0, L(q,q) = T − V (1.39)
which leads to the equations of motion for both discrete andcontinuum systems. In the above equation, (qi, i = 1, 2, 3...N)is a generalized coordinate.
Now we introduce the Hamiltonian, H, defined as
H =∑
qi∂L
∂qi− L
H =∑
qipi − L, via pi =∂L
∂qi
(1.40)
Observe that variation of the Lagrangian (L(q,q)) is given by
δL(q, q) =∑∂L∂qi
δqi +∂L
∂qiδqi+
∂L
∂tδt
⇓
δL(q, q) =∑piδqi +
∂L
∂qiδqi+
∂L
∂tδt, via pi =
∂L
∂qi
δL(q, q) =∑δ(piqi)− qiδpi +
∂L
∂qiδqi+
∂L
∂tδt
(1.41)
From (1.54) and the last expression of (1.55), we obtain variationof the Hamiltonian (δH) as
δH = δ(∑
qipi − L) = δ(∑
qipi)− δL
δH =∑qiδpi −
∂L
∂qiδqi −
∂L
∂tδt︸ ︷︷ ︸
via the third of (1.55)
(1.42)
On the other hand, from the definition of variation of the Hamil-tonian (H), we have
δH(q,p) =∑∂H∂qi
δqi +∂H
∂piδpi+
∂H
∂tδt (1.43)
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Comparing the above expression with (1.56), we obtain the fol-lowing relations:
∂H
∂qi= −∂L
∂qi
qi =∂H
∂pi∂H
∂t= −∂L
∂t
(1.44)
A critical missing link (perhaps the most important one) can beobtained by resorting to the Euler-Lagrange equation:
d
dt(∂L
∂qi)− ∂L
∂qi= 0
⇓
pi −∂L
∂qi= 0
⇓
pi =∂L
∂qi= −∂H
∂pivia the first of (1.44)
(1.45)
Finally, from the second of (1.44) and the last of (1.57), weobtain Hamilton’s equations:
pi = −∂H∂qi
, H = H(p,q)
qi =∂H
∂pi
(1.46)
A natural question that comes to our mind is that “why dowe need Hamilton’s equations?” Is it sufficient for us to utilizeHamilton’s principle along with the Euler-Lagrange equations?
To answer these questions, let us obtain Hamilton’s equations
21
for a single degree-of-freedom spring-mass system given by
L = T − V, T = 12mq2, V = 1
2kq2
p =∂L
∂q= mq
H = qp− L = T + V = 12pm(−1)p+ 1
2kq2
⇓
p = −∂H∂q
= −kq
q =∂H
∂p= m(−1)p ⇒ p = mq
(1.47)
which leads to the following matrix equation: qp =
0 1/m
−k 0
qp
(1.48)
Note that the Euler-Lagrange equation for this case is given by
d
dt(∂L
∂q)− ∂L
∂q= mq + kq = 0 (1.49)
From the viewpoint of purely algebraic manipulations, the Hamil-ton equations(1.60) can be obtained from the Euler-Lagrangeequation(1.62) by introducing the momentum (p = mq) andsubstitute mq by p. Hence, they are equivalent. The question is:is the momentum(p = mq) independent of the velocity (q)? Toanswer this question, let us examine their eigenvalues:−s1 1/m
−k −s1
qp
= 0,
qp =
qp es1t
(s22m+ k)q = 0, q = qes2t
(1.50)
Clearly, the momentum(p) and the displacement(q) in the Hamil-ton’s equation(1.60) are linearly independent as it has two dis-tinct eigenvalues, that is, its rank is equal to the degrees offreedom (2 for this example). Its homogeneous solution is thusgiven by p(t)q(t)
=
pq
1
ejωt +
pq
2
e−jωt, ω =√k/m (1.51)
22
Similarly, the homogeneous solution of the Euler-Lagrange equa-tion(1.62) is given by
q(t) = q1ejωt + q2e
−jωt, ω =√k/m (1.52)
Thus from an analytical viewpoint, there appears to be no ad-vantage of Hamilton’s equations(1.60) over the Euler-Lagrangeequation(1.62). However, this conclusion is short-sighted as thenext section illustrates for long-time integration endeavors.
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1.7 Method of Lagrange Multipliers
The Euler-Lagrange equations of motion are obtained from thesystem energy that consists of the kinetic energy, potential en-ergy and virtual work due to nonconservative forces. In otherwords, while enabling the derivation process simpler, the Euler-Lagrange formalism eliminate an important information: the re-action forces or moments on joints and kinematic constraintssuch as on the boundaries. Often, both the designers and ana-lysts need to know joint force levels so that necessary joint artic-ulations can be carried without jeopardizing the system safety. Acase in point is the human joints which, repeatedly over loaded,can cause excessive bone deformations and arthritis.
The question is: How do we re-introduce the constraint forces (re-action forces) within the Euler-Lagrange formalism? This wasoriginally developed by Lagrange and described in his book,Mecanique Analytique, published in 1788. This is accomplishedas follows.
In Section IV of his book, Lagrange introduces under the name of“a simple method (la methode tres-simple) to find the equationsnecessary for the equilibrium of systems.” Here is an excerpt ofwhat is now known as the method of Lagrange multipliers (takenfrom English translation version, J.L. Lagrange: Analytical Me-chanics, translated by A. Boissonanade and V. N. Vagliente,Kluwer Academic Publishers, 1997. pp.60-61):
Let L=0, M=0, N=0, etc. be the various equations of condi-tion (constraints) which are give n by the nature of the system.The quantities of L, M, N, etc. are finite functions of variables,x,y,z,x’,y’,z’, etc. By differentiation of these equations, the thefollowing equations results dL=0, dM=0, dN=0, etc. which willgive the relation which must exist between the differential of thesame variables. . . . .
Now, since these equations are only used to eliminate an equalnumber of differentia ls in the general formula of equilibrium,after which the coefficients of the remaining differentials mustindividually be set equal to zero, it is not difficult to prove by thetheory of elimination for linear equations that the same resultswill be obtained if the different equations of condition dL=0,dM=0, dN=0, etc. each multiplied by an undetermined coeffi-cient are simply added to this formula.
24
Then if the sum of all the terms which are multiplied by thesame differential are put to equal to zero, as many particularequations as there are differential will be obtained. Finally, fromthese latter equations the undetermined coefficients, by which theequation of condition have been multiplied, can be eliminated. .. .
The sum of moments(energy)of all the forces which must be inequilibrium can be gathered and to which can be added the variousdifferential functions which must be equal to zero from the con-ditions of the problem after having multiplied each of these func-tions by an undetermined coefficient. The whole can be equated tozero and thus a differential equation results which can be treatedas an ordinary equation of maxima and minima and from whichas many particular finite equations as there are variables can beextracted. These equations from which the undetermined coeffi-cients have been eliminated will give all the necessary conditionsfor equilibrium.
1.7.1 Step-by-step procedure for the method ofclassical Lagrange multipliers
Let’s consider a holonomic case, viz., the constraint conditionsthat can be explicitly stated in terms of position vectors, andconsider a one-dof spring-mass system shown in Fig. 6. Supposewe would like to know the reaction force between spring k1 andthe boundary. We now proceed with the procedure that leads tothe determination of the reaction force
Step 1: Partition the system into completely free sub-systems.
Step 2: Identify the conditions of constraints betweenthe completely free systems.
Step 3: Construct the energy of each of the completelyfree subsystems.
Step 4: Obtain the total energy by summing the en-ergy of each of the completely free subsystems.
Step 5: Append the conditions of constraints by mul-tiplying each with an unknown coefficient, λ (multiplier), to thetotal system energy (kinematic and potential).
25
Fig. 6. One-DOF Example Partitioned at the Boundary via ClassicalLagrange Multiplier
The system Lagrangian of the assembled system is given by
L = T − VT = 1
2mu21, V = 1
2k1u
21
δW = f1(t)δu1
(1.53)
Hence, the equations of motion for the assembled system aregiven via the Euler-Lagrange formalism as :
d
dt(∂L
∂u1)− ∂L
∂u1= f1(t)
⇓mu1(t) + ku1(t) = f(t)
(1.54)
1.7.2 Derivation of equations of motion for partitionedsystem via the method of classical Lagrange mul-tipliers
Step 1: Partition the system - done in Fig. 6.
26
Step 2: Identify the conditions of constraints:
at the fixed end : c01 = x1 − u0 = 0
⇓c01 = BT
clx + BT0 u0 = 0, xT = [x1 x2]
[BclT = [1 0], BT
0 = [−1]
(1.55)
Step 3: Energy of the completely free subsystem:
Now the system Lagrangian is given by
L = T − VT = 1
2mx21 + 1
2mx22 V = 1
2k1(x1 − x2)2
δW = f1(t)δu1
(1.56)
Step 5: Append the constraint functional
πcl = c01λ1 = (BTclx + BT
0 u0)λ01 = 0 (1.57)
The total Lagrangian of the classically partitioned system withthe appended constraint functional is thus given by
L(x1, x2,x1,x2, u0, λ01)
= T − V − πcl(1.58)
Classically partitioned equations of motion are obtained by uti-lizing the Euler-Lagrange equations:
δu0 − term: B0λ1 = 0
δx− term: [m]x + [k]x + Bclλ1 = f1(t)
δλ1 − term: BTclx + BT
0 u0 = 0
[m] =
m 0
0 m
, [k] = k1
1 −1
−1 1
(1.59)
Matrix form of the above partitioned equations of motion(1.59)
27
is given by(mD2 + k) 0 Bcl
0 0 B0
BTcl BT
0 0
x
u0
λ1
=
f
0
0
, D2 =
d2
dt2(1.60)
When the ground displacement (u0) is specified, the above equa-tion reduces to (mD2 + k) Bcl
BTcl 0
x
λ1
=
f
−B0u0
(1.61)
The above equation is applicable when the ground is movingsuch as the earthquake motion or when the spring-mass systemis attached to a lager structures such as the main structures ofa ship, airplane or vehicle.
Symbolically, the above equation can be expressed as A C
CT 0
x
λ
=
f
0
(1.62)
It is noted that solution of (1.62) provides both the displacementx and the reaction forces λ.
Question: Can one obtain the equations of motion for the as-sembled system (1.54)?
The answer is yes! To this end, we observe the following relationbetween the partitioned degrees of freedom and the assembledones as
u0
x1
x2
=
1 0
1 0
0 1
u0u1
⇒ x = Lug (1.63)
Substituting the above assembling operator L into (1.62), we
28
University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu
Classical Notion of Lagrange Multipliers!
Partitioning node, f, is lost!!
f!
Fig. 7. Classical Lagrange multipliers
obtain (LT A L) LTC
CT L 0
ug
λ
=
LT f
0
(1.64)
Finally, invoking the boundary condition
u0 = x1 = 0 (1.65)
we arrive at
(md2
dt2+ k1)u1 = f (1.66)
which is the same as (1.54). Note that solution of (1.60) providesthe reaction forces, whereas (1.66) (also (1.54)) does not!
1.8 Method of localized Lagrange multipliers
So far we have discussed the method of multipliers as Lagrangedescribed in Mecanique Analytique. Recently, a further localiza-tion of the interface constraints was proposed, which is labeledas the method of localized Lagrange multipliers. We will discussthe essential features of this method below.
Essential difference between the classical and localized Lagrangemultipliers are illustrated in Figs. 7, 8 and 9.
Comparing Figs. 7 and 8, and referring to 9, one find that the
29
University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu
New Interpretation of Lagrange Multipliers!
Partitioning node, f, is preserved!!
f!
Fig. 8. Localized Lagrange multipliers
University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu
Assembled!
Classical !Partitioning!
Localized !Partitioning!
Localization of Classical Lagrange Multipliers!
Split!!
Localization is achieved by introducing a frame node, f!
Fig. 9. Localization from Classical to Localized La-grange multipliers
crucial difference between the classical and localized lagrangemultipliers is in the use of frame node(s) and the associatedframe displacement(s) as denoted uf . A consequence is that theuse of localized Lagrange multipliers in the formulation of equa-tions of motion increases the number of unknowns, viz., uf andin general twice the number of Lagrange multipliers. However,it can be shown that the solution cost is the same for the twomethods in one exploits efficient sparse solution strategies. Tothis end, let’s revisit the one-dof spring-mass, this time parti-tioned at the boundary by the method of localized Lagrangemultipliers as shown in Fig.10. The system energy of this sys-tem is the same as that of the classically partitioned case (see(1.56)). The constraint functional is expressed by referring to
30
Fig. 10. One-DOF Example Partitioned at the Boundary via Local-ized Lagrange Multiplier
Fig. 10 as
π` = λT` (BT` xtotal − Lframeuframe) = 0
λT` =λ0 λ1
, xTtotal = [x0 x1 x2]
uTframe = [u0], B` =
B0 0
0 B1
B0 =
[1
], B1 =
1
0
, Lframe =
1
1
(1.67)
The total Lagrangian of the localized partitioned system withthe constraint functional is thus given by
L(xtotal,xtotal, λ`,uframe)
= T − V− λT` (BT
` xtotal − Lframeuframe)
(1.68)
Matrix form of the localized partitioned equations of motion can
31
be obtained as
(mD2 + k) B` 0
BT` 0 −Lframe
0 −LTframe 0
xtotal
λ`
uframe
=
f
0
0
k =
0 0 0
0 k1 −k10 −k1 k1
m = diag([0 m m])
fT = [0 fT1 ]
(1.69)
Observe that the benefit of employing the localized Lagrangemultipliers become evident when one considers multiple con-straints as shown in Fig. 11. As you can see, there are six pos-sible constraints; however, only three of them are linearly inde-pendent (why?). Hence, there are fifteen (15) different ways ofchoosing three independent constraints out of six constraints!This means the choice of three linearly independent constraintsis not unique. Figure 12 shows partitioning via the method of lo-calized Lagrange multipliers, which yields four linearly indepen-dent constraints. Hence, by increasing the number of constrainby one, the construction of localized constraint equations (fourin this example) is uniquely determined.
It is noted that the partitioned equations(1.69) can be special-ized to the classically partitioned equation(1.60) by seeking anull space of Lframe such that
λ` = N`clλcl
with property LTframe N`cl = 0
(1.70)
which, when introduced into (1.69), leads to
(mD2 + k) B`N`cl
NT`clB
T` 0
xtotal
λcl
=
f
0
m = diag([0 m m]), fT = [0 fT1 ]
(1.71)
32
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Classical !-Method (the CLM method) connects directly from !one substructural interface node to that of another interface node:!
Note that one has to choose 3 rank-sufficient conditions from!among 6 possible conditions.!
Fig. 11. Partitioning four bars via the method of clas-sical Lagrange multipliers
University of Colorado - Dept of Aerospace Engrg. Sci. & Center for Aerospace Structures - caswww.colorado.edu
Partitioning the assembled 4 rod-element system involves !four constraint conditions as shown below:!
Observe all four partitioned nodes refer to the one global node u5.!These four constraint equations are unique and rank-sufficient.!
Fig. 12. Partitioning four bars via the method of local-ized Lagrange multipliers
For the example problem, N`cl is given by
N`cl =
−1
1
(1.72)
which is a null-space matrix of Lframe.
33
References
1. Archimedes (circa 200 B. C.), Works of Archimedes, trans-lated by T. L. Heath, Cambridge University Press, 1897.
2. Galileo Galilei (1636), Two New Sciences, Translated by HenryCrew and Alfonso de Salvio, Macmillan Co., 1917.
3. Newton, I. (1687), Philosophiae Naturalis Principia Mathe-matica, translated and reprinted by Cambridge University Press,Cambridge, 1934.
4. Euler, L. (1736), Mechanica, Sive Motus Scientia AnalyticeExposita, see L. Euler Opera Omnia, Teubner, Leipzig (1911).
5. D’Alembert, J. le Rond (1743), Traite de Dynamique, Paris.
6. Euler, L. (1765), Theoria Motus Corporum Solidorum SeuRigidorum, Greifswald. See L. Euler Opera Omnia, Teubner,Leipzig(1911).
7. Lagrange, J. L., (1788), Mechanique Analytique, reprinted byGauthier-Villars, Paris (1888).
8. Hamilton, W. R. (1834), “On a general method in dynamics,”Phil. Trans. Royal Soc., 247-308.
9. Hamilton, W. R. (1835), “Second essay on a general methodin dynamics,” Phil. Trans. Royal Soc., 95-144.
10. Duhem, P. (1903), Evolution de la Mecanique, Joanin, Paris.
11. Lanczos, Cornelius, The Variational Principles of Mechanics,University of Toronto Press, 1970.
12. Dugars, Rene (1955), A History of Mechanics, Translated byJ. R. Maddox, Dover, 1988.
34
13. Mach, E., The Science of Mechanics, 6th ed., the Open CourtPub. Co., Lasalle, Ill., 1960.
14. Goldstein, H. (1980), Classical Mechanics, Addison-WesleyPublishing Co.
15. Landau. L. D. and Lifshitz, E. M. (1976), Mechanics (ThirdEdition), Pergamon Press.
16. Park, K. C. and Felippa, C. A., “A Variational Framework forSolution Method Developments in Structural Mechanics,”Journalof Applied Mechanics, March 1998, Vol. 65/1, 242-249.
17. Park, K. C. and Felippa, C. A., “A Variational Principlefor the Formulation of Partitioned Structural Systems,” Inter-national Journal of Numerical Methods in Engineering, vol. 47,2000, 395-418.
18. Felippa, C. A., Park, K. C. and Farhat, C., “PartitionedAnalysis of Coupled Mechanical Systems,” Computer Methodsin Applied Mechanics and Engineering, 190(24-25), 2001, 3247-3270.
35
Appendix A: What Isaac Newton (1642-1727)knew from his predecessors and Newton’s Prin-cipia
Or did he really get an inspiration to develop his theory of grav-itation when he observed an apple falling while he was sittingunder an apple tree?
A.1 A Historical Sketch
According to E. Mach and G. Kirchhoff, dynamics is a branchof science that attempts to explain the aggregate phenomena ofmass (force), space (geometry) and time (absolute). While thereis a growing consensus that statics can be subsumed into dy-namics as a special category, historically it was statics that firstreceived attention from many distinguished philosophers. Essen-tial concepts developed over the years in statics include: the prin-ciple of lever and pulley by Archytas (ca 400 B.C.) and Aristotle(384-322 B.C.), the center of gravity by Archimedes (287-212B.C.), the statical moments by Leonardo da Vinci (1452-1519),the principle of plane by Stebinus (1548-1620) and Ubaldo (1545-1607), the principle of virtual velocity by Jean Bernoulli (1667-1748), the rule of parallelogram by Daniel Bernoulli (1700-1782),and the state of stable or unstable equilibrium and the theoryof least action by Pierre L. M. de Maupertuis (1698-1759).
The development of these principles in statics was largely mo-tivated by the questions raised by Aristotle in his book entitledQuestions in Mechanics, perhaps the first known literature onmechanics. His ideas on the principle of balance and a fallingbody were then expanded by Archimedes who left us with amonumental document as compiled by the Cambridge UniversityPress in 1897. It is generally agreed, however, that the modernera of dynamics began with the work of Galileo Galilei (1564-1642) on the motion of projectiles contained in Two New Sci-ences published in 1636 by Elzevir at Leyden, Holland. Historyhas a special coincidence in that the day Michelangelo died on18 February 1564, Galileo was born and the year Galileo died in1642, Issac Newton was born.
Among the contributions of Galileo to dynamics are the conceptof uniformly accelerated motion from his famous experiment onfalling bodies conducted in 1583, the definition of time (or the
36
principle of relativity), the law of inertia which Newton laterincorporated as his first law, the principle of the superpositionof motions, and the establishment of the dynamics of a singlebody. Galileo is also credited to have proposed the design of apendulum clock after he observed the isochronism of a swingingbody. Unfortunately, he did not live to see his idea come true.Following these principles of dynamics due to Galileo, Huygens(1629-1695) introduced the concept of the center of oscillations,determined for the first time the acceleration of gravity by usinghis pendulum device in 1656, established the property of cen-trifugal forces, and the circle of curvature. his idea come true.
Following Galileo and Huygens, Newton (1642-1727) synthesizedthe works of Kepler, Galileo and Huygens to realize that a curvi-linear motion implies deflective acceleration. This observationpaved the way eventually leading to his discovery of universalgravitation. In addition, Newton generalized the idea of force,introduced the concept of mass, accomplished the general formu-lation of the principle of the parallelogram of forces, and estab-lished the law of action and reaction. Thus, he completed theformal enunciation of the mechanical principles now generallyaccepted. His contributions to dynamics, viz., the three laws ofmotions and the law of universal gravitation, are amalgamatedin Philosophiae Naturallis Principia Mathematica published in1687.
The awareness of the rotation of the Earth rekindled an in-terest in the problems of rotational mechanics during the mid-Eighteenth Century, which led both d‘Alembert (1717-1783) andEuler (1707-1783) to publish in 1749 on the problem of the pre-cession of the equinoxes. Subsequently, it was Euler who intro-duced the law of angular momentum which paved the way forthe solution of many rotational motions via the celebrated the-orem on the uniqueness of the axis of rotation. He was also thefirst to introduce the moment of inertia. His ideas and math-ematical treatise on rotational motions were published in 1736and 1765.
In 1788 Lagrange solved the so-called ‘heavy rigid body problem’including a revisitation of the torque-free top problem addressedby Euler. The mathematical treatment he employed therein wasthought to be simpler than that by Euler, which gave birth to
37
a new school called ‘Lagrangian Dynamics’ by his followers. Inaddition, he attacked the problem of the liberation of the Moon.He was able to combine Newton’s law of the forces of inertiaand Galileo’s principle of the composition of motions to a newprinciple that is now referred to as the principle of virtual workin dynamics: “The sum of the moments[that is, apart from sign,the virtual works] of all the powers[forces] which are in equilib-rium will be taken, and the differential functions which becomezero because of the conditions of the problem will be added toit, after each of these functions has been multiplied by an inde-terminate coefficient[Lagrange multiplier]; then the whole willbe equated to zero. Thus will be obtained a differential equa-tion which will be treated as an ordinary equation of maximis etminimis. From this will be deduced as many equations as thereare variables. These equations, being then rid of the indetermi-nate coefficients by elimination, will provide all the conditionsnecessary for equilibrium.
While Euler/Lagrange approaches continue to be influential inthe description of dynamical systems, Hamilton in his famouslecture of 1834 presented a unifying treatment by which boththe methods of Euler and Lagrange can be brought together todescribe the motions of a system of bodies. Hamilton made thefollowing remark regarding his law of varying action[Hamilton’sprinciple]: “In the method of the present essay, this problem[thedetermination of the motions of the planets] is reduced to thesearch and differentiation of a single function, which satisfiestwo partial differential equations of the first order and of thesecond degree: and every other dynamical problem, respectingthe motions of any system, however numerous, of attracting orrepelling points, is reduced, in like manner, to the study of onecentral function, of which the form marks out and characterizesthe properties of the moving system, and is to be determined bya pair of partial differential equations of the first order, com-bined with some simple consideration.” Since then, not onlyGalilean/Newtonian mechanics owes its dramatic advances toHamilton but also the modern developments of wave and quan-tum mechanics have been greatly facilitated by Hamilton’s prin-ciple.
Thus, the works of Galileo, Huygens, Newton, d‘Alembert, Eu-ler, Lagrange and Hamilton provided an impetus for practicalapplications of dynamics by the beginning of the Nineteenth
38
Century. Perhaps, the most significant application of the prin-ciples of dynamics is none other than the design of the sta-tionary reciprocating steam engine in the balancing of its crankmechanism. Another equally important application was in JamesWatt’s design of fly-ball governor in 1762 to maintain a steadyrotation for the generation of electricity from the steam engines.In modern parlance, these applications can be labeled as thecontrol of vibrations in machinery.
On the design of instruments beyond Huyghens’ pendulum, wecan mention an invention by a certain Englishman named Ser-son in 1742 of a spinning artificial horizon. Then, in 1817 vonBohnenberger in Tubingen invented a two-axis gyroscope, whichwas followed by Foucault’s demonstration of a modern gyro-scope that embodies a spinning body whose spin axis can changein direction. It was Foucault’s gyroscope that enabled for thefirst time to measure the rotation of the Earth. Today’s trans-portation systems such as automobile, train, ships, aircraft andeven sophisticated space vehicles all depend their crucial designfeatures on the sound applications of dynamics, hence our mo-tivation to study dynamics.
Let us recap individual contributions.
A.2 Notable Individual Contributions
1. Aristotle(384 − 322B .C .)
Mhqanik probl mata (Problems of Mechanics),
PerÈ oÌranoυ(Treatise on the Heavens)
Comments by Bertrand Russell (1872-1970): “Aristotle
maintained that women had fewer teeth than men: although he was
twice married, it never occurred to him to verify this statement by
examining his wives’ mouths.”
The four substances on Earth: earth, fire, air and water.
Concept of power ∝ m v;Falling bodies: time of fall ∝ 1
m;
Acceleration ∝ m;Resistance in the air ∝ density (ρ)
39
The natural place of heavy bodies is the center of theWorld. The natural place of light bodies is the regioncontiguous with the Sphere of the Moon. Every star isa body as it were divine, moved by its own divinity.
In kinematics, he advanced the composition of motion: Acurved path is generated when v1 and v2 are varied.
In dynamics, he advanced the composition of forces derivedfrom the composition of motions via the law of powers.
He believed the impossibility of a vacuum (no motionoccurs in a vacuum.
The Center of Universe and of the Earth coincide. (Heav-enly bodies are attracted to the center of the Universe,therefore they also are attracted to the Center of theEarth.)
He believed the Earth is a sphere (from the water shape,viz., “it is a property of water to run toward the lowestplaces.” Reasoning by Adrastus (360-317 B. C.): ”Often, dur-ing a voyage, one cannot see the Earth or an approaching shipfrom the deck, while sailors who climb to the top of a mast cansee these things because they are much higher and thus overcomethe convexity of the sea which is an obstacle.”
For all one can tell, Aristotle was ignorant of the conceptof the center of gravity.
2. Archimedes of Syracuse(287 − 212B .C ) , Author of On
the Equilibrium of Planes.
He introduced 8 axioms and 7 propositions to define thecenter of gravity. He employed the method of exhaustionto prove the center of gravity, the principle of lever,and Archimedes’ Principle of buoyancy. He used theword ”Eureka! Eureka! after he discovered the principleof buoyancy.
Other interesting mathematical problems he solved: 31071<
π < 317; 265/153 <
√3 < 1351/780;
∑∞n=0 4−n = 4/3 (for
computing the area of a parabola)
40
3. Euclid (circa300B .C .) , Elements (STOIQEIΩN)
4. Pappus of Alexandria (c.290 − 350 ) : He left the so-called
Arabic Manuscript, Synagoge. Of all the cotents writtenabout geometry, astronomy, philosophy, and mechanics,his writings on mechanics are largely preserved in BookVIII of Synagoge (Collection) that he wrote in a systematicarrangement, which both Newton and Lagrange quotein their books in reference to mechanics literature dur-ing the Antiquity.
5. William of Ockham and John I . Buridan
(rector of University of Paris) of XIVth Century:
They advanced the doctrine of Impetus and disputedAristotle’s axiom, viz., the continuous existence of a mo-tive agency in contact with, yet not part of, the projec-tile. That is, the idea of attributing a certain energy toa moving body solely on account of its motion was notaccepted in the Aristotelian dynamics. ”The existenceof impetus seems to be the cause by which the naturalfall of bodies accelerates indefinitely. At the beginningof the fall, indeed, the body is moved by the gravityalone. . . . But before long this gravity imparts a cer-tain impetus to the heavy body – an impetus which iseffective in moving the body at the same time as gravitydoes. ... The more rapid it becomes, the more intensethe impetus becomes. Therefore it can be seen that themotion will be accelerated continuously.”
6. Albert of Saxony and Nicole Oresme of XIV thCentury :He confirmed the sphericity of the Earth and the Oceans.Oresme articulated the concept of uniformly acceler-ated motion (a forerunner of gravity), which in modernmathematical language can be expressed as
e = 12gt2 = (1
2gt) · t = vmean · t
Nicole Oresme as a predecessor of Copernicus: ”No ob-servation could prove the heavens moved with a diurnal motion,and that the Earth did not.”
41
7. Leonard da Vinci (1452 − 1519 ) touched on many top-
ics on mechanics, including: The concept of moment, themotion of a heavy body on an inclined plane, the reso-lution of forces (parallelogram), the energy of a movingbody, the impossibility of perpetual motion, the figureof the Earth, the center of gravity,
8. Nicholas Copernicus (1472 − 1543 ) : The Sorbonne in
the XVI Century remain closed to Copernican ideas andcontinued to teach Ptolemy’s system (the Geocentricsystem of World). He challenged the Geocentric Sys-tem of the World, but in order not to anger the Vatican, hedidn’t use the Earth but mercury and Venus to placethe Sun at the Center of the Planets! This is what hestated, in dedicating his works to Pope Paul III, to pro-tect himself: ”I have believed that I would be readily permit-ted to examine whether, in supposing the motion of the Earth,something more conclusive might not be found in the motion ofcelestial bodies.” Thus he avoided the Vatican Inquisitionuntil 1616, 73 years after his death.
9. Guido Ubaldo (1545 − 1607 ) : He wrote a book enti-
tled, Mechanicorum Liber in which he clarified the defi-nition of the center of gravity by stating, ”All the force,all the gravity of the weight is massed and united at the cen-ter of gravity; it seems to run from all sides toward this point.Because of its gravity, indeed, the weight has a natural desireto pass through the center of the Universe. But it is the centerof gravity that properly tends to the center of the World.” Hisbook remained an authoritative source on mechanicsuntil the beginning of the XVIIIth Century and influ-enced Galileo, Descartes, Lagrange, among others.
J. Tycho Brahe (1546 − 1601 ) : He gathered and mea-
sured himself a vast amount of observation data withprecision, which were the foundation upon which Ke-pler’s laws were based.
10. Johannes Kepler (1571 − 1631 ) : Author of Astronomia
nova and Harmonices Mundi. A tireless calculator! Basedon careful observations and calculations using TychoBrahe’s data, he formulated the law of areas, the law of
42
ellipticity of planetary trajectories, and the third law,viz., ”One thing is absolutely certain and correct, thatthe ratio between the periods of any two planets is, tothe power 3
2, exactly that of their mean distances, that
is, of their orbits:
T ′
T= (
a′
a)32
11. Galilei Galileo (1564 − 1642 )
(http://en.wikipedia.org/wiki/Galileo Galilei):Early on he studied Aristotle’s De Caelo (On the Heav-ens; PERI OURANOU (PerÈ oÌranìs) . He is consideredthe father of modern science (dynamics). His contri-butions include: the notion of impeto or talento and mo-mento del discentere, the concept of uniform acceleration,the principle of inertia (”A body moving on a level surfacewill continue in the same direction at constant speed unless dis-turbed.”), the motion of projectiles (parabolic motion ofa falling body with initial horizontal velocity), the prin-ciple of virtual work, the isochoronous property of asimple pendulum, inventor of refracting telescope. . .He is credited with the basic principle of relativity, viz.,The laws of physics are the same in any system that is movingat a constant speed in a straight line, regardless of its particularspeed or direction.
He is author of The Starry Messenger (Sidereus Nuncius,1610), Discourse on the tides (Discorso sul flusso e il reflusso delmare, 1616), Discourse on the Comets (Discorso Delle Comete,1619), The Assayer (Il Saggiatore, 1623), Dialogue Concerningthe Two Chief World Systems (Dialogo sopra i due massimi sis-temi del mondo, 1632) which was banned by the Roman In-quisition; and Discourses and Mathematical DemonstrationsRelating to Two New Sciences (Discorsi e dimostrazioni matem-atiche, intorno a due nuove scienze, Leida, 1638) which dealswith small motions, hence was allowed to be published!
12. Rene Descartes (1596 − 1650 ) : the originator of the
Cartesian coordinate system, one of the best knownphilosophical statements (Cogito ergo sum, je pense, doncje suis, I think, therefore I am), the concept of infinitesi-mal calculus, an early form of the law of conservationof momentum, ...
43
13. Christiaan Huyghens(1629 − 1695 ) : He published his
work in Horologium Oscillatorium sive de motu pendulorum adhorologia aptato demonstrasiones geometricae (Paris, 1673). Inthat book he describes: Description of the clock, on thefall and motion of bodies on a cycloid, the evolution anddimensions of curved lines, on the center of oscillationor agitation, on the construction of a new clock witha circular pendulum, the theorems of the centrifugalforces, among others. Using the formula for the pendu-lum period
T = 2π
√`
g
he was able to compute the gravitational acceleration(g).
Also in 1659, he derived the formula for centripetal force
Fc =mv2
r
Equipped with these historical developmentsin mechanics, especially, inheriting the worksof Galileo and Huyghens, now comes IsaacNewton. It may be a historical coincidence:Galilei Galileo died in January 06, 1642. OnDecember 25 of the same year Isaac Newtonwas born.
44
EBOOK - 0
1
Philosophiae naturalis principia mathematica (): Sir Isaac Newton,Edmund Halley
•
Google Play
Fig. 13. Title page of Principia
A.3 Isaac Newton (1642-1727)
http://en.wikipedia.org/wiki/Isaac Newton
Rene Dugas’ ( author of A History of Mechanics) assess-ment of Newton’s place in the history of mechanics:Thanks to Galileo and Huyghens, mechanics had been emanci-pated from the scholastic discipline (occult and metaphysical).Essential problems like the motion of projectiles in the vacuumand the oscillations of a compound pendulum had been solved.Nevertheless, the task of constructing an organized corpus ofprinciples in dynamics remained. This was the task of Newton,
45
Fig. 14. English Title page of Principia
who set his seal on the foundation of classical mechanics at thesame time that he extended its field of application to celestialphenomena.
In the Author’s Preface (p. XVII) of Philosophiae Natu-ralis Principia Mathematica (Mathematical Principles of Natu-ral Philosophy) or simply Newton’s Principia (1687), he states. . . ”Therefore I offer this work as the mathematicalprinciples of philosophy, for the whole burden of phi-losophy seems to consist in this – from phenomena of
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Fig. 15. First page of English translation of Principia
motions to investigate the forces of nature, and thenfrom these forces to demonstrate the other phenomena;and to this end the general propositions in the first andsecond Book are directed.” . .
His description of gravity (Principia, p.547): ”But hith-erto I have not been able to discover the cause of thoseproperties of gravity from phenomena, and I frame no
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hypothesis; for whatever is not deduced from the phe-nomena is to be called a(an) hypothesis; and hypothe-sis, whether metaphysical or physical, whether of occultqualities or mechanical, have no place in experimentalphilosophy. In this philosophy particular propositionsare inferred from the phenomena, and afterwards ren-dered general by induction.”
Newton’s Eight Definitions (Principia, pp.1-6)
DEFINITION I: The quantity of matter[mass] is themeasure of the same, arising from its density and bulkconjointly.
DEFINITION II: The quantity of motion[linear momen-tum] is the measure of the same, arising from the veloc-ity and quantity of matter conjointly.
DEFINITION III: The vis insita, or the innate force ofmatter[inertia force], is a power of resisting, by whichevery body, as much as in it lies, continues in its presentstate, whether it be rest, or of moving uniformly for-wards in a right[straight] line.
DEFINITION IV: An impressed force is an action ex-erted upon a body, in order to change its state, eitherof rest, or of uniform motion in a right line.
DEFINITION V: A centripetal force is that by whichbodies are drawn or impelled, or any way tend, towardsa point as to a centre.
DEFINITION VI: An absolute centripetal force is themeasure of the same, proportional to the efficacy of thecause that propagates it from the centre, through thespaces round about.
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DEFINITION VII: The accelerative quantity of a cen-tripetal force is the measure of the same, proportionalto the velocity which it generates in a given time.
DEFINITION VIII: The motive[motion-causing] quan-tity of a centripetal force is the measure of the same,proportional to the motion which it generates in a giventime.
Newton’s Three Laws (Principia, pp.13-14)
Law I: Every body continues in its state of rest, or ofuniform motion in a right line, unless it is compelled to changethat state by forces impressed upon it.
Law II: The change of motion is proportional to themotive force impressed; and is made in the direction of the rightline in which the force is impressed.
Law III: To every action there is always opposed anequal reaction; or, the mutual actions of two bodies upon eachother are always equal, and directed to contrary parts.
In Book III of Principia (pp.398-400), he lists four rules ofreasoning in philosophy:
Rule 1. ”We are to admit no more causes of natural things thansuch as are both true and sufficient to explain their appearance.”(Do not assume other causes than those which are nec-essary to explain the phenomena.)
Rule 2. ”Therefore to the same natural effects we must, as faras possible, assign the same causes.” (Relate as completelyas possible analogous effects to the same cause.)
Rule 3. The qualities of bodies, which admit neither intensifica-tion nor remission of degrees, and which are found to belong toall bodies within the reach of our experiments, are to be esteemedthe universal qualities of all bodies whatsoever.” (Extend to allbodies the properties which are associated with those onwhich it is possible to make experiments.)
Rule 4. In experimental philosophy we are to look upon propo-
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sitions inferred by general induction from phenomena as accu-rately or very nearly true, till such time as other phenomenaoccur, by which they may either be made more accurate, or li-able to exceptions.” (Consider every proposition obtainedby induction from observed phenomena to be valid untila new phenomenon occurs and contradicts the proposi-tion or limits its validity.)
These four rules provide a guide for how to conductscientific research for subsequent generations. In par-ticular, it is Rule 3 that Newton relied on to formulatethe law of universal gravitation.
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